Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings

TL;DR

This work develops a comprehensive $SL(2,\mathbb Z)$ spectral framework to extract the full implications of S-duality for ${\cal N}=4$ SYM observables and its AdS/CFT dual. By treating modular-invariant observables as square-integrable on the modular domain, the authors deploy a Roelcke–Selberg decomposition into a modular average, Eisenstein overlaps, and cusp-form overlaps, which isolates perturbative data from non-perturbative instanton sectors. They demonstrate that ensemble averages over the conformal manifold coincide with modular averages and show that, at large $N$, the leading strong-coupling limit of observables equals their ensemble-averaged (supergravity) value, thus embedding averaging into holography. The analysis yields precise statements about instanton redundancy, Borel resummation in the $SL(2,\mathbb Z)$ setting, and non-perturbative corrections in both the ’t Hooft and very-strong-coupling limits, with clear bulk interpretations in terms of fundamental and D-string instantons. A novel perspective emerges: AdS$_5\times S^5$ supergravity can be viewed as the $N\to\infty$ limit of an ensemble of string theories over the conformal manifold, and wormhole-like contributions naturally arise within this averaged holographic framework. These results open avenues for applying spectral methods to broader holographic dualities and for exploring how arithmetic chaos and cusp forms shape non-perturbative physics in strongly coupled CFTs.

Abstract

We develop a new approach to extracting the physical consequences of S-duality of four-dimensional $\mathcal{N}=4$ super Yang-Mills (SYM) and its string theory dual, based on $SL(2,\mathbb{Z})$ spectral theory. We observe that CFT observables $\mathcal{O}$, invariant under $SL(2,\mathbb{Z})$ transformations of a complexified gauge coupling $τ$, admit a unique spectral decomposition into a basis of square-integrable functions. This formulation has direct implications for the analytic structure of $\mathcal{N}=4$ SYM data, both perturbatively and non-perturbatively in all parameters. For example, $k$-instanton sectors are uniquely determined by the zero- and one-instanton sectors, and Borel summable series around $k$-instantons have convergence radii with simple $k$-dependence. In large $N$ limits, we derive the existence and scaling of non-perturbative effects, which we exhibit for certain $\mathcal{N}=4$ SYM observables. An elegant benchmark for these techniques is the integrated four-point function conjecturally determined by [arXiv:2102.09537] for all $τ$ for $SU(N)$ gauge group; we derive and elucidate its form. These results have ramifications for holography. We explain how $\langle\mathcal{O}\rangle$, the ensemble average over the $\mathcal{N}=4$ supersymmetric conformal manifold with respect to the Zamolodchikov measure, is cleanly isolated by the spectral decomposition. We prove that the large $N$ limit of $\langle\mathcal{O}\rangle$ equals the large $N$, large 't Hooft coupling limit of $\mathcal{O}$. Holographically speaking, $\langle\mathcal{O}\rangle = \mathcal{O}_{\rm sugra}$, its value in type IIB supergravity on AdS$_5 \times$ S$^5$. This result, which extends to all orders in $1/N$, embeds ensemble averaging into the traditional AdS/CFT paradigm. The statistics of the $SL(2,\mathbb{Z})$ ensemble exhibit both perturbative and non-perturbative $1/N$ effects.

Figures (4)

• Figure 1: A cartoon depicting the two equivalent field theory duals of type IIB supergravity on AdS$_5\times S^5$, phrased in terms of the conformal manifold $\mathcal{M}$: as the limit of large 't Hooft coupling (depicted by the point approaching the cusp of $\mathcal{M}$) of planar $\mathcal{N} = 4$ SYM (left), and as the large $N$ limit of the ensemble average (denoted by the shading) of $\mathcal{N}=4$ SYM (right). On the left it is understood that the $N\to\infty$ limit is taken first.
• Figure 2: A plot of the zero mode of the square of the Eisenstein series $E_{{1\over 2}+i}(\tau)$ (rescaled by a factor of $1/2$) and of the first three even Maass cusp forms, defined as $(\phi_i^2)_0(y) \coloneqq \int_{-{1\over 2}}^{1\over 2} dx\, \phi_i(\tau)^2$, for $0 < y \leq 1$. The data for the cusp forms was obtained using the first 400 Fourier coefficients, known numerically to over 100 digits, available at lmfdb.
• Figure 3: A plot of ${2\pi^2\over 3N^3}{\cal V}(\mathcal{G}_N)$ as a function of $N$ for $N\leq 107$. This matches well onto large $N$ asymptotics \ref{['eq:VGNLargeN']}.
• Figure 4: Coupling-dependent observables ${\cal O}(\tau)$ have nonzero variance in the $\sl$ ensemble. This correlation may be represented by a "wormhole" in an abstract space containing two copies of the $\mathcal{N}=4$ SYM conformal manifold $\mathcal{M}$ (left). At large $N$, this invites a geometric re-interpretation, as a spacetime wormhole in AdS$_5 \times S^5$ with strongly coupled planar $\mathcal{N}=4$ SYM living on the conformal boundaries (right).