Geometric torsions and an Atiyah-style topological field theory

TL;DR

This paper generalizes the construction of topological invariants for 3D manifolds with triangulated boundaries using geometric torsions, employing Berezin's calculus of anti-commuting variables, and aligns with Atiyah's axioms.

Contribution

It extends previous invariants to multiple boundary components and introduces a framework based on anti-commuting variables consistent with Atiyah's axioms.

Findings

Invariants are constructed for any number of boundary components.

The invariants are based on torsion of acyclic complexes of geometric origin.

The theory aligns with a modified version of Atiyah's axioms.

Abstract

The construction of invariants of three-dimensional manifolds with a triangulated boundary, proposed earlier by the author for the case when the boundary consists of not more than one connected component, is generalized to any number of components. These invariants are based on the torsion of acyclic complexes of geometric origin. The relevant tool for studying our invariants turns out to be F.A. Berezin's calculus of anti-commuting variables; in particular, they are used in the formulation of the main theorem of the paper, concerning the composition of invariants under a gluing of manifolds. We show that the theory obeys a natural modification of M. Atiyah's axioms for anti-commuting variables.