Non-commutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves

TL;DR

This paper extends two-dimensional topological field theories to nonorientable surfaces, introduces structure algebras, classifies semisimple cases, and connects Hurwitz numbers for real algebraic curves to these theories.

Contribution

It introduces Klein topological field theories (KTFT) for nonorientable surfaces, establishes their algebraic classification, and links Hurwitz numbers for real curves to KTFT correlators.

Findings

KTFTs correspond to structure algebras with additional properties

Semisimple structure algebras are classified

Hurwitz numbers for real algebraic curves are correlators of KTFT

Abstract

It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend open-closed two-dimensional topological field theories to nonorientable surfaces. We call them Klein topological field theories(KTFT). We prove that KTFTs bijectively correspond to algebras with certain additional structures, called structure algebras. Semisimple structure algebras are classified. Starting from an arbitrary finite group, we construct a structure algebra and prove that it is semisimple. We define an analog of Hurwitz numbers for real algebraic curves and prove that they are correlators of a KTFT. The structure algebra of this KTFT is the structure algebra of the symmetric group.