Homotopy field theory in dimension 3 and crossed group-categories

TL;DR

This paper develops a framework linking homotopy quantum field theories in three dimensions with crossed group-categories, providing new invariants for pi-manifolds and a homotopy modular functor.

Contribution

It introduces modular crossed pi-categories and shows how they generate 3D HQFTs with target space K(pi,1), connecting algebraic structures with topological quantum field theories.

Findings

Constructed 3D HQFTs from modular crossed pi-categories.

Derived numerical invariants of 3-dimensional pi-manifolds.

Established a homotopy modular functor in 2D.

Abstract

A 3-dimensional homotopy quantum field theory (HQFT) can be described as a TQFT for surfaces and 3-cobordisms endowed with homotopy classes of maps into a given space. For a group $\pi$, we introduce a notion of a modular crossed $\pi$-category and show that such a category gives rise to a 3-dimensional HQFT with target space $K(\pi,1)$. This includes numerical invariants of 3-dimensional $\pi$-manifolds and a 2-dimensional homotopy modular functor. We also introduce and discuss a parallel notion of a quasitriangular crossed Hopf $\pi$-coalgebra.