Closed and open conformal field theories and their anomalies

TL;DR

This paper develops a comprehensive framework for understanding anomalies in both closed and open conformal field theories, extending existing notions of modular functors to higher categorical settings.

Contribution

It generalizes Segal's modular functor to 2-vector spaces and introduces a topological group completion for labeling conformal field theories with super-vector spaces.

Findings

Generalized modular functor to 2-vector spaces

Defined a topological group completion for vector spaces

Proposed a new labeling scheme for super-vector space CFTs

Abstract

In this paper, we give a general axiomatization of anomalies in closed and open conformal field theories. In particular, we generalize Segal's notion of modular functor to a setting where the ``set of labels'' is a 2-vector space. In the case of open conformal field theory, the ``set of $D$-branes'' is a 3-vector space. We also define a ``topological group completion'' of the symmetric bimonoidal category of finite-dimensional vector spaces, and propose it as a candidate for labelling conformal field theories whose modular functors are super-vector spaces.