Supersymmetry & String Theory

An Introduction

Johar M. Ashfaque

1 Origins

Attempts to understand the strong interactions gave birth to string theory. Empirical evidence forstring-like structure of hadrons comes from arranging mesons and baryons unto approximately linearRegge trajectories. Veneziano found the first and very simple expression for a manifestly dual 4-pointamplitude

A(s, t) ∼ Γ(−α(s))Γ(−α(t))

Γ(−α(s)− α(t))= B(−α(s),−α(t))

with an exactly linear Regge trajectory. Soon after Nambu, Nielsen and Susskind independently proposedits open string interpretation. This led to an explosion of interest in the early 70s in string theory adescription of strongly interacting particles. The idea is to think of a meson as an open string witha quark at one end-point and anti-quark at the other. Then various meson states arise as differentexcitations of the open string.

The string world dynamics is governed by the Nambu-Goto action area action

SNG = −T∫

dσdτ√−det ∂aXµ∂bXµ

where the indices a, b take two values ranging over σ and τ directions on the world sheet. The stringtension is related to the Regge slope by

T =1

2πα′.

The quantum consistency of the Veneziano model requires that the Regge intercept ia α(0) = 1 so thatspin-1 state is massless but spin-0 state is a tachyon. But the ρ meson is certainly not massless and thereis no tachyons in the real world. This is how the string theory of strong interactions started to run intoproblems.

Calculations of the string zero-point energy gives

α(0) =d− 2

24⇒ d = 26

meaning the model has to be defined in 26 space-time dimensions. Attempts to quantize such a stringmodel directly in 3+1 dimensions led to tachyons and problems with unitarity. Consistent supersymmetrictheories were discovered in 10 dimensions but their relation to the strong interactions was initially unclear.In fact, asymptotic freedom of strong interactions was discovered when QCD was singled out to be theexact field theory of strong interactions.

2 Shortcomings of Standard Model (SM) of Particle Physics

• quadratic divergences in scalar sector ⇒ fine-tuning

• explanation of masses

• origin of gauge symmetries and quantum numbers

• unification with gravity

• hierarchy problem

• cold dark matter - SM has no candidate

1

3 Motivations for Supersymmetry (SUSY)

• only possible extension of symmetry beyond Lie symmetries (Coleman-Mandula Theorem)

• correct unification of gauge couplings at MGUT

GUT assignment of quantum numbers (anomaly cancellation)

• Natural mechanism of EWSB, radiative symmetry breaking

• local SUSY enforces supergravity in String Theory

• SUSY has a weakly interacting massive particle (WIMP) if R-parity is conserved

4 Poincare Symmetry

Most important symmetry in relativistic QFT is

Poincare symmetry = Translation o Lorentz Transformation.

that is to sayxµ → xµ + aµ + ωµνx

ν = xµ + δxµ.

What are the properties?

Special relativity requires that space-time proper distance

∆s2 = ηµν∆xµ∆xν

is invariant.

Lorentz Generator⇒ ηµν

(∆xµ + ωµλ∆xλ

)(∆xν + ωνρ∆xρ

)= ∆s2 + (ωµν + ωνµ)∆xµ∆xν + ....

Clearly, ⇒ ωµν + ωνµ = 0 for ∆s2 to be invariant. Hence ⇒ ωµν = −ωνµ that is ω is anti-symmetric.

Concatenation of two Poincare generators should again be a Poincare generator

δ2δ1xµ = δ2(aµ1 + ωµ1 νx

ν)

= ωµ1 ν(δ2xν)

= ωµ1 ν(aν2 + ων2λxλ)

Similarly,δ1δ2x

µ = ωµ2 ν(aν1 + ων1λxλ).

The commutator of these is again a Poincare generator:

(δ1δ2 − δ2δ1)xµ = aµ + ωµλxλ,

where ω = [ω2, ω1] = ω2ω1 − ω1ω2 ∈ Lorentz Algebra.

More abstractly

δxµ = iaνPν(xµ) +i

2ωνλMµν(xµ)

where Pν is the translation generator and Mµν is the Lorentz generator and satisfy the Poincare algebra:

[Pµ, Pν ] = 0 ∀ µ, ν

[Pµ,Mνλ] = iηµνPλ − iηµλPν

[Mµν ,Mλρ] = iηνλMµρ + iηµρMνλ − iηµλMνρ − iηνρMµλ

2

4.1 Representations of the Poincare Algebra

There are two key representations the vector representation

(Mµν)λρ = iηµρδλν − iηνρδλµ

and the spinor representation

(Mµν)βα =i

4

(ΓµΓν − ΓνΓµ

)βα

where{Γµ,Γν} = 2ηµν .

5 The Coleman-Mandula No-Go Theorem

Can The Poincare algebra be non-trivially extended?

In any space-time dimension, D > 2, interacting quantum field theories have Lie algebra symmetries thatare

g× Poincare

where g is the Lie algebra generated by ta. This is to say that there is no Lie algebra that is a symmetryof interacting quantum field theories that is not a Lorentz scalar.

One key assumption of the Coleman-Mandula theorem is that the additional symmetry is a Lie algebrasymmetry. The No-Go theorem can be avoided by relaxing this assumption.

5.1 The Haag-Lopuszanski-Sohnius Theorem

In 1975, Haag, Lopuszanski and Sohnius presented their proof that by weakening the assumptions ofthe ColemanMandula theorem and allowing both commuting and anti-commuting symmetry generators,there is a non-trivial extension of the Poincare algebra, namely the supersymmetry (SUSY) algebra.

6 Graded Lie Algebras: Z2-Grading

Let g be a Lie algebra. Then g decomposes as

g = g0 ⊕ g1

where g0 represents even part and g1 represents the odd part.

For the linear map[ , ] : g× g→ g

we have

g0 × g0 → g0

g0 × g1 → g1

g1 × g0 → g1

g1 × g1 → g0

where it can be seen that the linear map on g0 acts as a commutator but on g1 acts as as ananti-commutator.

3

7 Conventions

ηµν = diag(+,−,−,−)

The left projection operator

PL =1

2

(1− γ5

)=

(1 00 0

)The right projection operator

PR =1

2

(1 + γ5

)=

(0 00 1

)Note. The states in one supermultiplet have the same mass.

Note. The number of bosonic degrees of freedom equals the number of fermionic degress of freedom ina supermultiplet, i.e.

# bosonic d.o.f = # fermionic d.o.f

Define F as an operator which counts the fermion number of a state. Then

(−1)F |boson〉 = +1|boson〉(−1)F |fermion〉 = −1|fermion〉

⇒ {(−1)F , Qα} = 0

where Qα is the Weyl spinor in D = 4.

Raising and lowering undotted and dotted indices can be done with the matrices:

εαβ = εαβ =

(0 1−1 0

)= iσ2

εαβ = εαβ =

(0 −11 0

)= −iσ2

satisfyingεγαεαλ = δγλ, εγαεαλ = δγ

λ.

Soχα = εαβχ

β

χα = εαβχβ

andψ†α = εαβψ†

β

ψ†α = εαβψ†β

For α a left chiral spinor and α a right chiral spinor and the vector µ, the Pauli matrices take the form

σµα α = (1, σi), σµα α = (1,−σi).

We have thatεψ = εαψα = εαεαβψ

β = ψβεβ = ψε

and similarly

ε†ψ† = εαψ†α = εαε

αβψ†β

= ψ†βεβ = ψ†ε†

4

7.0.1 Chiral Supermultiplet: Φ

Φ = (φ, ψα, F )

where

• φ is the complex scalar

• ψα is the Weyl fermion

• F is the auxiliary field (complex scalar)

7.0.2 Vector Supermultiplet: V

V = (ϑm, λα, D)

where

• ϑm is the spin-1 massless vector

• λα are the Weyl fermions (gauginos)

• D is the auxiliary field

7.1 Prefix: s-

The superpartners of Standard Model (SM) fermions are sfermions. Squarks are the superpartners ofquarks. Sleptons are the superpartners of leptons.

7.2 Suffix: -inos

The superpartners of Standard Model (SM) gauge bosons are the gauginos. Higgsino is the superpartnerof the Higgs boson.

8 N = 1 SUSY Algebra

Poincare +Qα, Q†α

where Qα and Q†α are Weyl spinors, left and right chiral respectively with

{Qα, Qβ} = 0 = {Q†α, Q†β}

{Qα, Q†α} = 2σµααPµ

where Pµ is the translation generator.

8.1 General Properties of Representations of this Algebra

The supersymmetric Hamiltonian operator is

P 0 = H =1

4(Q1Q1 +Q1Q1 +Q2Q2 +Q2Q2)

where Q1 and Q2 are the supersymmetric charges.

A sufficient condition for supersymmetry to be good is that vacuum should have zero energy whereinvariance of vacuum under a group of transformations is equivalent to the corresponding symmetrybeing unbroken.

5

• Bosonic and fermionic states in a supermultiplet (representation of N = 1 SUSY algebra) havesame mass

• States in a supermultiplet carry the same initial indices

•# bosonic degrees of freedom = # fermionic degrees of freedom

in a supermultiplet

• Vacuum: If SUSY is unbroken thenQα|0〉 = 0.

This has some slightly awkward implication:

Consider |0〉 such thatQα|0〉 6= 0

which means that SUSY is spontaneously broken which implies that vacuum has positive energy.

9 The Free Wess-Zumino Model: Theory of Chiral Multiplet

Let φ be a complex scalar and ψα be the Weyl fermion. The action contains only kinetic terms for φ andψα

S =

∫d4x (Lf + Ls)

where

Lf = ∂µφ∗∂µφ

Ls = iψ†σµ∂µψ

9.1 SUSY Transformations

fermions↔ bosons

δφ = εψ, δφ∗ = ε†ψ†

where ε is a constant (anti-commuting Grassmann variable),

⇒ δLs = ∂µδφ∗∂µφ+ ∂µφ∗∂µδφ

= ∂µ(ε†ψ†)∂µφ+ ∂µφ∗∂µεψ

= ε†∂µψ†∂µφ+ ε∂µφ∗∂µψ

δψα = −i(σνε†)α∂νφ, δψ†α = i(εσν)α∂νφ∗

⇒ δLf = iδψ†σµ∂µψ + iψ†σµ∂µδψ

= −εσν∂νφ∗σµ∂µψ + ψ†σµσνε†∂µ∂νφ

= −ε∂µψ∂µφ∗ − ε†∂µψ†∂µφ+ ∂µ

(εσµσνψ∂νφ

∗ − εψ∂µφ∗ + ε†ψ†∂µφ

)= −δLs + ∂µ

(εσµσνψ∂νφ

∗ − εψ∂µφ∗ + ε†ψ†∂µφ

)

where ∂µ

(εσµσνψ∂νφ

∗ − εψ∂µφ∗ + ε†ψ†∂µφ

)is the total derivative.

Note. The SUSY algebra closes for off-shell fermions and on-shell fermions.

6

10 The Interacting Wess-Zumino Model

φ, φ∗, ψ, ψ†, F and F ∗ were free so far.

The couplings considered must be renormalizable:

φ ψ F∫

d4x S L

1 32 2 -4 0 4

11 Supersymmetry Breaking

There is clearly a need for supersymmetry ot be broken in realistic models since we do not see scalarparticles accompanied by fermions degenerate in mass with them nor vice versa. The criterion forspontaneous supersymmetry breaking is that the physical vacuum state |0〉 should not be invariant underthe supersymmetry transformation.

The supersymmetric Hamiltonian operator is

P 0 = H =1

4(Q1Q1 +Q1Q1 +Q2Q2 +Q2Q2)

where Q1 and Q2 are the supersymmetric charges.

A sufficient condition for supersymmetry to be good is that vacuum should have zero energy whereinvariance of vacuum under a group of transformations is equivalent to the corresponding symmetrybeing unbroken.

The scalar potential, V , of the Hamiltonian H is given by

V = F ∗i Fi +1

2

∑DaDa.

It can be better expressed asV = VF + VD

where

VF = F ∗i Fi =∑n

∣∣∣∣ ∂W∂Φn

∣∣∣∣2where the sum runs over all the scalar fields Φn present in the theory and

VD =1

2g2(∑

n

qn|Φn|2)2

where qn is the charge of Φn under the U(1) symmetry.

Once a supersymmetry breaking takes place a massless Goldstone fermion, the goldstino, is expected dueto the supersymmetry generator being fermionic.

Note. Any spontaneous SUSY breaking theory has a tightly constrained mass spectrum.

12 D-Term SUSY Breaking: The Fayet-Iliopoulos Model

• U(1) vector superfield(Aµ, λ,D)

• use a non-zero D-term for U(1) gauge group

• idea is to add a term linear in auxiliary field to the theory with κ which is a constant parameterwith dimensions of mass

7

• Fayet-Iliopoulos term only invariant for abelian ideals

Note. D-term is not suitable to generate masses for the gauginos.

13 F -Term SUSY Breaking: The O’Raifeartaigh Model

The F -term is related to a function called the superpotential W(Φ1, ...,Φn) which much be a holomorphicfunction of order at most three(for renormalizability reasons)in the complex scalar fields Φi

The simplest example of a model without any supersymmetric minima is the O’Raifeartaigh model whichhas three chiral superfields Φ1, Φ2 and Φ3 with superpotential

W(Φ1,Φ2,Φ3) = λ1Φ1(Φ23 −M2) + µΦ2Φ3.

For this model there is no solution with F1, F2 and F3 all zero since

F ∗1 = −∂W∂φ1

⇒M2 = φ23

F ∗2 = −∂W∂φ2

⇒ µφ3 = 0

F ∗3 = −∂W∂φ3

⇒ 2λ1φ1φ3 + µφ2 = 0

As a consequence supersymmetry is spontaneously broken. The effective potential is given by

V =

3∑i=1

|Fi|2 = λ21|φ23 −M2|2 + µ2|φ3|2 + |µφ2 + 2λ1φ1φ3|2.

The absolute minimum of this effective potential occurs at

〈φ2〉 = 〈φ3〉 = 0

with 〈φ1〉 is undetermined that is to say that the potential has a flat direction. At this absolute minimum

F ∗1 = λ1M2, F ∗2 = F ∗3 = 0

andV = λ21M

4 > 0.

Since F 1 is non-zero, we expect ψ1 to be the Goldstino, which is a spinor in the chiral supermultiplet inΦ1 to which F1 belongs.

14 The Minimal Supersymmetric Standard Model: MSSM

14.1 Properties of MSSM

• Requires 2 Higgs doublets: Hu, Hd where

Hu =

(H+u

H0u

)

Hd =

(H0d

H−d

)These will generate masses via Yukawa couplings.

8

• Superpotential couplings for Higgs sector:

W = µHuHd = µ(H+u H

−d H

0uH

0d) SUSY mass term for Higgsinos

⇒ Lµ

∣∣∣∣quadratic

= −µ(...)− µ2

(|H0

u|2 + |H+u |2 + |H0

d |2 + |H−d |2

).

The µ problem is about the scale of the µ2 term appearing in the Lµ

∣∣∣∣quadratic

. Natural electroweak

symmetry breaking (EWSB) requires µ ∼ O(100GeV).

• Discrete symmetry: R-parity(−1)R(known particle) = 1

(−1)R(superpartners) = −1

15 Dynamical Supersymmetry Breaking

The mass sum rule forbids supersymmetry breaking directly in the MSSM otherwise some superpartnerswould have been observed. In other words, there is no gauge singlet for F -term breaking and D-termsupersymmetry breaking leads to an unacceptable spectrum.

Way Out. Supersymmetry breaking takes place in the hidden non-SM sector of the full theory andmediated through messenger fields to SM fields via some interaction.

There are three types of supersymmetry breaking in the MSSM

• Gauge Mediated Supersymmetry Breaking,

• Gravity Mediated Supersymmetry Breaking,

• Anomaly Mediated Supersymmetry Breaking.

16 A Note on R-Symmetry

The idea that there is a U(N) symmetry which rotates the supercharges amongst themselves. In N = 2SUSY, we can decompose

U(2)R = SU(2)R × U(1)R

where U(1)R acts on the anti-commuting superspace coordinates as

θI → eiαθI , θI → e−iαθI

for I = 1, 2 and SU(2)R symmetry subgroup rotates the index I of the supercharges.

In the case of N = 1 SUSY, R-symmetry is the discrete Z2 symmetry known best as R-parity which forSM particle content is +1 while supersymmetric particles have R-parity of 1.

17 The Bosonic StringThe Nambu-Goto & Polyakov Action

The world-sheet has coordinates ξα, α = 0, 1 with ξα = (τ, σ) embedded in a D-dimensional Minkowskispace-time according to xµ(ξα), µ = 0, ..., D − 1.

For the open string, take the range of σ0 ≤ σ ≤ π

while for the closed string−π < σ ≤ π.

9

Note. In the case of the closed string, it is natural to take the boundary condition

xµ(τ,−π) = xµ(τ, π)

as σ = −π and σ = π are the same point on the string.

17.1 The Nambu-Goto Action

A = − 1

2πα′

∫d2ξ√−det{∂αxµ∂bxνηµν}

where

∂αxµ =

∂xµ

∂ξα.

The dimension of xµ is [mass]−1. The Regge slope parameter α′ has dimension [mass]−2 in order for theaction to be dimensionless.

The Nambu-Goto action is invariant under

• Reparametrization Invariance

• Rigid Poincare Symmetry:xµ′(ξ) = Λµνx

ν(ξ) + aµ

and Lorentz transformations Λµν satisfy

ηµνΛµρΛντ = ηρτ .

17.2 The Polyakov Action:Classically Equivalent To The Nambu-Goto Action

An action like the Nambu-Goto action with a square root is not easy to handle especially when it comesto quantization.

Fortunately, by introducing the independent field gαβ , there exists another reparametrization invariantaction.

A = − 1

4πα′

∫d2ξ√−ggαβ∂αxµ∂βxνηµν

where gαβ can be identified as the two-dimensional metric on the world-sheet and g ≡ det gαβ .

Let us now discuss the symmetries of the Polyakov action:

• Poincare Invariance

δXµ = aµνXν + bµ, (aµν = −aνµ

δhαβ = 0

• Local Symmetries

– Reparametrization Invariance

δXµ = ξa∂aXµ

δhαβ = ξγ∂γhαβ + ∂aξγhγβ + ∂bξ

γhαγ

= ∇aξb +∇bξaδ√h = ∂a(ξa

√h)

10

– Weyl Scaling

δhαβ = 2Λhαβ

δXµ = 0

One immediately important consequence of the Weyl invariance of the Polyakov action is the vanishingthe energy-momentum tensor

hαβTαβ = 0.

17.3 The Bosonic Point Particle: An Equivalent Action

In D-dimensional Minkowski space-time with coordinates xµ, µ = 0, ..., D − 1, the point particle sweepsout a one-dimensional curve known as the world line. We express this world line as xµ(τ). The motionof the point particle is taken defined by the action

A = −m∫dτ√−xµxνηµν

where

xµ ≡ dxµ

dτ

and ηµν = (−1,+1,+1, ...,+1) is the Minkowski metric.

A classically equivalent way of the writing the action is

A =1

2

∫dτ{e−1xµxνηµν −m2e}

where xµ and e are independent fields. This action can be seen to be equivalent by taking computing theequation of motion for e gives

−e−2xµxνηµν −m2 = 0⇒ e−2xµxνηµν +m2 = 0⇒ xµxνηµν = −m2e2

which upon substituting back into the action yields

A =1

2

∫(−em2 − em2)dτ = −m

∫dτ√−xµxνηµν .

18 Superstrings

18.1 The Superstring Action: The Basics

In string theory, the string propagates in D-dimensional flat Minkowski spacetime. As this string movesthrough spacetime it traces out a worldsheet, which is a two-dimensional surface in spacetime. The pointson the worldsheet are parametrized by the two coordinates σ0 = τ , which is time-like and σ1 = σ, whichis spacelike. Therefore the action is given by

S = − 1

4πα′

∫d2σ

(∂αXµ∂

αXµ + iψµρα∂αψµ

)(1)

where Xµ = Xµ(σ, τ) and ψµ = ψµ(σ, τ) are the two-dimensional bosonic and fermionic fields respectivelyand µ = 0, 1, . . . , D − 1.Furthermore, the term ρα is a 2-dimensional Dirac matrix with α = 0, 1, which we can choose to be aMajorana representation, where a convenient choice will be

ρ0 =

(0 −11 0

), ρ1 =

(0 11 0

). (2)

11

These matrices can also be seen to satisfy the Clifford algebra. Moreover taking this choice of ρα, wededuce that

ψ = ψT ρ0. (3)

The equation of motions coming from the action is given as

∂α∂αXµ = 0 Wave equation (4)

ρα∂αψµ = 0 Dirac equation (5)

In addition, the action S is given to be globally invariant under the infinitesimal worldsheet supersym-metric transformations

δXµ = iεψµ , δψµ = ρα∂αXµε (6)

where ε is a constant infinitesimal Majorana spinor of the Grassmann nature.

18.2 Constraints and Light-Cone Coordinates

The constraint equations are the equations produced by the conserved current and supercurrent. Theconserved current is the energy momentum tensor given by the action’s worldsheet translation invariance.This is just a local infinitesimal transformation given by

δXµ = εα∂αXµ , δψµ = εα∂αψ

µ. (7)

Here εα is the infinitesimal translation. So the energy momentum tensor reads

Tαβ = ∂αXµ∂βXµ +

i

4ψµρα∂βψµ +

i

4ψµρβ∂αψµ −

1

2ηαβ

(∂δXµ∂δXµ +

i

2ψµγδ∂δψµ

). (8)

The conserved supercurrent is given by the supersymmetric property of the action. This follows from thelocal infinitesimal transformations

δXµ = iεψµ , δψµ = ρα∂αXµε (9)

where ε is non-constant. So now the supercurrent is given by

Jµα = −1

2ρβραψ

µ∂βXµ. (10)

Also both the conserved current and supercurrent have an interesting properly, which is useful whenquantizing the theory, that is

Tαβ = 0 (11)

Jα = 0. (12)

Quantizing string theory can be quite difficult with normal coordinates, so for convenience we to turn tolight-cone gauge coordinates. These are given by

σ+ = τ + σ , σ− = τ − σ. (13)

So now the Dirac equation becomes a two spinor components, that is

∂+ψµ− = ∂−ψ

µ+ = 0 (14)

where the Majorana field ψµ− describes right movers, while the Majorana field ψµ+ describes left movers.Also we can see that the left- and right-movers decoupled. Similarly the wave equation becomes

∂+∂−Xµ = 0. (15)

The constraint equations in this case is given by

T±± = ∂±Xµ∂±Xµ +

i

2ψµρ±∂±ψµ = 0 (16)

J± = ψµ±∂±Xµ = 0. (17)

12

18.3 Closed Strings and Mode Expansions

In closed strings, the boundary conditions are periodic or anti-periodic for the fermionic fields. Thereforethe boundary term vanishes when

ψ±(τ, σ) = ψ±(τ, σ + π) periodic boundary condition

ψ±(τ, σ) = −ψ±(τ, σ + π) anti-periodic boundary condition (18)

Where π is the length of the string. Furthermore the mode expansions are given by

ψµ±(τ, σ) =∑n∈Z

dµne−2in(σ±) (19)

ψµ±(τ, σ) =∑

r∈Z+ 12

bµr e−2ir(σ±). (20)

Similarly for the bosonic field, the boundary term vanishes when

X(τ, σ) = X(τ, σ + π) (21)

And the mode expansion is given as

Xµ(τ, σ) = xµ + α′pµτ + i

√α′

2

∑n 6=0

(aµnne−inσ

+

+aµnne−inσ

−). (22)

18.4 Quantization

Here we promote the fields Xµ and ψµ to operators Xµ and ψµ respectively. However we shall drop thehats for convenience. Also we will only consider fermions here, since we will be focusing on free-fermionicmodel building. The fermion anti-commutator relation is given by

{ψµA(τ, σ), ψνB(τ, σ′)} = πδ(σ − σ′)ηµνδA,B (23)

where A and B are the spin indices, ±. Using the above relation we deduce that the oscillators satisfies

{dµn, dνm} = ηµνδn+m,0 , {bµr , bνs} = ηµνδr+s,0

{dµn, dµm} = ηµνδn+m,0 , {bµr , bνs} = ηµνδr+s,0 (24)

where n,m ∈ Z and r,s ∈ Z + 12 . Now in order to get the spectrum of states the oscillators are acted on

the vacuum, so the anti-commutator relations in (2.4.2) become useful.Without going into detail the conserved current and supercurrent similarly to (2.4.2) result in the super-Virasoro algebra. Furthermore this results in the constraint

Ln|φ〉 = Ln|φ〉 = 0 n > 0 (25)

(L0 − a) |φ〉 = (L0 − a)|φ〉 = 0. (26)

Here |φ〉 is a physical state, a is a normal ordering constant and Ln is given as either

Ln =1

2

∞∑m=−∞

: an−man : +1

2

∞∑m=−∞

m : dn−mdn : (Periodic Modes) (27)

Or

Ln =1

2

∞∑m=−∞

: an−man : +1

2

∞∑r=−∞

r : bn−rbn : (Anti-Periodic Modes) (28)

where the semi-colons indicate normal ordering. Furthermore, the theory requires absence of negative-norm states, so this leads to specific normal ordering constants. In the case of periodic modes we havea = 0 and similarly for anti-periodic modes we get a = 1

2 . However both the value of the normal orderingconstants holds when D = 10. Also with bosonic strings, when D = 26 we get a single constant given asa = 1.

13

18.5 The Spectrum of States

In order to examine the spectrum of states, the number operator N is introduced. For the periodic sectorthis is given as

N =

∞∑n=1

ηµνaµ−na

νn +

∞∑m=1

mdµ−mdνmηµν

= Na +Nd. (29)

Similarly for the anti-periodic sector it is given as

N =

∞∑n=1

ηµνaµ−na

νn +

∞∑r= 1

2

rbµ−rbνrηµν

= Na +N b. (30)

Now using the constraint (L0 − a) |φ〉 = 0 and working with anti-periodic sector i.e.(L0 − 1

2

)|φ〉 = 0.

The L0 − 12 operator expansion in terms of oscillators gives

L0 −1

2=

1

2

∑n

: aµ−naνn : ηµν +

1

2

∑r

r : bµ−rbνr : ηµν −

1

2

=1

2aµ0a

ν0ηµν +

1

2

∞∑n=1

aµ−naνnηµν +

1

2

∞∑r= 1

2

rbµ−rbνrηµν −

1

2

=1

4α′pµpµ +N − 1

2(31)

where aµ0 =√

α′

2 pµ for closed strings. Moreover comparing (2.5.3) with the Klein-Gordan equation, the

spacetime mass-squared of a physical state is the eigenvalue of the operator

M2 =4

α′

(N − 1

2

)(32)

Similarly for the periodic sector, the spacetime mass-squared of a physical state is an eigenvalue of

M2 =1

α′N. (33)

18.6 Heterotic Strings and Compactification

The discussion up till now was on based on N = 2 world-sheet supersymmetry. However, closed stringsallow the supersymmetric transformations to decouple in the left and right-movers. For instance theinfinitesimal transformations become

δψµ− = −2∂−Xµε+ , δψµ+ = −2∂+X

µε−. (34)

Furthermore, this corresponds to two different supersymmetric transformations. So it is now possibleto get the N = 2 world-sheet supersymmetry and express it as an N = 1 world-sheet supersymmetryon the left-movers and on the right-movers. The idea here is that we can construct a model wherethe string has a N = 1 world-sheet supersymmetry on the left-movers and no supersymmetry on theright-movers, which will at the end generate a N = 1 space-time supersymmetry. Until now we workedon superstrings, but all the results apply to a non-supersymmetric strings (bosonic strings). This couldseem quite weird at first glance because cancelling the conformal anomaly requires that the space-timedimension for right-movers is 26 and 10 for the left-movers, although one would expect to have the samespace-time dimension. However, using compactification, we can reduce the two number of dimensionsto the same value, which can be taken as equal to four in order to match with our real 4-dimensional

14

space-time.Since the left-movers and right-movers decouple, the model can have more fermions added to it so that itwould be only right or left moving. The aim is that they would contribute to cancel the central charge,and thus reduce the space-time critical dimension. So if 44 right-moving fermions and 18 left-movingfermions are added, the conformal anomaly would become:

CL = −26 + 11 +DL +DL

2+

18

2(35)

CR = −26 +DR +44

2(36)

where DL and DR are the left and right space-time dimensions respectively. Moreover if DL = DR = 4the theory is conformally invariant.The action for the heterotic string in the fermionic formulation is

S =1

π

∫d2σ(2∂+Xµ∂−X

µ + i

18∑A=1

λA−∂+λA− + ψµ + i

44∑A=1

λA+∂−λA+). (37)

This action has a SO(18)L × SO(44)R global symmetry under which the internal fermions λA transformin the fundamental representation. The next step is to compute the one-loop partition function. This iscalled the free-fermionic construction.

18.7 The Partition Function

The partition function includes all the physical states and is sufficient to derive some constraints on themodel. It is an integration over all the possible world-sheets, in the case of the one-loop partition functionthe world-sheets are a torus. On the world-sheets, two boundary conditions need to be specified for thetwo non-contractible loops of the torus (poloidal and toroidal) for each of the free-fermionic fields. Theseconditions express the shifts in the phase of the fermionic fields under parallel transport around thesenon-contractible loops

f → −e− iπα(f)f (38)

where f is the fermionic field and α is the boundary condition of f . In other words, the fermionswhich propagate around the string have a boundary condition around the string in the direction of theσ coordinate and they also can pick up a phase by propagating along the τ dimension.The partition function is given as

Z =∑α,β

C

(αβ

)Z[α, β] (39)

where α and β are the boundary conditions (α, β = 1 denote the periodic boundary conditions andα, β = −1 denote the anti-periodic). Also any α or β corresponds to a space boundary condition and theother is the time. Furthermore, the partition function has 68 fields in total made of 64 internal fermionsand two each for Xµ

L, XµR and ψµ. The bosonic fields have no choice of boundary conditions, they are

only periodic. However the fermionic field ψµ can be periodic or anti-periodic, so the partition functionmust include all possible combinations of 64 boundary conditions of the fermions, and this is integratedover all the inequivalent tori. Thus, the boundary conditions take the values α, β = 1, ..., 64.

19 Dirac Quantization Condition By The Argument Of WuAnd Yang

Consider a source of magnetic field around which a 2-sphere is taken. The magnetic charge, g, given bythe magnetic flux leaving the 2-sphere is given by the formula

g =

∫S2

B · dS

We require two patches to cover S2, say S2N and S2

S to be taken around the North and South polesrespectively and overlapping at the equator. The magnetic charge is in the interior of the 2-sphere,

15

∇ · B = 0 on the two patches and so B can be expressed in terms of the vector gauge potential on eachpatch.

Let AN and AS be the vector gauge potentials on the two patches respectively. The magnetic field isglobally defined on the 2-sphere and therefore on the overlap the two vector gauge potentials must differby a gauge transformation with parameter say χ. Thus

intS2B · dS =

∫S2N

B · dS +

∫S2S

B · dS

=

∫S1

(AN −AS)dx

=

∫S1

∇chi dx

= χ(φ = 2π)− χ(φ = 0)

Note. If there was a globally defined vector gauge potential on S2 then

AN = AS

there would be no magnetic charge as the integral would vanish.

The wave-function must be single-valued on each patch and so around the equator but under the gaugetransformation

ψ → eiqχ~ ψ

This can only hold if

eiqχ(φ=0)

~ = eiqχ(φ=2π)

~

and as a result

χ(φ = 2π)− χ(φ = 0) = 2πm~q, m ∈ Z.

Hence

g =

∫S2

B · dS = χ(φ = 2π)− χ(φ = 0) = 2πm~q⇒ qg = 2πm~, m ∈ Z.

20 Superbranes

A superbrane can be viewed as a p+ 1 dimensional bosonic submanifold M with coordinates ξα

α = 0, 1, ..., p which moves the background superspace with coordinates

zN = (xµ,Θ)

where xµ is Grassmann even and Θ is Grassmann odd.

A simple superbrane has dynamics specified by

zN (ξα).

It moves to extremize the actionA = A1 +A2

where

A1 = −T∫dp+1ξ

√−detgαβ

is the Brink-Schwarz action andgαβ = ∂αz

N∂βzMgNM

where gNM is a metric in the background superspace and

A2 = q

∫dp+1ξεα1...αp+1∂α1

xµ1 ...∂αp+1xµp+1Aµ1...µp+1

+ ...

16

where Aµ1...µp+1 is the background form field. Every p + 1-rank background gauge field comes withD − p− 3-rank dual gauge field.

For the superbrane, q is always present and is fixed in terms of T .

In 10 dimensions there are 2 possible maximal supergravities, whereas in 11 there is only one.

A superbrane has world-volume supersymmetry which implies that there are equal numbers of bosonsand fermions on shell. There are D− p− 1 scalar degrees of freedom and 8 fermionic degrees of freedom.

When D = 10, we have8 = 10− p− 1⇒ p = 1.

Therefore, we have a string.

When D = 11, we have8 = 11− p− 1⇒ p = 2.

Therefore, we have a 2-brane or a membrane.

21 D-Branes & Type II Superstrings

D-branes are considered which are more precisely known as Dp-branes. D-branes are p-branes whichsatisfy von Neumann and Dirichlet boundary conditions on which ends of open strings are localized.

The von Neumann boundary condition is

∂σXµ, σ = 0, π

where the Dirichlet boundary condition is

δXµ = 0, σ = 0, π

where the end points of the string lie in some constant position in space.

More concretely

∂σXµ, a = 0, ..., p

XI = cI , I = p+ 1, ..., D − 1

Note. p denotes the dimension and D is for Dirichlet. D0-brane is the point particle, D1-brane is astring.

22 Modular Invariance & N = 1 SUSY

String theories are generally formulated in more than four dimensions and to make contact with the4-dimensional world, many different compactification schemes have been proposed where the basic ideais to express space-time as a manifold of the form

M10 = M4 ×K

where K is a compact 6-dimensional manifold. This idea can be taken further as our goal is to cancelthe Weyl anomaly to obtain a consistent theory. What we really want is to write down a decompositionof the form

CFTc=26 = CFTc=4 ⊕ CFTinternal

where the left hand side has been assumed to be bosonic.

For phenomenological reasons we are interested in the string theories based on heterotic strings anddemand N = 1 space-time SUSY. However, we immediately run into two problems on trying to writedown the modular invariant partition function for such a theory.

17

The first of these is that the only possible way of obtaining modular invariance in CFT is to have aleft-right symmetric spectrum that is an equal number of left movers and right movers in the spectrum.The other is that by demanding N = 1 SUSY, we don’t have a choice of modification of the left-movingsector as the space-time SUSY will arise from this very sector.

22.1 N = 2 SCFT & The Partition Function

In order to obtain N = 1 SUSY, we must begin from N = 2 world-sheet SUSY. Assume that left-movingsector has N = 2 SUSY. The relevant fields are the Laurent modes of the energy-momentum tensor

T (z) =∑

Lnz−n−2.

T (z) also has two fermionic superpartners

G± =∑

G±n z−n− 3

2 .

There is also a U(1) current

J(z) =∑

Jnz−n−1.

The values of the indices, in this case n, depend on the boundary conditions assumed for the superpartners

G±(exp(2πi)z) = exp(±2πiη)G±(z)

where η = 0 is NS and η = 12 is Ramond but 0 ≤ η ≤ 1 will be consistent.

The N = 2 SCFT is therefore defined by

[Lm, Ln] = (m− n)Lm+n +c

12(m3 −m)δm+n,0 (40)

[Lm, Jn] = −nJm+n (41)

[Lm, G±r ] =

(m

2− r)G±m+r (42)

[Jm, Jn] =c

3mδm+n,0 (43)

[Jm, G±r ] = ±G±m+r (44)

{G+r , G

−s } = 2Lr+s + (r − s)Jr+s +

c

3

(r2 − 1

4

)δr+s,0 (45)

{G+r , G

+s } = {G−r , G−s } = 0 (46)

(47)

The first equation is the usual Virasoro algebra for the Laurent modes of the energy-momentum tensor.The second and third commutation relations simply mean that G±(z) and J(z) are primary fields of theVirasoro algebra with dimensions 3

2 and 1 respectively. The fourth one implies that J(z) is a free bosonU(1) current. The Jns are the usual bosonic creation and annihilation operators. The fifth equationimplies that G±(z) have U(1) charges ±1. The last two relations are the new ones involving super-energy-momentum tensors. The appearing indices take integer values except r and s which take valuesin Z + 1

2 ± η.

18