[Paper Review] Decomposition of the Turaev-Viro TQFT

Jérôme Petit|ArXiv.org|Mar 26, 2009

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper introduces a decomposition of the Turaev-Viro topological quantum field theory (TQFT) by showing that it splits into blocks indexed by homotopy classes to the classifying space $B\Gamma_{\mathcal{C}}$, where $\Gamma_{\mathcal{C}}$ is a group derived from a spherical category $\mathcal{C}$ with invertible dimension. The key result is the construction of a new homotopical Turaev-Viro HQFT (higher-cobordism TQFT) that refines the original TQFT and provides a splitting of the Turaev-Viro invariant into components labeled by homotopy classes.

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.

Motivation & Objective

• To reformulate the Turaev-Viro TQFT in terms of higher-cobordism quantum field theories (HQFTs) with target space $B\Gamma_{\mathcal{C}}$.
• To define a homotopical Turaev-Viro invariant $HTV_{\mathcal{C}}(M,x)$ that splits the original Turaev-Viro invariant by grouping colorings according to homotopy classes $x \in [M, B\Gamma_{\mathcal{C}}]$.
• To construct a Turaev-Viro HQFT that extends the homotopical invariant and realizes the Turaev-Viro TQFT as a decomposition over these blocks.
• To provide a geometric and categorical framework for understanding twisted versions of the Turaev-Viro invariant through the lens of group graduations and classifying spaces.

Proposed method

• Define the group $\Gamma_{\mathcal{C}}$ as the graduator of a spherical category $\mathcal{C}$, generalizing the group of isomorphism classes of scalar objects in group categories.
• Assign to each edge coloring of a triangulation a homotopy class in $[M, B\Gamma_{\mathcal{C}}]$ via a universal grading, enabling a decomposition of the state-sum.
• Construct the homotopical Turaev-Viro invariant $HTV_{\mathcal{C}}(M,x)$ as a state-sum restricted to colorings yielding a fixed homotopy class $x$.
• Prove that the Turaev-Viro HQFT is an extension of the homotopical invariant and that the original Turaev-Viro TQFT arises as the direct sum over all homotopy classes.
• Use the classifying space $B\Gamma_{\mathcal{C}}$ as the target space for the HQFT, generalizing Turaev's $K(G,1)$-based HQFTs to non-abelian settings.
• Establish the splitting of the Turaev-Viro invariant as $TV_{\mathcal{C}}(M) = \sum_{x \in [M, B\Gamma_{\mathcal{C}}]} HTV_{\mathcal{C}}(M,x)$, showing the decomposition into HQFT blocks.

Experimental results

Research questions

• RQ1 Can the Turaev-Viro TQFT be decomposed into components indexed by homotopy classes of maps to a classifying space?
• RQ2 Does there exist an HQFT structure that refines the Turaev-Viro invariant by grouping colorings via homotopy classes?
• RQ3 How does the group $\Gamma_{\mathcal{C}}$, derived from the spherical category $\mathcal{C}$, control the splitting of the Turaev-Viro invariant?
• RQ4 Can the homological twisted version of the Turaev-Viro invariant be described via this HQFT framework?
• RQ5 Is the Turaev-Viro TQFT isomorphic to the direct sum of the Turaev-Viro HQFT over all homotopy classes?

Key findings

• The Turaev-Viro TQFT decomposes into blocks indexed by homotopy classes $x \in [M, B\Gamma_{\mathcal{C}}]$, with each block given by the homotopical Turaev-Viro invariant $HTV_{\mathcal{C}}(M,x)$.
• The Turaev-Viro HQFT is constructed as an extension of the homotopical invariant and realizes the original TQFT as a direct sum over all homotopy classes.
• For lens spaces $L(n,q)$, the Turaev-Viro invariant splits into components that depend on $n \mod m$, with explicit values given for $m=2$ to $9$, showing nontrivial block decomposition.
• When $n \equiv 0 \mod m$, the HTV invariant takes non-uniform values across blocks, such as $1/6(1+2\exp(\pm 4i\pi(2p+1)/6))$ for $m=6$, indicating nontrivial splitting.
• For $L(8p,q)$, the HTV invariant has non-zero values in multiple blocks, such as $(1/8, 1/8\exp(i\pi/4), (-1)^p/8, \dots)$ when $n \equiv 1 \mod 8$, confirming block decomposition.
• The construction generalizes Turaev’s $K(G,1)$-based HQFTs and provides a new framework for twisted invariants via the group $\Gamma_{\mathcal{C}}$.