Introduction to Superstring Theory

Carmen N´u˜nez

IAFE (UBA-CONICET) & PHYSICS DEPT. (University of Buenos Aires)

VI ICTP LASS 2015

Mexico, 26 October- 6 November 2015

...... Programme

Class 1: The classical fermionic string Class 2: The quantized fermionic string Class 3: Partition Function Class 4: Interactions

...... Outline

Class 1: The classical fermionic string The action and its symmetries Gauge fixing and constraints Equations of motion and boundary conditions Oscillator expansions

...... The spectra of bosonic strings contain a tachyon → it might indicate the vacuum has been incorrectly identified. The mass squared of a particle T is the quadratic term in the 2 ∂2V (T ) | − 4 ⇒ action: M = ∂T 2 T =0 = α′ = we are expanding around a maximum of V . If there is some other stable vacuum, this is not an actual inconsistency.

Why superstrings?

...... The mass squared of a particle T is the quadratic term in the 2 ∂2V (T ) | − 4 ⇒ action: M = ∂T 2 T =0 = α′ = we are expanding around a maximum of V . If there is some other stable vacuum, this is not an actual inconsistency.

Why superstrings?

The spectra of bosonic strings contain a tachyon → it might indicate the vacuum has been incorrectly identified.

...... If there is some other stable vacuum, this is not an actual inconsistency.

Why superstrings?

The spectra of bosonic strings contain a tachyon → it might indicate the vacuum has been incorrectly identified. The mass squared of a particle T is the quadratic term in the 2 ∂2V (T ) | − 4 ⇒ action: M = ∂T 2 T =0 = α′ = we are expanding around a maximum of V .

...... Why superstrings?

The spectra of bosonic strings contain a tachyon → it might indicate the vacuum has been incorrectly identified. The mass squared of a particle T is the quadratic term in the 2 ∂2V (T ) | − 4 ⇒ action: M = ∂T 2 T =0 = α′ = we are expanding around a maximum of V . If there is some other stable vacuum, this is not an actual inconsistency.

...... The tachyon potential around T = 0 looks like 1 V (T ) = M2T 2 + c T 3 + c T 4 + ··· 2 3 4 The T 3 term gives rise to a minimum, but the T 4 term destabilizes it again... Moreover tachyon exchange contributes IR divergences in loop diagrams

The critical dimension of the bosonic string is D=26

All physical d.o.f. of bosonic string are bosonic

Why superstrings?

Is there a good minimum elsewhere?

...... Moreover tachyon exchange contributes IR divergences in loop diagrams

The critical dimension of the bosonic string is D=26

All physical d.o.f. of bosonic string are bosonic

Why superstrings?

Is there a good minimum elsewhere? The tachyon potential around T = 0 looks like 1 V (T ) = M2T 2 + c T 3 + c T 4 + ··· 2 3 4 The T 3 term gives rise to a minimum, but the T 4 term destabilizes it again...

...... All physical d.o.f. of bosonic string are bosonic

Why superstrings?

Is there a good minimum elsewhere? The tachyon potential around T = 0 looks like 1 V (T ) = M2T 2 + c T 3 + c T 4 + ··· 2 3 4 The T 3 term gives rise to a minimum, but the T 4 term destabilizes it again... Moreover tachyon exchange contributes IR divergences in loop diagrams

The critical dimension of the bosonic string is D=26

...... Why superstrings?

Is there a good minimum elsewhere? The tachyon potential around T = 0 looks like 1 V (T ) = M2T 2 + c T 3 + c T 4 + ··· 2 3 4 The T 3 term gives rise to a minimum, but the T 4 term destabilizes it again... Moreover tachyon exchange contributes IR divergences in loop diagrams

The critical dimension of the bosonic string is D=26

All physical d.o.f. of bosonic string are bosonic ...... Associated with each bosonic d.o.f. X µ(σ, τ), world-sheet spinors are introduced: Ψµ(τ, σ) , µ = 0,..., D − 1 , ( ) ψ Ψ = + ψ− described by two-component Majorana spinors . Gliozzi, Scherk and Olive (1977): it is possible to get a model with no tachyons and equal masses and multiplicities for bosons and fermions Green and Schwarz (1980): this model had space-time susy

Why superstrings?

These shortcomings can be overcome by constructing a theory with world-sheet supersymmetry.

...... Gliozzi, Scherk and Olive (1977): it is possible to get a model with no tachyons and equal masses and multiplicities for bosons and fermions Green and Schwarz (1980): this model had space-time susy

Why superstrings?

These shortcomings can be overcome by constructing a theory with world-sheet supersymmetry. Associated with each bosonic d.o.f. X µ(σ, τ), world-sheet spinors are introduced: Ψµ(τ, σ) , µ = 0,..., D − 1 , ( ) ψ Ψ = + ψ− described by two-component Majorana spinors .

...... Green and Schwarz (1980): this model had space-time susy

Why superstrings?

These shortcomings can be overcome by constructing a theory with world-sheet supersymmetry. Associated with each bosonic d.o.f. X µ(σ, τ), world-sheet spinors are introduced: Ψµ(τ, σ) , µ = 0,..., D − 1 , ( ) ψ Ψ = + ψ− described by two-component Majorana spinors . Gliozzi, Scherk and Olive (1977): it is possible to get a model with no tachyons and equal masses and multiplicities for bosons and fermions

...... Why superstrings?

These shortcomings can be overcome by constructing a theory with world-sheet supersymmetry. Associated with each bosonic d.o.f. X µ(σ, τ), world-sheet spinors are introduced: Ψµ(τ, σ) , µ = 0,..., D − 1 , ( ) ψ Ψ = + ψ− described by two-component Majorana spinors . Gliozzi, Scherk and Olive (1977): it is possible to get a model with no tachyons and equal masses and multiplicities for bosons and fermions

Green and Schwarz (1980): this model had. . space-time...... susy...... As in the bosonic string theory, the action has to be formulated so as to avoid negative norm states µ SB couples D scalar fields X (σ, τ) to two-dim gravity hαβ

Ghosts are removed by physical state conditions:

δS T ∼ B = 0 αβ δhαβ =⇒ the absence of negative norm states depends crucially on reparametrization invariance

The action

In this superstring theory the one-loop diagrams are completely finite and free of ultraviolet divergences

...... µ SB couples D scalar fields X (σ, τ) to two-dim gravity hαβ

Ghosts are removed by physical state conditions:

δS T ∼ B = 0 αβ δhαβ =⇒ the absence of negative norm states depends crucially on reparametrization invariance

The action

In this superstring theory the one-loop diagrams are completely finite and free of ultraviolet divergences

As in the bosonic string theory, the action has to be formulated so as to avoid negative norm states

...... Ghosts are removed by physical state conditions:

δS T ∼ B = 0 αβ δhαβ =⇒ the absence of negative norm states depends crucially on reparametrization invariance

The action

In this superstring theory the one-loop diagrams are completely finite and free of ultraviolet divergences

As in the bosonic string theory, the action has to be formulated so as to avoid negative norm states µ SB couples D scalar fields X (σ, τ) to two-dim gravity hαβ

...... The action

In this superstring theory the one-loop diagrams are completely finite and free of ultraviolet divergences

As in the bosonic string theory, the action has to be formulated so as to avoid negative norm states µ SB couples D scalar fields X (σ, τ) to two-dim gravity hαβ

Ghosts are removed by physical state conditions:

δS T ∼ B = 0 αβ δhαβ =⇒ the absence of negative norm states depends crucially

on reparametrization invariance ...... Such an action does indeed provide a theory in which all negative norm states are removed by constraints arising as eom

Consistency of the theory requires D=10 for the superstring

The action

Treating X µ and Ψµ as susy partners for a world-sheet susy, and coupling them to two-dimensional supergravity is an appropriate construction

...... Consistency of the theory requires D=10 for the superstring

The action

Treating X µ and Ψµ as susy partners for a world-sheet susy, and coupling them to two-dimensional supergravity is an appropriate construction

Such an action does indeed provide a theory in which all negative norm states are removed by constraints arising as eom

...... The action

Treating X µ and Ψµ as susy partners for a world-sheet susy, and coupling them to two-dimensional supergravity is an appropriate construction

Such an action does indeed provide a theory in which all negative norm states are removed by constraints arising as eom

Consistency of the theory requires D=10 for the superstring

...... Applying GSO projection to the states of the theory, a D = 10 space-time supersymmetric theory without tachyons is obtained

This is the Neveu-Schwarz-Ramond superstring.

There is also the Green-Schwarz formalism in which space-time susy is manifest at the cost of world-sheet susy.

A covariant extension of the GS formalism is the pure spinor formulation.

Supersymmetry

Although this construction possesses world-sheet N=1 susy, it does not possess manifest space-time susy

...... This is the Neveu-Schwarz-Ramond superstring.

There is also the Green-Schwarz formalism in which space-time susy is manifest at the cost of world-sheet susy.

A covariant extension of the GS formalism is the pure spinor formulation.

Supersymmetry

Although this construction possesses world-sheet N=1 susy, it does not possess manifest space-time susy

Applying GSO projection to the states of the theory, a D = 10 space-time supersymmetric theory without tachyons is obtained

...... There is also the Green-Schwarz formalism in which space-time susy is manifest at the cost of world-sheet susy.

A covariant extension of the GS formalism is the pure spinor formulation.

Supersymmetry

Although this construction possesses world-sheet N=1 susy, it does not possess manifest space-time susy

Applying GSO projection to the states of the theory, a D = 10 space-time supersymmetric theory without tachyons is obtained

This is the Neveu-Schwarz-Ramond superstring.

...... A covariant extension of the GS formalism is the pure spinor formulation.

Supersymmetry

Although this construction possesses world-sheet N=1 susy, it does not possess manifest space-time susy

Applying GSO projection to the states of the theory, a D = 10 space-time supersymmetric theory without tachyons is obtained

This is the Neveu-Schwarz-Ramond superstring.

There is also the Green-Schwarz formalism in which space-time susy is manifest at the cost of world-sheet susy.

...... Supersymmetry

Although this construction possesses world-sheet N=1 susy, it does not possess manifest space-time susy

Applying GSO projection to the states of the theory, a D = 10 space-time supersymmetric theory without tachyons is obtained

This is the Neveu-Schwarz-Ramond superstring.

There is also the Green-Schwarz formalism in which space-time susy is manifest at the cost of world-sheet susy.

A covariant extension of the GS formalism is the pure spinor formulation...... Heterotic string: superstring modes for right-movers and bosonic string modes for left-movers. N = 1 susy in D = 10

Other possibilities?

Extended supersymmetry? N = 2 world-sheet supersymmetry → critical dimension D = 2 N = 4 world-sheet supersymmetry → negative critical dimension!

...... Other possibilities?

Extended supersymmetry? N = 2 world-sheet supersymmetry → critical dimension D = 2 N = 4 world-sheet supersymmetry → negative critical dimension!

Heterotic string: superstring modes for right-movers and bosonic string modes for left-movers. N = 1 susy in D = 10

...... X µ(τ, σ) are world-sheet scalars but space-time vectors

=⇒ their susy partners should be world-sheet spinors with a target space vector index

Superstring action

We want to find the susy extension of the Polyakov action Recall ∫ 1 √ S = − d2σ −hhαβ∂ X µ∂ X (1) P 4πα′ α β µ

Susy extension should be the coupling of supersymmetric ”matter” to two-dimensional supergravity

...... =⇒ their susy partners should be world-sheet spinors with a target space vector index

Superstring action

We want to find the susy extension of the Polyakov action Recall ∫ 1 √ S = − d2σ −hhαβ∂ X µ∂ X (1) P 4πα′ α β µ

Susy extension should be the coupling of supersymmetric ”matter” to two-dimensional supergravity

X µ(τ, σ) are world-sheet scalars but space-time vectors

...... Superstring action

We want to find the susy extension of the Polyakov action Recall ∫ 1 √ S = − d2σ −hhαβ∂ X µ∂ X (1) P 4πα′ α β µ

Susy extension should be the coupling of supersymmetric ”matter” to two-dimensional supergravity

X µ(τ, σ) are world-sheet scalars but space-time vectors

=⇒ their susy partners should be world-sheet spinors with a target space vector index

...... Let us consider the action ∫ [ ] 1 √ 1 S = − d2σ −h hαβ∂ X µ∂ X + iψ¯µρα∂ ψ (2) 4π α′ α β µ α µ

ψµ is a Majorana spinor

ψ¯ = ψ†ρ0 = (ψ∗)T C = ψT C Conjugate spinor

=⇒ Majorana spinors are real ρα are two dimensional Dirac matrices. A convenient basis is ( ) ( ) 0 1 0 1 ρ0 = iσ2 = , ρ1 = σ1 = (3) −1 0 1 0

...... Brief remainder on two-dimensional spinors

The two-dimensional Dirac matrices satisfy {ρα, ρβ} = 2hαβ (4) They transform under coordinate transformations and are related to the constant Dirac matrices ρa through the zweibein: ( ) −1 0 ρα = eαρa =⇒ {ρa, ρb} = 2ηab = 2 a 0 1

We define the analogue of γ5 in four-dimensions: ( ) 1 0 ρ¯ = ρ0ρ1 = σ3 = 0 −1 ...... Two-dimensional spinor indices take values A = , i.e. ( ) ψ+ ψA = (5) ψ−

+ − and ψ = −ψ−, ψ = ψ+

Brief remainder on two-dimensional spinors

Using spinor indices

A B A BA χ¯Γψ = χ ΓA ψB where χ = χB C where Γ is some combination of Dirac matrices. The charge conjugation matrix (CC † = 1) ( ) 0 1 C = ρ0 = −1 0

...... Brief remainder on two-dimensional spinors

Using spinor indices

A B A BA χ¯Γψ = χ ΓA ψB where χ = χB C where Γ is some combination of Dirac matrices. The charge conjugation matrix (CC † = 1) ( ) 0 1 C = ρ0 = −1 0

Two-dimensional spinor indices take values A = , i.e. ( ) ψ+ ψA = (5) ψ−

+ − and ψ = −ψ−, ψ = ψ ...... + ...... Useful relations (exercises)

Spin-flip property, valid for anticommuting Majorana spinors

α1 αn n αn α1 λ¯1ρ ··· ρ λ2 = (−1) λ¯2ρ ··· ρ λ1 (6)

Fierz identity, valid for anticommuting Majorana spinors 1 (ψλ¯ )(ϕχ¯ ) = − {(ψχ¯ )(ϕλ¯ ) + (ψ¯ρχ¯ )(ϕ¯ρ¯λ¯) + (ψρ¯ αχ)(ϕρ¯ λ)} 2 α

α ρ ρβρα = 0

ραρβ = hαβ + 1 ϵαβρ¯ with ϵ01 = 1 e ...... D real scalars X µ provide D bosonic degrees of freedom

D world-sheet Majorana fermions ψµ provide 2D fermionic µ degrees of freedom ψ±

=⇒ we have to introduce D real auxiliary scalar fields F µ

Together (X µ, ψµ, F µ) form an off-shell scalar multiplet of two-dimensional N=1 supersymmetry ∫ µ µ 2 µ On-shell (X , ψ ) suffice: SF ∝ d σe F Fµ

Back to the action for the supertring

Let us work out the balance between bosonic and fermionic degrees of freedom .

...... D world-sheet Majorana fermions ψµ provide 2D fermionic µ degrees of freedom ψ±

=⇒ we have to introduce D real auxiliary scalar fields F µ

Together (X µ, ψµ, F µ) form an off-shell scalar multiplet of two-dimensional N=1 supersymmetry ∫ µ µ 2 µ On-shell (X , ψ ) suffice: SF ∝ d σe F Fµ

Back to the action for the supertring

Let us work out the balance between bosonic and fermionic degrees of freedom .

D real scalars X µ provide D bosonic degrees of freedom

...... =⇒ we have to introduce D real auxiliary scalar fields F µ

Together (X µ, ψµ, F µ) form an off-shell scalar multiplet of two-dimensional N=1 supersymmetry ∫ µ µ 2 µ On-shell (X , ψ ) suffice: SF ∝ d σe F Fµ

Back to the action for the supertring

Let us work out the balance between bosonic and fermionic degrees of freedom .

D real scalars X µ provide D bosonic degrees of freedom

D world-sheet Majorana fermions ψµ provide 2D fermionic µ degrees of freedom ψ±

...... Together (X µ, ψµ, F µ) form an off-shell scalar multiplet of two-dimensional N=1 supersymmetry ∫ µ µ 2 µ On-shell (X , ψ ) suffice: SF ∝ d σe F Fµ

Back to the action for the supertring

Let us work out the balance between bosonic and fermionic degrees of freedom .

D real scalars X µ provide D bosonic degrees of freedom

D world-sheet Majorana fermions ψµ provide 2D fermionic µ degrees of freedom ψ±

=⇒ we have to introduce D real auxiliary scalar fields F µ

...... Back to the action for the supertring

Let us work out the balance between bosonic and fermionic degrees of freedom .

D real scalars X µ provide D bosonic degrees of freedom

D world-sheet Majorana fermions ψµ provide 2D fermionic µ degrees of freedom ψ±

=⇒ we have to introduce D real auxiliary scalar fields F µ

Together (X µ, ψµ, F µ) form an off-shell scalar multiplet of two-dimensional N=1 supersymmetry ∫ µ µ 2 µ On-shell (X , ψ ) suffice: SF ∝ d σe F Fµ ...... Back to the action ∫ [ ] 1 √ 1 S = − d2σ −h hαβ∂ X µ∂ X + iψ¯µρα∂ ψ 4π α′ α β µ α µ

The derivative is ordinary instead of covariant due to the Majorana µ α µ α spin-flip property: ψ¯ ρ ωαψµ = −ψ¯ ρ ωαψµ It is invariant under the infinitesimal transformations: √ 1 µ µ ′ δϵX = iϵψ¯ (7) α √ 2 1 δ ψµ = ρα∂ X µϵ (8) ϵ α′ 2 α with ϵ a constant anticommuting infinitesimal Majorana spinor.

Supersymmetry transformations mix bosonic and. . . fermionic...... fields...... A basic fact about susy is

µ µ α µ [δ1, δ2]X = δ1(¯ϵ2ψ ) − (1 ↔ 2) = a ∂αX the commutator of two supersymmetry transformations gives a spatial translation (here on the world-sheet) with

α α a = 2iϵ¯1ρ ϵ2

Here it is important that for Majorana spinors in two dimensions: α α ϵ¯1ρ ϵ2 = −ϵ¯2ρ ϵ1.

µ α µ [δ1, δ2]ψ = a ∂αψ Here it is necessary that ψµ obeys the Dirac equation: α µ ρ ∂αψ = 0...... The bein is necesary to describe spinors on a curved manifold as the group GL(n, R) does not have spinor representations whereas the tangent space group SO(n − 1, 1) does.

The zweibein has 4 components. There are two reparametrizations and one local Lorentz transformation as gauge symmetries. This leaves one bosonic degree of freedom in two-dimensions

The gravity sector

The supergravity multiplet consists of the a zweibein eα and the gravitino χα

...... The zweibein has 4 components. There are two reparametrizations and one local Lorentz transformation as gauge symmetries. This leaves one bosonic degree of freedom in two-dimensions

The gravity sector

The supergravity multiplet consists of the a zweibein eα and the gravitino χα The bein is necesary to describe spinors on a curved manifold as the group GL(n, R) does not have spinor representations whereas the tangent space group SO(n − 1, 1) does.

...... There are two reparametrizations and one local Lorentz transformation as gauge symmetries. This leaves one bosonic degree of freedom in two-dimensions

The gravity sector

The supergravity multiplet consists of the a zweibein eα and the gravitino χα The bein is necesary to describe spinors on a curved manifold as the group GL(n, R) does not have spinor representations whereas the tangent space group SO(n − 1, 1) does.

The zweibein has 4 components.

...... The gravity sector

The supergravity multiplet consists of the a zweibein eα and the gravitino χα The bein is necesary to describe spinors on a curved manifold as the group GL(n, R) does not have spinor representations whereas the tangent space group SO(n − 1, 1) does.

The zweibein has 4 components. There are two reparametrizations and one local Lorentz transformation as gauge symmetries. This leaves one bosonic degree of freedom in two-dimensions

...... [ n ] For N=1 supersymmetry there are 2 2 supersymmetry parameters, [ n ] leaving (n − 1)2 2 =2 fermionic degrees of freedom for n = 2

The complete off-shell sugra multiplet requires the introduction of an auxiliary real scalar field A.

a a The complete off-shell multiplet is (eα, χα, A). On-shell (eα, χα)

Sugra degrees of freedom

The gravitino is a world-sheet vector and a world-sheet Majorana [ n ] spinor. It has 2 2 n = 4 components in n = 2 dimensions

...... The complete off-shell sugra multiplet requires the introduction of an auxiliary real scalar field A.

a a The complete off-shell multiplet is (eα, χα, A). On-shell (eα, χα)

Sugra degrees of freedom

The gravitino is a world-sheet vector and a world-sheet Majorana [ n ] spinor. It has 2 2 n = 4 components in n = 2 dimensions [ n ] For N=1 supersymmetry there are 2 2 supersymmetry parameters, [ n ] leaving (n − 1)2 2 =2 fermionic degrees of freedom for n = 2

...... a a The complete off-shell multiplet is (eα, χα, A). On-shell (eα, χα)

Sugra degrees of freedom

The gravitino is a world-sheet vector and a world-sheet Majorana [ n ] spinor. It has 2 2 n = 4 components in n = 2 dimensions [ n ] For N=1 supersymmetry there are 2 2 supersymmetry parameters, [ n ] leaving (n − 1)2 2 =2 fermionic degrees of freedom for n = 2

The complete off-shell sugra multiplet requires the introduction of an auxiliary real scalar field A.

...... Sugra degrees of freedom

The gravitino is a world-sheet vector and a world-sheet Majorana [ n ] spinor. It has 2 2 n = 4 components in n = 2 dimensions [ n ] For N=1 supersymmetry there are 2 2 supersymmetry parameters, [ n ] leaving (n − 1)2 2 =2 fermionic degrees of freedom for n = 2

The complete off-shell sugra multiplet requires the introduction of an auxiliary real scalar field A.

a a The complete off-shell multiplet is (eα, χα, A). On-shell (eα, χα)

...... Local susy requires the additional term: ∫ (√ ) i 2 i S′ = d2σeχ¯ ρβραψµ ∂ X − χ¯ ψ 8π α α′ β µ 4 β µ

The auxiliary field A does not appear and the auxiliary matter√ µ | a | − scalars F can be eliminated via their eom. e = deteα = h. The kinetic term for the gravitino vanishes identically in two αβγ αβγ dimensionsχ ¯αΓ Dβχγ where Γ is the antisymmetrized product of three Dirac matrices which vanishes in two dimensions

The action

The action ∫ [ ] 1 √ 1 S = − d2σ −h hαβ∂ X µ∂ X + iψ¯µρα∂ ψ 4π α′ α β µ α µ is not locally susy.

...... The kinetic term for the gravitino vanishes identically in two αβγ αβγ dimensionsχ ¯αΓ Dβχγ where Γ is the antisymmetrized product of three Dirac matrices which vanishes in two dimensions

The action

The action ∫ [ ] 1 √ 1 S = − d2σ −h hαβ∂ X µ∂ X + iψ¯µρα∂ ψ 4π α′ α β µ α µ is not locally susy. Local susy requires the additional term: ∫ (√ ) i 2 i S′ = d2σeχ¯ ρβραψµ ∂ X − χ¯ ψ 8π α α′ β µ 4 β µ

The auxiliary field A does not appear and the auxiliary matter√ µ | a | − scalars F can be eliminated via their eom. e = deteα = h.

...... The action

The action ∫ [ ] 1 √ 1 S = − d2σ −h hαβ∂ X µ∂ X + iψ¯µρα∂ ψ 4π α′ α β µ α µ is not locally susy. Local susy requires the additional term: ∫ (√ ) i 2 i S′ = d2σeχ¯ ρβραψµ ∂ X − χ¯ ψ 8π α α′ β µ 4 β µ

The auxiliary field A does not appear and the auxiliary matter√ µ | a | − scalars F can be eliminated via their eom. e = deteα = h. The kinetic term for the gravitino vanishes identically in two αβγ αβγ dimensionsχ ¯αΓ Dβχγ where Γ is the antisymmetrized product of three Dirac matrices which vanishes. . in. . two. . . dimensions...... The action

The complete action is:

∫ [ 1 2 S = − d2σe hαβ∂ X µ∂ X + 2iψ¯µρα∂ ψ 8π α′ α β µ α µ (√ )] 2 i −iχ¯ ρβραψµ ∂ X − χ¯ ψ α α′ β µ 4 β µ

...... Symmetries

The action is invariant under the following local world-sheet symmetries Supersymmetry √ 2 µ µ ′ δϵX = iϵψ¯ , α (√ ) 1 2 i δ ψµ = ρα ∂ X µ − χ¯ ψµ ϵ , ϵ 2 α′ α 2 α i δ e a = ϵρ¯ αχ , ϵ α 2 α δϵχα = 2Dαϵ where ϵ(τ, σ) is a Majorana spinor which parametrizes susy transformations and Dα is a covariant derivative...... with. . . . torsion...... Symmetries

1 D ϵ = ∂ ϵ − ω ρϵ¯ α α 2 α 1 i ω = − ϵabω = ω (e) + χ¯ ρρ¯ βχ α 2 αab α 4 α β 1 ω (e) = − e ϵβγ∂ e a α e αa β γ

where ωα(e) is the spin connection without torsion

...... Symmetries

2 2 2Λ Weyl transformations: hαβ → Ω (τ, σ)hαβ for Ω = e

µ δΛX = 0 1 δ ψµ = − Λψµ Λ 2 a a δΛeα = Λeα 1 δ χ = Λχ Λ α 2 α Super-Weyl transformations

δηχα = ραη

δη(others) = 0

with η(τ, σ) a Majorana spinor parameter...... Symmetries

Two-dimensional Lorentz transformations µ δl X = 0 1 δ ψµ = − lρψ¯ µ l 2 a a b δl eα = lϵ beα 1 δ χ = − lρχ¯ l α 2 α Reparametrizations µ β µ δξX = −ξ ∂βX µ β µ δξψ = −ξ ∂βψ a β a a β δξeα = −ξ ∂βeα − eβ ∂αξ δ χ = −ξβ∂ χ − χ ∂ ξβ ξ α β α β. .α...... In addition to the local world-sheet symmetries, the action is also invariant under global space-time Poincar´etransformations:

µ µ ν µ δX = a νX + b , aµν = −aνµ

δhαβ = 0 µ µ ν δψ = a νψ

δχα = 0

Symmetries

The symmetry transformation rules can be obtained using the Noether method or superspace techniques.

...... Symmetries

The symmetry transformation rules can be obtained using the Noether method or superspace techniques.

In addition to the local world-sheet symmetries, the action is also invariant under global space-time Poincar´etransformations:

µ µ ν µ δX = a νX + b , aµν = −aνµ

δhαβ = 0 µ µ ν δψ = a νψ

δχα = 0

...... To do this we decompose the gravitino as ( ) β 1 β 1 β χα = hα − ραρ χβ + ραρ χβ ( 2 2) 1 1 = ρβρ χ + ρ ρβχ 2 α β 2 α β

=χ ˜α + ραλ (9)

1 β · 1 α whereχ ˜ = 2 ρ ραχβ is ρ-traceless: ρ χ˜ = 0 and λ = 2 ρ χα, corresponding to a decomposition of the spin 3/2 gravitino into helicity 3/2 and 1/2 components.

Gauge fixing

We can now use local susy, reparametrizations and Lorentz transformations to gauge away two d.o.f. of the zweibein and two d.o.f. of the gravitino.

...... Gauge fixing

We can now use local susy, reparametrizations and Lorentz transformations to gauge away two d.o.f. of the zweibein and two d.o.f. of the gravitino. To do this we decompose the gravitino as ( ) β 1 β 1 β χα = hα − ραρ χβ + ραρ χβ ( 2 2) 1 1 = ρβρ χ + ρ ρβχ 2 α β 2 α β

=χ ˜α + ραλ (9)

1 β · 1 α whereχ ˜ = 2 ρ ραχβ is ρ-traceless: ρ χ˜ = 0 and λ = 2 ρ χα, corresponding to a decomposition of the spin 3/2 gravitino into helicity 3/2 and 1/2 components...... The same decomposition can be made for the susy transformation of the gravitino:

δϵχα = 2Dαϵ β = 2(Πϵ)α + ραρ Dβϵ where ( ) 1 1 (Πϵ) = h β − ρ ρβ D ϵ = ρβρ D ϵ α α 2 α β 2 α β maps spin 1/2 fields to ρ-traceless spin 3/2 fields. Now we can write β χ˜α = ρ ραDβκ (10) α for some spinor κ (where we used ρ ρβρα = 0) =⇒ κ can be eliminated by a susy transformation →χα = ραλ ...... In this way we arrive at the superconformal gauge (a generalization of the conformal gauge): a ϕ a eα = e δα , χα = ραλ (12)

a a In the classical theory we can use Weyl (δΛeα = Λeα ) and super-Weyl (δηχα = ραη) transformations to gauge away ϕ and λ, a a leaving only eα = δα and χα = 0 In analogy to the bosonic case, these symmetries will be broken in the quantum theory except in the critical dimension.

Superconformal gauge

Reparametrizations and local Lorentz transformations allow to transform the zweibein into a ϕ a eα = e δα (11)

...... a a In the classical theory we can use Weyl (δΛeα = Λeα ) and super-Weyl (δηχα = ραη) transformations to gauge away ϕ and λ, a a leaving only eα = δα and χα = 0 In analogy to the bosonic case, these symmetries will be broken in the quantum theory except in the critical dimension.

Superconformal gauge

Reparametrizations and local Lorentz transformations allow to transform the zweibein into a ϕ a eα = e δα (11)

In this way we arrive at the superconformal gauge (a generalization of the conformal gauge): a ϕ a eα = e δα , χα = ραλ (12)

...... In analogy to the bosonic case, these symmetries will be broken in the quantum theory except in the critical dimension.

Superconformal gauge

Reparametrizations and local Lorentz transformations allow to transform the zweibein into a ϕ a eα = e δα (11)

In this way we arrive at the superconformal gauge (a generalization of the conformal gauge): a ϕ a eα = e δα , χα = ραλ (12)

a a In the classical theory we can use Weyl (δΛeα = Λeα ) and super-Weyl (δηχα = ραη) transformations to gauge away ϕ and λ, a a leaving only eα = δα and χα = 0

...... Superconformal gauge

Reparametrizations and local Lorentz transformations allow to transform the zweibein into a ϕ a eα = e δα (11)

In this way we arrive at the superconformal gauge (a generalization of the conformal gauge): a ϕ a eα = e δα , χα = ραλ (12)

a a In the classical theory we can use Weyl (δΛeα = Λeα ) and super-Weyl (δηχα = ραη) transformations to gauge away ϕ and λ, a a leaving only eα = δα and χα = 0 In analogy to the bosonic case, these symmetries will be broken in the quantum theory except in the critical dimension...... World-sheet indices are now raised and lowered with the flat metric αβ α α a η and ρ = δa ρ .

The action in superconformal gauge

In superconformal gauge the action simplifies to ∫ [ ] 1 1 S = − d2σ ∂ X µ∂αX + iψ¯µρα∂ ψ 4π α′ α µ α µ

This is the action of a free scalar superfield in two dimensions. To arrive at this action we have rescaled eϕ/2ψ → ψ.

...... The action in superconformal gauge

In superconformal gauge the action simplifies to ∫ [ ] 1 1 S = − d2σ ∂ X µ∂αX + iψ¯µρα∂ ψ 4π α′ α µ α µ

This is the action of a free scalar superfield in two dimensions. To arrive at this action we have rescaled eϕ/2ψ → ψ.

World-sheet indices are now raised and lowered with the flat metric αβ α α a η and ρ = δa ρ .

...... Equations of motion

The e.o.m. derived from the action ∫ [ ] 1 1 S = − d2σ ∂ X µ∂αX + iψ¯µρα∂ ψ 4π α′ α µ α µ are

α µ ∂α∂ X = 0 (13) α µ ρ ∂αψ = 0 . (14)

As in the bosonic theory, they have to be supplemented by boundary conditions (later)

...... For theories with fermions, the energy-momentum tensor is defined as 2π δS Tαβ = β eaα (15) e δea

We can analogously define the supercurrent as the response to variations of the gravitino: 2π δS T = (16) F α e iδχ¯α

Equations of motion

The e.o.m. for the zweibein and the gravitino are:

Tαβ = 0 , TF α = 0 . They are constraints on the system

...... We can analogously define the supercurrent as the response to variations of the gravitino: 2π δS T = (16) F α e iδχ¯α

Equations of motion

The e.o.m. for the zweibein and the gravitino are:

Tαβ = 0 , TF α = 0 . They are constraints on the system For theories with fermions, the energy-momentum tensor is defined as 2π δS Tαβ = β eaα (15) e δea

...... Equations of motion

The e.o.m. for the zweibein and the gravitino are:

Tαβ = 0 , TF α = 0 . They are constraints on the system For theories with fermions, the energy-momentum tensor is defined as 2π δS Tαβ = β eaα (15) e δea

We can analogously define the supercurrent as the response to variations of the gravitino: 2π δS TF α = α (16) e iδχ¯ ...... α Tracelessness Tα = 0 follows upon using the e.o.m. and as a consequence of Weyl invariance α The analogue ρ TF α = 0 follows from super-Weyl invariance

In the superconformal gauge they are ( ) 1 1 T = − ∂ X µ∂ X − η ∂γX µ∂ X αβ α′ α β µ 2 αβ γ µ ( ) i µ µ − ψ¯ ρα∂βψµ + ψ¯ ρβ∂αψµ = 0 4√ 1 2 T = − ρβρ ψµ∂ X = 0 F α 4 α′ α β µ

...... In the superconformal gauge they are ( ) 1 1 T = − ∂ X µ∂ X − η ∂γX µ∂ X αβ α′ α β µ 2 αβ γ µ ( ) i µ µ − ψ¯ ρα∂βψµ + ψ¯ ρβ∂αψµ = 0 4√ 1 2 T = − ρβρ ψµ∂ X = 0 F α 4 α′ α β µ

α Tracelessness Tα = 0 follows upon using the e.o.m. and as a consequence of Weyl invariance α The analogue ρ TF α = 0 follows from super-Weyl invariance

...... Conservation laws and conserved charges

The energy-momentum tensor and the supercurrent are conserved: α ∂ Tαβ = 0 (17) α ∂ TF α = 0 (18) These conservation laws lead to an infinite number of conserved charges. In light-cone coordinates on the world-sheet σ± = τ  σ (19) where ds2 = −dτ 2 + dσ2 = −dσ+dσ−

1 +− −+ η − = η− = − , η = η = −2 (20) + + 2 ++ −− 1 η++ = η−− = η = η = 0 , ∂± = (∂τ  ∂σ) (21) . 2...... µ ∂+∂−X = 0 , (22) µ µ ∂−ψ+ = ∂+ψ− = 0 . (23)

Analysis in light-cone coordinates on the world-sheet

The action and eom in light-cone coordinates are ∫ [ ] 1 2 2 µ µ µ S = − d σ ∂ X ∂−X + i(ψ ∂−ψ + ψ ∂ ψ− ) 2π α′ + µ + +µ − + µ

where ( ) ψ+ ψA = ψ−

and the eom

...... Analysis in light-cone coordinates on the world-sheet

The action and eom in light-cone coordinates are ∫ [ ] 1 2 2 µ µ µ S = − d σ ∂ X ∂−X + i(ψ ∂−ψ + ψ ∂ ψ− ) 2π α′ + µ + +µ − + µ

where ( ) ψ+ ψA = ψ−

and the eom

µ ∂+∂−X = 0 , (22) µ µ ∂−ψ+ = ∂+ψ− = 0 . (23) ...... Analysis in light-cone coordinates on the world-sheet

The energy-momentum tensor in light-cone coordinates is 1 i T = − ∂ X · ∂ X − ψ · ∂ ψ , ++ α′ + + 2 + + + 1 i T−− = − ∂−X · ∂−X − ψ− · ∂−ψ− , (24) α′ 2 T+− = T−+ = 0

with ∂−T++ = ∂+T−− = 0 And the supercurrent √ 1 2 T ± = − ψ± · ∂±X (25) F 2 α′ with

∂−TF + = ∂+TF − = 0 ...... (26)...... and from the conservation laws

∂−T++ = ∂+T−− = 0 , ∂−TF + = ∂+TF − = 0 =⇒

+ T++ and TF + are functions of σ only whereas − T−− and TF − are functions of σ only.

Solutions

From the e.o.m.

µ ⇒ µ µ + µ − ∂+∂−X = 0 = X (τ, σ) = XL (σ ) + XR (σ ) µ µ ⇒ µ µ + µ µ − ∂−ψ+ = ∂+ψ− = 0 = ψ+ = ψ+(σ ) , ψ− = ψ−(σ )

the fields can be split into left- and right-movers

...... Solutions

From the e.o.m.

µ ⇒ µ µ + µ − ∂+∂−X = 0 = X (τ, σ) = XL (σ ) + XR (σ ) µ µ ⇒ µ µ + µ µ − ∂−ψ+ = ∂+ψ− = 0 = ψ+ = ψ+(σ ) , ψ− = ψ−(σ )

the fields can be split into left- and right-movers and from the conservation laws

∂−T++ = ∂+T−− = 0 , ∂−TF + = ∂+TF − = 0 =⇒

+ T++ and TF + are functions of σ only whereas − T−− and TF − are functions of σ only.

...... The boundary term vanishes if µ µ ∂σX (τ, 0) = ∂σX (τ, l) = 0 These are Neumann boundary conditions on X µ: the ends of the open string move freely in space-time Surface term also vanishes if fields are periodic → closed string µ µ µ µ X (τ, l) = X (τ, 0) , ∂σX (τ, l) = ∂σX (τ, 0)

Boundary conditions

µ µ µ Varying X in the action such that δX (τ0) = 0 = δX (τ1) gives: ∫ ∫ 1 τ1 l √ δS = dτ dσ −hδX ∇2X µ P 2πα′ µ τ0 0 ∫ τ1 √ − 1 − µ|σ=l ′ dτ h δXµ∂σX σ=0 2πα τ0

...... Surface term also vanishes if fields are periodic → closed string µ µ µ µ X (τ, l) = X (τ, 0) , ∂σX (τ, l) = ∂σX (τ, 0)

Boundary conditions

µ µ µ Varying X in the action such that δX (τ0) = 0 = δX (τ1) gives: ∫ ∫ 1 τ1 l √ δS = dτ dσ −hδX ∇2X µ P 2πα′ µ τ0 0 ∫ τ1 √ − 1 − µ|σ=l ′ dτ h δXµ∂σX σ=0 2πα τ0 The boundary term vanishes if µ µ ∂σX (τ, 0) = ∂σX (τ, l) = 0 These are Neumann boundary conditions on X µ: the ends of the open string move freely in space-time

...... Boundary conditions

µ µ µ Varying X in the action such that δX (τ0) = 0 = δX (τ1) gives: ∫ ∫ 1 τ1 l √ δS = dτ dσ −hδX ∇2X µ P 2πα′ µ τ0 0 ∫ τ1 √ − 1 − µ|σ=l ′ dτ h δXµ∂σX σ=0 2πα τ0 The boundary term vanishes if µ µ ∂σX (τ, 0) = ∂σX (τ, l) = 0 These are Neumann boundary conditions on X µ: the ends of the open string move freely in space-time Surface term also vanishes if fields are periodic → closed string µ µ µ µ X (τ, l) = X (τ, 0) , ∂σX (τ, l) = .∂σ. X. . .(τ,. . 0)...... For the closed string this requires

(ψ+ · δψ+ − ψ− · δψ−)(σ) = (ψ+ · δψ+ − ψ− · δψ−)(σ + l) which is solved by µ  µ ψ+(σ) = ψ+(σ + l) (28) µ µ ψ−(σ) = ψ−(σ + l) (29)

and the same conditions on δψ±. Antiperiodicity of ψ is possible as they are fermions on the world-sheet.

Boundary conditions

To derive the e.o.m. for the fermions we impose µ µ δψ (τ0) = δψ (τ1) = 0 Further we have to impose bdry cond such that the boundary term ∫ τ1 1 · − · |σ=l δS = dτ(ψ+ δψ+ ψ− δψ−) σ=0 (27) 2π τ0 vanishes.

...... Boundary conditions

To derive the e.o.m. for the fermions we impose µ µ δψ (τ0) = δψ (τ1) = 0 Further we have to impose bdry cond such that the boundary term ∫ τ1 1 · − · |σ=l δS = dτ(ψ+ δψ+ ψ− δψ−) σ=0 (27) 2π τ0 vanishes. For the closed string this requires

(ψ+ · δψ+ − ψ− · δψ−)(σ) = (ψ+ · δψ+ − ψ− · δψ−)(σ + l) which is solved by µ  µ ψ+(σ) = ψ+(σ + l) (28) µ µ ψ−(σ) = ψ−(σ + l) (29)

and the same conditions on δψ±. Antiperiodicity of ψ is possible as they are fermions on the world-sheet...... Ramond and Neveu-Schwarz boundary conditions

Periodic bdy cond are called Ramond boundary conditions

µ µ ψ+(σ) = +ψ+(σ + l) µ µ ψ−(σ) = +ψ−(σ + l)

Anti-periodic bdy cond are called Neveu-Schwarz boundary conditions

µ − µ ψ+(σ) = ψ+(σ + l) µ µ ψ−(σ) = −ψ−(σ + l)

Space-time Poincar´einvariance requires that we impose the same boundary conditions in all directions µ.

This also guarantees that TF ± have definite periodicity...... Boundary conditions

Fermions on the world-sheet satisfy { µ 2πiϕ µ ϕ = 0 for the R sector ψ (σ + l) = e ψ (σ) 1 (30) ϕ = 2 for the NS sector More general phases are not allowed for real ψ. The conditions for the two spinor components ψ+ and ψ− can be chosen independently, leading to four possibilities

(R, R)(NS, NS)(NS, R)(R, NS)

We shall see that string states in the (R,R) and (NS,NS) sectors are space-time bosons while those in the (R, NS) and (NS, R)

sectors are space-time fermions...... To preserve space-time Poincar´einvariance we have to impose the same conditions on all µ.

Boundary conditions for the open string

For the open string the variation ∫ τ1 1 · − · |σ=l δS = dτ(ψ+ δψ+ ψ− δψ−) σ=0 2π τ0 has to be canceled on each boundary, i.e. at σ = 0 and σ = l, separately. This leads to

µ  µ µ  µ ψ+(0) = ψ−(0) , ψ+(l) = ψ−(l)

...... Boundary conditions for the open string

For the open string the variation ∫ τ1 1 · − · |σ=l δS = dτ(ψ+ δψ+ ψ− δψ−) σ=0 2π τ0 has to be canceled on each boundary, i.e. at σ = 0 and σ = l, separately. This leads to

µ  µ µ  µ ψ+(0) = ψ−(0) , ψ+(l) = ψ−(l)

To preserve space-time Poincar´einvariance we have to impose the same conditions on all µ.

...... We then have to distinguish beetween two sectors: η = +1 is the Ramond sector η = −1 is the Neveu Schwarz sector States in the R sector will turn out to be space-time fermions. States in the NS sector will turn out to be space-time bosons.

Boundary conditions for the open string

Without loss of generality we specify

µ µ µ µ ψ+(0) = ψ−(0) , ψ+(l) = ηψ−(l) (31)

where η = 1. Only the relative sign in the boundary conditions at σ = 0 and σ = l is relevant and by a redefinition ψ− → ψ−, which leaves the action invariant, we can always move the sign to the σ = l boundary.

...... Boundary conditions for the open string

Without loss of generality we specify

µ µ µ µ ψ+(0) = ψ−(0) , ψ+(l) = ηψ−(l) (31)

where η = 1. Only the relative sign in the boundary conditions at σ = 0 and σ = l is relevant and by a redefinition ψ− → ψ−, which leaves the action invariant, we can always move the sign to the σ = l boundary. We then have to distinguish beetween two sectors: η = +1 is the Ramond sector η = −1 is the Neveu Schwarz sector States in the R sector will turn out to be space-time fermions. States in the NS sector will turn out to be space-time bosons...... Superconformal algebra

To find the algebra satisfied by Tαβ and TF α we need the equal τ Poisson brackets. In conformal gauge [ ] X µ(σ), X˙ µ(σ′) = 2πα′ηµνδ(σ, σ′) [ ]PB [ ] X µ(σ), X µ(σ′) = X˙ µ(σ), X˙ µ(σ′) = 0 PB PB

{ µ ν ′ } { µ ν ′ } − − ′ µν ψ+(σ), ψ+(σ ) = ψ−(σ), ψ−(σ ) = 2πiδ(σ σ )η , { µ ν ′ } ψ+(σ), ψ−(σ ) = 0

Using these brackets one finds ...... Superconformal algebra

[ ] ( ) ′ ′ ′ ′ ′ ′ T±±(σ), T±±(σ ) =  2T±±(σ )∂ + ∂ T±±(σ ) 2πδ(σ − σ ) ( ) [ ] ′ 3 ′ ′ ′ ′ ′ T±±(σ), T ±(σ ) =  T ±(σ )∂ + ∂ T ±(σ ) 2πδ(σ − σ ) F 2 F F { } ′ i ′ ′ T ±(σ), T ±(σ ) =  T±±(σ )2πδ(σ − σ ) F F 2 We can also verify the supersymmetry transformations [ √ ] 2 µ ′ 1 µ ′ T ±(σ), X (σ ) = ψ (σ)2πδ(σ − σ ) F α′ 2 ± √ { } µ ′ i 2 µ ′ ′ T ±(σ), ψ (σ ) = ∂±X (σ )2πδ(σ − σ ) F ± 2 α′ ...... We do this for the unconstrained system. The constraints then have to be imposed on the solutions. We have to distinguish between closed and open strings The treatment for the bosonic coordinates is identical to the bosonic string. Let us briefly recall it.

Oscillator expansions

We now solve the classical equations of motion in conformal gauge taking into account the boundary conditions.

...... We have to distinguish between closed and open strings The treatment for the bosonic coordinates is identical to the bosonic string. Let us briefly recall it.

Oscillator expansions

We now solve the classical equations of motion in conformal gauge taking into account the boundary conditions. We do this for the unconstrained system. The constraints then have to be imposed on the solutions.

...... The treatment for the bosonic coordinates is identical to the bosonic string. Let us briefly recall it.

Oscillator expansions

We now solve the classical equations of motion in conformal gauge taking into account the boundary conditions. We do this for the unconstrained system. The constraints then have to be imposed on the solutions. We have to distinguish between closed and open strings

...... Oscillator expansions

We now solve the classical equations of motion in conformal gauge taking into account the boundary conditions. We do this for the unconstrained system. The constraints then have to be imposed on the solutions. We have to distinguish between closed and open strings The treatment for the bosonic coordinates is identical to the bosonic string. Let us briefly recall it.

...... Oscillator expansions: Closed bosonic string

The general solution of the two-dimensional wave equation µ ∂+∂−X = 0, compatible with the periodicity condition X µ(σ, τ) = X µ(σ + l, τ) is µ µ − µ X (σ, τ) = XR (τ σ) + XL (τ + σ) where √ ′ ′ ∑ µ 1 πα α 1 − 2π − − µ µ − µ l in(τ σ) XR (τ σ) = x + p (τ σ) + i αn e 2 l 2 ̸ n √ n=0 ′ ′ ∑ µ 1 µ πα µ α 1 µ − 2π in(τ+σ) X (τ + σ) = x + p (τ + σ) + i α˜ e l L 2 l 2 n n n=0̸ ...... Oscillator expansions: Closed bosonic string

If we define √ α′ αµ =α ˜µ = pµ 0 0 2 we can write √ ′ ∑+∞ µ 2π α − 2π − µ ˙ µ l in(τ σ) ∂−X = XR = αn e l 2 −∞ √ n= ′ ∑+∞ µ µ 2π α µ − 2π in(τ+σ) ∂ X = X˙ = α˜ e l + L l 2 n n=−∞

...... Oscillator expansions: Closed bosonic string

From the Poisson brackets for the X µ, we derive the brackets for µ µ µ µ the αn , α˜n , x , p µ ν µ ν − µν [αm, αn]PB = [˜αm, α˜n]PB = imδm+nη , µ ν [˜αm, αn]PB = 0 , µ ν µν [x , p ]PB = η

...... Oscillator expansions: Open bosonic string

For the open string we have to require X ′µ = 0 at σ = 0 and σ = l. The general solution of the wave equation subject to these bdry cond is ′ √ ∑ ( ) µ µ 2πα µ ′ 1 µ −i π nτ nπσ X (τ, σ) = x + p τ + i 2α α e l cos l n n l n=0̸ √ µ ′ µ from which we get, with α0 = 2α p , √ ′ ∑+∞ µ 1 µ ′µ π α µ − πin (τ±σ) ∂±X = (X˙  X ) = α e l 2 l 2 n n=−∞ Then µ ν − µν µ ν µν [αm, αn] = imδm+nη , [x , p ] = η PB . . . PB...... Oscillator expansions of fermionic fields: closed string

The fermionic fields require some care. We have to distinguish between two choices of boundary cond for each chirality. The general solutions of the two-dimensional Dirac equation with periodic (R) and antiperiodic (NS) bdy cond are √ ∑ µ 2π ˜µ −2πir(τ+σ)/l ψ+(σ, τ) = br e l ∈Z r +ϕ { ϕ = 0 (R) where ϕ = 1 (NS) √ 2 2π ∑ ψµ (σ, τ) = bµe−2πir(τ−σ)/l − l r r∈Z+ϕ The reality of the Majorana spinors translates into µ ∗ µ µ ∗ µ (br ) = b− , (b˜r ) = b˜− ...... r r ...... Oscillator expansions of fermionic fields: closed string

In terms of the fermionic oscillator modes, the anticommutators

{ µ ν ′ } { µ ν ′ } − − ′ µν ψ+(σ), ψ+(σ ) = ψ−(σ), ψ−(σ ) = 2πiδ(σ σ )η , { µ ν ′ } ψ+(σ), ψ−(σ ) = 0

translate to

{ µ ν} − µν br , bs = iδr+s η , {˜µ ˜ν} − µν br , bs = iδr+s η , { µ ˜ν} br , bs = 0

Next we decompose the generators of conformal and superconformal transformations into modes ...... Oscillator expansions of T and TF  The conservation equations

∂−T++ = 0 , ∂+T−− = 0 =⇒ the existence of an infinite number of conserved charges: for + + any function f (σ ) we have ∂−(f (σ )T++) = 0 and the corresponding charges are ∫ l 1 + + + Lf = ′ dσ f (σ )T++(σ ) πα 0 and similarly for T−−. We can choose for f (σ±) a complete set satisfying the periodicity condition appropriate for the closed string: ( ) 2πi f (σ±) = exp mσ± for all integers m m l ...... Energy-momentum tensor and supercurrent

We then define the super-Virasoro generators as the corresponding charges at τ = 0 ∫ l l − 2πi nσ L = − dσe l T−− , n 4π2 ∫0 l l 2πi ˜ − l nσ Ln = 2 dσe T++ , 4π 0

√ ∫ l 1 l −2πirσ/l G = − dσe T −(σ) r π 2π F √ ∫0 l 1 l 2πirσ/l G˜r = − dσe TF +(σ) π 2π 0 ...... µ From the definition we see TF ± has the same periodicity as the ψ± √ 1 2 T ± = − ψ± · ∂±X F 2 α′ periodic in the R-sector and antiperiodic in the NS-sector =⇒ the mode numbers are integer and half-integer respectively.

(α) (b) In terms of oscillators Lm = Lm + Lm ∑ (α) 1 L = α− · α n 2 m m+n m∈Z ∑ ( ) (b) 1 n L = r + b− · b n 2 2 r n+r ∑r Gr = α−m · br+m m ∑ Note b−r · bn+r = 0 as it corresponds to ∂−(ψ−ψ−). It was included to make the expression look more symmetric.

...... (α) (b) In terms of oscillators Lm = Lm + Lm ∑ (α) 1 L = α− · α n 2 m m+n m∈Z ∑ ( ) (b) 1 n L = r + b− · b n 2 2 r n+r ∑r Gr = α−m · br+m m ∑ Note b−r · bn+r = 0 as it corresponds to ∂−(ψ−ψ−). It was included to make the expression look more symmetric. µ From the definition we see TF ± has the same periodicity as the ψ± √ 1 2 T ± = − ψ± · ∂±X F 2 α′ periodic in the R-sector and antiperiodic in the NS-sector =⇒ the mode numbers are integer and half-integer respectively...... (α) (b) In terms of oscillators Lm = Lm + Lm ∑ (α) 1 L = α− · α n 2 m m+n m∈Z ∑ ( ) (b) 1 n L = r + b− · b n 2 2 r n+r ∑r Gr = α−m · br+m m ∑ Note b−r · bn+r = 0 as it corresponds to ∂−(ψ−ψ−). It was included to make the expression look more symmetric. µ From the definition we see TF ± has the same periodicity as the ψ± √ 1 2 T ± = − ψ± · ∂±X F 2 α′ periodic in the R-sector and antiperiodic in the NS-sector =⇒ the mode numbers are integer and half-integer respectively...... Classical super-Virasoro algebra

The generators Lm and Gr satisfy the following reality conditions

∗ ∗ Lm = L−m , Gr = G−r

Using the basic brackets, one can now verify

[Lm, Ln] = −i(m − n)Lm+n 1 [L , G ] = −i( m − r)G m r 2 m+r {Gr , Gs } = −2iLr+s

It can also be derived from the Poisson brackets for T±± and TF ± and the definitions of Lm, Gr , L˜m, G˜r . For the closed string there are two copies of this algebra, one for the left- and one for the right-movers...... Oscillator expansions of fermionic fields: open string

For the open string we also expand the fermionic fields in modes and implement the bdy cond

µ µ µ µ ψ+(0) = ψ−(0) , ψ+(l) = ηψ−(l)

The bdy cond relate the left- and right-moving modes and there is only one set of oscillators. √ ∑ { π Z (R) ψµ (σ, τ) = bµe−πir(τ±σ)/l where r ∈ ± l r Z + 1 (NS) r 2

...... The bdy cond mix left- and right-movers and consequently T++ and T−− ∫ l l iπ − l nσ Ln = 2 dσe T++(σ) 2π − √ ∫l l 1 l iπrσ/l Gr = − dσe TF +(σ) π π −l ( ) 1 ∑ 1 ∑ n =⇒ L = α− · α + r + b− · b n 2 m m+n 2 2 r m+r m∈Z r ∑ { r ∈ Z for R G = α− · b with r m r+m r ∈ Z + 1 for NS m 2

Energy-momentum tensor and supercurrent

We now derive the mode expansions of the Virasoro generators for the open string.

...... Energy-momentum tensor and supercurrent

We now derive the mode expansions of the Virasoro generators for the open string. The bdy cond mix left- and right-movers and consequently T++ and T−− ∫ l l iπ − l nσ Ln = 2 dσe T++(σ) 2π − √ ∫l l 1 l iπrσ/l Gr = − dσe TF +(σ) π π −l ( ) 1 ∑ 1 ∑ n =⇒ L = α− · α + r + b− · b n 2 m m+n 2 2 r m+r m∈Z r ∑ { r ∈ Z for R G = α− · b with r m r+m r ∈ Z + 1 for NS m 2 ...... Classical super-Virasoro algebra

The generators Lm and Gr satisfy the following reality conditions

∗ ∗ Lm = L−m , Gr = G−r

Using the basic brackets, one can now verify

[Lm, Ln] = −i(m − n)Lm+n 1 [L , G ] = −i( m − r)G m r 2 m+r {Gr , Gs } = −2iLr+s

It can also be derived from the Poisson brackets for T±± and TF ± and the definitions of Lm, Gr , L˜m, G˜r . For the closed string there are two copies of this algebra, one for the left- and one for the right-movers......