Decomposition of the Turaev-Viro TQFT

TL;DR

This paper demonstrates that the Turaev-Viro TQFT can be decomposed into blocks derived from an HQFT, providing a new perspective and reformulation in terms of classifying spaces and homotopical invariants.

Contribution

It introduces the Turaev-Viro HQFT, a novel extension that splits the Turaev-Viro TQFT into blocks associated with a classifying space, linking it to homotopical invariants.

Findings

Decomposition of Turaev-Viro TQFT into HQFT blocks.

Construction of Turaev-Viro HQFT with classifying space target.

Extension of the homotopical Turaev-Viro invariant.

Abstract

We show that for every spherical category $\C$ with invertible dimension, the Turaev-Viro TQFT admits a splitting into blocks which come from an HQFT, called the Turaev-Viro HQFT. The Turaev-Viro HQFT has the classifying space $B\grad$ as target space, where $\grad$ is a group obtained from the category $\C$. This construction gives a reformulation of the Turaev-Viro TQFT in terms of HQFT. Furthermore the Turaev-Viro HQFT is an extension of the \emph{homotopical Turaev-Viro invariant} which splits the Turaev-Viro invariant. An application of this result is a description of the homological twisted version of the Turaev-Viro invariant in terms of HQFT.