S-duality in Abelian gauge theory revisited

Definition of the partition function of U(1) gauge theory is extended to a class of four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) ones including the multi-Taub--NUT spaces. The partition function is calculated via zeta-function regularization with special attention to its modular properties. In the compact case, compared with the purely topological result of Witten, we find a non-trivial curvature correction to the modular weights of the partition function. But S-duality can be restored by adding gravitational counter terms to the Lagrangian in the usual way. In the ALF case however we encounter non-trivial difficulties stemming from original non-compact ALF phenomena. Fortunately our careful definition of the partition function makes it possible to circumnavigate them and conclude that the partition function has the same modular properties as in the compact case.Comment: LaTeX; 22 pages, no figure

A Global Uniqueness Theorem for Stationary Black Holes

A global uniqueness theorem for stationary black holes is proved as a direct consequence of the Topological Censorship Theorem and the topological classification of compact, simply connected four-manifolds.Comment: 9 pages, latex, journal reference adde

Explicit construction of the complex structure on the six dimensional sphere

New proof of existence of the novel complex structure on the six-sphere, followed by an explicit computation of its underlying integrable almost complex tensor by the aid of inner automorphisms of the octonions, is exhibited. Both are elementary and self-contained however the size and complexity of the emerging almost complex tensor field on the six-sphere is perplexing.Comment: 18 pages, no figures, LaTeX; all known facts on the Dolbeault cohomology of a hypothetical complex six-sphere has been reproduced within this constructio

Classification of 't Hooft instantons over multi-centered gravitational instantons

This work presents a classification of all smooth 't Hooft-Jackiw-Nohl-Rebbi instantons over Gibbons-Hawking spaces. That is, we find all smooth SU(2) Yang-Mills instantons over these spaces which arise by conformal rescalings of the metric with suitable functions. Since the Gibbons-Hawking spaces are hyper-Kahler gravitational instantons, the rescaling functions must be positive harmonic. By using twistor methods we present integral formulae for the kernel of the Laplacian associated to these spaces. These integrals are generalizations of the classical Whittaker-Watson formula. By the aid of these we prove that all 't Hooft instantons have already been found in a recent paper. This result also shows that actually all such smooth 't Hooft-Jackiw-Nohl-Rebbi instantons describe singular magnetic monopoles over the flat three-space with zero magnetic charge moreover the reducible ones generate the the full L^2 cohomology of the Gibbons-Hawking spaces.Comment: 18 pages, LaTeX, no figures; journal reference has been include