Classification and construction of unitary topological field theories in two dimensions

TL;DR

This work classifies unitary two-dimensional TQFTs by reducing the theory to the spectrum of a hermitian handle-creation operator. It proves that the genus-$g$ partition functions take the form $Z_g=\sum_{i=1}^n \lambda_i^{\,g-1}$ with $\lambda_i>0$, and that the theory is uniquely determined (up to equivalence) by these eigenvalues. The authors provide an explicit triangulated-surface construction that realizes arbitrary positive eigenvalues as the handle-operator spectrum, thereby enabling direct-sum realizations of general 2D unitary TQFTs. They further discuss equivalence criteria, irreducibility, and potential extensions to higher dimensions, highlighting the relative simplicity of 2D in contrast to higher-dimensional classifications.

Abstract

We prove that unitary two-dimensional topological field theories are uniquely characterized by $n$ positive real numbers $λ_1,\ldots λ_n$ which can be regarded as the eigenvalues of a hermitean handle creation operator. The number $n$ is the dimension of the Hilbert space associated with the circle and the partition functions for closed surfaces have the form $$ Z_g=\sum_{i=1}^{n}λ_i^{g-1} $$ where $g$ is the genus. The eigenvalues can be arbitary positive numbers. We show how such a theory can be constructed on triangulated surfaces.