Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin's invariant

TL;DR

This paper introduces a new gauge-theoretic integer invariant for certain 4-manifolds, combining Seiberg-Witten solution counts and Dirac operator indices, with dependencies canceling out under metric and perturbation variations.

Contribution

It presents a novel integer lift of the Rohlin invariant for 4-manifolds with specific homology, integrating Seiberg-Witten theory and end-periodic Dirac operators.

Findings

Defined a gauge-theoretic integer invariant for 4-manifolds.

Proved the invariant's independence from metric and perturbation choices.

Connected the invariant to classical topological invariants.

Abstract

We introduce a gauge-theoretic integer lift of the Rohlin invariant of a smooth 4-manifold X with the homology of $S^1 \times S^3$. The invariant has two terms; one is a count of solutions to the Seiberg-Witten equations on X, and the other is essentially the index of the Dirac operator on a non-compact manifold with end modeled on the infinite cyclic cover of X. Each term is metric (and perturbation) dependent, and we show that these dependencies cancel as the metric and perturbation vary in a 1-parameter family.