Homological Topological Quantum Field Theories

Abstract

We develop a new framework for quantum invariants of $3$-manifolds by extending to cobordisms a homological construction of mapping class group representations. More specifically, we construct a $(2+1)$-dimensional topological quantum field theory (TQFT) that assigns to each surface the twisted homology of its unordered configuration space. The construction requires a choice of local systems on configuration spaces together with additional data. We formulate sufficient conditions on these data that guarantee the TQFT axioms, and we show that there are at least two useful examples satisfying them. One of them yields a homological construction of the projective Kerler--Lyubashenko TQFT, while the other recovers the Frohman--Nicas--Donaldson TQFT. In contrast to the classical algebraic constructions of quantum invariants, our approach is purely topological and relies on multi-trajectory spaces of cobordisms.