Electromagnetic Duality for Line Defect Correlators in $\mathcal{N}=4$ Super Yang-Mills Theory

We study particular integrated correlation functions of two superconformal primary operators of the stress tensor multiplet in the presence of a half-BPS line defect labelled by electromagnetic charges $(p,q)$ in $\mathcal{N}=4$ supersymmetric Yang-Mills theory (SYM) with gauge group $SU(N)$. An important consequence of ${\rm SL}(2,\mathbb{Z})$ electromagnetic duality in $\mathcal{N}=4$ SYM is that correlators of line defect operators with different charges $(p,q)$ must be related in a non-trivial manner when the complex coupling $\tau=\theta/(2\pi)+4\pi i /g_{_{\rm YM}}^2$ is transformed appropriately. In this work we introduce a novel class of real-analytic functions whose automorphic properties with respect to ${\rm SL}(2,\mathbb{Z})$ match the expected transformations of line defect operators in $\mathcal{N}=4$ SYM under electromagnetic duality. At large $N$ and fixed $\tau$, the correlation functions we consider are related to scattering amplitudes of two gravitons from extended $(p,q)$-strings in the holographic dual type IIB superstring theory. We show that the large-$N$ expansion coefficients of the integrated two-point line defect correlators are given by finite linear combinations with rational coefficients of elements belonging to this class of automorphic functions. On the other hand, for any fixed value of $N$ we conjecture that the line defect integrated correlators can be expressed as formal infinite series over such automorphic functions. The resummation of this series produces a simple lattice sum representation for the integrated line defect correlator that manifests its automorphic properties. We explicitly demonstrate this construction for the cases with gauge group $SU(2)$ and $SU(3)$. Our results give direct access to non-perturbative integrated correlators in the presence of an 't Hooft-line defect, observables otherwise very difficult to compute by other means.