Classification and construction of unitary topological field theories in two dimensions

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.