We propose holographic dualities between higher spin gravity theories extended with Chan-Paton factor on AdS 3 and a large N limit of two dimensional Grassmannian models with and without supersymmetry. These proposals are natural extensions of the duality without Chan-Paton factor, and the extensions are motivated by a higher dimensional version of the duality, which implies a possible relation to superstring theory via ABJ theory. As evidence for the proposals, we show that the free limit of the Grassmannian models have the higher spin symmetry expected from the dual gravity theory. Furthermore, we construct currents in the 't Hooft limit of the supersymmetric Grassmannian model and compare them with the currents from the bulk theory. One-loop partition function of the supergravity theory is reproduced by the 't Hooft limit of the Grassmannian model after decoupling a gauge sector.

Let V be a vertex operator algebra with a category C of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let A be a vertex operator (super)algebra extension of V . We employ tensor categories to study untwisted (also called local) Amodules in C, using results of Huang-Kirillov-Lepowsky that show that A is a (super)algebra object in C and that generalized A-modules in C correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a C-algebra and (under suitable conditions) of generalized A-modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang.Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of V -modules to A-modules is a vertex tensor functor. Two applications are given:First, we derive Verlinde formulae for regular vertex operator superalgebras and regular 1 2 Z-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and Z-graded components, respectively.Second, we analyze parafermionic cosets C = Com(VL, V ) where L is a positive definite even lattice and V is regular. If the vertex tensor category of either V -modules or C-modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples. * T. C. is supported by the Natural Sciences and Engineering Research Council of Canada (RES0020460).

We analyze the asymptotic symmetry of higher spin gravity with M × M matrix valued fields, which is given by rectangular W-algebras with su(M ) symmetry. The matrix valued extension is expected to be useful for the relation between higher spin gravity and string theory. With the truncation of spin as s = 2, 3, . . . , n, we evaluate the central charge c of the algebra and the level k of the affine currents with finite c, k. For the simplest case with n = 2, we obtain the operator product expansions among generators by requiring their associativity. We conjecture that the symmetry is the same as that of Grassmannian-like coset based on our proposal of higher spin holography. Comparing c, k from the both theories, we obtain the map of parameters. We explicitly construct low spin generators from the coset theory, and, in particular, we reproduce the operator product expansions of the rectangular W-algebra for n = 2. We interpret the map of parameters by decomposing the algebra in the coset description.

ABSTRACT. This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with the remarkable observation of Cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory. Cardy's derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators. For this, a short introduction to the fusion algorithm of Nahm, Gaberdiel and Kausch is provided.While the percolation logarithmic conformal field theory is still not completely understood, there are several examples for which the formalism familiar from rational conformal field theory, including bulk partition functions, correlation functions, modular transformations, fusion rules and the Verlinde formula, has been successfully generalised. This is illustrated for three examples: The singlet model M 1,2 , related to the triplet model W 1,2 , symplectic fermions and the fermionic bc ghost system; the fractional level Wess-Zumino-Witten model based on sl(2) at k = − n . These examples have been chosen because they represent the most accessible, and most useful, members of the three best-understood families of logarithmic conformal field theories: The logarithmic minimal models W q, p , the fractional level Wess-ZuminoWitten models, and the Wess-Zumino-Witten models on Lie supergroups (excluding OSP(1|2n)).In this review, the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations. Each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum, which turns out to be continuous, as well as a selection of natural reducible, but indecomposable, modules. This is followed by a detailed description of how to obtain character formulae for each irreducible, a derivation of the action of the modular group on the characters, and an application of the Verlinde formula to compute the Grothendieck fusion rules. In each case, the (genuine) fusion rules are known, so comparisons can be made and favourable conclusions drawn. In addition, each example admits an infinite set of simple currents, hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed. The spectrum of such an extended theory is typically discrete and this is how the triplet model W 1,2 arises, for example. Moreover, simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory. Finally, each example is concluded by a brief description of the computation of some bulk correlators, a discussion of the structure of the bulk state space, and remarks concerning more advanced developments and generalisations.The final part gives a very short account of the theory of staggered modules, the (simplest class of) representations that are respons...

Gravity, itself a gauge theory of a spin 2 field, can be extended to a higher spin gauge theory on AdS spaces. Recently, Gaberdiel and Gopakumar conjectured that a large N limit of a 2d minimal model is dual to a bosonic subsector of a higher spin supergravity theory on 3d AdS space. We propose and test the untruncated supersymmetric version of this conjecture where the dual CFT is a large N limit of the N = 2 CP N Kazama-Suzuki model.Keywords: Higher spin gauge theory; AdS/CFT correspondence; W symmetry.PACS numbers: 11.25.Hf, 11.25.Tq BackgroundRecently, higher spin gravity theories have attracted a lot of attention due to their application to the AdS/CFT correspondence. Higher spin gravity theories extend ordinary gravity with gauge fields having spin s ≥ 3. On anti-de Sitter space, a higher spin gauge theory with an infinite number of spins and non-trivial interactions can be constructed, the so-called Vasiliev theory. In Ref. 1 it was proposed that a 4d Vasiliev theory is dual to the 3d O(N ) vector model. Very recently, a 3d Vasiliev theory was conjectured to be dual to a large N limit of a 2d minimal model.2 In Ref. 3, we have extended the conjecture to the case including supersymmetry. * The poster presentation was given by Y. Hikida.