Item

Abstract

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.