Quantum invariants of flat 2-bundles over 3-manifolds

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 $χ$ and using the classification of such 2-bundles via the classifying space $Bχ$, this amounts to constructing a homotopy invariant of maps from 3-manifolds to $Bχ$. The construction of the invariant relies on a combinatorial description of such maps by $χ$-colored Heegaard diagrams. When the corresponding map to $Bχ$ 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.