Fake R4s, Einstein spaces and Seiberg-Wittenmonopole equations

Abstract

We discuss the possible relevance of some recentmathematical results and techniques on 4-manifolds tophysics. We first suggest that the existence ofuncountably many R4s with non-equivalent smoothstructures, a mathematical phenomenon unique to fourdimensions, may be responsible for the observedfour-dimensionality of spacetime. We then point out theremarkable fact that self-dual gauge fields and Weylspinors can live on a manifold of Euclidean signaturewithout affecting the metric. As a specific example, weconsider solutions of the Seiberg-Witten monopoleequations in which the U(1) fields are covariantlyconstant, the monopole Weyl spinor has only a singleconstant component, and the 4-manifold ℳ4 isa product of two Riemann surfacesΣp1 and Σp2. There are p1-1(p2-1)magnetic (electric) vortices onΣp1 (Σp2), withp1 + p2⩾2(p1 = p2 = 1 being excluded). When the two genera are equal, theelectromagnetic fields are self-dual and one obtains the Einstein spaceΣp×Σp, the monopole condensate serving as thecosmological constant.