S-duality transformation of $\mathcal{N}$ $=4$ SYM theory at the operator level

TL;DR

This paper investigates the S-duality transformation in $ =4$ SYM theory at the operator level, defining how gauge invariant operators and states transform under $SL(2,Z)$ and establishing criteria for S-duality invariance.

Contribution

It introduces an operator $S$ implementing S-duality as an $SL(2,Z)$ transformation in loop space, and provides a criterion for the theory's S-duality invariance based on superconformal charges.

Findings

Defined S-duality transformation at the operator level.

Established criterion for S-duality invariance involving supercharges.

Connected supercharges of BPS operators via a chiral rotation.

Abstract

We consider the S-duality transformation of gauge invariant operators and states in $\mathcal{N}$ $=4$ SYM theory. The transformation is realized through an operator $ S $ which is the $ SL(2,Z) $ canonical transformation in loop space with the gauge invariant electric and the magnetic flux operators composing the canonical variables. Based on $ S $, S-duals for all of the physical operators and states can be defined. The criterion for the theory to be S-duality invariant is that the superconformal charges and their S-duals differ by a $ U(1)_{Y} $ phase. The verification can be done by checking the S transformation for supersymmetry and special supersymmetry variations of the loop operators. The fact that supercharges preserved by BPS Wilson operators and the S-dual BPS 't Hooft operators differ by a $4d $ chiral rotation could in some sense serve as a proof.