Noncommutative Topological Quantum Field Theory, Noncommutative Floer Homology, Noncommutative Hodge Theory

TL;DR

This paper explores the development of a noncommutative topological quantum field theory and Floer homology, aiming to unify physics and mathematics by extending these theories to broader contexts including gravity and all 3-manifolds.

Contribution

It introduces novel concepts of noncommutative TQFT and Floer homology, extending their applicability and proposing new noncommutative Hodge theory frameworks.

Findings

Proposes a noncommutative TQFT framework for all known interactions including gravity.

Suggests noncommutative Floer homology applies to all 3-manifolds, not just homology spheres.

Develops noncommutative Hodge theory for differential forms and foliations.

Abstract

We present some ideas for a possible Noncommutative Topological Quantum Field Theory (NCTQFT) and Noncommutative Floer Homology (NCFH). Our motivation is two-fold and it comes both from physics and mathematics: On the one hand we argue that NCTQFT is the correct mathematical framework for a quantum field theory of all known interactions in nature (including gravity). On the other hand we hope that a possible NCFH will apply to practically every 3-manifold (and not only to homology 3-spheres as ordinary Floer Homology currently does). The two motivations are closely related since, at least in the commutative case, Floer Homology Groups constitute the space of quantum observables of (3+1)-dim Topological Quantum Field Theory. Towards this goal we present some "Noncommutative" Versions of Hodge Theory for noncommutative differentail forms and tangential cohomology for foliations.