Topological Frobenius algebras

TL;DR

This paper introduces new algebraic structures called topological Frobenius algebras, explores their properties, and connects them to various homological theories and topological quantum field theories.

Contribution

It defines unital, counital, biunital infinitesimal anti-symmetric bialgebras and coFrobenius bialgebras, and introduces graded 2D open-closed TQFTs within the context of Tate vector spaces.

Findings

Structures arise in Rabinowitz Floer homology

Applicable to loop space and quantum homology

Involves infinite-dimensional Tate vector spaces

Abstract

We define the notions of unital/counital/biunital infinitesimal anti-symmetric bialgebras and coFrobenius bialgebras and discuss their algebraic properties. We also define the notion of a graded 2D open-closed TQFT. These structures arise in Rabinowitz Floer homology, loop space homology, quantum homology, and the homology of finite dimensional manifolds. The underlying vector spaces, which are typically infinite dimensional, belong to a class of topological vector spaces known as Tate vector spaces.