Christoph Schweigert

TFT construction of RCFT correlators. I: Partition functions

Abstract We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of Moore–Seiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are A-modules, and (generalised) defect lines are A–A-bimodules. The relation with three-dimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in three-manifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIM-rep properties. We suggest that our results can be interpreted in terms of non-commutative geometry over the modular tensor category of Moore–Seiberg data.

TFT CONSTRUCTION OF RCFT CORRELATORS III: SIMPLE CURRENTS

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 theories.

From Dynkin diagram symmetries to fixed point structures

Any automorphism of the Dynkin diagram of a symmetrizable Kac-Moody algebra g induces an automorphism of g and a mappingτω between highest weight modules of g. For a large class of such Dynkin diagram automorphisms, we can describe various aspects of these maps in terms of another Kac-Moody algebra, the “orbit Lie algebra” ğ. In particular, the generating function for the trace ofτω over weight spaces, which we call the “twining character” of g (with respect to the automorphism), is equal to a character of ğ. The orbit Lie algebras of untwisted affine Lie algebras turn out to be closely related to the fixed point theories that have been introduced in conformal field theory. Orbit Lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory.

SYSTEMATIC APPROACH TO CYCLIC ORBIFOLDS

We introduce an orbifold induction procedure which provides a systematic construction of cyclic orbifolds, including their twisted sectors. The procedure gives counterparts in the orbifold theory of all the current-algebraic constructions of conformal field theory and enables us to find the orbifold characters and their modular transformation properties.

SYMMETRY BREAKING BOUNDARIES I. GENERAL THEORY

Abstract We study conformally invariant boundary conditions that break part of the bulk symmetries. A general theory is developed for those boundary conditions for which the preserved subalgebra is the fixed algebra under an abelian orbifold group. We explicitly construct the boundary states and reflection coefficients as well as the annulus amplitudes. Integrality of the annulus coefficients is proven in full generality.

DUALITY AND DEFECTS IN RATIONAL CONFORMAL FIELD THEORY

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 Potts model, is treated as an illustrative example.

BOUNDARIES, CROSSCAPS AND SIMPLE CURRENTS

Universal formulas for the boundary and crosscap coefficients are presented, which are valid for all symmetric simple current modifications of the charge conjugation invariant of any rational conformal field theory.

Correspondences of ribbon categories

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.

Correlation functions and boundary conditions in RCFT and three-dimensional topology

We give a general construction of correlation functions in rational conformal field theory on a possibly nonorientable surface with boundary in terms of three-dimensional topological field theory. The construction applies to any modular category in the sense of Turaev. It is proved that these correlation functions obey modular invariance and factorization rules. Structure constants are calculated and expressed in terms of the data of the modular category.

A matrix

Abstract A formula is presented for the modular transformation matrix S for any simple current extension of the chiral algebra of a conformal field theory. This provides in particular an algorithm for resolving arbitrary simple current fixed points, in such a way that the matrix S we obtain is unitary and symmetric and furnishes a modular group representation. The formalism works in principle for any conformal field theory. A crucial ingredient is a set of matrices SJab, where J is a simple current and a and b are fixed points of J. We expect that these input matrices realize the modular group for the torus one-point functions of the simple currents. In the case of WZW models these matrices can be identified with the S-matrices of the orbit Lie algebras that were introduced recently [J. Fuchs et al., preprint hep-th/9506135, Commun. Math. Phys., in press]. As a special case of our conjecture we obtain the modular matrix S for WZW theories based on group manifolds that are not simply connected, as well as for most coset models.