Brane Topological Field Theory and Hurwitz numbers for CW-complexes

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, graded by CW-complexes of smaller dimensions. We define general and regular Hurwitz numbers of brane complexes and prove that they generate Brane Topological Field Theories. For general Hurwitz numbers, the corresponding algebra is an algebra of coverings of smaller dimension. For regular Hurwitz numbers, the Frobenius algebra is an algebra of families of subgroups of finite groups.