ABSTRACT. We find a nonsemisimple fusion algebra F p associated with each (1, p) Virasoro model. We present a nonsemisimple generalization of the Verlinde formula which allows us to derive F p from modular transformations of characters.

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.

The set of modular invariants that can be obtained from Galois transformations is investigated systematically for WZW models. It is shown that a large subset of Galois modular invariants coincides with simple current invariants. For algebras of type B and D infinite series of previously unknown exceptional automorphism invariants are found.--------

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 curvature regime by means of CFT on surfaces with boundary.

Mappings between certain infinite series of N = 2 superconformal coset models are constructed. They make use of level-rank dualities for B, C and D type WZW theories, which are described in some detail. The WZW level-rank dualities do not constitute isomorphisms of the theories; for example, for B and D type WZW theories, only simple current orbits rather than individual primary fields are mapped onto each other. Nevertheless they lead to level-rank dualities of N = 2 coset models that preserve the conformal field theory properties in such a man-finding the physical fields by a so-called 'fixed point resolution.' Every fixed point of length N f has to be resolved in N 0 /N f distinct physical fields.As has been shown in [26], coset theories C[g/h] K with