Fake R^4's, Einstein Spaces and Seiberg-Witten Monopole Equations

TL;DR

This paper explores the mathematical structures of four-manifolds, particularly R^4's and Seiberg-Witten equations, and their potential implications for understanding the nature of four-dimensional spacetime and Einstein spaces.

Contribution

It connects recent mathematical results on four-manifolds and monopole equations to physical theories, proposing a link between exotic smooth structures and spacetime dimensionality.

Findings

Existence of uncountably many R^4's may explain four-dimensionality of spacetime.

Solutions of Seiberg-Witten equations on product manifolds yield Einstein spaces with specific vortex configurations.

Self-dual electromagnetic fields on certain manifolds relate to Einstein spaces and cosmological constants.

Abstract

We discuss the possible relevance of some recent mathematical results and techniques on four-manifolds to physics. We first suggest that the existence of uncountably many R^4's with non-equivalent smooth structures, a mathematical phenomenon unique to four dimensions, may be responsible for the observed four-dimensionality of spacetime. We then point out the remarkable fact that self-dual gauge fields and Weyl spinors can live on a manifold of Euclidean signature without affecting the metric. As a specific example, we consider solutions of the Seiberg-Witten Monopole Equations in which the U(1) fields are covariantly constant, the monopole Weyl spinor has only a single constant component, and the 4-manifold M_4 is a product of two Riemann surfaces Sigma_{p_1} and Sigma_{p_2}. There are p_{1}-1(p_{2}-1) magnetic(electric) vortices on \Sigma_{p_1}(\Sigma_{p_2}), with p_1 + p_2 \geq 2 (p_1=p_2= 1 being excluded). When the two genuses are equal, the electromagnetic fields are self-dual and one obtains the Einstein space \Sigma_p x \Sigma_p, the monopole condensate serving as the cosmological constant.