Harnessing S-duality in $$ \mathcal{N} $$ = 4 SYM & supergravity as SL(2, ℤ)-averaged strings

Abstract

A bstract We develop a new approach to extracting the physical consequences of S-duality of four-dimensional $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> = 4 super Yang-Mills (SYM) and its string theory dual, based on SL(2, ℤ) spectral theory. We observe that CFT observables 𝒪, invariant under SL(2, 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> = 4 SYM data, both perturbatively and non-perturbatively in all parameters. These are especially constraining for the structure of instantons: 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, in both N and the ‘t Hooft coupling, which we exhibit for certain $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> = 4 SYM observables. An elegant benchmark for these techniques is the integrated stress tensor multiplet four-point function, conjecturally determined by [1] for all τ for SU( N ) gauge group; we elucidate its form, and explain how the SU(2) case is the simplest possible observable consistent with SL(2, ℤ)-invariant perturbation theory. These results have ramifications for holography. We explain how $$ \left\langle \mathcal{O}\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfenced><mml:mi>O</mml:mi></mml:mfenced></mml:math> , the ensemble average of 𝒪 over the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> = 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 $$ \left\langle \mathcal{O}\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfenced><mml:mi>O</mml:mi></mml:mfenced></mml:math> equals the large N , large ‘t Hooft coupling limit of 𝒪. Holographically speaking, $$ \left\langle \mathcal{O}\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfenced><mml:mi>O</mml:mi></mml:mfenced></mml:math> = 𝒪 sugra , its value in type IIB supergravity on AdS 5 × 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, ℤ) ensemble exhibit both perturbative and non-perturbative 1/ N effects. We discuss further implications and generalizations to other AdS compactifications of string/M-theory.

Topics & Concepts