inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-product 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 twodimensional 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. 1 CFT boundary conditions Boundary conditions in conformal field theory have various physical applications, ranging from the study of defects in condensed matter physics to the theory of open strings. Such boundary conditions are partially characterized by the maximal vertex operator subalgebra A of the bulk chiral algebra Abulk that they respect [43, 75]. That A is respected by a boundary condition means that the

The correlators of two-dimensional rational conformal field theories that are obtained in the TFT construction of [FRS I, FRS II, FRS IV] are shown to be invariant under

We compute the fundamental correlation functions in two-dimensional rational conformal field theory, from which all other correlators can be obtained by sewing: the correlators of three bulk fields on the sphere, one bulk and one boundary field on the disk, three boundary fields on the disk, and one bulk field on the cross cap. We also consider conformal defects and calculate the correlators of three defect fields on the sphere and of one defect field on the cross cap. Each of these correlators is presented as the product of a structure constant and the appropriate conformal two- or three-point block. The structure constants are expressed as invariants of ribbon graphs in three-manifolds.

We use simple currents to construct symmetric special Frobenius algebras in modular tensor categories. We classify such simple current type algebras with the help of abelian group cohomology. We show that they lead to the modular invariant torus partition functions that have been studied by Kreuzer and Schellekens. We also classify boundary conditions in the associated conformal field theories and show that the boundary states are given by the formula proposed in hep-th/0007174. Finally, we investigate conformal defects in these

We study topological defect lines in two-dimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting onedimensional phase boundaries can be used to extract symmetries and order-disorder dualities of the CFT. The case of central charge c = 4/5, for which there are two inequivalent world sheet phases corresponding to the tetra-critical Ising model and the critical three-states

Starting from first principles, a constructive method is presented to obtain boundary states in conformal field theory. It is demonstrated that this method is well suited to compute the boundary states of logarithmic conformal field theories. By studying the logarithmic conformal field theory with central charge c = −2 in detail, we show that our method leads to consistent results. In particular, it allows to define boundary states corresponding to both, indecomposable representations as well as their irreducible subrepresentations.

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.

We introduce the notions of open-closed field algebra and open-closed field algebra over a vertex operator algebra V. In the case that V satisfies certain finiteness and reductivity conditions, we show that an open-closed field algebra over V canonically gives an algebra over a C-extension of Swiss-cheese partial operad. We also give a tensor-categorical formulation and constructions of open-closed field algebras over V. 0

Abstract: The coset construction is the most important tool to construct rational conformal field theories with known chiral data. For some cosets at small level, so-called maverick cosets, the familiar analysis using selection and identification rules breaks down. Intriguingly, this phenomenon is linked to the existence of exceptional modular invariants. Recent progress in CFT, based on studying algebras in tensor categories, allows for a universal construction of the chiral data of coset theories which in particular also applies to maverick cosets. 1 Coset conformal field theories The coset construction is among the oldest [1] tools for obtaining rational two-dimensional conformal field theories and has been very successful. It has been used to construct prominent classes of models, such as (super-)Virasoro minimal models and Kazama-Suzuki models. Still, it presents a number of mysteries, even in the case of unitary conformal field theories, to which we will restrict ourselves in this contribution. The coset construction is based on the following data: A (finite-dimensional, complex, reductive) Lie algebra g together with a choice k of levels, i.e. a positive integer for each

In this thesis, a constructive method is presented to obtain boundary states in conformal field theory. It is compatible to the usual approach via Ishibashi states in ordinary conformal field theories but extendible to cases that have a more complicated structure, such as rank-2 indecomposable Jordan cells as in logarithmic conformal field theories. In particular, it allows to study boundary states keeping the structure of the underlying bulk theory visible. Using this method the logarithmic conformal field theory with central charge c = −2 is studied in detail, deriving the maximal set of boundary states in this case. The analysis shows the existence of states corresponding to indecomposable representations as well as their irreducible subrepresentations. Furthermore, a new kind of boundary states emerges. So-called mixed boundary states glue together the two different irreducible representations of the c = −2 theory at the boundary. A relation between the boundary states is deduced that implies a deeper connection to the unique local logarithmic conformal field theory studied by M.R. Gaberdiel and H.G. Kausch. Both, the three-dimensional and the five-dimensional representation of the modular group are found when calculating the cylinder amplitudes, the latter one by introducing additional states