One-dimensional topological theories with defects: the linear case

TL;DR

This paper explores the structure of one-dimensional topological theories with defects, revealing their connection to symmetric Frobenius algebras and their role in describing open-closed 2D topological quantum field theories.

Contribution

It characterizes the Karoubi envelope of such theories via symmetric Frobenius algebras and relates it to the Frobenius-Brauer category, extending the understanding of defect theories.

Findings

Karoubi envelope determined by symmetric Frobenius algebra K

Equivalence to quotient of Frobenius-Brauer category modulo negligible morphisms

Coupling K to universal construction yields a topological theory of open-closed cobordisms

Abstract

The paper studies the Karoubi envelope of a one-dimensional topological theory with defects and inner endpoints, defined over a field. It turns out that the Karoubi envelope is determined by a symmetric Frobenius algebra K associated to the theory. The Karoubi envelope is then equivalent to the quotient of the Frobenius-Brauer category of K modulo the ideal of negligible morphisms. Symmetric Frobenius algebras, such as K, describe two-dimensional TQFTs for the category of thin flat surfaces, and elements of the algebra can be turned into defects on the side boundaries of these surfaces. We also explain how to couple K to the universal construction restricted to closed surfaces to define a topological theory of open-closed two-dimensional cobordisms which is usually not an open-closed 2D TQFT.