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On the gauge invariance of the Kuperberg invariant of certain high genus framed 3-manifolds

Liang Chang, Yilong Wang, Saifei Zhai

TL;DR

This work addresses generating gauge invariants of finite-dimensional Hopf algebras from topological data by examining the Kuperberg invariant for framed 3-manifolds. It develops a framework using framed Heegaard diagrams, integrals, and Drinfeld twists to study gauge transformations and their impact on invariants, culminating in a main result that for any framing $f$ on the Weeks manifold $W$ and any finite-dimensional Hopf algebra $H$, the invariant $Z(W,f,H)$ remains unchanged under gauge equivalence. The authors also prove a parallel gauge invariance result for the 3-torus, $T^3$, with framing $f_1$, strengthening the case that hyperbolic and flat geometries can systematically yield gauge-invariant invariants for Hopf algebras, including non-semisimple ones. These results support CNW25’s program of constructing categorical 3-manifold invariants from topological methods and have potential implications for non-semisimple quantum invariants and related topological field theories.

Abstract

We show that the Kuperberg invariant of the Weeks manifold with any framing is a gauge invariant of finite-dimensional Hopf algebras, which provides the first example of gauge invariants of general finite-dimensional Hopf algebras via hyperbolic 3-manifolds. We also show that the Kuperberg invariant of the 3-torus is gauge invariant, which further supports the idea of systematically producing gauge invariants of Hopf algebras via topological methods proposed in \cite{CNW25}.

On the gauge invariance of the Kuperberg invariant of certain high genus framed 3-manifolds

TL;DR

This work addresses generating gauge invariants of finite-dimensional Hopf algebras from topological data by examining the Kuperberg invariant for framed 3-manifolds. It develops a framework using framed Heegaard diagrams, integrals, and Drinfeld twists to study gauge transformations and their impact on invariants, culminating in a main result that for any framing on the Weeks manifold and any finite-dimensional Hopf algebra , the invariant remains unchanged under gauge equivalence. The authors also prove a parallel gauge invariance result for the 3-torus, , with framing , strengthening the case that hyperbolic and flat geometries can systematically yield gauge-invariant invariants for Hopf algebras, including non-semisimple ones. These results support CNW25’s program of constructing categorical 3-manifold invariants from topological methods and have potential implications for non-semisimple quantum invariants and related topological field theories.

Abstract

We show that the Kuperberg invariant of the Weeks manifold with any framing is a gauge invariant of finite-dimensional Hopf algebras, which provides the first example of gauge invariants of general finite-dimensional Hopf algebras via hyperbolic 3-manifolds. We also show that the Kuperberg invariant of the 3-torus is gauge invariant, which further supports the idea of systematically producing gauge invariants of Hopf algebras via topological methods proposed in \cite{CNW25}.
Paper Structure (5 sections, 8 theorems, 57 equations, 3 figures)

This paper contains 5 sections, 8 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Hopf algebra and gauge invariant
  3. The Kuperberg invariant
  4. The Weeks manifold
  5. 3-torus

Key Result

Theorem 1

Let $H$ be a finite-dimensional Hopf algebra, $\Lambda$ and $\lambda$ is a pair of normalized integrals such that $\lambda(\Lambda)=1$, $X\in \mathrm{End}(H)$$(1)$$S(a)\Lambda_{(1)}\otimes \Lambda_{(2)}=\Lambda_{(1)}\otimes a\Lambda_{(2)}$. $(2)$$\mathrm{Tr}(X)=\lambda(S\circ X(\Lambda_{(2)})\Lambda

Figures (3)

  • Figure 1: Heegaard diagram of the Weeks manifold
  • Figure 2: Framed Heegaard diagram of the Weeks manifold
  • Figure 3:

Theorems & Definitions (13)

  • Theorem 1
  • Proposition 1: CNW25
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Proposition 2
  • proof
  • ...and 3 more