A bstract When the SU( N ) $$ \mathcal{N} $$ N = 4 superYangMills (SYM) theory with complexified gauge coupling τ is placed on a round foursphere and deformed by an $$ \mathcal{N} $$ N = 2preserving mass parameter m , its free energy F ( m, τ, $$ \overline{\tau} $$ τ ¯ ) can be computed exactly using supersymmetric localization. In this work, we derive a new exact relation between the fourth derivative $$ {\partial}_m^4F\left(m,\tau, \overline{\tau}\right)\left{{}_m}_{=0}\right. $$ ∂ m 4 F m τ τ ¯ m = 0 of the sphere free energy and the integrated stresstensor multiplet fourpoint function in the $$ \mathcal{N} $$ N = 4 SYM theory. We then apply this exact relation, along with various other constraints derived in previous work (coming from analytic bootstrap, the mixed derivative $$ {\partial}_{\tau }{\partial}_{\overline{\tau}}{\partial}_m^2F\left(m,\tau, \overline{\tau}\right)\left{{}_m}_{=0}\right. $$ ∂ τ ∂ τ ¯ ∂ m 2 F m τ τ ¯ m = 0 , and type IIB superstring theory scattering amplitudes) to determine various perturbative terms in the large N and large ’t Hooft coupling λ expansion of the $$ \mathcal{N} $$ N = 4 SYM correlator at separated points. In particular, we determine the leading large λ term in the $$ \mathcal{N} $$ N = 4 SYM correlation function at order 1 /N 8 . This is three orders beyond the planar limit.
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Modular invariance in superstring theory from $$ \mathcal{N} $$ = 4 superYangMills
A bstract We study the fourpoint function of the lowestlying halfBPS operators in the $$ \mathcal{N} $$ N = 4 SU( N ) superYangMills theory and its relation to the flatspace fourgraviton amplitude in type IIB superstring theory. We work in a large N expansion in which the complexified YangMills coupling τ is fixed. In this expansion, nonperturbative instanton contributions are present, and the SL(2 , ℤ) duality invariance of correlation functions is manifest. Our results are based on a detailed analysis of the sphere partition function of the massdeformed SYM theory, which was previously computed using supersymmetric localization. This partition function determines a certain integrated correlator in the undeformed $$ \mathcal{N} $$ N = 4 SYM theory, which in turn constrains the fourpoint correlator at separated points. In a normalization where the twopoint functions are proportional to N 2 − 1 and are independent of τ and $$ \overline{\tau} $$ τ ¯ , we find that the terms of order $$ \sqrt{N} $$ N and $$ 1/\sqrt{N} $$ 1 / N in the large N expansion of the fourpoint correlator are proportional to the nonholomorphic Eisenstein series $$ E\left(\frac{3}{2},\tau, \overline{\tau}\right) $$ E 3 2 τ τ ¯ and $$ E\left(\frac{5}{2},\tau, \overline{\tau}\right) $$ E 5 2 τ τ ¯ , respectively. In the flat space limit, these terms match the corresponding terms in the type IIB Smatrix arising from R 4 and D 4 R 4 contact interactions, which, for the R 4 case, represents a check of AdS/CFT at finite string coupling. Furthermore, we present striking evidence that these results generalize so that, at order $$ {N}^{\frac{1}{2}m} $$ N 1 2 − m with integer m ≥ 0, the expansion of the integrated correlator we study is a linear sum of nonholomorphic Eisenstein series with halfinteger index, which are manifestly SL(2 , ℤ) invariant.
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 Award ID(s):
 1820651
 NSFPAR ID:
 10311711
 Date Published:
 Journal Name:
 Journal of High Energy Physics
 Volume:
 2020
 Issue:
 11
 ISSN:
 10298479
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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