Coends in conformal field theory

TL;DR

This paper explores the use of coend constructions to formalize the summation over intermediate states in conformal field theory, highlighting subtleties in their application within functor categories.

Contribution

It clarifies the role and subtleties of coend constructions in conformal field theory, connecting abstract categorical concepts with physical locality principles.

Findings

Identifies key subtleties in coend applications for conformal blocks.

Bridges categorical formalism with physical locality in CFT.

Provides insights into the mathematical structure underlying conformal blocks.

Abstract

The idea of "summing over all intermediate states" that is central for implementing locality in quantum systems can be realized by coend constructions. In the concrete case of systems of conformal blocks for a certain class of conformal vertex algebras, one deals with coends in functor categories. Working with these coends involves quite a few subtleties which, even though they have in principle already been understood twenty years ago, have not been sufficiently appreciated by the conformal field theory community.