Surface operators in N = 2 abelian gauge theory

Abstract

We generalise the analysis in [arXiv:0904.1744] to superspace, and explicitly prove that for any embedding of surface operators in a general, twisted N = 2 pure abelian theory on an arbitrary spin (or non-spin) four-manifold, the parameters transform naturally under the SL(2,Z) (or Γ_0(2)) duality of the theory. However, for nontrivially-embedded surface operators, exact S-duality holds if and only if the "quantum" parameter effectively vanishes, while the overall SL(2,Z) (or Γ_0(2)) duality holds up to a c-number at most, regardless. Nevertheless, this observation sets the stage for a physical proof of a remarkable mathematical result by Kronheimer and Mrowka [1]—that expresses a "ramified" analog of the Donaldson invariants solely in terms of the ordinary Donaldson invariants — which, will appear, among other things, in forthcoming work [2]. As a prelude to that, the effective interaction on the corresponding u-plane will be computed. In addition, the dependence on second Stiefel-Whitney classes and the appearance of a Spinc structure in the associated low-energy Seiberg-Witten theory with surface operators, will also be demonstrated. In the process, we will stumble upon an interesting phase factor that is otherwise absent in the "unramified" case.

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