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

Understand

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.