A connection between lattice and surgery constructions of three-dimensional topological field theories

TL;DR

This paper explores the deep connections between lattice and surgery approaches to constructing three-dimensional topological field theories, revealing how certain theories are interrelated through representation theory and topological methods.

Contribution

It establishes a formal relationship between lattice-based and surgery-based constructions of 3D topological field theories, linking Chung-Fukuma-Shapere and Altschuler-Coste theories, as well as Turaev-Viro and Reshetikhin-Turaev theories.

Findings

Lattice Chung-Fukuma-Shapere theory has a representation theoretic reformulation.

The reformulation is closely related to the surgery-based Altschuler-Coste theory.

A similar relation exists between Turaev-Viro and Reshetikhin-Turaev theories.

Abstract

We study the relation between lattice construction and surgery construction of three-dimensional topological field theories. We show that a class of the Chung-Fukuma-Shapere theory on the lattice has representation theoretic reformulation which is closely related to the Altschuler-Coste theory constructed by surgery. There is a similar relation between the Turaev-Viro theory and the Reshetikhin-Turaev theory.