We present an algorithm for an efficient calculation of the fusion rules of twisted representations of untwisted affine Lie algebras. These fusion rules appear in WZW orbifold theories and as annulus coefficients in boundary WZW theories; they provide NIM-reps of the WZW fusion rules.
Topological field theory in three dimensions provides a powerful tool to construct correlation functions and to describe boundary conditions in two-dimensional conformal field theories.
We formulate two-dimensional rational conformal field theory as a natural generalization of two-dimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central
ingredient is a special Frobenius algebra object A in the modular category that encodes the Moore-Seiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of T-duality. We present a construction of correlators, based on a triangulation of the world sheet, that generalizes the one in lattice TFTs. These correlators are modular invariant and satisfy factorization rules. The construction works for arbitrary orientable world sheets, in particular for surfaces with boundary. Boundary conditions correspond to representations of the algebra A. The partition functions on the torus and on the annulus provide modular invariants and NIM-reps of the fusion rules, respectively.
We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-prod
uct we obtain results about the Frobenius-Schur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in two-dimensional conformal field theory and present illustrative examples. We show that when the module category is tensor, then it gives rise to a NIM-rep of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras.
We investigate the mathematical structure of the world sheet in two-dimensional conformal field theories.
The role of automorphisms of infinite-dimensional Lie algebras in conformal field theory is examined. Two main types of applications are discussed; they are related to the enhancement and reduction of symmetry, respectively. The structures one encoun
ters also appear in other areas of physics and mathematics. In particular, they lead to two conjectures on the sub-bundle structure of chiral blocks, and they are instrumental in the study of conformally invariant boundary conditions.
This is an introduction to two-dimensional conformal field theory and its applications in string theory. Modern concepts of conformal field theory are explained, and it is outlined how they are used in recent studies of D-branes in the strong curvatu
re regime by means of CFT on surfaces with boundary.
The correlation functions of a two-dimensional rational conformal field theory, for an arbitrary number of bulk and boundary fields and arbitrary world sheets can be expressed in terms of Wilson graphs in appropriate three-manifolds. We present a sys
tematic approach to boundary conditions that break bulk symmetries. It is based on the construction, by `alpha-induction, of a fusion ring for the boundary fields. Its structure constants are the annulus coefficients and its 6j-symbols give the OPE of boundary fields. Symmetry breaking boundary conditions correspond to solitonic sectors.
