Sutured Floer homology, sutured TQFT and noncommutative QFT

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 $\mathbb{Z}$ of sutured manifolds $\left(D^2\times S^1,F\times S^1\right)$ where $F$ is finite. This generalises previous results of the author over ${\mathbb{Z}}_2$ coefficients. Our approach elaborates upon the quantum field theoretic aspects of sutured Floer homology, building a noncommutative 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.