S-duality in Abelian gauge theory revisited

Abstract

Definition of the partition function of U(1) gauge theory is extended to a class of fourmanifolds containing all compact spaces and the asymptotically locally flat (ALF) ones including the multi-Taub-NUT sopaces. The partition function is calculated via zetafunction regularization and heat kernel techniques 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.

Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality

“…We have found the discussion in [30] and [14] in three and four dimensional abelian Yang-Mills theory, respectively, to be very useful. Our discussion is a generalization to six dimensions and an extension to include deformations.…”

“…We now go back to our problem and consider the action as in (1.6) but drop the term involving C 6 for simplicity because it does not affect the modularity argument. Assuming as before that 3-curvature is self-adjoint H † B = H B with respect to the deformed inner product, the action is 14) as in the case for analyzing the partition function in the Yang-Mills case. The second term in the action (3.14) is a topological invariant, involving the Dixmier-Douady classes of the underlying gerbe.…”

Electric-magnetic duality of Abelian gauge theory on the four-torus, from the fivebrane on T 2 × T 4, via their partition functions

“…We find the two quantizations distribute zero and oscillator mode contributions differently, and thus these factors transform differently under the action of SL(2, Z). We summarize the path integral quantization results from [9], [14], [15], [32]. Following [9], [15], the two-form zero mode part, F 2π is the harmonic representative and can be expanded in terms of the basis α I = 1 (2π) 2 dθ 1 ∧ dθ 2 , etc., I = 1, 2, .., 6 namely…”

The four dimensional Yang--Mills partition function in the vicinity of the vacuum

“…In this paper, as a continuation of our earlier work on the Abelian case [4], we make an attempt to return to the original setup and to compute the partition function of the non-supersymmetric, four dimensional Euclidean, non-Abelian pure gauge theory. The sacrifice we make for not using any supersymmetric, etc.…”

“…consists of a derivative dA and a quadratic (interacting) term A ∧ A of the gauge potential. In four dimensions there is a delicate balance between these terms as a consequence of the Sobolev embedding L 2 1 ⊂ L 4 which is on the borderline in four dimensions. Indeed, this embedding allows one to compare the L 2 -norm of the dA and A ∧ A terms.…”