Supersymmetry and String Theory

Page 1

Supersymmetry & String Theory An Introduction Johar M. Ashfaque

1

Origins

Attempts to understand the strong interactions gave birth to string theory. Empirical evidence for string-like structure of hadrons comes from arranging mesons and baryons unto approximately linear Regge trajectories. Veneziano found the first and very simple expression for a manifestly dual 4-point amplitude Γ(−α(s))Γ(−α(t)) = B(−α(s), −α(t)) A(s, t) ∼ Γ(−α(s) − α(t)) with an exactly linear Regge trajectory. Soon after Nambu, Nielsen and Susskind independently proposed its open string interpretation. This led to an explosion of interest in the early 70s in string theory a description of strongly interacting particles. The idea is to think of a meson as an open string with a quark at one end-point and anti-quark at the other. Then various meson states arise as different excitations of the open string. The string world dynamics is governed by the Nambu-Goto action area action Z p SN G = −T dσdτ − det ∂a X µ ∂b Xµ where the indices a, b take two values ranging over σ and τ directions on the world sheet. The string tension is related to the Regge slope by 1 . T = 2πα0 The quantum consistency of the Veneziano model requires that the Regge intercept ia α(0) = 1 so that spin-1 state is massless but spin-0 state is a tachyon. But the ρ meson is certainly not massless and there is no tachyons in the real world. This is how the string theory of strong interactions started to run into problems. Calculations of the string zero-point energy gives d−2 ⇒ d = 26 24 meaning the model has to be defined in 26 space-time dimensions. Attempts to quantize such a string model directly in 3+1 dimensions led to tachyons and problems with unitarity. Consistent supersymmetric theories 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 the exact field theory of strong interactions. α(0) =

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 MGU T 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

Poincaré Symmetry

Most important symmetry in relativistic QFT is Poincaré symmetry = Translation o Lorentz Transformation. that is to say xµ → xµ + aµ + ωνµ xν = xµ + δxµ . What are the properties? Special relativity requires that space-time proper distance ∆s2 = ηµν ∆xµ ∆xν is invariant.

µ

Lorentz Generator ⇒ ηµν ∆x +

ωλµ ∆xλ

ν

∆x +

ωρν ∆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 Poincaré generators should again be a Poincaré generator δ 2 δ 1 xµ

= δ2 (aµ1 + ω1µν xν ) = ω1µν (δ2 xν ) = ω1µν (aν2 + ω2ν λ xλ )

Similarly, δ1 δ2 xµ = ω2µν (aν1 + ω1ν λ xλ ). The commutator of these is again a Poincaré generator: (δ1 δ2 − δ2 δ1 )xµ = âµ + ω̂λµ xλ , where ω̂ = [ω2 , ω1 ] = ω2 ω1 − ω1 ω2 ∈ Lorentz Algebra. More abstractly i δxµ = iaν Pν (xµ ) + ω νλ Mµν (xµ ) 2 where Pν is the translation generator and Mµν is the Lorentz generator and satisfy the Poincaré algebra: [Pµ , Pν ] = 0

∀ µ, ν

[Pµ , Mνλ ] = iηµν Pλ − iηµλ Pν [Mµν , Mλρ ] = iηνλ Mµρ + iηµρ Mνλ − iηµλ Mνρ − iηνρ Mµλ 2

4.1

Representations of the Poincaré Algebra

There are two key representations the vector representation (Mµν )λρ = iηµρ δνλ − iηνρ δµλ and the spinor representation (Mµν )βα =

β i Γµ Γν − Γν Γµ 4 α

where {Γµ , Γν } = 2ηµν .

5

The Coleman-Mandula No-Go Theorem

Can The Poincaré algebra be non-trivially extended? In any space-time dimension, D > 2, interacting quantum field theories have Lie algebra symmetries that are g × Poincaré where g is the Lie algebra generated by ta . This is to say that there is no Lie algebra that is a symmetry of 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 algebra symmetry. The No-Go theorem can be avoided by relaxing this assumption.

5.1

The Haag-Lopuszanski-Sohnius Theorem

In 1975, Haag, Lopuszański and Sohnius presented their proof that by weakening the assumptions of the ColemanMandula theorem and allowing both commuting and anti-commuting symmetry generators, there is a non-trivial extension of the Poincaré 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 an anti-commutator.

3

7

Conventions η µν = diag(+, −, −, −)

The left projection operator PL =

1 1 0 1 − γ5 = 0 0 2

PR =

1 0 1 + γ5 = 0 2

The right projection operator 0

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 in a 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 |bosoni = F

(−1) |fermioni =

+1|bosoni −1|fermioni

⇒ {(−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: 0 1 αβ = α̇β̇ = = iσ 2 −1 0

αβ

=

α̇β̇

=

0 1

−1 0

= −iσ 2

satisfying γα αλ = δλγ ,

γ̇ α̇ α̇λ̇ = δλ̇γ̇ .

So χα = αβ χβ χα = αβ χβ and ψ †α̇ = α̇β̇ ψβ̇†ψα̇†= α̇β̇ ψ †β̇ For α a left chiral spinor and α̇ a right chiral spinor and the vector µ, the Pauli matrices take the form σ µα α̇ = (1, −σ i ).

Ïƒαµ α̇ = (1, σ i ), We have that

εψ = εα ψα = εα αβ ψ β = ψ β εβ = ψε and similarly ε†ψ †= εα̇ ψ †α̇ = εα̇ α̇β̇ ψβ̇†= ψβ̇†εβ̇ = ψ †εâ€

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 of quarks. Sleptons are the superpartners of leptons.

7.2

Suffix: -inos

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

8

N = 1 SUSY Algebra Poincaré + Qα , Q†α̇

where Qα and Q†α̇ are Weyl spinors, left and right chiral respectively with {Qα , Qβ }

=

0 = {Q†α̇ , Q†β̇ }

{Qα , Q†α̇ }

=

2σαµα̇ Pµ

where Pµ is the translation generator.

8.1

General Properties of Representations of this Algebra

The supersymmetric Hamiltonian operator is P0 = H =

1 (Q Q1 + Q1 Q1 + Q2 Q2 + Q2 Q2 ) 4 1

where Q1 and Q2 are the supersymmetric charges. A sufficient condition for supersymmetry to be good is that vacuum should have zero energy where invariance of vacuum under a group of transformations is equivalent to the corresponding symmetry being unbroken.

5

• Bosonic and fermionic states in a supermultiplet (representation of N = 1 SUSY algebra) have same 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 then Qα |0i = 0. This has some slightly awkward implication: Consider |0i such that Qα |0i = 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 and ψα Z S = d4 x (Lf + Ls ) where

9.1

Lf

= ∂ µ φ∗ ∂ µ φ

Ls

= iψ †σ µ ∂µ ψ

SUSY Transformations fermions ↔ bosons δφ∗ = ε†ψ â€

δφ = εψ,

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

δLs

=

∂ µ δφ∗ ∂µ φ + ∂ µ φ∗ ∂µ δφ

=

∂ µ (ε†ψ †)∂µ φ + ∂ µ φ∗ ∂µ εψ

=

ε†∂ µ ψ †∂µ φ + ε∂ µ φ∗ ∂µ ψ

δψα̇†= i( σ ν )α̇ ∂ν φ∗

δψα = −i(σ ν †)α ∂ν φ, ⇒

δLf

= iδψ †σ µ ∂µ ψ + iψ †σ µ ∂µ δψ = −εσ ν ∂ν φ∗ σ µ ∂µ ψ + ψ †σ µ σ ν ε†∂µ ∂ν φ = −ε∂ µ ψ∂µ φ∗ − ε†∂ µ ψ †∂µ φ + ∂µ εσ µ σ ν ψ∂ν φ∗ − εψ∂ µ φ∗ + ε†ψ †∂ µ φ = −δLs + ∂µ εσ µ σ ν ψ∂ν φ∗ − εψ∂ µ φ∗ + ε†ψ †∂ µ φ

where ∂µ εσ µ σ ν ψ∂ν φ∗ − εψ∂ µ φ∗ + ε†ψ †∂ µ φ 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:

11

φ

ψ

F

1

3 2

2

R

d4 x

S

L

-4

0

4

Supersymmetry Breaking

There is clearly a need for supersymmetry ot be broken in realistic models since we do not see scalar particles accompanied by fermions degenerate in mass with them nor vice versa. The criterion for spontaneous supersymmetry breaking is that the physical vacuum state |0i should not be invariant under the supersymmetry transformation. The supersymmetric Hamiltonian operator is P0 = H =

1 (Q Q1 + Q1 Q1 + Q2 Q2 + Q2 Q2 ) 4 1

where Q1 and Q2 are the supersymmetric charges. A sufficient condition for supersymmetry to be good is that vacuum should have zero energy where invariance of vacuum under a group of transformations is equivalent to the corresponding symmetry being unbroken. The scalar potential, V , of the Hamiltonian H is given by V = Fi∗ Fi +

1X a a D D . 2

It can be better expressed as V = VF + VD where VF = Fi∗ Fi =

X

∂W

2

∂Φn

n

where the sum runs over all the scalar fields Φn present in the theory and 2 1 2 X 2 VD = g qn |Φn | 2 n 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 due to 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 parameter with 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 holomorphic function 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 which has three chiral superfields Φ1 , Φ2 and Φ3 with superpotential W(Φ1 , Φ2 , Φ3 ) = λ1 Φ1 (Φ23 − M 2 ) + µΦ2 Φ3 . For this model there is no solution with F1 , F2 and F3 all zero since

F1∗

=

F2∗

=

F3∗

=

∂W ⇒ M 2 = φ23 ∂φ1 ∂W ⇒ µφ3 = 0 − ∂φ2 ∂W − ⇒ 2λ1 φ1 φ3 + µφ2 = 0 ∂φ3 −

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

3 X

|Fi |2 = λ21 |φ23 − M 2 |2 + µ2 |φ3 |2 + |µφ2 + 2λ1 φ1 φ3 |2 .

i=1

The absolute minimum of this effective potential occurs at hφ2 i = hφ3 i = 0 with hφ1 i is undetermined that is to say that the potential has a flat direction. At this absolute minimum F1∗ = λ1 M 2 , F2∗ = F3∗ = 0 and V = λ21 M 4 > 0. Since F 1 is non-zero, we expect ψ1 to be the Goldstino, which is a spinor in the chiral supermultiplet in Φ1 to which F1 belongs.

14 14.1

The Minimal Supersymmetric Standard Model: MSSM Properties of MSSM

• Requires 2 Higgs doublets: Hu , Hd where

Hu+ Hu0

Hd0 Hd−

Hu = Hd = These will generate masses via Yukawa couplings.

8

• Superpotential couplings for Higgs sector: W = µHu Hd = µ(Hu+ Hd− Hu0 Hd0 ) SUSY mass term for Higgsinos

= −µ(...) − µ2 |Hu0 |2 + |Hu+ |2 + |Hd0 |2 + |Hd− |2 . ⇒ Lµ

quadratic

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

2

. Natural electroweak quadratic

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 superpartners would have been observed. In other words, there is no gauge singlet for F -term breaking and D-term supersymmetry breaking leads to an unacceptable spectrum. Way Out. Supersymmetry breaking takes place in the hidden non-SM sector of the full theory and mediated 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 = 2 SUSY, 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 → eiα θI , θI → e−iα θI 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 for SM particle content is +1 while supersymmetric particles have R-parity of 1.

17

The Bosonic String The Nambu-Goto & Polyakov Action

The world-sheet has coordinates ξ α , α = 0, 1 with ξ α = (τ, σ) embedded in a D-dimensional Minkowski space-time according to xµ (ξ α ), µ = 0, ..., D − 1. For the open string, take the range of σ 0≤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 2πα0

Z

d2 ξ

q

− det{∂α xµ ∂b xν ηµν }

where ∂α xµ =

∂xµ . ∂ξ α

The dimension of xµ is [mass]−1 . The Regge slope parameter α0 has dimension [mass]−2 in order for the action to be dimensionless. The Nambu-Goto action is invariant under • Reparametrization Invariance • Rigid Poincaré Symmetry:

0

xµ (ξ) = Λµν xν (ξ) + aµ and Lorentz transformations Λµν satisfy ηµν Λµρ Λντ = 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 comes to quantization. Fortunately, by introducing the independent field gαβ , there exists another reparametrization invariant action. A=−

1 4πα0

Z

√ d2 ξ −gg αβ ∂α xµ ∂β xν ηµν

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: • Poincaré Invariance δX µ

= aµν X ν + bµ ,

δhαβ

=

(aµν = −aνµ

0

• Local Symmetries – Reparametrization Invariance δX µ

=

ξ a ∂a X µ

δhαβ

=

ξ γ ∂γ hαβ + ∂a ξ γ hγβ + ∂b ξ γ hαγ

= δ h = √

∇a ξ b + ∇ b ξ a √ ∂a (ξ a h)

10

– Weyl Scaling δhαβ

=

2Λhαβ

δX µ

=

0

One immediately important consequence of the Weyl invariance of the Polyakov action is the vanishing the energy-momentum tensor hαβ Tαβ = 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 sweeps out a one-dimensional curve known as the world line. We express this world line as xµ (τ ). The motion of the point particle is taken defined by the action Z p A = −m dτ −ẋµ ẋν ηµν where

dxµ dτ = (−1, +1, +1, ..., +1) is the Minkowski metric. ẋµ ≡

and ηµν

A classically equivalent way of the writing the action is Z 1 A= dτ {e−1 ẋµ ẋν ηµν − m2 e} 2 where xµ and e are independent fields. This action can be seen to be equivalent by taking computing the equation of motion for e gives −e−2 ẋµ ẋν ηµν − m2 = 0 ⇒ e−2 ẋµ ẋν ηµν + m2 = 0 ⇒ ẋµ ẋν ηµν = −m2 e2 which upon substituting back into the action yields Z Z p 1 (−em2 − em2 )dτ = −m dτ −ẋµ ẋν ηµν . A= 2

18 18.1

Superstrings The Superstring Action: The Basics

In string theory, the string propagates in D-dimensional flat Minkowski spacetime. As this string moves through spacetime it traces out a worldsheet, which is a two-dimensional surface in spacetime. The points on the worldsheet are parametrized by the two coordinates σ 0 = τ , which is time-like and σ 1 = σ, which is spacelike. Therefore the action is given by S=−

1 4πα0

Z

µ d2 σ ∂α Xµ ∂ α X µ + iψ ρα ∂α ψµ

(1)

where X µ = X µ (σ, τ ) and ψ µ = ψ µ (σ, τ ) are the two-dimensional bosonic and fermionic fields respectively and 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 a Majorana representation, where a convenient choice will be ρ0 =

0 1

−1 0

,

11

ρ1 =

0 1

1 0

.

(2)

These matrices can also be seen to satisfy the Clifford algebra. Moreover taking this choice of Ď Îą , we deduce that Ďˆ = Ďˆ T Ď 0 .

(3)

The equation of motions coming from the action is given as ∂ Îą âˆ‚Îą X Âľ = 0 Îą

Âľ

Ď âˆ‚Îą Ďˆ = 0

Wave equation

(4)

Dirac equation

(5)

In addition, the action S is given to be globally invariant under the infinitesimal worldsheet supersymmetric transformations δX Âľ = i Ďˆ Âľ

δĎˆ Âľ = Ď Îą âˆ‚Îą X Âľ

,

(6)

where 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. The conserved current is the energy momentum tensor given by the action’s worldsheet translation invariance. This is just a local infinitesimal transformation given by δX Âľ = Îą âˆ‚Îą X Âľ

δĎˆ Âľ = Îą âˆ‚Îą Ďˆ Âľ .

,

(7)

Îą

Here is the infinitesimal translation. So the energy momentum tensor reads i Âľ δ i Âľ i Âľ 1 δ Âľ Tιβ = âˆ‚Îą X ∂β XÂľ + Ďˆ Ď Îą ∂β ĎˆÂľ + Ďˆ Ď Î˛ âˆ‚Îą ĎˆÂľ − Ριβ ∂ X ∂δ XÂľ + Ďˆ Îł ∂δ ĎˆÂľ . (8) 4 4 2 2 The conserved supercurrent is given by the supersymmetric property of the action. This follows from the local infinitesimal transformations Âľ

δX Âľ = iÂŻ Ďˆ Âľ

δĎˆ Âľ = Ď Îą âˆ‚Îą X Âľ

,

(9)

where is non-constant. So now the supercurrent is given by 1 (10) Jι¾ = − Ď Î˛ Ď Îą Ďˆ Âľ ∂β XÂľ . 2 Also both the conserved current and supercurrent have an interesting properly, which is useful when quantizing the theory, that is Tιβ = 0 (11) JÎą = 0.

(12)

Quantizing string theory can be quite difficult with normal coordinates, so for convenience we to turn to light-cone gauge coordinates. These are given by Ďƒ+ = Ď„ + Ďƒ

,

Ďƒ − = Ď„ − Ďƒ.

(13)

So now the Dirac equation becomes a two spinor components, that is Âľ Âľ =0 ∂+ Ďˆâˆ’ = ∂− Ďˆ+ Âľ Ďˆâˆ’

(14) Âľ Ďˆ+

where the Majorana field describes right movers, while the Majorana field describes left movers. Also we can see that the left- and right-movers decoupled. Similarly the wave equation becomes ∂+ ∂− X Âľ = 0.

(15)

The constraint equations in this case is given by i Âľ T¹¹ = âˆ‚Âą X Âľ âˆ‚Âą XÂľ + Ďˆ Ď Âą âˆ‚Âą ĎˆÂľ = 0 2 Âľ J Âą = ĎˆÂą âˆ‚Âą XÂľ = 0. 12

(16) (17)

18.3

Closed Strings and Mode Expansions

In closed strings, the boundary conditions are periodic or anti-periodic for the fermionic fields. Therefore the 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 µ ψ± (τ, σ) =

X

dµn e−2in(σ

±

)

(19)

n∈Z

X

µ ψ± (τ, σ) =

±

bµr e−2ir(σ ) .

(20)

r∈Z+ 12

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 r µ

µ

0 µ

X (τ, σ) = x + α p τ + i

α0 X aµn −inσ+ ãµn −inσ− + . e e 2 n n

(22)

n6=0

18.4

Quantization

Here we promote the fields X µ and ψ µ to operators X̂ µ and ψ̂ µ respectively. However we shall drop the hats for convenience. Also we will only consider fermions here, since we will be focusing on free-fermionic model building. The fermion anti-commutator relation is given by µ ν {ψ̂A (τ, σ), ψ̂B (τ, σ 0 )} = πδ(σ − σ 0 )η µν δA,B

(23)

where A and B are the spin indices, ±. Using the above relation we deduce that the oscillators satisfies {dµn , dνm } = η µν δn+m,0 {d˜µ , d˜µ } = η µν δn+m,0 n

m

,

{bµr , bνs } = η µν δr+s,0

,

{b̃µr , b̃νs } = η µν δr+s,0

(24)

where n,m ∈ Z and r,s ∈ Z + 21 . 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 superVirasoro algebra. Furthermore this results in the constraint Ln |φi = L̂n |φi = 0

n>0

(L0 − a) |φi = (L̂0 − a)|φi = 0.

(25) (26)

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

Ln =

∞ ∞ 1 X 1 X : an−m an : + m : dn−m dn : (Periodic Modes) 2 m=−∞ 2 m=−∞

(27)

∞ ∞ 1 X 1 X : an−m an : + r : bn−r bn : 2 m=−∞ 2 r=−∞

(28)

Or Ln =

(Anti-Periodic Modes)

where the semi-colons indicate normal ordering. Furthermore, the theory requires absence of negativenorm states, so this leads to specific normal ordering constants. In the case of periodic modes we have a = 0 and similarly for anti-periodic modes we get a = 12 . However both the value of the normal ordering constants holds when D = 10. Also with bosonic strings, when D = 26 we get a single constant given as a = 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 sector this is given as N=

∞ X

Ρ¾ν aÂľâˆ’n aνn +

n=1 a

∞ X

mdÂľâˆ’m dνm Ρ¾ν

m=1 d

=N +N .

(29)

Similarly for the anti-periodic sector it is given as N=

∞ X

Ρ¾ν aÂľâˆ’n aνn +

n=1

∞ X

rbÂľâˆ’r bνr Ρ¾ν

r= 21

= N a + N b.

(30)

Now using the constraint (L0 − a) |φi = 0 and working with anti-periodic sector i.e. The L0 − 12 operator expansion in terms of oscillators gives

L0 −

L0 −

1 2

|φi = 0.

1X 1 1X Âľ ν 1 : a−n an : Ρ¾ν + r : bÂľâˆ’r bνr : Ρ¾ν − = 2 2 n 2 r 2 =

∞ ∞ 1 Âľ ν 1X Âľ ν 1 1X Âľ ν a0 a0 Ρ¾ν + a−n an Ρ¾ν + rb b Ρ¾ν − 2 2 n=1 2 1 −r r 2 r= 2

1 1 = Îą0 pÂľ pÂľ + N − 4 2

(31)

q 0 where aÂľ0 = Îą2 pÂľ 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 Îą0

N−

1 2

(32)

Similarly for the periodic sector, the spacetime mass-squared of a physical state is an eigenvalue of M2 =

18.6

1 N. Îą0

(33)

Heterotic Strings and Compactification

The discussion up till now was on based on N = 2 world-sheet supersymmetry. However, closed strings allow the supersymmetric transformations to decouple in the left and right-movers. For instance the infinitesimal transformations become Âľ δĎˆâˆ’ = −2∂− X Âľ +

,

Âľ δĎˆ+ = −2∂+ X Âľ − .

(34)

Furthermore, this corresponds to two different supersymmetric transformations. So it is now possible to get the N = 2 world-sheet supersymmetry and express it as an N = 1 world-sheet supersymmetry on the left-movers and on the right-movers. The idea here is that we can construct a model where the string has a N = 1 world-sheet supersymmetry on the left-movers and no supersymmetry on the right-movers, which will at the end generate a N = 1 space-time supersymmetry. Until now we worked on superstrings, but all the results apply to a non-supersymmetric strings (bosonic strings). This could seem quite weird at first glance because cancelling the conformal anomaly requires that the space-time dimension for right-movers is 26 and 10 for the left-movers, although one would expect to have the same space-time dimension. However, using compactification, we can reduce the two number of dimensions to 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 it would 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-moving fermions are added, the conformal anomaly would become: CL = −26 + 11 + DL + CR = −26 + DR +

18 DL + 2 2

(35)

44 2

(36)

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

1 π

Z

d2 σ(2∂+ Xµ ∂− X µ + i

18 X A=1

A λA − ∂+ λ− + ψµ + i

44 X

A λA + ∂− λ+ ).

(37)

A=1

This action has a SO(18)L × SO(44)R global symmetry under which the internal fermions λA transform in the fundamental representation. The next step is to compute the one-loop partition function. This is called 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 the model. It is an integration over all the possible world-sheets, in the case of the one-loop partition function the world-sheets are a torus. On the world-sheets, two boundary conditions need to be specified for the two non-contractible loops of the torus (poloidal and toroidal) for each of the free-fermionic fields. These conditions express the shifts in the phase of the fermionic fields under parallel transport around these non-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 fermions which propagate around the string have a boundary condition around the string in the direction of the σ coordinate and they also can pick up a phase by propagating along the τ dimension. The partition function is given as X α Z= C Z[α, β] (39) β α,β

where α and β are the boundary conditions (α, β = 1 denote the periodic boundary conditions and α, β = −1 denote the anti-periodic). Also any α or β corresponds to a space boundary condition and the other is the time. Furthermore, the partition function has 68 fields in total made of 64 internal fermions µ and two each for XLµ , XR and ψ µ . The bosonic fields have no choice of boundary conditions, they are only periodic. However the fermionic field ψ µ can be periodic or anti-periodic, so the partition function must include all possible combinations of 64 boundary conditions of the fermions, and this is integrated over all the inequivalent tori. Thus, the boundary conditions take the values α, β = 1, ..., 64.

19

Dirac Quantization Condition By The Argument Of Wu And Yang

Consider a source of magnetic field around which a 2-sphere is taken. The magnetic charge, g, given by the magnetic flux leaving the 2-sphere is given by the formula Z g= B · dS S2 2 We require two patches to cover S 2 , say SN and SS2 to be taken around the North and South poles respectively 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 each patch. Let AN and AS be the vector gauge potentials on the two patches respectively. The magnetic field is globally defined on the 2-sphere and therefore on the overlap the two vector gauge potentials must differ by a gauge transformation with parameter say χ. Thus Z intS 2 B ¡ dS

Z B ¡ dS +

= 2 SN

Z

B ¡ dS 2 SS

(AN − AS )dx

= S1

Z ∇chi dx

= S1

= χ(φ = 2Ď€) − χ(φ = 0) Note. If there was a globally defined vector gauge potential on S 2 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 gauge transformation iqχ Ďˆâ†’e ~ Ďˆ This can only hold if e

iqχ(φ=0) ~

=e

iqχ(φ=2π) ~

and as a result χ(φ = 2Ď€) − χ(φ = 0) = 2Ď€ Hence

m~ , q

Z B ¡ dS = χ(φ = 2Ď€) − χ(φ = 0) = 2Ď€

g= S2

20

m ∈ Z.

m~ ⇒ qg = 2πm~, q

m ∈ Z.

Superbranes

A superbrane can be viewed as a p + 1 dimensional bosonic submanifold M with coordinates Ξ Îą Îą = 0, 1, ..., p which moves the background superspace with coordinates z N = (xÂľ , Θ) where xÂľ is Grassmann even and Θ is Grassmann odd. A simple superbrane has dynamics specified by z N (Ξ Îą ). It moves to extremize the action A = A1 + A2 where

Z A1 = −T

dp+1 Ξ

p

−detgιβ

is the Brink-Schwarz action and gιβ = âˆ‚Îą z N ∂β z M gN M where gN M is a metric in the background superspace and Z A2 = q dp+1 Ξ Îą1 ...Îąp+1 âˆ‚Îą1 xÂľ1 ...âˆ‚Îąp+1 xÂľp+1 AÂľ1 ...Âľp+1 + ... 16

where Aµ1 ...µp+1 is the background form field. Every p + 1-rank background gauge field comes with D − 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 bosons and fermions on shell. There are D − p − 1 scalar degrees of freedom and 8 fermionic degrees of freedom. When D = 10, we have 8 = 10 − p − 1 ⇒ p = 1. Therefore, we have a string. When D = 11, we have 8 = 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 which satisfy von Neumann and Dirichlet boundary conditions on which ends of open strings are localized. The von Neumann boundary condition is ∂σ X µ , σ = 0, π where the Dirichlet boundary condition is δX µ = 0, σ = 0, π where the end points of the string lie in some constant position in space. More concretely ∂σ X µ ,

a = 0, ..., p

I

I = p + 1, ..., D − 1

I

X =c ,

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

22

Modular Invariance & N = 1 SUSY

String theories are generally formulated in more than four dimensions and to make contact with the 4-dimensional world, many different compactification schemes have been proposed where the basic idea is 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 cancel the Weyl anomaly to obtain a consistent theory. What we really want is to write down a decomposition of 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 and demand N = 1 space-time SUSY. However, we immediately run into two problems on trying to write down 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 a left-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-moving sector 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-moving sector has N = 2 SUSY. The relevant fields are the Laurent modes of the energy-momentum tensor X T (z) = Ln z −n−2 . T (z) also has two fermionic superpartners G± =

X

3

−n− 2 G± . nz

There is also a U (1) current J(z) =

X

Jn z −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 =

1 2

is Ramond but 0 ≤ η ≤ 1 will be consistent.

The N = 2 SCFT is therefore defined by [Lm , Ln ] = (m − n)Lm+n +

c (m3 − m)δm+n,0 12

[Lm , Jn ] = −nJm+n m [Lm , G± ] = − r G± r m+r 2 c [Jm , Jn ] = mδm+n,0 3 ± [Jm , G± r ] = ±Gm+r − {G+ r , Gs } = 2Lr+s + (r − s)Jr+s + + − − {G+ r , Gs } = {Gr , Gs } = 0

(40) (41) (42) (43) (44)

c 2 1 r − δr+s,0 3 4

(45) (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 the Virasoro algebra with dimensions 23 and 1 respectively. The fourth one implies that J(z) is a free boson U (1) current. The Jn s are the usual bosonic creation and annihilation operators. The fifth equation implies that G± (z) have U (1) charges ±1. The last two relations are the new ones involving superenergy-momentum tensors. The appearing indices take integer values except r and s which take values in Z + 21 ± η.

18

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