Classification and construction of unitary topological field theories in two dimensions

TL;DR

This paper characterizes two-dimensional unitary topological field theories by positive real eigenvalues and provides a construction method on triangulated surfaces, linking algebraic data to geometric models.

Contribution

It establishes a complete classification of 2D unitary topological field theories via eigenvalues of a handle operator and offers a construction approach on triangulated surfaces.

Findings

Theories are characterized by n positive eigenvalues.

Partition functions depend on eigenvalues as sum of powers.

Construction method on triangulated surfaces is provided.

Abstract

We prove that unitary two-dimensional topological field theories are uniquely characterized by $n$ positive real numbers $\lambda _1,\ldots \lambda _n$ 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 $$ Z_g=\sum_{i=1}^{n}\lambda _i^{g-1} $$ where $g$ is the genus. The eigenvalues can be arbitary positive numbers. We show how such a theory can be constructed on triangulated surfaces.