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Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook 

World-sheet supersymmetry and 

supertargets 

Peter Browne Rønne 

University of the Witwatersrand, Johannesburg 

Chapel Hill, August 19, 2010 

arXiv:1006.5874 Thomas Creutzig, PR


Sigma-models on supergroups/cosets 

The Wess-Zumino-Novikov-Witten (WZNW) model of a Lie 

supergroup is a CFT with additional affine Lie superalgebra 

symmetry. 

Important role in physics 

• Statistical systems 

• String theory, AdS/CFT 

Notoriously hard: Deformations away from WZNW-point 

Use and explore the rich structure of dualities and 

correspondences in 2d field theories


Sigma models on supergroups: AdS/CFT 

The group of global symmetries of the gauge theory and also of 

the dual string theory is a Lie supergroup G. 

The dual string theory is described by a two-dimensional sigma 

model on a superspace. 

AdS 5 × S 5 PSU(2, 2|4) 

SO(4, 1) × SO(5) 

AdS 2 × S 2 PSU(1, 1|2) 

U(1) × U(1) 

supercoset 

supercoset 

AdS 3 × S 3 PSU(1, 1|2) supergroup 

AdS 3 × S 3 × S 3 D(2, 1; α) supergroup 

Obtained using GS or hybrid formalism. What about RNS 

formalism? And what are the precise relations?


AdS 3 × S 3 × T 4 

D1-D5 brane system on T 4 with near-horizon limit 

AdS 3 × S 3 × T 4 . After S-duality we have N = (1, 1) WS SUSY 

WZNW model on SL(2) × SU(2) × U 4 . It has N = (2, 2) 

superconformal symmetry (as we will see). 

Hybrid formalism of string theory on AdS 3 × S 3 gives a sigma 

model on PSU(1,1|2) . 

RR-flux is deformation away from WZ-point. 

[Berkovits, Vafa, Witten] 

Dual 2-dimensional CFT is deformation of symmetric orbifold 

Sym N (T 4 ) = (T 4 ) N /S N . It has N = (4, 4) superconformal 

symmetry.


Recent progress: Chiral ring 

[Kirsch, Gaberdiel][Dabholkar, Pakman] 

Computed chiral ring in the NS-formulation. 

Results match the boundary computations, but at a different 

point in moduli space. 

Correlators are very simple 

〈O 1 (z 1 , ¯z 1 )O 2 (z 2 , ¯z 2 )O 3 (z 3 , ¯z 3 )〉 = 

C 123 

∏ |zi − z j | ∆ ij 

where C 123 = 0 or C 123 = 1. 

• Why?


Correspondence 

S N =(2,2) 

GL(N)×GL(N) + 1 ∫ 

2π 

dτdσ Φ 

G + (z) 

G − (z) 

U(z) 

Def. + B-twist 

−→ 


−→ 


−→ 

S GL(N|N) 


−→ 

J B (z) . 

J F (z) 

∑ 

J F (z)J B (z) + ∂J F


Motivation 

Motivation 

Outline 

World-Sheet SUSY 

N = (1, 1) world-sheet SUSY 

N = (2, 2) world-sheet SUSY 

Lie superalgebras 

Superalgebras and supergroups 

Free fermion resolution 

From World-Sheet SUSY to Supertargets 

Twisting and TCFT 

Relation for GL(N|N) 

Deformations 

String theory on AdS 3 × S 3 × T 4 

Outlook 

Outlook


N = (1, 1) world-sheet SUSY WZNW model 

[di Vecchia, Knizhnik, Petersen, Rossi] 

• Bosonic Lie group H 

• Introduce superspace in 2d: θ 1 , θ 2 

• Map G : Σ 2|2 ↦→ H 

• Components 

G(z, ¯z, θ α ) = g + i ¯θψ + 1 2 i ¯θθF 

• N = (1, 1) WZNW Lagrangian 

S N =(1,1) 

WZNW [G] = 1 ∫ 

4λ 2 d 2 zd 2 θ tr( ¯DG −1 DG) 

+ k ∫ 

d 2 xd 2 θdt tr(G −1 ∂ t G 

16π 

¯DG −1 γ 5 DG)


• N = (1, 1) for all λ, k 

Decoupling the fermions 

• Enhanced affine symmetry at the WZNW point 

λ 2 = 8π 

k 

• WZNW model is “rigid”: Fermions decouple 

• {t a } basis for g 

• Redefine ψ ↦→ χ 

χ = χ a t a , 

¯χ = ¯χ a t a 

G = exp(iθχ) g exp(−i ¯θ ¯χ) 

S N =(1,1) 

WZNW [G] = S′ WZNW [g] + 1 ∫ 

2π 

d 2 z (χ, ¯∂χ) + (¯χ, ∂ ¯χ)


Quantum measure 

• ( , ) non-degenerate, invariant bi-linear form 

• Absorbed k into ( , ) 

• Quantum measure effect from chiral decoupling of 

fermions: S 

WZNW ′ [g] has bilinear form 

( , ) ↦→ ( , ) + 1 2 < , > Killing


• Ĝ × Ĝ affine algebra 

Current symmetries 

J a (z)J b (w) ∼ (ta , t b ) + 1 2 < ta , t b > Killing 

(z − w) 2 + f ab cJ c (w) 

(z − w) 

¯J a (¯z)¯J b ( ¯w) ∼ (ta , t b ) + 1 2 < ta , t b > Killing 

(¯z − ¯w) 2 

• Free fermions 

χ a (z)χ b (w) ∼ (ta , t b ) 

(z − w) 

+ f ab c¯J c ( ¯w) 

(¯z − ¯w) 

¯χ a (¯z)¯χ b ( ¯w) ∼ (ta , t b ) 

(¯z − ¯w) 

• Implies N = (1, 1) superconformal symmetry 

(schematically) 

T (z) = (J(z), J(z)) + (χ(z), ∂χ(z)) 

G(z) = (J(z), χ(z)) + ([χ(z), χ(z)], χ(z))


N = (2, 2) superconformal symmetry 

• When can we construct the N = (2, 2) current algebra? 

G + (z)G − (w) ∼ 

G ± (z)G ± (w) ∼ 0 

U(z)G ± (w) ∼ ±G± (w) 

(z − w) 

c/3 

U(z)U(w) ∼ 

(z − w) 2 

• Fermionic currents G ± , ∆(G ± ) = 3/2 

• Bosonic U(1) current U, ∆(U) = 1 

c/3 

(z − w) 3 + U(w) 

(z − w) 2 + T (w) + 1 2 ∂U(w) 

(z − w)


[Getzler] 

Manin decomposition 

• Current construction possible if algebra has Manin 

decomposition 

g = a + ⊕ a − 

• a ± isotropic Lie subalgebras 

• {x i } basis a + 

• {x i } dual basis a − 

(x i , x j ) = δ j i 

[x i , x j ] = c k ij x k 

[x i , x j ] = f ij kx k 

[x i , x j ] = c j ki x k + f jk ix k 

• Fermions now ( 1 2 , 1 2 

) ghost systems 

χ j (z)χ i (w) ∼ δj i /(z − w)


N = (2, 2) construction 

• Sugawara Virasoro tensor 

T (z) = 1 2 (:Ji J i : + :J i J i : + :∂χ i χ i : − :χ i ∂χ i :) 

• Fermionic currents 

G + (z) = J i χ i − 1 2 c ij k :χ i χ j χ k : 

G − (z) = J i χ i − 1 2 f ij k :χ i χ j χ k : 

• U(1) current 

U(z) = :χ i χ i : +ρ k J k + ρ k J k + c i mn f mn j :χ j χ i : . 

where 

ρ = −[x i , x i ]


Deformations 

• Wide range of deformations with background charges 

• Define a 0 

a 0 = {x ∈ g | (x, y) = 0 ∀ y ∈ [a + , a + ] ⊕ [a − , a − ]} 

• a = p i x i + q i x i ∈ a 0 

G a 

+ = G + + q i ∂χ i 

• Implies 

G − a 

= G − + p i ∂χ i 

U a (z) = U(z) + p i I i (z) − q i I i (z) 

T a (z) = T (z) + 1 2 (pi ∂I i (z) + q i ∂I i (z)) 

• Super-version of affine currents 

I i = J i − c k ij :χ j χ k : − 1 2 f jk i :χ j χ k : 

I i = J i − f ij k :χ j χ k : − 1 2 c jk i :χ j χ k :


Deformations 

• Chiral ring unchanged as vector space 

• Actually, unchanged as ring: Deformation acts as spectral 

flow on the zero modes 

• Does change conformal dimension 

c a = c − 6q i p i


Lie superalgebras and supergroups 

• Lie Superalgebra G = G 0 ⊕ G 1 

• Z 2 graded product 

[G 0 , G 0 ] ⊂ G 0 ( subalgebra) 

[G 0 , G 1 ] ⊂ G 1 

[G 1 , G 1 ] ⊂ G 0 

• Graded anti-symmetry 

[X, Y ] = −[Y , X] 

[X, Y ] = −[Y , X] 

[X, Y ] = [Y , X] 

• Generalised Jacobi identity


Lie superalgebras and supergroups 

• Fundamental supermatrix representation, gl(M|N) 

( ) A B 

M = 

C D 

• Supertrace (invariant, supersymmetric bilinear form) 

strM = tr(A) − tr(D) 

• Grassmann envelope η (0) 

a T (0) 

a + η (1) 

b T (1) 

b 

η (0) 

a , η (1) 

b 

Grassmann elements 

T (0) 

a , T (1) 

b 

basis for G = G 0 ⊕ G 1 

• Supergroup 

( 

) 

exp η (0) 

a T (0) 

a + η (1) 

b T (1) 

b



[Quella, Schomerus] 

• Introduce generators for super Lie algebra 

( K 

i 

S a 1 

S 2a K i ) 

• Non-degenerate invariant bi-linear form 〈A, B〉 = kstr(AB) 

〈S a 1 , S 2b〉 = kδ a b 

〈K i , K j 〉 ≡ κ ij 

• Fermions transform in representation of bosonic subgroup 

• Fermionic fields c, ¯c 

[K i , S a 1 ] = −(Ri ) a b Sb 1 

c = c a S 2a , ¯c = ¯c a S a 1 

• Parametrization (Type I supergroup) 

g = e c g B e¯c



• Free fermion resolution: Introduce auxiliary fields 

b = b a S a 1 , 

¯b = ¯ba S 2a 

• Action 

• Screening charges 

S pert = − 1 

4πk 

S WZNW [g] = S 0 + S pert 

∫ 

Σ 

d 2 σ str(Ad(g B )(¯b)b) 

• Bosonic WZNW, background charges, (1, 0) bc-ghosts 

S 0 = Sren WZNW [g B ] − 1 ∫ 

dz 2√ hR (2) 1 

4π 

2 ln det R(g B) 

+ 1 ∫ 

dz 2 ( b a ¯∂c a − 

2π 

¯b a ∂¯c a)



• Quantum measure (again) 

〈K i B , K j B 〉 ren = κ ij − γ ij , γ ij = trR i R j , 

• Free fermion currents 

K i (z) = KB i + b a(R i ) a b cb 

S1 a(z) = k∂ca + k(R i ) a b κ ijc b K j B − k 2 (Ri ) a b κ ij(R j ) c d b cc b c d 

S 2a (z) = −b a 

• Stress-energy tensor 

( 

) 

TG FF = 1 2 

KB i Ω ijK j B + tr(ΩRi )κ ij ∂K j B 

− b a ∂c a , 

(Ω −1 ) ij = κ ij − γ ij + 1 2 f im nf jn m, 

(Ω −1 ) a b = δa b + (Ri κ ij R j ) a b .


Topological conformal field theory 

• BRST-currents/charges Q, ¯Q 

• Cohomology by Q or Q + ¯Q 

• Conformal tensor exact 

H phys = kernel(Q) 

image(Q) 

T (z) = [Q, G(z)] 

¯T (¯z) = [Q, Ḡ(¯z)] 

• Cohomology ring: Position independent structure 

constants 

φ i φ j ∼ a ij k φ k


TCFT from N = (2, 2) by twisting 

• Positive B-twist 

• Gives TCFT with c = 0 

T + 

twisted (z) = T (z) + 1 2 ∂U(z) , 

¯T + 

twisted (¯z) = ¯T (¯z) + 1 2 ¯∂Ū(¯z) . 

• ∆(G + , G − , U) = (3/2, 3/2, 1) ↦→ (1, 2, 1) 

• G + ↦→ Q 

• Chiral ring maps to cohomology 

• G − cohomology partner to T + 

twisted 

• U partner to Q 

T + 

twisted = [Q, G− ]


From WS SUSY to target superspaces 

• Idea: Consider Lie supergroup G 

• Look at free fermion resolution of WZNW model for G 

• Construct N = (2, 2) WS SUSY theory on the bosonic 

subgroup G B 

• By twisting reach TCFT with T + 

twisted = T G 

FF and Q one of 

the fermionic currents 

• Immediate constraint sdimg = 0 

• PSL(N|N) does not work, and no topological sectors were 

found 

• What about GL(N|N)?


• Generators ( 

GL(N|N) 

E αβ 

+ F αβ 

+ 

F αβ 

− 

E αβ 

− 

• Invariant form kstr (positive/negative on bosonic part) 

kstr(E αβ 

ɛ 

) 

E γδ 

ɛ 

) = κ ′ 

(αβ ɛ )( γδ 

ɛ ′ ) = kɛδ ɛɛ ′δ αδ δ βγ 

• Can calculate free fermion currents and stress-energy 

tensor 

• Consider GL(N) × GL(N) bosonic base generated by E αβ 

± 

• Choose initial metric on bosonic base 

κ start = κ ij − γ ij + 1 2 f im nf jn m 

• Extra terms only changes metric on U(1) factors 

κ start = (E αβ 

ɛ 

, E γδ 

ɛ 

) = kɛδ ′ ɛɛ ′δ αδ δ βγ − ɛɛ ′ δ αβ δ γδ 

• Simply corresponds to field redefinition of U(1) factors


Choose Manin decomposition 

• The right choice turns out to be 

gl(N) ⊕ gl(N) = a + ⊕ a − 

a + = span{E αβ 

+ + E αβ 

− } 

a − = span{ E + αα − E− 

αα 

2k 

+ 1 

2k 

2 Id, E αβ 

+ 

k 

• Can then calculate SUSY currents G ± , U, T N =(2,2) 

, E βα 

− 

−k | α > β}


• Deformation parameter 

Fix deformations 

a 0 = span{x αα , ∑ α 

x αα } 

a = p αα x αα + q ∑ α 

x αα 

• Twist 

T + 

twisted = T N =(2,2) 

a + 1 2 ∂U a 

• p αα fixed by 

T + 

twisted = T GL(N|N) 

• The single remaining parameter q fixed by 

G + a is fermionic current 

• c = 3N 2 ↦→ c a = 0 (before twisting) 

• Whole deformation can be seen as background charge


WS SUSY currents as super Lie currents 

• After the twist χ i , χ i ghost system (1/2, 1/2) ↦→ (1, 0) 

• Fix dictionary-automorphism of ghosts 

c αβ = −χ βα , 

b αβ = −χ βα 

• We got what we aimed for 

G + a = ∑ α 

F αα 

+ 

• ... and a surprise 

U a = 1 

2k 

∑α 

Ga − = 1 

1 

k 

2k 

∑α>β 

(k + N + 1 − 2α)Ẽ 

αα 

∑α F αα 

− 

(Ẽ 

αα 

αβ ˜F − Ẽ βα 

− + 1 k 

+ + 1 

2k 

+ 1 ββ 

k 

∑β 

(Ẽ 

+ − Ẽ − 

αα 

∑α


Deformations with chiral fields 

• N = (2, 2) WS SUSY: chiral and twisted chiral 

supermultiplets (and anti-) 

• Classical SUSY-preserving deformation by F- and D-terms 

• Chiral multiplet generated by field in (++)-ring 

• Remaining components 

[G + −1/2 , φ ++] = [Ḡ+ −1/2 , φ ++] = 0 

ψ ++ = −[G − −1/2 , φ ++], ¯ψ ++ = −[Ḡ− −1/2 , φ ++] 

F ++ = −{G − −1/2 , [Ḡ− −1/2 , φ ++]} 

• F-term perturbation non-exact in twisted theory 

• Anti-chiral and twisted (anti-)chiral F-term perturbations 

are exact in G + + Ḡ+ cohomology


Screening charges in supergroup model 

• We achieved T + 

twisted = T GL(N|N) ⇒ 

S GL(N)×GL(N) 

N =(1,1) 

+ deformations + twist = S GL(N|N) 

0 

• Free fermion resolution boson-fermion interaction terms 

S pert = − 1 

4πk 

• Are screening charges: 

∫ 

Σ 

d 2 σ str(Ad(g B )(¯b)b) 

J(z)L pert ∼ total derivatives 

• Conclusion: Is (classical) SUSY deformation 

• Is in fact chiral F-term perturbation: Take g B = 

( ) A 0 

0 B 

• φ = tr(A −1 B) in (+, +)-ring, has dimension (1/2, 1/2) 

F ++ = −{G − −1/2 , [Ḡ− −1/2 , φ ++]} = 4π 

k L pert


Bosonic screening charges 

• Free fermion resolution gave us 

S 0 = Sren WZNW [g B ] − 1 ∫ 

dz 2√ hR (2) 1 

4π 

2 ln det R(g B) 

+ 1 ∫ 

dz 2 ( b a ¯∂c a − 

2π 

¯b a ∂¯c a) 

• GL(2) Gauss decomposition 

exp γ 

( ) ( ( ) ( )) 

0 1 

1 0 1 0 

exp φ 

0 0 

0 + φ 

0 1 z exp ¯γ 

0 −1 

• First order formalism: Introduce auxiliary fields β and ¯β 

∫ 

Sren WZNW [g B ] = S kin+b.g. [φ 0 , φ z , β, γ] + d 2 zβ ¯βe φz/√ k 

• Screening charges are chiral F-terms. 

( ) 0 0 

1 0


Exactness 

• Free fermion resolution AND bosonic first order formalism 

S GL(2|2) = S kin+b.g. [φ ± 0 , φ± z , β ± , γ ± ] 

} {{ } 

G + +Ḡ+ exact 

+ S bos. screening + S bos.-ferm. interaction 

} {{ } 

G − +Ḡ− exact


Principal chiral field deformations 

• Principal chiral field 

S geom. = − k ∫ 

∫ 

d 2 z 〈g −1 ∂g, g −1 ¯∂g〉 ≡ 

4π 

d 2 z Φ principal 

• PSU(1, 1|2) deformation of coefficient principal chiral field 

to turning on RR-flux in string theory 

• Can show this is D-term 

• Is G + -exact 

Φ principal = −{G + −1/2 , [Ḡ+ −1/2 , φ]} 

φ = − 1 ( 0 Id 

4πk 〈 Id 0 

) ( 

g −1 0 Id 

Id 0 

) 

Jg¯J〉



• The procedure works for string theory on AdS 3 × S 3 × T 4 

• AdS 3 × S 3 

N = (2, 2) WZNW on SU(1, 1) × SU(2) 

• T 4 

N = (2, 2) WZNW on U(1) 4 

• String ghosts – can be seen as 

N = (2, 2) WZNW on U(1) 2 

• Divide into 

U(1, 1) × U(2) ↦→ U(1, 1|2) 

U(1) × U(1) ↦→ U(1|1) 

U(1) × U(1) ↦→ U(1|1) 

• String theory has total central charge zero



• Our method: Before twisting, after deformation 

c = 0 

• Comparison to string theory 

G − string = G− , 

U string = U, 

G + string = G+ + extra terms 

• Novel choice of N = (2, 2) 

• G − chiral ring is the same as in string theory!


Conclusions and Outlook 

• Start from bosonic group, add background charges plus 

twist −→ Supergroup 

• Interaction terms are G − -exact F-terms 

• G − chiral ring is the same as in string theory 

• Calculate extremal correlators in string theory on 

AdS 3 × S 3 × T 4 , higher genus! 

• Gives proposal for boundary action via analogy with 

Landau-Ginzburg model – this should be checked explicitly 

• Also works for SL(2|1) 

• Other groups? Supergroups? 

• N = (4, 4) supersymmetric point 

• Is it possible to use equivariant localisation in this context?

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