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 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 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 Motivation World-Sheet SUSY Lie superalgebras From World-Sheet SUSY to Supertargets Outlook 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. PSU(2; 2j4) AdS × S5 supercoset 5 SO(4; 1) × SO(5) PSU(1; 1j2) AdS × S2 supercoset 2 U(1) × U(1) 3 AdS3 × S PSU(1; 1j2) 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 brane 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(1,1j2) . RR-flux is deformation away from WZ-point. [Berkovits, Vafa, Witten] Dual 2-dimensional CFT is deformation of symmetric orbifold 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 moduli space. Correlators are very simple C hO ( ; ¯ )O ( ; ¯ )O ( ; ¯ )i = 123 1 z1 z1 2 z2 z2 3 z3 z3 Q ∆ jzi − zj j 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(NjN) + 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 fermion resolution From World-Sheet SUSY to Supertargets Twisting and TCFT Relation for GL(NjN) 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 Lie group H • Introduce superspace in 2d: θ1; θ2 • Map G :Σ2j2 7! H • Components ¯ α ¯ 1 ¯ G(z; z; θ ) = g + iθ + 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 fermions • N = (1; 1) for all λ, k • Enhanced affine symmetry at the WZNW point 8π λ2 = k • WZNW model is “rigid”: Fermions decouple • ftag basis for g • Redefine 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 • G^ × G^ 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 • fx g basis a+ • fxi g 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 = fx 2 g j (x; y) = 0 8 y 2 [a+; a+] ⊕ [a−; a−]g i i • a = p xi + qi x 2 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 • Lie Superalgebra 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(MjN) 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 Lie algebra 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 p 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+;