Invariants of Four-Manifolds with Flows Via Cohomological Field Theory

TL;DR

This paper extends cohomological field theory to define new invariants of four-manifolds with smooth flows, incorporating dynamical systems, supersymmetric gauge theories, and string theory insights.

Contribution

It introduces a novel framework for four-manifold invariants that include flow data, generalizing Donaldson-Witten invariants using higher-dimensional asymptotic cycles and supersymmetric theories.

Findings

Invariants encode flow and manifold topology information.

Seiberg-Witten invariants are incorporated into the framework.

Connections to string theory provide interpretative insights.

Abstract

The Jones-Witten invariants can be generalized for non-singular smooth vector fields with invariant probability measure on 3-manifolds, giving rise to new invariants of dynamical systems [22]. After a short survey of cohomological field theory for Yang-Mills fields, Donaldson-Witten invariants are generalized to four-dimensional manifolds with non-singular smooth flows generated by homologically non-trivial p-vector fields. These invariants have the information of the flows and they are interpreted as the intersection number of these flow orbits and constitute invariants of smooth four-manifolds admitting global flows. We study the case of Kahler manifolds by using the Witten's consideration of the strong coupling dynamics of N=1 supersymmetric Yang-Mills theories. The whole construction is performed by implementing the notion of higher dimensional asymptotic cycles a la Schwartzman [18]. In the process Seiberg-Witten invariants are also described within this context. Finally, we give an interpretation of our asymptotic observables of 4-manifolds in the context of string theory with flows.