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

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 $\tau$, 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 $\tau$ for $SU(N)$ gauge group; we derive and elucidate its form. These results have ramifications for holography. We explain how $⟨\mathcal{O}⟩$, 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 $⟨\mathcal{O}⟩$ equals the large $N$, large 't Hooft coupling limit of $\mathcal{O}$. Holographically speaking, $⟨\mathcal{O}⟩={\mathcal{O}}_{\mathrm{sugra}}$, 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.

arXiv comment77 pages + refs. v2: added refs, fixed typos, some reorganization/clarifications in Sections 11 and 12. v3: added refs, minor changes

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