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

World-sheet and supertargets

Peter Browne Rønne

University of the Witwatersrand, Johannesburg

Chapel Hill, August 19, 2010 arXiv:1006.5874 Thomas Creutzig, PR Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Sigma-models on supergroups/cosets

The Wess-Zumino-Novikov-Witten (WZNW) model of a Lie is a CFT with additional affine Lie .

Important role in physics • Statistical systems • theory, AdS/CFT Notoriously hard: Deformations away from WZNW-point

Use and explore the rich structure of dualities and correspondences in 2d field theories Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Sigma models on supergroups: AdS/CFT The of global of the and also of the dual is a Lie supergroup G.

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

PSU(2, 2|4) AdS × S5 supercoset 5 SO(4, 1) × SO(5) PSU(1, 1|2) AdS × S2 supercoset 2 U(1) × U(1)

3 AdS3 × S PSU(1, 1|2) supergroup

3 3 AdS3 × S × S D(2, 1; α) supergroup Obtained using GS or hybrid formalism. What about RNS formalism? And what are the precise relations? Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

3 4 AdS3 × S × T D1-D5 system on T 4 with near-horizon limit 3 4 AdS3 × S × T . After S-duality we have N = (1, 1) WS SUSY WZNW model on SL(2) × SU(2) × U4. It has N = (2, 2) superconformal symmetry (as we will see).

3 Hybrid formalism of string theory on AdS3 × S 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 4 4 N SymN (T ) = (T ) /SN . It has N = (4, 4) superconformal symmetry. Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

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 .

Correlators are very simple

C hO ( , ¯ )O ( , ¯ )O ( , ¯ )i = 123 1 z1 z1 2 z2 z2 3 z3 z3 Q ∆ |zi − zj | ij

where C123 = 0 or C123 = 1. • Why? Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Correspondence

1 Z SN =(2,2) + dτdσ Φ Def. + B-twist S GL(N)×GL(N) 2π −→ GL(N|N) + Def. + B-twist F G (z) −→ J (z) − Def. + B-twist X F B F G (z) −→ J (z)J (z) + ∂J Def. + B-twist B U(z) −→ J (z) . Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Outline Motivation Motivation World-Sheet SUSY N = (1, 1) world-sheet SUSY N = (2, 2) world-sheet SUSY Lie superalgebras Superalgebras and supergroups Free resolution From World-Sheet SUSY to Supertargets Twisting and TCFT Relation for GL(N|N) Deformations 3 4 String theory on AdS3 × S × T Outlook Outlook Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

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

[di Vecchia, Knizhnik, Petersen, Rossi] • Bosonic H • Introduce superspace in 2d: θ1, θ2

• Map G :Σ2|2 7→ H • Components

¯ α ¯ 1 ¯ G(z, z, θ ) = g + iθψ + 2 iθθF • N = (1, 1) WZNW Lagrangian

1 Z SN =(1,1)[G] = d 2zd 2θ tr(DG¯ −1DG) WZNW 4λ2 k Z + d 2xd 2θdt tr(G−1∂ GDG¯ −1γ DG) 16π t 5 Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Decoupling the • N = (1, 1) for all λ, k • Enhanced affine symmetry at the WZNW point

8π λ2 = k • WZNW model is “rigid”: Fermions decouple • {ta} basis for g • Redefine ψ 7→ χ

a a χ = χat , χ¯ =χ ¯at

G = exp(iθχ) g exp(−iθ¯χ¯)

1 Z SN =(1,1)[G] = S0 [g] + d 2z (χ, ∂χ¯ ) + (¯χ, ∂χ¯) WZNW WZNW 2π Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Quantum measure

• ( , ) non-degenerate, invariant bi-linear form • Absorbed k into ( , ) • Quantum measure effect from chiral decoupling of 0 fermions: SWZNW [g] has bilinear form

1 ( , ) 7→ ( , ) + 2 < , >Killing Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Current symmetries • Gˆ × Gˆ affine algebra

a b 1 a b ab c (t , t ) + < t , t >Killing f J (w) Ja(z)Jb(w) ∼ 2 + c (z − w)2 (z − w) a b 1 a b ab c (t , t ) + < t , t >Killing f J¯ (w¯ ) J¯a(z¯)J¯b(w¯ ) ∼ 2 + c (z¯ − w¯ )2 (z¯ − w¯ ) • Free fermions (ta, tb) (ta, tb) χa(z)χb(w) ∼ χ¯a(z¯)¯χb(w¯ ) ∼ (z − w) (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)) Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

N = (2, 2) superconformal symmetry

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

c/3 U(w) T (w) + 1 ∂U(w) G+(z)G−(w) ∼ + + 2 (z − w)3 (z − w)2 (z − w) G±(z)G±(w) ∼ 0 ±G±(w) U(z)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 Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Manin decomposition [Getzler] • Current construction possible if algebra has Manin decomposition g = a+ ⊕ a−

• a± isotropic Lie subalgebras i • {x } basis a+

• {xi } dual basis a−

j j (xi , x ) = δi k [xi , xj ] = cij xk i j ij k [x , x ] = f k x j j k jk [xi , x ] = cki x + f i xk

• 1 1 Fermions now ( 2 , 2 ) ghost systems i i χj (z)χ (w) ∼ δj /(z − w) Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

N = (2, 2) construction • Sugawara Virasoro tensor

1 T (z) = (:Ji J : + :J Ji: + :∂χi χ : − :χi ∂χ :) 2 i i i i • Fermionic currents 1 G+(z) = J χi − c k :χi χj χ : i 2 ij k 1 G−(z) = Ji χ − f ij :χ χ χk: i 2 k i j • U(1) current

i k k i mn j U(z) = :χ χi: +ρ Jk + ρk J + cmn f j :χ χi: .

where i ρ = −[x , xi ] Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Deformations • Wide range of deformations with background charges • Define a0 a0 = {x ∈ g | (x, y) = 0 ∀ y ∈ [a+, a+] ⊕ [a−, a−]} i i • a = p xi + qi x ∈ a0 + + i Ga = G + qi ∂χ − − i Ga = G + p ∂χi • Implies i i Ua(z) = U(z) + p Ii (z) − qi I (z) 1 T (z) = T (z) + (pi ∂I (z) + q ∂Ii (z)) a 2 i i • Super-version of affine currents 1 I = J − c k :χj χ : − f jk :χ χ : i i ij k 2 i j k 1 Ii = Ji − f ij :χ χk: − c i :χj χk: k j 2 jk Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Deformations

• Chiral ring unchanged as vector space • Actually, unchanged as ring: Deformation acts as spectral flow on the zero modes • Does change conformal dimension

i ca = c − 6qi p Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Lie superalgebras and supergroups

• G = G0 ⊕G 1 • Z2 graded product

[G0, G0] ⊂G 0 ( subalgebra)

[G0, G1] ⊂G 1

[G1, G1] ⊂G 0

• Graded anti-symmetry

[X, Y ] = −[Y , X] [X, Y ] = −[Y , X] [X, Y ] = [Y , X]

• Generalised Jacobi identity Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

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)

(0) (0) (1) (1) • Grassmann envelope ηa Ta + ηb Tb (0) (1) ηa ,η b Grassmann elements (0) (1) Ta , Tb basis for G = G0 ⊕G 1 • Supergroup  (0) (0) (1) (1) exp ηa Ta + ηb Tb Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Free fermion resolution [Quella, Schomerus] • Introduce generators for super  i a  K S1 i S2a K • Non-degenerate invariant bi-linear form hA, Bi = kstr(AB) a a hS1, S2bi = kδb hK i , K j i ≡ κij • Fermions transform in representation of bosonic subgroup i a i a b [K , S1] = −(R ) bS1 • Fermionic fields c, c¯ a a c = c S2a, c¯ = c¯aS1 • Parametrization (Type I supergroup) c c¯ g = e gBe Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Free fermion resolution • Free fermion resolution: Introduce auxiliary fields

a ¯ ¯a b = baS1, b = b S2a • Action WZNW S [g] = S0 + Spert • Screening charges Z 1 2 ¯ Spert = − d σ str(Ad(gB)(b)b) 4πk Σ • Bosonic WZNW, background charges, (1, 0) bc-ghosts 1 Z √ S = SWZNW[g ] − dz2 hR(2) 1 ln det R(g ) 0 ren B 4π 2 B 1 Z + dz2 b ∂¯ca − b¯ ∂c¯a 2π a a Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Free fermion resolution • Quantum measure (again)

i j ij ij ij i j hKB, KBiren = κ − γ , γ = trR R , • Free fermion currents i i i a b K (z) = KB + ba(R ) bc a a i a b j k i a j c b d S1(z) = k∂c + k(R ) bκij c KB − 2 (R ) bκij (R ) d bcc c S2a(z) = −ba • Stress-energy tensor

FF 1  i j i j  a TG = 2 KBΩij KB + tr(ΩR )κij ∂KB − ba∂c ,

−1 ij ij ij 1 im jn (Ω ) = κ − γ + 2 f nf m, −1 a a i j a (Ω ) b = δb + (R κij R ) b. Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Topological conformal field theory

• BRST-currents/charges Q, Q¯ • Cohomology by Q or Q + Q¯

kernel(Q) H = phys image(Q)

• Conformal tensor exact

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

• Cohomology ring: Position independent structure constants k φi φj ∼ aij φk Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

TCFT from N = (2, 2) by twisting

• Positive B-twist 1 T + (z) = T (z) + ∂U(z) , twisted 2 1 T¯ + (z¯) = T¯(z¯) + ∂¯U¯ (z¯) . twisted 2 • Gives TCFT with c = 0 • ∆(G+, G−, U) = (3/2, 3/2, 1) 7→ (1, 2, 1) • G+ 7→ Q • Chiral ring maps to cohomology − + • G cohomology partner to Ttwisted

+ − Ttwisted = [Q, G ] • U partner to Q Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

From WS SUSY to target

• 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 GB + FF • By twisting reach TCFT with Ttwisted = TG 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)? Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

GL(N|N) • Generators αβ αβ ! E+ F+ αβ αβ F− E− • Invariant form kstr (positive/negative on bosonic part)

γδ αβ γδ αβ (  )( 0 ) αδ βγ kstr(E E0 ) = κ = kδ0 δ δ • Can calculate free fermion currents and stress-energy tensor αβ • Consider GL(N) × GL(N) bosonic base generated by E± • Choose initial metric on bosonic base ij ij 1 im jn κstart = κ − γ + 2 f nf m • Extra terms only changes metric on U(1) factors αβ γδ αδ βγ 0 αβ γδ κstart = (E , E0 ) = kδ0 δ δ −  δ δ • Simply corresponds to field redefinition of U(1) factors Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Choose Manin decomposition

• The right choice turns out to be

gl(N) ⊕ gl(N) = a+ ⊕ a− αβ αβ a+ = span{E+ + E− } Eαα − Eαα 1 Eαβ Eβα a = span{ + − + Id, + , − | α > β} − 2k 2k 2 k −k

• Can then calculate SUSY currents G±, U, T N =(2,2) Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Fix deformations • Deformation parameter

X αα a0 = span{xαα, x } α

αα X αα a = p xαα + q x α • Twist 1 T + = T N =(2,2) + ∂U twisted a 2 a • pαα fixed by + Ttwisted = TGL(N|N) • The single remaining parameter q fixed by + Ga is fermionic current 2 • c = 3N 7→ ca = 0 (before twisting) • Whole deformation can be seen as background charge Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

WS SUSY currents as super Lie currents

i • After the twist χ , χi ghost system (1/2, 1/2) 7→ (1, 0) • Fix dictionary-automorphism of ghosts

αβ βα c = −χ , bαβ = −χβα

• We got what we aimed for

+ X αα Ga = F+ α • ... and a surprise 1 P ˜ αα 1 P ˜ αα Ua = 2k α(k + N + 1 − 2α)E+ + 2k α(−k + N + 1 − 2α)E− − 1 P αα ˜ αα ˜ αα 1 P ˜ ββ ˜ ββ  Ga = 2k α F− E+ − E− + k β(E+ + E− ) − 1 P ˜ αβ ˜ βα 1 P αβ ˜ βα 1 P αα k α>β F− E− + k α<β F− E+ − 2k α(2N − 2α + 1)∂F− Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

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

+ ¯ + [G−1/2, φ++] = [G−1/2, φ++] = 0

• Remaining components

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

− ¯ − F++ = −{G−1/2, [G−1/2, φ++]} • F-term perturbation non-exact in twisted theory • Anti-chiral and twisted (anti-)chiral F-term perturbations are exact in G+ + G¯ + cohomology Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Screening charges in supergroup model + • We achieved Ttwisted = TGL(N|N) ⇒ GL(N)×GL(N) GL(N|N) SN =(1,1) + deformations + twist = S0 • Free fermion resolution -fermion interaction terms Z 1 2 ¯ Spert = − d σ str(Ad(gB)(b)b) 4πk Σ • Are screening charges:

J(z)Lpert ∼ total derivatives • Conclusion: Is (classical) SUSY deformation A 0 • Is in fact chiral F-term perturbation: Take g = B 0 B • φ = tr(A−1B) in (+, +)-ring, has dimension (1/2, 1/2) 4π F = −{G− , [G¯ − , φ ]} = L ++ −1/2 −1/2 ++ k pert Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Bosonic screening charges • Free fermion resolution gave us 1 Z √ S = SWZNW[g ] − dz2 hR(2) 1 ln det R(g ) 0 ren B 4π 2 B 1 Z + dz2 b ∂¯ca − b¯ ∂c¯a 2π a a • GL(2) Gauss decomposition

0 1  1 0 1 0  0 0 exp γ exp φ + φ exp γ¯ 0 0 0 0 1 z 0 −1 1 0

• First order formalism: Introduce auxiliary fields β and β¯ Z √ WZNW 2 ¯ φz / k Sren [gB] = Skin+b.g.[φ0, φz , β, γ] + d zββe

• Screening charges are chiral F-terms. Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Exactness

• Free fermion resolution AND bosonic first order formalism

GL(2|2) ± ± ± ± S = Skin+b.g.[φ0 , φz , β , γ ] | {z } G++G¯ + exact

+ Sbos. screening + Sbos.-ferm. interaction | {z } G−+G¯ − exact Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

Principal chiral field deformations

• Principal chiral field

k Z Z S = − d 2z hg−1∂g, g−1∂¯gi ≡ d 2z Φ geom. 4π 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

+ ¯ + Φprincipal = −{G−1/2, [G−1/2, φ]}

1  0 Id   0 Id  φ = − h g−1 JgJ¯i 4πk Id 0 Id 0 Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

3 4 String theory on AdS3 × S × T

3 4 • The procedure works for string theory on AdS3 × S × T 3 • AdS3 × S N = (2, 2) WZNW on SU(1, 1) × SU(2) • T4 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) 7→ U(1, 1|2) U(1) × U(1) 7→ U(1|1) U(1) × U(1) 7→ U(1|1) • String theory has total zero Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

3 4 String theory on AdS3 × S × T

• Our method: Before twisting, after deformation

c = 0

• Comparison to string theory

− − Gstring = G ,

Ustring = U, + + Gstring = G + extra terms

• Novel choice of N = (2, 2) • G− chiral ring is the same as in string theory! Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook

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 3 4 AdS3 × S × T , 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?