Quantum invariants of flat 2-bundles over 3-manifolds

TL;DR

This paper introduces a new scalar invariant for flat principal 2-bundles over 3-manifolds derived from involutory Hopf algebras, extending classical invariants via homotopy and combinatorial methods.

Contribution

It constructs a homotopy invariant of maps from 3-manifolds to classifying spaces using $ ext{G}$-colored Heegaard diagrams, generalizing the Kuperberg invariant.

Findings

The invariant reduces to the Kuperberg invariant for trivializable bundles.

The construction uses a combinatorial description of maps via $ ext{G}$-colored Heegaard diagrams.

The invariant is expressed in terms of an involutory Hopf algebra graded by the structure 2-group.

Abstract

We construct a scalar invariant of flat principal 2-bundles over 3-manifolds, with structure 2-group $\mathcal{G}$, from an involutory Hopf algebra graded by $\mathcal{G}$. Expressing $\mathcal{G}$ in terms of a crossed module $\chi$ and using the classification of such 2-bundles via the classifying space $B\chi$, this amounts to constructing a homotopy invariant of maps from 3-manifolds to $B\chi$. The construction of the invariant relies on a combinatorial description of such maps by $\chi$-colored Heegaard diagrams. When the corresponding map to $B\chi$ is nullhomotopic or, equivalently, when the associated flat principal $\mathcal{G}$-bundle is trivializable, the invariant reduces to the Kuperberg invariant of the underlying 3-manifold.