Relations between integrated correlators in $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory

Abstract

A bstract Integrated correlation functions in $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 supersymmetric Yang-Mills theory with gauge group SU( N ) can be expressed in terms of the localised S 4 partition function, Z N , deformed by a mass m . Two such cases are $$ {\mathcal{C}}_N={\left(\operatorname{Im}\tau \right)}^2{\partial}_{\tau }{\partial}_{\overline{\tau}}{\partial}_m^2\log {\left.{Z}_N\right|}_{m=0} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mfenced> <mml:mrow> <mml:mo>Im</mml:mo> <mml:mi>τ</mml:mi> </mml:mrow> </mml:mfenced> <mml:mn>2</mml:mn> </mml:msup> <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:mo>log</mml:mo> <mml:msub> <mml:mfenced> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mfenced> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> and $$ {\mathcal{H}}_N={\partial}_m^4\log {\left.{Z}_N\right|}_{m=0} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>∂</mml:mi> <mml:mi>m</mml:mi> <mml:mn>4</mml:mn> </mml:msubsup> <mml:mo>log</mml:mo> <mml:msub> <mml:mfenced> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mfenced> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> , which are modular invariant functions of the complex coupling τ . While $$ {\mathcal{C}}_N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> was recently written in terms of a two-dimensional lattice sum for any N and τ , $$ {\mathcal{H}}_N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> has only been evaluated up to order 1/ N 3 in a large- N expansion in terms of modular invariant functions with no known lattice sum realisation. Here we develop methods for evaluating $$ {\mathcal{H}}_N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> to any desired order in 1/ N and finite τ . We use this new data to constrain higher loop corrections to the stress tensor correlator, and give evidence for several intriguing relations between $$ {\mathcal{H}}_N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> and $$ {\mathcal{C}}_N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> to all orders in 1/ N . We also give evidence that the coefficients of the 1/ N expansion of $$ {\mathcal{H}}_N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> can be written as lattice sums to all orders. Lastly, these large N and finite τ results are used to accurately estimate the integrated correlators at finite N and finite τ .

Topics & Concepts