Non-chiral current algebras for deformed supergroup WZW models

We study deformed WZW models on supergroups with vanishing Killing form. The deformation is generated by the isotropic current-current perturbation which is exactly marginal under these assumptions. It breaks half of the global isometries of the original supergroup. The current corresponding to the remaining symmetry is conserved but its components are neither holomorphic nor anti-holomorphic. We obtain the exact two- and three-point functions of this current and a four-point function in the first two leading orders of a 1/k expansion but to all orders in the deformation parameter. We further study the operator product algebra of the currents, the equal time commutators and the quantum equations of motion. The form of the equations of motion suggests the existence of non-local charges which generate a Yangian. Possible applications to string theory on Anti-de Sitter spaces and to condensed matter problems are briefly discussed.Comment: 43 pages, Latex, one eps figure; v.2: minor corrections, a reference adde

Branching rules of semi-simple Lie algebras using affine extensions

We present a closed formula for the branching coefficients of an embedding p in g of two finite-dimensional semi-simple Lie algebras. The formula is based on the untwisted affine extension of p. It leads to an alternative proof of a simple algorithm for the computation of branching rules which is an analog of the Racah-Speiser algorithm for tensor products. We present some simple applications and describe how integral representations for branching coefficients can be obtained. In the last part we comment on the relation of our approach to the theory of NIM-reps of the fusion rings of WZW models with chiral algebra g_k. In fact, it turns out that for these models each embedding p in g induces a NIM-rep at level k to infinity. In cases where these NIM-reps can be be extended to finite level, we obtain a Verlinde-like formula for branching coefficients.Comment: 11 pages, LaTeX, v2: one reference added, v3: Clarified proof of Theorem 2, completely rewrote and extended Section 5 (relation to CFT), added various references. Accepted for publication in J. Phys.

Geometric construction of D-branes in WZW models

The geometric description of D-branes in WZW models is pushed forward. Our starting point is a gluing condition\, $J_{+}=FJ_-$ that matches the model's chiral currents at the worldsheet boundary through a linear map $F$ acting on the WZW Lie algebra. The equivalence of boundary and gluing conditions of this type is studied in detail. The analysis involves a thorough discussion of Frobenius integrability, shows that $F$ must be an isometry, and applies to both metrically degenerate and nondegenerate D-branes. The isometry $F$ need not be a Lie algebra automorphism nor constantly defined over the brane. This approach, when applied to isometries of the form $F=R$ with $R$ a constant Lie algebra automorphism, validates metrically degenerate $R$-twined conjugacy classes as D-branes. It also shows that no D-branes exist in semisimple WZW models for constant\, $F=-R$.Comment: 23 pages, discussion of limitations of the gluing condition approach adde

Matrix difference equations for the supersymmetric Lie algebra sl(2,1) and the `off-shell' Bethe ansatz

Based on the rational R-matrix of the supersymmetric sl(2,1) matrix difference equations are solved by means of a generalization of the nested algebraic Bethe ansatz. These solutions are shown to be of highest-weight with respect to the underlying graded Lie algebra structure.Comment: 10 pages, LaTex, references and acknowledgements added, spl(2,1) now called sl(2,1

't Hooft Operators in Gauge Theory from Toda CFT

We construct loop operators in two dimensional Toda CFT and calculate with them the exact expectation value of certain supersymmetric 't Hooft and dyonic loop operators in four dimensional \Ncal=2 gauge theories with SU(N) gauge group. Explicit formulae for 't Hooft and dyonic operators in \Ncal=2^* and \Ncal=2 conformal SQCD with SU(N) gauge group are presented. We also briefly speculate on the Toda CFT realization of arbitrary loop operators in these gauge theories in terms of topological web operators in Toda CFT.Comment: 49 pages, LaTeX. Typos fixed, references adde

Supersymmetric t-J Gaudin Models and KZ Equations

Supersymmetric t-J Gaudin models with both periodic and open boundary conditions are constructed and diagonalized by means of the algebraic Bethe ansatz method. Off-shell Bethe ansatz equations of the Gaudin systems are derived, and used to construct and solve the KZ equations associated with $sl(2|1)^{(1)}$ superalgebra.Comment: LaTex 21 page

Fusion algebra of critical percolation

We present an explicit conjecture for the chiral fusion algebra of critical percolation considering Virasoro representations with no enlarged or extended symmetry algebra. The representations we take to generate fusion are countably infinite in number. The ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of these representations decomposes into a finite direct sum of these representations. The fusion rules are commutative, associative and exhibit an sl(2) structure. They involve representations which we call Kac representations of which some are reducible yet indecomposable representations of rank 1. In particular, the identity of the fusion algebra is a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the recent results of Eberle-Flohr and Read-Saleur. Notably, in agreement with Eberle-Flohr, we find the appearance of indecomposable representations of rank 3. Our fusion rules are supported by extensive numerical studies of an integrable lattice model of critical percolation. Details of our lattice findings and numerical results will be presented elsewhere.Comment: 12 pages, v2: comments and references adde

Higher string functions, higher-level Appell functions, and the logarithmic ^sl(2)_k/u(1) CFT model

We generalize the string functions C_{n,r}(tau) associated with the coset ^sl(2)_k/u(1) to higher string functions A_{n,r}(tau) and B_{n,r}(tau) associated with the coset W(k)/u(1) of the W-algebra of the logarithmically extended ^sl(2)_k conformal field model with positive integer k. The higher string functions occur in decomposing W(k) characters with respect to level-k theta and Appell functions and their derivatives (the characters are neither quasiperiodic nor holomorphic, and therefore cannot decompose with respect to only theta-functions). The decomposition coefficients, to be considered ``logarithmic parafermionic characters,'' are given by A_{n,r}(tau), B_{n,r}(tau), C_{n,r}(tau), and by the triplet \mathscr{W}(p)-algebra characters of the (p=k+2,1) logarithmic model. We study the properties of A_{n,r} and B_{n,r}, which nontrivially generalize those of the classic string functions C_{n,r}, and evaluate the modular group representation generated from A_{n,r}(tau) and B_{n,r}(tau); its structure inherits some features of modular transformations of the higher-level Appell functions and the associated transcendental function Phi.Comment: 34 pages, amsart++, times. V2: references added; minor changes; some nonsense in B.3.3. correcte

Rolling tachyon in anti-de Sitter space-time

We study the decay of the unstable D-particle in three-dimensional anti-de Sitter space-time using worldsheet boundary conformal field theory methods. We test the open string completeness conjecture in a background for which the phase space available is only field-theoretic. This could present a serious challenge to the claim. We compute the emission of closed strings in the AdS(3) x S^3 x T^4 background from the knowledge of the exact corresponding boundary state we construct. We show that the energy stored in the brane is mainly converted into very excited long strings. The energy stored in short strings and in open string pair production is much smaller and finite for any value of the string coupling. We find no "missing energy" problem. We compare our results to those obtained for a decay in flat space-time and to a background in the presence of a linear dilaton. Some remarks on holographic aspects of the problem are made.Comment: JHEP style, 45 pages, one figure; v2: typos corrected, references added, version to appear in JHE

Factorizable ribbon quantum groups in logarithmic conformal field theories

We review the properties of quantum groups occurring as Kazhdan--Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models.Comment: 27pp., amsart++, xy. v2: references added, some other minor addition