Surface Operators in N=2 Abelian Gauge Theory

TL;DR

This paper extends the analysis of surface operators in N=2 abelian gauge theories to superspace, demonstrating their transformation properties under SL(2,Z) duality and exploring implications for Donaldson invariants and Seiberg-Witten theory.

Contribution

It provides a detailed proof of the transformation behavior of surface operators under duality in superspace and connects these to mathematical invariants and low-energy effective theories.

Findings

Surface operators transform under SL(2,Z) duality in N=2 theories.

Exact S-duality holds only when the quantum parameter vanishes.

Dependence on Stiefel-Whitney classes and Spin^c structures is demonstrated.

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 four-manifold, the parameters transform naturally under the SL(2,Z) 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) 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--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. 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 Spin^c 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.