On 3-dimensional homotopy quantum field theory III: comparison of two approaches

TL;DR

This paper proves the equivalence of two different 3-dimensional homotopy quantum field theories derived from a spherical G-fusion category, using state sum and surgery approaches, respectively.

Contribution

It establishes an isomorphism between state sum and surgery-based 3D HQFTs for spherical G-fusion categories, linking two foundational constructions.

Findings

State sum and surgery HQFTs are isomorphic for given categories.

The G-center of a spherical G-fusion category plays a key role.

Provides a bridge between two approaches to 3D HQFTs.

Abstract

Let G be a discrete group and C be an additive spherical G-fusion category. We prove that the state sum 3-dimensional HQFT derived from C is isomorphic to the surgery 3-dimensional HQFT derived from the G-center of C.