On extended Frobenius structures

TL;DR

This paper systematically studies extended Frobenius algebras, which generalize classical Frobenius algebras, to classify unoriented 2-dimensional topological quantum field theories across various mathematical frameworks.

Contribution

It provides a comprehensive analysis, classification, and construction methods for extended Frobenius algebras in multiple settings, expanding the understanding of their role in quantum topology.

Findings

Classification results for extended Frobenius algebras over fields

Examples of extended Frobenius algebras in monoidal categories

Construction techniques for extended Frobenius algebras

Abstract

A classical result in quantum topology is that oriented 2-dimensional topological quantum field theories (2-TQFTs) are fully classified by commutative Frobenius algebras. In 2006, Turaev and Turner introduced additional structure on Frobenius algebras, forming what are called extended Frobenius algebras, to classify 2-TQFTs in the unoriented case. This work provides a systematic study of extended Frobenius algebras in various settings: over a field, in a monoidal category, and in the framework of monoidal functors. Numerous examples, classification results, and general constructions of extended Frobenius algebras are established.