A finite-dimensional TQFT for three-manifolds based on group PSL(2, C) and cross-ratios

TL;DR

This paper introduces a new finite-dimensional TQFT for three-manifolds using PSL(2,C) and cross-ratios, constructing manifold invariants via acyclic chain complexes and their torsion, with promising nontrivial examples.

Contribution

It develops a novel TQFT framework based on group PSL(2,C) and chain complex torsion, creating new manifold invariants for manifolds with boundary.

Findings

Constructed manifold invariants using chain complex torsion.

Invariants are highly nontrivial in example cases.

Extended invariants to manifolds with boundary.

Abstract

In this paper, we begin constructing a new finite-dimensional topological quantum field theory (TQFT) for three-manifolds, based on group PSL(2,C) and its action on a complex variable by fractional-linear transformations, by providing its key ingredient -- a new type of chain complexes. As these complexes happen to be acyclic often enough, we make use of their torsion to construct different versions of manifold invariants. In particular, we show how to construct a large set of invariants for a manifold with boundary, analogous to the set of invariants based on Euclidean geometric values and used in a paper by one of the authors for constructing a "Euclidean" TQFT. We show on examples that our invariants are highly nontrivial.