Semiclassical spectrum of a Jordaniandeformation of AdS5 × S5

Riccardo Borsato,a Sibylle Driezen,a Juan Miguel Nieto Garcıa b and Leander Wyss b

a Instituto Gallego de Fısica de Altas Energıas (IGFAE) and Departamento de Fısica de Partıculas,

Universidade de Santiago de Compostela

b Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK

riccardo.borsato@usc.es, sib.driezen@gmail.com,

j.nietogarcia@surrey.ac.uk, l.wyss@surrey.ac.uk

Abstract

We study a Jordanian deformation of the AdS5×S5 superstring. It is an example of homogeneousYang-Baxter deformations, a class which generalises TsT deformations to the non-abelian case.Many of the attractive features of TsT carry over to this more general class, from the possibilityof generating new supergravity solutions to the preservation of worldsheet integrability. Inthis paper, we exploit the fact that the deformed σ-model with periodic boundary conditionscan be reformulated as an undeformed one with twisted boundary conditions, to discuss theconstruction of the classical spectral curve and its semi-classical quantisation. First, we findglobal coordinates for the deformed background, and identify the global time corresponding tothe energy that should be computed in the spectral problem. Using the curve of the twistedmodel, we obtain the one-loop correction to the energy of a particular solution, and we find thatthe charge encoding the twisted boundary conditions does not receive an anomalous correction.Finally, we give evidence suggesting that the unimodular version of the deformation (giving riseto a supergravity background) and the non-unimodular one (whose background does not solvethe supergravity equations) have the same spectrum at least to one-loop.

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Contents

1 Introduction 2

2 A Jordanian-deformed background 5

2.1 The deformed σ-model and the background fields . . . . . . . . . . . . . . . . . . 5

2.2 Residual isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Global coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 On the generators of time translations . . . . . . . . . . . . . . . . . . . . 10

2.3.2 Analysis of geodesic completeness . . . . . . . . . . . . . . . . . . . . . . . 11

3 BMN-like solution of the σ-model 14

4 Map to an undeformed yet twisted model 16

4.1 Review of the twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Manifest symmetries of the twisted model . . . . . . . . . . . . . . . . . . . . . . 18

4.3 A simpler twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.4 Classical solution in the twisted model . . . . . . . . . . . . . . . . . . . . . . . . 20

4.4.1 Map from the classical solution of the deformed model . . . . . . . . . . . 20

4.4.2 More general classical solutions of the twisted model . . . . . . . . . . . . 22

5 Classical spectral curve 24

5.1 Asymptotics of the quasimomenta . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Classical algebraic curve for the BMN-like solution . . . . . . . . . . . . . . . . . 29

6 Quantum corrections to the BMN-like solution 30

6.1 Corrections to the classical quasimomenta . . . . . . . . . . . . . . . . . . . . . . 30

6.2 Anomalous correction to the twist . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.3 One-loop correction to the energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 The unimodular case and the spectral equivalence 36

8 Conclusions 39

A Relation to embedding coordinates of AdS 41

B Geodesic incompleteness of Poincare coordinates 42

C Identifying the Cartan subalgebra of isometries 44

D Analytic structure of the perturbations 45

1

D.1 Excitations associated to the sphere . . . . . . . . . . . . . . . . . . . . . . . . . 45

D.2 Excitations associated to the deformed AdS space . . . . . . . . . . . . . . . . . 47

D.3 Fermionic excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

D.4 Composition and inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

E Bosonic sector of the quadratic fluctuations 51

1 Introduction

We currently know various examples of deformations of σ-models that preserve integrability,see [1] for a review. Important representatives are the λ-deformation [2–4], the inhomogeneousdeformation (also called η-deformation) [5–8] and the class of homogeneous Yang-Baxter de-formations [9, 10]. These are being studied with various motivations, for example to clarify apossible link between integrability and renormalisability of the σ-models [11–13]. When thesedeformations are applied to string σ-models, they lead to deformations of the correspondingtarget-space backgrounds. By now, the conditions when they give rise to backgrounds thatsolve the supergravity equations are also well understood [14–19]. Interestingly, they were alsoreformulated in the language of Double Field Theory (see e.g. [19–21]), which in turn can beused to identify their α′-corrections, at least to the first order (i.e. 2 loops in the σ-model) in theα′-expansion [22], see also [23–25] for the generalisation to other solution-generating techniques.One reason why they have attracted so much attention is that they can be used to deform a verylarge class of string backgrounds, including the maximally symmetric ones with AdS factors,which were paramount in the development of the AdS/CFT correspondence. The possibility ofdeforming such backgrounds (e.g. AdS5×S5 or AdS3×S3×T 4) while preserving the underlyingintegrability is an exciting direction to identify possible generalisations of the AdS/CFT corre-spondence, with new gauge theories that are potentially exactly solvable (in the same sense asN = 4 is exactly solvable in the large-N limit).

In this article, we will focus on homogeneous Yang-Baxter deformations (and for simplicitywe will always omit the prefix “homogeneous” from now on). This is a class of integrabledeformations that can be understood as a generalisation of TsT deformations (which in theDouble Field Theory language would be called β-shifts) [26]. While TsT deformations [27–29] can be applied whenever we have two commuting isometries of the “seed” undeformedbackground, Yang-Baxter deformations generalise this possibility to the non-abelian case. Thecentral ingredient in the construction is a constant and antisymmetric R-matrix that must solvethe classical Yang-Baxter equation on the Lie (super)algebra of isometries. When restricting tosupercosets (for example the one of AdS5 × S5), this family of deformations has a rich list ofpossibilities, and everything is known about their supergravity embedding. In particular, thedeformed background (which in general can have all NSNS and RR fields non-trivially turnedon) is a solution of the type IIB supergravity equations if and only if the R-matrix solves a verysimple and algebraic “unimodularity” condition [14,30].1

On the other hand, not much has been done to extend the methods of integrability, that were

1See [31] for the generic treatment of a Green-Schwarz string with a semisimple Lie algebra of isometries.Recently, Yang-Baxter deformations have also been constructed for symmetric space sigma models with non-semisimple symmetry algebra [32].

2

developed for the undeformed AdS5 × S5 case, to the Yang-Baxter deformed models.2 For thevery restricted subclass of “diagonal” TsT deformations (i.e. combinations of TsT deformationsthat involve only the Cartan isometries of AdS5 × S5), the solution to the spectral problemis understood in terms of a simple deformation of the Bethe equations and ThermodynamicBethe Ansatz [35, 36] or Quantum Spectral Curve [37]. But as soon as one goes beyond thisspecial subclass, various obstructions do not yet allow for the application of the integrabilitytechniques. This happens already for the rather simple class of “non-diagonal” TsT deformations(i.e. involving at least one non-Cartan isometry) [38].

A natural strategy to try to extend the methods of integrability to the full class of Yang-Baxter deformations is to do what worked in the case of diagonal TsT deformations, where onereplaces the study of the spectral problem of the deformed σ-model with that of an undeformedmodel which, however, has twisted boundary conditions on the worldsheet. The two σ-models(the deformed one and the undeformed yet twisted one) are equivalent on-shell, i.e. there is amap that relates solutions of the σ-model equations of motion on the two sides. In the case ofTsT deformations, the twisted boundary conditions for the alternative picture are very easy towrite down, essentially because they are linear in terms of the fields [28, 29]. For more genericYang-Baxter deformations, writing down the equivalent twisted boundary conditions is morecomplicated. Previous expressions were written in terms of a path-ordered exponential, and thenon-localities that this introduces were making it impossible to make any progress [39–41]. Thisproblem was solved in [42], where the twisted boundary conditions were rewritten in terms ofthe convenient degrees of freedom, namely those of the twisted model itself. While the twistedboundary conditions can lead to complicated non-linear relations among the fields—a fact whichis related to the non-abelian nature of the deformations—the expressions are local.

Among the methods of integrability that we can apply, the classical spectral curve hasthe right balance of complexity and computational power. The curve is defined as the N -sheeted Riemann surface obtained from the eigenvalue-problem of a monodromy matrix of sizeN × N [43]. Its power was originally applied in the context of string theory in AdS5 × S5 byreformulating the construction of classical string solutions as a Riemann-Hilbert problem [44–48],as all their information is encoded in terms of cuts and poles on the Riemann surface. It gainedeven more traction after it was realised that the classical curve can also be used to obtain 1-loopinformation related to these classical solutions [49–53]. This information is retrieved by addingmicroscopic cuts and poles to the Riemann surface, which will behave as quantum fluctuationsaround the classical solution. The classical spectral curve was soon generalised from AdS5×S5 toother AdSd backgrounds, see [54] and references therein, as well as deformed backgrounds, suchas the flux-deformed AdS3×S3 [55–57] or the Schrodinger background obtained from TsT [58].

In this article, we will use the equivalence between deformed models and undeformed yettwisted model to apply the method of the classical spectral curve and its semiclassical quan-tisation to a particular example of a Yang-Baxter deformation of AdS5 × S5 of non-abeliantype. It belongs to the so-called class of Jordanian deformations, which make use of an sl(2,R)subalgebra of the Lie algebra of superisometries [9,10,59,60]. Importantly, the simplest versionof Jordanian deformations (i.e. the one needing just the sl(2,R)) is non-unimodular. Therefore,the deformed background does not lead to a solution of the type IIB supergravity equations, butrather to one of the modified supergravity equations of [61,62]. It is however possible to cure thisproblem by constructing “extended” Jordanian deformations [63] that exploit a superalgebra ofisometries containing sl(2,R). They lead to unimodular R-matrices, and to backgrounds that aresolutions of the standard type IIB supergravity equations. Notice that the unimodular and non-

2We stress that in this paper we focus on homogeneous Yang-Baxter deformations. For the case of theinhomogeneous deformation, the integrability methods have been pushed to the level of the thermodynamicBethe ansatz [33] and Quantum Spectral Curve [34].

3

unimodular versions share the same background metric and Kalb-Ramond field.3 Interestingly,both cases are closely related to non-abelian T-duality [60,64,65]: a non-vanishing deformationparameter can be rescaled away in the background by a simple coordinate transformation, andthe solution is equivalent to the background resulting from doing non-abelian T-duality on the2-dimensional Borel subalgebra of sl(2,R) (or of its supersymmetric extension if considering theunimodular case).

This article is organised as follows. In section 2, we review the construction of Yang-Baxterdeformations of semi-symmetric space σ-models, and we specify the particular Jordanian defor-mation which we will study. After displaying its deformed metric and Kalb-Ramond field, weidentify in section 2.2 the residual bosonic isometries that survive. In section 2.3 we then proposea coordinate system for which we explicitly identify the time coordinate. We prove that thissystem provides global coordinates of the deformed spacetime for every value of the deformationparameter by analysing its geodesic completeness. In section 3 we derive a particular pointlikestring solution of the deformed σ-model, which can be interpreted as the analog of the BMNsolution of AdS5 × S5.

In section 4, we transfer from the deformed Yang-Baxter picture to the undeformed modelwith twisted boundary conditions. We review the results of [42] regarding the on-shell equiv-alence in section 4.1 and identify the symmetries of the twisted model in 4.2. In section 4.3,we show how the twist of the Jordanian model can be simplified. The transformation of ourBMN-like solution to the twisted model is then performed in section 4.4, where we pay particularattention to the gauge ambiguities of the supercoset. Additionally, in this section we analysealso solutions of the twisted model in more generality and show, under certain assumptions, thatthey lead to extended string solutions with profiles involving Airy functions.

We start the program of the classical spectral curve and its quantisation in section 5 with areview of the construction of the classical curve in terms of the quasimomenta of the (general)twisted supercoset model on PSU(2, 2|4). We show in section 5.1 how the asymptotics of thequasimomenta give rise to local Cartan charges, involving the energy, of the symmetry algebra ofour Jordanian twisted model. We then construct the algebraic curve for our BMN-like solution insection 5.2. In section 6, we study the semi-classical quantum corrections to the correspondingBMN-like quasimomenta by applying the recipe of [49, 53] to our twisted case. We identifythe frequencies of all possible excitations whose resummation, which we perform in section 6.3,gives the one-loop correction to the energy of our string solution. In section 6.2 we furthermoreshow that the charge identifying the twist of the boundary conditions is protected—it doesnot receive any anomalous correction—at one-loop. Finally, in section 7, we discuss how ourresults regarding the classical spectral curve and its semi-classical quantisation are insensitiveto whether we consider the unimodular or non-modular version of our Jordanian deformation.

We end in section 8 with a conclusion and outlook. In appendix A, we show how the globalcoordinates defined in section 2.3 are related to the embedding coordinates of (undeformed) AdS.For completeness, we discuss in appendix B why the geodesic incompleteness of the Poincarecoordinate system persists along the deformation. In appendix C we identify the possible Cartansubalgebras of the residual isometry algebra of the deformed model. In appendix D we presentthe details regarding the calculation of the frequencies of excitations presented in section 6. Wecross-check these results in appendix E by computing explicitly, in the picture of the deformedmodel, the frequencies of the quadratic bosonic fluctuations around the classical solution.

3In the unimodular case these are supplemented by a dilaton and RR fluxes that solve the type IIB equations.In the non-unimodular case the RR fields are replaced by fields that do not satisfy the standard Bianchi identities,and the background is supplemented by a non-dynamical Killing vector that enters the modified supergravityequations.

4

2 A Jordanian-deformed background

2.1 The deformed σ-model and the background fields

In order to set up the notation, let us review the construction of the homogeneous Yang-Baxterdeformation of a string σ-model on a semi-symmetric space. The starting point is a Lie su-pergroup G with a corresponding Lie superalgebra g that admits a Z4-graded decomposition,i.e. g = ⊕3

i=0g(i) such that [[g(i), g(j)]] ⊂ g(i+j mod 4). Here we are using [[·, ·]] to denote the

graded bracket on g. The action of the string σ-model on the supercoset G\G(0) can be writtenas

S0 = −√λ

4π

∫dτdσ Παβ

(−) STr(Jα dJβ

), (2.1)

where J = g−1dg ∈ g is the Maurer-Cartan form constructed from the group element g(τ, σ) ∈ Gthat depends on the worldsheet coordinates τ, σ. The linear operator d : g → g is a linearcombination d = 1

2P(1) + P (2) − 1

2P(3) of projectors P (i), which by definition project onto the

subspace g(i).4 Moreover, STr denotes the supertrace on g, which we use to obtain an ad-invariantgraded-symmetric non-degenerate bilinear form on g. The worldsheet indices α, β in the actionare contracted with the projector Παβ

(±) = 12(√|h|hαβ ± εαβ), where hαβ is the worldsheet metric

and ετσ = −εστ = −1. Finally, we use√λ

4π to denote the string tension.

To construct the homogeneous Yang-Baxter deformation of the above action, one needs alinear operator R : g→ g that is antisymmetric with respect to the supertrace

STr(Rx y) = −STr(x Ry), ∀x, y ∈ g, (2.2)

and that solves the classical Yang-Baxter equation (CYBE) on g

[[Rx,Ry]]−R ([[Rx, y]] + [[x,Ry]]) = 0, ∀x, y ∈ g. (2.3)

If we choose a basis TA for g such that [[TA,TB]] = fABCTC , then we can think of R as a matrix

identified by RTA = RABTB. Moreover, if KAB = STr(TATB) is the metric on g and KAB

its inverse, we can define the dual generators TA = TBKBA that satisfy5 STr(TAT

B) = δBA .We can then equivalently think of R as r = −1

2RABTA ∧ TB ∈ g ∧ g where we use the graded

wedge product x ∧ y = x ⊗ y − (−1)deg(x)∗deg(y)y ⊗ x, and RAB = KACR

CB, so that Rx =STr2(r(1⊗x)) in which STr2 denotes the supertrace on the second factor of the tensor product.Naturally, we only consider R-matrices that do not mix even and odd gradings, meaning that

r = −12R

abTa∧Tb− 12R

αβTα∧Tβ, where a, b are indices of even (bosonic) generators while α, β

are indices of odd (fermionic) generators.

Notice that (2.2) implies that Rab is antisymmetric while Rαβ is symmetric. With theseingredients the deformed action is now given by [8]

Sη = −√λ

4π

∫dτdσ Παβ

(−) STr

(Jα d

1

1− ηRgdJβ

), (2.4)

where Rg = Ad−1g RAdg (with Adg x = gxg−1) and η is a deformation parameter. To have a

real action, we take η ∈ R and RAB antihermitian, so that Rab is real while Rαβ has imaginarycomponents.

4For algebra elements x ∈ g we will use the notation x(i) = P (i)x. When there is no risk of ambiguity, we willalways use a notation such that a linear operator on the Lie algebra acts on what sits to its right, e.g. dJ = d(J).

5Extra care must be taken when the elements are of odd grading, because in that case STr(xy) = −STr(yx).

5

The space defined by the image of the R-matrix, f = Im(R), will play a central role in ourcomputations. We can check that it is a subalgebra of g as a consequence of the CYBE. We willdenote its corresponding Lie group by F . Later we will use indices I, J for generators TI ∈ f,and indices i, j when restricting to the bosonic subalgebra. Using the metric on g induced bythe supertrace, we can also define the dual to f, which we denote by f∗. It is a subspace spannedby TI = TAK

AI such that STr(TITJ) = δJI .

So far we have reviewed the main ingredients — in particular, eqs. (2.2) and (2.3) — thatmake the action (2.4) an integrable deformation of the seed supercoset action (2.1) [6–8]. Seelater for more details. In general, extra conditions are necessary in order to make sure thatthe deformed model can be interpreted also as a consistent string σ-model. Since we are de-forming a σ-model on a semi-symmetric space, the necessary and sufficient condition for thebackground fields of (2.4) to satisfy the type IIB supergravity equations, is that R also solvesthe unimodularity condition [14]

0 = KAB[[TA, RTB]] = RABfABDTD. (2.5)

When this is not satisfied, the background fields solve the more general equations of “modifiedsupergravity” [61,62].

In this article we are interested in deformations of the superstring on AdS5 × S5 and thuswe will take g = psu(2, 2|4). The original supercoset is then PSU(2, 2|4) \ (SO(1, 4)× SO(5)).6

Furthermore, we want to study deformations that are not interpretable as simple TsT trans-formations. This implies that we must deform the AdS factor of the background. Therefore,it will be sufficient to state only the commutation relations for the so(2, 4) ∼= su(2, 2) confor-mal algebra, which is a subalgebra of g and corresponds to the isometries of AdS5. Given theLorentz indices µ, ν = 0, . . . , 3 in 3 + 1 dimensions, we will use the Lorentz generators Mµν , thetranslations pµ, the special conformal transformations kµ and the dilatation d that close intothe following commutation relations

[Mµν , pρ] = ηνρpµ − ηµρpν , [d, pµ] = +pµ,

[Mµν , kρ] = ηνρkµ − ηµρkν , [d, kµ] = −kµ,[Mµν , d] = 0, [pµ, kν ] = 2Mµν + 2ηµνd,

[Mµν ,Mρσ] = −ηµρMνσ + ηνρMµσ + ηµσMνρ − ηνσMµρ,

(2.6)

with the Minkowski metric ηµν = diag(−1, 1, 1, 1).

An interesting and rich class of deformations that deform AdS are given by Jordanian ones [9,10,59,60]. Here we choose to work with7 [63]

r = e ∧ h− i2ζ(Q1 ∧ Q1 + Q2 ∧ Q2) (2.7)

where8

h =d+M01

2, e =

p0 + p1

√2

(2.8)

6We will not need to give the full set of (anti)commutation relations of this superalgebra, nor provide anexplicit matrix realisation of its elements. Conventions for these can be found in various places in the literature,see e.g. [66].

7Comparing to [63] one should rescale the deformation parameter there as η → −η/2.8If we denote Th = h,T+ = e as generators of f then we can identify the duals Th = 2Th and T+ = k1−k0

2√

2= T−

as generators of f∗, so that STr(TaTb) = δab . Moreover, we can check that they form an sl(2,R) algebra, [Th,T±] =±T± and [T+,T−] = 2Th. According to our conventions the R-matrix acts as R(Th) = T+ and R(T+) = −Th.

6

and they satisfy the commutation relation

[h, e] = e. (2.9)

The supercharges Q1,Q2 complete the (anti)commutation relations to an N = 1 super Weylalgebra in one dimension. In this article we will not need to specify them further, and moreinformation can be found in [63]. For our convenience we have also introduced the parameterζ. It is worth to stress that r solves the CYBE only for ζ = 0 or ζ = 1. When ζ = 0 the purelybosonic r-matrix is not unimodular, and it gives rise to background fields that solve only themodified supergravity equations; when ζ = 1, we have an additional “fermionic tail” such thatr is also unimodular, and therefore one can obtain a standard supergravity background [14,63].In this article we will find it interesting to consider both cases and compare them.

Let us identify the bosonic part of (2.4) (i.e. when setting fermions to zero) with Sη =

−√λ

4π

∫dτdσ Παβ

(−) ∂αXm∂βX

n(Gmn + Bmn). Obviously, the background metric Gmn and theKalb-Ramond field Bmn do not depend on ζ. In order to find an explicit expression, we can pa-rameterise the group element as g = gags where ga ∈ SO(2, 4) parameterises the AdS spacetimein the undeformed limit, and gs ∈ SO(6) the sphere factor. We take

ga = exp(θM23) exp(ρp3 + x0p0 + x1p1) exp(log(z)d), (2.10)

which gives9

ds2 =dz2 + dρ2 + ρ2dθ2 − 2dx+dx−

z2− η2(z2 + ρ2)dx−2

4z6+ ds2

S5 ,

B = 12BmndX

m ∧ dXn =η

2

(ρ dρ ∧ dx−

z4+dz ∧ dx−

z3

) (2.11)

with x± = (x0 ± x1)/√

2 and ds2S5 the metric of the undeformed 5-sphere.

Notice that when η 6= 0, this deformation parameter can be reabsorbed by a simple coordinateredefinition, which is however singular in the η → 0 limit. This property is in fact a consequenceof the fact that the Jordanian deformation is related to a non-abelian T-duality transformationof the original background [64,65].

We will not need to write explicitly the other background fields, and we refer to [63] for theirexpression. As remarked above, it is only for ζ = 1 that one can find a set of Ramond-Ramondfields and dilaton that complete the background to a standard supergravity one.

2.2 Residual isometries

As is well known, the isometries are realised in the generic σ-model action (2.1) as global transfor-mations which act as g → gLg with gL constant, in this case an element of G = PSU(2, 2|4). TheLie algebra of bosonic isometries of the undeformed AdS5×S5 background is so(2, 4)⊕ so(6) ∼=su(2, 2)⊕ su(4), corresponding respectively to the AdS5 and S5 factor of the spacetime. The de-formed action (2.4) is invariant only under a subset of these global transformations, in particularthose which are preserved by the operator R

Ad−1gLRAdgL = R . (2.12)

9In terms of the usual Poincare coordinates from ga = exp(x2p2 + x3p3 + x0p0 + x1p1) exp(log(z)d) we wouldget the metric z−2(dz2 + (dx2)2 + (dx3)2− 2dx+dx−)− 1/4z−6(η2(z2 + (x2)2 + (x3)2)dx−2 for the deformed AdSfactor. Instead we choose to use polar coordinates x2 = ρ sin θ and x3 = ρ cos θ in the 2−3 plane such that a U(1)isometry corresponding to shifts of θ, which survives the deformation, becomes manifest. In the next subsectionwe will elaborate on the residual isometries of the deformed model.

7

At the level of the algebra this condition translates to

[[TA, R(x)]] = R[[TA, x]], ∀x ∈ g (2.13)

where TA denotes a generator of a (super)isometry. Note that using the Jacobi identity it iseasy to show that the subset span(TA) ⊂ g forms a subalgebra. Since the R-matrix acts triviallyon the sphere, all of the so(6) isometries will be preserved by the deformation, and its Noethercurrents correspond as usual to the Maurer-Cartan forms g−1

s dgs, see e.g. [66]. More generally,in the deformed model the Noether currents corresponding to the residual (super)isometries aregiven by10

JA± = STr(J±TA), J± = Adg

(A

(2)± ∓ 1

2(A(1)± −A

(3)± )), (2.14)

where the worldsheet one-form A = A+dσ+ + A−dσ

−, here written in terms of the worldsheetlightcone coordinates σ± = 1

2(τ ± σ), is defined as

A± =1

1± ηRgdg−1∂±g . (2.15)

From these Noether currents one may also define the Noether charges

QA =1

2π

∫ 2π

0dσJAτ . (2.16)

For a generic σ-model with background metric and Kalb-Ramond field, the Killing vectors kaassociated to bosonic isometries can then be found from solving (see e.g. [67])

Ja± = kma (Gmn ∓Bmn)∂±Xn ± 2ωan∂±X

n, (2.17)

where LkaG = 0 and LkaB = dωa, with L the Lie derivative and ωa an arbitrary one-form.Since the deformation only acts non-trivially on AdS, we will focus on this subsector for mostof the remainder of this paper.

Solving (2.13) for the R-matrix (2.7) in question then gives five residual isometries of so(2, 4),forming the subalgebra11

k = span(Ta, a = 1, . . . , 5) = span(d−M01, p0, p1, k0 + k1,M23) . (2.18)

The corresponding Killing vectors in the (polar) Poincare coordinates read

km1 ∂m = 2x−∂x− , km2 ∂m =∂x+ − ∂x−√

2, km3 ∂m =

∂x+ − ∂x−√2

,

km4 ∂m =√

2(z2 + ρ2)∂x+ +√

22(x−)2∂x− +√

22x−ρ∂ρ +√

22x−z∂z , km5 ∂m = ∂θ ,

(2.19)

which shows that we have in particular three translation isometries in x+, x− and θ.12

10Because of the factorisation of the AdS and sphere algebra, if we are interested only in the residual AdSisometries, we may simply substitute g with ga in the formulas above.

11From an algebraic perspective we thus see that indeed M23 is a residual U(1) isometry which correspondsto shifts in an angular coordinate.

12One can also derive the one-forms ωa needed to relate the Noether currents Ja to the Killing vectors ka. Inthis case, they are found to be non-vanishing but closed for generic values of η. The B-field itself is thereforeisometric with respect to the residual isometries.

8

2.3 Global coordinates

In view of the applications of the Jordanian deformation of AdS5 × S5 in holography, it isimportant to identify global coordinates for the background for generic values of η. As it iswell known, the Poincare coordinates are globally incomplete already for the undeformed AdSspacetime. Thus, our first task is to find an appropriate coordinate transformation. A firstobvious attempt is to transform to the usual global coordinates of AdS, and then check if theyremain global coordinates also when η 6= 0. However, we find that the deformed metric in thesecoordinates is complicated and does not exhibit any of the manifest residual isometries. Wehave therefore not pursued this coordinate system further, as it would not be useful for practicalpurposes.

In fact, similar issues were found in [68], which analysed the geometry of Schodinger space-times Schz. Such geometries possess an anisotropic scale invariance (t, xi)→ (λzt, λxi) charac-terised by a critical exponent z > 1, where the case z = 1 corresponds to usual AdS. A globalcoordinate system different from the usual global AdS coordinates was then obtained for z = 2.Conveniently, the Jordanian deformed background (2.11) is extremely reminiscent of the Sch2

case and, in fact, exhibits the same anisotropic scale symmetry.13 It is therefore conceivablethat a good candidate for a global coordinate system of our Jordanian background coincideswith that found in [68]. The transformation from polar Poincare coordinates to the global onesXm = (T, V, P,Θ, Z) is obtained from14

x+ = V +1

2(Z2 + P 2) tanT, z =

Z

cosT, ρ =

P

cosT,

θ = Θ, x− = tanT.(2.20)

We can cover the full spacetime if we take the coordinates in the ranges T, V ∈]−∞,+∞[,Θ ∈ [0, 2π[ and P,Z ∈]0,+∞[. The only coordinate that we allow to be periodically identified isΘ, with period 2π. These ranges have been derived from the relation between these coordinatesand the embedding coordinates of AdS, which we discuss in appendix A. In the new coordinatesthe background fields (2.11) then read

ds2 =dZ2 + dP 2 + P 2dΘ2 − 2dTdV

Z2− 4Z4(Z2 + P 2) + η2(Z2 + P 2)

4Z6dT 2 + ds2

S5 ,

B =η

2

(PdP ∧ dT

Z4+dZ ∧ dTZ3

).

(2.21)

Clearly, this coordinate system will be useful for practical purposes, since it manifestly displaysthree translational isometries, namely in T , V , and Θ. Because the Cartan subalgebra of thealgebra of residual isometries is three-dimensional (see appendix C) this is in fact the maximalnumber of manifest translational isometries that a coordinate system could display. Explicitly,

13In particular, when defining (x2, x3) = (ρ sin θ, ρ cos θ), the Jordanian background is invariant underz → λz, x2 → λx2, x3 → λx3, x− → λ2x− and x+ → x+, which translates to eq. (2.1) of [68] by iden-tifying (z, x2, x3, x−, x+) with (r, x2, x3, t, ξ) for the critical exponent z = 2. Notice that the only differ-ence between the Jordanian metric and the Schrodinger Sch2 metric, given in eq. (1.1) of [68], is the termρ2

4z6dx−2 = (x2)2+(x3)2

4z6dx−2.

14Strictly speaking, the relation to the global Schrodinger coordinates of [68] is obtained by transforming (P,Z)to (X2, X3) = (P sin Θ, P cos Θ).

9

the Killing vectors introduced earlier read

km1 ∂m = sin(2T )∂T −(P 2 + Z2

)sin(2T )∂V + P cos(2T )∂P + Z cos(2T )∂Z ,

km2 ∂m =cos2(T )√

2∂T −

(P 2 + Z2

)cos(2T )− 2

2√

2∂V −

P sin(T ) cos(T )√2

∂P −Z sin(T ) cos(T )√

2∂Z ,

km3 ∂m = −cos2(T )√2

∂T +

(P 2 + Z2

)cos(2T ) + 2

2√

2∂V +

P sin(T ) cos(T )√2

∂P +Z sin(T ) cos(T )√

2∂Z ,

km4 ∂m = 2√

2 sin2(T )∂T +√

2(P 2 + Z2

)cos(2T )∂V +

√2P sin(2T )∂P +

√2Z sin(2T )∂Z ,

km5 ∂m = ∂Θ.(2.22)

Note that (km2

+ km3

)∂m =√

2∂V and (2km2− 2km

3+ km

4)∂m = 2

√2∂T .

Following the above proposal, in the rest of this section, we will prove that the coordinatesystem Xm = (T, V,X2, X3, Z) with (X2, X3) = (P sin Θ, P cos Θ) gives rise to a geodesicallycomplete spacetime.

2.3.1 On the generators of time translations

Before performing the full analysis of geodesic completeness of the deformed spacetime, letus first identify the possible Hamiltonians (generators of time translations) within the isometryalgebra k in (2.18). We will look for generators of this kind that are part of the Cartan subalgebraof k, because in that case one can find an adapted coordinate system where the generator oftime translations simply shifts one of the coordinates, which can be identified as time.

The first natural question is thus to identify all the possible Cartan subalgebras of k, asexplained in appendix C. Up to automorphisms, there are two possible inequivalent choices ofCartan subalgebras, namely

(I) : spand−M01, p0 + p1, M23,(II) : spanp0 − p1 + α(k0 + k1), p0 + p1, M23,

(2.23)

where we leave a possible α > 0 coefficient for later convenience.

Identifying the possible generators of time translations is now a very simple task. One canconstruct a generic linear combination of the Cartan generators and demand that the corre-sponding Killing vector km has a strictly negative norm in every region of spacetime. Notice,however, that p0 + p1 and M23 are not timelike and, being central elements, it is easy to removetheir contribution by a spacetime-coordinate redefinition. We can therefore simply compute thenorm of the Killing vectors associated to the two possible Cartans of the sl(2,R) subalgebra ofk. We find

(I) : km = km1 , kmkm =(P 2 + Z2)(4Z4 − η2 sin(2T )2)

4Z6

(II) : km = km(2) − km(3) + 1

2km4 , kmkm = −(Z2 + P 2)(η2 + 4Z4)

2Z6,

(2.24)

where in the latter case we took α = 1/2. While other choices of α > 0 are possible, we makethis choice here to match with the expression (2km(2) − 2km(3) + km

4)∂m = 2

√2∂T found above

for the generator of T translations. Whilst in case (I) the norm of the Killing vector is noteverywhere negative (set e.g. T = 0), in case (II) we do have a strictly timelike Killing vectorfor generic values of the coordinates and of the deformation parameter η. The relevant Cartan

10

subalgebra is thus that of class (II) with α = 1/2. Concluding, the coordinate T will be ourpreferred global time coordinate and the Hamiltonian is15

H = 1√2(p0 − p1 + 1

2(k0 + k1)) . (2.25)

Interestingly, when taking the undeformed limit η → 0, the above Hamiltonian does not reduceto something equivalent to the BMN generator of time translations [69], because it is possibleto check that there is no inner automorphism of SU(2, 2) that relates H to p0 + k0.

2.3.2 Analysis of geodesic completeness

In this section we will analyse whether the coordinate system Xm = (T, V, P,Θ, Z), whichdescribes Sch2 spacetimes globally, are also well-defined global coordinates for the Jordaniandeformation. This analysis is necessary because when comparing the Sch2 metric, given ineq. (3.18) of [68], and the Jordanian metric, given in (2.21), we see that the Jordanian case has

an additional term P 2

4Z6dT2, which may lead to pathologies.

There are typically two common ways to analyse whether coordinates are global or not.First, one can try to find the embedding in a higher dimensional spacetime16 and argue whetheror not the embedding coordinates cover the full hypersurface. Another possibility is to considergeodesics γ(t), parameterised by a geodesic parameter t ∈ R, and investigate geodesic complete-ness of the coordinate system. We will follow the latter method. Before doing so explicitly, letus recall the criterions concerning geodesic (in)completeness and singular spacetimes.

A coordinate system is called geodesically complete if all geodesics γ(t) are defined for allvalues of the geodesic parameter t ∈ R, within the ranges of coordinates that parameterise themanifold. A spacetime is said to be geodesically incomplete if there exists a finite value t0of the geodesic parameter such that the geodesic γ(t0) hits an extremal value allowed by therange of spacetime coordinates, and the geodesic γ(t) cannot be extended to t > t0. Hitting an(apparent) metric singularity would not be considered pathological — and therefore would notclassify as geodesic incompleteness — if the geodesic “bounces back” (i.e. if it can be continuedto allowed ranges of the spacetime coordinates at later t) or if the singularity is reached only inthe limit t→∞.17

Rather than solving the five second-order geodesic equations for Xm(t), we can make use ofthe five residual Killing isometries to perform a single integration and simplify the calculations

15In the literature, in similar setups, sometimes the Hamiltonian is identified with d −M01 ∼ h. While thisis of course possible when considering the complexified algebra (where any choice of Cartan can be brought tod −M01, see e.g. (C.1)), here we insist on respecting reality conditions. Notice that the square of d −M01 hasa positive trace, and thus cannot correspond to a timelike Killing vector. In fact, if we take for the sake of theargument the case of group manifolds, the Killing vectors can be identified by kma = Tr(Tb Ad−1

g Ta)`bm where

`bm is the inverse of `m

b and dXm`aµTa = g−1dg. When moving to the identity of the group (i.e. at g = 1), thenorm kma kam of the Killing vector reduces to Tr(TaTa). Therefore, a necessary condition for ka to be timelike isthat Tr(TaTa) < 0.

16In the case of undeformed AdS5 this is of course R2,4, which is invariant under the conformal algebra SO(2, 4).We show the relation between the coordinate system Xm = (T, V, P,Θ, Z) and the AdS embedding coordinatesin appendix A.

17When at least one geodesic ends in a point p of the manifold, then p is a singularity. In this case one shouldfurther study whether p is a physical singularity or just a coordinate singularity. It is only in the latter casethat one speaks of geodesic incompleteness. To distinguish the nature of the singularity one typically computescurvature scalars or tidal forces. The point p is an honest curvature singularity when at least one of these becomeinfinite at p. If all of them remain finite and the geodesics appear to end at p, then the coordinate system isconsidered to be incomplete. A fully general criterion about whether or not an apparent singularity is honest,however, does not exist. Fortunately, we will not find such singularities and therefore we will not need to performthis analysis.

11

considerably. In particular, recall that for each Killing vector field kma , there is an independentcharge

Qa = kma GmnXn , (2.26)

which is conserved along the geodesics.18 In other words, since the full system of geodesicequations can be rewritten as Qa = 0 and we have as many Killing vectors as independentcoordinates, the geodesics can be analysed by studying the system (2.26) rather than the usualgeodesic equations. To do so, it will be useful to introduce the conserved quantities related tothe manifest translational isometries in T , V and Θ,19

kT = ∂T : QT = −4Z4(Z2 + P 2) + η2(Z2 + P 2)

4Z6T − 1

Z2V ,

kV = ∂V : QV = − 1

Z2T ,

kΘ = ∂Θ : QΘ =P 2

Z2Θ.

(2.27)

An interesting feature now emerges when analysing the system (2.26): instead of five first orderdifferential equations for Xm(t), one finds that (2.26) decouples into four first order differentialequations and one algebraic relation among the coordinates, which restricts the geodesics to ahypersurface. We choose to decouple (2.26) as

T (t) = −QV Z(t)2 ,

V (t) =QV4

(η2 + 4Z(t)4)

(1 +

P (t)2

Z(t)2

)−QTZ(t)2 ,

Θ(t) =Z(t)2

P (t)2QΘ ,

P (t) =Z(t)

P (t)

(Z(t)

(Q1 + sin 2T (t)

(QV (Z(t)2 + P (t)2)−QT

))cos 2T (t)

− Z(t)

),

(2.28)

subjected to the algebraic relation, which we will call the “hypersurface equation”,

2QV P2(t) = 2QT − (2QT −

√2Q4) cos 2T (t)− 2Q1 sin 2T (t)− 2QV Z(t)2 . (2.29)

In fact this equation defines a circle in the (P,Z) plane with a varying (but bounded) radiusdepending on T (t), i.e. P (t)2 +Z(t)2 = R2(T (t)).20 In addition, there is another useful constantof motion along the geodesics, namely

ε = GmnXmXn , (2.30)

with ε = 0,−1, 1 for null, timelike, or spacelike paths. Substituting the system (2.28) into (2.30)leads, in general, to a complicated differential equation for Z subjected to the hypersurfaceequation.

Let us now analyse potential pathological regions of the deformed background. Notice thatwe can completely exclude the coordinates T, V,Θ from our analysis, as translations in these di-rections are manifest isometries and, thus, will not give rise to pathologies. From the background

18Here the dot refers to derivation with respect to the geodesic parameter t.19The relation to the previously defined Noether charges is Q2 = 1

4

(2√

2QT + 2√

2QV −Q4

), Q3 =

14

(−2√

2QT + 2√

2QV +Q4

)and Q5 = QΘ.

20There are a number of conditions that we can derive for the values of the charges such that R2 ≥ 0, whichshould be interpreted as reality conditions for Z and P . They should be further subjected to a condition ensuringalso that T is real when T remains constant along the geodesic.

12

fields (2.21) we see the only regions that can potentially be pathological are Z,P = 0,+∞,as the (inverse) spacetime metric appears to blow up there. First, let us consider the regionsZ → ∞ or P → ∞, in which case the hypersurface equation plays an important role in theanalysis. In particular, the hypersurface bounds the geodesic motion in Z and P , and forcesboth to a maximal absolute value, |Z|, |P | ≤ |R|, which is always finite for a finite value ofthe conserved charges Qa and for any finite value of t ∈ R. The only exception to this is whenthe charge QV = 0, for which the dependence of P and Z drops out of (2.29) (equivalently, theradius R becomes infinite). However, we can argue that also in this case there is no pathology.First, for QV = 0 the hypersurface equation forces T (t) to be a constant in t, which is consis-tent with its geodesic equation T = −QV Z2. Hence, the coordinate time does not evolve inthe geodesic parameter. This should therefore be an extreme spacelike path, as we also see byanalysing the equation for Z obtained from (2.30). Remarkably, for QV = 0, this equation infact greatly simplifies to Z2(t) = εZ(t)2 which is immediately solved by

Z(t) = Ae±√εt , (2.31)

where A is an integration constant. Reality conditions thus tell us that paths with QV = 0are only possible when they are spacelike or null (the latter being trivial in that case). Thespacelike paths can in fact reach Z = ∞ (as well as Z = 0, see also later), but they do so onlyat an infinite value of the geodesic parameter t. We can thus conclude that Z → ∞ is not apathological region. Concerning the behaviour at P →∞, one finds that taking QV = 0 wouldlead to paths that are not defined due to reality conditions. Given that for generic QV themotion in P is bounded, we can again conclude that also the region P →∞ is not pathological.In fact, even though P can become very large for small QV , the region P → ∞ will never bereached by any geodesic in finite t.

Next, let us consider the Z → 0 region and analyse the small Z behaviour. Note that wecan just focus on the system T, P, Z (the solutions for Θ and V will be given once solutions forT, P, Z are found). We will write Z(t) = λZ(t) and expand the equations around λ = 0 up toO(λ), unless stated otherwise. From the first equation in (2.28) and (2.29), we will find that Tand P are constant in t up to linear order in λ. In particular21

T (t) = T0 +O(λ2),

P (t) =√R2(T0) +O(λ2)

=

√2QT − (2QT −

√2Q4) cos 2T0 − 2Q1 sin 2T0

2QV+O(λ2) ,

(2.32)

with T0 the constant value of time.22 Now let us analyse the equation for Z obtained from (2.30)and (2.28) for small Z. Up to subleading order in λ, where T (t) = T0 + λ2T (t) + O(λ3) andP 2(t) = R2(T (t))−λ2Z2(t) +O(λ3) = R2(T0) +λ2(∂T0R

2(T0)T (t)− Z2(t)) +O(λ3), where T (t)is such that T (t) = −QV Z(t)2 and T (t = 0) = 0, we find

ε =1

λ2

(η2Q2

VR2(T0) + Z(t)2

Z(t)2

)+

(Z(t)2

R2(T0)+η2Q2

V ∂T0R2(T0)T (t)

Z(t)2

)+O(λ) , (2.33)

which one can analyse order by order in λ. The leading term thus gives

Z(t)2 = −η2Q2VR(T0)2 +O(λ2) , (2.34)

21Recall that P ∈]0,∞[.22Note that if T0 is such that R2(T0) = 0, then (2.29) implies that P (t) will have a linear in λ contribution,

namely P (t) =

√−Z(t)2λ + O(λ2). However, due to reality conditions, the latter is only possible when Z = 0

and thus P = O(λ2). Hence, this is consistent with (2.32) at R2(T0) = 0.

13

which in general would give an imaginary solution for Z, i.e. Z(t) = Z0 ± t√−η2Q2

VR(T0)2 +

O(λ2). This means that we cannot get arbitrarily close to Z = 0 except if (i) η = 0 (theundeformed limit, in which case the spacetime is geodesically complete, see also appendix A),(ii) QV = 0 (a case which we already analysed in generality above and where it was found thatgeodesics can reach Z = 0 only in an infinite amount of the geodesic time), and (iii) R2(T0) = 0(in which case P = O(λ2) and Z = 0 (see also footnote 22) and thus all the coordinates areconstant in t). Finally, we must analyse consistency with the equation for P given in (2.28),which can be interpreted as RR−ZZ = PP . From the leading solution for P in (2.32) we knowthat P = O(λ2). Therefore also PP = O(λ2) and we can rewrite RR− ZZ = PP as

λ2 Z(t)2(Q1 + sin 2T0(QVR2(T0)−QT ))

cos 2T0− λZ(t)Z(t) = O(λ2) (2.35)

i.e. Z(τ) = O(λ). This is consistent with what we have found previously, i.e. eq. (2.34), requiring(i) η = 0, (ii) QV = 0, or (iii) R(T0)2 = 0. We can thus conclude that the geodesics will neverreach Z = 0 by an evolution in t and that this region is not pathological. Finally, analysing theregion P → 0 using a similar strategy does not give us a conclusive answer. However, this is ofcourse only the apparent pathology of polar coordinates: neither the metric nor its inverse issingular for X2 = X3 = 0.

The arguments above prove that the coordinate system Xm = (T, V,X2, X3, Z) is globallywell-defined for any value of the deformation parameter. We will however prefer to work interms of Xm = (T, V, P,Θ, Z), in which the shift isometry in Θ is manifest. This concludes ourdiscussion and, combined with the previous section, shows that the coordinate T is our globaltime coordinate.

3 BMN-like solution of the σ-model

In this section, we give the derivation of a particular pointlike string solution to the equationsof motion of our Jordanian deformation. As we will show in section 5, it will lead to relativelysimple quasimomenta. For us it will be the analog of the BMN solution valid in AdS5×S5, andwe will call it a BMN-like solution.

For a general bosonic string configuration Xm(τ, σ), let us recall that the equations of motion

of the action Sη = −√λ

4π

∫dτdσ Παβ

(−) ∂αXm∂βX

n(Gmn +Bmn) can be written as

ηαβ∂α∂βXm +

(ηαβΓmnl −

1

2εαβHm

nl

)∂αX

n∂βXl = 0 , (3.1)

which is a generalisation of the geodesic equation of a point particle. Here Γmnl are the usualChristoffel symbols, andHmnl are the components of the torsion 3-formH = dB. Interestingly, itcan be checked that these equations evaluated for the Jordanian deformation admit a consistenttruncation on x2(τ, σ) = x3(τ, σ) = 0, or equivalently P (τ, σ) = Θ(τ, σ) = 0,23 for any value ofthe deformation parameter.

Let us now consider the following σ-independent ansatz on the coordinates

Xm = amτ + bm , (3.2)

23In the latter case, the limit P (τ, σ) → 0 on the equations of motion (in particular, the equation of motionfor Θ) should be taken with care. First one must set P (τ, σ) to a constant P to then find that in the resultingequations of motion the limit P → 0 can indeed be taken smoothly.

14

with am and bm constant variables which will be determined upon imposing (3.1). Furthermore,we will work on the consistent truncation P (τ, σ) = Θ(τ, σ) = 0, equivalently we set aP = aΘ =bP = bΘ = 0. Since there is no σ-dependence, this pointlike string will not couple to the Kalb-Ramond two-form, and its equations of motion are therefore simply the geodesic equations.Solving them for the variables am and bm while imposing that fields are real singles out thesolution24

aZ = 0, and aV = −η2aT4b2Z

, (3.3)

while the other variables remain free. Because of the manifest isometries in T , V and Θ wecan however set bT = bV = bΘ = 0 without loss of generality. In summary, our pointlike stringsolution thus evolves as

T = aT τ, V = −η2aT4b2Z

τ, Z = bZ , P = 0, Θ = 0 . (3.4)

This solution is in fact exactly the same as the one studied for the Schrodinger Sch5 × S5

background (at least, in the deformed AdS subsector) in [38]. The reason is that on the consistenttruncation P = 0, the Jordanian metric given in (2.11) coincides precisely with the Schrodingermetric. Given that this solution is pointlike, the isometry charges defined in (2.26) will coincidewith the Noether charges defined in (2.16). In terms of (2.26) and (2.27) we get

QT = −aT , QV = −aT b−2Z , QΘ = 0, Q1 = 0, Q4 = −

√2aT . (3.5)

Note that these charges are independent of the deformation parameter. As a consistency checkone can furthermore verify that the hypersurface equation (2.29) is satisfied, as it should.

The solution we described here is further subjected to the Virasoro constraints which willcouple the deformed AdS with the undeformed sphere subsector. For Yang-Baxter deformationsthe Virasoro constraints, obtained by varying the action (2.4) with respect to the worldsheetmetric, can be written as [8]

STr(Aα(2)(±) A

β(2)(±)

)= 0 , (3.6)

where we defined the worldsheet projections Aα(±) = Παβ(±)Aβ of the one-form A defined in (2.15).

In conformal gauge the Virasoro constraints would read STr(A(2)± A

(2)± ) = 0, and parametrising

the group element as before as g = gags, we can identify an AdS and sphere contribution as

STr(A

(2)a±A

(2)a±

)+ STr

(A

(2)s±A

(2)s±

)= 0 ⇐⇒ Tr

(A

(2)a±A

(2)a±

)= Tr

(A

(2)s±A

(2)s±

), (3.7)

where we used that the supertrace is defined as the trace in the AdS algebra while it is minusthe trace in the sphere algebra, and where Aa± and As± are the projections of A on the AdSand sphere subalgebra respectively. On our solution, (3.4) the Virasoro constraints then requirethe following relation between sphere and AdS variables(

1− η2

4b4Z

)a2T = −Tr

(A

(2)s±A

(2)s±

). (3.8)

24When one does not impose the consistent truncation P (τ, σ) = Θ(τ, σ) = 0, but still assumes the pointlike

ansatz (3.2), there is one other real solution namely aZ = aP = 0, aV = − η2aT (b2Z+b2P )

4b4Z

and aΘ = aT√η2 + 4b4Z2b2Z

while the other variables remain free. Carrying the analysis of this solution through (again using the isometriesto set bT = bV = bΘ = 0) one will find the associated quasimomenta to be quite complicated and we will thereforenot consider this possibility further.

15

4 Map to an undeformed yet twisted model

Starting from this section, for simplicity we specify our discussion to the non-unimodular case.We therefore consider a deformation generated by an R-matrix as in (2.7) with ζ = 0. Weremind that this case gives rise to background fields that do not satisfy the type IIB super-gravity equations, but only the “modified” supergravity equations. This study will be howeverpreparatory to the unimodular case, to which we will go back in section 7.

4.1 Review of the twist

Homogeneous Yang-Baxter deformed σ-models are known to be on-shell equivalent to unde-formed yet twisted models, see e.g [40,41]. This is due to the fact that it is possible to identifythe Lax connections of the two models, while achieving a map of the equations of motion of thetwo sides. In other words, there is an on-shell relation between the σ-model on the η-dependentdeformed background and the σ-model on the undeformed one, and there is a one-to-one map ofthe solutions to the equations of motion of the two sides. This map, however, implies that if wechoose to have periodic boundary conditions on the worldsheet for the deformed model (whichmay be motivated by the choice of studying closed strings on the deformed background), then wemust generically have twisted boundary conditions for the undeformed one. This situation is ageneralisation of what happens for TsT transformations, where the twisted boundary conditionscan be derived and written explicitly in a local and linear way [28, 29]. From now on, we willuse tildes on all objects of the undeformed yet twisted model. For example, if g ∈ G denotes thegroup element used to construct the periodic Yang-Baxter model, we will use g ∈ G to denotethe group element of the η-independent twisted model. The map between the two sets of degreesof freedom can be written as

g = F g h, (4.1)

where F ∈ F is called the “twist field”, and is responsible for translating between the twoformulations. Recall that F is the Lie group of f = Im(R). In these expressions we are alsoallowing for a possible gauge transformation, implemented as a right-multiplication by h ∈ G(0).This possibility is always present in the (super)coset case, given the local G(0)-invariance ofthe σ-model action. The twist field F is fixed by demanding that the Lax connections of thedeformed and undeformed models can be identified (see also (5.1)). This amounts to requirethat the “modified current” A± of the deformed model defined in (2.15) can be identified withthe Maurer-Cartan current J± = g−1∂±g of the undeformed model, up to a possible gaugetransformation h, so that

A± = h−1J±h+ h−1∂±h. (4.2)

It is important to remark that the Virasoro constraints are compatible with the on-shell identifi-cation. In fact, it is easy to see that using (4.2) the Virasoro conditions for the deformed model

written in (3.6) give STr(Jα(2)(±) J

β(2)(±)

)= 0, which are precisely the Virasoro conditions for the

undeformed model. This is actually a crucial point to claim on-shell equivalence in the case ofa string σ-model.

The fact that g has periodic worldsheet boundary conditions now indeed implies that g isnot periodic in σ ∈ [0, 2π] but rather satisfies the following twisted boundary conditions

g(2π) = W g(0)h(0)h−1(2π), (4.3)

16

where W = F−1(2π)F(0) is called the twist.25 Both the twist W and the twist field F are gauge-invariant under the right-multiplication by h. Moreover, it is clear that it is always possible touse a compensating gauge transformation to fix h = 1 in (4.1). In this sense, only W controlsthe physically relevant twisted boundary conditions, and we can talk about a left twist only. Atthe same time, we want to include the possibility of having a non-trivial h 6= 1 because it maybe important in certain calculations. It becomes crucial, for example, if we want to insist onusing the same parametrisation for g and g in terms of coordinates.26 The reader may worrythat the need of determining h will introduce unnecessary complications in the calculations, butin section 4.4.1 we will in fact explain how to efficiently by-pass it and, if needed, easily derivethe relevant h.

By identifying the Lax connections, it is in principle straightforward to write down an ex-pression for F and W in terms of the degrees of freedom g of the deformed model, see [40, 41].In addition to being an unnatural expression for the twisting of the undeformed model, thatsolution is however written in terms of a path-order exponential. It is therefore plagued bynon-localities that make it unusable for practical purposes. In [42] this problem was solved byrewriting the solution for W in terms of the degrees of freedom g instead. Depending on thechoice of the R-matrix, the twisted boundary conditions may correspond to complicated non-linear relations among the coordinates evaluated at σ = 2π and σ = 0 (a fact which is relatedto the non-abelian nature of the deformation) but W itself takes a simple expression in termsof g that only involves single (as opposed to nested) integrals. Importantly, W is constant and,therefore, it is written in terms of conserved charges of the twisted σ-models. These do not needto be Noether charges, as they may correspond to some hidden symmetries. We refer to [42] fora more detailed explanation on how to construct W from a given R-matrix, where the derivationwas done for the case of the PCM as well as that of (super)cosets on (semi)symmetric spaces.In fact, the generalisation from the former to the latter only involves some decorations of theformulas with certain linear combinations of projectors on the Zn-graded subspaces of g (withn = 2, 4). In particular, the derivation of the Jordanian twist of section 4.3 of [42] is still validalso in the supercoset case, and it is therefore given by27

W = exp (Q(h− q e)) , (4.4)

which takes values in the Lie subgroup F , and where we have the following expressions for theconserved charges

Q ≡ log1− η Y+(τ, 0)

1− η Y+(τ, 2π), q ≡ Yh(τ, 2π)− Yh(τ, 0)

Y+(τ, 2π)− Y+(τ, 0), (4.5)

with Yh, Y+ projections of the Lie-algebra valued field

Y (τ, σ) = Yh(τ, σ)Th + Y+(τ, σ)T+

= P T(∫ σ

0dσ′dg−1

(∂τ gg

−1)

(τ, σ′) +

∫ τ

0dτ ′dg−1

(∂σ gg

−1)

(τ ′, 0)

)+ Y (0, 0).

(4.6)

As anticipated, the above expression differs from the one valid in the PCM case (which can befound in [42]) just by the insertion of the linear operator dg−1 = Adg dAd−1

g , where d was defined

25Notice that although g, g, h,F depend on both worldsheet coordinates τ and σ, when writing the boundaryconditions we will omit the explicit τ -dependence and write down only the values that σ takes, to have a lighternotation.

26For example, as we will do later, this happens if we parametrise g as in (2.10) in terms of coordinatesx0, x1, ρ, θ, z and g in the same way but with coordinates x0, x1, ρ, θ, z (or equivalently in terms of thecorresponding global coordinates under the transformation (2.20)). In this case the on-shell identification conditionrequires h 6= 1. A gauge transformation can be used to reabsorb h in g or g, but that would imply that theparameterisation of the coset representative would not be given anymore by (2.10).

27Compared to [42] here we use the notation Q = logQA and q = QB .

17

in section 2.1. Note that also Y is gauge-invariant under the local right-multiplication of g by h.The above formula for the Jordanian twist W will be our starting point for the construction ofthe classical spectral curve of our model, as well as for the computation of the one-loop quantumcorrections to its spectrum.

4.2 Manifest symmetries of the twisted model

Before continuing with the study of the Jordanian deformation and the corresponding twistedmodel, it is useful to analyse the manifest symmetries of the latter and understand how theyare related to those of the former, which were discussed in section 2.2. The following discussionis valid for the whole family of homogeneous Yang-Baxter deformations, so we do not needto specify to the Jordanian case. We give the presentation for the supercoset, but the sameconsiderations can be made already in the simpler PCM setup.

Let us define

Q =1

2π

∫ 2π

0dσ Jτ , (4.7)

where28

Jα = Adg

(J (2)α − 1

2γαβεβγ(J (1)

γ − J (3)γ )). (4.8)

In the undeformed and periodic (i.e. W = 1) supercoset σ-model, (4.7) are the conserved Noethercharges and (4.8) the Noether currents corresponding to the (left) G invariance of the action [66].In fact, the Noether charges Q of the deformed model defined in (2.16) agree with Q in the limitη → 0. In the twisted case (W 6= 1), Q in (4.7) is not necessarily a conserved charge. Infact, although the Noether current is still conserved (∂αJ α = 0) simply as a consequence of theequations of motion, the twisted boundary conditions may in general break the constancy intime of Q, since

∂τ Q =1

2π

∫ 2π

0dσ ∂τ Jτ =

1

2π

∫ 2π

0dσ ∂σJσ = (AdW −1)(Jσ)|σ=0, (4.9)

where (4.3) was used. In particular, if we consider the projection of Q along a given generatorTA ∈ g as QA = STr(QTA) we have ∂τ QA = STr[(Jσ)|σ=0(W−1TAW − TA)]. It is now clearthat if some TA ⊂ TA commutes with the whole subalgebra f, then QA is a conserved chargefor the twisted model. Note that span(TA) forms a subalgebra of g. For the specific Jordaniandeformation we consider in this paper, we have that the manifest bosonic symmetries are

Ta = d−M01, p0 − p1, k0 + k1,M23 . (4.10)

Importantly, if TA commutes with f, then it is also a symmetry of the R-matrix and of thedeformed model because it automatically solves (2.13).29 Therefore, manifest symmetries of thetwisted model correspond to isometries of the YB model.

It is worth stressing that the opposite is not always true: one may have an isometry of theYB model satisfying (2.13), which is not a manifest symmetry of the twisted model because TAdoes not commute with f. In our case of a Jordanian R-matrix, an example of this kind is givenby e = (p0 + p1)/

√2. In fact, e is a symmetry of the R-matrix and therefore an isometry of

the YB model, but it does not commute with f, so Qe = STr(Qe) is not a conserved charge (inparticular W−1eW 6= e).

28In conformal gauge it reads Jτ = Adg(J

(2)τ − 1

2(J

(1)σ − J(3)

σ ))

.29In fact, [[TA, RX]] = R[[TA, X]] because the left-hand-side is obviously 0, and because using Rx =

− 12RIJSTr(xTJ)TI with TI ∈ f the right-hand-side is − 1

2RIJSTr([TA, X]TJ)TI = − 1

2RIJSTr([TJ ,TA]X)TI = 0.

18

Notice that when TA commutes with f, the expression for the charges of the twisted modelagree with those of the Noether charges of the deformed one, and they are related by the on-shellmap (4.2). In fact, let us start from

Q =1

2π

∫ 2π

0dσ Jτ , (4.11)

where J was defined in (2.14), so that QA = STr(QTA) is a conserved Noether charge of thedeformed model already defined in (2.16) if TA satisfies (2.13). Then it is straightforward to seethat

[[TA, f]] = 0 =⇒ QA =1

2π

∫ 2π

0dσ STr[g[J (2)

τ − 12(J (1)

σ − J (3)σ )]g−1TA]

=1

2π

∫ 2π

0dσ STr[F−1g[A(2)

τ − 12(A(1)

σ −A(3)σ )]g−1FTA]

=1

2π

∫ 2π

0dσ STr[g[A(2)

τ − 12(A(1)

σ −A(3)σ )]g−1FTAF−1]

=1

2π

∫ 2π

0dσ STr[g[A(2)

τ − 12(A(1)

σ −A(3)σ )]g−1TA]

= QA, (4.12)

where we used P (i)(J) = hP (i)(A)h−1 for i 6= 0. According to the discussion above, all manifestsymmetries QA can be written as isometries QA of the deformed model, but not all isometries

QA can be written as a manifest symmetry QA, so that schematically QA ⊂ QA. Because ofthe classical equivalence between the two models, we expect that the isometries of the deformedmodel that are not manifest symmetries of the twisted model should correspond to a moregeneral family of hidden symmetries, which is always there for integrable models.

4.3 A simpler twist

In general, there is not a unique way to write the twist W that controls the boundary conditions.We should rather think in terms of equivalent classes of twists. For example, if g satisfies thetwisted boundary conditions (4.3) and we define a new field g′ related to g by the simple fieldredefinition g′ = ug with u ∈ G constant, then it follows that the new field satisfies the boundaryconditions

g′(2π) = W ′g′(0)h(0)h−1(2π), W ′ = uWu−1. (4.13)

The fields g and g′ are equivalent representations of the same model in terms of different variables,so that we are free to choose whether we want to work with g or g′. Therefore, we should notthink of the twisted boundary conditions as being identified by a unique expression W , sinceW ′ = uWu−1 belonging to the same equivalence class actually describes the same physics.

Notice that if W ∈ F then in general W ′ belongs to F ′ = uFu−1, in other words it belongsto an adjoint orbit of G on F . The relation to the degrees of freedom of the YB model nowis g = Fu−1g′h. In [42] the solution for the twist field was constructed by fixing the initialcondition F(0, 0), and while the above field redefinition breaks this choice, it can be restored bythe compensating field redefinition g′ = ug, so that we can write

g′ = F ′g′h, F ′ = uFu−1. (4.14)

Notice that in general the YB model written in terms of these new degrees of freedom g′ will beconstructed not in terms of R but rather by the R-matrix R′ = AduRAd−1

u . This is obviously

19

a physically equivalent antisymmetric solution of the CYBE.30 It is worth remarking that theLax connections constructed out of g and g′ are equal because u is assumed to be constant, andtherefore also the two corresponding monodromy matrices (and not just their eigenvalues) areequal to each other.

We now want to exploit the above possibility to define a new twist W ′ for the Jordaniandeformation, that has the advantage of having a simpler expression compared to W . We startfrom (4.4) and we rewrite it in a “factorised” form by using the following identities coming fromthe Baker-Campbell-Hausdorff (BCH) formula

exp(A) exp(B) = exp

(A+

s

1− e−sB

), and exp(A) exp(B) exp(−A) = exp (esB) , (4.15)

which hold when [A,B] = sB for s 6= 2πin. First, we parameterise W in (4.4) as

W = exp

(A+

s′

1− e−s′B′), (4.16)

where we have to identify

A = Qh, B′ = q(e−Q − 1)e, s′ = Q, (4.17)

which is consistent with [h, e] = e. Now, using the above identities we can write

exp

(A+

s′

1− e−s′B′)

= exp(s′B′) exp(A) = exp((es − 1)B) exp(A)

= exp(−B) exp(esB) exp(A) = exp(−B) exp(A) exp(B),

(4.18)

where we defined B ≡ es′B′/(es − 1). For consistency with [A,B] = sB we must have s = s′.31

Notice that in our case we simply have B = −qe. To conclude, the twist can be written as

W = u−1W ′u, where W ′ = exp(Qh), and u = exp(−qe). (4.19)

This is remarkable, because the new twist is identified by the Lie algebra element h only, andall information regarding e is lost in the g′-model. The above arguments also imply that thespectrum of both the g- and the g′-models do not depend on the charge q, at least at the classicaland one-loop level. Naively the charge q controls the boundary conditions of the twisted model,but as we just showed this dependence can be removed by a simple field redefinition. In the nextsection, we will see this independence of the spectrum on q more explicitly. Notice that in thederivation of factorising the twist there is a subtlety when Q→ 0, because then some formulasdiverge. We will come back to this point in section 7, where we will discuss the factorisation ofthe twist in a more general set-up that encompasses also the case of unimodular deformations.

4.4 Classical solution in the twisted model

4.4.1 Map from the classical solution of the deformed model

In this section, we return to our discussion of the particular point-like solution (3.4) of theJordanian deformed model and show how it can be mapped to a twisted solution of undeformed

30While the above considerations are valid for a generic u ∈ G, if u is also a symmetry of R then R′ = R, andboth F ′ and W ′ are still elements of the unprimed F subgroup, see also [42]. It turns out that the u that we areabout to consider is of this type.

31In fact, as soon as we have [A,B] = sB and we define B′ = αB (for any non-vanishing α) while requiringthat [A,B′] = s′B′, it is then obvious that we must have s′ = s.

20

AdS5. This means that we need to apply the reverse logic of [42] and derive the twist field Fstarting from the deformed variables. As we mentioned in 4.1, its expression is then in general acomplicated path-ordered exponential, but on our specific simple solution (3.4) it can howeverbe evaluated explicitly. In this case the easiest way to obtain F is actually to work with theassociated differential system, namely

∂±F = V±F , V± = ±ηRAdg dA±, (4.20)

which is obtained from identifying the Lax connections of the two models. When using theinitial condition F(0, 0) = 1, we have F ∈ F and thus in full generality we can write

F = exp (fh(τ, σ)h + fe(τ, σ)e) . (4.21)

Solving the differential equations (4.20) on the solution (3.4) is then rather straightforward andafter imposing F(0, 0) = 1 we obtain

fh(τ, σ) =ηaTσ

b2Z, fe(τ, σ) =

η3a2T τσ

4

(1− e

ηaT σ

b2Z

)b4Z

. (4.22)

Note that the twist field F reduces to the identity element in the undeformed η → 0 limit, as itshould.

Given the solution for F , the transformation to the undeformed variables is done in principleby employing the map g = F gh, although this may be complicated by possible gauge transfor-mations. In fact, in our case, if we insist on parameterising both g and g in the same way, thenwe cannot relate the two models if h = 1. Guessing the appropriate gauge transformation is notstraightforward, if not unsatisfactory, and thus the translation between deformed and twistedvariables begs a gauge-invariant procedure in the case of (super)coset models. To do this, weintroduce the gauge-invariant objects

G = gKgt , G = gKgt , (4.23)

where t denotes usual matrix transposition and K satisfies32 hKht = K for h ∈ SO(1, 4) (seee.g. §1.5.2 of [66]). Then indeed G and G are invariant under the local right-multiplicationsg → gh and g → gh respectively. Using these objects the map g = F gh can then be rewrittenin the following gauge-invariant way

G = F GF t , (4.25)

and also the twisted boundary conditions (4.3) can be rewritten as

G(2π) = W G(0)W t . (4.26)

None of these formulae now suffer from any possible complications arising from gauge ambigui-ties.

Using (4.25), it is now very easy to translate to the undeformed variables. We find, for gparametrised in terms of the global coordinates, that (3.4) is mapped to

T = aT τ, V = 0, Z = exp

(−ηaTσ

2b2Z

)bZ , P = 0, (4.27)

32In the matrix realisation of su(2, 2|4) that is typically used (see e.g. [66]) one takes

K =

(J2 00 J2

), J2 =

(0 −11 0

). (4.24)

21

while Θ remains free. The latter is however only a redundancy of the chosen parametrisation andwe may set Θ = 0 without loss of generality. Having already determined g, g,F , the unknowngauge transformation h can now be simply derived as h = g−1F−1g. Furthermore, knowing Fwe can also obtain the twist W of the solution, i.e.

W = F−1(2π)F(0) = exp

(−2πηaT

b2Zh

). (4.28)

Consequently, we can identify the Jordanian charges Q and q from (4.4) giving precisely

Q = −2πηaTb2Z

, q = 0 . (4.29)

Thus, for this particular classical solution, there is no need to simplify the twist followingsection 4.3, because q is already zero. Translating the twisted boundary conditions for g interms of explicit expressions for the associated coordinates, we see that all of them are periodicexcept Z, which satisfies Z(2π) = exp(Q/2)Z(0).

There are now several consistency checks of the calculations just performed. First, one canverify that (4.26) holds, while the twisted boundary conditions for g have a non-trivial gaugetransformation h in their expression. Secondly, knowing the solution g, one can calculate thegauge-invariant object Y defined in (4.6). We find

Yh(τ, σ) = −ηaT τ4b2Z

, Y+(τ, σ) = η−1

(1− exp

(ηaTσ

b2Z

)), (4.30)

which satisfies Y (0, 0) = 0, consistent with F(0, 0) = 1. Comparing now the Jordanian chargesQ and q obtained from (4.5) with the expressions (4.29) obtained directly from the twist field,we can check that the two results agree. At last, one can verify that (4.27) is a solution to theequations of motion of the undeformed σ-model.

4.4.2 More general classical solutions of the twisted model

We now want to explore the question of finding more general solutions of the undeformedtwisted model. Although the boundary conditions will depend on the deformation parameter η,the equations of motion will not, and thus one may hope to find more solutions than (4.27). Tomaintain some level of simplicity, however, we will continue to work on the consistent truncationP (τ, σ) = Θ(τ, σ) = 0 and assume that the global coordinate time T evolves linearly with theworldsheet time τ , i.e. as before T (τ, σ) = aT τ .

On these assumptions, let us first determine the most general solution to the twisted bound-ary conditions (4.26) for the Jordanian twist W (4.4) in terms of the coordinates Z(τ, σ) andV (τ, σ). It is not difficult to show that in general they must satisfy

Z(τ, 2π) = exp(Q/2)Z(τ, 0), V (τ, 2π) = exp(Q)V (τ, 0) + (1− exp(Q))q . (4.31)

As before one will have, however, g(2π) 6= Wg(0) implying that the group element used toconstruct the Lax connection of the twisted model must be dressed with a gauge transformation.

Next, we compute the equations of motion of the undeformed model, and find that theyreduce to the following three differential equations

0 = ∂τ Z, (4.32)

0 = 2aT∂τ V − (∂σZ)2 + Z∂2σZ, (4.33)

0 = 2∂σV ∂σZ − Z∂2σV . (4.34)

22

From (4.32) and (4.33) we find that V must be linear in τ as V = 12aT

V1τ + V0 with

V1 ≡ (∂σZ)2 − Z∂2σZ , (4.35)

and V0 ≡ V0(σ) an unknown function of σ. Now (4.34) holds two equations, one at O(τ0) andone at O(τ), namely

−2∂σV0∂σZ + Z∂2σV0 = 0 , and − 2∂σV1∂σZ + Z∂2

σV1 = 0 , (4.36)

respectively. They can be rewritten as

2∂σ log Z = ∂σ log V ′0 = ∂σ log V ′1 , (4.37)

where the the prime denotes derivation to σ, V ′0 ≡ ∂σV0 and V ′1 ≡ ∂σV1. Since all the objectsinvolved only depend on σ, we can perform one integration to find

V ′0 = c0Z2, V ′1 = c1Z

2, (4.38)

where c0 and c1 are integration constants. Only the latter equation of (4.38) is now a differentialequation for Z, and once we obtain its solution, the solution for V0 (and consequently of V ) isobtained immediately by performing a straightforward integration of the first equation of (4.38).First note that we can rewrite the expression for V ′1 as

V ′1 = Z ′Z ′′ − ZZ ′′′ = −Z2

(Z ′′

Z

)′. (4.39)

Then the second equation of (4.38) becomes(Z ′′Z−1

)′= −c1 which means we can do another

simple integration to findZ ′′ + (c1σ + c2)Z = 0 , (4.40)

with c2 another integration constant. Here the experienced reader may recognise the Airy (orStokes) differential equation which has as linearly independent solutions the Airy functions Ai(x)and Bi(x). Concluding, within our assumptions given in the beginning of this section, the mostgeneral solution for the variable Z of the undeformed model is

Z(τ, σ) = αAi

(− c1σ + c2

(−c1)2/3

)+ βBi

(− c1σ + c2

(−c1)2/3

), (4.41)

before imposing any boundary condition. These are oscillatory functions that at a certainturning point become exponential. In principle, we can now also write down an explicit solutionfor V , as mentioned before, however we will refrain from doing so because it is not particularlyenlightening. Nevertheless, note that, because of the additional integration performed to obtainV0, the system has one additional integration constant (say c3). Before imposing boundaryconditions, the full solution thus has in total 7 free parameters, namely aT , c0, c1, c2, c3, α andβ.

Imposing the twisted boundary conditions (4.31) gives three equations for Z, V1 and V0,obtained order by order in τ . The equation for Z fixes an expression for Q while the one forV0 fixes q. The remaining condition on V1 becomes a constraint between the parameters,33

reducing the number of free parameters to 6. Two further constraints between the parametersmay arise when comparing the expressions of Q and q with those obtained from (4.5) and (4.6),as these must of course coincide for consistency. We have not looked into this further.

33On the solution for Z and V1 the constraint simply reads V1(2π)Z(0)2 = V1(0)Z(2π)2.

23

We close this section by discussing the special c1 = 0 case. From (4.40) it is obvious thatthe most generic solution for Z then is simply

Z(σ) = α exp(√−c2σ) + β exp(−

√−c2σ). (4.42)

Depending on the sign of c2 these are exponential or oscillatory functions. The simple so-lution given in (4.27), which is the one we study in the remainder of this paper, falls intothe former class. It has α = 0, β = bZ and

√−c2 = ηaT

2b2Z. In general for V we then get

V = −c0

b2Z exp

(− ηaT σ

b2Z

)ηaT

+ c3 which seems more general than (4.27). Consistency with the bound-ary conditions (4.31) however sets c0 = 0 and c3 = q. Here we see explicitly that a possiblecontribution from the q charge can be eliminated by a field redefinition, which is in fact atranslational isometry in the V coordinate.

5 Classical spectral curve

Knowing a Lax connection that is flat on the σ-model equations of motion allows one to applythe methods of classical integrability. While in principle it is possible to work from the pointof view of the Yang-Baxter deformed σ-model, we find it more convenient to work from thepoint of view of the undeformed yet twisted σ-model, which is on-shell equivalent to the former.In fact, this is in analogy to how the classical and quantum spectrum of TsT deformations isobtained, see e.g. [28, 35]. Here we want to give a brief summary of facts that are valid in theundeformed and periodic case and that carry over also to the twisted one. For more details werefer to the original literature developed in the periodic case, see e.g. [45].

The Lax connection of the twisted σ-model on the AdS5 × S5 supercoset is34

Lα = J (0)α +

1

2

(z2 +

1

z2

)J (2)α −

1

2

(z2 − 1

z2

)γαβε

βγ J (2)γ + z J (1)

α +1

zJ (3)α , (5.1)

which agrees with the Lax connection of the undeformed and periodic case [70, 71] becauseboundary conditions play no role in its construction. Here J (i) = P (i)J are projections ofthe currents on the Z4-graded subspaces of psu(2, 2|4), and γαβ =

√|h|hαβ. From the Lax

connection one can construct the monodromy matrix

Ω(z, τ) = P exp

(−∫ 2π

0dσ′Lσ(z, τ, σ′)

), (5.2)

with z ∈ C the spectral parameter. In the undeformed and periodic case the eigenvalues λ(z)of Ω(z, τ) are conserved because Lτ (2π) = Lτ (0) as a consequence of the periodic boundaryconditions. When considering twisted boundary conditions as in (4.3), Lτ remains periodic aslong as we take into account the compensating gauge transformation. Therefore the eigenvaluesλ(z) of Ω(z, τ) are again time-independent, and encode the infinite number of conserved quan-tities (when expanding in powers of the spectral parameter z), rendering the model classicallyintegrable. The eigenvalues depend analytically on z except at z = 0,∞ (where the poles ofthe Lax connection imply essential singularities for λ(z)) and at the points where two (or more)eigenvalues degenerate. Instead of working with the eigenvalues, it is simpler to work with the

34In conformal gauge this is just L±(z) = J(0)± + zJ

(1)± + z∓2J

(2)± + z−1J

(3)± . The Lax connection of the

Yang-Baxter deformed model is obtained simply by replacing J by A.

24

quasimomenta p(z), defined as λ = eip, as the essential singularities of λ at z = 0,∞ becomepoles. For the quasimomenta we use the same notation as for the undeformed and periodic case

p1(z), p2(z), p3(z), p4(z)||p1(z), p2(z), p3(z), p4(z), (5.3)

where pi(z) with i = 1, . . . , 4 are the quasimomenta of the (deformed/twisted) AdS factor cor-responding to SU(2, 2) ⊂ PSU(2, 2|4) and pi(z) with i = 1, . . . , 4 are those of the (in thiscase undeformed/untwisted) sphere factor corresponding to SU(4) ⊂ PSU(2, 2|4). When twoeigenvalues of the same type (i.e. both p or both p) degenerate, they give rise to branch pointsof square-root cuts, that correspond to collective bosonic excitations. When two eigenvalues ofdifferent type (i.e. one p and the other p) degenerate, they give rise to poles, that correspondto fermionic excitations. The quasimomenta are everywhere analytic except at these bosonicbranch points and fermionic poles, and at z = 0,∞. Because of its definition, the quasimomen-tum has a 2πZ ambiguity, but this ambiguity is lost when considering dp, which can be thoughtof as an abelian differential. The spectral problem can therefore be reformulated in terms of theclassification of the admissible algebraic curves with abelian differential dp. These curves can beunderstood as a collection of 4 + 4 sheets (one for each of the quasimomenta) connected by theabove cuts and poles. We follow the usual conventions and employ an alternative useful param-

eterisation of the spectral parameter, taking z =√

1+x1−x so that the points z = 0,∞ correspond

to x = −1,+1.

Let us continue with the list of the facts that are valid in the undeformed and periodicAdS5 × S5 case and that continue to be valid also in the twisted one. The Lax connection issupertraceless, which implies that the superdeterminant of the monodromy matrix is 1. In turn,this implies the “supertraceless condition”35∑

i

pi =∑i

pi. (5.4)

Moreover, the Z4 automorphism of psu(2, 2|4) is still implemented by sending z → iz or equiv-alently x→ 1/x at the level of the Lax connection. We will call this the “inversion symmetry”.At the level of the quasimomenta it implies [45]

pk(1/x) = −pk′(x), pk(1/x) = −pk′(x) + 2πmεk, (5.5)

where k = (1, 2, 3, 4) and k′ = (2, 1, 4, 3). Notice that this permutation is related to the choice ofcharge conjugation matrix in psu(2, 2|4). For the sphere quasimomenta, the inversion symmetryallows for a possible shift with εk = (1, 1,−1,−1) and m ∈ Z is related to winding around the S5

space. For the AdS quasimomenta, this possibility is not allowed by the requirement of absenceof winding for the time coordinate.

Combining the supertraceless condition, the inversion symmetry, and the Virasoro con-straints, one has a synchronisation36 of the poles of the quasimomenta at x = ±1 also in thetwisted case

p(x) =diag

(α(±), α(±),−α(±),−α(±)|α(±), α(±),−α(±),−α(±)

)x± 1

+O(x± 1)0 . (5.6)

35In principle one can modify the above equation by adding a shift by 2πn with n ∈ Z. This can be removedby the freedom of shifting the quasimomenta by any integer multiple of 2π. In the periodic case one can use thisfreedom so that at large x the quasimomenta go like p(x) ∼ O(1/x). In the twisted case the twist can introduceadditional finite terms in the strict x→∞ limit. However, equation (5.4) remains valid because these finite shiftscancel each other in the above equation, as a consequence of detW = 1.

36When we later study one-loop corrections, we will slightly relax this synchronisation of the poles to p(x) ∼diag(α(±),α(±),β(±),β(±)|α(±),α(±),β(±),β(±))

x±1+ O(x ± 1)0, because in that case one should consider also fermionic

contributions and the tracelessness condition of su(2, 2) and su(4) of the purely bosonic classical case is no longerenforced.

25

The bosonic cuts and the fermionic poles must respect the inversion symmetry, which means thatin both cases they must come in even number. For example, we can write the gluing conditionof two quasimomenta on a cut (or more generally on a collection of cuts) C(k,l) or C(k,l) as

pk(x+ iε)− pl(x− iε) = 2πn(k,l), for x ∈ C(k,l),

pk(x+ iε)− pl(x− iε) = 2πn(k,l), for x ∈ C(k,l),(5.7)

and similar equations must hold for the cuts which are the images of C(k,l) and C(k,l) underx → 1/x. Similarly, if there is a fermionic pole at x∗(k,l), by inversion symmetry there will be

another pole for quasimomenta at 1/x∗(k,l). Because of this property, we can declare a “physical”

region as the one for which |x| > 1. The cuts in the physical region (i.e. half of the cuts in thepair of cuts connected by inversion symmetry) will be considered the “fundamental” ones.

To conclude, let us also say that we can still define the filling fractions for sphere cuts C(k,l)

connecting the sheets k and l as

K(k,l) = −√λ

8π2i

∮dx

(1− 1

x2

)pk =

√λ

8π2i

∮ (x+

1

x

)dpk = −

√λ

8π2i

∮ (x+

1

x

)dpl , (5.8)

where the integration path is defined by a closed loop that encircles the fundamental cut C(k,l).

Similarly, we define the filling fractions for AdS cuts C(k,l) connecting the sheets k and l as

K(k,l) = −√λ

8π2i

∮dx

(1− 1

x2

)pl =

√λ

8π2i

∮ (x+

1

x

)dpl = −

√λ

8π2i

∮ (x+

1

x

)dpk. , (5.9)

where the integration path is defined by a closed loop that encircles the fundamental cut C(k,l).In the periodic case these filling fractions are identified as the action variables of the integrablemodel [46], and they are taken to be quantised as integers in a semiclassical treatment [45]. Inthe twisted case we can do the same, because the identification of the definition of the fillingfractions comes from the position of the poles of the Lax connection, and these are still atx = ±1. The filling fractions can also be written in terms of the so-called twist function [72],and this is invariant under an homogeneous Yang-Baxter deformation [40].

In the periodic case, the expansion of the monodromy matrix around z = 1 (or equivalentlyx =∞) yields (at the lowest non-trivial order) the conserved Noether charges coming from thepsu(2, 2|4) superisometry. In the twisted case this picture is modified by the presence of W . Infact, after considering a gauge transformed Lax connection L′σ = gLσ g−1 − ∂σ gg−1, one finds37

g(0)Ω(z)g(0)−1 = W−1P exp

(−∫ 2π

0dσ L′σ

), (5.10)

which, when writing z = 1 + ε, leads to

g(0)Ω(z = 1 + ε)g(0)−1 = W−1 + 4πε W−1Q+O(ε2), (5.11)

where Q was defined in (4.7). In the periodic case (W = 1), Q is the Lie-algebra valued conservedNoether charge, and one can use a PSU(2, 2|4) automorphism to put it in the Cartan subalgebra.After doing that, the monodromy matrix is diagonal at least to order ε and the eigenvalues aretherefore written in terms of the Cartan charges. In the twisted case, the monodromy matrix

37For simplicity, we are writing these formulas in the case that the twisted boundary conditions are justg(2π) = Wg(0), i.e. h = 1 in (4.3). If h 6= 1 one simply has to use a gauge transformation by gh rather than justg, and the right-hand-side of (5.11) is still given by the same expression.

26

is non-trivial already at order zero in the ε-expansion, and the quasimomenta will depend alsoon the eigenvalues of W . Generically, Q on its own will not be conserved in the twisted case.As we will see in the next section, when computing the eigenvalues of the above expansion theyare not necessarily written in terms of the standard Cartan charges of psu(2, 2|4): there is a“polarisation” induced by the presence of W , so that instead the quasimomenta are writtenin terms of the conserved Cartan charges of the algebra of symmetries of the twisted model(which we discussed in section 4.2). The symmetries of the twisted model form a subalgebra ofpsu(2, 2|4), and the corresponding choice of Cartan subalgebra may not be equivalent to the oneof the periodic case. We will now be more explicit about the computation of the eigenvalues ofthe monodromy matrix around z = 1.

Let us emphasise the great advantage of making the above discussion from the point of viewof the twisted rather than deformed model. In the latter case the expansion of the monodromymatrix around z = 1 would give rise to complicated non-local expressions that would be veryhard to work with.

5.1 Asymptotics of the quasimomenta

To obtain the expression for the quasimomenta around z = 1 we must compute the eigenvaluesof the expression (5.11). Since the Jordanian twist W given in (4.4) is diagonalisable, we canrestrict our discussion to diagonalisable matrices. The explanation that we are about to give isactually valid for a generic diagonalisable twist.

The diagonalisation of (5.11) to order ε follows the same route as the diagonalisation of aHamiltonian in perturbation theory of quantum mechanics. Let us rephrase that procedure ina different language. First, consider the diagonalisation of W−1 as D = S−1W−1S. To be asgeneral as possible, we allow eigenvalues to be degenerate

D = diag(d1, . . . , d1, d2, . . . , d2, . . .), (5.12)

where the eigenvalue di comes with multiplicity mi. We will introduce projectors P [i] =diag(0, 0, . . . , 0, 1, 1, . . . , 1, 0, 0, . . . , 0) that have only mi entries with value 1 in correspondencewith the eigenvalues di, and 0 otherwise. Note that P [i]D = DP [i] = diP

[i]. We can then writefor Ω′(z) = g(0)Ω(z)g(0)−1

Ω′(z = 1 + ε) = S(D + εX)S−1 +O(ε2), X ≡ 4πDS−1QS . (5.13)

From simple reasoning in linear algebra, we know that to compute the ε correction to theeigenvalues of D, we must first project

X [i] ≡ P [i]XP [i], (5.14)

and then compute the eigenvalues of X [i]. If there is a block corresponding to eigenvalues allof which have multiplicity 1, then the above is equivalent to projecting on this non-degenerateblock, and the diagonal part of this block will give the correction to the eigenvalues.

With this procedure it is easy to compute the expansion of the eigenvalues. In our case wefind in the AdS sector

d1 = 1 + 2πε(−C1 − C3) +O(ε2),

d2 = e+Q/2(1 + 2πε(C1 − C2)) +O(ε2),

d3 = e−Q/2(1 + 2πε(C1 + C2)) +O(ε2),

d4 = 1 + 2πε(−C1 + C3) +O(ε2),

(5.15)

27

whereC1 = iQ[M23],

C2 = Q[d+M01] −√

2qQ[p0+p1],

C3 =√

(Q[d−M01])2 + Q[−p0+p1]Q[k0+k1],

(5.16)

and Qi = STr(QTi). The explicit expression for q given in (4.5) can be written precisely asq = Qh/Qe = Q[d+M01]/(

√2Q[p0+p1]) which implies C2 = 0. This fact is very important. It

means that the only components of Q that contribute to the eigenvalues of the monodromymatrix are those which correspond to the symmetries of the twisted model, see (4.10). In fact,we managed to get rid of both d+M01 and p0 +p1, which do not have this property. We remindthat all of the symmetries of the twisted model are also isometries of the YB model, and in termsof the isometric charges Qa = STr(TaQ) with Q defined in (4.11) and the isometric generatorsTa in (2.18) we can write

C1 = iQ5, C3 =√

(Q1)2 + (−Q2 +Q3)Q4. (5.17)

Again, the combination Q2 + Q3 does not appear: it would be an isometry of the deformedmodel, but it is not a symmetry of the twisted model.

The appearance of a square root is archetypal in the computations of eigenvalues, and in factit appears also in the undeformed periodic case. It is, however, an awkward feature for practicalcomputations, especially when computing semi-classical corrections to the conserved charges ofthe model. Nevertheless, as in the periodic case see e.g. [44], one can use an automorphism to getrid of the square root and in fact transform the expressions to ones which only involve Cartansof the symmetry algebra. First, as done in appendix C, we identify h = 1

2(d − M01), e+ =12(p0 − p1), e− = −1

2(k0 + k1) which close into a sl(2,R) subalgebra of the algebra of isometriesk. We can construct dual generators h? = 1

2h, e± = e∓ spanning a dual sl(2,R) algebra. We can

then construct Qsl(2,R) = Qhh? + Q+e

+ + Q−e− = (Q1h

? + (Q2 − Q3)e+ − Q4e−)/2. It is easy

to check that 2STr[(Qsl(2,R))2] = (Q1)2 + (−Q2 +Q3)Q4 and thus

C3 =√

2STr[(Qsl(2,R))2] . (5.18)

Now recall that the generator of global time translations is (2.25) which we can write as H =1√2(2e+ − e−). The global energy is thus E = STr(Qsl(2,R)H) = 1√

2(Q2 −Q3 + 1

2Q4). The other

generators in sl(2,R) can then be rotated such that they close into an so(2, 1) ∼ sl(2,R) algebra.Indeed, define TE ≡ H/2, TP ≡ (e+ + e−/2)/

√2 then TJ ≡ h [TJ ,TE ] = TP , [TJ ,TP ] = TE

and [TE ,TP ] = TJ hold.38 We can now define the dual generator H? of global time translationsas STr(H?TP ) = STr(H?TJ) = 0 and STr(H?H) = 1 which fixes

H? = − e+√2

+e−

2√

2= −T2 − T3

2√

2− T4

4√

2. (5.19)

At this point we use the classification of adjoint orbits in sl(2,R) (see e.g. appendix C). Depend-ing on the algebra element, the adjoint action may bring it to something proportional to h, e+ ore+−e−. When discussing the classical spectral curve, we always assume that we are consideringclassical solutions which correspond to highest weight states belonging to the same orbit as thegenerator of time translations H. In other words, we assume that by an inner automorphism, wecan transform Qsl(2,R) to the form Qsl(2,R) = EH?. In that case one has STr[(Qsl(2,R))

2] = −E2/2such that we simply get

C3 = iE . (5.20)

38This is the algebra of AdS2, or the Lorentz algebra in 3 dimensions. Hence TP can be interpreted as amomentum generator, and TJ as a boost.

28

The AdS quasimomenta in the x variable now simply expand as

p1 ∼ −2π(QΘ + E)

x+O(x−2),

p2 ∼ −i

2Q +

2πQΘ

x+O(x−2),

p3 ∼ +i

2Q +

2πQΘ

x+O(x−2),

p4 ∼2π(−QΘ + E)

x+O(x−2).

(5.21)

where we defined QΘ ≡ Q5, since it is the angular momentum charge related to rotations by theangle Θ. Notice that the asymptotics of the AdS quasimomenta depend only on three conservedcharges, namely the energy E and the angular momentum QΘ at O(x−1), and the charge Q atorder O(x0), which controls the twisted boundary conditions. This situation should be comparedto the three charges (energy and two spins) that appear in the asymptotics of the quasimomentain the periodic case at O(x−1).

5.2 Classical algebraic curve for the BMN-like solution

Let us now consider a specific classical solution and derive the corresponding quasimomenta.We will do this for the BMN-like solution that we have constructed. We can compute thequasimomenta using either the deformed or undeformed description. On the one hand, inthe deformed picture (when the solution is given by (3.4)), the Lax connection is clearly σ-independent, so that the path-ordered exponential in the definition of the monodromy matrixreduces to a simple matrix exponential of Lσ. On the other hand, in the twisted picture (whenthe solution is given by (4.27)), the solution depends on σ, making the computation potentiallymore involved. However, this dependence can be eliminated via a gauge transformation andleads to the same result. This is in fact not surprising, since the two models share the same Laxconnection, so that the computation is bound to be the same. Given that we can ignore thepath-ordering in the exponential, to obtain the quasimomenta we can then simply compute theeigenvalues of Lσ itself. In terms of the spectral parameter x we find

p1(x) = −p4(x) =2πaT

√x2−β2

(x−1)2

(x+ 1), p2(x) = −p3(x) =

2πaTx√

1−β2x2

(x−1)2

(x+ 1), (5.22)

where we defined β ≡ η2b2Z

, and we remind that aT and bZ are parameters that enter the

classical solution, see e.g. (3.4). In the vanishing deformation limit, which now correspondsto the β → 0 limit, they reduce to the meromorphic quasimomenta associated to the BMNsolution of undeformed AdS5×S5 [49] and in that sense, the solution (3.4) (equivalently (4.27))is a “BMN-like” solution. The deformation, however, introduces a cut between sheet 1 and 4and a cut between sheet 2 and 3. Note that the inversion symmetry (5.5) is satisfied. For laterconvenience, let us write the quasimomenta in the physical region |x| > 1

p1(x) = −p4(x) =2πaT

√x2 − β2

x2 − 1, p2(x) = −p3(x) =

2πaTx√

1− x2β2

x2 − 1. (5.23)

Matching these explicit expressions with the general asymptotics of the quasimomenta (5.21)gives us the following expressions of the charges involved

QΘ = 0, E = QT = −aT , Q = −4πaTβ , (5.24)

29

with matches with (3.5) and (4.29).

In addition to the kinematics in the deformed AdS5 space, we will take our solution tohave non-trivial kinematics in the S5 space as well. Specifically, we consider a pointlike stringtravelling with constant velocity around a big circle of S5, φ = ωτ . Here ω is proportional tothe conserved charge associated to an angular momentum on S5. As the S5 part of our space isnot deformed, we can borrow the quasimomenta from the usual BMN solution [49]

p1 = p2 = −p3 = −p4 =2πωx

x2 − 1. (5.25)

In the solution that we are considering, the Virasoro condition (3.8) reads

ω = aT√

1− β2 . (5.26)

This implies that if want both ω and aT to be real we must have |β| ≤ 1. Notice that, as wecommented above, the Virasoro constraint can be reinterpreted as a synchronisation conditionon the poles of the quasimomenta at x = ±1.

6 Quantum corrections to the BMN-like solution

In this section we study the quantum corrections to the quasimomenta related to our specificBMN-like solution. Specifically, we apply the recipe presented in [49,53] to compute the one-loopcorrection to its energy. Despite the fact that those articles originally present this technique forundeformed AdS5 × S5, it has been successfully applied also to deformed backgrounds, such asthe flux-deformed AdS3 × S3 [55–57] or the Schrodinger background [58].

6.1 Corrections to the classical quasimomenta

Although we are not working on undeformed AdS5 × S5, most of the road map to constructquantum fluctuations described in [49, 53] applies to our deformed background. The main ideabehind this computation is to introduce (quantum) excitations in the form of cuts so small thatwe can treat them as poles. As in the macroscopic case, if the microscopic excitation we areadding connects two sheets associated to hatted quasimomenta or two sheets associated to atilded quasimomenta, the excitation is bosonic. If the excitation we are adding connects a tildedquasimomenta with a hatted quasimomenta, the excitation is fermionic.

These new cuts will modify the quasimomenta we computed in the previous section, but theproperties that they must fulfil are restrictive enough to fully constrain how they are altered.The starting point is the fact that the corrections to the quasimomenta cannot alter the gluingcondition on a cut (5.7), that is

(pk + δpk)(x+ iε)− (pl + δpl)(x− iε) = 2πn(k,l), for x ∈ C(k,l) or x ∈ C(k,l) . (6.1)

This condition is true for the macroscopic cuts associated to the classical solution, which imposesthe following condition on the perturbations

δpk(x+ iε)− δpl(x− iε) = 0, for x ∈ C(k,l) or x ∈ C(k,l) . (6.2)

A similar condition is imposed for the microscopic cuts that we are introducing to compute thequantum corrections, and this fixes the positions, xkln , where these cuts can be placed by

pk(xkln )− pl(xkln ) = 2πn , (6.3)

30

where the mode number n takes integer values. Notice that, due to inversion symmetry (5.5),we only focus on solutions with |xkln | > 1. We say that an excitation is in the physical region ifthe position of the pole fulfils that property.

Now that we know where to place the new cuts/poles, we also need information regardingtheir residues at xkln . We will not discuss the explicit expression of the residues at this point,but instead will only talk about the relative sign between the residues in the two sheets. Wewill provide more details on the residues in appendix D. For bosonic excitations, the polesare actually infinitesimally small square-root cuts connecting two sheets. This means that theresidues on the two sheets have to have opposite signs. For fermionic excitations, the poles areactual poles, and they have to have the same residue. This can be summarised as

Resx=xijn

pk = (δik − δjk)α(xijn )N ijn , Res

x=xijn

pk = (δjk − δik)α(xijn )N ijn , (6.4)

with i < j taking values 1, 2, 3, 4, 1, 2, 3, 4. Here N ijn indicates the number of quantum

excitations in the physical region that connect pi with pj and have mode number n, and α(xijn )is the residue of the pole associated to that excitation. A more detailed explanation on theorigin of these signs can be found in [45].

The number of excitationsN ijn are not generically free: they have to satisfy the level-matching

condition ∑n

n∑all ij

N ijn = 0 . (6.5)

This condition can be understood from a mathematical perspective as a consequence of theRiemann bilinear identity [45]. Physically, because the number of poles N ij

n correct the fillingfractions (see also footnote 39), they are interpreted as the amplitudes of the physical mode,which are related to mode numbers through conventional string level-matching.

In addition to these pieces of information, these corrections have some important featuresthat they inherit from properties of the classical quasimomenta. Among those, the most relevantfor us are the inversion symmetry of (5.5)

δpk(x) = −δpk′(1/x) , (6.6)

and the synchronisation of the poles at ±1

δp(x) ∼diag

(δα(±), δα(±), δβ(±), δβ(±)|δα(±), δα(±), δβ(±), δβ(±)

)x± 1

+ . . . . (6.7)

Notice that we are using the relaxed version of the synchronisation of poles as commented onin footnote 36, because we will have to consider also fermionic excitations when computingquantum corrections.

The final piece of information that we need to compute the perturbations is their asymptoticbehaviour for large values of x. This can be obtained by analysing the asymptotic behaviour of

31

the quasimomenta (5.21). We require39

δp1

δp4

δp1

δp2

δp3

δp4

=4π

x√λ

+ δ∆2 +N14 +N13 +N13 +N14

− δ∆2 −N14 −N24 −N24 −N14

−N13 −N14 −N13 −N14

−N23 −N24 −N23 −N24

+N13 +N23 +N13 +N23

+N14 +N24 +N14 +N24

+O(x−2) , (6.8)

where δ∆ is the anomalous correction of the energy and where Nij =∑

nNijn is the total number

of poles in the physical region connecting sheet i and j. The above formula can be justified in thesame way as in the undeformed periodic case. In particular, one can assume that the additionof quantum excitations produces integer shifts in the charges that appear at order 1/x in thelarge-x asymptotics of the quasimomenta (5.21), which justifies the presence of Nij . Here weare also assuming that only the energy is allowed to receive anomalous corrections, and we willsee that the results will be consistent with this. Notice that above we have not included theasymptotics of p2 and p3. The reason is that the classical asymptotic behaviour (5.21) suggeststhat their form is non-standard (i.e. they might be finite in the x→∞ limit) as a consequenceof the twist. We will actually not need them to compute δ∆, and we refer to [58] for a similarstrategy applied in the case of the dipole deformation. Nevertheless, we discuss the asymptoticbehaviour of p2 in detail in section 6.2 and, importantly, we will show that the charge Q doesnot get any anomalous correction.

Combining all the properties we have enumerated, we possess enough information aboutthe analytic structure of these corrections to be able to reconstruct them. As this is a tediousand sometimes repetitive process, we have relegated the details regarding this reconstruction toappendix D, while here we collect the final results. If we denote each contribution to δ∆ as

δ∆ =∑n

Ωij(xn)N ijn , (6.9)

we find the following expressions

Ω13(xn) = Ω14(xn) = Ω23(xn) = Ω24(xn) = 2K(1)

x2n − 1

, (6.10)

Ω14(xn) = 2x2nK(1/xn)

x2n − 1

− 2 , (6.11)

Ω23(xn) =2K(xn)

x2n − 1

, (6.12)

Ω13(xn) = Ω24(xn) =x2nK(1/xn) +K(xn)

x2n − 1

− 2 , (6.13)

Ω13(xn) = Ω14(xn) = Ω14(xn) = Ω24(xn) =x2nK(1/xn) +K(1)

x2n − 1

− 1 , (6.14)

Ω23(xn) = Ω24(xn) = Ω13(xn) = Ω23(xn) =K(xn) +K(1)

x2n − 1

, (6.15)

39In all the following expressions we rescale our conserved charges with a factor of√λ, so that the reader can

compare them with the previous literature more easily. We want to stress that the normalisation factor 4π√λ

canbe modified in the sense that it does not affect the computation of the anomalous contribution to the energy δ∆,at the condition that we also modify the normalisation of the residues α(xijn ) consistently. However, the naturalway to fix this normalisation factor is by demanding that the correction to the filling fractions K(i,j) is given

by Nij , i.e., −√λ

8π2i

∮dx(1− 1

x2

)(pi + δpi) = K(i,j) + Nij . In this sense, the normalisation chosen for the filling

fractions fixes also the normalisation of the asymptotics of δp.

32

where K(x) =√

1− x2β2.

The final step is to use equation (6.3) to find the value of the spectral parameter at which wehave to place the microscopic cuts, xn, and substitute it in the above equations. As the processis the same for all the cases mutatis mutandis, we will illustrate it in the case 13. The algebraicequation p1(xn)− p3(xn) = 2πn has two solutions

xn =aT√

1− β2 ±√a2T + n2

n. (6.16)

Because√a2T + n2 ≥ aT ≥ aT

√1− β2 for real values of n and |β| ≤ 1 (as required from reality

conditions, see below (5.26)) the solution with minus sign has modulus smaller than one and we

will disregard it. Substituting this result into Ω13(xn), we find

Ω13(xn) =x2nK(1/xn) +K(1)

x2n − 1

− 1 =

√a2T + n2

a2T

− 1 . (6.17)

The remaining contributions take the following form

Ω13(xn) = Ω14(xn) = Ω23(xn) = Ω24(xn) = −√

1− β2 +

√1− β2 +

n2

a2T

, (6.18)

Ω14(xn) = −2 +

√√√√2 +n2

a2T

+ 2

√1 +

n2

a2T

(1− β2) , (6.19)

Ω23(xn) =

√√√√2 +n2

a2T

− 2

√1 +

n2

a2T

(1− β2) , (6.20)

Ω13(xn) = Ω24(xn) = −1 +

√1 + β2 +

n2

a2T

, (6.21)

Ω13(xn) = Ω14(xn) = Ω14(xn) = Ω24(xn) = −1 +

√1 +

n2

a2T

, (6.22)

Ω23(xn) = Ω24(xn) = Ω13(xn) = Ω23(xn) = −√

1− β2 +

√1 +

n2

a2T

. (6.23)

In order to have an independent check of our results, we have computed the bosonic AdScontributions using the method of quadratic fluctuations in the picture of the deformed model.The details of that computation are collected in appendix E, and the final results are consistentwith the ones obtained using the classical spectral curve of the twisted model.

Another non-trivial check that our results pass is the fact that we recover the usual BMN

contributions in the undeformed limit (see, for example, [73]), i.e. Ω = −1 +√

1 + n2

ω2 , upon

using the Virasoro constraint (5.26).

6.2 Anomalous correction to the twist

Before computing the one-loop correction to the dispersion relation of the BMN-like solution,we want to make some comments on the asymptotic behaviour of δp2 and its effect on the chargeQ of the twist. In particular, we will show that there is no contribution to the O(x0) term of

33

the expansion. To that end, we will use the inversion symmetry (6.6) to compute δp2 from thedifferent δp1 excitations that we constructed in appendix D.

Let us begin with the excitation 23. Using inversion symmetry on

δp1(x) =4π√λ

K(1)

K(1/x)

xnx2n − 1

2x

x2 − 1, (6.24)

we can check that

δp2(x) = −δp1

(1

x

)≈ 8π

x√λ

K(1)√−β2

xnx2n − 1

+O(x−2) . (6.25)

Despite getting a non-trivial O(x−1) term, we can check that it actually vanishes when weimpose the level-matching condition. The other three excitations associated to the S5 have thesame behaviour, as δp1 has a similar form for all four.

The situation is very similar for both the excitations associated to 14 and 13. In particular,we have

δp2 ≈∑n

K(1/xn)√−β2

α(xn)Nn

xnx+O(x−2) , (6.26)

δp2 ≈∑n

α(xn)

xn√−β2

K(1/xn) +K(1)

2xNn +O(x−2) , (6.27)

respectively. In both cases, the 1/x term is proportional to the pole-fixing condition (6.3) and,thus, proportional to n. This means that both the x0 and x−1 term vanish once the level-matching condition is imposed.

The situation for the excitations associated to 13 and 23 is slightly different. For these kindsof excitations, we find that

δp2 ≈ −∑n

(4π√λ

+α(xn)

xn

K(xn) +K(1/xn)√−β2

)Nn

2x+O(x−2) , (6.28)

δp2 ≈∑n

(4π√λ

+α(xn)

xn

K(xn) +K(1)√−β2

)Nn

2x+O(x−2) , (6.29)

respectively. The analysis in both cases is the same: the x0 contribution vanishes automatically,while the second term in the x−1 contribution vanishes due to the level-matching condition.However, in both cases we have a non-vanishing ∓

∑n

2πNn√λ

at order x−1.

The behaviour of the remaining excitations can be inferred using the different tricks describedin appendix D, e.g. the composition of excitations described in D.4 or the fact that the classicalsolution has pairwise symmetric quasimomenta (D.26).

Let us summarise and compare our results with what we would expect from the asymptoticbehaviour of the quasimomenta (5.21). First, we find that the x0 term of the expansion of δp2

for large values of x vanishes in all the cases. In the asymptotic expansion of the quasimomenta,this term corresponds to the conserved charge Q, associated to the twist. Thus, we can claimthat it does not receive any anomalous correction, δQ = 0. The interpretation of the x−1 termsis more subtle. Depending on the excitation we are considering, δp2 either vanishes at this order(after imposing the level-matching condition) or gives a constant contribution times the numberof excitations that is half the one we would find in undeformed AdS5 × S5. In particular, 13,24, 23, 24, 13 and 23 fall in this second category, where the two bosonic excitations contribute

34

with −Nn/2 and the four fermionic excitations with +Nn/2. If we try to interpret this from

the lens of undeformed AdS5 × S5, the appearance of, e.g., a contribution proportional to N 13n

in δp2 is surprising. Due to the presence of the twist, however, the contribution of certainexcitations to the asymptotics of δp2 and δp3 may in fact mix. Note in particular that the onlysubtle excitations are those which connect one sheet affected by the twist with a sheet that isnot affected by the twist. Comparing with (5.21) they suggest a shift of the spin charge QΘ.Nevertheless, this contribution is non-dynamical, as it only depends on the number of excitationsN ijn but not on the position of the poles xijn . Assigning an integer value to the bosonic quanta and

half an integer to the fermionic quanta, their total contribution to the sum over all excitationswill vanish. We believe, however, further examination of these non-vanishing individual termsare needed.

6.3 One-loop correction to the energy

Now we have enough information to compute the one-loop correction to the energy of ourclassical string. This is done by considering that the excitations we are introducing behave asharmonic oscillators, and computing their ground state energy. Mathematically, this means that

E ≈ Eclass. + E1−loop = Eclass. +1

2

∑n∈Z

∑ij

(−1)FijΩij(xn) , (6.30)

where we have Fij = 0 for bosonic and Fij = 1 for fermionic contributions, respectively, andwhere Eclass. = −E = aT .

Before attempting to perform the sum over n, we should check if the series of the addendsconverges. We can check that it is the case, as∑

ij

(−1)FijΩij(xn) = −2β2

n3+O(n−4) , (6.31)

meaning that the partial sums over the mode numbers have to converge, and we are allowed tocompute the sum over integer numbers.

First of all, we should notice that the mode number n always appears divided by aT . Wecan safely assume that aT 1 because it is related to the energy of our classical solution. Thisallows us to approximate our sum over n ∈ Z by an integral over n ∈ R. Here, we can usethe integration method proposed in [74] to argue that the error in our approximation has to beexponentially small in aT , as none of our contributions have a branch cut for real values of n.Then

1

2

∑n∈Z,ij

(−1)FijΩij(xn) ≈ aT2

∫ ∞−∞

dn∑ij

(−1)FijΩij(xaTn) = aT

∫ ∞0

dn∑ij

(−1)FijΩij(xaTn) ,

(6.32)where by xaTn we mean that we are rescaling n by a factor aT . Notice that, to get the secondequality, we have used that the frequencies only depend on n quadratically and we are thereforedealing with an even integrand.

At this point, it is useful to divide our contributions into two different types

Ωnested = Ω14(xn) + Ω23(xn) =√

2

√2 + n2 +

√n2(n2 + 4β2) , (6.33)

Ωother = −8√

1 + n2 + 4√

1 + n2 − β2 + 2√

1 + n2 + β2 . (6.34)

35

We do so because the nested square root structure makes the integration of Ωnested a bit involved.Let us first address the integral of Ωother. Instead of integrating from 0 to ∞, we shall integrateonly up to a positive cut-off, Λ. We do so because, while the total series is convergent, the twoseparate contributions Ωnested and Ωother are not. Obviously, they have divergences that laterwill cancel each other. After some algebra, we find

Iother

aT= 2 lim

Λ→∞

∫ Λ

0dnΩother = −2Λ2 − 2(1 + β2) log(2Λ)

−[(1 + β2) + (1 + β2) log

(1 + β2

)+ 2

(1− β2

)log(1− β2

)] (6.35)

The integral over Ωnested is more intricate, but it can be simplified using a change of variablesinspired by the one proposed in Appendix C of [75]. In particular, after we change into a variabley(n) that eliminates the nested integral, i.e. Ωnested =

√4 + y(n)2, we can perform the integral

as

limΛ→∞

∫ Λ

0dnΩnested = lim

Λ→∞

∫ Λ′

0dy√

4 + y2y3 + 8yβ2

2√y2 + 4β2

3 , (6.36)

where Λ′ = 2Λ + β2

Λ + O(Λ−3). Notice that, although we are interested in the limit of verylarge Λ, we have kept the next-to-leading order of the map between the two cut-offs. We havedone that because the integral will diverge as Λ′2, which means that this subleading order of thetransformation will give rise to a non-trivial contribution to the integral. After evaluating theintegral, and with a healthy dose of algebra, we arrive at

IAdSaT

= 2 limΛ→∞

∫ Λ

0dnΩAdS = 2Λ2 + 2(1 + β2) log(2Λ)

+ 1 + 2β − β2 − (1 + β2) log(1− β2

)+(1 + β2

)log

(1− β1 + β

).

(6.37)

Putting both integrals together, it is immediate to check that the divergent parts perfectlycancel, giving us

2E1−loop

aT=IAdS + Iother

aT= 2β − 2β2 − (3− β2) log

(1− β2

)− (1 + β2) log

(1 + β2

)+(1 + β2

)log

(1− β1 + β

).

(6.38)

As a check of our expression, we can examine two particularly interesting values of β. Onthe one hand, we can check that E1−loop = 0 for β = 0. This is consistent with the fact thatwe recover the BMN solution in the undeformed limit, which does not receive corrections. Onthe other hand, in the β = 1 limit the form of Ωnested simplifies, as we can get rid of the nestedsquare root

limβ→1

Ωnested =

√4 + 2n2 + 2

√n2(n2 + 4) =

√n2 +

√n2 + 4 . (6.39)

Thus, in this limit all the contributions have the form of a square root and there is no needto separate the integral into two divergent contributions, avoiding the possible issues that acut-off may introduce. When taking β → 1, the limit of the integral and the integral of the limitcoincide and give E1−loop = −aT log(8).

7 The unimodular case and the spectral equivalence

So far we have considered the non-unimodular deformation, i.e. we have set ζ = 0 in (2.7). Herewe wish to show that these results can be extended to unimodular deformations (ζ = 1) thanks

36

to the observation that even in this case the twist can be factorised as explained in section 4.3 forthe non-unimodular case. First, an extended Jordanian R-matrix of the type (2.7) with ζ = 1solves the classical Yang-Baxter equation if the odd elements satisfy the (anti)commutationrelations [14,63,76]

[Qi, e] = 0, [h,Q1] = 12Q1 − ξQ2, [h,Q2] = 1

2Q2 + ξQ1, Qj ,Qk = −iδjke, (7.1)

where ξ is a free real parameter. From [42] we know that for a generic homogeneous Yang-Baxterdeformation the twist can be taken to be in the subgroup F , and therefore it can be written asW−1 = exp(ηRQ), where Q is a conserved charge that takes values in the dual (with respect tothe bilinear form of g) of f. That means that we can write it as

W−1 = exp(ηRQ) = exp [η (−Qeh +Qhe− i(Q1Q1 +Q2Q2))] , (7.2)

where QA are projections of Q. We will not calculate them explicitly because this will not beneeded for the following argument, but notice that, when setting ζ = 1, in principle all of themcan receive contributions from the fermionic degrees of freedom. In other words, when goingfrom the non-unimodular to the unimodular case, the difference is not only the presence of thenew (fermionic) charges Q1,Q2, because we can also have different expressions for Qh,Qe.

First notice that Q1,Q2 are Grassmann variables. We are therefore working with the Grass-mann enveloping algebra, in which case anticommutators become standard commutators. Inparticular, notice that if we define Z = −iη(Q1Q1 + Q2Q2), then [Z,Z] = 0, because the onlynon-vanishing anticommutator in (7.1) is when we take the same odd generator, but then Q2

i = 0because it is Grassmann. The non-unimodular twist is recovered by formally setting Z = 0.

When ξ = 0 in the commutation relations, the generators Q1,Q2 do not mix and we canrepeat the argument of section 4.3 to show that W = v−1W ′v, where W ′ = exp(ηQeh). In

fact, first we can write the twist in the form W−1 = exp(A+ s

1−e−sB′), and repeat the steps

in (4.18) if we now identify

A = η(−Qeh +Qhe),s

1− e−sB′ = Z. (7.3)

Using that [h,Z] = 12Z and imposing [A,B′] = sB′ one finds s = −ηQe/2. This shows that the

unimodular twist is related by a Z-dependent similarity transformation to exp[η(−Qeh+Qhe)].This has the same form as the non-unimodular twist and, as explained in section 4.3, it thenfollows that it is also related by a similarity transformation to W ′.

To prove the factorisation of the twist for generic values of ξ we will now consider a differentand even simpler method, that works both for ζ = 0 and ζ = 1 and for any ξ. For definitenesswe will provide the discussion in the ζ = 1 case. First, let us construct

v = exp(αee) exp(α1Q1) exp(α2Q2), (7.4)

where α1, α2 are Grassmann, and compute

v−1hv = h + αee + (12α1 + ξα2)Q1 + (1

2α2 − ξα1)Q2 − iξα1α2e (7.5)

The above formula implies that

v−1 exp(ch)v = exp[c(h + (αe − iξα1α2)e + (1

2α1 + ξα2)Q1 + (12α2 − ξα1)Q2

)]. (7.6)

Now we want to match the right-hand-side of the above equation with the right-hand-sideof (7.2). We can do this only if we assume c 6= 0, and then

c = −ηQe, αe = ηQh/c+ iξα1α2 (7.7)

37

and if we also require

12α1 + ξα2 = −iηQ1/c,

12α2 − ξα1 = −iηQ2/c. (7.8)

This system has always a solution except if ξ2 = −1/4 but this possibility is excluded becauseit must be ξ ∈ R to respect the real form of psu(2, 2|4). To conclude, we can always solve alsofor α1, α2, αe, and then the twist can be put in the wanted form

W = v−1W ′v, W ′ = exp(ηQeh). (7.9)

Let us make a comment on the assumption c 6= 0 made above. Given the solution for c in (7.7),there are only two possibilities in which c can vanish. The undeformed limit η → 0 is obviouslynot problematic: when η → 0 then W → 1 so the factorisation is trivially true. The only realworry should be the case Qe = 0 with η 6= 0. Let us analyse this situation in the non-unimodularcase, when Qe = −Q/η and Q was given in (4.5). Then, Qe can vanish (at finite η) only ifQ = 0. Given (4.4), this leads to a trivial twist W = 1, unless q compensates the zero of Qwith a divergence. This is possible only if the field configuration is such that

Yh(τ, 2π)− Yh(τ, 0) 6= 0, Y+(τ, 2π)− Y+(τ, 0) = 0. (7.10)

In other words, the degrees of freedom in Ye are periodic while those in Yh are not. Therefore,in this sector of the theory the non-unimodular twist cannot be put into the form exp(ch),and it is instead equal to W = exp(−we) with w = Qq finite and non-zero. This is in fact anon-diagonalisable twist, with all eigenvalues equal to 1. While the leading-order asymptoticsof the quasimomenta around z = 1 (or x = ∞) would match those of the untwisted case, thenext-to-leading order will change. A similar analysis is obviously valid also for the unimodulartwist.

We conclude this section by arguing the spectral equivalence of the unimodular and the non-unimodular models, up to certain caveats that we are about to point out. In the sector where thetwists of the unimodular and non-unimodular models are both equivalent to W ′ = exp(ηQeh),the fact that the classical spectra of the two models are the same is rather straightforward.First, in both cases one can implement field redefinitions as explained in 4.3 to obtain boundaryconditions controlled just by W ′ = exp(ηQeh). As remarked above, the explicit expressions forQe in the two cases differ, because in the unimodular case there are additional contributions fromthe fermionic degrees of freedom. However, these vanish when considering classical solutions,and therefore the classical spectral curves of the two models are indistinguishable. One mayworry that the non-diagonalisable sector (where the twist is instead W = exp(−we)) may beproblematic from the point of view of the spectral equivalence. But also in this case the twistsof the two models would differ only by contributions of the fermionic degrees of freedom, thatdo not contribute at the classical level.

We can push this argument even to the one-loop level. In fact, one may use the methodof [49, 53] to compute quantum corrections to the classical spectrum, as we have done in theprevious section for the non-unimodular case. The advantage of this method is that the onlydata that is needed to compute the one-loop shift is the classical spectral curve itself. Becausethe two classical spectral curves agree, the unimodular and non-unimodular models will alsohave the same 1-loop corrections to the spectrum. It would be interesting to understand if itis possible to make any statement beyond one-loop, since it is always possible that the spectralequivalence breaks down at higher loops. However, we want to point out that even at one-looplevel there may be extra subtleties not discussed so far. In fact, the whole argument is doneunder the assumption that it is possible to reformulate the Yang-Baxter deformed model as theundeformed yet twisted one. While this is certainly true at the classical level because the two

38

σ-models are equivalent on-shell, it is possible that the equivalence breaks down at the quantumlevel. An anomaly may occur especially in the non-unimodular case, when the background fieldsdo not satisfy the standard supergravity equations and the σ-model is not Weyl invariant. Inthe presence of such an anomaly, even if the (quantum) spectra of the two twisted models werethe same, the spectra of the corresponding Yang-Baxter deformations (with ζ = 0 and withζ = 1) may not be related in an obvious way. It would be very interesting to investigate thesepossible scenarios in more details.

8 Conclusions

In this paper we have considered a Jordanian deformation of the AdS5 × S5 superstring back-ground, that preserves the (classical) integrability. We have identified global coordinates for thedeformed background, and in particular a global time coordinate. This fact is crucial for thecorrect identification of the energy when considering the spectral problem, and we believe that itwill be important in order to have insights into an AdS/CFT interpretation for the deformation.

Importantly, we have reformulated the deformed model in terms of an undeformed yet twistedmodel. In this twisted picture we were able to obtain a general class of solutions to the σ-modelequations of motion written in terms of Airy functions. Our further study, however, focused on aparticular simple solution which we called BMN-like. Given that the difference compared to thestandard (periodic) case of AdS5 × S5 is only in a twist appearing in the boundary conditions,we could borrow several methods of integrability that had already been used in the past. Wedid this not just at the level of the classical spectral curve, but also when including its firstquantum corrections since, despite the twist, the methods are essentially unaltered.

The only place where the construction of the classical spectral curve is altered with respectto the standard case is in the form that the quasimomenta take at large values of the spectralparameter. In particular, we are forced to re-evaluate the ansatz for the asymptotic behaviourof the quantum corrections for some of our quasimomenta. For example, in the standard casewe can extract the contribution to the anomalous correction to the energy from any of thecorrections to the quasimomenta associated to the AdS directions. However, in the deformedbackground we have studied, this contribution does not appear in those quasimomenta whichare affected by the twist.

The advantage of working in the language of the twisted model resides also in the factthat the conserved charges that label the full spectrum admit local expressions in terms of thevariables of the twisted models, while they would be non-local in the variables of the deformedone. Moreover, only (the Cartans of) the manifest symmetries of the twisted model appear aslabels of the spectrum, and this is crucial because these are only a subset of the isometries ofthe deformed one.

An interesting outcome of our results is that we find no anomalous quantum correction for thecharges that control the twisted boundary conditions. The only charge that receives quantumcorrections is the energy, and we interpret this as the spectral problem being well-posed. Thesituation is reminiscent to the β-deformation [27] where the twist appears in the form of phasesthat enter the Bethe equations [35], for example. Also in that situation, it is the energy ofthe string (or the dual anomalous dimension of single-trace operators) that receives quantumcorrections, while the charges controlling the twist appear as external data, so that the Betheequations can be solved consistently.

In the deformed picture, we know that the deformation parameter can be reabsorbed, al-though this is done in such a way that the undeformed limit η → 0 becomes subtle. One can

39

actually think in terms of two possible cases to be considered, namely η = 0 and η 6= 0. This isrelated to the fact that Jordanian deformation (at η 6= 0) are essentially non-abelian T-duality,without any continuous deformation parameter. However, in the twisted picture we do not seeevidence for a similar interpretation, because the one-loop spectrum seems to depend continu-ously on the deformation parameter η through the combination β = η

2b2z. It would be interesting

to understand this further.

Let us also comment on the fact that we were able to consider both the unimodular Jordaniandeformation (giving rise to a type IIB supergravity background) and the non-unimodular one(which does not correspond to a supergravity solution). In this respect, the main messagefrom our results is that the corresponding twisted models share the same spectrum, at leastto one-loop. Unless the argument regarding the equivalence of the deformed model to thetwisted one at the quantum level fails, this seems to suggest that the unimodular and non-unimodular deformations share the same spectrum at least to that order. It would be interestingto understand these subtleties further, and see whether we can make any statement beyond one-loop. A first non-trivial check was obtained already in appendix E, where we found that thebosonic (AdS) frequencies obtained from the analysis of quadratic fluctuations of the deformedσ-model matches with the bosonic (AdS) frequencies obtained from the curve of the twistedσ-model. In addition, let us mention that the equivalence of the quantum spectrum betweena unimodular and a non-unimodular Yang-Baxter deformation has been noted also in [77],although in that case the inhomogeneous Yang-Baxter deformation was considered, for whichthe reinterpretation as a twist of the boundary conditions in the undeformed model does nothold.

Our results offer a first step towards the understanding of how to tackle the spectral problemfor Jordanian deformations (and more generally diagonalisable Yang-Baxter deformations) ofAdS5×S5. It would be interesting to start working from the other side of the AdS/CFT duality,and construct a corresponding deformation of N = 4 super Yang-Mills. The expectation is thatthere should be a notion of a deformed/twisted spin-chain in that case, and it is worth exploringthe possibility of constructing this spin-chain directly, to obtain insights on the potentially moredifficult construction of the deformation of the gauge theory.

From a broader perspective, it would be very interesting to reinterpret the more genericsupergravity solution-generating techniques of [18,19] (which include also non-abelian T-duality,and are integrability-preserving) in terms of twisted models within a first-order formulation. Thismay open the possibility of tackling the spectral problem in this more generic class of deformedand dual σ-models.

Acknowledgements

We thank Tristan Mc Loughlin, Olof Ohlsson Sax, Roberto Ruiz and Stijn van Tongerenfor discussions, and we are grateful to Roberto Ruiz and Stijn van Tongeren for commentson the manuscript. RB and SD are supported by the fellowship of “la Caixa Foundation”(ID 100010434) with code LCF/BQ/PI19/11690019, by AEI-Spain (under project PID2020-114157GB-I00 and Unidad de Excelencia Marıa de Maetzu MDM-2016-0692), Xunta de Gali-cia (Centro singular de investigacion de Galicia accreditation 2019-2022, and project ED431C-2021/14), and by the European Union FEDER. LW is funded by a University of Surrey DoctoralCollege Studentship Award. JMNG is supported by the EPSRC-SFI grant EP/S020888/1 Solv-ing Spins and Strings. No data beyond those presented and cited in this work are needed tovalidate this study.

40

A Relation to embedding coordinates of AdS

In this appendix we want to explain how the global coordinates that we use to parameterisethe Jordanian-deformed spacetime are related to the embedding coordinates of AdS. This isboth to offer additional geometric intuition on the coordinates used, and to give an alternativeargument to the fact that they are global coordinates, although here this applies only in theundeformed η = 0 limit.

AdSD is defined as the hyperboloid X20 + X2

D −∑D−1

i=1 X2i = R2 (with R the AdS radius)

in R2,D−1 with metric ds2 = −dX20 − dX2

D +∑D−1

i=1 dX2i . Global coordinates in AdS can

be obtained by setting X0 = R cosh ρ cos τG, XD = R cosh ρ sin τG, Xi = R sinh ρ Ωi with theconstraint

∑D−1i=1 Ω2

i = 1. This gives the AdS metric ds2 = R2(− cosh2 ρ dτG+dρ2+sinh2 ρ dΩ2),with dΩ2 the metric of the (D − 2)-dimensional sphere. For D > 2 the hyperboloid is coveredonce if we take ρ ≥ 0, 0 ≤ τG < 2π, and the universal cover is obtained by decompactifying theglobal time τG. The metric diverges at ρ → ∞ but that is not a problematic place: geodesicseither reach it in an infinite time, or if they do it in a finite time (as it happens for some nullgeodesics) then they come back to finite ρ at later time. Therefore, the spacetime is geodesicallycomplete in these coordinates.

Alternatively, the identification X0 = 12z(1 + z−2(R2 + xixi − t2)), XD = Rt/z, Xi =

Rxi/z(i = 1, . . . , D − 2), XD−1 = 12z(1− z

−2(R2 − xixi + t2)) gives the metric

ds2 = R2(dz2 + dxidxi − dt2

z2) , (A.1)

in the so-called Poincare coordinates. These coordinates do not cover the whole hyperboloidand in fact, as we will show in the more generic deformed case in appendix B, some geodesicsare not complete in this patch (e.g. they reach z = ∞ in a finite proper time, and cannot becontinued to z < ∞). In these coordinates the boundary of AdS is at z = 0, where the metricdiverges.

In the coordinates (T, V,Θ, P, Z) of (2.20) we see that the AdS metric (i.e. when settingη = 0) diverges at Z = 0, and because of the relation to the previous coordinates we canidentify that as the boundary of AdS. The composition of relations between the different setsof coordinates leads to the relation of the global coordinates (T, V,Θ, P, Z) to the embeddingcoordinates. Setting R = 1 we find

X0 =

(1 + Z2 + P 2

)cosT − 2V sinT

2Z,

X1 =

(P 2 + Z2 − 2

)sinT + 2V cosT

2√

2Z,

X2 =P sin Θ

Z,

X3 =P cos Θ

Z,

X4 =

(−1 + Z2 + P 2

)cosT − 2V sinT

2Z,

X5 =

(P 2 + Z2 + 2

)sinT + 2V cosT

2√

2Z.

(A.2)

41

These expressions can be inverted as

P =

√2√X2

2 +X23√

2(X0 −X4)2 + (X1 −X5)2,

Θ = arctan(X2/X3),

V =X0(3X1 +X5)−X4(X1 + 3X5)√

2 (2(X0 −X4)2 + (X1 −X5)2),

Z =

√2√

2(X0 −X4)2 + (X1 −X5)2,

T = arctan

(X1 −X5√2(X4 −X0)

).

(A.3)

From these expressions we see that we can take P > 0 and Z > 0, and moreover40 Θ ∈ [0, 2π[and T ∈ [0, 2π[, which in both cases are periodically identified with periods 2π. For V we haveinstead V ∈]−∞,+∞[. While τG is an angle in the (X0, X5) plane, T is instead an angle in theplane (X4 −X0, X1 −X5). Also in this case, we decompactify it and take T ∈]−∞,+∞[.

We will now argue that these coordinates cover the whole hyperboloid. First, P and Θcover all the (X2, X3) plane, with a “warping” depending on the other 4 coordinates. If wenow define Y ±0 = X0 ± X4 and Y ±1 = (X1 ± X5)/

√2, then Z and T take care of the whole

(Y −0 , Y −1 ) plane (notice that Z is the inverse of the radial coordinate in that plane). Finally,we need to understand if we can cover the whole (Y +

0 , Y +1 ) plane. Notice that (unlike other

coordinates) V depends also on the combinations Y +0 , Y +

1 , not just Y −0 , Y −1 . One may worrythat we have just one degree of freedom (i.e. V ) to cover this (Y +

0 , Y +1 ) plane, but in fact this is

not an issue, because we should remember that we are constraining the coordinates to satisfy 1 =X2

0 +X2D−

∑D−1i=1 X2

i = −X22−X2

3 +Y +0 Y −0 −2Y +

1 Y −1 , so that if for example Y +0 , Y −0 , Y −1 , X2, X3

have been fixed (because for example one solves for Y +0 in terms of V ), then the remaining Y +

1

is uniquely identified. Notice that the hyperboloid constraint is quadratic in XM , but it is linear(for example) in Y +

1 , so when solving for Y +1 the sign is unambiguous. Naively, an issue is

present when Y −1 = 0, because then Y +1 is undetermined. But when Y −1 = 0 then V = Y +

1 /Y −0 ,and we can just swap Y +

0 and Y +1 in the previous reasoning. Then one can solve the hyperboloid

constraint for Y +0 instead. The issue remains when both Y −0 = Y −1 = 0, because then V, P and

Z diverge. This is not problematic because Y −0 = Y −1 = 0 is a codimension-2 subspace, whichis not an open set in spacetime.

B Geodesic incompleteness of Poincare coordinates

For completeness, we show in this appendix that the geodesic incompleteness of the parametri-sation by Poincare coordinates persists also when turning on the deformation η. As usual, thestrategy is to look for solutions to the geodesic equations that are pathological, i.e. the boundaryvalues of the coordinates are reached in a finite amount of geodesic time.

To start, let us construct the conserved quantities Qa = kµaGµνXν along the geodesics. They

40Given that Θ and T are obtained from the arctan, normally they would take values in [−π/2, π/2]. Inparticular, arctan(y/x) is insensitive to a simultaneous change of sign of x and y. To avoid this problem, one candefine a function arctan(x, y) that gives the value of the angle identified by the point in the (x, y) plane takinginto account the quadrants in which x and y are placed.

42

are equivalent to the following equations

θ =Q5z

2

ρ2,

x− = −(Q2 +Q3) z2

√2

,

x+ =η2 (Q2 +Q3) ρ2 − 4 (Q2 −Q3) z4 + η2 (Q2 +Q3) z2

4√

2z2,

ρ =z(z(√

2 (Q3 −Q2)x− +Q1

)− z′

)ρ

,

(Q2 +Q3)(ρ2 + z2

)= Q4 − 2

√2Q1x

− + 2 (Q2 −Q3) (x−)2.

(B.1)

At the same time we also have the conserved quantity

ε = XmXmGmn = −8z4x−x+ + η2 (x−)

2 (ρ2 + z2

)− 4z4

(ρ2θ2 + ρ2 + z2

)4z6

. (B.2)

Let us consider null geodesics, so that ε = 0. Now the strategy is to look for solutions that atlarge z go like z ∝ (τ − τ0)−A for some A > 0. These are pathological because z =∞ is reachedat finite τ . To simplify the analysis we fix some of the values of the Qa. At the same time,we must be careful and make sure that the last equation in (B.1) does not reduce to the formCzz

2 + Cρρ2 + C−(x−)2 = C with Cz, Cρ, C−, C > 0: if that happened, the geodesic motion

would be bounded and no pathological behaviour at z = ∞ would be possible. To be concretewe take

Q1 = 0, Q2 6= 0, Q3 = 0, Q4 = 0, Q5 = 0. (B.3)

It is of course possible to take more generic situations, but this will already be enough to identifypathological behaviour. In this case (B.1) reduce to

θ = 0, x− = −Q2z2

√2, x+ =

Q2

(η2ρ2 − 4z4 + η2z2

)4√

2z2,

ρ =z(−√

2Q2zx− − z

)ρ

, 0 = z2 − 2(x−)2 + ρ2.

(B.4)

Notice that θ is constant, and that a solution for x+ can be found once a solution for z is given.Therefore, in the following we focus on the equations for the remaining coordinates. At thispoint we look for geodesic solutions that at large values of z have

x− ∼ α−z, for z 1, (B.5)

with α− a constant. Within this ansatz, the algebraic constraint given in the last equationof (B.4) reduces to (1− 2α2

−)z2 + ρ2 ∼ 0, which is compatible with large z, ρ only if

α2− > 1/2, =⇒ ρ ∼ z

√2α2− − 1. (B.6)

Demanding compatibility with the equation for ρ then gives

z ∼ − Q2√2 α−

z2. (B.7)

This is what we were seeking, since such an equation is solved by41 z ∝ (τ − τ0)−1. Thepathological behaviour of this solution is enough to conclude that the Poincare coordinates arenot global coordinates for the deformed spacetime.

41Importantly, this solution is compatible with (B.2) in the large-z expansion.

43

C Identifying the Cartan subalgebra of isometries

In this appendix, we identify the possible Cartan subalgebras of the algebra of isometries k givenin (2.18). Since M23 and p0 + p1 are central elements, i.e. they commute with all generators ink, we can focus on the remaining generators which span an sl(2,R) = span(h, e+, e−) subalgebrawith [h, e±] = ±e± and [e+, e−] = 2h as is found by identifying h = 1

2(d−M01), e+ = 12(p0−p1),

and e− = −12(k0 + k1). Inequivalent choices of the Cartan subalgebra are then obtained by

classifying the inequivalent adjoint orbits of sl(2,R), and then requiring that the representativeelements have a diagonalisable adjoint action. Although this is a standard exercise, we repeatit explicitly to clarify certain comments.

Let us write a generic element of the group SL(2,R) as g = exp(chh) exp(c+e+) exp(c−e−)and of the algebra sl(2,R) as x = αhh + α+e+ + α−e−. For each element x of sl(2,R) one canthen identify its adjoint orbit gxg−1. One finds three inequivalent possibilities. First, in fact,we can transform the generic element x to the element h whilst preserving reality conditions as

gxg−1 =√γ h, with γ = α2

h + 4α−α+, (C.1)

by specifying the element g with the coefficients

c+ =α+√γ, c− =

αh(√γ − αh

)− 4α−α+

2α+√γ

, (C.2)

if γ > 0.42 The second case is that for γ = 0, which can be obtained when at least α+ or α−are non-zero (otherwise x would also be zero). For definiteness say that α+ 6= 0. One can thentransform x into e+ as

gxg−1 = echα+ e+, with c− =αh

2α+. (C.3)

Alternatively, we can take α− 6= 0, which would transform x into e−. However, e± are related byan inner automorphism as ge+g

−1 = −e− with g = exp(π/2(e+ − e−)) so these two possibilitiesare in fact equivalent. The third and final case is that for γ < 0, which can be obtained onlywhen both α+ and α− are non-vanishing. In this case one can transform x into e+ − e− asfollows

gxg−1 =1

2α+

√− γ

α2+

(e+ − e−), (C.4)

by using

ch =1

2log

(− γ

4α2+

), c+ = 0, c− =

αh2α+

. (C.5)

Thus, given a generic x ∈ sl(2,R) which we want to simplify by means of inner automorphisms,we have three inequivalent possibilities43

h, or e+, or e+ − e−. (C.6)

To identify the possible Cartan subalgebras, we must now verify whether or not the adjointactions adx for the above inequivalent elements are diagonalisable. A simple computation showsthat adh is diagonalisable with eigenvalues (0, 1,−1), ade+ is not diagonalisable, and ade+−e− is

42Although the case α+ = 0 seems singular, it can be analysed on its own (for αh 6= 0) leading to the sameconclusion.

43It is immediate to see that these choices cannot be related by inner automorphisms, given that Tr(h2) =1,Tr(e2

+) = 0,Tr((e+ − e−)2) = −2.

44

diagonalisable with eigenvalues (0, 2i,−2i).44 To conclude, up to automorphisms, there are twopossible choices of Cartan subalgebras of k, namely

(I) : spand−M01, p0 + p1, M23,(II) : spanp0 − p1 + α(k0 + k1), p0 + p1, M23,

(C.7)

where we leave a possible α > 0 coefficient for later convenience. The choice e+ − e− wouldcorrespond to α = 1, but in the main text we find it convenient to choose α = 1/2.

D Analytic structure of the perturbations

In this appendix, we detail how to compute the correction to the quasimomenta δpi that arisefrom adding small excitations to our classical solution using only their analytic properties, asdescribed in section 6.1. Most of our computations mimic those of the sl(2,R) circular stringin [49] due to the presence of a single cut in the pi quasimomenta, but the asymptotic behaviourof our classical quasimomenta and some of their corrections is very different.

D.1 Excitations associated to the sphere

Excitations that connect two quasimomenta pi and pj are related to considering quadratic fluc-tuations around a classical solution for the modes associated to the sphere. Of those, the onlyones that give rise to physical excitations are those that connect either 1 or 2 with either 3 or4. Similarly to the case described in [49], the two remaining possible combinations, 12 and 34,do not give rise to physical excitations.

For concreteness, here we will focus on the excitation that connects p2 and p3, which we willdenote as 23, and turn off any other excitation. The computations for the other three excitationsare the same mutatis mutandis. Nevertheless, we will make some comments on 24 and we willdiscuss the remaining two in section D.4.

Let us focus first on the correction δp2. We know that it has to have poles at ±1 and at x23n .

Thus, we propose the following ansatz

δp2 =δα+

x− 1+

δα−x+ 1

−∑n

α(x23n )N 23

n

x− x23n

. (D.1)

As there will be no ambiguity, we will drop the superindex of xn and Nn to alleviate our notation.

We also know that δp2 is associated to δp1 by inversion symmetry. Thus, we can use theinformation about their behaviour at large values of x to fix the residues of the poles. On theone hand, from equation (6.8), we have

δp1 ≈ O(x−2) , δp2 ≈ −4π√λ

∑n

Nn

x+O(x−2) . (D.2)

On the other hand, from our ansatz and inversion symmetry (6.6), we have

δp1 ≈

(δα+ − δα− −

∑n

α(xn)Nn

xn

)+δα+ + δα− −

∑nα(xn)Nn

x2n

x+O(x−2) , (D.3)

δp2 ≈δα+ + δα− −

∑n α(xn)Nn

x+O(x−2) . (D.4)

44Notice that ade+−e− is real, only its diagonalised version has imaginary coefficients.

45

Consistency between the two expansions gives us enough information to fix the three residues

α(xn) =4π√λ

x2n

x2n − 1

, (D.5)

2δα± =∑n

α(xn)Nn

(1

x2n

± 1

xn

). (D.6)

As we anticipated in section 6.1, the function α(x) does not need to be fixed beforehand. Theconsistency between the positions and signs of the poles with the asymptotic behaviour of thecorrections is enough to fix its form as a function of the position of the pole. As expected, thefunction α(x) is unchanged from the undeformed case [49]. This is consistent with the fact thatit is related to the weight appearing in the definition of the filling fractions (5.9). In particular,we require that

−Nn = −√λ

8π2i

∮Cdx

(1− 1

x2

)δp2 = −

√λ

4πNnα(xn)

(1− 1

x2n

), (D.7)

where C is the integration contour that encircles the small cut we are putting in. As theexpressions of the filling fractions are not affected by the deformation, neither are the residuesα(x).

If we now use the condition (6.3) to find the position of the cut, we get

p2(xn)− p3(xn) = 2πn =⇒ xn =ω ±√ω2 + n2

n. (D.8)

Here, we are only interested in the solution with the plus sign, as it is the one in the physicalregion. Substituting the explicit expression for xn in α(xn), we find that (D.6) gives

2δα± =4π√λ

∑n

Nn

(1± xnx2n − 1

)=

4π√λ

∑n

nNn

2ω

(n

ω −√ω2 + n2

± 1

). (D.9)

The second term in this expression actually vanishes when we impose the level-matching condi-tion

∑n nNn = 0. This means that δα+ = δα−, although we will not make use of this relation

in the following equations.

Now that we have computed δα±, we can consider the effect that the excitation has onthe quasimomenta associated to the (deformed) AdS part of our space and compute δ∆. Theansatze for these corrections are a bit more involved, as we have to consider the possibility of ashift of the branch cuts arising from a backreaction of the excitation. Thus, we will assume thefollowing ansatze

δp1(x) = f(x) +g(x)

K(1/x), δp4(x) = f(x)− g(x)

K(1/x), (D.10)

where K(x) =√

1− x2β2 and the functions f(x) and g(x) are functions to be determined. Thesecond term of our ansatz is inspired by the fact that ∂βK(x) ∝ 1

K(x) and by eq. (5.23), as thequasimomentum p4 still has to be the analytic continuation of p1 through a square-root cut.

Let us start by constraining the function f(x) = δp1(x)+δp4(x)2 . As we are considering a

bosonic excitation, the synchronisation of the poles at ±1 forces δp1(x) and δp4(x) to haveresidues with opposite signs, meaning that f(x) has no poles at ±1. In addition, δp1 ≈ −δp4 ≈

4πx√λδ∆2 +O(x−2) for large values of x. As we have not added any new poles to the AdS directions,

this means that f(x) is a holomorphic function that approaches zero at infinity. According toLiouville’s theorem, the only function that fulfils these requirements is f(x) = 0.

46

Now, we can fix the function g(x) by demanding that δp1 has the correct properties. Asδp1(x) only has poles at ±1, we propose the following ansatz for g(x)

g(x) =K(1)δα+

x− 1+K(1)δα−x+ 1

, (D.11)

where we have used the synchronisation of the poles at ±1 as in eq. (6.7) to fix the residues.Matching the asymptotic behaviour of δp1(x) with its expected behaviour gives us

δ∆

2=

√λ

4π

K(1)

K(0)(δα+ + δα−) =

√1− β2

∑n

Nn

x2n − 1

. (D.12)

Notice that δ∆ can only be real if |β| ≤ 1, which is exactly what we found from reality conditionsimplied by the Virasoro constraint.

We should remark that this construction is blind to having the second pole in the sheet 3 orthe sheet 4, as we did not use that information at any point of the reconstruction. This impliesthat the contributions to δ∆ of 23 and 24 have the same form as a function of xn. We will lateruse the inversion symmetry to show that the other two give us the same result.

D.2 Excitations associated to the deformed AdS space

Excitations that connect two quasimomenta pi and pj are related to modes associated to thedeformed AdS space. Similarly to the case of the sphere, the only ones that give rise to physicalexcitations are those that connect either 1 or 2 with either 3 or 4. Due to the presence of thetwist, the excitations associated to 14 and 23 behave differently to those associated to 13 and24, and thus we will treat them separately. We will consider only one case for each respective setbecause the other one can be obtained through inversion symmetry, as we will discuss in D.4.

Let us start with the case associated to 14 and turn off any other excitation. First, notethat we are not modifying the quasimomenta associated to the sphere. This means that all δpidecay at infinity as O(x−2). The only possible configuration with this property is the one withδα± = δβ± = 0, which allows us to set all the sphere corrections to zero, δpi = 0. Consequently,none of the corrections δpi have poles at ±1 due to their synchronisation.

Similarly to what happened for the excitation 23, the residues at xn of δp1 and δp4 have the

opposite value. Furthermore, for large values of x, we have that δp1 ≈ −δp4 ≈ 4πx√λ

δ∆+2∑nNn

2 +

O(x−2). If we consider the ansatz

δp1(x) = f(x) +g(x)

K(1/x), δp4(x) = f(x)− g(x)

K(1/x), (D.13)

these considerations imply that f(x) = 0 and fix g(x) to

g(x) =∑n

K(1/xn)α(xn)Nn

x− xn. (D.14)

Here we have used the insight from our sphere computation to argue that the residue at xn hasto be the same as the undeformed one, so we can use the function α(x) we computed above in(D.5). Matching the asymptotic behaviour of our ansatz with its expected behaviour gives us

δ∆ =∑n

(2x2nK(1/xn)

x2n − 1

− 2

)Nn . (D.15)

47

Let us consider now the case associated to 13 and turn off any other excitation. Again,we are not modifying the quasimomenta associated to the sphere, implying that none of thecorrections δpi have poles at ±1.

Similarly to our previous cases, we start by considering the ansatz

δp1(x) = f(x) +g(x)

K(1/x), δp4(x) = f(x)− g(x)

K(1/x). (D.16)

However, in contrast to previously, δp1(x) has a pole at xn but δp4(x) does not, meaning thatthe function f(x) is not holomorphic and thus does not vanish in this case. Notice that, instead,δp4(x) has a pole at 1/xn inherited from δp3(x) due to the inversion symmetry. In fact, assumingthat δp3(x) has a pole at xn with residue −α(xn), inversion symmetry forces δp4(x) to have a

pole at 1/xn with residue −α(xn)x2n

= α(

1xn

).45 Substituting this information into an ansatz of

the form f(x) = a+ bx−xn + c

x−1/xn, we arrive to

f(x) =∑n

4π√λ

xNn

2(x− xn)(x− 1/xn), (D.17)

which fulfils all our requirements, including consistency with the asymptotic behaviour of δp1(x)and δp4(x), which reads f(x) ≈

∑n

4πx√λNn2 + O(x−2). Similarly to the function f(x), we can

assume that g(x) can be written as g(x) = a+ bx−xn + c

x−1/xn. These constants can be fixed by

demanding that δp1 has the properties we want, i.e. it has a pole with residue α(xn) at xn andno pole at 1/xn. After some algebra, we get that

g(x) =∑n

4π√λ

K(xn)(x− xn) + x2nK(1/xn)(x− 1/xn)

2(x2n − 1)(x− xn)(x− 1/xn)

Nn . (D.18)

Matching the asymptotic of our ansatz with the expected asymptotic behaviour of δp1 gives us

δ∆ =∑n

(x2nK(1/xn) +K(xn)

x2n − 1

− 1

)Nn . (D.19)

D.3 Fermionic excitations

Let us move finally to fermionic excitations. Excitations that connect a quasimomentum pi anda quasimomentum pj or vice-versa are related to fermionic modes. Here we will focus only onthe excitation associated to 13 and turn off all other excitations. We will make some commentson the remaining excitations at the end of this section.

We should first focus on the corrections to the quasimomenta associated to the sphere. Aswe are adding an excitation to the sheet 3, we will consider the following ansatz

δp3(x) =δβ+

x− 1+

δβ−x+ 1

+∑n

α(xn)Nn

x− xn= −δp4(1/x) , (D.20)

δp1(x) = δp2(x) = 0 . (D.21)

Notice that, as we are considering a fermionic excitation, these corrections fulfil the relaxedsynchronisation condition (6.7). If we impose the correct value of the residues and the correct

45Notice that, although δp2 and δp4 have a pole, they are not in the physical region and they do not contribute

to N 24n .

48

asymptotic behaviour, we get exactly the same equations as the ones for the 23 excitations (upto replacing δα± with δβ± and Nn with −Nn). Thus, we can borrow the results we found forδβ± and the residue α(xn) given in (D.5) and (D.9).

For the correction to the AdS quasimomenta, we will assume the same ansatz as in theprevious cases. Similarly to the case 13, we cannot set the function f(x) to zero because δp1 hasa pole at xn but δp4 does not. In addition, δp4 has poles at ±1 from the synchronisation con-dition, while δp1 does not. After imposing these restrictions, as well as the required asymptoticbehaviour, we get

f(x) =∑n

4π√λ

x2Nn

2(x− xn)(x2 − 1). (D.22)

With this information, we can now compute the function g(x) by demanding that δp1(x) hasthe correct residues at ±1 and xn, which gives us

g(x) =∑n

4π√λ

x2n(x2 − 1)K(1/xn) + (x2 − x2

n)K(1)

2(x− xn)(x2 − 1)(x2n − 1)

Nn . (D.23)

Finally, matching δp1(x) with its required asymptotic behaviour, we obtain that

δ∆ =∑n

(x2nK(1/xn) +K(1)

x2n − 1

− 1

)Nn . (D.24)

In the next section we will discuss how to obtain the contribution of 14 from this one, butit is easy to see that the computation has to be exactly the same as the one presented above,giving us the same δ∆. The same happens for the excitations 14 and 24.

The steps to compute the contribution of 23 are relatively similar. If fact, by analysing thepole structure it is easy to reach the conclusion that the expression of δp2 is exactly the sameone as the expression of δp1 for the 13 excitation after substituting K(1/x) by K(x) and addinga constant contribution to g(x). Applying the inversion symmetry to get δp1 and matching itwith its required asymptotic behaviour, we obtain that

δ∆ =∑n

(K(xn) +K(1)

x2n − 1

)Nn . (D.25)

Although at this point we would have to study also the fermionic excitations that involveeither 3 or 4, we can argue that this is not necessary. This happens because our classical solutionhas pairwise symmetric quasimomenta

p1 = −p4 , p2 = −p3 , p1 = −p4 , p2 = −p3 . (D.26)

This implies that, after all the appropriate computations, we will find that

Ω14 = Ω14 , Ω24 = Ω13 , Ω13 = Ω24 , Ω23 = Ω23 , (D.27)

both for Ω understood as a function of the position of the poles and as a function of the modenumber n.

D.4 Composition and inversion

Up to this point we have computed δ∆ for some selected excitations. This is enough for us, asthe contribution of the other excitations can be computed using properties such as compositionof poles and the inversion property.

49

p1

p3

p4

=

Figure 1: Illustration of the composition rule (D.30). The fermionic frequency Ω13(x) and the bosonic frequency

Ω34(y) can be composed, as the two excitations have a pole with opposite residue on sheet p3. In particular, therecomposition vanishes in the limit y → x, giving us the pole structure (and asymptotic behaviour) associated with

the fermionic excitation 14. Thus, we can claim that off-shell Ω13(x) + Ω34(x) = Ω14(x).

The idea is that the procedure we used to compute δ∆ does not care about the exactposition xn of the poles/microscopic cuts we are adding. Before substituting the expressionsof xn obtained from (6.3) in the frequency of the corresponding excitation, we can thus thinkof Ω(x) as formal, off-shell, functions of x. Once we substitute x = xn the frequency becomeson-shell. We can use the off-shell expressions of Ω(x) in our advantage in two different ways. Thefirst one is the inversion symmetry: if we consider an excitation connecting a given pair of sheets,we can move the pole associated to it to the interior of the unit circle, making it non-physical,and a new physical pole will emerge in the sheets connected to the original ones by inversionsymmetry. The second one is composing two excitations that share a pole with opposite residueon the same sheet, say j, but connect to different sheets, say i and k respectively. The pole onsheet j will cancel, making the composition of the two excitations ij and jk indistinguishablefrom having only one excitation ik. In mathematical terms, these two ideas imply

Ω14(x) = −Ω23

(1

x

)− 2 , (D.28)

Ω14(x) = −Ω23

(1

x

)+ Ω23(0) , (D.29)

andΩij(x)± Ωjk(x) = Ωik(x) , (D.30)

where the ± is chosen appropriately to cancel the residue in sheet j. We will not reproduce herethe derivation of these expressions, which can be found in [53]. Let us stress again that theserelations between the different contributions only hold when they are understood as functionsof x, not as functions of n.

Let us now compute some of the remaining frequencies using the ones we obtained above.The process is relatively similar for most of them, so we will consider only some specific cases:the excitations 23, 14 and 14. Both 23 and 14 can be obtained by immediate application of theabove formulas

Ω23(xn) = −Ω14

(1

xn

)− 2 =

(−2K(xn)

1− x2n

+ 2

)− 2 =

2K(xn)

x2n − 1

, (D.31)

Ω14(xn) = −Ω23

(1

xn

)+ Ω23(0) = −2

√1− β2

x−2n − 1

− 2√

1− β2 = 2

√1− β2

x2n − 1

. (D.32)

The last one is obtained by considering first the composition of the excitations 23 and 24. Thisgives us that the excitation 34 has Ω34 = 0, which is consistent with the fact that it is anunphysical excitation. This result can then be used to show, e.g., that Ω14 = Ω13 + Ω34 = Ω13.

50

E Bosonic sector of the quadratic fluctuations

In order to cross-check the results we obtained using the classical spectral curve method, we cancompute the contribution of the bosonic excitations to the one-loop correction to the energy byconsidering the effective Lagrangian of small fluctuations around the classical solution we areinterested in. We will perform this computation in the deformed periodic picture, rather thanthe undeformed twisted one. The expansion of the deformed AdS5 sector of the Lagrangian isgiven by

L(xclas. + εx) ≈ L(xclas.) + ε(E.o.M.) +ε2

2b6Z

[4b4Z(P 2 − P ′2 + Z4 − Z ′2)

− (4b6Z + η2b2Z)(T 2 − T ′2)− 8b4Z(T V − T ′V ′) + 4aT bZZ(η2T + 4b2Z V + 3ηZ ′)

−a2T (4b4Z + η2)P 2 − 4a2

T η2Z2 − 4ηaT b

2ZPP

′ + 4bZη(T ′Z − TZ ′)]

+O(ε3) ,

(E.1)

where, by a slight abuse of notation, we have denoted the fluctuation around the coordinateswith the same symbols as the coordinates. Here the dot and prime represent derivatives withrespect to τ and σ respectively. Notice that neither the coordinate Θ nor the Kalb-Ramond fieldterms contribute at quadratic order. The fact that Θ does not appear is simply a consequenceof the non-canonical kinetic term that it has, which would appear at higher orders in the fieldexpansion.46

The next step is to compute the equations of motion of the fluctuations. We will use the factthat we do not want to spoil the periodicity condition of our classical solution by consideringthe following ansatze for each of the coordinates

Xm =∑n

Am cos(Ωnτ + nσ) +Bm sin(Ωnτ + nσ) . (E.2)

After some algebra, the equations of motion become the system of linear equations Mv = 0 withv the vector given by v = (AZ AT AV AP BZ BT BV BP ) and M is the matrix

√η3

2β3 (n2 + 4a2Tβ

2 − Ω2) 0 0 0 0 −aT η2Ω −2aT ηΩβ 0

0 −1+β2

β2 η2(n2 − Ω2) −2ηβ (n2 − Ω2) 0

√23η3βaTΩ 0 0 0

0 −(n2 − Ω2) 0 0√

23βη aTΩ 0 0 0

0 0 0√

2η3

β3 (a2T (1 + β2) + n2 − Ω2) 0 0 0 0

0 atη2Ω 2aT ηΩ

β 0√

η3

2β3 (n2 + 4a2Tβ

2 − Ω2) 0 0 0

−√

23η3βaTΩ 0 0 0 0 −1+β2

β2 η2(n2 − Ω2) −2ηβ (n2 − Ω2) 0

−√

23βη aTΩ 0 0 0 0 −(n2 − Ω2) 0 0

0 0 0 0 0 0 0√

2η3

β3 (a2T (1 + β2) + n2 − Ω2)

.

(E.3)If we want this system of equations to have a non-trivial solution, we need the matrix of coeffi-cients to have vanishing determinant. This gives us the condition

(n2 − Ω2)2(n2 + a2T (1 + β2)− Ω2)2(n4 − 4a2

Tβ2n2 − 2(2a2

T + n2)Ω2 + Ω4)2 = 0 . (E.4)

We can check that the solutions for Ω to this equation, divided by QT = −aT , perfectly match(6.19), (6.20) and (6.21) up to a constant term. Additionally, we also get solutions of the formΩ = ±n. The modes associated to these are not physical, and are cancelled by conformalghosts [78].

Instead of repeating the same computation for the S5 sector of the Lagrangian, we can arguethat this part of the space is blind to the deformation and hence the final result has to be the

46If we want to do the counting of degrees of freedom correctly, we should change from P and Θ to Cartesiancoordinates. However, as we are only interested in independently checking our computations in appendix D, thiscomputation is sufficient.

51

same as the one for the undeformed background. In fact, we can check that

Ω =ω ±√ω2 + n2

QT= −

√1− β2 ∓

√1− β2 +

n2

a2T

, (E.5)

which perfectly matches (6.18).

The mismatch by a constant between the contributions computed using the quadratic fluctu-ations and the ones computed using the classical spectral curve may be uncomfortable. However,they are also present for undeformed AdS5 × S5 and arise due to each method describing theperturbations around the classical solution using a different frame of reference [49].

Even though these shifts have a non-physical origin, this does not mean at all that they areharmless, as they can give rise to ambiguities. To discuss those ambiguities, first we have todistinguish between two kinds of shifts that may appear: a constant shift of the full contribution(usually proportional to the energy or an angular momentum), and a shift of the mode number n(usually proportional to a winding number). The ones of the first kind are completely harmless,but the ones of the second kind give rise to an ambiguity. To see that, we can compare whatwould be the contribution to the one-loop energy we obtain with and without performing a shiftof the mode number. Although the sum

∑∞n=−∞ [2Ω(xn)− Ω(xn−m)− Ω(xn+m)] might initially

seem to vanish because the terms cancel each other after a relabelling, this is not entirely correctif we consider a (large enough) partial sum up to Λ. Using that Ω(xn) ≈ n for large n we seethat, up to subleading orders in Λ,

Λ∑n=−Λ

[2Ω(xn)− Ω(xn−m)− Ω(xn+m)] = 2Λ∑

n=Λ−m+1

[Ω(xn)− Ω(xn−m)] ≈ 2m . (E.6)

Luckily, we do not have shifts of this second kind in our problem. In fact, they are usuallyassociated to winding in the classical solution, which ours does not possess. We thereforeconclude that the final result for E1−loop is unambiguous.

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