A note on TQFTs for orientable 2-dimensional cobordisms

TL;DR

This paper explores a classification framework for 2D topological quantum field theories focusing on orientable cobordisms, bridging the gap between oriented and unoriented cases with applications in skein theory and Khovanov homology.

Contribution

It introduces an intermediate classification framework for 2D TQFTs that handles orientable cobordisms, extending the classical Frobenius algebra approach.

Findings

Provides a new classification scheme for 2D TQFTs for orientable cobordisms.

Connects TQFT classification to skein-theoretic models and Khovanov homology.

Bridges the gap between oriented and unoriented TQFTs.

Abstract

Topological quantum field theories (TQFTs) are symmetric monoidal functors out of cobordism categories. In dimension two, oriented TQFTs are famously classified by commutative Frobenius algebras. In the unoriented setting, the classification requires additional data: an involution and a value assigned to the M\"obius strip. In this work, we describe an intermediate framework that classifies 2-dimensional TQFTs for orientable cobordisms, in an appropriate sense. Our motivation arises from skein-theoretic models of surfaces embedded in 3-manifolds and Khovanov homology, where surfaces are often treated as unoriented, even though the associated 2-dimensional TQFTs themselves need not be fully unoriented.