Far beyond the planar limit in strongly-coupled $$ \mathcal{N} $$ = 4 SYM
Shai M. Chester, Silviu S. Pufu
Abstract
A bstract When the SU( N ) $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 super-Yang-Mills (SYM) theory with complexified gauge coupling τ is placed on a round four-sphere and deformed by an $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2-preserving mass parameter m , its free energy F ( m, τ, $$ \overline{\tau} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> ) can be computed exactly using supersymmetric localization. In this work, we derive a new exact relation between the fourth derivative $$ {\partial}_m^4F\left(m,\tau, \overline{\tau}\right)\left|{{}_m}_{=0}\right. $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>∂</mml:mi> <mml:mi>m</mml:mi> <mml:mn>4</mml:mn> </mml:msubsup> <mml:mi>F</mml:mi> <mml:mfenced> <mml:mi>m</mml:mi> <mml:mi>τ</mml:mi> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mfenced> <mml:mfenced> <mml:msub> <mml:msub> <mml:mrow/> <mml:mi>m</mml:mi> </mml:msub> <mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mfenced> </mml:math> of the sphere free energy and the integrated stress-tensor multiplet four-point function in the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 $$ {\partial}_{\tau }{\partial}_{\overline{\tau}}{\partial}_m^2F\left(m,\tau, \overline{\tau}\right)\left|{{}_m}_{=0}\right. $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> <mml:msub> <mml:mi>∂</mml:mi> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> <mml:msubsup> <mml:mi>∂</mml:mi> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mi>F</mml:mi> <mml:mfenced> <mml:mi>m</mml:mi> <mml:mi>τ</mml:mi> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mfenced> <mml:mfenced> <mml:msub> <mml:msub> <mml:mrow/> <mml:mi>m</mml:mi> </mml:msub> <mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mfenced> </mml:math> , 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 SYM correlator at separated points. In particular, we determine the leading large- λ term in the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 SYM correlation function at order 1 /N 8 . This is three orders beyond the planar limit.