Sutured Floer homology, sutured TQFT and non-commutative QFT

TL;DR

This paper introduces a sutured topological quantum field theory framework that links sutured Floer homology with non-commutative quantum field theory, enabling new computations and revealing algebraic structures.

Contribution

It develops a sutured TQFT approach that connects contact elements in sutured Floer homology to a non-commutative Fock space, extending previous results to integer coefficients.

Findings

Computed contact elements in sutured Floer homology over Z for certain manifolds.

Established an isomorphism between sutured TQFT of discs and a non-commutative Fock space.

Revealed algebraic structures underlying sutured Floer homology and TQFT.

Abstract

We define a "sutured topological quantum field theory", motivated by the study of sutured Floer homology of product 3-manifolds, and contact elements. We study a rich algebraic structure of suture elements in sutured TQFT, showing that it corresponds to contact elements in sutured Floer homology. We use this approach to make computations of contact elements in sutured Floer homology over $\Z$ of sutured manifolds $(D^2 \times S^1, F \times S^1)$ where $F$ is finite. This generalises previous results of the author over $\Z_2$ coefficients. Our approach elaborates upon the quantum field theoretic aspects of sutured Floer homology, building a non-commutative Fock space, together with a bilinear form deriving from a certain combinatorial partial order; we show that the sutured TQFT of discs is isomorphic to this Fock space.