Fake R(4)s, Einstein spaces and Seiberg-Witten monopole equations

We discuss the possible relevance of some recent mathematical results and techniques on 4-manifolds to physics. We first suggest that the existence of uncountably many (RS)-S-4 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 (p1) and Sigma (p2). There are p(1) - 1(p(2) - 1) magnetic (electric) vortices on Sigma (p1) (Sigma (p2)), with p(1) + p(2) greater than or equal to 2 (p(1) = p(2) = 1 being excluded). When the two genera 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.