Lattice Topological Field Theory in Two Dimensions

TL;DR

This paper provides a comprehensive lattice formulation of two-dimensional topological field theories, explicitly solving for their structure and classifying them via associative algebras, with applications to various models.

Contribution

It establishes a one-to-one correspondence between lattice TFTs and associative algebras, explicitly solves for their structure, and explores perturbations within this framework.

Findings

Lattice TFTs correspond to associative algebras R.

The physical Hilbert space is the center Z(R) of R.

All TFTs can be generated from a basic TFT via perturbations.

Abstract

The lattice definition of a two-dimensional topological field theory (TFT) is given generically, and the exact solution is obtained explicitly. In particular, the set of all lattice topological field theories is shown to be in one-to-one correspondence with the set of all associative algebras $R$, and the physical Hilbert space is identified with the center $Z(R)$ of the associative algebra $R$. Perturbations of TFT's are also considered in this approach, showing that the form of topological perturbations is automatically determined, and that all TFT's are obtained from one TFT by such perturbations. Several examples are presented, including twisted $N=2$ minimal topological matter and the case where $R$ is a group ring.