Solutions of Yang–Mills theory in four-dimensional de Sitter ...

Solutions of Yang–Mills theory infour-dimensional de Sitter space

Von der Fakultät für Mathematik und Physik

der Gottfried Wilhelm Leibniz Universität Hannover

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr. rer. nat.

genehmigte Dissertation von

M.Sc. Kaushlendra Kumar

2022

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Referent: Prof. Dr. Olaf Lechtenfeld

Korreferent: Prof. Dr. Domenico Giulini

Korreferent: Prof. Dr. Gleb Arutyunov

Tag der Promotion: 04.05.2022

Printed and/or published with the support of the German Academic Exchange Service.

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ResourcesThis thesis is based on following published research articles:

1. Kaushlendra Kumar, Olaf Lechtenfeld, and Gabriel Picanço Costa,Trajectories of charged particles in knotted electromagnetic fields,Journal of Physics A: Mathematical & Theoretical 55 (2022) 315401.

2. Kaushlendra Kumar, Gabriel Picanço Costa, and Lukas Hantzko,Conserved charges for rational electromagnetic knots,European Physical Journal Plus 137 (2022) 407.

3. Kaushlendra Kumar, Olaf Lechtenfeld, and Gabriel Picanço Costa,Instability of cosmic Yang-Mills fields,Nuclear Physics B 973 (2021) 115583.

4. Kaushlendra Kumar and Olaf Lechtenfeld,On rational electromagnetic fields,Physics Letters A 384 (2020) 126445.

Following notebooks validate the data presented in this thesis:• Kaushlendra Kumar and Gabriel Picanço Costa,

“Yang-Mills Theory in 4-Dimensional de Sitter Space"from the Notebook Archive (2022), https://notebookarchive.org/2022-04-dbzrbqg.

• Kaushlendra Kumar, Olaf Lechtenfeld and Gabriel Picanco Costa,“Trajectories of Charged Particles in Knotted Electromagnetic Fields"from the Notebook Archive (2022), https://notebookarchive.org/2022-05-7es6sj9.

https://doi.org/10.1088/1751-8121/ac7c49

https://doi.org/10.1140/epjp/s13360-022-02563-4

https://doi.org/10.1016/j.nuclphysb.2021.115583

https://doi.org/10.1016/j.physleta.2020.126445

https://notebookarchive.org/2022-04-dbzrbqg

https://notebookarchive.org/2022-05-7es6sj9

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ZusammenfassungDiese Doktorarbeit befasst sich mit der Analyse einiger Yang-Mills-Lösungen auf vier di-mensionalen de Sitter Raum dS4. Die konforme Äquivalenz dieses Raums mit einemendlichen Lorentz-Zylinder über der 3-Sphäre S3 und auch mit Teilen des Minkowski-Raums—zusammen mit der Tatsache, dass die Yang-Mills-Theorie in der 4-dimensionalenRaumzeit konform invariant ist—hat kürzlich zur Entdeckung einer Familie rational ver-knoteter elektromagnetischer Feldkonfigurationen geführt. Diese „Basisknoten“-Lösungender Maxwell-Gleichungen, auch bekannt als Abelsche Yang-Mills-Theorie, sind mit denhypersphärischen Harmonischen Yj,m,n auf der S3 gekennzeichnet und haben nette Eigen-schaften wie endliche Energie, endliche Wirkung und das Vorhandensein einer konserviertentopologischen Größe namens Helizität. Ihre Feldlinien bilden geschlossene Knotenschleifenim dreidimensionalen Euklidischen Raum. Diese könnten in der Kosmologie des frühenUniversums eine Rolle spielen, um das symmetrische Higgs-Vakuum zu stabilisieren.

Wir untersuchen Symmetrieaspekte dieser elektromagnetischen Knotenkonfigurationenund berechnen alle erhaltenen Ladungen für die konforme Gruppe SO(2, 4) im Zusam-menhang mit dem 4-dimensionalen Minkowski-Raum für eine komplexe Linearkombi-nation dieser Basislösungen für ein festes j. Die Berücksichtigung einer solchen kom-plexen linearen Kombination ist wichtig, da diese rationalen Basislösungen verwendetwerden können, um auf diese Weise jede endliche Energiefeldkonfiguration zu erzeugen;wir demonstrieren diese Tatsache mit einigen bekannten Ergebnissen für bestimmte mod-ifizierte Hopf–Ranãda-Knoten. Wir finden, dass die skalaren Ladungen entweder ver-schwinden oder proportional zur Energie sind. Für die nicht verschwindenden Vektor-ladungen finden wir eine schöne geometrische Struktur, die auch die Berechnung ihrersphärischen Komponenten erleichtert. Wir finden auch, dass die Helizität mit der Energiezusammenhängt. Darüber hinaus charakterisieren wir den Unterraum von Nullfeldernund präsentieren einen Ausdruck für den elektromagnetischen Fluss bei null unendlich,der mit der Gesamtenergie übereinstimmt, wodurch die Energieerhaltung validiert wird.Schließlich untersuchen wir die Trajektorien von Punktladungen im Hintergrund solcherBasisknotenkonfigurationen. Dazu finden wir je nach Feldkonfiguration und verwende-tem Parametersatz unterschiedliche Verhaltensweisen. Dazu gehören eine Beschleuni-gung von Teilchen durch das elektromagnetische Feld aus dem Ruhezustand auf ultra-relativistische Geschwindigkeiten, eine schnelle Konvergenz ihrer Flugbahnen in wenigeschmale Kegel, asymptotisch für einen ausreichend hohen Wert der Kopplung, und einausgeprägtes Verdrehen und Wenden der Bahnen in kohärenter Weise.

Wir analysieren die lineare Stabilität der „kosmischen Yang-Mills-Felder“ der SU(2) gegen-über allgemeinen Eichfeldstörungen, während wir die Metrik eingefroren halten, indemwir den (zeitabhängigen) Yang-Mills-Fluktuationsoperator um sie herum diagonalisierenund die Floquet-Theorie auf ihre Eigenfrequenzen und normale Modi anwenden. MitAusnahme der exakt lösbaren SO(4)-Singulett-Perturbation, die linear marginal stabil,aber nichtlinear begrenzt ist, wachsen generische Normalmoden aufgrund von Resonanz-effekten häufig exponentiell an. Selbst bei sehr hohen Energien werden alle kosmischenYang-Mills-Hintergründe linear instabil gemacht.

Schlüsselwörter: Yang–Mills-Theorie, vierdimensionaler de Sitter-Raum, elektromagnetis-che Knoten, kosmische Eichfelder.

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AbstractThis doctoral work deals with the analysis of some Yang–Mills solutions on 4-dimensionalde Sitter space dS4. The conformal equivalence of this space with a finite Lorentzian cylin-der over the 3-sphere S3 and also with parts of Minkowski space—together with the factthat Yang–Mills theory is conformally invariant in 4-dimensional spacetime—has recentlyled to the discovery of a family of rational knotted electromagnetic field configurations.These “basis-knot" solutions of the Maxwell equations, aka Abelian Yang–Mills theory, arelabelled with the hyperspherical harmonics Yj,m,n of the S3 and have nice properties suchas finite-energy, finite-action and presence of a conserved topological quantity called he-licity. Their field lines form closed knotted loops in the 3-dimensional Euclidean space.Moreover, in the non-Abelian case of the gauge group SU(2) there exist time-dependentsolutions of Yang–Mills equation on dS4 in terms of Jacobi elliptic functions that are of cos-mological significance. These might play a role in early-universe cosmology for stabilizingthe symmetric Higgs vacuum.

We study symmetry aspects of these electromagnetic knot configurations and computeall conserved charges for the conformal group SO(2, 4) associated with the 4-dimensionalMinkowski space for a complex linear combination of these basis solutions for a fixed j.Consideration of such complex linear combination is important because these rational ba-sis solutions can be used to generate any finite energy field configuration in this way; wedemonstrate this fact with some known results for certain modified Hopf–Ranãda knots.We find that the scalar charges either vanish or are proportional to the energy. For thenon-vanishing vector charges we find a nice geometric structure that facilitates compu-tation of their spherical components as well. We also find that helicity is related to theenergy. Moreover, we characterize the subspace of null fields and present an expressionfor the electromagnetic flux at null infinity that matches with the total energy, thus vali-dating the energy conservation. Finally, we investigate the trajectories of point charges inthe background of such basis-knot configurations. To this end, we find a variety of behav-iors depending on the field configuration and the parameter set used. This includes anacceleration of particles by the electromagnetic field from rest to ultrarelativistic speeds,a quick convergence of their trajectories into a few narrow cones asymptotically for suffi-ciently high value of the coupling, and a pronounced twisting and turning of trajectoriesin a coherent fashion.

We analyze the linear stability of the SU(2) “cosmic Yang–Mills fields” against generalgauge-field perturbations while keeping the metric frozen, by diagonalizing the (time-dependent) Yang–Mills fluctuation operator around them and applying Floquet theoryto its eigenfrequencies and normal modes. Except for the exactly solvable SO(4) singletperturbation, which is found to be marginally stable linearly but bounded nonlinearly,generic normal modes often grow exponentially due to resonance effects. Even at veryhigh energies, all cosmic Yang–Mills backgrounds are rendered linearly unstable.

Keywords: Yang–Mills theory, four dimensional de Sitter space, electromagnetic knots,cosmic gauge fields.

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AcknowledgementsFirst, I would like to thank my supervisor, Olaf Lechtenfeld, for his constant support andencouragements at all stages of my PhD in his research group. He was always availablefor discussions and clarifications and, over the years, has propelled me to think criticallyfor myself on a given research question. He has also very kindly helped me to improvevarious manuscripts not just related to our joint projects but also otherwise. I also thankhim for suggesting the wonderful “Saalburg Summer Schools" where I learned a lot aboutthe state of the art in various research areas.

I would also like to thank Biswajit Chakraborty who helped me at a critical juncture of mycareer when I made a transition to theoretical physics and who introduced me to the beau-tiful world of noncommutative geometry. I am also thankful to many teachers like IndreshParihar, Diwakar Nath Jha, Anand Kumar (Super 30), Anand Dasgupta and SunandanGangopadhyay for their wonderful mentoring at various stages of learning and makingphysics/mathematics so interesting for me.

I am grateful to Domenico Giulini and Gleb Arutyunov for revieing my thesis and sug-gesting valuable improvements. I thank Rolf Haug for agreeing to be the head of my PhDcommittee.

Furthermore, I thank my colleagues Gleb Zhilin, Gabriel Picanço Costa, Joshua Cork andTill Bargheer for several interesting and valuable discussions (both academic and other-wise) at ITP (Institute for Theoretical Physics), Leibniz University Hannover (LUH).

I am grateful to the DAAD (Deutscher Akademischer Austauschdienst) for the doctoralresearch grant 57381412 that enabled me to pursue my PhD research work at LUH. I amalso grateful to the ITP for providing excellent working environment and equipments. Iam thankful to Tanja Wießner and Brigitte Weskamp for bureaucratic support.

Last, but not the least, I thank my family and friends for all their love and support. Theculmination of this thesis is also a testament to their patience and hard work. I wouldattribute my keen interest in mathematics from an early age to the teaching environmentat home created by my father (being a mathematics teacher).

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Contents

Resources ii

Zusammenfassung iii

Abstract iv

Acknowledgements v

1 Introduction 11.1 Electromagnetic knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Cosmic SU(2) Yang–Mills fields . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outline and summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Geometry & Symmetry 72.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6.1 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6.2 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6.3 Maurer–Cartan form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.8 Hodge duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.9 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.10 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Gauge theory 243.1 Principal G-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Associated bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.1 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Yang-Mills equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Yang–Mills equations on dS4 354.1 The de Sitter-Minkowski correspondence via S3-cylinder . . . . . . . . . . . 35

4.1.1 Structure of 3-sphere and its harmonics . . . . . . . . . . . . . . . . . 384.2 Yang-Mills field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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5 Abelian solutions: U(1) 435.1 Family of "knot" solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1.1 Generic solution on the S3-cylinder . . . . . . . . . . . . . . . . . . . 435.1.2 Pulling the solution back to Minkowski space . . . . . . . . . . . . . 45

5.2 Symmetry analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.3 Conformal group and Noether charges . . . . . . . . . . . . . . . . . . . . . . 49

5.3.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3.2 Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3.3 Dilatation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3.4 Special conformal transformations . . . . . . . . . . . . . . . . . . . . 545.3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4 Null fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.5 Flux transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.6 Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Non-Abelian solutions: SU(2) 686.1 SO(4)-symmetric cosmic Yang–Mills solution . . . . . . . . . . . . . . . . . . 686.2 Natural perturbation frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 716.3 Stability analysis: stroboscopic map and Floquet theory . . . . . . . . . . . . 786.4 Singlet perturbation: exact treatment . . . . . . . . . . . . . . . . . . . . . . . 85

7 Conclusion & Outlook 89

A Carter-Penrose transformation 91

B Rotation of indices 92

C Polynomials in the characteristic equation 93

1

Chapter 1

Introduction

The 4-dimensional de Sitter space dS4 plays an important role in gravity. It is one of thethree (topological) types1 of Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimethat can model a homogeneous and isotropic universe like ours (at distance scales of 100Mpsec). This has a positive global scalar curvature of the underlying 3-space (viewed asglobal foliation) that is consistent with the observed positive cosmological constant Λ akadark energy that is argued to be fueling the accelerated expansion of our universe. It isbelieved that our universe is asymptotically de Sitter which means in the future, when thedark energy dominates, our universe would become de Sitter.

Gauge theory, in particular Yang–Mills theory, is of central importance in the classical de-scription of fundamental forces of nature like electromagnetism, weak and strong nuclearforces. Classical Yang–Mills theory also has applications in other physics areas such asQCD confinement of high energy physics and spin-orbit interaction in condensed matterphysics. Even gravity can be understood as a gauge theory. It is, therefore, natural to seeksolutions of Yang–Mills theory on four-dimensional de Sitter space. Furthermore, owingto the conformal relation of dS4 with Minkowski space R1,3, it turns out that even AbelianYang–Mills theory aka electromagnetism studied on the former has nice application sincethe solutions can be pulled back to Minkowski space (of our laboratory) owing to the con-formal invariance of the Yang–Mills theory in 4-dimensions.

1.1 Electromagnetic knots

Theoretical discovery of electromagnetic knots dates back to 1989 when Rañada [1] con-structed them using the Hopf map. These finite-energy finite-action vacuum solutions ofMaxwell’s equations are constructed from a pair of complex scalar fields φ and θ on the 4-dimensional spacetime where the 3-space is compatified to S3 with the addition of a pointat infinity. These solutions are thus characterised by a topological quantity called the Hopfindex of the following Hopf map (A = 1, 2, 3, 4):

h : S3 → S2 , ωA 7→ x1, x2, x3 (1.1.1)

where S2, arising from the compactification of the complex plane C, has coordinates xi,satisfying x2

1 + x22 + x2

3 = 1, that are constructed from the S3 coordinates ωA , satisfyingω2

1 + ω22 + ω2

3 + ω24 = 1, as follows

x1 := 2(ω1ω2 +ω3ω4) , x2 := 2(ω1ω4−ω2ω3) , x3 := (ω21 +ω2

3)− (ω22 +ω2

4) . (1.1.2)

The level curves of these complex-valued functions θ and φ are identified with electricand magnetic field lines. Several other approaches to construct such electromagnetic knots

1There are 3 types of FLRW spacetime viz. Minkowski space (κ=0), de Sitter space (κ=1), and Anti-deSitter space (κ=−1) according to the global topology of the backgroud 3-space (labelled by κ).

Chapter 1. Introduction 2

have been developed since then such as Bateman’s complex Euler potentials, conformalinversion and Penrose twistors (see [2] for a full review).

Apart from these there exists another way of constructing these knotted electromagneticfields via the conformal correspondence between de Sitter space dS4 and Minkowski spaceR1,3, while passing through a finite Lorentzian S3-cylinder, as was shown in [3]. In thismethod, one obtains a complete family2 of such electromagnetic knotted configurationsthat are labelled with hyperspherical harmonics Yj;m,n of the 3-sphere. The de Sitter spaceenjoys a larger symmetry group in SO(1, 4) whose subgroup SO(4) ∼= (SU(2)×SU(2))/Z2is made use of in this construction by working with the S3-cylinder. Here one employs theright-action of SU(2)—the group manifold of S3—on the 3-sphere to write down the gaugefield in terms of the left-invariant one-forms of S3. The resulting Maxwell’s equation canbe solved analytically and are then pulled back to the Minkowski space using the confor-mal map. This “de Sitter" method of construction has advantage over the others becauseof the SO(4) covariant treatment of Maxwell theory.

Knotted electromagnetic fields might become important for future applications because oftheir unique topological properties. It is, therefore, important to seek experimental set-tings to generate those fields and to study scenarios with them. Irvine and Bouwmeester[4] discuss the generation of knotted fields using Laguerre–Gaussian beams and predictpotential applications in atomic particle trapping, the manipulation of cold atomic en-sembles, helicity injection for plasma confinement, and in the generation of soliton-likesolutions in a nonlinear medium. Moreover, laser beams with knotted polarization sin-gularities were recently employed to produce some simple knotted field configurations,including the one with figure-8 topology in the lab [5].

1.2 Cosmic SU(2) Yang–Mills fields

Finding analytic solution to the Einstein–Yang–Mills system of equations arising from thefollowing action (without topological term ∼ F∧F)

S = SYM + SEH =1

2κ

∫M

d4x√−g (R + 2Λ) +

18g2

∫M

Tr(F∧ ∗ F) , (1.2.1)

where Λ is the cosmological term and κ, g are coupling constants, is not possible in general.There is, however, one scenario where such a solution can be obtained and that is FLRWcosmology. Here the Yang–Mills equation decouple from Einstein equation due to theconformal invariance of the former in 4-dimensional spacetime M. This means that givena solution of Yang–Mills equation in one of these FLRW spacetime, the corresponding scalefactor can be obtained via Friedmann equations. Such a solution with finite energy andaction do exist for the de Sitter case together with the SU(2) gauge group [6–8].

A crucial ingredient of the Standard Model of cosmology called inflation can also be tackledwith homogeneous and isotropic non-Abelian Yang–Mills fields in the Minkowski type(spatially flat) FLRW background in theories of gauge-flation or chromo-natural inflation(see [9] for a review). Another more minimalistic approach towards tackling this issuewas recently put forth by Daniel Friedan [10]. He considers a coupled Einstein–Yang–Mills–Higgs system where a rapidly oscillating isotropic SU(2) gauge field stabilizes thesymmetric Higgs vacuum in a de Sitter type (spatially closed) FLRW spacetime.

Based on these, it is only natural to analyze the stability behaviour of such “cosmic Yang–Mills fields" under generic linear perturbation of the Yang–Mills field equation. For the

2In the sense that any given finite-energy rational Maxwell solution can be expanded in terms of thesebasis configurations.


gauge-flation scenario such an analysis has been done before, but for the later scenariothere only exits result for spin-j=0 case of these SU(2) gauge fields [11].

In light of this, we present here a complete stability analysis of these SU(2) solutions ina closed FLRW universe. This analysis is in contrast with that of the guage-flation onewhere conformal invariance is broken; our homogeneous and isotropic gauge field wouldgive rise to inhomogeneous Yang–Mills fields on flat FLRW spacetime. For the sake ofsimplicity we keep the background metric fixed in this perturbation analysis. While thisdoes mean that our analysis is still partial, we can argue for the relevance of our analysisas follows:

(a) In 4-dimensional spacetime, the gauge fields decouple with the background metric.Therefore, fluctuation of the latter does not affect the gauge fields of our theory.

(b) The fluctuation of the gauge fields is extremely rapid when compared to the evolu-tion of the background metric and its subsequent fluctuations. The former, therefore,do not experience any significant effect due to such slow metric fluctuations.

The only known family of finite-energy SU(2) Yang-Mills field configuration on FLRWspacetime are obtained, in an efficient manner, by employing the following conformalcorrespondence between de Sitter space and the cylinder I × S3 for I := (0, π). Thisconformal map arises via a temporal reparametrization and Weyl rescaling [10–13],

ds2dS4

= −dt2 + `2 cosh2 t` dΩ2

3 = `2

sin2τ

(−dτ2 + dΩ2

3)

for t ∈ (−∞,+∞) ⇔ τ ∈ (0, π) ,(1.2.2)

where dΩ 23 is the round metric on S3, and ` is the de Sitter radius. We observe that the

relation between the conformal time τ and co-moving time t in (1.2.2) fixes the cosmolog-ical constant to Λ = 3/`2. At this point, we can employ an S3-symmetric ansatz for thegauge field by noting that SU(2) is the group manifold of S3. This yields an ODE for somescalar function ψ(τ) parametrized by the conformal time, that is nothing but a Newton’sequation for a classical point particle under the influence of a double-well potential

V(ψ) = 12 (ψ

2 − 1)2 . (1.2.3)

The solution for these anharmonic oscillators are well known in terms of Jacobi ellipticfunctions that depend on time. Although these solutions exists for all three FLRW metrics,only spatially closed one admits an isotropic solution under the above conformal transfor-mation. For de Sitter type FLRW spacetime we have

ds2 = −dt2 + a(t)2 dΩ 23 = a(τ)2(−dτ2 + dΩ 2

3)

for t ∈ (0, tmax) ⇔ τ ∈ I ≡ (0, T′) ,(1.2.4)

where we impose a big-bang initial condition a(0)=0, so that

dτ =dt

a(t)with τ(t=0) = 0 and τ(t=tmax) =: T′ < ∞ . (1.2.5)

The lifetime tmax of the universe can be infinite (big rip, a(tmax)=∞) or finite (big crunch,a(tmax)=0). Moreover, bouncing cosmologies as in (1.2.2) are also allowed but will not bepursued here.

We notice that the contribution of these SO(4)-symmetric Yang–Mills fields, and theirstress-energy tensor, in the one-way coupling with the background de Sitter FLRW metricvia Friedmann equation (as discussed before) is such that it only modifies the scale factor


a(τ). It is well known that the equation of motion governing this scale factor arises as aNewton’s equation with the following (cosmological) potential

W(a) = 12 a2 − Λ

6 a4 , (1.2.6)

which is another anharmonic oscillator (although inverted).

The pair of solutions (ψ, a) corresponding to (1.2.3) and (1.2.6) yields an exact classicalEinstein–Yang–Mills configuration. The conserved mechanical energy E for ψ fixes thesame i.e. E′ for a via the Wheeler–DeWitt constraint

E′ = εE ; E :=12

ψ2 + V(ψ) and E′ :=12

a2 + W(a) , (1.2.7)

where the overdot denotes a derivative with respect to conformal time and ε depends oncoupling constants.

One can also introduce a complex scaler Higgs field φ in the fundamental SU(2) represen-tation to the Standard Model of cosmology with Higgs potential

U(φ) = 12 λ2 (φ†φ− 1

2 v2)2 , (1.2.8)

where v/√

2 is the Higgs vev and λv is the Higgs mass. It turns out that imposing anSO(4)-invariance makes the Higgs field φ ≡ 0, which provides us with a definite positivecosmological constant of

Λ = κ U(0) = 18 κ λ2v4 , (1.2.9)

where κ is the gravitational coupling. The full Einstein–Yang–Mills–Higgs action (in stan-dard notation),

S =∫

d4x√−g

1

2κ R + 18g2 trFµνFµν − Dµφ†Dµφ−U(φ)

, (1.2.10)

reduces in the SO(4)-invariant sector to

S[a, ψ, Λ] = 12π2∫ T′

0dτ

1κ

(− 1

2 a2 + W(a))+ 1

2g2

( 12 ψ2 −V(ψ)

), (1.2.11)

where g is the gauge coupling.

If the Yang–Mills energy E is large enough it could propel an eternal expansion of the uni-verse that is accompanied by rapid fluctuations of the gauge field. The coupling of thisYang–Mills field with the Higgs field stabilizes the symmetric vacuum φ≡0 at the localmaximum of U through a parametric resonance effect, as long as a is not too large. Even-tually, when a exceeds a critical value of electro-weak symmetry breaking scale aEW, theHiggs field will begin to roll down towards a minimum of U, thus breaking the SO(4) sym-metry. The corresponding time tEW signifies the electroweak phase transition in the earlyuniverse. This is a rather unconventional scenario put forward recently by Friedan [10].

1.3 Outline and summary of results

In the next two chapters we review the mathematical preliminaries that builds up gaugetheory. In Chapter 2 we present a brief but thorough review of the mathematics of back-ground geometry such as manifold, fibre bundles, etc. and symmetry in physics such asLie groups, Lie algebra and their representations. Next, in Chapter 3 we review the con-struction of gauge theory via principal bundle formalism.


In Chapter 4 we discuss the geometry of and calculus on the 3-sphere apart from demon-strating the conformal equivalence of the S3-cylinder with Minkowski space. We thenpresent Yang–Mills equation for a SO(4)-asymmetric SU(2) Yang–Mills theory on the S3-cylinder and discuss its two limiting cases where analytic solutions can be obtained.

We present the construction of electromagnetic knotted field configurations in Chapter 5and study their symmetry feature and other properties. We analyze the effect of the deSitter group SO(1, 4), i.e. the isometry group of dS4, on these solutions. To that end, wedemonstrate the emergence of the Poincare group ISO(1, 3) of R1,3 from SO(1, 4) in thelimit `→ ∞. We observe that only the subgroup SO(3) is common in the two cases.

We then proceed, also in Chapter 5, to compute all the Noether charges associated withthe conformal group SO(2, 4) viz. energy, momentum, angular momentum, boost, dilata-tion and special conformal transformations (SCT) for a linear combination—in terms ofcomplex coefficients Λj,m,n—of these basis-knot configurations. These conserved chargedensities are evaluated on de Sitter space at τ=0 where considerable simplifications oc-cur demonstrating the usefulness of this “de Sitter method". We find that the dilatationvanishes while the scalar SCT charge V0 is proportional to the energy E. Furthermore, theboosts Ki vanish and the vector SCT charges Vi are proportional to the momenta Pi. Inter-estingly, for the vector charge densities viz. momenta pi, angular momenta li and vectorSCT vi we find that the one-form, e.g. pidxi constructed on the spatial slice R3 → R1,3 isproportional to a similar one on de Sitter space. This correspondence allows us to computeadditional charges (pr, pθ , pφ) by the action of such one-forms on spherical vector fields(∂r, ∂θ , ∂φ). At j=0 it turns out that there are only four independent non-zero charges: theenergy E and momenta Pi. The situation for higher spin j is more complicated, but someof the components of the charges in spherical coordinates are found to vanish for arbitraryj. The action of so(3) generators Da on the indices of Λj,m,n can easily be obtained for afixed j owing to the SO(4) isometry. This allows for an action of these generators on thecharges. For (the Cartesian components of) the vector charges this action is found to in-herit the original so(3) Lie algebraic structure, as expected. We also compute the correctcoefficients Λ0;0,n corresponding to two interesting generalisations of the Hopfian solutionobtained via Bateman’s construction in [14], which allow us to validate our generic formu-lae of these charges. We also demonstrate the relationship of energy with the conservedhelicity.

Furthermore, in Chapter 5 we characterize the moduli space of null solutions which turnsout to be a complete-intersection projective complex variety of complex dimension 2j+1.We also demonstrate how the energy flux is radiated to infinity with an energy profile thatis concentrated along the lightcone situated at the origin (selected by these solutions). Fi-nally, we study the trajectories of multiple identical charged particles in the backgroundof these basis knot configurations. We employ several initial conditions for these chargedparticles and find interesting features of the trajectories like coherent twisting, ultrarela-tivistic acceleration of particles starting from rest and a quick convergence of their trajec-tories into a few narrow cones asymptotically for sufficiently high value of the coupling.

In Chapter 6 we first review the classical configurations (Aµ, gµν) in terms of Newtoniansolutions (ψ, a) for the anharmonic oscillator pair (V, W). We then investigate arbitrarysmall perturbations of the gauge field departing from the time-dependent background Aµ

parametrized by the “gauge energy” E. Later on we linearize the Yang–Mills equationaround it and diagonalize the fluctuation operator to obtain a spectrum of time-dependentnatural frequencies. To decide about the linear stability of the cosmic Yang–Mills configu-rations we have to analyze the long-time behavior of the solutions to Hill’s equation for allthese normal modes. To that end, we employ Floquet theory to learn that their growth rate


is determined by the stroboscopic map or monodromy, which is easily computed numer-ically for any given mode. We do so for a number of low-frequency normal modes andfind, when varying E, an alternating sequence of stable (bounded) and unstable (expo-nentially growing) fluctuations. The unstable bands roughly correspond to the parametricresonance frequencies. With growing “gauge energy” the runaway perturbation modesbecome more prominent, and some of them persist in the infinite-energy limit, where wedetect universal natural frequencies and monodromies. A special role is played by theSO(4)-invariant fluctuation of j=0 mode, which merely shifts the parameter E of the back-ground. We treat it exactly and beyond the linear regime. This “singlet” mode turns outto be marginally stable, i.e. it has a vanishing Lyapunov exponent. Its linear growth,however, gets limited by nonlinear effects of the full fluctuation equation, whose analyticsolutions exhibit wave beat behavior.

Finally we present a brief summary of our work in Chapter 7 and present the future out-look. We also collect several relevant data in various appendices and present a direct mapbetween the cylinder and Minkowski space via Carter–Penrose transformation, avoidingthe de Sitter detour, in appendix A.

7

Chapter 2

Geometry & Symmetry

Two of the most important concepts that play a fundamental role in physics are geometryof background space and presence of symmetries. They have well understood mathemat-ical foundation in differential geometry and Lie groups/algebras respectively. Here wepresent a short exposition of these mathematical topics that is essential towards buildingthe subsequent Gauge theory. The following contents are built upon some basic mathe-matical structures like vector spaces, groups and topological spaces all of which can befound in [15]. We refrain from presenting proofs of the statements in this chapter and sug-gest [16, 17], which are the source for most of the contents in this chapter. For details onLie groups/algebras and their representations we refer to classic texts [18] and [19].

2.1 Manifolds

Remark 2.1.1 The idea of a Manifold generalizes the notion of differentiation, or moreprecisely calculus on Rn, in the same way as a Topological space allows one to study thenotion of continuity in a more abstract way. In this thesis, we will only be concerned with(finite-dimensional) smooth manifolds, and subsequently, smooth structures on it.

Definition 2.1.1 An n-dimensional manifold M is a topological space1 equipped with anatlas consisting of charts (Ui, φi) such that

• Ui are open sets and the maps φi are homeomorphism between Ui and some openball in Rn, and

• the transition maps φ2 φ−11 : φ1(U1 ∩U2) → φ2(U1 ∩U2) is infinitely differentiable

for any two charts (U1, φ1) and (U2, φ2).

Example 2.1.1 A nice example of a smooth manifold is the round n-sphere

Sn := x ∈ Rn+1 | x · x = 1 (2.1.1)

embedded in Rn+1 that requires two charts: UN = Sn − (0, 0, . . . , 1) and US := Sn −(0, 0, . . . ,−1), along-with their respective stereographic projections:

φN(x) :=(

x11−xn+1 , x2

1−xn+1, . . . , xn

1−xn+1

)and

φS(x) :=(

x11+xn+1

, x21+xn+1

, . . . , xn1+xn+1

) (2.1.2)

with xn+1 = 1− x21 − x2

2 − . . .− x2n.

Definition 2.1.2 Given a manifold M a map f : M → R is called smooth or C∞ if it isinfinitely differentiable. The set of all such smooth maps are denoted as C∞(M).

1Some required technicalities like paracompactness and Haussdorffness have been assumed.

Chapter 2. Geometry & Symmetry 8

Definition 2.1.3 Diffeomorphism f : M → N between any two manifolds M and N is abijection such that both f and f−1 are smooth. The set of all diffeomorphisms from M toitself forms a group called Diff(M).

Remark 2.1.2 Notice here that the differentiability of such a map f is decided using localcharts: given a chart (U, φ) in M containing p ∈ U and another chart (V, ψ) in N contain-ing f (p) ∈ V, one can differentiate the map ψ f φ-1 using standard calculus. A crucialnotion in differential geometry is that of a tangent vector. It can intrinsically be defined interms of a curve on a manifold.

Definition 2.1.4 A curve σ : (−ε, ε) ⊂ R → M on a manifold M is defined as a smoothmap from some open interval of the real line to M.

Definition 2.1.5 The pull-back of a map f by another map φ is defined as

φ∗ f := f φ . (2.1.3)

Definition 2.1.6 A tangent to a curve σ at a point p = σ(0) on a manifold M is defined bythe map

σ′(t) : C∞(M)→ R , σ′(t=0)[ f ] := ∂∂t(σ∗ f )

∣∣t=0 (2.1.4)

This idea can be generalized as follows.

Definition 2.1.7 A tangent vector at a point m ∈ M is a map vm : C∞(M)→ R that satisfiesthe following properties

• vm[ f + g] = vm[ f ] + vm[g] ∀ f , g ∈ C∞(M) ,

• vm[a f ] = a vm[ f ] ∀ a ∈ R & f ∈ C∞(M) , and

• vm[ f g] = f (p) vm[g] + g(p) vm[ f ] ∀ f , g ∈ C∞(M).

Definition 2.1.8 The set of all such tangent vectors at m ∈ M is known as the tangent spaceat m and is denoted by Tm M, which forms a vector space over R.

Remark 2.1.3 This vector space is spanned by vectors

∂∂xi

∣∣m ≡ ∂i|m

where xi are the

coordinate functions in some chart (U, φ) containing m, i.e. φ(m) = (x1, x2, . . . , xn) andwhose actions is defined by

∂∂xi

∣∣m[ f ] := ∂(φ∗ f )

∂xi ∀ f ∈ C∞(N) . (2.1.5)

The dimension of Tm M is, therefore, equal to the dimension of the manifold M.

Definition 2.1.9 Given a map ϕ : M → N between two manifolds M and N and a vectorv ∈ Tm M, the push-forward of v by ϕ is defined by

ϕ∗v[ f ] := v[φ∗ f ] ∀ f ∈ C∞(N) . (2.1.6)

Remark 2.1.4 Note that the push-forward induces a map between following tangent spaces:

φ∗ : Tm M→ Th(m)N (2.1.7)

for manifolds M and N.

Definition 2.1.10 The vector space dual to Tm M is called cotangent space and is denoted byT∗m M.

Remark 2.1.5 The vector space T∗m M is spanned by covectors dxi|m defined as

〈dxi|m , v〉 := v[xi] ∀ v ∈ Tm M (2.1.8)


and has the same dimension as Tm M.

Definition 2.1.11 The pull-back ϕ∗ω ∈ T∗m M of a covector ω ∈ T∗n N by a map ϕ : M → Nbetween two manifolds M and N is defined via

〈ϕ∗ω , v〉 := 〈ω , ϕ∗v〉 (2.1.9)

with n = ϕ(m).

2.2 Fibre bundles

Remark 2.2.1 The theory of bundles has proved to be the correct way of studying clas-sical gauge theories like general relativity and Yang-Mills theory. Here we shall confineourselves to bundles constructed on/with manifolds.

Definition 2.2.1 A bundle is a triple (E, M, π) consisting of a manifold E, aka the targetspace, a manifold M, aka the base space, and a continuous surjection π : E → M, aka the

projection map. It is denoted diagrammatically asE

M

π

Definition 2.2.2 A bundle (E, M, π) is called a fibre bundle with typical fibre F if the inverseimage of all p ∈ M under π is isomorphic to some space F, i.e. π−1(p) ∼= F. If F is a vectorspace then the bundle (E, M, π) becomes a vector bundle.

Definition 2.2.3 A pair of fibre bundles (E, M, π) and (E, M, π) are called isomorphic (asbundles) if there exist a pair of diffeomorphisms ϕ : E → E and ψ : M → M such thatπ ϕ = ψ π and π ϕ−1 = ψ−1 π. Diagrammatically this means that the following

diagram, and its inverse, commutesE E

M M

ϕ

π π

ψ

Definition 2.2.4 A vector bundle (E, M, π) with typical fibre F is called locally trivial if forany U ⊂ M the induced bundle (π−1(U), U, π|π−1(U)) is isomorphic to the product bundle(U× F, U, π1), where π1 is the projection in the first slot. The set (U, ϕ)2 is known as a localtrivialization of the vector bundle. A trivial bundle is one where E = M × F and π = π1(projection in the first slot).

Remark 2.2.2 We shall only deal with locally trivial bundles here. A vector bundle (E, M, π)will, sometimes, be simply denoted E. A classic example of a bundle that is not globallytrivial is the following.

Example 2.2.1 A Möbius strip (E, S1, π) with fibre [0, 1] is locally isomorphic3 to the trivialbundle (S1 × [0, 1], S1, π1). The former, however, is not trivial as transporting any vectoracross a loop yields the corresponding vector, albeit inverted.

Definition 2.2.5 Given a fibre bundle (E, M, π) with typical fibre F and a pair of localtrivializations (Ui, ϕi) and (Uj, ϕj) with Ui ∪ Uj 6= 0 on it, we obtain transition functionsgij(x) with x ∈ Ui ∩Uj of the vector bundle (Ui ∩Uj)× F by realizing that ϕj ϕi(x, v) =(x, gij(x)v) for any vector v ∈ F. The set of such transition functions gij(x) forms a groupG ⊂ End(F) called the structure group.

2Notice, here, that ψ = IdU .3Meaning that they share local trivialization (U, ϕ) for any x ∈ U.


Definition 2.2.6 A (local) section σ (for some local trivialization (U, ϕ)) of a fibre bundle(E, M, π) is a smooth map σ : (U)M → E such that π σ = (IdU)IdM. The set of all such(local) global sections are denoted (Γ∞(U, E)) Γ∞(M, E). The space of global sections isalso denoted, more simply, as Γ∞(E).

2.3 Lie groups

Remark 2.3.1 The notion of symmetry in modern physics is analyzed with tools from thetheory of Lie groups and Lie algebras. We will only consider finite, matrix Lie groups inwhat follows.

Definition 2.3.1 A Lie group (aka continuous group) G is both a group and a smooth, finite-dimensional manifold where the group operation, · : G× G → G, as well as the inversionmap, −1 : G → G satisfying g−1 · g = g · g−1 = e ∀ g ∈ G with identity element e, aresmooth.

Remark 2.3.2 We will omit the group multiplication symbol · and use 1 interchangeablywith e for the matrix Lie groups (to be considered below) from now onward.

Definition 2.3.2 A Lie group homomorphism ϕ : G → H between Lie groups G and His a smooth map that is also a group homomorphism. The map ϕ becomes an isomor-phism if it is bijective and its inverse ϕ−1 is smooth; the Lie groups G and H then becomesisomorphic.

Definition 2.3.3 A positive-definite inner product is a bilinear map 〈·, ·〉 : V × V ∼−→ R for avector space V 3 x, y that is

(a) symmetric: 〈x, y〉 = 〈y, x〉 ,

(b) positive-definite: 〈x, x〉 > 0 , and

(c) nondegenerate: i.e. if 〈x, y〉 = 0 for all y =⇒ x = 0 .

If 〈x, x〉 6> 0 then the inner product is called indefinite.

Remark 2.3.3 Most of the important Lie groups in physics emerges as matrix Lie groupsby considering invertible linear maps V ∼−→ V on some finite-dimensional vector space Vthat preserves a given inner product on V. Such general linear groups are denoted GL(V).Furthermore, these are Lie subgroups4 of the general Linear groups GL(n,R) or GL(n,C)consisting of invertible linear maps on vector field R or C respectively. This becomesevident when we consider Riemannian manifolds which contains a metric (see Section2.6).

Example 2.3.1 The special linear groups SL(n,R) (over R) and SL(n,C) (over C) are n× ninvertible matrices with unit determinant, i.e.

SL(n,R/C) = A ∈ GL(n,R/C) | detA = 1 . (2.3.1)

Example 2.3.2 The Unitary group U(n) is the set of n× n complex-valued matrices whoseinverse is the same as its conjugate transpose:

U(n) = A ∈ GL(n,C) | A−1 = AT ≡ A† . (2.3.2)

The special unitary group is defined as

SU(n) = A ∈ U(n) | detA = 1 . (2.3.3)

4A Lie subgroup H ⊂ G is a subgroup as well as a submanifold of G. It turns out to be a Lie group in itself.


Remark 2.3.4 The special unitary groups SU(n) preserves the standard inner product onCn 3 x, y given by

〈x, y〉C =n

∑i=1

xiyi (2.3.4)

and also the norm of a given vector (induced from this inner product).

Example 2.3.3 The orthogonal group O(k, n) are the (k + n) × (k + n) matrices that pre-serve the following inner product (x, y ∈ Rk+n):

〈x, y〉k,n := −x1y1 − . . .− xkyk + xk+1yk+1 + . . . + xk+nyk+n . (2.3.5)

One can show that A ∈ O(k, n) if and only if

ATη(k,n)A = η(k,n) , (2.3.6)

where η(k,n) is the diagonal matrix diag(1, . . . 1,−1, . . . ,−1). The special orthogonal groupis defined as

SO(k, n) := A ∈ O(k, n) | detA = 1 . (2.3.7)

The orthogonal group O(0, n) is denoted O(n) and is more famously defined as

O(n) := A ∈ GL(n,R) | A−1 = AT . (2.3.8)

Similarly, the special orthogonal group SO(0, n) is denoted as SO(n).

Remark 2.3.5 The group SO(n) consists of rotations in n-dimensions and is thus knowsas isometry group of the n-sphere Sn. The group O(n) consists of rotations as well asreflections. The groups SO(1, 3) and SO(2, 4) known, respectively, as the Lorentz groupand the Conformal group are of special significance in physics. Another important groupfor us in this thesis is the de Sitter group SO(1, 4).

Example 2.3.4 The Poincaré group or the inhomogeneous Lorentz group ISO(1, 3) consists ofLorentz transformation together with translations and is defined as

ISO(1, 3) := Ix A | A ∈ SO(1, 3) , (2.3.9)

where the translations Ix acts on a given vector y ∈ R1,3 as

Ixy = x + y . (2.3.10)

Remark 2.3.6 It can be shown that the group SO(4) decomposes into two copies of SU(2)and that there exist a 2−1 (2 to 1) homomorphism

SU(2)× SU(2) 2−1−−→ SO(4) . (2.3.11)

This is a rather generic fact that arises when one considers spin-groups (e.g. Spin(4) here),which provide universal cover5 to some group (e.g. SO(4) here). To this end, we notedown the following relevant facts here:

Spin(4) ∼= SU(2)× SU(2) , Spin(3) ∼= SU(2) , and Spin(1, 3) ∼= SL(2,C) . (2.3.12)

5This is a topological term that refers to a connected, simply connected group that projects down to thegiven group G via a smooth surjection π such that any open U ∈ G lifts to a disjoint union π−1(U) whosemembers are isomorphic (individually) to U.


Definition 2.3.4 The left action of a Lie group G (aka left G-action) on a set M is defined bythe map

ϕl : G×M→ M , (g, m) 7→ ϕl(g, m) ≡ gm (2.3.13)

that satisfies the following properties:

• em = m for the identity element e ∈ G and every m ∈ M, and

• g1(g2m) = (g1g2)m for all g1, g2 ∈ G and m ∈ M.

The set M is known as a homogeneous space, that splits into an orbit space M/G of equiva-lence classes of orbits

Om := n ∈ M | ∃ g ∈ G : n = ϕl(g, m) . (2.3.14)

Definition 2.3.5 The left translations or left multiplications of a Lie group G is its diffeomor-phism lg ∈ Di f f (G) defined by

lg : G → G , h 7→ gh . (2.3.15)

Remark 2.3.7 An equivalent notion of right action defined by ϕr(g, m) := mg also exits andis of prime importance for the principle G-bundles that we will discuss in the next chapter.It is important to note here that one can induce a right-action ϕr from a given left-actionϕl as follows:

ϕr(g, m) := ϕl(g−1, m) ∀ m ∈ M . (2.3.16)

Similarly, we have the notion of right translations rg for the Lie group G but we will onlydeal with left translations here.

Definition 2.3.6 The coset space G/H for a Lie group G and its subgroup H ⊂ G is definedas

G/H := gH | g ∈ G where gH := gh | h ∈ H . (2.3.17)

Remark 2.3.8 There exist a natural left G-action (and hence, a left H-action) on G/H de-fined by

ϕl(g′, gH) = g′gH ∀ g, g′ ∈ G . (2.3.18)

Definition 2.3.7 A left G-action on M is called free if, for all m ∈ M, gm = m implies thatg = e. The action would be transitive if for all m, m′ ∈ M there exists g ∈ G such thatm = gm′.

Remark 2.3.9 If the left action of G on M is free then every orbit Om is diffeomorphic to theLie group G.

Definition 2.3.8 The stability/isotropy subgroup Gm of a left G-action on M 3 m for a Liegroup G is its closed subgroup defined by

Gm := g ∈ G | gm = m . (2.3.19)

Theorem 2.3.1 For a transitive left G-action there exist an isomorphism6 G/Gm ∼= M be-tween the coset space G/Gm and the homogeneous space M, for any m ∈ M, given by

jp : G/Gp → M , gGp 7→ gp . (2.3.20)

6Some technicalities like G be locally compact and M be locally compact and connected are required.


Remark 2.3.10 For a Lie group G and its closed subgroup H (e.g. Gm) the homogeneousspace G/H can be canonically endowed with the structure of a smooth manifold. More-over, there exist a canonical projection from G to G/H given by

π0 : G → G/H , g 7→ gH . (2.3.21)

Example 2.3.5 The round n-sphere Sn is diffeomorphic to the following homogenoeusspace:

Sn ∼= SO(n+1)/SO(n) . (2.3.22)

In particular, we have that S3 ∼= SO(4)/SO(3).

Example 2.3.6 A (2n+1)-sphere can also be realized as the following homogeneous space:

S2n+1 ∼= SU(n+1)/SU(n) . (2.3.23)

A special case is S3 ∼= SU(2), where the group SU(1) is trivial.

2.4 Vector fields

Definition 2.4.1 The tangent bundle (TM, M, π) over a manifold M is nothing but a unionof tangent spaces at all point of the manifold i.e.

TM =⋃

p∈M

Tp M , (2.4.1)

where the projection π just picks out the base point of a given vector in TM.

Remark 2.4.1 The dimension of the tangent bundle for an n-dimensional manifold is 2n, asits members are (n-dimensional) vectors of some Tp M labelled by the coordinate (n-tuple)of a base point p ∈ M.

Definition 2.4.2 A vector field X is a smooth section of the tangent bundle (TM, M, π). Theset of all such vector fields are denoted as X(M) := Γ∞(TM).

Remark 2.4.2 Given a vector field X ∈ X(M) and a smooth function f ∈ C∞(M), one canshow that X f , defined by

X f (p) := Xp[ f ] for Xp ∈ Tp M , (2.4.2)

is also smooth, i.e. X f ∈ C∞(M). The set of vector fields X(M) do not form a vectorspace rather a module over the algebra of smooth functions C∞(M)7. Nevertheless, onecan choose a basis of vector fields

∂

∂xi ≡ ∂i

on some local chart (U, ϕ) with coordinate

functions xi (see Remark 2.1.3) to write X ∈ X(M) as

X =n

∑i=1

(Xxi) ∂i . (2.4.3)

Definition 2.4.3 There is a well defined notion of Lie bracket associated with vector fieldsdefined as

[X, Y] f := X(Y f )−Y(X f ) , (2.4.4)

which satisfy the following properties

7An algebra is just a vector space V equipped with a multiplication operation V × V → V, which forfunctions is just composition. A module over an algebra is the same thing as a vector space over a field.


• bilinearity: [·, ·] : X(M)×X(M)∼−→ X(M),

• anti-symmetry: [X, Y] = −[Y, X], and

• Jacobi identity: [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0,

for all X, Y, Z ∈ X(M).

Remark 2.4.3 In general, it is not possible to induce a push-forward (see Definition 2.1.9)between vector fields X(M) 3 X and X(N) 3 Y of two manifolds M and N with a givenmap ϕ : M → N (it works if ϕ is a diffeomorphism). Nevertheless, it is useful to definethe following relation.

Definition 2.4.4 A vector field X ∈ X(M) is called h-related to another vector field Y ∈X(M) and is written as Y = ϕ∗X if for all p ∈ M we have that

ϕ∗Xp = Yϕ(p) . (2.4.5)

Remark 2.4.4 If X1 and X2 is h-related to Y1 and Y2 respectively then [X1, X2] is h-relatedto [Y1, Y2] for all X1, X2, Y1, Y2 ∈ X(M) i.e.

ϕ∗[X1, X2] = [Y1, Y2] . (2.4.6)

Definition 2.4.5 Given a vector field X ∈ X(M) its integral curve through p ∈ M is a curve

σX : (−ε, ε)→ M , (2.4.7)

such thatσX(0) = p and σ′X(t) = XσX(t) (2.4.8)

for all t ∈ (−ε, ε).

Definition 2.4.6 If the interval (−ε, ε) can be extended to whole of R for any p ∈ M thenthe vector field, and also the underlying manifold, is called complete.

Remark 2.4.5 Presence of singularities (e.g. a black hole) on a manifold makes it incom-plete. We will only consider complete manifolds in this thesis.

2.5 Lie algebra

Definition 2.5.1 A left-invariant vector field X for a Lie group G is defined as the vectorfield which is lg-related to itself for all g ∈ G:

lg∗X = X or lg∗Xg′ = Xgg′ (2.5.1)

for all g′ ∈ G.

Definition 2.5.2 The set of all left-invariant vector fields—forming a vector space that isdenoted as L(G)—together with the Lie bracket [·, ·] i.e. (L(G), [·, ·]) is known as the Liealgebra of G8.

Theorem 2.5.1 We have a Lie algebra isomorphism9 (TeG, [·, ·]) ∼= (L(G), [·, ·]) given by

j : TeG → L(G) , A 7→ j(A) ≡ LA , (2.5.2)

8Remark 2.4.4 is crucial here.9It is an invertible map that preserves the Lie algebraic structure.


where the left-invariant vector field LA is defined by

LAg := lg∗A ∈ TgG . (2.5.3)

Remark 2.5.1 Note that the Lie bracket on TeG is induced from the one on L(G) via themap j, i.e.

[A, B] := j−1[LA, LB] = lg−1∗[LAg , LB

g ] (2.5.4)

for any g ∈ G. This takes the form of usual matrix commutator for matrix Lie groups thatwe are dealing with. Thus we find that dimL(G) = dimTeG = dimG.

Example 2.5.1 We note down below in Table 2.1 Lie algebras TeG of some Lie groups Gthat we encountered before.

G TeGGL(n,R/C) M(n,R/C) := The set of n× n real/complex matricesSO(n) so(n) := A ∈ M(n,R) | AT = −ASU(n) su(n) := A ∈ M(n,C) | A = −A & trA = 0

TABLE 2.1: A list of Lie groups and their Lie algebras.

Theorem 2.5.2 A Lie group homomorphism ϕ : G → H between Lie groups G and Hinduces a Lie algebra homomorphism

dϕ ≡ ϕ∗ : TeG → TeH , i.e. ϕ∗[A, B] = [ϕ∗A, ϕ∗B] . (2.5.5)

Remark 2.5.2 Given a basis E1, . . . , En of L(G) ∼= TeG for an n-dimensional Lie group G,its structure constants f k

ij are defined by

[Ei, Ej] =n

∑k=1

f kij Ek . (2.5.6)

Example 2.5.2 The structure constants for the Lie algebra su(2) are given by f kij = 2 ε k

ij ,where ε k

ij is the 3-dimensional Levi-Civita symbol.

Theorem 2.5.3 A left-invariant vector field X on a Lie group G is complete.

Remark 2.5.3 A consequence of the above theorem is that there exist a unique integralcurve

t 7→ σLA(t) (2.5.7)

for all t ∈ R and left-invariant vector field LA constructed from a given A ∈ TeG of the Liegroup G.

Definition 2.5.3 The exponential map for a Lie group G is defined by

exp : TeG → G , A 7→ exp(A) := σLA(t=1) . (2.5.8)

Remark 2.5.4 The exponential map is locally diffeomorphic. Furthermore, it lifts TeG to aconnected component of G (connected to e) and is a surjection when G is compact.

Definition 2.5.4 A one-parameter subgroup of a Lie group G is a smooth homomorphismµ : R→ G from the additive group R into G i.e.

µ(t1 + t2) = µ(t1)µ(t2). (2.5.9)


Theorem 2.5.4 If µ : R→ G is a one-parameter subgroup of G then, for all t ∈ R,

µ(t) = exp(tA) with A := µ∗ddt

∣∣0 . (2.5.10)

Remark 2.5.5 The above theorem shows that there exist a one-to-one correspondence be-tween one-parameter subgroups of a Lie group G and its Lie algebra TeG.

Definition 2.5.5 Given a vector field X ∈ X(M) one defines the flow generated by X as theone-parameter group ϕt : t ∈ R where the set of maps ϕt : M → M are nothing but theintegral curves

ϕt(m) = σX(t) with σX(0) = m . (2.5.11)

One can define the Lie derivative of Y along X, for Y ∈ X(F), by

LXY =ddt

(ϕ−1t )∗(Y)

∣∣∣t=0

. (2.5.12)

Remark 2.5.6 Interestingly, one can show that LXY = [X, Y].

2.6 Tensor fields

Definition 2.6.1 A (p,q)-tensor Tp,q(V) for a vector space V and its dual V∗ is the space ofmultilinear functions

f : V∗ ×p· · · ×V∗ ×V ×

q· · · ×V ∼−→ R (2.6.1)

and is, alternatively, denoted using tensor products as V ⊗q· · · ⊗V ⊗V∗ ⊗

p· · · ⊗V∗.

Remark 2.6.1 We note down have the following results/observations:

T0,0(V) := R , T0,1(V) ∼= V∗ and T1,0(V) ≡ (V∗)∗ ∼= V (2.6.2)

for any finite-dimensional vector space V. Elements of Tp,q(V) can be easily expanded interms of a given basis of V and the corresponding dual basis of V∗.

Definition 2.6.2 A (p,q)-tensor bundle (Tp,q M, M, π) over a manifold M is the followingdisjoint union of tensors:

Tp,q M =⋃

m∈M

Tp,q(Tm M) (2.6.3)

where the projection π associates the base point m ∈ M of a given vector in Tp,q M.

Remark 2.6.2 The tangent bundle TM is isomorphic to T1,0M, while the bundle T0,1M isisomorphic to the so called cotangent bundle T∗M defined analogously to Definition 2.4.1before.

Definition 2.6.3 The exterior algebra over a vector space V – denoted as∧

V – is the alge-bra10 generated by the so called wedge product ∧ satisfying

v1 ∧ v2 = −v2 ∧ v1 (2.6.4)

for all v1, v2 ∈ V. A subspace of∧

V consisting of linear combinations of k–fold wedgeproducts of vectors in V is denoted

∧k(V).

10Plainly speaking, it is a linear combination of all possible (finitely many) anti-symmetrized tensor prod-ucts of vectors in V.


2.6.1 Differential forms

Definition 2.6.4 A k-form ω is a (0, k)-tensor field ω ∈ Γ∞(M, T0,k M) such that ωm ∈∧k(Tm M) for all m ∈ M.

Remark 2.6.3 The space of k-forms is denoted Ωk(M). Notice that Ω0(M) ≡ C∞(M) andthe only member of Ωn(M) (upto to a scalar multiple), known as the volume form, is de-noted µ or dV. The space Ωn(M) splits into two equivalence classes [µ+], [µ−] with therelation µ ∼ µ′ defined by µ = λµ′ such that µ ∈ [µ+] if λ > 0 and µ ∈ [µ−] if λ < 0.

Remark 2.6.4 On a chart (U, ϕ) of M with coordinates ϕ(m) = (x1, . . . , xn) one can expanda k–form ω as

ω = 1k!

n

∑i1,...,ik=1

ωi1,...,ik dxi1 ∧ . . . ∧ dxik (2.6.5)

where the coefficients ωi1,...,ik ∈ C∞(M) are totally anti-symmetric in its indices.

Definition 2.6.5 The exterior derivative d is the linear map d : Ωk(M)∼−→ Ωk+1(M) defined,

for all ω ∈ Ωk(M), by

dω = 1k!

n

∑i1,...,ik=1

∂i(ωi1,...,ik)dxi ∧ dxi1 ∧ . . . ∧ dxik , (2.6.6)

where the partial derivative ∂i is taken with respect to coordinate xi. It can also be definedin a coordinate independent way, for all X1, . . . , Xk+1 ∈ X(M), as

dω(X1, . . . , Xk+1) =k+1

∑i=1

(−1)i+1 Xi(ω(Xi, . . . , Xi, . . . , Xk+1))

+ ∑1≤i<j≤n

(−1)i+j ω([Xi, Xj], X1, . . . , Xi, . . . , Xj, . . . , Xk+1) ,(2.6.7)

where the circumflex means that the symbol beneath it is omitted.

Remark 2.6.5 The action of exterior derivative follow graded Leibniz rule:

d(α ∧ β) = dα ∧ β + (−1)p α ∧ dβ (2.6.8)

for all α ∈ Ωp(M) and β ∈ Ωq(M). Moreover, it can easily be shown (using, e.g. the firstdefinition above) that

d2 ≡ d d = 0 . (2.6.9)

Definition 2.6.6 For a map ϕ : M→ N and a k-form ω ∈ Ωk(N) its pull-back ϕ∗ω ∈ Ωk(M)is defined via

(ϕ∗ω)m(X1, . . . , Xk) := ωϕ(m)(ϕ∗X1, . . . , ϕ∗Xk) (2.6.10)

for all X1, . . . , Xk ∈ X(M).

Remark 2.6.6 It can be shown that the exterior derivative is natural, i.e. it is compatiblewith the pull-back:

ϕ∗(dω) = d(ϕ∗ω) , (2.6.11)

for any ω ∈ Ωk(M). Furthermore, one can prove that

ϕ∗(α ∧ β) = ϕ∗α ∧ ϕ∗β and (ϕ ϕ)∗ω = ϕ∗(ϕ∗ω) . (2.6.12)


Definition 2.6.7 The Lie derivative of ω along X, for ω ∈ Ωk(M) and X ∈ X(M), can bedefined analogous to Definition 2.5.5:

LXω =ddt

(ϕ−1t )∗(ω)

∣∣∣t=0

. (2.6.13)

Remark 2.6.7 There exists a nice formula by Élie Cartan for the Lie derivative of differentialforms given, for any X ∈ X(M), by

LX = d ιX + ιX d , (2.6.14)

where the linear map ιX : Ωp(M)∼−→ Ωp−1(M) is the interior product defined by

(ιXω)(X1, . . . , Xp−1) = ω(X, X1, . . . , Xp−1) (2.6.15)

for ω ∈ Ωp(M) and X1, . . . , Xp−1 ∈ X(M).

2.6.2 Metric

Definition 2.6.8 A metric g is a (0, 2)-tensor field g ∈ Γ∞(T0,2M) where gm for every m ∈ Mdefines an inner product on the vector space Tm M. If this inner product is positive definitethen the manifold is called Riemannian. If the metric g has an underlying inner productthat is indefinite then the manifold is called pseudo-Riemannian.

Remark 2.6.8 A metric g on an n-dimensional manifold M can be written, in local coordi-nates, as

g =n

∑i=1

gij dxi ⊗ dxj with gij := g(∂i, ∂j) (2.6.16)

being the metric components for the basis vector fields ∂i, ∂j ∈ X(M). Often times, in physicsliterature, the tensor product sign is ignored. The fact that g is non-degenerate means thatthe matrix g can be inverted to yield another matrix g−1 with components g−1

ij =: gij that,in turn, defines a (2, 0)-tensor field called the induced metric. These can be used to raise orlower indices of any tensor component, e.g.

T j′k′

i′ := gii′gjj′gkk′Tijk ∀ T ∈ Γ∞(M, T1,2M) . (2.6.17)

Example 2.6.1 A standard example of a Reimannian manifold is Rn with metric

g = (dx1)2 + . . . + (dxn)2 . (2.6.18)

A prominent example of a psuedo-Riemannian manifold is Minkowski spaceR1,n with met-ric

g = −(dx1)2 + (dx2)2 + . . . + (dxn)2 . (2.6.19)

Definition 2.6.9 The signature of a pseudo-Riemannian manifold (M, g) of dimension n isa count of the number of positive and negative eigenvalues of the matrix gij and is denotedby an n-tuple of + and −. A 4-dimensional Lorentzian manifold has one of the signaturedifferent from the rest three.

Example 2.6.2 The signature of the Minkowski spaceR1,3 as presented above is (−,+,+,+),and is thus a Lorentzian manifold.


Definition 2.6.10 Two (pseudo-)Riemannian manifolds (M, gM) and (N, gN) are called con-formal if their metrices are related by some smooth function Ω2 ∈ C∞(N):

gM = Ω2 gN . (2.6.20)

Definition 2.6.11 A locally orthonormal basis of 1-forms ei (aka coframe) with i = 1, . . . , nfor an n-dimensional Riemannian manifold M satisfy, for all ei, ej ∈ Ω1(M),

g(ei, ej) = gij = δij . (2.6.21)

Similarly, one defines locally orthonormal basis of vector fields (aka frame) X1, . . . , Xnby demanding that they satisfy

gij = g(Xi, Xj) = δij (2.6.22)

for all Xi, Xj ∈ X(M).

2.6.3 Maurer–Cartan form

Definition 2.6.12 A k-form ω ∈ Ωk(G) of a Lie group G is said to left-invariant if, for allg ∈ G,

l∗gω = ω , i.e. l∗g(ωg′) = ωg−1g′ ∀ g′ ∈ G . (2.6.23)

The set of all left-invariant 1-forms on G is denoted L∗(G).

Remark 2.6.9 From Remark 2.6.6 we see that if ω is left-invariant then so is its exteriorderivative:

l∗g(dω) = dl∗gω . (2.6.24)

Remark 2.6.10 Similar to j of Theorem 2.5.1, there exist an isomorphism j between T∗e Gand L∗(G) given by

j : T∗e G → L∗(G) , ω 7→ j(ω) ≡ λω , (2.6.25)

where the left-invariant 1-forms λω is defined as

λωg := l∗g−1(ω) ∈ T∗g G . (2.6.26)

Notice that these 1-forms are dual to the left-invariant vector fields LA for A ∈ TeG:

〈λω, LA〉g = 〈ω, A〉 (2.6.27)

for all g ∈ G.

Remark 2.6.11 For a basis E1, . . . , En of L(G) with dimG = n (see Remark 2.5.2) we candefine the dual basis ω1, . . . , ωn of L∗(G) by

〈ωi, Ej〉 = δij , (2.6.28)

which satisfy the Maurer–Cartan structure equation

dωi +12

f ijk ω j ∧ωk . (2.6.29)


Definition 2.6.13 The Maurer–Cartan 1-form Ωl is the L(G)-valued 1-form on G that, forany A ∈ TgG, gives a left-invariant vector field as follows

〈Ωl , A〉g′ := lg′∗(lg−1∗A) (2.6.30)

for any g′ ∈ G.

Remark 2.6.12 It is more useful to consider the Maurer–Cartan 1-form to be valued inTeG ∼= L(G), which yields a nice result:

〈Ωl , LA〉 = A . (2.6.31)

It can be shown that the Maurer-Cartan 1-form takes the following explicit form for thematrix Lie groups that we are intereseted in,

Ωijl =

n

∑k=1

(g−1)ik dgkj , (2.6.32)

where gij are the coordinates for the matrix Lie group G on a given chart.

2.7 Integration

Definition 2.7.1 A manifold M is called orientable if it is equipped with a nowhere van-ishing volume form µ. An orientation of M refers to a choice made for one of the the twoequivalence classes in Remark 2.6.3; this is usually chosen to be the positive one.

Example 2.7.1 The standard volume form on Rn, which is an oriented manifold, is givenby

µ = dx1 ∧ . . . ∧ dxn . (2.7.1)

Möbius strip provides a classic example of a nonorientable manifold.

Definition 2.7.2 Integration of a compactly supported11 volume form µ of an oriented man-ifold M that is covered with charts (U1, φ1), . . . , (UN , φN) can be defined, using a partitionof unity12 fi, as ∫

Mµ =

N

∑i=1

∫φ(Ui)

(φ−1i )∗( fiµ) . (2.7.2)

Remark 2.7.1 It can be shown that this definition of integration is independent of the choiceof a chart.

Theorem 2.7.1 For an oriented manifold M with boundary ∂M (that inherits an inducedorientation from M) the integral of a compactly supported (n−1)-form ω ∈ Ωn−1(M) isrelated to an integral of its exterior derivative dω:∫

Mdω =

∫∂M

ω . (2.7.3)

Remark 2.7.2 This is the famous Stockes’ theorem. Notice that if the manifold M has noboundary then the above integral vanishes.

11This simply means that it vanishes outside of a compact subset in M12It is a set of smooth functions fi ∈ C∞(M) that vanish outside of Ui, lies within [0, 1], and satisfies the

condition: ∑Ni=1 fi(m) = 1 ∀ m ∈ M.


Definition 2.7.3 For an n-dimensional, oriented, Riemannian manifold (M, g) the Rieman-nian volume form is given by

µg =√|g|dx1 ∧ · · · ∧ dxn , (2.7.4)

where |g| := |det(gij)| for metric components gij.

Remark 2.7.3 Integration on Riemannian manifolds are performed using volume form µg.

2.8 Hodge duality

Remark 2.8.1 On an n-dimensional pseudo-Riemannian manifold (M, g) one can inducean inner product on Ωk(M)—generated by k-forms dxi1 ∧ · · · ∧ dxik with i1, . . . , ik ∈1, . . . , n—that is given by⟨

dxi1 ∧ · · · ∧ dxik , dxj1 ∧ · · · ∧ dxjk⟩

= gi1 j1 · · · gik jk , (2.8.1)

where gij := g(dxi, dxj).

Definition 2.8.1 The Hodge star operator ∗ : Ωk(M)∼−→ Ωn−k(M) is a linear map that is

defined, on an oriented n-dimensional pseudo-Riemannian manifold (M, g) with volumeform µ, as

α ∧ ∗β = 〈α, β〉 µ (2.8.2)

for all α, β ∈ Ωk(M). Here ∗β is called the Hodge dual of β.

Remark 2.8.2 For an n-dimensional oriented pseudo-Riemannian manifold (M, g) withsignature (−, s. . .,−,+, n−s. . . ,+) the following condition holds true on Ωp(M):

∗2 = (−1)p(n−p)+s . (2.8.3)

Example 2.8.1 For the Minkowski space R1,3 with signature (−,+,+,+), coordinates(x0, x1, x2, x3) =: (t, x, y, z) and volume form dV = dt ∧ dx ∧ dy ∧ dz we have the follow-ing results:

∗ dz = −dt ∧ dx ∧ dy , ∗ dt ∧ dx = −dy ∧ dz , ∗ dt ∧ dy = dx ∧ dz ,∗ dt ∧ dz = −dx ∧ dy , ∗ dx ∧ dy = −dt ∧ dz , ∗ dx ∧ dz = −dt ∧ dy ,∗ dy ∧ dz = dt ∧ dx , ∗ dt ∧ dx ∧ dy = −dz , ∗ dt ∧ dx ∧ dz = dy ,∗ dt ∧ dy ∧ dz = −dx , ∗ dx ∧ dy ∧ dz = −dt , ∗ dt ∧ dx ∧ dy ∧ dz = −1 .

2.9 Representations

Definition 2.9.1 A representation of a Lie group G is a Lie group homomorphism

Π : G → GL(V) (2.9.1)

for a finite dimensional vector space V.

Definition 2.9.2 A representation of a Lie algebra g, for a finite dimensional vector space V, isa Lie algebra homomorphism

π : g→ gl(V) , (2.9.2)

where gl(V) := End(V)13.13The endomorphism of V denoted End(V) is the space of linear maps V ∼−→ V.


Example 2.9.1 The standard/fundamental representation of a Lie group G 3 g is Π(g) = gand of a Lie algbra g 3 X is π(X) = X.

Example 2.9.2 For matrix Lie groups G with Lie algebra g, its adjoint representation is thefollowing homomorphism

Ad : G → GL(g) , g 7→ Adg , (2.9.3)

where the adjoint map AdA is defined by

Adg : g∼−→ g , X 7→ Adg(X) := gXg−1 . (2.9.4)

Remark 2.9.1 It can be shown that the map Ad induces (see Theorem 2.5.2) the followinglie algebra homomorphism

Ad∗ ≡ ad : g→ gl(g) , X 7→ adX , (2.9.5)

where the action of the map adX can be shown to be the following

adX : g→ g , Y 7→ adX(Y) := [X, Y] . (2.9.6)

This is known as the adjoint representation of a finite-dimensional Lie algebra g.

2.10 Maxwell equations

Remark 2.10.1 For the nice14 manifolds that we are interested in this thesis the Maxwelltheory boils down to a choice of the gauge potential A, which is a 1-form on the manifold.

Definition 2.10.1 For Minkowski space R1,3 we choose the gauge potential A as

A = A0 dt + Ai dxi , (2.10.1)

and the corresponding field strength F is organized as

F := dA = Ei dxi ∧ dt +12

Bi εijk dxj ∧ dxk (2.10.2)

with electric field E and magnetic field B.

Remark 2.10.2 The source free Maxwell equations viz.

∇ · B = 0 and ∇× E +∂B∂t

= 0 (2.10.3)

are given bydF = 0 . (2.10.4)

Notice that, due to Remark 2.6.5, the above condition is trivially satisfied here.

Remark 2.10.3 The other two Maxwell equations with source viz.

∇ · E = ρ and ∇× B− ∂E∂t

= J (2.10.5)

with charge density ρ and current density J is given by

δF = j with j = −ρ dt + Ji dxi ∈ Ω1(R1,3) , (2.10.6)14Here it means a connected and simply connected manifold.


where the codifferential δ := ∗d∗.

Remark 2.10.4 An important feature of the Maxwell theory is that it possesses gauge sym-metry, i.e. the transformation

A→ A + d f , (2.10.7)

for all f ∈ C∞(R1,3), leaves the field strength F invariant. There are many ways to fixthis redundancy by employing some kind of gauge-fixing. On Minkowski space we canalways work in the so called “temporal gauge" where A0 = 0 (see Chapter 6 of [16]).

Remark 2.10.5 Noticing that δ2 = ± ∗ d2∗ = 0 and applying δ on the Maxwell equationswith source j, we arrive at the following continuity equation:

0 = δ2F = δj =⇒ ∂ρ

∂t+∇ · J = 0 . (2.10.8)

24

Chapter 3

Gauge theory

All four known forces of nature viz. gravity, electromagnetism, weak and strong nuclearforces can be described (at least classically) in terms of a gauge theory. The study of mod-ern gauge theory requires the notion of principal bundles and various structures on it.While it is possible to study a lot of physics—including Yang–Mills theory—with just vec-tor bundles alone, e.g. as in [16], a lot of deep physics arising from the underlying topol-ogy of the base manifold can not be fully appreciated without following principle bundlesapproach. We review, in this chapter, the construction of gauge theory from principal bun-dles without bothering about proof of any statement, all of which can be found in [20].For a quick review of Yang–Mills theory one may refer to [17] or a nice review article byDaniel and Viallet [21].

3.1 Principal G-bundles

Definition 3.1.1 A bundle (P, M, π) is called a principle G-bundle and depicted, diagram-

matically, asG P

M

rg

π if there exist a free right action of the Lie group G on P and,

moreover, if there exist a bundle isomorphism between (P, M, π) and (P, P/G, π0), with

the canonical projection π0 (2.3.21), meaning that the diagramP P

M P/G

π

ϕ

π0

ψ

commutes.

Remark 3.1.1 Notice that the structure group in this case is G. Furthermore, the fibreπ−1(m), for any m ∈ M, is diffeomorphic to G, but it does not have a canonical groupstructure. Another way to put this would be to say that the Manifold M has a G-fibreattached to all its points, wherein the identity element is forgotten. Also, notice here thatthe projection map is insensitive to the G-action.

Example 3.1.1 The simplest example of a principle bundle is (G×M, M, π) with the rightaction given by (x, g0)g := (x, g0g).

Example 3.1.2 An important example of a principle G-bundle is the so called frame bundleLM over an n-dimensional manifold M, which is a collection of basis-frames

Lm M := (b1, . . . , bn) | 〈b1, . . . , bn〉 = Tm M ∼= GL(n,R) (3.1.1)

corresponding to the tangent bundle TM attached at each point m of the manifold, i.e.

LM :=⋃

m∈M

Lm M . (3.1.2)

Chapter 3. Gauge theory 25

The free right action of any g ∈ GL(n,R) on a given (b1, . . . , bn) ∈ Lm M is given by

(b1, . . . , bn)g := (bi gi1, . . . , bi gi

n) . (3.1.3)

Example 3.1.3 Another important example of a principle bundle is (G, G/H, π) where Hacts freely from right on G via group multiplication resulting into orbits of cosets G/H. Afamous example of this kind is the Hopf bundle, represented diagrammatically as follows:

U(1) SU(2)

SU(2)/U(1)

≡S1 S3

S2

Definition 3.1.2 A principle morphism between a pair of bundles (P, M, π) and (P′, M′, π′)is a bundle morphism (ϕ, ψ) that, additionally, satisfies

ϕ(pg) = ϕ(p)g ∀ p ∈ P and g ∈ G . (3.1.4)

Remark 3.1.2 The notion of trivialization, both local (U, ϕ) as well as global (G×M, M, π1),follows similar to the general theory of fibre bundles that we saw before, albeit with thisextra condition of principle morphism. In a similar way, the idea of transition functions gijbetween any two such overlapping local trivializations (Ui, ϕi) and (Uj, ϕj) and the corre-sponding structure group carries over. One can also define smooth sections on a principlebundle P just like before.

Remark 3.1.3 The set of all principal morphisms between a bundle P to itself forms a groupAut(P) called the automorphism group of the principle bundle P. In the case of a trivialbundle P = G×M we have that Aut(P) ∼= C∞(M, G), where the latter is the well-knowngroup of gauge transformations.

Theorem 3.1.1 A principle G-bundle (P, M, π) is trivial if and only if it possesses a smoothsection σ : M→ P.

Remark 3.1.4 An illustrative counter-example of the above theorem is the fact that theframe bundle LS2 over the 2-sphere is not trivial because there does not exist nowherevanishing smooth vector fields, and hence a basis, on TS2 (look at the north or southpoles). This result is famously summarized as “sphere can not be combed".

Remark 3.1.5 The topological properties, such as twisting, of the base manifold M is in-trinsically linked with that of the principal bundle P and also carries over to the belowdefined associated bundles.

3.2 Associated bundles

Definition 3.2.1 The G-product, denoted X ×G Y, of two spaces X and Y, both admittingright action of G, is the space of orbits under this action on the Cartesian product X × Y.In other words, it is defined via the following equivalence relation

(x, y) ∼ (x′, y′) iff ∃ g ∈ G : x′ = xg and y′ = yg , (3.2.1)

where the equivalence class is denoted as [x, y].


Definition 3.2.2 For a given principle G-bundle (P, M, π) and a manifold F, which admitsa left G-action, one defines the associated bundle PF by1

PF := P×G F where (p, v)g := (pg, g−1v) (3.2.2)

and the projection πF by

πF : PF → M , πF([p, v]) := π(p) . (3.2.3)

Remark 3.2.1 It can be shown that the associated bundle (PF, M, πF) has the structure of afibre bundle with typical fibre F.

Example 3.2.1 An important example of a fibre associated with the frame bundle LM withLie group GL(n,R) is the tensor bundle Tp,q M with fibres F = (Rn)×p × (Rn∗)×q wherethe left action of a given g ∈ GL(n, R) on some v ∈ F is given by the following representa-tion ρ

(ρ(g)v)i1,...,ipj1,...,jq := v

i′1,...,i′pj′1,...,j′q

(g)i1i′1

. . . (g)ipi′p(g−1)

j′1j1

. . . (g−1)j′qjq . (3.2.4)

Another very useful generalization of this is the so called tensor of density ω that admits aleft-action via the following representation

(ρ(g)v)i1,...,ipj1,...,jq := (detg)ω v

i′1,...,i′pj′1,...,j′q

(g)i1i′1

. . . (g)ipi′p(g−1)

j′1j1

. . . (g−1)j′qjq . (3.2.5)

Definition 3.2.3 Given a principle morphism (ϕ, ψ) between principal G-bundles (P, M, π)and (P′, M′, π′) one defines an associated bundle morphism ϕF between associated bundlesPF ×G F and P′ ×G F by

ϕF([p, v]) := [ϕ(p), v] , (3.2.6)

which is well-defined since

ϕF([pg, g−1v]) = [ϕ(pg), g−1v] = [ϕ(p)g, g−1v] = [p, v] . (3.2.7)

Remark 3.2.2 An associated bundle PF is called trivial if the underlying principal bundleP is trivial. An important point to note here is that a trivial associated bundle is a trivialfibre bundle, but the converse is not true.

Definition 3.2.4 Let H ⊂ G be a closed subgroup of G while P respectively P′ are princi-pal G- respectively H-bundle defined over the same base space. If there exist a principalmorphism ϕ with respect to H, i.e.

ϕ(ph) = ϕ(p)h (3.2.8)

for all h ∈ H, then P is called a G-extension of H while P′ is called a H-restriction of P.

Remark 3.2.3 While there always exists an extension P of a given P′ as defined above,the converse is not always true. This has important ramifications both in Riemanniangeometry as well as in Yang–Mills theory; for the latter this is related to the importantquestion of spontaneous breakdown of the internal symmetry group from G down to H.

Theorem 3.2.1 A principle G-bundle (P, M, π) can be restricted to a closed subgroup H ⊂G iff the bundle (P/H, M, π0) admits a smooth section.

1Notice how we have employed left G-action to define right action on F (see Remark 2.3.7).


Remark 3.2.4 An important application of the above theorem is that one can always havea Riemannian metric defined on any n-diemensional manifold M as LM/SO(n), for theclosed subgroup SO(n) ⊂ GL(n,R), always admits a smooth section; the same is notalways true for a pseudo-Riemannian metric as, LM/SO(1, n−1) (with closed subgroupSO(1, n) ⊂ GL(n,R)), for instance, does not always admits a smooth section because ofpossible topological restrictions.

Theorem 3.2.2 There exists a one-to-one G-equivariant correspondence between a sectionσ ∈ Γ∞(PF) of an associated bundle (PF, M, πF) and map φ : P→ F satisfying

φ(pg) = g−1φ(p) ∀ p ∈ P and g ∈ G (3.2.9)

where the section σφ, corresponding to φ, arises as

σφ(x) := [p, φ(p)] for p ∈ π−1(x) . (3.2.10)

Definition 3.2.5 Given two local trivializing sections2 σi : Ui ⊂ M→ P and σj : Uj ⊂ M→P on a given principal G-bundle P with Ui ∩Uj 6= 0 there exists some local gauge functionsΛij : Ui ∩Uj → G such that

σj(x) = σi(x)Λij(x) ∀ x ∈ Ui ∩Uj . (3.2.11)

One defines local representatives si : Ui → F, for a section s ∈ Γ∞(PF) corresponding to suchlocal sections σi, by

si(x) := φsi(σi(x)) . (3.2.12)

Remark 3.2.5 The local representatives also satisfy the same gauge transformation rule asabove, i.e.

sj(x) = si(x)Λij(x) . (3.2.13)

It turns out that these gauge functions Λij are nothing but transitions functions gij of localtrivializations arising from σi and σj. For this thesis, we will identify the gauge group withthe structure group.

3.3 Connections

Remark 3.3.1 Let (P, M, π) be a principal G-bundle. There exists a Lie algebra homomor-phism between L(G) ∼= TeG and Γ∞(TP) given by

i : TeG → Γ∞(TP) , A→ XA , (3.3.1)

where the induced vector-field XA arises from the right action of G on P as follows

XA[ f ] := f ′(p exp(tA))(t=0) . (3.3.2)

Definition 3.3.1 For a given principal G-bundle (P, M, π) one defines the vertical subspaceat p ∈ P, denoted by VpP, as follows

VpP := X ∈ TpP | π∗(X) = 0 (3.3.3)

The horizontal subspace HpP arises as the orthogonal complement of VpP in the tangentspace TpP:

TpP = VpP⊕ HpP . (3.3.4)

2This refers to the existence of a local trivialization (Ui, ϕi) such that ϕ−1i (x, g) := σi(x)g.


Remark 3.3.2 It can be shown that

XAp ∈ VpP ∀ p ∈ P . (3.3.5)

This means that the above map i induces an isomorphism ip between TeG and VpP.

Definition 3.3.2 A connection on a principle G-bundle (P, M, π) is a smooth assignment ofHpP to each point p ∈ P such that

(a) TpP = VpP⊕ HpP,

(b) (rg)∗(HpP) = HpgP (see Remark 2.3.7) and

(c) every Xp ∈ TpP has a unique decomposition according to (a) as follows

Xp = ver(Xp) + hor(Xp) , (3.3.6)

where ver(Xp) ∈ VpP and hor(Xp) ∈ HpP.

Remark 3.3.3 A more technically convenient way to deal with connections is by associatingthem with a Lie-algebra valued one-form ω in the following way

ωp(X) = i−1p (ver(X)) , (3.3.7)

which, in turn, imposes following conditions on ω:

(i) ωp(XA) = A for all p ∈ P and A ∈ L(G),

(ii) (rg)∗ω = Adg−1 ω, i.e. (r∗gω)p(X) = Adg−1(ωp(X)) for all X ∈ TpP and

(iii) X ∈ HpP iff ωp(X) = 0.

Definition 3.3.3 For a local trivializing section σ : U ⊂ M → P of a principal G-bundle P,its local gauge fields arise from the following local representative of a Lie-algebra valuedone-form ω:

ωU := σ∗ω . (3.3.8)

We label the 1-form components of such local gauge fields in Yang–Mills theory as

Aµ ≡ (ωU)µ , (3.3.9)

and in general relativity asΓµ ≡ (ωU)µ . (3.3.10)

Theorem 3.3.1 An explicit form of the local gauge field ωU for the local trivialization ϕ :π−1(U)→ U × G arising from σ with (α, β) ∈ T(x,g)(U × G) ∼= TxU ⊕ TgG is given by

(ϕ∗ω)(x,g) = Adg−1(ωUx (α)) + 〈Ωl , β〉g , (3.3.11)

where Ωl is the Maurer–Cartan one-form.

Example 3.3.1 An important example of such a local representative ωU is in the case of theframe bundle LM for an n-dimensional manifold M with a given chart (U, φ). For a givenlocal section σ : U ⊂ M→ LM defined by

σ(m) := ((∂1)m, . . . , (∂n)m, ) (3.3.12)

the corresponding ωU has following components (the three indices below are divided intotwo indices α, β for the Lie-algebra L(GL(n,R)) and one 1-form index µ):

((ωU)µ)αβ =: Γα

µβ with α, β, µ = 1, . . . , n (3.3.13)


where Γαµβ is the famous Christoffel symbol of this Levi-Civita or affine connection3 that is

widely used in Riemannian geometry and general relativity.

Theorem 3.3.2 Let (P, M, π) be a principal G-bundle with Ui, Uj ⊂ M such that Ui ∩Uj 6=0. Further, let A(i)

µ and A(j)µ be local gauge functions arising from given local trivializing

sections σi : Ui → P and σj : Uj → G respectively. The transformation of these fields underthe action of gauge functions Λij : Ui ∩Uj → G is, for any m ∈ M, given by

A(j)µ (m) = AdΛij(m)−1

(A(i)

µ (m))+ (Λ∗ijΩl)µ(m) . (3.3.14)

Remark 3.3.4 We notice that for matrix Lie groups the above transformation rule takes thefollowing simple form:

A(j)µ (m) = Λij(m)−1

(A(i)

µ (m))

Λij(m) + Λij(m)−1∂µΛij(m) . (3.3.15)

Remark 3.3.5 This gauge transformation behaviour is what prevents a gauge field ωU frombeing global, i.e. A or Γ 6∈ Ω1(M). In particular, this explains why the Christoffel symbolis not a tensor! This is because of the following transformation rule between any two givencharts (U, φ) and (U′, φ′) on a 4-dimensional manifold M 3 m with φ(m) = x0, . . . , x4and φ′(m) = x′0, . . . , x′4 where the indices, such as α below, takes temporal α=0 as wellas spatial α=1, 2, 3 values:

Γ′αµβ =

∂x′α

∂xα

∂xµ

∂x′µ∂xβ

∂x′βΓα

µβ+

∂x′α

∂xλ

∂xλ

∂x′µ∂x′β. (3.3.16)

An explicit expression for Γαµβ for a given basis of vector fields ∂0, . . . , ∂4 on M equipped

with a pseudo-Riemannian metric g with components g(∂µ, ∂ν) = gµν is given by

Γαµβ =

12

gαλ(

gµλ,β + gλβ,µ − gµβ,λ)

, (3.3.17)

where gµβ,λ := ∂λ[gµβ].

3.4 Parallel transport

Definition 3.4.1 Owing to the fact that π∗ : HpP→ Tπ(p)M is an isomorphism, there existsthe notion of a unique vector field for a given X ∈ X(M) known as the horizontal lift of Xand is denoted as X↑. This satisfies, for all p ∈ P, the following conditions

(i) ver(X↑p) = 0 and

(ii) π∗(X↑p) = Xπ(p).

Remark 3.4.1 The act of horizontal lifting is G-equivariant i.e. (rg)∗(X↑p) = X↑pg.

Definition 3.4.2 A horizontal lift of a smooth path γ : [a, b] ⊂ R → M is another pathγ↑ : [a, b] → P which is horizontal, i.e. ver(γ↑) = 0 such that π(γ↑(t)) = γ(t) for allt ∈ [a, b].

Theorem 3.4.1 For each point p ∈ π−1(γ(a)) there exist a unique horizontal lift γ↑ :[a, b]→ P of γ such that γ↑(a) = p.

3This relies on a choice of the natural metric on M, which is akin to choosing a basis G ab of the Lie-algebra

L(GL(n,R)) with components (G ab )c

d := δcbδa

d.


Remark 3.4.2 Given a path γ : [a, b] → M and another path β : [a, b] → P which projectsdown to γ, i.e. π(β(t)) = γ(t) for all t ∈ [a, b], there exits some unique function g : [a, b]→G such that

γ↑(t) = β(t) g(t) ∀ t ∈ [a, b] . (3.4.1)

Theorem 3.4.2 The unique path g : [a, b] → G defined above satisfies the following firstorder ODE in terms of a Lie-algebra valued 1-form ω:

0 = Adg(t)−1∗

(ωβ(t)(Xβ,β(t))

)+ 〈Ωl , Xg〉g(t) , (3.4.2)

where Xβ,β(t) ≡ β(t) is the tangent vector to the curve β at point β(t) (see Definition 2.1.6).

Remark 3.4.3 For a matrix Lie group G and a local gauge field Aµ (3.3.9), the above ODEtakes the following simple form

g(t) = −Aµ(γ(t)) γµ(t) g(t) , (3.4.3)

where the components of the curve γ in a local chart has been denoted as γµ. The solutionto this ODE, for an initial condition g(0) = g0, is obtained as a path-ordered exponentialin the following way

g(t) =

P exp

− t∫a

Aµ(γ(s)) γµ(s)ds

g0

= g0 −

t∫a


g0

+

t∫a

ds1

∫ s1

ads2 Aµ1(γ(s1)) Aµ2(γ(s2)) γµ1(s1) γµ2(s2)

g0 − . . . .

(3.4.4)

Remark 3.4.4 From the above result we observe that the local expression for the horizontallift γ↑, for the path γ : [a, b]→ U ⊂ M, is given by

γ↑(t) = σ(γ(t))

P exp

− t∫a


g0 . (3.4.5)

Definition 3.4.3 The parallel transport along a path γ : [a, b] → M is defined by the follow-ing map

Tγ : π−1(γ(a))→ π−1(γ(b)) , p 7→ γ↑p(b) , (3.4.6)

where γ↑ is the unique horizontal lift of γ passing through p ∈ π−1(γ(a)).

Remark 3.4.5 We observe that Tγ is a bijection on fibres and thus on G. An interestingthing happens when γ : [a, b] → M is a loop i.e. γ(a) = γ(b); one obtains a natural mapfrom loops based at γ(a) ∈ M to elements of G. The subgroup of all elements of G that canbe obtained in this way is called the holonomy group of the principle bundle P. This playsan important role in understanding the relation between certain topological properties ofM with the connection Aµ.

Definition 3.4.4 Let (P, M, π) be a principal G-bundle equipped with a connection 1-formω. Furthermore, let (PF, M, πF) be associated with P via the left action of G on F. A vertical


subspace of T[p,v]PF is defined, analogous to that of principal bundle, as

V[p,v]PF := X ∈ T[p,v]PF | πF∗X = 0 . (3.4.7)

Similarly, the horizontal subspace H[p,v]PF can be defined as the orthogonal complement:

T[p,v]PF = V[p,v]PF ⊕ H[p,v]PF . (3.4.8)

Definition 3.4.5 The horizontal lift of a path γ : [a, b] → M to the associated bundle PF andpassing through [p, v] ∈ π−1

F (γ(a)) is defined as

γ↑F(t) := [γ↑(t), v] (3.4.9)

where γ↑(a) = p.

Remark 3.4.6 One can define parallel transport Tγ along a path γ : [a, b] → M on the as-sociated bundle PF analogous to the principal bundle case using the above definition ofthe horizontal lifting. For associated vector bundles PV with the vector space V admit-ting a linear representation of G, this notion then facilitates the following definition of thecovariant derivative.

3.4.1 Covariant derivative

Definition 3.4.6 The covariant derivative of a section ψ : M → PV of an associated vectorbundle PV along a path γ : [0, ε]→ M with ε > 0 at m0 =: γ(0) is defined by

DXγ,γ(0)ψ := lim

t→0

(Tγ(ψ(γ(t)))− ψ(m0)

t

)∈ π−1

V (m0) . (3.4.10)

Remark 3.4.7 The covariant derivative DX, for X ∈ Tm M, has the following algebraicproperties for any section ψ, ψ ∈ Γ∞(PV):

(i) D f X+Yψ = f DXψ + DYψ for all f ∈ C∞(M) and X, Y ∈ Tm M,

(ii) DX(ψ + ψ) = DXψ + DXψ for all X ∈ Tm M and

(iii) DX( f ψ) = X[ f ]ψ + f DXψ for all f ∈ C∞(M) and X ∈ Tm M.

This following more generic notion of a covariant derivative (3.4.12) is related to this one,as we will see below.

Definition 3.4.7 The exterior covariant derivative of a k-form ω ∈ Ωk(P) on a principal bun-dle P is a horizontal (k+1)-form defined by

Dω(X1, . . . , Xk+1) := dω(hor(X1), . . . , hor(Xk+1)) (3.4.11)

for a given set of vector fields X1, . . . , Xk+1 ∈ X(P).

Remark 3.4.8 This notion of covariant derivative can be extended to an associated vectorbundle PV discussed before with the aid of a given G-equivariant map φ : P → F that hasan associated section σφ ∈ Γ∞(PV) (see Theorem 3.2.2) as follows

Dφ := dφ hor . (3.4.12)

One can show, for any X ∈ X(P) and a given connection ω on P, that

F 3 Dφ(X) ≡ DXφ := dφ(X) + ω(X)φ , (3.4.13)


where we notice that φ are F-valued functions on P. Furthermore, one can pull this defini-tion back to M using any local trivializing map σ : U ⊂ M → P and noticing the fact thatthe pull-back operation is natural:

σ∗(Dφ)(X) := dσ∗φ(X) + σ∗(ω)(X)(σ∗φ) ∀ X ∈ X(P) . (3.4.14)

Example 3.4.1 For a given local orthonormal coframe of vector fields ∂µ with µ =0, 1, 2, 3 on a 4-dimensional Lorentzian manifold M, e.g. the Minkowski space R1,3, thecovariant derivative D∂µ

=: Dµ of the above-mentioned section φ (or σφ to be precise) isgiven in terms of local gauge fields Aµ as

Dµφ := ∂µφ + Aµφ . (3.4.15)

Example 3.4.2 The covariant derivative Dµ =: ∇µ of the tensor bundle Tp,q(M) associatedwith the frame bundle LM on M is given in terms of the Levi–Civita connection and canbe expressed in terms of the Chritoffel symbols. For example, covariant derivative of T ∈T1,2(M) with components Tµ

αβ := T(dxµ, ∂α, ∂β) is given by

Dρ Tµαβ = ∂ρTµ

αβ + ΓµρλTλ

αβ − ΓλραTµ

λβ − ΓλρβTµ

λα . (3.4.16)

Remark 3.4.9 The Levi–Civita connection Γ is metric compatible:

∇g = 0∇µ or gαβ = 0 . (3.4.17)

Moreover this connection is torsion-free, i.e.

0 = T(X, Y) := ∇XY−∇YX− [X, Y] (3.4.18)

for all X, Y ∈ X(M). Choosing vector fields X = ∂α and Y = ∂β, the torsion-free conditionensures that the Christoffel symbol is symmetric in the subscript indices:

Γµαβ = Γµ

βα . (3.4.19)

3.4.2 Curvature

Definition 3.4.8 If ω is a connection 1-form on a principal G-bundle P then Dω =: Ω isthe Lie-algebra valued curvature 2-form of ω.

Remark 3.4.10 It can be shown that the curvature 2-form Ω satisfies the Bianchi identity:

DΩ = 0 . (3.4.20)

Theorem 3.4.3 For arbitrary pair of vector fields X, Y ∈ X(P) the curvature 2-form Dωsatisfies the following Cartan structure equation

Dω(X, Y) := dω(X, Y) + [ω(X), ω(Y)] , (3.4.21)

where we have employed the Lie bracket on L(G).

Example 3.4.3 For the Levi–Civita connection Γ the curvature 2-form DΓ =: R—valuedin L(GL(n,R)) for an n-dimensional pseudo-Riemannian manifold M—is known as theRiemann curvature and is given by

R = dΓ + Γ ∧ Γ . (3.4.22)


This turns out to be a (1, 3)-tensor, i.e. R ∈ T1,3(M) and, in local coframe ∂µ and framedxµ with µ = 0, 1, 2, 3, of M, admits the following expression in terms of Christoffelsymbol

Rρσµν = ∂µΓρ

νσ − ∂νΓρµσ + Γρ

µλ Γλνσ − Γρ

νλ Γλµσ . (3.4.23)

Remark 4.3.11 Riemann curvature tensor gives rise to the so called Ricci tensor Rµν :=Rρ

µρν ∈ T0,2(M) through index contraction, which yields the scalar curvature R := gµνRµν.With these tools and the insight of equivalence principle:“For any point m ∈ M there always exists a local chart (U, φ) with m ∈ U admittinga smooth local orthonormal frame σ ∈ Γ∞(LU) of the local frame bundle LU; in otherwords, every (spacetime) manifold is locally flat in a Minkowski sense"Albert Einstein was able to construct the following field equation relating the curvature ofspacetime with its matter content

Gµν := Rµν −12

gµνR =8πG

c4 Tµν , (3.4.24)

where G is the Newton constant, c is the speed of light and Tµν is the stress-energy tensorarising from the variation of the matter action Sm with respect to the matric:

Tµν = − 2√−det g

δSm

δgµν. (3.4.25)

Example 3.4.4 A trivial solution of the vacuum Einstein equation is the Minkowski metricηµν which is globally flat. Another very important solution is the FLRW metric for homo-geneous and isotropic universe, where the evolution of the scale factor a(t) is governed byan appropriate stress-energy tensor Tµν, is given by

g = −c2dt2 + a(t)2(

dr2

1− κr2 + r2dθ2 + r2 sin2 θdφ2)

, (3.4.26)

where r, θ, φ are coordinates on celestial spheres with respect to an observer (like us)and the parameter κ = −1, 0 or 1 denotes the topology of the 3-dimensional Euclideanspace as being open, flat or closed respectively.

Example 3.4.5 For local Yang–Mills field A := σ∗ω the curvature 2-form Dω =: F is givenby

F = dA + A ∧ A . (3.4.27)

3.5 Yang-Mills equation

Remark 3.5.1 A local expression for the Yang–Mills curvature F on a Lorentzian manifoldM with local orthonormal coframe ∂µ is given by

Fµν = ∂µ Aν − ∂ν Aµ − [Aµ, Aν] . (3.5.1)

Remark 3.5.2 The Yang–Mills curvature transforms under a local gauge transformationΛij(m) (see Definition 3.2.5) in the following way

F(j)µν (m) = Λij(m)−1 F(i)

µν (m)Λij(m) , (3.5.2)

and is thus global, i.e. F ∈ Ω2(M).


Remark 3.5.3 The Yang–Mills equation on a Lorentzian manifold M (with D as in 3.4.12)is given by

∗ D(∗F) = J , (3.5.3)

where the 1-form J ∈ Ω1(M) is the current that arises from the presence of any source“charge" on the manifold.

Remark 3.5.4 The Yang–Mills action (coupling constant g)

SYM =1

2g2

∫M

Tr(F ∧ ∗F) (3.5.4)

together with the following action for the source J:

SJ =1g2 Tr(A ∧ ∗J) (3.5.5)

gives rise to the above Yang–Mills equation by variational principle.

Remark 3.5.5 Maxwell’s theory of electromagnetism that we saw in the last chapter is aspecial case of Yang–Mills theory where the gauge group is U(1).

35

Chapter 4

Yang–Mills equations on dS4

Here we review the conformal relation of the 4-dimensional de Sitter space with a finiteLorentzian cylinder with S3-slicing, study calculus on S3 and present the Yang–Mills fieldequations on the cylinder. The content of this chapter is partially taken from [22–24]1.

4.1 The de Sitter-Minkowski correspondence via S3-cylinder

The de Sitter space in four dimensions dS4 has a natural embedding as a single-sheetedhyperboloid in five-dimensional Minkowski space R1,4 with coordinates (q0, qA); A =1, 2, 3, 4 and global length scale ` and is given by

− q20 + q2

1 + q22 + q2

3 + q24 = `2. (4.1.1)

One can use the flat metric on R1,4:

ds2(1,4) = −dq2

0 + dq21 + dq2

2 + dq23 + dq2

4 (4.1.2)

to construct the corresponding metric on dS4 using the coordinate constraint (4.1.1). Themetric thus obtained is conformally equivalent to the metric on a finite Lorentzian cylin-der over the 3-sphere S3: I × S3 with I = (0, π). To see this, we employ the followingcoordinates

qA = `ωA csc τ and q5 = −` cot τ with τ ∈ (0, π) , (4.1.3)

where ωA are the natural embedding coordinates of S3 → R4: ωA ωA = 12. A naturalhyperspherical parametrisation of ωA is given by

ω1 = sin χ sin θ cos φ , ω2 = sin χ sin θ sin φ , ω3 = sin χ cos θ , ω4 = cos χ ,(4.1.4)

with 0 ≤ χ, θ ≤ π and 0 ≤ φ ≤ 2π. The modified metric has the following form:

ds2 = ds2(1,4)|dS4 =

`2

sin2 τ

(−dτ2 + dΩ2

3)

; dΩ 23 = dωA dωA|dS4 . (4.1.5)

By gluing two copies of such Lorentzian cylinders at τ=0 by taking τ ∈ I := (−π, π)

one finds that half of the resultant cylinder I × S3 is conformally equivalent to the 4-dimensional Minkowski space via following parametrization of (t, x, y, z) ∈ R1,3:

cot τ =r2−t2+`2

2 ` t, ω1 = γ

x`

, ω2 = γy`

, ω3 = γz`

, ω4 = γr2−t2−`2

2 `2 , (4.1.6)

1The role of La and Ra in [22] is interchanged as compared to this thesis.2Repeated indices are summed over.

Chapter 4. Yang–Mills equations on dS4 36

where we have abbreviated

FIGURE 4.1: An illustration of the map between a cylinder 2I × S3 and Minkowski space R1,3.The Minkowski coordinates cover the shaded area. The boundary of this area is given by thecurve ω4= cos χ= cos τ. Each point is a two-sphere spanned by ω1, ω2, ω3, which is mapped to

a sphere of constant r and t.

r2 = x2 + y2 + z2 & γ =2 `2√

4 `2t2 + (r2 − t2 + `2)2=

2 `2√4 `2r2 + (t2 − r2 + `2)2

. (4.1.7)

A straightforward computation yields

cos τ − cos χ = γ > 0 , (4.1.8)

which shows that only half of the cylinder—constrained by cos τ < cos χ—is allowed bythe map (4.1.6). Plugging this map back into the de Sitter metric (4.1.5) we obtain

ds2 =`2

t2

(−dt2 + dx2 + dy2 + dz2) with (x, y, z) ≡ (x1, x2, x3) ∈ R3 & t ∈ R , (4.1.9)

which is the Minkowski metric up to a conformal factor3. A smooth gluing of the twocylinders across the time slice t = τ = 0 is nicely depicted in Figure 4.1. Now, looking atthe maps (4.1.4) and (4.1.6) we see a SO(3)-symmetry that we can exploit by writing

ωa = sin χ ωa and xa = r ωa for a = 1, 2, 3 and r = `γ sin χ (4.1.10)

where the unit S2 coordinates ωa are given by

ω1 = sin θ cos φ , ω2 = sin θ sin φ and ω3 = cos θ . (4.1.11)

We can identify the unit S2 between the Minkowski space and the de Sitter space using(4.1.6) and (4.1.10), so that, we have an effective map between coordinates (t, r) and (τ, χ):

t`

=sin τ

cos τ − cos χand

r`

=sin χ

cos τ − cos χfor χ > |τ| , (4.1.12)

which reveals that the triangular (τ, χ) domain (where the points represent unit S2) isnothing but the Penrose diagram of Minkowski space (See Figure 4.2). The special linesand points in Figure 4.2 are given by

3In fact, the map (4.1.6) covers only the positive half of the Minkowski space, i.e. t ∈ R+ for the originalcylinder I × S3.


FIGURE 4.2: Penrose diagram of Minkowski space R1,3. Each point hides a two-sphere S23θ, φ.Blue curves indicate t=const slices while brown curves depict the world volumes of r=const

spheres. The lightcone of the Minkowski-space origin is drawn in red.

τ=0 south pole boundary — north pole τ=±π χ±τ=π

(τ, χ) (0, χ) (τ, π) (±χ, χ) (0, π) (0, 0) (±π, π) (±π∓χ, χ)

(t, r) (0, r) (t, 0) (±∞, ∞) (0, 0) (t, ∞) (±∞, r) (±r, r)t=0 r=0 I ± origin i0 i± lightcone

with the Minkowski spatial and temporal infinity i0 and i± corresponding to the edgeswhile the Minkowski null infinity I ± corresponding to the conformal boundary χ=|τ| ofthe Penrose diagram.

Equation (4.1.12) can be used to obtain the Jacobian for the transformation between thecoordinates ym ∈ τ, χ, θ, φ and xµ ∈ t, r, θ, φ:

(Jm

µ

):=

∂(τ, χ, θ, φ)

∂(t, r, θ, φ)=

1`

(p −q

q −p

)⊕ 12 , (4.1.13)

where the polynomials p, q are given by

p = γ2

`2 (r2+t2+`2)/2 = 1− cos τ cos χ and q = γ2

`2 t r = sin τ sin χ . (4.1.14)

A more direct way of visualizing the conformal correspondence between Minkwoski spaceand the S3-cylinder is via Carter-Penrose transformation that readily produces the light-cone picture (see Appendix A for details).


4.1.1 Structure of 3-sphere and its harmonics

The presence of S3 in the metric (4.1.5) is an added advantage which can be exploited towrite an SO(4)-invariant gauge connection and obtain the corresponding fields by solvingYang–Mills equation on the Lorentzian cylinder. These quantities can be later exported tothe Minkowski spacetime owing to the conformal invariance of the vacuum Yang–Millsequation in 4-dimensions. To this end, we start with the group SO(4)—which is isomor-phic to two copies of SU(2) (up to a Z2 grading). Each of these SU(2) has a group ac-tion that generates a left (right) multiplication (aka translation) on S3. This can easily bechecked using the map

g : S3 → SU(2); (ω1, ω2, ω3, ω4) 7→ −i(

β α∗

α −β∗

), (4.1.15)

with α := ω1 + iω2 and β := ω3 + iω4. This parameterization of g ensures that theidentity element e = 12 of the group SU(2) can be obtained from (0, 0, 0, 1), i.e. the Northpole of S3. It is well known that S3 is the group manifold of SU(2). Keeping this in mind,we consider the Maurer–Cartan one-form

Ωl(g) := g−1 dg = ea Ta, with Ta = −iσa (4.1.16)

being the SU(2) generators. The left-invariant one-form4 ea can, alternatively, be expressedusing the so called self-dual ’t Hooft symbol ηa

BC:

ea = −ηaBC

ωB dωC where ηabc = εabc and ηa

b4 = −ηa4b = δa

b . (4.1.17)

They satisfy following useful identities

δab ea eb = dΩ23 and dea + εabc eb ∧ ec = 0. (4.1.18)

The left-invariant vector fields La—generating the right translations—are dual to ea andare given by

La = −ηaBC

ωB∂

∂ωC⇒ [La, Lb] = 2 εabc Lc . (4.1.19)

In a similar way, the right-invariant vector fields Ra (generating the left translations) aregiven by

Ra = −ηaBC

ωB∂

∂ωC⇒ [Ra, Rb] = 2 εabc Rc , (4.1.20)

where the anti self–dual ’t Hooft symbols ηaBC

are obtained from (4.1.17) by flipping thesign in the previous defintion whenever B, C = 4. Furthermore, the vector fields La andRa act on the one-forms ea via their Lie derivative, which can be performed using Cartanformula (see Section 2.6):

La eb := LLa eb = d ιLa eb + ιLa deb

= d(

eb(La))− εb

ij ιLa

(ei ∧ ej

)= d

(δb

a

)− εb

ij

(ei(La)ej − ej(La)ei

)= 2 εabc ec ,

(4.1.21)

where in the second line we have used (4.1.18). A similar calculation for the action of Rayields

Ra eb = 0 . (4.1.22)4Named so because they remain invariant under the dragging induced by the left SU(2) multiplication.


We can now write differential d of the functions f ∈ C∞ (I × S3) using La5 as

d f = dτ ∂τ f + ea La f . (4.1.23)

Functions on S3 can be expanded in a basis of harmonics Yj(χ, θ, φ) with 2j ∈ N0, whichare eigenfunctions of the scalar Laplacian6:

−43Yj = 2j(2j+2)Yj = 4j(j+1)Yj = − 12 (L2 + R2)Yj = − 1

4 (D2 + P2)Yj (4.1.24)

where L2 = LaLa and R2 = RaRa are (minus four times) the Casimirs of su(2)L and su(2)R,respectively,

− 14 L2 Yj = − 1

4 R2 Yj = − 14 43Yj = j(j+1)Yj . (4.1.25)

We have also introduced D2 = DaDa and P2 = PaPa with

Da := Ra + La = −2 ε bca ωb ∂c and Pa := Ra − La = 2 ω[a ∂4] (4.1.26)

with ∂A ≡ ∂∂ωA

so that

[Da,Db] = 2 ε cab Dc , [Da,Pb] = 2 ε c

ab Pc , [Pa,Pb] = 2 ε cab Dc . (4.1.27)

Hence, Da spans the diagonal subalgebra su(2)D ⊂ so(4), which generates the stabilizersubgroup SO(3) in the coset representation S3 ∼= SO(4)/SO(3). Therefore, D2 is (minusfour times) the Casimir of su(2)D, with eigenvalues l(l+1) for l = 0, 1, . . . , 2j, and 1

4D2 =42 is the scalar Laplacian on the S2 slices traced out in S3 by the SO(3)D action.

To further characterize a complete basis of S3 harmonics, there are two natural options,corresponding to two different complete choices of mutually commuting operators to bediagonalized. First, the left-right (or toroidal) harmonics Yj;m,n are eigenfunctions of L2 =R2, L3 and R3:

i2 L3 Yj;m,n = n Yj;m,n and i

2 R3 Yj;m,n = m Yj;m,n , (4.1.28)

and hence the corresponding ladder operators

L± = (L1 ± iL2)/√

2 and R± = (R1 ± iR2)/√

2 (4.1.29)

act, in their Hermitian avatar, as

i2 L± Yj;m,n =

√(j∓n)(j±n+1)/2 Yj;m,n±1 ,

i2 R± Yj;m,n =

√(j∓m)(j±m+1)/2 Yj;m±1,n .

(4.1.30)

The normalized harmonics Yj;m,n can be expanded in terms of functions α, β and theircomplex conjugates as

Yj;m,n(ω) =2j

∑k=0

Kj,m,n(k) αn+m+k αk βj−m−k βj−n−k with

Kj,m,n(k) = (−1)m+n+k

√2j + 12π2

√(j + m)!(j−m)!(j + n)!(j− n)!

(n + m + k)!(j− n− k)!(j−m− k)!k!.

(4.1.31)

5One can use Ra and its dual 1-form as well.6The SO(4) spin of these functions is actually 2j, but we label them with their half-spin for reasons to be

clear below.


They satisfy the orthonormality condition∫d3Ω3 Yj;m,n Yj′;m′,n′ = δj,j′ δm,m′ δn,n′ with d3Ω3 = sin2 χ sin θ dχ dθ dφ . (4.1.32)

Second, the adjoint (or hyperspherical) harmonics Yj;l,M are eigenfunctions of L2 = R2, D2

and D3 (its Hermitian version to be precise):

− 14 D

2 Yj;l,M = l(l+1) Yj;l,M and i2 D3 Yj;l,M = M Yj;l,M , (4.1.33)

with the ladder-operator actions [18]

i2 D± Yj;l,M =

√(l∓M)(l±M+1)/2 Yj;l,M±1 ,

i2 P± Yj;l,M = ∓

√(l∓M−1)(l∓M)/2 cj,l Yj;l−1,M±1

±√(l±M+1)(l±M+2)/2 cj,l+1 Yj;l+1,M±1 ,

i2 P3 Yj;l,M =

√l2−M2 cj,l Yj;l−1,M +

√(l+1)2−M2 cj,l+1 Yj;l+1,M ,

(4.1.34)

wherecj,l =

√((2j+1)2 − l2

)/((2l−1)(2l+1)

). (4.1.35)

In this case, there exists a recursive construction for harmonics on Sk+1 from those on Sk:

Yj;l,M(χ, θ, φ) = Rj,l(χ)Yl,M(θ, φ) ; Rj,l(χ) = i2j+l√

2j+1sin χ

(2j+l+1)!(2j−l)! P−l− 1

22j+ 1

2(cos χ) , (4.1.36)

where Yl,M are the standard S2 spherical harmonics and Pba denote associated Legendre

polynomials of the first kind.7 The two bases of harmonics are related via the standardClebsch–Gordan series for the angular momentum addition j⊗ j = 0⊕ 1⊕ . . .⊕ 2j,

Yj;m,n =2j

∑l=0

l

∑M=−l

Cl,Mm,n Yj;l,M , with Cl,M

m,n = 〈2j; l, M|j, m; j, n〉 (4.1.37)

being the Clebsch–Gordan coefficients enforcing m+n=M and l ∈ 0, 1, . . . , 2j.

A somewhat cumbersome calculation involving (4.1.6), (4.1.7) and (4.1.17) gives us thecylinder one–forms in Minkowski coordinates:

e0 := dτ = γ2

`3

( 12 (t

2+r2+`2)dt− t xkdxk)= γ2

`3

( 12 (t

2+r2+`2)dt− t r dr)

and

ea = γ2

`3

(t xadt−

[ 12 (t

2−r2+`2) δak + xaxk + ` εajkxj]dxk)= γ2

`3

(xa[r t dt− 1

2 (t2+r2+`2)dr

]− 1

2 (t2−r2+`2) r dxa − ` r2εajk xjdxk) .

(4.1.38)

7With fractional indices, it is rather a Gegenbauer polynomial, but also a hypergeometric function (seeeq. (2.8) of [25, 26]).


We further note down the expressions for the vector fields La in terms of S3-angles (χ, θ, φ),using (4.1.4) and (4.1.19), for later purposes

L1 = sin θ cos φ ∂χ + (cot χ cos θ cos φ + sin φ) ∂θ − (cot χ csc θ sin φ− cot θ cos φ) ∂φ ,L2 = sin θ sin φ ∂χ + (cot χ cos θ sin φ− cos φ) ∂θ + (cot χ csc θ cos φ + cot θ sin φ) ∂φ ,L3 = cos θ ∂χ − cot χ sin θ ∂θ − ∂φ .

(4.1.39)

4.2 Yang-Mills field equations

One can make use of the left-invariant one forms ea to expand a generic Yang–Mills gaugepotential A (in the temporal gauge Aτ=0) as

A = Xa(τ, g) ea, (4.2.1)

where the three SU(2) matrices Xa depends, in general, on both the cylinder parameter τand internal S3-coordinates ω via the map g (4.1.15). The corresponding field strength F ,for this gauge connection A, is given by

F = dA+A∧A = Xa e0 ∧ ea +12

(L[b Xc] − 2 εabc Xa + [Xb, Xc]

)eb ∧ ec , (4.2.2)

where e0 := dτ and the dot on Xa refers to its derivative w.r.t. τ. The Yang–Mills equation,for this gauge field A,

d ∗ F +A∧ ∗F − ∗F ∧A = 0 , (4.2.3)

after a straightforward calculation, yields the constraint condition

− 2 i Ja Xa + [Xa, Xa] = 0 , (4.2.4)

along-with the field equation

Xa =− 4(

J2+1)

Xa − 4 i εabc Jb Xc + 4 Ja Jb Xb + 3 εabc [Xb, Xc]

+ 2 i [Xa, Jb Xb] + 2 i [Xb, Ja Xb] + 4 i [Jb Xa, Xb]− [Xb, [Xa, Xb]] ,(4.2.5)

where the operators Ja and J2 are given by

Ja :=i2

La and J2 := Ja Ja . (4.2.6)

Finding a general solution for this equation is a daunting task, but progress can be madein two limiting cases as follows.

One of the limiting case of (4.2.5) is the Abelian one, where commutators vanish to yield

Xa = −4(

J2+1)

Xa + 2 i εabc Jb Xc . (4.2.7)

Furthermore the constraint (4.2.4) in this case becomes

Ja Xa = 0 . (4.2.8)

One thing to note here is that the above constraint, along-with the temporal gauge Aτ = 0,is not the usual Coulomb gauge on Minkowski space. In fact, we can make use of theinverse Jacobian (4.1.13), while promoting the gauge potential to the Minkowski space


A= Aa ea = Aµ dxµ = A, to get

0 = Aτ = A (∂τ) = A(

`2

γ2 (p ∂t + q ∂r))

=⇒(r2 + t2 + `2) At + 2 r t Ar = 0 . (4.2.9)

Solution to (4.2.7) was obtained in [3] using hyperspherical harmonics Yj;m,n as we will seein the Chapter 5.

Another limiting case of (4.2.5) is when, a more symmetric, SO(4)-equivariant conditionis imposed yielding [7, 10]:

Xa =12(1 + ψ(τ)) Ta (4.2.10)

with some function ψ : I → R and SU(2) generators Ta satisfying the Lie algebra

[Ta, Tb] = 2 εabc Tc . (4.2.11)

Moreover, we work in the adjoint representation for the Lie algebra genertors Ta wheretr(Ta Tb)=−8 δab. The constraint (4.2.4) for this symmetric ansatz, i.e.

[Xa, Xa] = 0 (4.2.12)

is automatically satisfied and the field-equation (4.2.5) becomes

Xa = −4 Xa + 3 εabc [Xb, Xc]− [Xb, [Xa, Xb]] . (4.2.13)

Solution to this equation was obtained in [27] as we will see in Chapter 6.

43

Chapter 5

Abelian solutions: U(1)

In this chapter we present the knotted electromagnetic fields arising via the “de Sitter"method and study its various properties. The content of this chapter has generated threepublished works: [22, 23, 28]. Section 5.4 on null solutions is due to Olaf Lechtenfeldwith verification by Colin Becker and some clarifications from Harald Skarke. I have beeninvolved at all stages of research works for rest of the sections in this chapter.

5.1 Family of "knot" solutions

It was shown in [3], that the general solution of (4.2.7) decomposes into spin-j representa-tions of so(4) and are labelled with hyperspherical harmonics of S3. We review the con-struction of these solutions below in two steps: (a) we first solve (4.2.7) on the S3-cylinderand (b) we then pull these solution back to Minkowski space using (τ, χ) → (t, r) coordi-nate transformation.

5.1.1 Generic solution on the S3-cylinder

To solve (4.2.7) we first write it down in terms of ladder operators J± and J3 (4.2.6) togetherwith the redefined functions

X± :=1√2(X1 ± iX2) (5.1.1)

as follows:∂2

τ X+ = −4 (J2 − J3 + 1)X+ − 4 J+ X3 ,

∂2τ X3 = −4 (J2 + 1)X3 + 4 J+ X− − 4 J− X+ ,

∂2τ X− = −4 (J2 + J3 + 1)X− + 4 J− X3 .

(5.1.2)

Similarly, the constraint condition (4.2.8) takes becomes

0 = J3 X3 + J+ X− + J− X+ . (5.1.3)

We can solve (5.1.2) by the following ansatz

X+ = ∑j,m,n

Zj;m,n+ eiΩj,m,nτ ; Zj;m,n

+ = c+ Yj;m,n+1 ,

X3 = ∑j,m,n

Zj;m,n3 eiΩj,m,nτ ; Zj;m,n

3 = c3 Yj;m,n , and

X− = ∑j,m,n

Zj;m,n− eiΩj,m,nτ ; Zj;m,n

− = c− Yj;m,n−1 ,

(5.1.4)

Chapter 5. Abelian solutions: U(1) 44

where X∗(j, m, n) with ∗ ∈ +, 3,− has been expanded in terms of S3-harmonics. Plug-ging this ansatz back in (5.1.2) and using (4.1.30) we find, for every mode (j, m, n), aneigenvalue equation for the vector (c+ c3 c−)T:

M

c+c3c−

= Ω(j, m, n)2

c+c3c−

, with

M =

4(j2+j−n) 2√

2(j−n)(j+n+1) 0

2√

2(j−n)(j+n+1) 4(j2+j+1) −2√

2(j+n)(j−n+1)

0 −2√

2(j+n)(j−n+1) 4(j2+j+n)

,

(5.1.5)

which admits an eigensystem with 3 distinct eigenvalues Ω2j (that turns out to be indepen-

dent of m and n) and their corresponding eigenvectors. One of these eigenvectors doesnot satisfy the constraint (5.1.3) and is, therefore, discarded. We label the remaining twoeigensystems as type I, with eigenfrequency Ω2

j = 4(j+1)2, and type II, with eigenfre-quency Ω2

j = 4 j2, as follows

• type I : j≥0 , m = −j, . . . ,+j , n = −j−1, . . . , j+1 , Ωj = ±2 (j+1) ,

Z(j;m,n)+ I (ω) =

√(j−n)(j−n+1)/2 Yj;m,n+1(ω) ,

Z(j;m,n)3 I (ω) =

√(j+1)2 − n2 Yj;m,n(ω) ,

Z(j;m,n)− I (ω) = −

√(j+n)(j+n+1)/2 Yj;m,n−1(ω) .

(5.1.6)

• type II : j≥1 , m = −j, . . . ,+j , n = −j+1, . . . , j−1 , Ωj = ±2 j ,

Z(j;m,n)+ II (ω) = −

√(j+n)(j+n+1)/2 Yj;m,n+1(ω) ,

Z(j;m,n)3 II (ω) =

√j2 − n2 Yj;m,n(ω) ,

Z(j;m,n)− II (ω) =

√(j−n)(j−n+1)/2 Yj;m,n−1(ω) .

(5.1.7)

We can take a linear combination of these basis solutions and write down a real-valuedconnection 1-form as

A = ∞

∑2j=0

X ja I(ω) e2(j+1)iτ + c.c.

ea +

∞

∑2j=2

X ja II(ω) e2j iτ + c.c.

ea (5.1.8)

where we have reorganized the complex angular functions X ja as

X j1 = 1√

2

(Zj+ + Zj

−)

, X j2 = i√

2

(Zj− − Zj

+

), X j

3 = Zj3 (5.1.9)

for both types and expanded the functions Zj± and Zj

3 into the above spin-j basis solutionsof type I (5.1.6) and type II (5.1.7) (for ∗ ∈ +, 3,−),

Zj∗ I(ω) =

j

∑m=−j

j+1

∑n=−j−1

λIj;m,n Z(j;m,n)

∗ I (ω) ,

Zj∗ II(ω) =

j

∑m=−j

j−1

∑n=−j+1

λIIj;m,n Z(j;m,n)

∗ II (ω)

(5.1.10)


with (2j+1)(2j+3) arbitrary complex coefficients λIj;m,n and (2j+1)(2j−1) coefficients λII

j;m,n1.

Inserting (5.1.6) and (5.1.7) into (5.1.10) and the resulting expression into (5.1.9) provides aharmonic expansion

X ja(ω) =

j

∑m=−j

j

∑n=−j

X j;m,na Yj;m,n(ω) (5.1.11)

for both types of angular functions in (5.1.8)2.

It is useful for later purposes to introduce here the “sphere-frame" electric Ea and magneticBa fields,

F = Ea ea∧eτ + 12 Ba εa

bc eb∧ec . (5.1.12)

For a fixed type (I or II) and spin j, we may eliminate L[b Ac] in (4.2.2) (without the com-mutator term) by using (4.2.7) and employ

∂2τA(j) = −Ω2

j A(j) and L2A(j) = −4j(j+1)A(j) (5.1.13)

to obtainE (j)

a = −∂τX(j)a and B(j)

a = ∓Ωj X(j)a , (5.1.14)

where the upper sign pertains to type I and the lower one to type II. We note in pass-ing that, due to the compactness of the Lorentzian cylinder, the sphere-frame energy andaction are always finite.

Due to the linearity of Maxwell theory, the overall scale of any solution is arbitrary. Fur-thermore, the parity transformation L ↔ R—that corresponds to m ↔ n—interchangesa spin-j solution of type I with a spin-(j+1) solution of type II. Finally, electromagneticduality at fixed j is realized by shifting Ωjτ by π

2 for type I or by −π2 for type II, which

maps A to a dual configuration AD and likewise F to FD.

5.1.2 Pulling the solution back to Minkowski space

We have completely solved the vacuum Maxwell equations on the Lorentzian cylinder I ×S3, which, by conformal invariance, carries over to any conformally equivalent spacetimeincluding de Sitter space dS4 and Minkowski space R1,3. We can translate our Maxwellsolutions from I × S3 to R1,3 simply by the coordinate change

τ = τ(t, x, y, z) and ωA = ωA(t, x, y, z)

or τ = τ(t, r) and χ = χ(t, r) .(5.1.15)

In other words, abbreviating x ≡ xµ and y ≡ yρ and expanding

A = Xa(τ(x), g(x)

)ea(x) = Aµ(x)dxµ = Aρ(y)dyρ and

dA = ∂τXa e0∧ea +(

Lb Xc − Xa εabc)

eb∧ec

= 12 Fµν(x)dxµ∧dxν = 1

2 Fρλ(y)dyρ∧dyλ

(5.1.16)

using (4.1.38) we may read off Aµ (note that At 6= 0, as discussed before) and Fµν and thusthe electric and magnetic fields

F = F = Ea dxa ∧ dt +12

Ba εabc dxb∧dxc (5.1.17)

1Note that type-II solutions are absent for j=0 and j= 12 .

2Note the different range of n for X j;m,na and Zj;m,n

∗ ; they are not easily related as X ja and Zj

∗ are in (5.1.9).


in Cartesian or in spherical coordinates. In general, the knotted electromagnetic fieldsarising from the basis configurations (5.1.6) are complex, and thus the basis configurationson Minkowski space will also be complex. Hence, they combine two physical solutions,namely the real and imaginary parts3, which we denote as

(j; m, n)R configuration and (j; m, n)I configuration ,

respectively. Such knotted electromagnetic fields (5.1.17) for the basis configurations (5.1.6)and (5.1.7) increase in complexity with increasing j, as shown in Figure 5.1.

Furthermore, it comes in handy thatA, after computation, contains only even powers of γand depends on τ only through integral powers of

exp(2i τ) =[(`+ it)2 + r2]2

4 `2t2 + (r2 − t2 + `2)2 . (5.1.18)

Therefore, our Minkowski solutions have the remarkable property of being rational func-tions of (t, x, y, z). More precisely, their electric and magnetic fields are of the form

type I:P2(2j+1)(x)Q2(2j+3)(x)

, type II:P2(2j−1)(x)Q2(2j+1)(x)

(5.1.19)

where Pr and Qr denote polynomials of degree r. Thus, as expected, their energy andaction are finite. Indeed, the fields fall off like r−4 at spatial infinity for fixed time, butthey decay merely like (t±r)−1 along the light-cone. Hence, the asymptotic energy flow isconcentrated on past and future null infinity I ±, as it should be, but peaks on the light-cone of the spacetime origin. Since our basis solutions (5.1.6) and (5.1.7) form a completeset on de Sitter space, their Minkowski relatives are also complete on the space of finite-action configurations.

FIGURE 5.1: Electric (red) and magnetic (green) field lines at t=0 with 4 fixed seed points. Left:(j; m, n) = (0; 0, 1)I configuration, Center: (j; m, n) = ( 1

2 ;− 12 , 3

2 )R configuration, Right: (j; m, n) =(1; 1, 2)I configuration. More self-knotted field lines start appearing with additional seed points in

the simulation.

For illustration, we display a type I basis solution with (j; m, n) = (1; 0, 0) obtained from

X± ∝ 1√2(ω1±iω2)(ω3±iω4) cos 4τ and X3 ∝ (ω2

1+ω22−ω2

3−ω24) cos 4τ . (5.1.20)

3The notation should not be confused with the type I configuration (5.1.6).


It bodes well to combine these field configurations into the Riemann–Silberstein vector

S = E + iB (5.1.21)

whose components, for (5.1.20), read (up to some overall scale)

Sx = − 2iN

2y + 3ity− xz + 2t2y + 2itxz− 8x2y− 8y3 + 4yz2 + 4it3y

− 6t2xz− 8itx2y− 8ity3 + 4ityz2 + 10x3z + 10xy2z− 2xz3

+ 2(itxz + yr2)(−t2+r2) + (ity− xz)(−t2+r2)2

,

(5.1.22)

Sy =2iN

2x + 3itx + yz + 2t2x− 2ityz− 8x3 − 8xy2 + 4xz2 + 4it3x

+ 6t2yz− 8itx3 − 8itxy2 + 4itxz2 − 10x2yz− 10y3z + 2yz3

+ 2(−ityz+xr2)(−t2+r2) + (itx + yz)(−t2+r2)2

, &

(5.1.23)

Sz =iN

1 + 2it + t2 − 11x2 − 11y2 + 3z2 + 4it3 − 16itx2 − 16ity2 + 4itz2 − t4

− 2t2x2 − 2t2y2 − 2t2z2 + 11x4 + 22x2y2 − 10x2z2 + 11y4 − 10y2z2

+ 3z4 + 2it(t2−3r2+2z2)(t2−r2)− (t2+r2)(−t2+r2)2 (5.1.24)

with N =((t−i)2 − r2)5. We can also obtain electromagnetic field configurations corre-

sponding to the gauge field (5.1.8) that consists of complex linear combinations of thesebasis solutions. We demonstrate this for the famous Hopf–Ranãda field configuration,which was first discovered by Ranãda in 1989 [1] using the Hopf fibration. The Riemann–Silberstein vector S for this field configuration is given by [3]

S =1

((t−i)2−r2)3

(x−i y)2−(t−i−z)2

i(x−i y)2+i(t−i−z)2

−2(x−i y)(t−i−z)

. (5.1.25)

We find that this solution, in our construction, is related to (0, 0, 1)I basis configurationsand is obtained by using following coefficients in (5.1.10)

λI0;0,−1 = 0 , λI

0;0,0 = 0 , and λI0;0,1 =

π

4. (5.1.26)

Moreover, we find that some of our basis configurations are related to the (p, q)-torusknots arising from Bateman’s construction [2]. We illustrate this point in Figure 5.1 wherewe find the following correspondences:

Hopfion↔ (1, 1) , ( 12 ,− 1

2 , 32 )R ↔ (2, 1) , (1, 1, 2)I ↔ (1, 3) . (5.1.27)

5.2 Symmetry analysis

The main advantage of constructing Minkowski-space electromagnetic field configura-tions via the detour over de Sitter space is the enhanced manifest symmetry of our con-struction. The isometry group SO(1,4) of dS4 is generated by (A, B = 1, 2, 3, 4 and a, b, c =1, 2, 3, abbreviate ∂

∂qB≡ ∂B )

MAB≡−q[A∂B] , M0B≡ q(0∂B) = Mab=εabcDc , M4a=Pa , M04=P0 , M0b=Kb ,(5.2.1)


which can be contracted (with `→∞) to the isometry group ISO(1,3) of R1,3 (the Poincarégroup) generated by (µ, ν = 0, 1, 2, 3 and i, j, k = 1, 2, 3)

Mµν , Pµ = Mij=ε ijkDk , Pi , P0 , M0j=Kj , (5.2.2)

where the two sets are ordered likewise, and we employ (as aleady earlier) calligraphicsymbols for de Sitter quantities and straight symbols for Minkowskian ones. Here, Ddenotes spatial rotations, P are translations, and K stand for boosts in Minkowski space.

Since the two spaces are conformally equivalent already at `<∞ via (4.1.6), the correspond-ing generators should be related. Indeed, the common SO(3) subgroup in

SO(1, 4) ⊃ SO(4) ⊃ SO(3) and ISO(1, 3) ⊃ SO(1, 3) ⊃ SO(3) (5.2.3)

is identified, Di=Di=−2ε kij xj∂k. However, any other generator becomes nonlinearly real-

ized when mapped to the other space via (4.1.12). For example, the would-be translationP3 defined in (4.1.26) reads

P3 = L3 − R3 = −2 cos θ ∂χ + 2 cot χ sin θ ∂θ

= 1` cos θ

(2 r t ∂t + (t2+r2+`2) ∂r

)− 1

` r (t2−r2+`2) sin θ ∂θ

→ 2`(cos θ ∂r − 1

r sin θ ∂θ

)= 2` ∂z = ` Pz for `→ ∞

(5.2.4)

as it should be. Similarly, P0 → ` P0 andKb → Kj for `→∞ when expanded around (t, r) =(`, 0) corresponding to the S3 south pole at q0=0. Nevertheless, the de Sitter constructionenjoys an SO(4) covariance (generated by Da and Pa) which extends the obvious SO(3)covariance in Minkowski space. It allows us to connect all solutions of a given type (I orII) with a fixed value of the spin j by the action of SO(4) ladder operators L± and R± orD± and P±, which is non-obvious on the Minkowski side. On the other hand, Minkowskiboosts and translations have no simple realization on de Sitter space.

Actually, Maxwell theory on either space is also invariant under conformal transforma-tions. These may be generated by the isometry group together with a conformal inversion.On the Minkowski side, the latter is

J : xµ 7→ xµ

x · x with x · x = r2 − t2 . (5.2.5)

We have to distinguish two cases:

spacelike: t2 < r2 ⇒ J> :(t, r, θ, φ

)7→

( tr2−t2 , r

r2−t2 , θ, φ)

,

timelike: t2 > r2 ⇒ J< :(t, r, θ, φ

)7→

( −tt2−r2 , r

t2−r2 , π−θ, φ+π)

.(5.2.6)

On the de Sitter side, this is either (spacelike) a reflection on the S3 equator χ=π2 or (time-

like) a π-shift in cylinder time τ plus an S2 antipodal flip,

spacelike: |τ|+χ < π ⇒ J> :(τ, χ, θ, φ

)7→

(τ, π−χ, θ, φ

),

timelike: |τ|+χ > π ⇒ J< :(τ, χ, θ, φ

)7→

(τ±π, χ, π−θ, φ+π

).

(5.2.7)

In the spacelike case, merely the sign of ω4≡ cos χ gets flipped, which amounts to a par-ity flip L ↔ R. In the timelike case, both cos τ and sin τ change sign, which combinesa time reversal with a reflection at τ=π

2 or τ=−π2 . Note that it is different from the S3

antipodal map, which is not a reflection but a proper rotation, ωA 7→ −ωA or (χ, θ, φ) 7→


(π−χ, π−θ, φ+π). The lightcone is singular under the inversion; it is mapped to the con-formal boundary r=±t=∞ or χ=±τ. We infer that the conformal inversion allows us torelate type-I and type-II solutions of the same spin. It is easily checked that the spatialfall-off behavior of our rational solutions is not modified by the inversion.

Finally, one may consider dilatations in Minkowski space,

xµ 7→ λ xµ for λ ∈ R+ . (5.2.8)

However, this amounts to a trivial rescaling also achieved by changing the de Sitter radius,` 7→ λ `, as the scale ` was removed on the Lorentzian cylinder.

5.3 Conformal group and Noether charges

It is well known [29] that free Maxwell theory on R1,3 arising from the action

S[Aµ

]=∫

d4xL ; L = − 14 FµνFµν (5.3.1)

is invariant under the conformal group SO(2, 4). Furthermore, the above action is alsoinvariant under the gauge transformations: Aµ(x) → Aµ(x) + ∂µλ(x). The conformalgroup is generated by transformations xµ → xµ + ζµ(x), where the vector fields ζµ obeythe conformal Killing equations:

ζµ,ν + ζν,µ = 12 ηµν ζα

,α with ηµν = diag(−1, 1, 1, 1) . (5.3.2)

The conserved Noether current Jµ is obtained by equating the “on-shell variation" ofthe action where the variations δAµ are arbitrary and the fields Aµ satisfies the Euler-Langrange equations with its “symmetry variation" where the fields Aµ are arbitrary butvariations δAµ satisfy the symmetry condition. The correct variation δAµ is obtained byimposing the gauge invariance on the Lie derivative of Aµ w.r.t. the vector field ζµ:

Lζα Aµ := A′µ(x)− Aµ(x) = −ζα ∂α Aµ − Aα ∂µζα −→ δAµ = Fµνζν . (5.3.3)

Finally, the conserved current is obtained as

Jµ =∂L

∂Aρ,µδAρ + ζµL = ζρ

(FµαFρα − 1

4 δµρ F2) with F2 = FβγFβγ , (5.3.4)

which satisfies the continuity equation and gives the conserved (in time) charge Q4:

∂µ Jµ = 0 =⇒ dQdt

= −∫

∂Vd2s · J = 0 ; Q =

∫V

d3x J0 . (5.3.5)

Before proceeding further, a couple of remarks pertaining to the subsequent calculationsare in order:

• All the charges Q are computed at t=0 owing to the simple J0 expressions on thistime-slice. To that end, we record the following useful identities at t=τ=0:

eai = 1

` (γ ω4 δai −ωa ωi + εaic γ ωc) , ea

i ebi = γ2

`2 δab

γ = 1−ω4 , d3x = `3

γ3 d3Ω3 ; d3Ω3 := e1 ∧ e2 ∧ e3 .(5.3.6)

4For source-free fields the current J is assumed to vanish when the surface ∂V is taken to infinity.


Moreover, the electromagnetic fields at t=0 are given in terms of tetrads eτ=eτµ dxµ

and ea=eaµ dxµ as follows:

Ei = eτ0 ea

i Ea and Bi = 12 εijk εabc eb

j eck Ba (5.3.7)

using the 2-form (5.1.12) and its Minkowski counterpart (5.1.17).

• The charges are computed for type I (5.1.6) solutions only (except for the energy andthe related helicity) and we relabel the complex coefficients (5.1.10) λI

j;m,n as Λj;m,n

for convenience.

• Furthermore, these charges Q are computed for a fixed spin-j and, thus, we willsuppress the index j from now onward, unless necessary. Note that the sphere-frameEM fields for fixed j can be obtained by using the expansion (5.1.8) (for type I only)in (5.1.14) as

Aa = Xa(ω) eΩ iτ + Xa(ω) e−Ω iτ ⇒ Ea = −i Ω Xa eΩ iτ + i Ω Xa e−Ω iτ

Ba = −Ω Xa eΩ iτ − Ω Xa e−Ω iτ

(5.3.8)

with Xa denoting the complex conjugate of Xa. Notice that for type II the overall signin Ba flips.

• The simplifications below for the charge density J0 are carried out using the har-monic expansion (5.1.11) for the type I solution (5.1.6).

• We frequently use below the well known fact that an odd ωA integral over S3

vanishes because of the opposite contributions coming from the antipodal points onthe sphere. In particular, it can be checked that the following integral vanishes5:∫

d3Ω3 (ω1)a (ω2)

b (ω3)c (ω4)

d Yj;m,n Y j;m′,n′ = 0 for a + b + c + d ∈ 2N0 + 1 .

(5.3.9)

Having made these remarks, we now proceed to compute the charges Q for various con-formal transformations ζµ obeying (5.3.2) in following four categories.

5.3.1 Translations

An easily seen solution to (5.3.2) is the set of four constant translations

ζµ = εµ , (5.3.10)

which also partly generates the Poincaré group and give rise to the usual stress-energytensor of electrodynamics

Jµ = Tµν = FµαFνα − 1

4 δµν F2 , (5.3.11)

corresponding to the µ-component of the translation for an arbitrary εν. The correspond-ing charges are the energy E and the momentum P.

Energy. The expression of the energy density e := T00 simplifies to

e :=12(E2

i + B2i)=(γ

`

)4ρ with ρ = 1

2

(E2

a + B2a)

, (5.3.12)

5Note that the power of ωA in Yj;m,nYj;m′ ,n′ is always even. One way to check this is by employing thetoroidal coordinates: ω1 = cos η cos κ1, ω2 = cos η sin κ1, ω3 = sin η cos κ2, ω4 = sin η sin κ2 with η ∈ (0, π

2 )and κ1, κ2 ∈ (0, 2π) in (4.1.31). The resultant selection rules coming from κ1, κ2 integral would yield m−m′ ∈2k+1

2 with k ∈ N0, which is not feasible for fixed j.


which, in turn, simplifies the expression for the energy E to

E =∫

Vd3x e = 1

2`

∫S3

d3Ω3 (1− cos χ)(E2

a + B2a)

. (5.3.13)

Notice here that the orientation of the S3 volume measure d3Ω3 is chosen to providea positive result. From (5.3.8) we find that the “sphere-frame” energy density is time-independent and has similar expression for both solution types (with appropriate eigen-frequency Ω):

12

(E2

a + B2a)= 2 Ω2 XaXa(ω) . (5.3.14)

The resultant expression for E in terms of complex parameters λm,n (for both solutiontypes) is given by

E =1`(2j + 1)Ω3 ∑

m,n|λm,n|2 . (5.3.15)

Notice again here that for type I solutions we would have Λm,n in the above expression.

Helicity. Although helicity is not a Noether charge of the conformal group, it is neverthe-less a conserved quantity for the Maxwell system and turns out to be related to the energy.The expression for the helicity is metric-free and can thus be evaluated over any spatialslice. Choosing again t=τ=0,

h = hmag + hel =12

∫R3

(A ∧ F + AD ∧ FD

)= −1

2

∫S3

d3Ω3 (1− cos χ)(AaBa +AD

a Ea)

,

(5.3.16)where the subscript/superscript ‘D′ refers to the dual fields. Once again, taking type I(upper sign) or type II (lower sign) and fixing the spin j we obtain

ADa = ±i Xa(ω) eΩ iτ ∓ i Xa(ω) e−Ω iτ , (5.3.17)

which yields a constant “sphere-frame” helicity density

− 12

(AaBa +AD

a Ea)= ±2 Ω XaXa(ω) . (5.3.18)

As a result, even before performing the S3 integration, we find a linear helicity-energyrelation

Ω h = ±` E for fixed spin and type . (5.3.19)

Since the helicity measure an average of the linking numbers of any two electric or mag-netic field lines [30–32], the latter must be related to the value j of the spin. The individuallinking number of two field lines, however, appears neither to be independent of the lineschosen nor constant in time, as our observations indicate. An exception are the Rañada–Hopf knots discussed before, which display a conserved linking number of unity betweenany pair of electric or magnetic field lines.

Momentum. For the momentum densities pi = T0i = (E× B)i we obtain an interestingcorrespondence relating the one-form p := pi dxi on Minkowski space with a similar oneon de Sitter space:

p = (γ` )

3 Pa ea =: (γ` )

3 P with Pa := εabc Eb Bc . (5.3.20)

A straightforward calculation then yields the expression of momenta Pi:

Pi =∫

Vd3x pi =

∫S3

d3Ω3 Pa eai = 2i Ω2 εabc

∫S3

d3Ω3 eai Xb Xc (5.3.21)


j = 1/2 j = 1

P3

9`

(|Λ−1/2,−3/2|2 − |Λ−1/2,1/2|2

− 2|Λ−1/2,3/2|2 + 2|Λ1/2,−3/2|2

+ |Λ1/2,−1/2|2 − |Λ1/2,3/2|2)

24`

(|Λ−1,−2|2 − |Λ−1,0|2 − 2|Λ−1,1|2 − 3|Λ−1,2|2

+ 2|Λ0,−2|2 + |Λ0,−1|2 − |Λ0,1|2 − 2|Λ0,2|2

+ 3|Λ1,−2|2 + 2|Λ1,−1|2 + |Λ1,0|2 − |Λ1,2|2)

Pφ

9(

2|Λ−1/2,−3/2|2 + |Λ−1/2,−1/2|2

− |Λ−1/2,3/2|2 + |Λ1/2,−3/2|2

− |Λ1/2,1/2|2 − 2|Λ1/2,3/2|2)

24(

3|Λ−1,−2|2 + 2|Λ−1,−1|2 + |Λ−1,0|2 − |Λ−1,2|2

+ 2|Λ0,−2|2 + |Λ0,−1|2 − |Λ0,1|2 − 2|Λ0,2|2

+ |Λ1,−2|2 − |Λ1,0|2 − 2|Λ1,1|2 − 3|Λ1,2|2)

L3

9`(− 2|Λ−1/2,−3/2|2 − |Λ−1/2,−1/2|2

+ |Λ−1/2,3/2|2 − |Λ1/2,−3/2|2

+ |Λ1/2,1/2|2 + 2|Λ1/2,3/2|2)

−24`(

3|Λ−1,−2|2 + 2|Λ−1,−1|2 + |Λ−1,0|2

− |Λ−1,2|2 + 2|Λ0,−2|2 + |Λ0,−1|2

− |Λ0,1|2 − 2|Λ0,2|2 + |Λ1,−2|2

− |Λ1,0|2 − 2|Λ1,1|2 − 3|Λ1,2|2)

TABLE 5.1: Expressions of P3, Pφ and L3 for j = 1/2 and 1.

with eai given by (5.3.6). The results for j = 0 are

P(j=0)1 = −

√2` ((Λ0,−1 + Λ0,1)Λ0,0 + Λ0,0 (Λ0,−1 + Λ0,1)) ,

P(j=0)2 = i

√2

` ((−Λ0,−1 + Λ0,1)Λ0,0 + Λ0,0 (Λ0,−1 −Λ0,1)) ,

P(j=0)3 = 2

`

(|Λ0,−1|2 − |Λ0,1|2

).

(5.3.22)

As a consistency requirement, we check that the vector charges Pi are rotated according tothe algebra of Da (4.1.26) (See appendix B):

Da Pb = 2 εabc Pc . (5.3.23)

We also note down P3 for j = 1/2 and 1 in table 5.1. One can compute the correspondingP1 and P2 for j = 1/2 and j = 1 by employing the action of an appropriate Da of the tablein Appendix B.

We can additionally compute the spherical components of the momentum (Pr, Pθ , Pφ) byletting the one-form ea in (5.3.20) act on the vector fields (∂r, ∂θ , ∂φ). In practice, we firstwrite ∂r = −γ

` ∂χ using (4.1.13) at t=0 and then invert the vector fields (∂r, ∂θ , ∂φ) in termsof the left invariant vector fields (L1, L2, L3) using (4.1.39). Finally, using the duality rela-tion ea(Lb) = δa

b we obtain

Pr = −1`

∫S3

d3Ω3 (1− cos χ) (sin θ cos φP1 + sin θ sin φP2 + cos θ P3) ,

Pθ =∫

S3d3Ω3 sin χ cos χ

((cos θ cos φ + tan χ sin φ)P1

+ (cos θ sin φ− tan χ cos φ)P2 − sin θ P3

),

Pφ =∫

S3d3Ω3 sin2 χ sin θ

((cos θ cos φ− cot χ sin φ)P1

+ (cos θ sin φ + cot χ cos φ)P2 − sin θ P3

).

(5.3.24)


From the expression of Pr we see that the integrand over S2 i.e. ωaPa (4.1.10) is an oddfunction6, which makes Pr vanish. We also find with explicit calculations (verified for upto j = 1) that Pθ vanishes. For j = 0 we find that Pφ is proportional to P3:

P(j=0)φ = ` P(j=0)

3 = 2(|Λ0,−1|2 − |Λ0,1|2

). (5.3.25)

The expressions of Pφ for j = 1/2 and 1 has been recorded in the table 5.1.

5.3.2 Lorentz transformations

Another solution of (5.3.2) is given by

ζµ(x) = εµ

ν xν where εµν = −ενµ , (5.3.26)

which correspond to the six generators of the Lorentz group SO(1, 3). These six togetherwith the above four translations generates the full Poincaré group. The corresponding sixcharges are grouped into the boost K and the angular momentum L.

Boost. The conserved charge densities arising from J0 (5.3.4) corresponding to ε0i are theboost densities k = ρ x− p t, which simplify for t=0 to

ki = (γ` )

3Ki with Ki = ρ ωi . (5.3.27)

The corresponding charges Ki vanishes because of the odd integrand as discussed in anearlier remark:

Ki =∫

Vd3x ki =

∫S3

d3Ω3Ki = 2∫

S3d3Ω3 ωi X j

a X ja = 0 . (5.3.28)

Angular momentum. The other three conserved charge densities J0 corresponding to εijin (5.3.4) are the angular momentum densities l = p× x, which takes a simple form justlike in momentum (5.3.20):

li dxi = (γ` )

2 La ea with La = εabc Pb ωc . (5.3.29)

The expressions for the charges Li simplify to

Li =∫

Vd3x li =

∫S3

d3Ω3 (`γ )La ea

i = 2i` Ω2∫

S3d3Ω3

1γ ea

i (Xa ωbXb − Xa ωbXb) .

(5.3.30)Explicit calculation show that for j=0 the angular momenta Li are proportional to themomenta Pi:

L(j=0)i = −`2 P(j=0)

i . (5.3.31)

This is, however, not true for higher spin j. The angular momenta Li has the same rotationbehaviour as for the momenta Pi (5.3.23). We therefore note down the results of L3 forj = 1/2 and 1 in table 5.1, from which the corresponding expressions of L1 and L2 can beobtained using the table in Appendix B.

We can again compute the spherical components of the angular momentum (Lr, Lθ , Lφ)using the relations (5.3.24) by replacing P with L in it. We realize that the S2 integrand forLr would only have terms like ωaωbPc, which are all odd functions7 of φ and, therefore,

6The terms of Pa using (4.1.36) are proportional to Yl,MYl′ ,M′ , which is even in ωa for a fixed j.7Here again the terms of Pc, which goes like Yl,MYl′ ,M′ (4.1.36), are all even functions of φ for a fixed j.


the φ integral over the domain (0, 2π) would make Lr vanish. We also find, with explicitcomputations, that the charges Lθ and Lφ for j=0 are proportional to P3:

L(j=0)θ = 4

3 `2 P(j=0)

3 and L(j=0)φ = − 1

3 `2 P(j=0)

3 . (5.3.32)

As a non trivial example, we collect these charges for j = 1/2 below:

L(j=1/2)θ = 12

5 `(

9|Λ−1/2,−3/2|2 + 6|Λ−1/2,−1/2|2 + |Λ−1/2,1/2|2 − 6|Λ−1/2,3/2|2

+ 6|Λ1/2,−3/2|2 − |Λ1/2,−1/2|2 − 6|Λ1/2,1/2|2 − 9|Λ1/2,3/2|2)

,(5.3.33)

L(j=1/2)φ = − 3

5`(

6|Λ−1/2,−3/2|2 − |Λ−1/2,−1/2|2 − 6|Λ−1/2,1/2|2 − 9|Λ−1/2,3/2|2

+ 9|Λ1/2,−3/2|2 + 6|Λ1/2,−1/2|2 + |Λ1/2,1/2|2 − 6|Λ1/2,3/2|2

+√

3(Λ1/2,−3/2Λ−1/2,−1/2 − Λ1/2,1/2Λ−1/2,3/2

+ Λ−1/2,−1/2Λ1/2,−3/2 − Λ−1/2,3/2Λ1/2,1/2))

.

(5.3.34)

5.3.3 Dilatation

It is easy to verify that a constant rescaling by λ:

ζµ = λ xµ (5.3.35)

is also a solution of (5.3.2). The charge density corresponding to this single generator ofthe conformal group is p · x− e t, which for t=0 simplifies to

pi xi = (γ` )

3 Pa ωa . (5.3.36)

The corresponding charge D vanishes because of the odd integrand:

D =∫

Vd3x pi xi = 2i Ω2 εabc

∫S3

d3Ω3 ωa Xb Xc = 0 . (5.3.37)

5.3.4 Special conformal transformations

A fairly straightforward calculation shows that the following not so obvious transforma-tion

ζµ = 2 xµ bνxν − bµ xνxν (5.3.38)

also satisfies (5.3.2). The four generators corresponding to bµ give rise to four differentcharges V0 and V.

Scalar SCT. The charge density J0 corresponding to b0 is

v0 = (x2 + t2) e− 2t p · x , (5.3.39)

which for t=0 simplifies tov0 = x2 e = (γ

` )2ρ ω2

a . (5.3.40)

The expression for the corresponding charge V0 takes the following simple form:

V0 =∫

Vd3x v0 =

∫S3

d3Ω3 (`γ ) (1−ω2

4) ρ = 2`Ω2∫

S3d3Ω3 (1 + ω4) Xa Xa . (5.3.41)


Here again the ω4 term of the integral, being odd, vanishes and yields

V0 = `2 E = 8 ` (j + 1)3(2j + 1) ∑m,n|Λm,n|2 . (5.3.42)

Vector SCT. The charge densities J0 that correspond to bi read:

v = 2x(x · p)− 2t x e− (x2 − t2)p . (5.3.43)

This simplify at t=0 and takes a structure similar to the momentum densities pi (5.3.20):

vi dxi = (γ` )Va ea with Va = 2ωa (Pb ωb)−ω2

b Pa . (5.3.44)

The expressions for the charges Vi then simplify to

Vi =∫

Vd3x vi = 2i `2 Ω2

∫S3

d3Ω3 γ−2eai(2εbcd ωa ωb Xc Xd − εabc (1−ω2

4) Xb Xc)

.

(5.3.45)With explicit computation we observe that the charges Vi are proportional to the momentaPi (verified explicitly for up to j=1)

Vi = `2 Pi . (5.3.46)

As before, we can compute the spherical components (Vr, Vθ , Vφ) by using the expressions(5.3.24) and replacing P with V in it. We notice that the charge Vr vanishes owing to theodd S2 integrand8 just like in the case of Pr. However, unlike Pθ here the charge Vθ is non-vanishing. Explicit calculations show that for j = 0 the charges Vθ and Vφ are proportionalto the momentum P3:

V(j=0)θ = − 4

3`3 P(j=0)

3 and V(j=0)φ = − 5

3`3 P(j=0)

3 . (5.3.47)

Additionally, we record below the charges Vθ and Vφ for the non-trivial case of j=1/2

V(j=1/2)θ = − 6

5`2(

9|Λ−1/2,−3/2|2 + |Λ−1/2,−1/2|2 − 9|Λ−1/2,1/2|2 − 21|Λ−1/2,3/2|2

+ 21|Λ1/2,−3/2|2 + 9|Λ1/2,−1/2|2 − |Λ1/2,1/2|2 − 9|Λ1/2,3/2|2)

,(5.3.48)

V(j=1/2)φ = − 3

5`2(

42|Λ−1/2,−3/2|2 + 33|Λ−1/2,−1/2|2 + 8|Λ−1/2,1/2|2 − 33|Λ−1/2,3/2|2

+ 33|Λ1/2,−3/2|2 − 8|Λ1/2,−1/2|2 − 33|Λ1/2,1/2|2 − 42|Λ1/2,3/2|2

+ 8√

3(− Λ1/2,−3/2Λ−1/2,−1/2 + Λ1/2,1/2Λ−1/2,3/2

− Λ−1/2,−1/2Λ1/2,−3/2 + Λ−1/2,3/2Λ1/2,1/2))

.(5.3.49)

One can compute these charges for higher spin-j by following the same strategy.

5.3.5 Applications

The method of constructing rational electromagnetic fields presented in this paper has theadded advantage that it produces a complete set labelled by (j, m, n); any electromagneticfield configuration having finite energy can, in principle, be obtained from an expansion

8Observe that the terms in Va (5.3.44) are all even in ωa.


like in (5.1.8-5.1.10), albeit with a varying j. The operational difficulty involved in thisprocedure has to do with the fact that this set is infinite as j ∈ N

2 . There are, however,many important cases where only a finite number of knot-basis solutions (sometimes onlywith a fixed j) need to be combined to get the desired EM field configuration. One suchvery important case is that of the Hopfian solution discussed before. Below we analyse twovery interesting generalisations of the Hopfian solution presented in [14] in the context ofpresent construction. It is imperative to note here that while the scope of construction ofa new solution from the known ones as presented in [14] is limited, the same is not truefor the method presented in this paper, which by design can produce arbitrary number ofnew field configurations. Some of these possible new field configurations obtained fromthe j=0 sector (possibly from j=1/2 or 1 as well) could find experimental application withimproved experimental techniques like in [5].

FIGURE 5.2: Sample electric (red) and magnetic (green) field lines at t=0. Left: time-translatedHopfian with c = 60, Right: Rotated Hopfian with θ = 1.

Bateman’s construction, employed in [14], hinges on a judicious ansatz for the Riemann-Silberstein vector (5.1.21) satisfying Maxwell’s equations:

S = ∇α×∇β =⇒ ∇ · S = 0 & i ∂tS = ∇× S , (5.3.50)

using a pair of complex functions (α, β). An interesting generalization of the Hopfiansolution is obtained in equations (3.16-3.17) of [14] using (complex) time-translation (TT)to obtain the following (α, β) pair (up to a normalization)

(α, β)TT =

(A− 1− iz

A + i(t + ic),

x− iyA + i(t + ic)

); A = 1

2

(x2 + y2 + z2 − (t + ic)2 + 1

),

(5.3.51)where c is a constant real parameter. The corresponding EM field configuration is ob-tained, in our case, by choosing `=1−c and only the following j=0, and hence type I,complex coefficients in (5.1.10)

(Λ0;0,1 , Λ0;0,0 , Λ0;0,−1)TT =(

0 , 0 , −iπ

2`2

). (5.3.52)

A sample electromagnetic knot configuration of this modified Hopfian is illustated in fig-ure 5.2. Another interesting generalization of the Hopfian is constructed in equations (3.20-3.21) of [14] using a (complex) rotation (R) to get the following (α, β) pair (again, up to anormalization)

(α, β)R =

(A− 1 + i(z cos iθ + x sin iθ)

A + it,

x cos iθ − z sin iθ − iyA + it

), (5.3.53)


Time-translated Hopfian Rotated Hopfian

Energy (E) 2π2

(1−c)5 =: ETT 2π2 cosh2 θ =: ER

Momentum (P)(0 , 0 , 1

4

)ETT

(0 , − 1

4 tanh θ , 14 sech θ

)ER

Boost (K) (0 , 0 , 0) (0 , 0 , 0)

Ang. momentum (L)(0 , 0 , − 1

4 (1− c)2) ETT(0 , 1

4 tanh θ , − 14 sech θ

)ER

Dilatation (D) 0 0

Scalar SCT (V0) (1− c)2ETT ER

Vector SCT (V)(0 , 0 , 1

4 (1− c)2) ETT(0 , − 1

4 tanh θ , 14 sech θ

)ER

TABLE 5.2: Conformal charges for the time-translated and rotated Hopfianconfigurations.

where A = 12

(x2 + y2 + z2 − t2 + 1

). To get this particular EM field configuration we need

to set `=1 and use the following combination of only j=0 type I coefficients in (5.1.10)

(Λ0;0,1 , Λ0;0,0 , Λ0;0,−1)R =

(iπ

4(cosh θ − 1) , − π

2√

2sinh θ , −i

π

4(cosh θ + 1)

).

(5.3.54)We illustrate the EM field lines for a sample θ value of this modified Hopfian in figure5.2. We note down the conformal charges corresponding to these solutions in table 5.2by plugging the j=0 coefficients (5.3.52) and (5.3.54) in the appropriate formulae of theprevious section. The results matches with the ones given in [14] up to a rescaling of theenergy, which can be achieved by an appropriate choice of normalization.

5.4 Null fields

An interesting subset of vacuum electromagnetic fields are those with vanishing Lorentzinvariants,

~E2 − ~B2 = 0 and ~E · ~B = 0 ⇐⇒(~E± i~B

)2= 0 . (5.4.1)

As a scalar equation it must equally hold on the de Sitter side, and so we can try to char-acterize such configurations with our SO(4) basis above. For a given type and spin, theexpressions in (5.3.8) immediately give the Riemann-Silberstein vector on the S3 cylinder,

Ea ± iBa = −2i Ω Xa(ω) eΩ iτ , (5.4.2)

where the upper (lower) sign pertains to type I (II). Note that the negative-frequency partof this field has cancelled. The vanishing of (Ea±iBa)(Ea±iBa) is then equivalent to acondition on the angular functions,

0 = X1(ω)2 + X2(ω)2 + X3(ω)2 = 2 Z+(ω)Z−(ω) + Z3(ω)2 . (5.4.3)

When expanding the angular functions Zj∗ I or Zj

∗ II into basis solutions with (5.1.10), onearrives at a system of homogeneous quadratic equations for the free coefficients λI/II

j;m,n.

Let us analyze the situation for type I and spin j. The functions Zj∗(ω) transform under


a (j, j) representation of su(2)L ⊕ su(2)R. The null condition (5.4.3) then yields a repre-sentation content of (0, 0)⊕ (1, 1)⊕ . . .⊕ (2j, 2j) and may thus be expanded into the corre-sponding harmonics. Furthermore, The independent vanishing of all coefficients produces16 (4j+1)(4j+2)(4j+3) equations for the (2j+1)(2j+3) parameters λj;m,n (note the rangesof m and n for type I). Clearly, this system is vastly overdetermined. However, it turns outthat only 4j2+6j+1 equations are independent, still leaving 2j+2 free complex parametersfor the solution space. The independent equations can be organized as (suppressing j)

λ2m,n ∼ λm,n−1 λm,n+1 for m, n = −j . . . , j ,

λm,j+1 λm+1,−j−1 = λm+1,j+1 λm,−j−1 for m = −j, . . . , j−1 .(5.4.4)

We have checked for j≤5 that the upper equations are solved by 9

λ2j+2m,n =

√( 2j+2

j+1−n) λj+1−nm,−j−1 λ

j+1+nm,j+1 for m = −j, . . . , j and n = −j−1, . . . , j+1 ,

(5.4.5)while the lower ones imply that the highest weights n=j+1 are proportional to the lowestweights n=−j−1 (independent of m),

λm,−j−1 = w λm,j+1 for w ∈ C∗ . (5.4.6)

Therefore, the full (generic) solution reads

λm,n =√( 2j+2

j+1−n) wj+1−n2j+2 e2πikm

j+1−n2j+2 zm with zm ∈ C and km ∈ 0, 1, . . . , 2j+1 ,

(5.4.7)containing 2j+2 complex parameters zm and q as well as 2j discrete choices km (one ofthem can be absorbed into zm). This completely specifies the type-I null fields for a givenspin. Type-II null fields are easily obtained by applying electromagnetic duality to type-Inull fields.

In the simplest case of j=0, the single equation λ20,0 = 2λ0,−1λ0,1 describes a generic rank-3

quadric inCP2, or a cone over a sphereCP1 inside the parameter spaceC3. For higher spin,the moduli space of type-I null fields remains a complete-intersection projective variety ofcomplex dimension 2j+1.

We conclude the Section with a display of typical field lines (see Figure 5.3) for a type Ij= 1

2 and j=1 null field at t=0. For t 6=0 the pictures get smoothly distorted.

5.5 Flux transport

We have seen that electromagnetic energy is radiated away along the light-cones. Let ustry to quantify its amount over future null infinity I +. Before proceeding further we notedown the determinant of the Jacobian J (4.1.13)

|detJ| = p2−q2

`2 =γ2

`2 =sin2τ

t2 =sin2χ

r2 with γ2 = p2−q2 (5.5.1)

and the spherical Minkowski components

At = Aτ Jτt +Aχ Jχ

t , Ar = Aτ Jτr +Aχ Jχ

r , Aθ = Aθ , Aφ = Aφ , (5.5.2)

9These are the generic solutions. There exist also special solutions given by (5.4.6) and λm,n = 0 for |n| 6=j+1, for arbitrarily selected choices of m ∈ −j, . . . , j.


FIGURE 5.3: Sample electric (red) and magnetic (green) field lines at t=0. Left: a pair of electricand a pair of magnetic field lines for the (j; m, n) =

(12 ; 1

2 , 32

)R

field configuration. Right: a pair ofelectric field lines, a magnetic field line of self-linking one and a magnetic field line of self-linking

seven for the (j; m, n) = (1; 0, 2)R field configuration.

or any other such tensor component arising due to (4.1.13). For later use, we also note herethe transformation of the volume form

d4x = dt r2dr d2Ω2 = r2 |detJ|−1dτ dχ d2Ω2 = sin2χ |detJ|−2dτ dχ d2Ω2 =`4

γ4 dτ d3Ω3 .

(5.5.3)

The energy flux at time t0 passing through a two-sphere of radius r0 centered at the spatialorigin is given by

Φ(t0, r0) =∫

S2(r0)d2~σ ·

(~E× ~B

)(t0, r0, θ, φ) =

∫S2

r20 d2Ω2 T(M)

t r (t0, r0, θ, φ) , (5.5.4)

where d2Ω2 = sin θ dθ dφ, and T(M)t r is the (t, r) component of the Minkowski-space stress-

energy tensor

T(M)µ ν = FµρFνλ gρλ − 1

4 gµνF2 with (gµν) = diag(−1, 1, r2, r2 sin2θ) (5.5.5)

for µ, ν, . . . ∈ t, r, θ, φ. We carry out this computation in the S3-cylinder frame by usingthe conformal relations

`2 T(dS)µ ν = t2 T(M)

µ ν = sin2τ T(cyl)µ ν = sin2τ T(cyl)

m n Jmµ Jn

ν for m, n ∈ τ, χ, θ, φ (5.5.6)

with the Jacobian (4.1.13) and the fact that r sin τ = t sin χ so that

Φ(τ0, χ0) =∫

S2sin2χ d2Ω2 T(cyl)

t r (τ0, χ0, θ, φ) =∫

S2sin2χ d2Ω2 T(cyl)

m n Jmt Jn

r . (5.5.7)

A straightforward computation using (gmn) = diag(−1, 1, sin2χ, sin2χ sin2θ) then yields

Φ(τ0, χ0) =p q`2

∫d2Ω2

((Fτθ)

2 + (Fχθ)2 + 1

sin2θ

[(Fτφ)

2 + (Fχφ)2])

+p2+q2

`2

∫d2Ω2

(Fτθ Fχθ +

1sin2θFτφ Fχφ

).

(5.5.8)


The sphere-frame components Fmn can be computed by expanding ea = eam dξm in

F = Ea ea∧eτ + 12 Ba εa

bc eb∧ec = Fmn dξm∧dξn with ξn ∈ τ, χ, θ, φ . (5.5.9)

The expression for the flux in sphere-frame fields then becomes

`2 Φ = p q sin2χ∫

S2d2Ω2

[(sin φ E1 − cos φ E2)

2 + (cos θ cos φ E1 + cos θ sin φ E2

− sin θ E3)2 + (sin φB1 − cos φB2)

2

+ (cos θ cos φB1 + cos θ sin φB2 − sin θ B3)2]

+ (p2+q2) sin2χ∫

S2d2Ω2

[(sin φB1 − cos φB2)(cos θ cos φ E1

+ cos θ sin φ E2 − sin θ E3)− (sin φ E1 − cos φ E2)

· (cos θ cos φB1 + cos θ sin φB2 − sin θ B3)]

.(5.5.10)

The total energy flux across future null infinity is obtained by evaluating this expressionon I + and integrating over it. Introducing cylinder light-cone coordinates

u = τ+χ and v = τ−χ so that t+r = −` cot v2 and t−r = −` cot u

2(5.5.11)

we characterize I + as t+r → ∞

t−r ∈ R

⇔

u ∈ (0, 2π)

v = 0

⇒ p = q = sin2χ and γ = 0 .

(5.5.12)Further noticing that

d(t−r) =` dup+q

=` du

1− cos uand sin2χ = sin2 u−v

2 = 12

(1− cos(u−v)

),

(5.5.13)we may express this total flux as

Φ+ =∫ ∞

−∞d(t−r) Φ

∣∣I + =

∫ 2π

0

` du1− cos u

Φ( u2 , u

2 ) (5.5.14)

to obtain

Φ+ =18`

∫du (1− cos u)2

∫d2Ω2

[cos θ cos φ E1 + cos θ sin φ E2 − sin θ E3 + sin φB1

− cos φB22

+

cos θ cos φB1 + cos θ sin φB2 − sin θ B3 − sin φ E1 + cos φ E22]

(5.5.15)The square bracket expression above can be further simplified for a fixed spin and type byemploying (5.3.8) along with (5.1.9), (5.1.10), (5.1.6) and (5.1.7) to get

Φ(j)+ =

Ω2

4`

∫du (1− cos u)2

∫d2Ω2

∣∣±Zj+e−iφ(1± cos θ)∓Zj

−eiφ(1∓ cos θ)−√

2Zj3 sin θ

∣∣2 ,

(5.5.16)where the upper (lower) sign corresponds to a type-I (type-II) solution. In the special caseof j = 0 (Ω=2) the contribution to the two-sphere integral only comes from the part whichis independent of (θ, φ), i.e. 4

3

(|Z0

+|2 + |Z0−|2 + |Z0

3 |2), so that the integration can easily be


performed by passing to the adjoint harmonics Yj;l,M (4.1.37) and using (4.1.36) to get

Φ(0)+ =

163 `

2π∫0

du sin4 u2

∣∣R0,0(u2 )∣∣2 1

∑n=−1

|λ0,n|2 =8`

1

∑n=−1

|λ0,n|2 = E(0) . (5.5.17)

The same equality Φ+ = E continues to hold true as we go up in spin j (we verified it forj= 1

2 and j=1), thus validating the energy conservation ∂µTµ0 = 0.

5.6 Trajectories

Given a knotted electromagnetic field configuration, a natural question that arises is howdo charged particles propagate in the background of such a field? We proceed to addressthis issue here by analyzing, with numerical simulations, the trajectories of several (iden-tical) charged point particles for the family of knotted field configurations (5.1.17) that weencountered before. We will consider type I (5.1.6) basis field configurations (up to j=1 forsimplicity) and the Hopf–Rañada field configuration (5.1.25).

In some of the simulations we employ the maximum of the energy density at time t, i.e.Emax(t) (that occurs at several points xmax that are located symmetrically with respect tothe origin), for different initial conditions and field configurations. In such cases we haveemployed a parameter Rmax(t) of “maximal” radius defined via

E (t, xmax(t)) = Emax(t) =⇒ Rmax(t) := |xmax(t)| . (5.6.1)

A characteristic feature of these basis knot electromagnetic fields is that they have a pre-ferred z-axis direction due to our convention to diagonalize the J3 action in (4.1.28); noticehere that the SO(3) isometry subgroup, and hence its generators Ja, are identified on thecylinder and the Minkowski side as shown in (5.2.3). This is clearly exemplified in Figure5.4, where the energy density E := 1

2 (E2 + B2) decreases along the z-axis. As a result, the

basis fields along the z-axis (i.e. E(t, x=0, y=0, z) and B(t, x=0, y=0, z)) are either directedin the xy-plane or along the z-axis. In fact, for extreme field configurations (j;±j,±(j+1)),for any j>0, the fields along the z-axis vanish for all times.

FIGURE 5.4: Contour plots for energy densities at t=0 (yellow), t=1 (cyan) and t=1.5 (purple) withcontour value 0.9Emax(t=1.5). Left: Hopf–Ranãda configuration, Center: (j; m, n) = ( 1

2 ;− 12 ,− 3

2 )Rconfiguration, Right: (j; m, n) = (1;−1, 1)R configuration.

The trajectories of these particles are governed by the relativistic Lorentz equation

dpdt

= q(E` + v× B`) , (5.6.2)


where q is the charge of the particle, p = γmv is the relativistic three-momentum, v is theusual three-velocity of the particle, m is its mass, γ = (1− v 2)−1/2 is the Lorentz factor,and E` and B` are dimensionful electric and magnetic fields respectively. With the energyof the particle Ep = γm and dEp/dt = q v · E, one can rewrite (5.6.2) in terms of thederivative of v [33] as

dvdt

=q

γm(E` + v× B` − (v · E`) v) . (5.6.3)

Equations (5.6.2) and (5.6.3) are equivalent, and either one can be used for a simula-tion purpose; they only differ by the position of the nonlinearity in v. In natural unitsh=c=ε0=1, every dimensionful quantity can be written in terms of a length scale. Werelate all dimensionful quantities to the de Sitter radius ` from equation (4.1.1) and workwith the corresponding dimensionless ones as follows:

T :=t`

, X :=x`

, V :=dXdT≡ v , E := `2E`, and B := `2B` . (5.6.4)

Moreover, the fields are solutions of the homogeneous (source-free) Maxwell equations, sothey can be freely rescaled by any dimensionless constant factor λ. Combining the aboveconsiderations, one can rewrite (5.6.2) (or analogously (5.6.3)) fully in terms of dimension-less quantities as

d(γV)

dT= κ(E + V× B) , (5.6.5)

where κ = q`3λm is a dimensionless parameter. One consequence of this parameter is that

we can tune the values of each of the constants separately. In particular, we can make thecharge as small as needed without changing κ such that the effect of the backreaction onthe trajectories becomes negligible. As for the initial conditions, we mostly work in thefollowing two main scenarios:

(1) N identical charged particles with V0≡V(T=0)=0 located symmetrically (with re-spect to the origin), or

(2) N identical charged particles with X0≡X(T=0)=0 with particle velocities directedradially outward in a symmetric fashion (with respect to the origin; shown in coloredarrows),

with the following 3 sub-cases for both of these conditions:

(A) along a line,

(B) on a circle of radius r,

(C) on a sphere of radius r.

We vary several parameters including the initial conditions with different directions oflines and planes for each configuration, the value of κ, and the simulation time in orderto study the behavior of the trajectories. In several field configurations studied below,we find that Rmax(0) = 0, so we use a small radius r for the initial condition of kind(1) to be able to probe the particles around a region of maximum energy of the field. Inthis scenario, the effect of the field on the trajectories of the particles is more prominent, asexpected, and this helps us understand small perturbations of the trajectories as comparedto a particle starting at rest from the origin. The effect of the fields on particles starting nearthe maximum of the energy density is also more prominent for Rmax(0) 6=0, as illustrated inFigure 5.5. Moreover, for the initial condition of kind (2) we use the particle initial speeds


in the range where it is (i) non-relativistic, (ii) relativistic (usually between 0.1 and 0.9),and (iii) ultrarelativistic (here, 0.99 or higher).

FIGURE 5.5: Simulation of N=18 particles in scenario (1C) for ( 12 ;− 1

2 ,− 32 )R with κ=10 and for

t ∈ [0, 1]. Left: r=Rmax(0)≈0.447. Right: r=Rmax(0)/3.

We observe a variety of different behaviors for these trajectories, some of which we sum-marize below with the aid of figures. Firstly, it is worth noticing that, even with all fieldsdecreasing as powers of both space and time coordinates, in most field configurations weobserve particles getting accelerated from rest up to ultrarelativistic speeds. The limit ofthese ultrarelativistic speeds for higher times depend on the magnitude of the fields (see,for example, Figure 5.6).

-3 -2 -1 1 2 3t

0.2

0.4

0.6

0.8

|v|

-3 -2 -1 1 2 3t

0.2

0.4

0.6

0.8

1.0

|v|

FIGURE 5.6: Trajectory of a charged particle for ( 12 ;− 1

2 ,− 32 )R configuration with initial conditions

X0=(0.01, 0.01, 0.01) and V0=0 simulated for t ∈ [−1, 1]. Left: Particle trajectory. Center: absolutevelocity profile for κ=10. Right: absolute velocity profile for κ=100.

With fixed initial conditions (of kind (1) or (2)) and for higher values of κ one can expect, ingeneral, that the initial conditions may become increasingly less relevant. For some fieldsconfigurations we indeed found that, with increasing κ, the particles get more focusedand accumulate like a beam of charged particles along some specific region of space andmove asymptotically for higher simulation times. This is exemplified below with twoj=0 configurations: the (0, 0,−1)I configuration in Figure 5.7, and the HR configurationin Figure 5.8. We have verified this feature not just with symmetric initial conditions ofparticles like that with initial conditions (1) and (2) (as in Figure 5.7), but also in severalinitial conditions asymmetric with respect to the origin, like particles located randomlyinside a sphere of fixed radius about the origin with zero initial velocity, and particleslocated at the origin but with different magnitudes of velocities. Figure 5.8 is an illustrativeexample for both of these latter scenarios of asymmetric initial conditions.


FIGURE 5.7: Simulation of N=18 particles for (0, 0,−1)I configuration, with κ=100 and t ∈ [0, 1].Left: scenario (1C) with r=0.1. Right: scenario (2C) with r=0.75.

FIGURE 5.8: Simulation of N=20 particles for (0, 0, 1)I configuration, with κ=1000 and t ∈ [0, 1].Left: particles starting from rest and located randomly inside a solid ball of radius r=0.01 (Rmax =0). Right: particles located at origin and directed randomly (shown with colored arrows) with

|V0|=0.45.

This is not always the case though. For some j= 12 and j=1 configurations, and with initial

particle positions in a sphere of very small radius about the origin, we are able to observethe splitting of particle trajectories (starting in some specific solid angle regions aroundthe origin) into two, three or even four such asymptotic beams that converge along someparticular regions of space (depending on the initial location of these particles in one ofthese solid angle regions). Trajectories generated by two such j=1 configurations havebeen illustrated in Figure 5.9.

FIGURE 5.9: Simulation of N=18 particles in scenario (1C) with r=0.01 and for t ∈ [0, 3]. Left:(1,−1,−2)R configuration with κ=500. Right: (1,−1,−1)R configuration with κ=10.


Naturally, there are also regions of unstable trajectories for particles starting between thesesolid angle regions (see Figure 5.10), which generally include the preferred z-axis, since insome cases trajectories that start at rest in the z-axis never leave it.

FIGURE 5.10: Simulation of N=18 particles in scenario (1C) with κ=10, r=0.01 and for t ∈ [0, 3].Left: ( 1

2 ;− 12 ,− 3

2 )R configuration. Right: (1, 0,−2)I configuration.

We employ the parameter Rmax (5.6.1) in the the following Figures 5.11, 5.12, 5.13, 5.14,5.15, and 5.16 for both kinds of initial conditions viz. (1) and (2) (it is especially relevantfor the former) to understand the effect of field intensity on particle trajectories.

FIGURE 5.11: Simulation of N=11 particles in scenario (1A) with |X0| ∝ 0.025 (including one at theorigin), for ( 1

2 ; 12 , 1

2 )I configuration (Rmax = 0) with κ=10 and t ∈ [−1, 1]. Left: particles initiallylocated along z-axis (blue line). Right: particles initially located along some (blue) line in xy-plane.

One very interesting feature of trajectories for some of these field configurations is thatthey twist and turn in a coherent fashion owing to the symmetry of the background field.For particles with initial condition of kind (2), we see that their trajectories take sharpturns, up to two times, with mild twists before going off asymptotically. This has to dowith the presence of strong background electromagnetic fields with knotted field lines.This is clearly demonstrated below in Figures 5.12, 5.13, and 5.14. It is worthwhile tonotice in Figure 5.12 that the particle which was initially at rest moves unperturbed alongthe z-axis; again, this has to do with the fact that these fields have preferred z-direction.This feature is even more pronounced in Figure 5.13 and (the right subfigure of) Figure 5.14where we see that particles with ultrarelativistic initial speeds are forced to turn (almostvertically upwards) due to the strong electromagnetic field. These particles later take veryinteresting twists in a coherent manner. This twisting feature is much more refined for thecase where initial particle velocities were directed along the xy-plane. Here also, we can


safely attribute this behavior of the particle trajectories to the special field configurations,with preferred z-direction, that we are working with.

FIGURE 5.12: Simulation of N=11 particles in scenario (2A) with |V0| ∝ 0.025 in the direction of(0, 1, 0) (including one at rest), for (1, 0, 0)I configuration (Rmax = 0), with κ=10. Left: t ∈ [0, 1].

Right: t ∈ [0, 3].

FIGURE 5.13: Simulation of N=10 particles in scenario (2B) with r=0.99 for (1,−1, 1)R configura-tion with κ=100 and t ∈ [0, 3]. Left: normal direction is y-axis. Right: normal direction is z-axis.

FIGURE 5.14: Simulation of N=18 particles in scenario (2C) for (1, 0, 0)I configuration and t ∈ [0, 2].Left: r=0.1 and κ=10. Right: r=0.99 and κ=100.

We see in Figure 5.15 that the trajectories of particles that were initially located on a circlewhose normal is along the z-axis flow quite smoothly with mild twists for some time be-fore they all turn symmetrically in a coherent way and go off asymptotically. Comparing


this with the other case in Figure 5.15, where particles split into two asymptotic beams, werealize that this is yet another instance of the preferred choice of direction for the electro-magnetic fields influencing the trajectories of particles.

FIGURE 5.15: Simulation of N=10 particles in scenario (1B) with r=0.1 for (1, 0, 0)I configuration(Rmax = 0) with κ=10 and t ∈ [0, 2]. Left: normal direction is x-axis. Right: normal direction is

z-axis.

In Figures 5.16 and 5.13 we find examples of kind (1) and (2) respectively where bothtwisting as well as turning of trajectories is prominant. We see in Figure 5.16 that theparticles that start very close to the origin take a longer time to show twists as comparedto the ones that start off on a sphere of radius Rmax. This is due to the fact that the fieldis maximal at Rmax and hence its effect on particles is prominent, as discussed before. Wealso notice here that the particles sitting along the z-axis at T=0 (either on the north pole oron the south pole of this sphere) keep moving along the z-axis without any twists or turns.This exemplifies again the fact that these background fields have a preferred direction.

FIGURE 5.16: Simulation of N=18 particles in scenario (1C) for (1,−1, 1)R configuration with t ∈[0, 1]. Left: r=0.01 and κ=10. Right: r=0.1 and κ=100.

For higher-spin configurations the maximum of the energy density increases but it gets lo-calized into an increasing number of lobes centered around the origin, due to the presenceof higher-spin harmonics. Thus, only particles located very close to the tip of these lobes ofmaximum energy density get accelerated to ultrarelativistic speeds, while particles locatedoutside (which effectively means most of the space) remain unaffected.

68

Chapter 6

Non-Abelian solutions: SU(2)

Here we present cosmic Yang–Mills solution with SU(2) gauge group and study their sta-bility behaviour. The contents of this chapter, in large parts, are taken from the publishedwork [24]. All the graphics in this chapter is due to Gabriel Picanço Costa. I have beenthoroughly involved at all stages of this research project.

6.1 SO(4)-symmetric cosmic Yang–Mills solution

The Yang–Mills action on this ansatz simplifies to

S =−14g2

∫I×S3

tr F ∧ ∗F =6π2

g2

∫I

dτ[ 1

2 ψ2−V(ψ)]

with V(ψ) = 12 (ψ

2−1)2 , (6.1.1)

where g here denotes the gauge coupling. Due to the principle of symmetric criticality [34],solutions to the mechanical problem

ψ + V ′(ψ) = 0 (6.1.2)

will, via (4.2.10), provide Yang–Mills configurations which extremize the action. Conser-vation of energy implies that

12 ψ2 + V(ψ) = E = constant , (6.1.3)

and the generic solution in the double-well potential V is periodic in τ with a period T(E).

Hence, fixing a value for E and employing time translation invariance to set ψ(0) = 0uniquely determines the classical solution ψ(τ) up to half-period shifts. Its explicit formis

ψ(τ) =

kε cn

(τε , k)

with T = 4 ε K(k) for 12 < E < ∞

0 with T = ∞ for E = 12

±√

2 sech(√

2 τ)

with T = ∞ for E = 12

± kε dn

( k τε , 1

k

)with T = 2 ε

k K( 1k ) for 0 < E < 1

2

±1 with T = π for E = 0

, (6.1.4)

where cn and dn denote Jacobi elliptic functions, K is the complete elliptic integral of thefirst kind, and

2 ε2 = 2k2−1 = 1/√

2E with k = 1√2, 1, ∞ ⇔ E = ∞, 1

2 , 0 . (6.1.5)

Chapter 6. Non-Abelian solutions: SU(2) 69

For E 12 , we have k2→ 1

2 , and the solution is well approximated by 2ε cos

( 2√

π3

Γ(1/4)2τε

). At

the critical value of E= 12 (k=1), the unstable constant solution coexists with the celebrated

bounce solution, and below it the solution bifurcates into oscillations in the left or rightwell of the double-well potential, which halfens the oscillation period. The two constantminima ψ = ±1 correspond to the vacua A = 0 and A = g−1dg. Actually, the timetranslation freedom is broken by the finite range of I , so that time-shifted solutions differin their boundary values and also in their values for the total energy and action.

0.001 0.002 0.003 0.004 0.005 0.006 0.007τ

-600

-400

-200

200

400

600

ψ(τ)

5 10 15 20 25τ

-1.5

-1.0

-0.5

0.5

1.0

1.5

ψ(τ)

0 2 4 6 8 10 12τ

0.5

1.0

1.5

ψ(τ)

0.0 0.5 1.0 1.5 2.0 2.5 3.0τ

0.5

1.0

1.5

ψ(τ)

FIGURE 6.1: Plots of ψ(τ) over one period, for different values of k2: 0.500001 (top left),0.9999999 (top right), 1.0000001 (bottom left) and 2 (bottom right).

The corresponding color-electric and -magnetic field strengths read

Ea = F0a = 12 ψ Ta and Ba = 1

2 ε bca Fbc = 1

2 (ψ2−1) Ta , (6.1.6)

which yields a finite total energy (on the cylinder) of 6π2E/g2 and a finite action [13, 27]

g2S[ψ] = 6π2∫I

dτ[E− (ψ2−1)2] = 6π2

∫I

dτ[ψ2 − E

]≥ −3π2|I| . (6.1.7)

The energy-momentum tensor of our SO(4)-symmetric Yang–Mills solutions is readilyfound as

T =3 Eg2a2

(dτ2 + 1

3 dΩ23)

, (6.1.8)

which is traceless as expected.

The Einstein equations for a closed FLRW universe with cosmological constant Λ reduce totwo independent relations, which can be taken to be its trace and its time-time component.


In conformal time one gets, respectively, −R + 4 Λ = 0

Rττ +12 R a2 −Λ a2 = κ Tττ

⇔

a + W ′(a) = 012 a2 + W(a) = κ

2g2 E =: E′

(6.1.9)

with a gravitational coupling κ = 8πG, a gravitational energy E′ and a cosmological po-tential

W(a) = 12 a2 − Λ

6 a4 . (6.1.10)

The two anharmonic oscillators, with potential V for the gauge field and potential W forgravity, are coupled only via the balance of their conserved energies,

1κ

[ 12 a2 + W(a)

]=

12g2

[ 12 ψ2 + V(ψ)

], (6.1.11)

which is nothing but the Wheeler–DeWitt constraint H = 0.

-2 -1 1 2a

-0.2

0.2

0.4

0.6

0.8

1.0W (a)

-1.5 -1.0 -0.5 0.5 1.0 1.5ψ

-0.2

0.2

0.4

0.6

0.8

1.0V(ψ)

FIGURE 6.2: Plots of the cosmological potential W(a) for λ=1 and the double-well potential V.

The Friedmann equation (6.1.9), being a mechanical system with an inverted anharmonicpotential (6.1.10), is again easily solved analytically,

a(τ) =

√3Λ

12ε′

√1−cn

(τε′ ,k

′)

1+cn(

τε′ ,k

′) with T′ = 2 ε′K(k′) for 3

8Λ < E′ < ∞√3

2Λ tanh(τ/√

2)

with T′ = ∞ for E′ = 38Λ√

3Λ

12ε′

√1−dn

( k′τε′ , 1

k′)

1+dn( k′τ

ε′ , 1k′) with T′ = 2 ε′

k′K(1k′ ) for 0 < E′ < 3

8Λ

0 with T′ = π for E′ = 0

,

(6.1.12)where we abbreviated 1

2 ε′2= 2k′2−1 = 1/

√8Λ3 E′ so that k′ = 1√

2, 1, ∞ ⇔ E′ = ∞, 3

8Λ , 0 . (6.1.13)

For E′ 38Λ , we have k′2→ 1

2 , and the solution is well approximated by√

3Λ

12ε′ tan

( √π3

Γ(1/4)2τε′).

We only listed solutions with initial value a(0) = 0 (big bang). There exist also (forE′ < 3

8Λ ) bouncing solutions, where the universe attains a minimal radius amin =[ 3

2Λ (1 +

1Our k′2 should not be confused with the dual modulus 1−k2, which is often denoted this way.


0 1 2 3τ

0.2

0.4

0.6

0.8

1.0

1.2a(τ)

0 2 4 6 8 10 12τ

2

4

6

8

10a(τ)

2 4 6 8 10 12τ

0.2

0.4

0.6

0.8

1.0

1.2

a(τ)

0.00 0.05 0.10 0.15 0.20 0.25τ

20

40

60

80

100a(τ)

FIGURE 6.3: Plots of a(τ) over one lifetime, for Λ=1 and different values of k′2: 0.505 (top left),0.9999999 (top right), 1.0000001 (bottom left) and 1.1 (bottom right).

√1− 8Λ

3 E′)]1/2 between infinite extension in the far past (t=−∞ ↔ τ=0) and the far fu-

ture (t=+∞ ↔ τ=T′). For E′>0 they are obtained by sending dn → −dn in (6.1.12)above. The quantity T′ listed there is the (conformal) lifetime of the universe, from the bigbang until either the big rip (for E′ > 3

8Λ ) or the big crunch of an oscillating universe (forE′ < 3

8Λ ). The solution relevant to our Einstein–Yang–Mills system is entirely determinedby the Newtonian energy E characterizing the cosmic Yang–Mills field: above the criticalvalue of

Ecrit =2 g2

κ

38 Λ

(6.1.14)

the universe expands forever (until tmax=∞), while below this value it recollapses (attmax=

∫ T′

0 dτ a(τ)). It demonstrates the necessity of a cosmological constant (whose rolemay be played by the Higgs expectation value) as well as the nonperturbative nature of thecosmic Yang–Mills field, whose contribution to the energy-momentum tensor is of O(g−2).

6.2 Natural perturbation frequencies

Our main task in this paper is an investigation of the stability of the cosmic Yang–Mills so-lutions reviewed in the previous section. For this, we should distinguish between globaland local stability. The former is difficult to assess in a nonlinear dynamics but clear fromthe outset in case of a compact phase space. The latter refers to short-time behavior in-duced by linear perturbations around the reference configuration. We shall look at thisfirstly, in the present section and the following one. Here, we set out to diagonalize thefluctuation operator for our time-dependent Yang–Mills backgrounds and find the naturalfrequencies.


Even though our cosmic gauge-field configurations are SO(4)-invariant, we must allow forall kinds of fluctuations on top of it, SO(4)-symmetric perturbations being a very specialsubclass of them. A generic gauge potential “nearby” a classical solution A on I × S3 canbe expanded as

A+ Φ = A(τ, g) +3

∑p=1

Φp0(τ, g) Tp dτ +

3

∑a=1

3

∑p=1

Φpa (τ, g) Tp ea(g) (6.2.1)

with, using (µ) = (0, a),

Φpµ(τ, g) = ∑

j,m,nΦp

µ|j;m,n(τ)Yj;m,n(g) , (6.2.2)

on which we notice the following actions (suppressing the τ and g arguments),

(LaΦpµ)j;m,n = Φp

µ|j;m,n′(

La)n′n , (SaΦ)

p0 = 0 ,

(SaΦ)pb = −2εabc Φp

c , (TaΦ)pµ = −2εapq Φq

µ ,(6.2.3)

where La matrix elements are determined from (4.1.28) and (4.1.30), while Sa are spin op-erator components. The (metric and gauge) background-covariant derivative reads

DτΦ = ∂τΦ and DaΦ = LaΦ + [Aa, Φ] with Aa = 12

(1 + ψ(τ)

)Ta , (6.2.4)

which is equivalent to

DaΦpb = LaΦp

b − εabcΦpc + [Aa, Φb]

p since Da eb = La eb − εabc ec = εabc ec . (6.2.5)

The background A obeys the Coulomb gauge condition,

Aτ = 0 and LaAa = 0 , (6.2.6)

but we cannot enforce these equations on the fluctuation Φ. However, we may impose theLorenz gauge condition,

DµΦpµ = 0 ⇒ ∂τΦp

0 − LaΦpa − 1

2 (1+ψ)(TaΦa)p = 0 , (6.2.7)

which is seen to couple the temporal and spatial components of Φ in general. We thenlinearize the Yang–Mills equations around A and obtain

DνDνΦµ − RµνΦν + 2[Fµν, Φν] = 0 (6.2.8)

with the Ricci tensorRµ0 = 0 and Rab = 2δab . (6.2.9)

After a careful evaluation, the µ=0 equation yields[∂2

τ − LbLb + 2(1+ψ)2]Φp0 − (1+ψ)Lb(TbΦ0)

p − ψ(TbΦb)p = 0 , (6.2.10)

while the µ=a equations read[∂2

τ − LbLb + 2(1+ψ)2+4]Φp

a − (1+ψ)Lb(TbΦ)pa − Lb(SbΦ)

pa

− 12 (1+ψ)(2−ψ)(SbTbΦ)

pa − ψ(TaΦ0)

p = 0 .(6.2.11)


It is convenient to package the orbital, spin, isospin, and fluctuation triplets into formalvectors,

~L = (La) , ~S = (Sa) , ~T = (Ta) , ~Φ = (Φa) , (6.2.12)

respectively, but they act in different spaces, hence on different indices, such that ~S2 =~T2 = −8 on Φ. In this notation, (6.2.7), (6.2.10) and (6.2.11) take the compact form (sup-pressing the color index p)

∂τΦ0 −~L·~Φ− 12 (1+ψ)~T·~Φ = 0 , (6.2.13)[

∂2τ−~L2 + 2(1+ψ)2]Φ0 − (1+ψ)~L·~T Φ0 − ψ~T·~Φ = 0 , (6.2.14)[

∂2τ−~L2− 1

2~S2+2(1+ψ)2]Φa − (1+ψ)~L·~T Φa −~L·(~S Φ)a

− 12 (1+ψ)(2−ψ)~T·(~S Φ)a − ψ TaΦ0 = 0 .

(6.2.15)

A few remarks are in order. First, except for the last term, (6.2) is obtained from (6.2) bysetting ~S = 0, since Φ0 carries no spin index. Second, both equations can be recast as[∂2

τ −1−ψ

2~L2 − 1+ψ

2 (~L+~T)2 − 2(1+ψ)(1−ψ)]Φ0 = ψ~T · ~Φ , (6.2.16)[

∂2τ −

(1−ψ)(2+ψ)4

~L2 − ψ(1+ψ)4 (~L+~T)2 + ψ(1−ψ)

4 (~L+~S)2 − (1+ψ)(2−ψ)4 (~L+~T+~S)2

− 2(1+ψ)(1−ψ)]~Φ = ψ~TΦ0 ,

(6.2.17)

which reveals a problem of addition of three spins and a corresponding symmetry under

ψ ↔ −ψ , ~L ↔ ~L+~T+~S and ~L+~S ↔ ~L+~T . (6.2.18)

Third, for constant backgrounds (ψ=0) the temporal fluctuation Φ0 decouples and maybe gauged away. Still, the fluctuation operator in (6.2) is easily diagonalized only whenthe coefficient of one of the first three spin-squares vanishes, i.e. for~L=0 (j=0), for the twovacua ψ=±1, or for the “meron” ψ=0. The latter case has been analyzed by Hosotani [11].

Let us decompose the fluctuation problem (6.2)–(6.2) into finite-dimensional blocks ac-cording to a fixed value of the spin j ∈ 1

2N,

~L2 Φpµ|j = −4 j(j+1)Φp

µ|j (6.2.19)

and suppress the j subscript. We employ the following coupling scheme,2

~L+~T =: ~U then ~U+~S = (~L+~T)+~S =: ~V . (6.2.20)

Clearly, ~U and ~V act on ~Φ in su(2) representations j ⊗ 1 and j ⊗ 1⊗ 1, respectively. OnΦ0, we must put ~S=0 and have just ~V=~U act in a j ⊗ 1 representation. Combining thecoupled equations (6.2) and (6.2) to a single linear system for (Φp

µ) = (Φp0 , Φp

a ), we get a12(2j+1)× 12(2j+1) fluctuation matrix Ω2

(j),[δ

pqµν ∂2

τ + (Ω2(j))

pqµν

]Φq

ν = 0 . (6.2.21)

Actually, there is an additional overall (2j+1)-fold degeneracy present due to the triv-ial action of the su(2)R generators Ra, which plays no role here and will be suppressed.

2Another (less convenient) scheme couples~L+~S, then (~L+~S)+~T =: ~V.


Roughly speaking, the 3(2j+1) modes of Φ0 are related to gauge modes,3 and we stillmust impose the gauge condition (6.2), which also has 3(2j+1) components. Therefore,a subspace of dimension 6(2j+1) inside the space of all fluctuations will represent thephysical gauge-equivalence classes in the end.

Our goal is to diagonalize the fluctuation operator (6.2.21) for a given fixed value of j. Ithas a block structure,

Ω2(j) =

(N −ψ T>

−ψ T N

), (6.2.22)

where N and N are given by the left-hand sides of (6.2) and (6.2), respectively. We intro-duce a basis where ~U2, ~V2 and V3 are diagonal, i.e.

~U2 |uvm〉 = −4 u(u+1) |uvm〉 and ~V2 |uvm〉 = −4 v(v+1) |uvm〉 , (6.2.23)

with m = −v, . . . , v and denote the irreducible su(2)v representations with those quantumnumbers as

[ vu]. On the Φ0 subspace, u is redundant since u=v as ~S=0. Working out the

tensor products, we encounter the values[ vu]=[ j−2

j−1

];[ j−1

j−1

],[ j−1

j

];[ j

j−1

],[ j

j

],[ j

j+1

];[ j+1

j

],[ j+1

j+1

];[ j+2

j+1

]on ~Φ ,[ v

u]=

[ j−1j−1

];

[ jj

];

[ j+1j+1

]on Φ0 ,

(6.2.24)with some representations obviously missing for j<2.

Let us treat the ψ T term in (6.2.22) as a perturbation and momentarily put it to zero, sothat Ω2

(j) is block-diagonal for the time being. Then, it is easy to see from (6.2) and (6.2)

that [~V, N] = [~U, N] = 0 and [~V, N] = 0, even though [~U, N] 6= 0 because (~L+~S)2 is notdiagonal in our basis. Therefore, we have a degeneracy in m. Furthermore, both N andN decompose into at most three respectively five blocks with fixed values of v rangingfrom j−2 to j+2 and separated by semicolons in (6.2.24). Moreover, the N blocks areirreducible and trivially also carry a value of u=v. In contrast, N is not simply reducible;its ~V representations have multiplicity one, two or three. Only the N blocks with extremalv values in (6.2.24) are irreducible. The other ones are reducible and contain more than one~U representation, hence the u-spin distinguishes between their (two or three) irreduciblev subblocks. The only non-diagonal term in N is the (~L+~S)2 contribution, which couplesdifferent copies of the same v-spin to each other, but of course not to any u=v block of N,and does not lift the V3=m degeneracy. As a consequence, the unperturbed fluctuationequations for Φ0=Φ(v) and ~Φ=Φ(v,α) take the form (suppressing the m index)

1(v)[∂2

τ + ω2(v)]

Φ(v) = 0 and 1(v)[∂2

τ + ω2(v,α)

]Φ(v,α) = 0

for ψ = 0 with v ∈ j−1, j, j+1 and v ∈ j−2, j−1, j, j+1, j+2 ,(6.2.25)

where 1(v) denotes a unit matrix of size 2v+1, and α counts the multiplicity of the v-spin representation in N (between one and three). According to (6.2) the unperturbedfrequency-squares for N are the eigenvalues

ω2(v) = 2(1−ψ) j(j+1) + 2(1+ψ) v(v+1)− 2(1+ψ)(1−ψ) (6.2.26)

3Strictly, they are gauge modes only when ψ=0. Otherwise, the gauge modes are mixtures with the Φamodes.


with multiplicity 2v+1, hence we get

ω2(j−1) = 2 ψ2 − 4j ψ + 2(2j2−1) ,

ω2(j) = 2 ψ2 + 2(2j2+2j−1)

ω2(j+1) = 2 ψ2 + 4(j+1)ψ + 2(2j2+4j+1) .

(6.2.27)

Considering N in (6.2), we can read off the eigenvalues at v = j±2 because in these twoextremal cases (~L+~S)2 = ~U2 is already diagonal in the

|uvm〉

basis. For the other v-

values we must diagonalize a 2×2 or 3×3 matrix to find

ω2(j−2) = root of Qj−2(λ) = −2(2j−1)ψ + 2(2j2−2j+1) ,

ω2(j−1,α) = two roots of Qj−1(λ) ,

ω2(j,α) = three roots of Qj(λ) ,

ω2(j+1,α) = two roots of Qj+1(λ) ,

ω2(j+2) = root of Qj+2(λ) = 2(2j+3)ψ + 2(2j2+6j+5) ,

(6.2.28)

each with multiplicity 2v+1, where Qv denotes a linear, quadratic or cubic polynomial.4

Let us now turn on the perturbation ψ T, which couples N with N, and consider the char-acteristic polynomial Pj(λ) of our fluctuation problem,

Pj(λ) := det( N−λ −ψT>

−ψT N−λ

)= det(N−λ) · det

[(N−λ)− ψ2 T>(N−λ)−1T

]=[∏v det(N(v)−λ)

]· det

[(N−λ)− ψ2 T>⊕v(N(v)−λ)−1 T

],

(6.2.29)

where we made use of

〈u v m|N |u′v′m′ > =(

N(v))

uu′ δvv′δmm′ . (6.2.30)

Since T furnishes an su(2) representation (and not an intertwiner) it must be representedby square matrices and thus cannot connect different v representations. Hence the pertur-bation does not couple different v sectors but only links N and N in a common v=v sector.Therefore, it does not affect the extremal sectors v = j±2. Moreover, switching to a diago-nal basis |αvm〉 for N we can simplify to

T>⊕v(N(v)−λ)−1 T]=⊕

vT>(N−λ)−1T(v)=⊕

v

∑α(ω

2(v,α)−λ)−1(T>|α〉〈α| T)

(v)

.

(6.2.31)

Observing that(T>|α〉〈α| T

)(v) = −tv,α

(~T2)

(v) = 8 tv,α1(v) with some coefficient functionstv,α(ψ), with ∑α tv,α = 1, we learn that the V3 degeneracy remains intact and arrive at

4For j<2 some obvious modifications occur due to the missing of v<0 representations.


(v ∈ j−1, j, j+1)

Pj(λ) =[∏vQv(λ)

2v+1] ·∏v(ω2

(v)−λ)− 8ψ2∑αtv,α(ω2(v,α)−λ)−12v+1

= (ω2(j−2)−λ)2j−3 · (ω2

(j+2)−λ)2j+5 ·∏v(ω2

(v)−λ) Qv(λ)− 8ψ2Pv(λ)2v+1

= (ω2(j−2)−λ)2j−3 · (ω2

(j+2)−λ)2j+5 ·∏vRv(λ)2v+1 ,

(6.2.32)where Pv = Qv ∑α tv,α(ω2

(v,α)−λ)−1 is a polynomial of degree one less than Qv since allpoles cancel, and Rv is a polynomial of one degree more. We list the polynomials Qv, Pvand Rv for j≤2 in the Appendix.

To summarize, by a successive basis change (m′ = −j, . . . , j and m = −v, . . . , v)|µ p m′〉

⇒

|vm〉, |uvm〉

⇒

|vm〉, |αvm〉

⇒

|βvm〉

(6.2.33)

we have diagonalized (6.2.21) to[∂2

τ + Ω2(j,v,β)

]Φ(v,β) = 0 with v ∈ j−2, j−1, j, j+1, j+2 , (6.2.34)

where Ω2(j,v,β) are the distinct roots of the characteristic polynomial Pj in (6.2.32), and (for

j≥2) the multiplicity β takes 1, 3, 4, 3, 1 values, respectively:

Ω2(j,j±2) = ω2

(j±2) , Ω2(j,j±1,β) = three roots of Rj±1(λ) , Ω2

(j,j,β) = four roots of Rj(λ) .(6.2.35)

The reflection symmetry (6.2.18) implies that Ω2(v,j,·)(ψ) = Ω2

(j,v,·)(−ψ). For j<2, obviousmodifications occur due to the absence of some v representations.

We still have to discuss the gauge condition (6.2), which can be cast into the form

0 = ∂τΦ0 −[ 1

2 (1−ψ)~L + 12 (1+ψ)~U

]· ~Φ = ∂τΦ(v,m) − K v,m,α

v,m (ψ)Φ(v,m,α) (6.2.36)

with a 3(2j+1)×7(2j+1) linear (in ψ) matrix function K.5 Here the v sum runs over(j−1, j, j+1) only, since the gauge condition (6.2) has components only in the middlethree v sectors, like the gauge-mode equation (6.2). It does not restrict the extremal v sec-tors v = j±2, since these fluctuations do not couple to the gauge sector Φ0 and are entirelyphysical. For the middle three v sectors (labelled by v), the ψ T perturbation leads to amixing of the N modes with the N gauge modes, so their levels will avoid crossing. Per-forming the corresponding final basis change, the gauge condition takes the form[

L v′,m′,βv,m (ψ) ∂τ −M v′,m′,β

v,m (ψ)]

Φ(v′,m′,β) = 0 (6.2.37)

with certain 3(2j+1)×10(2j+1) matrix functions L and M. This linear equation representsconditions on the normal mode functions Φ(v,m,β) and defines a 7(2j+1)-dimensional sub-space of physical fluctuations, which of course still contains a 3(2j+1)-dimensional sub-space of gauge modes. For j<1, these numbers are systematically smaller. Together withthe two extremal v sectors, we end up with (7− 3 + 2)(2j+1) = 6(2j+1) physical degreesof freedom for any given value of j(≥2), as advertized earlier.

We conclude this section with more details for the simplest examples, which are constantbackgrounds and j=0 backgrounds. For the vacuum background, say ψ = −1, which is

5We have to bring back the m indices because the gauge condition is not diagonal in them.


isospin degenerate, one gets(∂2

τ − 12~L2 − 1

2 (~L+~S)2) ~Φ = 0, ~L·~Φ = 0 , Φ0 = 0 . (6.2.38)

It yields the positive eigenfrequency-squares

ω2(j,u′) = 2j(j+1) + 2u′(u′+1) =

4j2 at j≥1 for u′ = j−14j(j+1) for u′ = j4(j+1)2 for u′ = j+1

(6.2.39)

for j = 0, 12 , 1, . . ., but the ~L·~Φ = 0 constraint removes the u′=j modes. Clearly, all (con-

stant) eigenfrequency-squares are positive, hence the vacuum is stable.

For the “meron” background, ψ ≡ 0, one has(∂2

τ − 12~L2 − 1

2 (~L+~T+~S)2 − 2

)~Φ = 0,

(~L + 1

2~T)· ~Φ = 0 , Φ0 = 0 . (6.2.40)

In this case, we read off

ω2(j,v) + 2 = 2j(j+1) + 2v(v+1) =

4(j2−j+1) for v = j−2 (0 to 1×)4j2 for v = j−1 (0 to 2×)4j(j+1) for v = j (1 to 3×)4(j+1)2 for v = j+1 (1 to 2×)4(j2+3j+3) for v = j+2 (1×)

,

(6.2.41)but the constraint removes one copy from each of the three middle cases (and less whenj<1). We end up with a spectrum ω2 = −2, 1, 6, 7, 10, . . . with certain degenera-cies [11]. The single non-degenerate negative mode ω2

(0,0)=−2 is a singlet, Φpa = δ

pa φ(τ),

and it corresponds to rolling down the local maximum of the double-well potential. Themeron is stable against all other perturbations.

For a time-varying background, the natural frequencies Ω(j,v,β) inherit a τ dependencefrom the background ψ(τ). Direct diagonalization is still possible for j=0, where weshould solve

∂τΦ0 − 12 (1+ψ)~T·~Φ = 0 ,[

∂2τ + 2(1+ψ)2]Φ0 − ψ~T·~Φ = 0 ,[

∂2τ + 2(3ψ2−1)− 1

4 (1+ψ)(2−ψ)(~S+~T)2] ~Φ− ψ~T Φ0 = 0 ,

(6.2.42)

with

(~S+~T)2 = ~V2 = −4 v(v+1) = 0,−8,−24 for v = 0, 1, 2 . (6.2.43)

It implies the unperturbed frequencies (suppressing the j index)

ω2(1) = 2(ψ+1)2 (3×) , ω2

(0) = 2(3ψ2−1) (1×) ,

ω2(1) = 2(2ψ2+ψ+1) (3×), ω2

(2) = 2(3ψ+5) (5×)(6.2.44)

for(Φ0)

p ≡(Φ(v=1)

)p=: δpbφb ,

(~Φ)pa ≡

(Φ(0) + Φ(1) + Φ(2)

)pa =: φ δ

pa + ε

pab φb + (φ(ab)−δabφ)δbp ,

(6.2.45)


as long as ψ is ignored. There are no v-spin multiplicities (larger than one) here. Turningon ψ and observing that (~T·~Φ)p ∼ δpbφb, the characteristic polynomial of the coupled12×12 system in the |uvm〉 basis reads

P0(λ) = det

(ω2

(1)−λ)13 0 −ψ T>(1) 0

0 (ω2(0)−λ)11 0 0

−ψ T(1) 0 (ω2(1)−λ)13 0

0 0 0 (ω2(2)−λ)15

. (6.2.46)

Specializing the general discussion above to j=0, we find just t1=1 so that P1=1 and arriveat

P0(λ) = (ω2(0)−λ)1(ω2

(1)−λ)3(ω2(2)−λ)5[(ω2

(1)−λ)− 8ψ2(ω2(1)−λ)−1]3

= (ω2(0)−λ)(ω2

(2)−λ)5(ω2(1)−λ)(ω2

(1)−λ)− 8ψ23 .(6.2.47)

We see that the frequencies Ω2(0)=ω2

(0) and Ω2(2)=ω2

(2) are unchanged and given by (6.2.44),while the gauge mode ω2

(1) gets entangled with the (unphysical) v=1 mode to produce thepair

Ω2(1,±) = 1

2 (ω2(1)+ω2

(1)) ±√

14 (ω

2(1)+ω2

(1))2 − ω2

(1)ω2(1) + 8ψ2

= 3ψ2+3ψ+2 ±√

ψ2(ψ−1)2 + 8ψ2(6.2.48)

with a triple degeneracy. There are avoided crossings at ψ=0 and ψ=1. Removing theunphysical and gauge modes in pairs, we remain with the singlet mode Ω2

(0,0) and thefivefold-degenerate Ω2

(0,2). For all higher spins j>0, analytic expressions for the naturalfrequencies Ω(j,v,β) now require merely solving a few polynomial equations of order fourat worst. We have done so up to j=2 and list them in the Appendix but refrain fromgiving further explicit examples here. From the cases of j=0 and j=2 displayed below onecan see that some of the normal modes dip into the negative regime, i.e. their frequency-squares become negative, for a certain fraction of the time τ. Because of this and, quitegenerally, due to the τ variability of the natural frequencies, it is not easy to predict thelong-term evolution of the fluctuation modes. Clearly, the stability of the zero solutionΦ≡0, equivalent to the linear stability of the background Yang–Mills configuration, is notsimply decided by the sign of the τ-average of the corresponding frequency-square.

6.3 Stability analysis: stroboscopic map and Floquet theory

The diagonalized linear fluctuation equation (6.2.34) represents a bunch of Hill’s equa-tions, where the frequency-squared is a root of a polynomial of order up to four withcoefficients given by a polynomial of twice that order in Jacobi elliptic functions. A uniquesolution requires fixing two initial conditions, and so for each fluctuation Φ(j,v,β) there is atwo-dimensional solution space. It is well known that Hill’s equation, e.g. in the limit ofMathieu’s equation, displays parametric resonance phenomena, which can stabilize oth-erwise unstable systems or destabilize otherwise stable ones.

For oscillating dynamical systems with periodically varying frequency, there exist somegeneral tools to analyze linear stability. Switching to a Hamiltonian picture and to phasespace, it is convenient to transform the second-order differential equation into a system of


0.1 0.2 0.3 0.4 0.5 0.6 0.7τ

-100

100

200

300

v=0

v=1

v=1

v=2

2 4 6 8 10τ

5

10

15 v=0

v=1

v=1

v=2

1 2 3 4 5τ

5

10

15 v=0

v=1

v=1

v=2

0.5 1.0 1.5 2.0 2.5 3.0τ

5

10

15

v=0

v=1

v=1

v=2

FIGURE 6.4: Plots of Ω2(0,v,β)(τ) over one period, for different values of k2: 0.51 (top left),

0.99 (top right), 1.01 (bottom left) and 5 (bottom right).

0.1 0.2 0.3 0.4 0.5τ

-200

200

400

600

v=0

v=1

v=1

v=1

v=2

v=2

v=2

v=2

v=3

v=3

v=3

v=4

0.5 1.0 1.5τ

-20

20

40

60

80

100

v=0

v=1

v=1

v=1

v=2

v=2

v=2

v=2

v=3

v=3

v=3

v=4

2 4 6 8 10 12τ

10

20

30

40

50

60

70 v=0

v=1

v=1

v=1

v=2

v=2

v=2

v=2

v=3

v=3

v=3

v=4

1 2 3 4 5 6 7τ

10

20

30

40

50

60

70 v=0

v=1

v=1

v=1

v=2

v=2

v=2

v=2

v=3

v=3

v=3

v=4

FIGURE 6.5: Plots of Ω2(2,v,β)(τ) over one period, for different values of k2: 0.505 (top left),

0.550 (top right), 0.999 (bottom left) and 1.001 (bottom right).

two coupled first-order equations (suppressing all quantum numbers),

[∂2

τ + Ω2(τ)]Φ(τ) = 0 ⇔ ∂τ

(Φ

Φ

)=

(0 1

−Ω2 0

)(Φ

Φ

)=: i Ω(τ)

(Φ

Φ

), (6.3.1)


where the frequency Ω(τ) is T-periodic (sometimes T2 -periodic) in τ. The solution to this

first-order system is formally given by(Φ

Φ

)(τ) = T exp

∫ τ

0dτ′ i Ω(τ′)

(Φ

Φ

)(0) , (6.3.2)

where T denotes time ordering. Because of the time dependence of Ω, the time evolutionoperator above is not homogeneous thus does not constitute a one-parameter group, ex-cept when the propagation interval is an integer multiple of the period T. For τ=T, onespeaks of the stroboscopic map [35]

M := T exp∫ T

0dτ i Ω(τ)

⇒

(Φ

Φ

)(nT) = Mn

(Φ

Φ

)(0) . (6.3.3)

The linear map M is a functional of the chosen background solution ψ and hence dependson its parameter E or k. This background is Lyapunov stable if the trivial solution Φ≡0 is,which is decided by the two eigenvalues µ1 and µ2 of M. Since the system is Hamiltonian,det M=1, we have three cases:

|tr M| > 2 ⇔ µi ∈ R ⇔ hyperbolic/boost ⇔ strongly unstable ,|tr M| = 2 ⇔ µi = ±1 ⇔ parabolic/translation ⇔ marginally stable ,|tr M| < 2 ⇔ µi ∈ U(1) ⇔ elliptic/rotation ⇔ strongly stable .

(6.3.4)Clearly, |tr M| determines the linear stability of our classical solution.

Let us thus try to evaluate the trace of the stroboscopic map M, making use of the specialform of the matrix Ω,

tr M =∞

∑n=0

in∫ T

0dτ1

∫ τ1

0dτ2 . . .

∫ τn−1

0dτn tr

[Ω(τ1) Ω(τ2) · · · Ω(τn)

]= 2 +

∞

∑n=1

(−1)n∫ T

0dτ1

∫ τ1

0dτ2 . . .

∫ τn−1

0dτn Hn(τ1, τ2, . . . , τn)Ω2(τ1)Ω2(τ2) · · ·Ω2(τn)

with Hn(τ1, τ2, . . . , τn) = (τ1−τ2)(τ2−τ3) · · · (τn−1−τn)(τn−τ1+1) and H1(τ1) = 1 .(6.3.5)

It is convenient to scale the time variable such as to normalize the period to unity,

τ = T x and Ω2(Tx) =: ω2(x) , H(Tx) =: h(x) , (6.3.6)

hence

tr M = 2 +∞

∑n=1

(−T2)n

∫ 1

0dx1

∫ x1

0dx2 . . .

∫ xn−1

0dxn hn(x1, x2, . . . , xn)ω

2(x1)ω2(x2) · · ·ω2(xn)

=∞

∑n=0

2(2n)!

Mn(−T2)n

=: 2−M1T2 + 112 M2T4 − 1

360 M3T6 + 120160 M4T8 − . . . .

(6.3.7)

It is impossible to evaluate the integrals Mn without explicit knowledge of ω2(x). As acrude guess, we replace the weight function by its (constant) average value

〈hn〉 :=1n!

∫ 1

0dx1

∫ x1

0dx2 . . .

∫ xn−1

0dxn hn(x1, x2, . . . , xn) =

2 n!(2n)!

(6.3.8)


and obtain

Mn =(2n)!

2〈hn〉

∫ 1

0dx1

∫ x1

0dx2 . . .

∫ xn−1

0dxn

n

∏i=1

ω2(xi) =(∫ 1

0dx ω2(x)

)n=: 〈ω2〉n ,

(6.3.9)which yields

tr M = 2∞

∑n=0

(−1)n

(2n)!〈ω2〉n T2n = 2 cos

(√〈ω2〉 T

). (6.3.10)

This expression indicates stability as long as 〈ω2〉 > 0. However, the result for the j=0singlet mode ω2 = Ω2

(0,0) in (6.3.11) already showed that the averaged frequency-squaredmay turn negative in certain domains thus changing the cos into a cosh there.

To do better, let us look at the individual terms Mn in (6.3.7) for the simplest case of theSO(4) singlet fluctuation, i.e. Ω2

(0,0) = 6ψ2−2 in (6.2.44). Its average frequency-square iseasily computed to be

〈Ω2(0,0)〉 =

1ε2

(6

E(k)K(k)

+ 4k2 − 5)

, (6.3.11)

where E(k) and K(k) denote the second and first complete elliptic integrals, respectively.Plotting this expression as a function of the modulus k, we see that it becomes negativeonly in a very narrow range around k=1, namely for |k−1| . 0.00005. We have only been

0.8 1.0 1.2 1.4 1.6k

-2

2

4

6

<Ω2 (0,0)>

0.99995 1.00000 1.00005 1.00010k

-2.0

-1.5

-1.0

-0.5

<Ω2 (0,0)>

FIGURE 6.6: Plot of 〈Ω2(0,0)〉 as a function of k, with detail on the right.

able to analytically evaluate (with k<1 for simplicity)

M1 = 〈Ω2(0,0)〉 and M2 = 〈Ω2

(0,0)〉2 − 1

ε4

(9

2−k2

K(k)2 − 27E(k)K(k)3 +

9 π2

4 K(k)4

), (6.3.12)

which does not suffice to rule out instability. Indeed, numerical studies show that Mn asa function of k looses its positivity in a range around k=1 which increases with n, wherethe series (6.3.7) ceases to be alternating. Moreover, even in the limit of a very large back-ground amplitude, k2 → 1

2 , we find that

〈Ω2(0,0)〉 →

24 π2

ε2 Γ( 14 )

4≈ 1.37

ε2 ⇒√〈Ω2〉 T →

√24 π ≈ 8.68 , (6.3.13)

implying that we must push the series in (6.3.7) at least to O(M10T20), even though it turnsout that Mn < 〈Ω2

(0,0)〉n at k2 = 1

2 for n > 1.

For a more complete analysis of linear stability in an oscillating system with time-dependentfrequency we can take recourse to Floquet theory. It tells us that a general fundamental


matrix solution

Φ(τ) =

(Φ1 Φ2

Φ1 Φ2

)(τ) ⇒ ∂τΦ(τ) = i Ω(τ) Φ(τ) (6.3.14)

of our system (6.3.1) with some initial condition Φ(0) = Φ0 can be expressed in so-calledFloquet normal form as

Φ(τ) = Q(τ) eτR with Q(τ+2T) = Q(τ) , (6.3.15)

where Q(τ) and R are real 2×2 matrices, so that the time dependence of the frequency canbe transformed away by a change of coordinates,

Ψ(τ) := Q(τ)−1Φ(τ) ⇒ ∂τΨ(τ) = R Ψ(τ) . (6.3.16)

Due to the identity

Φ(τ+T) = Φ(τ) Φ(0)−1 Φ(T) = Φ(T) Φ(0)−1 Φ(τ) = M Φ(τ) (6.3.17)

we see that our stroboscopic map M is nothing but the monodromy, and

M2 = Φ(2T) Φ(0)−1 = Q(0) Φ(0)−1 Φ(2T) Q(0)−1 = Q(0) e2RT Q(0)−1 , (6.3.18)

so that its eigenvalues (or characteristic multipliers)

µi = eρiT for i = 1, 2 (6.3.19)

define a pair of (complex) Floquet exponents ρi whose real parts are the Lyapunov expo-nents. Since µ1µ2 = 1 implies that ρ1+ρ2 = 0, our system is linearly stable if and only ifboth eigenvalues ρi of R are purely imaginary (or zero).

Generally it is impossible to find analytically the monodromy pertaining to a normalmode Φ(j,v,β).6 However, we can get a qualitative understanding by looking numericallyat some examples. Before numerically integrating Hill’s equation, however, let us estimateat which energies E or, rather, moduli k, possible resonance frequencies might occur. Tothis end, we determine the period-average of the natural frequency Ω(j,v,β) and compare itto its modulation frequency 2π

T . If we model

Ω2(τ) = 〈Ω2〉(1 + h(τ)

)with 〈Ω2〉 = 1

T

T∫0

dτ Ω2(τ) and h(τ) ∝ cos(2πτ/T) ,

(6.3.20)where T = 4 ε K(k), then the resonance condition is met for√

〈Ω2〉 = `π

T⇒ k = k`(j, v, β) for ` = 1, 2, 3, . . . . (6.3.21)

Since this model reproduces only the rough features of Ω2(τ), we expect potential insta-bility due to parametric resonance effects in a band around or near the values k`.

Below we display, together with the would-be resonant values k`, the function trM(k) forthe sample cases of (j, v) = (2, 0) and (2, 2). One sees that, on both sides of the criticalvalue of E= 1

2 (or k=1), corresponding to the double-well local maximum, the k` valuesaccumulate at the critical point. But while for k>1 (energy below the critical point) trM(k)

6An exception is the SO(4) singlet perturbation Φ(0,0), to be treated in the following section.


0.8 0.9 1.0 1.1 1.2k

-8

-6

-4

-2

2

4

6tr(M(2,0) )

0.9995 1.0000 1.0005 1.0010k

-2

-1

1

2

tr(M(2,0) )

FIGURE 6.7: Plot of tr M(k) for (j, v) = (2, 0), with detail on the right. Would-be resonances markedin red.

0.8 0.9 1.0 1.1 1.2k

-2

2

4

6

8

tr(M(2,2,1) )

0.8 0.9 1.0 1.1 1.2k

-2

-1

1

2

3

tr(M(2,2,2) )

0.8 0.9 1.0 1.1 1.2k

-2

-1

1

2

tr(M(2,2,3) )

0.8 0.9 1.0 1.1 1.2k

-2

-1

1

2

tr(M(2,2,4) )

FIGURE 6.8: Plots of tr M(k) for (j, v) = (2, 2) and β = 1, 2, 3, 4. Would-beresonances marked in red.

oscillates between values close to 2 in magnitude and thus exponential growth is rare andmild, for k<1 (energy above the critical point) the oscillatory behavior of trM(k) comeswith an amplitude exceeding 2 and growing with energy. Hence, in this latter regimestable and unstable bands alternate. This is supported by long-term numerical integration,as we demonstrate by plotting Φ(τ) for (j, v, β) = (2, 2, 1) with initial values Φ(0)=1 andΦ(0)=0 on both sides very close to the end of the first instability (at the highest value of Eor the lowest value of k).

Most relevant for the cosmological application is the regime of very large energies, E→ ∞(or k → 1/

√2). In this limit, we observe the following universal behavior. Because the

period T collapses with ε=√

k2−1/2, we rescale

τε = z ∈ [0, 4K( 1

2 )] , ε ψ = ψ , ε2ψ = ∂zψ , ε2Ω2 = Ω2 , ε2λ = λ (6.3.22)


10 20 30 40 50τ

5

10

15

20

25

Φ(τ)

10 20 30 40 50τ

0.2

0.4

0.6

0.8

1.0

Φ(τ)

FIGURE 6.9: Plot of Φ(τ) for (j, v, β) = (2, 2, 1) and k=0.73198 (left) and k=0.73199 (right).

so that the tilded quantities remain finite in the limit, and find, with ω2(v) → 2ψ2,7

Q(v±2) ∼ λ ,

Q(v±1) ∼ λ (λ− 4ψ2) , R(v±1) ∼ λ[(λ− 2ψ2)(λ− 4ψ2)− 8(∂zψ)2] ,

Q(v) ∼ λ (λ− 4ψ2)(λ− 6ψ2) , R(v) ∼ λ[(λ− 2ψ2)(λ− 4ψ2)− 8(∂zψ)2](λ− 6ψ2) ,

(6.3.23)because all j-dependent terms in the polynomials are subleading and drop out in the limit.Factorizing the R polynomials, we find the four universal natural frequency-squares

Ω21 = 0 , Ω2

2 = 3 ψ2 −√

ψ4 + 8(∂zψ)2 , Ω23 = 3 ψ2 +

√ψ4 + 8(∂zψ)2 , Ω2

4 = 6 ψ2 .(6.3.24)

One must pay attention, however, to the fact that the avoided crossings disappear in theε→ 0 limit. Therefore, the correct limiting frequencies to input into[

∂2z + Ω2

(j,v,β)

]Φ(j,v,β) = 0 (6.3.25)

are

Ω2(j,j±2) = 0 ,

Ω2(j,j±1,β) ∈

min(Ω2

1, Ω22), max(Ω2

1, Ω22), Ω2

3

,

Ω2(j,j,β) ∈

min(Ω2

1, Ω22), max(Ω2

1, Ω22), min(Ω2

3, Ω24), max(Ω2

3, Ω24)

,

(6.3.26)

of which we show below the last list as a function of z. The monodromies are easilycomputed numerically,8

tr M(j,j±2)(E→∞) = 2 ,

tr M(j,j±1,β)(E→∞) ∈

306.704, −1.842, −1.659

,

tr M(j,j,β)(E→∞) ∈

306.704, −1.842, 2.462, −1.067

,

(6.3.27)

in agreement with the figures above. In particular, the extremal v-values become marginallystable, while part of the non-extremal cases are unstable for high energies.

Of course, for each non-extremal value of v we still have to project out unphysical modes

7For the cases (j, v) = (0, 1) and (1, 0), the factor λ is missing; for (j, v) = (0, 0), one only has R = Q ∼(λ−6ψ2).

8For the cases (j, v) = (0, 1) and (1, 0) one gets 56.769, −1.659; for (j, v) = (0, 0) we have tr M = 2.


1 2 3 4 5 6 7z

-1

1

2

3

Ω(j,j)2

FIGURE 6.10: Plot of the universal limiting natural frequency-squares Ω2(j,v,β) for v=j and

β = 1, 2, 3, 4.

by imposing the gauge condition (6.2.37). However, in the 12(2j+1)-dimensional fluc-tuation space the gauge condition has rank 3(2j+1) while we see that (for j≥2) in total4(2j+1) normal modes are unstable at high energy. Therefore, the projection to physicalmodes cannot remove all instabilities. We must conclude that, for sufficiently high en-ergy E, some fluctuations grow exponentially, implying that the solution Φ≡0 is linearlyunstable, and thus is the Yang–Mills background.

6.4 Singlet perturbation: exact treatment

Even though the Floquet representation helped to reduce the long-time behavior of theperturbations to the analysis of a single period T, it normally does not give us an exactsolution to Hill’s equation. However, for the SO(4) singlet fluctuation around ψ(τ), wecan employ the fact that ψ trivially solves the fluctuation equation,

(ψ)·· = (ψ)· = −(V ′(ψ)

)·= −V ′′(ψ) ψ = −(6ψ2−2) ψ = −Ω2

(0,0)(τ) ψ , (6.4.1)

with a frequency function which is T2 -periodic. This implies that all fluctuation modes are

T-periodic. With the knowledge of an explicit solution to the fluctuation equation we canreduce the latter to a first-order equation and solve that one to find a second solution. Thenormalizations are arbitrary, so we choose

Φ1(τ) = − ε3

k ψ(τ) and Φ2(τ) = Φ1(τ)∫ τ dσ

Φ1(σ)2 = − kε3 ψ(τ)

∫ τ dσ

ψ2(σ), (6.4.2)

which are linearly independent since

W(Φ1, Φ2) ≡ Φ1Φ2 −Φ2Φ1 = 1 . (6.4.3)


For simplicity, we restrict ourselves to the energy range 12<E<∞, i.e. 1>k2> 1

2 . Explicitly,we have

Φ1(τ) = ε sn(

τε , k)

dn(

τε , k)

,

Φ2(τ) = 11−k2 cn

(τε , k)[(2k2−1)dn2( τ

ε , k)− k2]

+ sn(

τε , k)

dn(

τε , k)[

τε +

2k2−11−k2 E

(am( τ

ε , k), k)]

,

(6.4.4)

where am(z, k) denotes the Jacobi amplitude and E(z, k) is the elliptic integral of the sec-ond kind. As can be checked, the initial conditions are

10 20 30 40τ

-0.3

-0.2

-0.1

0.1

0.2

0.3

Φ1(τ)

10 20 30 40τ

-50

50

Φ2(τ)

FIGURE 6.11: Plot of the SO(4) singlet fluctuation modes Φ1 and Φ2 over eight periods for k2=0.81.

Φ1(0) = 0 , Φ1(0) = 1 and Φ2(0) = −1 , Φ2(0) = 0 , (6.4.5)

which fixes the ambiguity of adding to Φ2 a piece proportional to Φ1. Hence,

Φ(0) =(

0−11 0

)⇒ M = Φ(T)

(0 1−1 0

). (6.4.6)

We know that Φ1 ∼ ψ is T-periodic, and so is Φ1, but not the second solution,

Φ2(τ+T) = Φ2(τ) + γ T Φ1(τ) with γ =1T

∫ T

0

dσ

Φ1(σ)2

∣∣∣∣reg

=: k2

ε6

⟨ψ−2⟩

reg , (6.4.7)

where the integral diverges at the turning points and must be regularized by subtractingthe Weierstraß ℘ function with the appropriate half-periods. Since Φ1 has periodic zeros,Φ2 does return to −1 at integer multiples of T. It follows that the Φ2 oscillation linearlygrows in amplitude with a rate (per period) of

γ =1ε2

[1 +

2k2−11−k2

E(k)K(k)

], (6.4.8)

which is always larger than 7.629, attained at k ≈ 0.882.

In essence, we have managed to compute the monodromy

M =

(−Φ2(T) Φ1(T)

−Φ2(T) Φ1(T)

)=

(1 0

−γ T 1

)= exp

−γ T

( 0 01 0

)(6.4.9)

and thus easily obtain the Floquet representation,

R =

(0 γ

0 0

)⇒ eτR =

(1 γ τ

0 1

)and Q(τ) =

(Φ1 Φ2−Φ1γ τ

Φ1 Φ2−Φ1γ τ

). (6.4.10)


0.75 0.80 0.85 0.90 0.95 1.00k

10

20

30

40

50γ(k)

FIGURE 6.12: Plot of the linear growth rate γ as a function of k.

Obviously, we have encountered a marginally stable situation, since M is of parabolic type.There is no exponential growth, and Φ1 is periodic thus bounded, but Φ2 grows withoutbound as long as one stays in the linear regime. Note that we never made use of the formof our Newtonian potential. In fact, this behavior is typical for a conservative mechanicalsystem with oscillatory motion.

What to make of this linear growth? It can be (and actually is) easily overturned by non-linear effects. Going beyond the linear regime, though, requires expanding the Yang–Millsequation to higher orders about our classical Yang–Mills solution (4.2.10). While this is aformidable task in general, it can actually be done to all orders for the singlet perturba-tion! The reason is that a singlet perturbation leaves us in the SO(4)-symmetric subsector,thus connecting only to a neighboring “cosmic background”, ψ → ψ. Since (6.1.4) givesus analytic control over all solutions ψ(τ), the full effect of such a shift can be computedexactly. Splitting an exact solution ψ into a background part and its (full) deviation,

ψ(τ) = ψ(τ) + η(τ) , (6.4.11)

and inserting ψ into the equation of motion (6.1.2), we obtain

0 = η + V ′′(ψ) η + 12 V ′′′(ψ) η2 + 1

6 V ′′′′(ψ) η3 = η + (6ψ2−2) η + 6ψ η2 + 2 η3 , (6.4.12)

extending the linear equation (6.4.1) by two nonlinear contributions. Perturbation theoryintroduces a small parameter ε and formally expands

η = εη(1) + ε2η(2) + ε3η(3) + . . . , (6.4.13)

which yields the infinite coupled system[∂2

τ + (6ψ2−2)]

η(1) = 0 ,[∂2

τ + (6ψ2−2)]

η(2) = −6ψ η2(1) ,[

∂2τ + (6ψ2−2)

]η(3) = −12ψ η(1)η(2) − 2 η3

(1) ,

. . . ,

(6.4.14)

which could be iterated with a seed solution η(1) of the linear system.


However, we know that the exact solutions to the full nonlinear equation (6.4.12) is simplygiven by the difference

η(τ) = ψ(τ)− ψ(τ) (6.4.15)

of two analytically known backgrounds. The SO(4)-singlet background moduli space isparametrized by two coordinates, e.g. the energy E (or elliptic modulus k) and the choiceof an initial condition which fixes the origin τ=0 of the time variable. In (6.1.4), we selectedψ(0) = 0, but relaxing this we can reintroduce this collective coordinate by allowing shiftsin τ. We may then parametrize the SO(4)-invariant Yang–Mills solutions as

ψk,`(τ) = ψ(τ−`) with 2E = 1/(2k2−1)2 and ` ∈ R (6.4.16)

where ψ is taken from (6.1.4). Note that ψk,` solves the background equation (6.4.1) witha frequency-squared ω2

k,` = 6ψ2k,`−2. Without loss of generality we assign ψ = ψk,0 and

ψ = ψk+δk,δ`, hence

η(τ) = δk ∂kψ(τ)− δ` ψ(τ) + 12 (δk)2 ∂2

kψ(τ)− δkδ` ∂kψ(τ) + 12 (δ`)

2 ψ(τ) + . . .

= δk ∂kψ(τ−δ`) + 12 (δk)2 ∂2

kψ(τ−δ`) + 16 (δk)3 ∂3

kψ(τ−δ`) + . . . ,(6.4.17)

because ∂`ψ = −ψ. Clearly, a shift in ` only shifts the time dependence of the frequencyand does not alter the energy E, which is not very interesting. Its linear part corresponds tothe mode Φ1 ∼ ψ of the previous section. A change in k, in contract, will lead to a solutionwith an altered frequency and energy. Its linear part is given by Φ2, which grows linearlyin time. However, due to the boundedness of the full motion, the nonlinear correctionshave to limit this growth and ultimately must bring the fluctuation back close to zero.This is the familiar wave beat phenomenon: the difference of two oscillating functions, ψand ψ, with slightly different frequencies, will display an amplitude oscillation with a beatfrequency given by the difference. This is borne out in the following plots. As a result, we

5 10 15 20 25 30τ

-1.5

-1.0

-0.5

0.5

1.0

1.5

η(τ)

50 100 150 200τ

-3

-2

-1

1

2

3

η(τ)

FIGURE 6.13: Plots of the full perturbation η at k=0.95 for η(0)=0.02, η(0)=0, giving a beat ratioof ∼19.

can assert a long-term stability of the cosmic Yang–Mills fields against the SO(4) singletperturbation, even though on shorter time scales an excursion to a nearby solution is notmet with a linear backreaction.

89

Chapter 7

Conclusion & Outlook

We have studied stability behaviour of some well known solutions of SU(2) Yang–Millsfields in 4-dimensional de Sitter space dS4 under generic gauge perturbation. These so-lutions could be of relevance to early time cosmology (before the electro-weak symmetrybreaks down) in a scenario recently presented by Friedan [10]. An SO(4) symmetric sec-tor is analytically solvable and reduces to three coupled anharmonic oscillators (for themetric, an SU(2) Yang–Mills field and the Higgs field, the latter being frozen to its vac-uum state). We have presented a complete analysis of the linear gauge-field perturbationsof the time-dependent Yang–Mills solution, by diagonalizing the fluctuation operator andstudying the long-time behavior of the ensuing Hill’s equations using the stroboscopicmap and Floquet theory. For parametrically large gauge-field energy (as is required inFriedan’s setup) the natural frequencies and monodromies become universal, and someunstable perturbation modes survive even in this limit. This provides strong evidencethat such oscillating cosmic Yang–Mills fields are unstable against small perturbations, al-though we have not yet included metric fluctuations here. Their influence will be analyzedin follow-up work.

We have also analyzed a family of electromagenetic knot configurations recently devel-oped by making use of a conformal equivalence between dS4, Minkwoski spaceR1,3 and afinite Lorentzian S3-cylinder. These solutions are constructed on the cylinder in an SO(4)covariant way and are then pulled back to the Minkowski space using the conformal map(that leaves the Maxwell’s theory invariant). These “basis-knot" solutions of Maxwell’sequation are labelled with S3 harmonics Yj,m,n and give rise to field configurations of knot-ted field lines when pulled back to the Minkowski space. We have analyzed the symmetryfeature of these basis knotted electromagnetic field configurations with the isometry groupSO(1, 4) of the de Sitter space. We have further studied, numerically, the effect of these ba-sis configurations on the trajectories of multiple identical charged particles with differentinitial conditions. Various behaviors were obtained, including a separation of trajectoriesinto different “solid angle regions” that converge asymptotically into a beam of chargedparticles along a few particular regions of space, an ultrarelativistic acceleration of parti-cles and coherent twists/turns of the trajectories before they go off asymptotically. Theresults contribute to an effort to better understand the interactions between electromag-netic knots and charged particles [36]. This becomes increasingly relevant as laboratorygeneration of knotted fields progresses [5]. Further work in this direction could be to an-alyze a single Fourier mode of these solutions to understand its experimental realizationvia monochromatic laser beams.

Furthermore, we have considered complex linear combinations of these basis-knot fieldconfigurations (that can model any finite-energy field configuration) and characterizedthe corresponding moduli space of null fields. We have also computed Noether chargesof such a linearly combined configuration with fixed j for the conformal group SO(2, 4),

Chapter 7. Conclusion & Outlook 90

which is the largest symmetry group for the Maxwell theory. Here again the “de Sitter"method proved to be advantageous in that the expressions of these charge densities sim-plifies immensely on the cylinder and, thus, can be computed with ease. We found thatmany of these charges vanish owing to the orthogonality of the harmonics and that the en-ergy and momentum are the only independent charged in many cases. A nice geometricstructure of 1-forms facilitated the computation of spherical components of vector chargesas well. We verified our results against the results for some modified Hopf–Ranãda fieldconfigurations of [14]. We would like to further check the validity of our results by com-paring them with other solutions presented in [14].

91

Appendix A

Carter-Penrose transformation

The metric on the Minkowski space R1,3 in polar coordinates is given by

ds2Mink = −dt2 + dr2 + r2 dΩ2

2 ; dΩ22 = dθ2 + sin2 θ dφ2, (A.0.1)

where −∞ < t < ∞, 0 ≤ r < ∞, 0 ≤ θ ≤ π, and 0 ≤ φ ≤ 2π. We first employ thelight-cone coordinates (u, v) to transform the metric in the following way

u := t− r , v := t + r ; −∞ < u ≤ v < ∞

=⇒ ds2Mink = −dv du + 1

4 (v− u)2 dΩ22 .

(A.0.2)

In the second step we compactify the spacetime with the help of the coordinate (U,V) asfollows:

U := arctan u , V := arctan v ; −π2 < U ≤ V < π

2

=⇒ ds2Mink =

14 cos2 V cos2 U

[−4 dV dU + sin2(V −U)dΩ2

2]

.(A.0.3)

Finally, we rotate the coordinate system back using (τ, χ) to obtain the desired form of themetric:

τ := V + U , χ := π + U −V ; 0 < χ ≤ π , |τ| < χ

=⇒ ds2Mink = γ−2 [−dτ2 + dχ2 + sin2 χ dΩ2

2]

; γ = cos τ − cos χ .(A.0.4)

We realize that the Minkowski metric is conformally equivalent to a Lorentzian cylinderI × S3 ; I = (−π, π) with a conformal factor that can be recasted in terms of Minkowskicoordinates using the above transformations (A.0.2-A.0.4):

ds2cyl := −dτ2 + dΩ2

3 = γ2 ds2Mink ; γ =

2`2√4 t2 `2 + (r2 − t2 + `2)2

, (A.0.5)

where we have made γ dimensionless using the de Sitter radius `. The lightcone structureof the spacetime as presented in the Penrose diagram (see Figure 4.2) is a direct conse-quence of (A.0.4).

92

Appendix B

Rotation of indices

By construction the gauge potential A is SO(4) invariant, which means it is also invari-ant under the action of SO(3) generators Da (4.1.26). For a complex-valued A = Aa ea

expanded using (5.1.8) (for type I) and (5.1.11) at a fixed j this means

0 = Da(A)

= ∑m,n,n

Cn,nj,b eiΩτ

(Da(Λm,n) eb Yj;m,n + Λm,nDa(eb)Yj;m,n + Λm,n ebDa(Yj;m,n)

) (B.0.1)

where Da(eb) are determined from (4.1.21) and (4.1.22) while Da(Yj;m,n) are determinedfrom (4.1.28-4.1.30). By collecting the coefficients of various linearly independent eb andYj;m,n terms in the above expansion for a fixedDa one gets a set of coupled linear equationsfor Da(Λm,n), which can be easily solved. The action of the generators Da on Λm,n forj = 0, 1/2 and 1 is given in the following table.

D1 D2 D3

j = 0 :Λ0,−1 7→Λ0,0 7→Λ0,1 7→

√2iΛ0,0√2i(Λ0,1 + Λ0,−1)√2iΛ0,0

−√

2Λ0,0√2(Λ0,−1 −Λ0,1)√2Λ0,0

−√

2iΛ0,−1

0√

2iΛ0,1

j = 12 :

Notation︸︷︷︸± 1

2 ≡ ±± 3

2 ≡↑↓

Λ−,↓ 7→Λ−,− 7→Λ−,+ 7→Λ−,↑ 7→Λ+,↓ 7→Λ+,− 7→Λ+,+ 7→Λ+,↑ 7→

i(√

3Λ−,− + Λ+,↓)

i(√

3Λ−,↓ + 2Λ−,+ + Λ+,−)

i(2Λ−,− +√

3Λ−,↑ + Λ+,+)

i(√

3Λ−,+ + Λ+,↑)

i(Λ−,↓ +√

3Λ+,−)

i(Λ−,− +√

3Λ+,↓ + 2Λ+,+)

i(Λ−,+ + 2Λ+,− +√

3Λ+,↑)

i(Λ−,↑ +√

3Λ+,+)

−√

3Λ−,− −Λ+,↓√3Λ−,↓ − 2Λ−,+ −Λ+,−

2Λ−,− −√

3Λ−,↑ −Λ+,+√3Λ−,+ −Λ+,↑

Λ−,↓ −√

3Λ+,−

Λ−,− +√

3Λ+,↓ − 2Λ+,+

Λ−,+ + 2Λ+,− −√

3Λ+,↑

Λ−,↑ +√

3Λ+,+

− 4iΛ−,↓

− 2iΛ−,−

0

2iΛ−,↑

− 2iΛ+,↓

0

2iΛ+,+

4iΛ+,↑

j = 1 :Notation︸︷︷︸± 1 ≡ ±± 2 ≡↑↓

Λ−,↓ 7→Λ−,− 7→Λ−,0 7→Λ−,+ 7→Λ−,↑ 7→Λ0,↓ 7→Λ0,− 7→Λ0,0 7→Λ0,+ 7→Λ0,↑ 7→Λ+,↓ 7→Λ+,− 7→Λ+,0 7→Λ+,+ 7→Λ+,↑ 7→

i(2Λ−,− +√

2Λ0,↓)

i(2Λ−,↓ +√

6Λ−,0 +√

2Λ0,−)

i(√

6Λ−,− +√

6Λ−,+ +√

2Λ0,0)

i(√

6Λ−,0 + 2Λ−,↑ +√

2Λ0,+)

i(2Λ−,+ +√

2Λ+,↑)

i(√

2Λ−,↓ + 2Λ0,− +√

2Λ+,↓)

i√

2(Λ−,− +√

2Λ0,↓ +√

3Λ0,0 + Λ+,−)

i√

2(Λ−,0 +√

3Λ0,− +√

3Λ0,+ + Λ+,0)

i√

2(Λ−,+ +√

3Λ0,0 +√

2Λ0,↑ + Λ+,+)

i(√

2Λ−,↑ + 2Λ0,+ +√

2Λ+,↑)

i(√

2Λ0,↓ + 2Λ+,−)

i(√

2Λ0,− + 2Λ+,↓ +√

6Λ+,0)

i(√

2Λ0,0 +√

6Λ+,− +√

6Λ+,+)

i(√

2Λ0,+ +√

6Λ+,0 + 2Λ+,↑)

i(√

2Λ0,↑ + 2Λ+,+)

− 2Λ−,− −√

2Λ0,↓

2Λ−,↓ −√

6Λ−,0 −√

2Λ0,−√6Λ−,− −

√6Λ−,+ −

√2Λ0,0√

6Λ−,0 − 2Λ−,↑ −√

2Λ0,+

2Λ−,+ −√

2Λ0,↑√2Λ−,↓ − 2Λ0,− −

√2Λ+,↓√

2(Λ−,− +√

2Λ0,↓ −√

3Λ0,0 −Λ+,−)√2(Λ−,0 +

√3Λ0,− −

√3Λ0,+ −Λ+,0)√

2(Λ−,+ +√

3Λ0,0 −√

2Λ0,↑ −Λ+,+)√2Λ−,↑ + 2Λ0,+ −

√2Λ+,↑√

2Λ0,↓ +−2Λ+,−√2Λ0,− + 2Λ+,↓ −

√6Λ+,0√

2Λ0,0 +√

6Λ+,− −√

6Λ+,+√2Λ0,+ +

√6Λ+,0 − 2Λ+,↑√

2Λ0,↑ + 2Λ+,+

− 6iΛ−,↓

− 4iΛ−,−

− 2iΛ−,0

0

2iΛ−,↑

− 4iΛ0,↓

− 2iΛ0,−

0

2iΛ0,+

4iΛ0,↑

− 2iΛ+,↓

0

2iΛ+,0

4iΛ+,+

6iΛ+,↑

93

Appendix C

Polynomials in the characteristicequation

j , v P(λ) Q(λ) R(λ)

0 , 0 N/A (−2 + 6ψ2)− λ −(2− 6ψ2) + λ

0 , 1 1 (2 + 2ψ + 4ψ2)− λ (4 + 12ψ + 20ψ2 + 20ψ3 + 8ψ4 − 8ψ2)− (4 +6ψ + 6ψ2)λ + λ2

0 , 2 N/A (10 + 6ψ)− λ N/A12 , 1

2 −(1 + 6ψ2) +λ

−(1 + 2ψ2 + 24ψ4) + (2 +10ψ2)λ− λ2

−(1 + 4ψ2 + 28ψ4 + 48ψ6 − 8ψ2 − 48ψ2ψ2) +(3 + 16ψ2 + 44ψ4 − 8ψ2)λ− (3 + 12ψ2)λ2 + λ3

12 , 3

2 −(7 + 3ψ) + λ −(49 + 42ψ + 32ψ2 +12ψ3) + (14 + 6ψ +

4ψ2)λ− λ2

−(343 + 588ψ + 574ψ2 + 360ψ3 + 136ψ4 +24ψ5− 56ψ2− 24ψψ2)+ (147+ 168ψ+ 124ψ2 +48ψ3 + 8ψ4 − 8ψ2)λ− (21 + 12ψ + 6ψ2)λ2 + λ3

12 , 5

2 N/A (17 + 8ψ)− λ N/A

1 , 0 1 (2− 2ψ + 4ψ2)− λ (4− 12ψ + 20ψ2 − 20ψ3 + 8ψ4 − 8ψ2)− (4−6ψ + 6ψ2)λ + λ2

1 , 1 (36 + 36ψ2)−(12 + 6ψ2)λ +

λ2

(216 + 192ψ2 + 104ψ4)−(108 + 92ψ2 + 24ψ4)λ +

(18 + 10ψ2)λ2 − λ3

(1296 + 1584ψ2 + 1008ψ4 + 208ψ6 − 288ψ2 −288ψ2ψ2)− (864 + 960ψ2 + 432ψ4 + 48ψ6 −96ψ2 − 48ψ2ψ2)λ + (216 + 188ψ2 + 44ψ4 −

8ψ2)λ2 − (24 + 12ψ2)λ3 + λ4

1 , 2 −(14 + 4ψ) +λ

−(196 + 112ψ + 60ψ2 +16ψ3) + (28 + 8ψ +

4ψ2)λ− λ2

−(2744 + 3136ψ + 2128ψ2 + 928ψ3 + 248ψ4 +32ψ5 − 112ψ2 − 32ψψ2) + (588 + 448ψ +

236ψ2 + 64ψ3 + 8ψ4 − 8ψ2)λ− (42 + 16ψ +6ψ2)λ2 + λ3

1 , 3 N/A (26 + 10ψ)− λ N/A32 , 1

2 −(7− 3ψ) + λ −(49− 42ψ + 32ψ2 −12ψ3) + (14− 6ψ +

4ψ2)λ− λ2

−(343− 588ψ + 574ψ2 − 360ψ3 + 136ψ4 −24ψ5− 56ψ2 + 24ψψ2)+ (147− 168ψ+ 124ψ2−48ψ3 + 8ψ4 − 8ψ2)λ− (21− 12ψ + 6ψ2)λ2 + λ3

32 , 3

2 (169 +78ψ2)− (26 +

6ψ2)λ + λ2

(2197 + 962ψ2 + 216ψ4)−(507 + 204ψ2 + 24ψ4)λ +

(39 + 10ψ2)λ2 − λ3

(28561 + 16900ψ2 + 4732ψ4 + 432ψ6 −1352ψ2 − 624ψ2ψ2) + (−8788− 4628ψ2 −

936ψ4 − 48ψ6 + 208ψ2 + 48ψ2ψ2)λ + (1014 +412ψ2 + 44ψ4 − 8ψ2)λ2 − (52 + 12ψ2)λ3 + λ4

32 , 5

2 −(23 + 5ψ) +λ

−(529 + 230ψ + 96ψ2 +20ψ3) + (46 + 10ψ +

4ψ2)λ− λ2

−(12167 + 10580ψ + 5566ψ2 + 1880ψ3 +392ψ4 + 40ψ5 − 184ψ2 + 40ψψ2) + (1587 +

920ψ + 380ψ2 + 80ψ3 + 8ψ4 − 8ψ2)λ + (−69−20ψ− 6ψ2)λ2 + λ3

32 , 7

2 N/A (37 + 12ψ)− λ N/A

2 , 0 N/A (10− 6ψ)− λ N/A

2 , 1 −(14− 4ψ) +λ

−(196− 112ψ + 60ψ2 −16ψ3) + (28− 8ψ +

4ψ2)λ− λ2

−(2744− 3136ψ + 2128ψ2 − 928ψ3 + 248ψ4 −32ψ5 − 112ψ2 + 32ψψ2) + (588− 448ψ +

236ψ2 − 64ψ3 + 8ψ4 − 8ψ2)λ− (42− 16ψ +6ψ2)λ2 + λ3

2 , 2 (484 +132ψ2)−

(44 + 6ψ2)λ +λ2

(10648 + 2816ψ2 +360ψ4)− (1452 + 348ψ2 +24ψ4)λ + (66 + 10ψ2)λ2 −

λ3

(234256 + 83248ψ2 + 13552ψ4 + 720ψ6 −3872ψ2 − 1056ψ2ψ2)− (42592 + 13376ψ2 +

1584ψ4 + 48ψ6 − 352ψ2 − 48ψ2ψ2)λ + (2904 +700ψ2 + 44ψ4 − 8ψ2)λ2 − (88 + 12ψ2)λ3 + λ4

2 , 3 −(34 + 6ψ) +λ

−(1156 + 408ψ + 140ψ2 +24ψ3) + (68 + 12ψ +

4ψ2)λ− λ2

−(39304 + 27744ψ + 11968ψ2 + 3312ψ3 +568ψ4 + 48ψ5 − 272ψ2 − 48ψψ2) + (3468 +

1632ψ + 556ψ2 + 96ψ3 + 8ψ4 − 8ψ2)λ− (102 +24ψ + 6ψ2)λ2 + λ3

2 , 4 N/A (50 + 14ψ)− λ N/A

94

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Curriculum Vitae

1. PersonalName Kaushlendra KumarDate of Birth 06/05/1993Place of Birth Varanasi, IndiaNationality IndianEmail [email protected] https://k-kumar.netlify.app/

2. Education.10/2018 – 09/2022 PhD physics, Institute for Theoretical Physics, Leibniz

University Hannover. Examination date: 04.05.2022

07/2015 – 05/2018 MSc Physics, Indian Institute of Science Education and Research,Kolkata. Examination date: 20.07.2017

07/2010 – 05/2015 BTech & MTech Biochemical Engineering, Indian Institute ofTechnology, BHU, Varanasi. Examination date: 27.05.2015

05/2010 Senior School Certificate, science stream (Physics, Chemistry,Mathematics), Jesus & Mary Academy, Darbhanga.

05/2008 Secondary School Certificate, Jawahar Navodaya Vidayalaya,Piprakothi, East Champaran.

3. List of publications.

3.1 Refereed.

1. K. Kumar, O. Lechtenfeld and G. Picanço Costa, Trajectories of charged particles inknotted electromagnetic fields, J. Phys. A: Math. Theor. 55 (2022) 315401.

2. L. Hantzko, K. Kumar and G. Picanço Costa, Conserved charges for rational electro-magnetic knots, Eur. Phys. J. Plus 137 (2022) 407.

3. K. Kumar, O. Lechtenfeld and G. Picanço Costa, Instability of cosmic Yang–Millsfields, Nucl. Phys. B 973 (2021) 115583.

4. K. Kumar and O. Lechtenfeld, On rational electromagnetic fields, Phys. Lett. A 384(2020) 126445.

5. K. Kumar and B. Chakraborty, Spectral distances on the doubled Moyal plane usingDirac eigenspinors, Phys. Rev. D 97 (2018) 086019.

6. Y.C. Devi, K. Kumar, B. Chakraborty and F.G. Scholtz, Revisiting Connes’ finite spec-tral distance on noncommutative spaces: Moyal plane and fuzzy sphere, Int. J. Geom.Methods Mod. Phys. 15 (2018) 1850204.

7. K. Kumar, S. Prajapat and B. Chakraborty, On the role of Schwinger’s SU(2) genera-tors for simple harmonic oscillator in 2D Moyal plane, Eur. Phys. J. Plus 130 (2015)120.

3.2 Preprint.

• K. Kumar, O. Lechtenfeld, G. Picanço Costa and J. Röhrig, Yang–Mills solutions onMinkowski space via non-compact coset spaces, arXiv:2206.12009 [hep-th].

https://k-kumar.netlify.app/

http://arxiv.org/abs/2206.12009

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