Surface Operators in Abelian Gauge Theory

TL;DR

This paper studies how surface operators in abelian gauge theories transform under dualities, revealing conditions for exact S-duality and the modular properties of partition functions and correlators on four-manifolds.

Contribution

It explicitly demonstrates the transformation properties of surface operator parameters under SL(2,Z) duality and introduces a duality wall formalism for their analysis.

Findings

Surface parameters transform under SL(2,Z) duality.

Exact S-duality holds only when quantum parameters vanish.

Partition functions and correlators behave as modular forms.

Abstract

We consider arbitrary embeddings of surface operators in a pure, non-supersymmetric abelian gauge theory on spin (non-spin) four-manifolds. For any surface operator with a priori simultaneously non-vanishing parameters, we explicitly show that the parameters transform naturally under an SL(2, Z) (or a congruence subgroup thereof) duality of the theory. However, for non-trivially-embedded surface operators, exact S-duality holds only if the quantum parameter effectively vanishes, while the overall SL(2, Z) (or a congruence subgroup thereof) duality holds up to a c-number at most, regardless. Via the formalism of duality walls, we furnish an alternative derivation of the transformation of parameters - found also to be consistent with a switch from Wilson to 't Hooft loop operators under S-duality. With any background embedding of surface operators, the partition function and the correlation functions of non-singular, gauge-invariant local operators on any curved four-manifold, are found to transform like modular forms under the respective duality groups.