Brane Topological Field Theories and Hurwitz numbers for CW-complexes

TL;DR

This paper develops a new class of topological field theories based on CW-complexes called brane complexes, linking them to Frobenius algebras and defining Hurwitz numbers that generate these theories.

Contribution

It introduces Brane Topological Field Theories associated with CW-complexes and establishes their correspondence with infinite dimensional Frobenius algebras, along with defining related Hurwitz numbers.

Findings

Brane Topological Field Theories correspond to infinite dimensional Frobenius algebras.

Hurwitz numbers for brane complexes generate these topological theories.

Different types of Hurwitz numbers relate to specific algebraic structures.

Abstract

We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they generate Brane Topological Field Theories. For general Hurwitz numbers corresponding algebra is an algebra of coverings of lesser dimension. For regular Hurwitz numbers the Frobenius algebra is an algebra of families of subgroups of finite groups.