Twin TQFTs and Frobenius algebras

TL;DR

This paper characterizes the category of singular 2D cobordisms algebraically using twin Frobenius algebras and introduces a generalized 2D TQFT framework based on this structure.

Contribution

It provides a complete algebraic description of singular 2D cobordisms via twin Frobenius algebras and establishes their equivalence to a generalized 2D TQFT.

Findings

Category of singular 2D cobordisms is freely generated by twin Frobenius algebras.

Introduces a generalized 2D TQFT based on twin Frobenius algebra structures.

Establishes an equivalence between singular cobordisms and algebraic data.

Abstract

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (C, W, z, z^*) consists of a commutative Frobenius algebra C, a symmetric Frobenius algebra W, and an algebra homomorphism z: C - > W with dual z^*: W -> C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.