Supergroup Wess-Zumino-Witten Models a Geometric Road to Logarithmic Conformal ﬁeld Theories

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Supergroup Wess-Zumino-Witten models A geometric road to logarithmic conformal ﬁeld theories

Thomas Quella University of Amsterdam [email protected]

Introduction Supergroup WZW models Symmetries WZW models are the most important building block in the Supergroup WZW models describe maps from a 2D Rie- The WZW model has an obvious g ⊕ g symmetry corre- toolkit of 2D conformal field theories. Their features: mann surface into a supergroup. The WZW Lagrangian for sponding to the regular action of G on itself. This action maps g :Σ → G from a two-dimensional Riemann surface extends to an action of the full, centrally extended, loop Lie • Description of string propagation on a Lie group such as Σ to the supergroup G is fully specified in terms of an in- superalgebra gˆ ⊕ gˆ, giving the state space H the structure of SU(2) or AdS3, the universal cover of SLf (2, R). variant metric on g and it reads a gˆ ⊕ gˆ-module. • Geometric interpretation. Z Z ⊕ WZW 1 −1 −1 i The WZW models also possesses a Vir Vir Virasoro al- •∞ -dimensional affine Kac-Moody algebra symmetry. S [g] = − hg dg, g ∗ dgi − Ω3 . 4π Σ 24π B gebra symmetry which is embedded into the universal en- veloping algebra U(gˆ ⊕ gˆ). Hence, there is also the structure Replacing the Lie group by a Lie supergroup leads to a new −1 −1 −1 In the second term, the 3-form Ω3 = hg dg, [g dg, g dg]i of a vertex superalgebra. phenomenon: Logarithmic correlation functions. is integrated over an auxiliary three-manifold B which sat- From far away a small string resembles a point-like parti- isfies ∂B = Σ. If Ω is an “integral” cohomology class then This poster focuses on two aspects: 3 cle, i.e. the dynamics of point-particles on supergroups is exp(2πiSWZW[g]) B h·, ·i • The reduction of supergroup WZW models to WZW mod- is independent of . If one expresses included in the description of the WZW model. In fact, k els on the bosonic base. as times the Killing form this is the usual quantization in this case one can turn the argument around and use the k • The supergeometric origin of the logarithms. of the level . (However, for simple Lie superalgebras the affine symmetry to lift the results for point particles to the Killing form might vanish.) full WZW model. In other words, one has: Finally, I will sketch relations to other topics of recent inter- The Lagrangian S[g] describes the propagation of strings on est, especially to twisted K-theory. ∼ 3 ⇒ the supergroup G, such as PSU(1, 1|2) = SUSY(AdS3 × S ). Harmonic analysis full CFT

Left action Right action Left-right action D-brane annihilation g g g L K ⊗ K∗ µ µ µ K-theory: Supergeometry: S3 S3 Typical - Harmonic analysis on supergroups Generalization of Freed-Hopkins-Teleman? L K dimK∗ L dimK K∗ sector µ µ µ µ µ µ ? g g g - Coadjoint orbits (“D-branes”) Verlinde algebra ↔ twisted K-theory

g g g I[σ] M σ Atypical dim λ × (µ) −→ Nλµ (σ) L ∗ L ∗ Pµ dim L dim Lµ P sector | {z } µ µ µ µ Afﬁne Lie superalgebras: superposition σ g g g String theory: - Representation theory K-theory = D-branes/Dynamical relations Harmonic analysis on GL(1|1) - Strings in ﬂux backgrounds WZW Models - Character formulas - AdS/CFT Duality - Verlinde algebra

Open and The non-diagonalizability of the energy operator L0 implies D-branes closed strings Vertex operator algebras: logarithmic behavior of correlation functions - Logarithmic modules h 1 1 0 0 1 L = ⇒ zL0 = zh + ln(z) - Non-semisimple fusion categories 0 0 h 0 1 0 0 Open and closed strings on SU(2)

Decoupling bosons and fermions Harmonic analysis Representations and characters Assuming the existence of a -compatible -grading g = The structure of the modules H and can be recovered ˆ ren ˆ Z2 Z H Let Lµ be a simple gˆ0 -module and let Lpθ be the unique g1 ⊕ g0 ⊕ g−1 allows to make a Gauss-like decomposition from harmonic analysis on the supergroup G. module of the fermions. Then one can define the natural ¯ gˆ-module g = eθ g eθ (θ, g , θ¯) ↔ (g , g , g ) . Conformal Field Theory Geometry ˆ ˆ ˆ 0 with 0 1 0 −1 Kµ = Lµ ⊗ Lpθ , Ground states of H Algebra of functions F(G) Furthermore, introducing canonical momenta p and p¯ the g ⊕ g ⊂ gˆ ⊕ gˆ Regular action of g ⊕ g whose characters are easy to write down. Moreover, for generic choice of µ these modules are simple. The modules Lagrangian may be written as a sum Energy operator L0 Laplacian ∆ = Casimir Kˆµ are thus the natural analogue of Kac modules in the con- WZW WZW ¯ S [g0, p, θ] = Sren [g0] + Sfree[θ, p, θ, p¯] + Sint[g0, p, p¯] In fact, just as one has to distinguish the modules H and H text of affine Lie superalgebras. one also has to distinguish two algebras of functions F(G) The structure of atypical Kac modules Kˆµ of gˆ can be un- of a (metric-)renormalized bosonic WZW model on G0, free and (G) which possess a different g ⊕ g-module structure F K g (symplectic) fermions (of spin 0) and an interaction term. In and different Laplacians. Their “typical part” agrees, derstood in terms of that of the Kac modules µ of using fact, one has the following embedding of vertex algebras, spectral flow (aka the affine Weyl group) (see e.g. [3]). M F(G) ∼ (G) ∼ L ⊗ L∗ , typ = F typ = µ µ U(gˆ) ,→ U gˆren ⊕ pθ-fermions , 0 µ typical Logarithmic behavior which is crucial for all what follows. By dropping the inter- being a sum over typical simple modules. However, there By definition, in a logarithmic CFT there are modules on S g ⊕ g action int one obtains another theory with symmetry. is also an “atypical sector” with significant differences: which the energy operator L0 is not diagonalizable. Since ¯ ) ( L0 − L0 should still be diagonalizable this implies a com- Full WZW model Decoupled theory F(G) (G) ←→ atyp F atyp plicated entanglement between the two chiral halfs in a full gˆ ⊕ gˆ-module H gˆ ⊕ gˆ-module H M M ∗ LCFT which had not been understood previously. I[σ] Kµ ⊗ Kµ [σ] atypical block µ atypical In supergroup WZW models, L is proportional to the Lapla- The spaces H and H are isomorphic as gˆ0 ⊕ gˆ0 modules but 0 cian when restricted to ground states. Consequently, these different as gˆ ⊕ gˆ-modules. At the same time they are also K ∞ The µ are Kac modules. In contrast, the -dimensional models are logarithmic. Moreover, the harmonic analysis g ⊕ g-modules. I non-chiral representations [σ] are built from projective cov- on the supergroup precisely determines how the two chiral Remark: The construction above resembles the Feigin-Frenkel construction which, ers Pσ. The Laplacian is not diagonalizable on I[σ]. halfs have to be coupled to define a full logarithmic CFT. A however, makes use of a Z-grading corresponding to the root space decomposition key role is played by projective covers of simple modules. The derivation employs a BGG-like duality Pµ : Kλ = and thus breaks the manifest symmetry down to the Cartan subalgebra. Kλ : Lµ between simple modules Lµ, Kac modules Kλ and projective covers Pµ. Conclusions Symbols and assumptions Supergroup WZW models provide a convenient framework

g : finite dimensional Lie superalgebra, g = g0 ⊕ g1 D-branes and K-theory to address geometric and algebraic questions related to supergroups and affine Lie superalgebras from a unified per- h·, ·i : supersymmetric non-degenerate invariant form D-branes carry charges which are conserved under annihi- spective. gˆ : loop superalgebra based on g, centrally extended using h·, ·i lation processes. For bosonic WZW models it is known that G : Lie supergroup associated with g these charges are classified by twisted K-theory [1]. Since In addition, using their geometric interpretation they give

G0 : base of G (Lie group associated with g0) D-brane annihilation is described by the Verlinde algebra valuable insights into the chiral and non-chiral structure of L, K, P : simple module, Kac module, projective cover this provides a link to a recent theorem of Freed-Hopkins- logarithmic conformal ﬁeld theories. Teleman. It is an open problem whether this connection More work is required in order to clarify the structure of Assumption: g should admit a Z-grading g = g1 ⊕ g0 ⊕ g−1 which is generalizes to Lie supergroups. fusion rings of afﬁne Lie superalgebras and to relate them consistent with the Z2-grading. G is assumed to be simply-connected. to a suitable version of twisted K-theory.

Literature [1] Peter Bouwknegt and Varghese Mathai, “D-branes, B-ﬁelds and twisted K-theory”, JHEP 03 (2000) 007 [arXiv: hep-th/0002023]. [2] Thomas Quella and Volker Schomerus, “Free fermion resolution of supergroup WZNW models”, JHEP 09 (2007) 085 [arXiv:0706.0744]. [3] Gerhard Gotz,¨ Thomas Quella and Volker Schomerus, “The WZNW model on PSU(1, 1|2)”, JHEP 03 (2007) 003 [arXiv: hep-th/0610070]. [4] Thomas Creutzig, Thomas Quella, Volker Schomerus, “Branes in the GL(1|1) WZNW-Model.”, Nuclear Physics B792 (2008) 257-283 [arXiv:0708.0583]