Frobenius algebras and planar open string topological field theories

TL;DR

This paper establishes a categorical framework linking Frobenius algebras to planar open string topological field theories, generalizing classical results to a higher-dimensional membrane context.

Contribution

It defines a new category 2Thick and shows it is the free monoidal category on a noncommutative Frobenius algebra, extending the classical cobordism-Frobenius algebra correspondence.

Findings

2Thick is the free monoidal category on a noncommutative Frobenius algebra.

Introduces a 2-category of open strings and membranes, related to categorified Frobenius algebras.

Provides a categorical formalism for open string topological field theories.

Abstract

Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the category of 2-dimensional cobordisms can be described as the free symmetric monoidal category on a commutative Frobenius algebra, 2Thick is shown to be the free monoidal category on a noncommutative Frobenius algebra, hence justifying this choice of data in the Moore-Segal axioms. Our formalism is inherently categorical allowing us to generalize this result. As a stepping stone towards topological membrane theory we define a 2-category of open strings, planar open string worldsheets, and isotopy classes of 3-dimensional membranes defined by diffeomorphisms of the open string worldsheets. This 2-category is shown to be the free (weak monoidal) 2-category on a `categorified Frobenius algebra', meaning that categorified Frobenius algebras determine invariants of these 3-dimensional membranes.