The n-Category Café

Introduction:

Two-dimensional conformal field theory plays a fundamental role in the theory of two-dimensional critical systems of classical statistical mechanics, in quasi one-dimensional condensed matter physics and in string theory. The study of defects in systems of condensed matter physics, of percolation probabilities and of (open) string perturbation theory in the background of certain solitonic solutions of string theory, the so-called D-branes, forces one to analyze conformal field theories on surfaces that may have boundaries and/or can be non-orientable. This study has recently led to much new insight in the mathematical structure of conformal field theory. Many mathematical disciplines have contributed to a better understand of conformal field theory and received stimulating input from questions arising in conformal field theories.

There are two major approaches to chiral conformal field theory: one that is based on operator algebras and one based on vertex algebras. Both structures lead to representation categories that are tensor categories and, in the case of rational chiral conformal field theories, more specifically modular tensor categories. They also encode the monodromy representations of the vector bundles of conformal blocks for rational vertex algebras, objects that are of interest for algebraic geometry. Moreover, modular tensor categories are a crucial ingredient in the construction of three-dimensional topological field theories.

While chiral conformal field theories have certain physical applications in the description of quantum Hall systems, full local conformal field theories are relevant for the physical applications referred to in the first paragraph. Recently, it has been understood that the construction of a full local conformal field theory is best described using the structure of a module category over the tensor category that describes the chiral data.

In this Arbeitsgemeinschaft, we will explain this mathematical notion, related concepts and some applications. The category theoretic framework allows to emphasize those aspects that are common to all approaches to chiral conformal field theory.