3-manifolds and Vafa-Witten theory

TL;DR

This paper initiates explicit calculations of Vafa-Witten invariants for 3-manifolds, exploring their properties and relations to Floer homology, with computations on specific examples involving surgery operations.

Contribution

It provides the first explicit computations of Vafa-Witten invariants for 3-manifolds and discusses their structural properties and connections to existing theories.

Findings

Computed Vafa-Witten invariants for specific 3-manifolds involving surgery operations

Identified structural properties such as modular group action and relation to Floer homology

Established the infinite-dimensionality and absence of instantons in general cases

Abstract

We initiate explicit computations of Vafa-Witten invariants of 3-manifolds, analogous to Floer groups in the context of Donaldson theory. In particular, we explicitly compute the Vafa-Witten invariants of 3-manifolds in a family of concrete examples relevant to various surgery operations (the Gluck twist, knot surgeries, log-transforms). We also describe the structural properties that are expected to hold for general 3-manifolds, including the modular group action, relation to Floer homology, infinite-dimensionality for an arbitrary 3-manifold, and the absence of instantons.