Laplace-difference equation for integrated correlators of operators with general charges in $$ \mathcal{N} $$ = 4 SYM
Augustus Brown, Congkao Wen, Haitian Xie
Abstract
A bstract We consider the integrated correlators associated with four-point correlation functions $$ \left\langle {\mathcal{O}}_2{\mathcal{O}}_2{\mathcal{O}}_p^{(i)}{\mathcal{O}}_p^{(j)}\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>O</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>p</mml:mi> <mml:mfenced> <mml:mi>i</mml:mi> </mml:mfenced> </mml:msubsup> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>p</mml:mi> <mml:mfenced> <mml:mi>j</mml:mi> </mml:mfenced> </mml:msubsup> </mml:mrow> </mml:mfenced> </mml:math> in four-dimensional $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 supersymmetric Yang-Mills theory (SYM) with SU( N ) gauge group, where $$ {\mathcal{O}}_p^{(i)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>p</mml:mi> <mml:mfenced> <mml:mi>i</mml:mi> </mml:mfenced> </mml:msubsup> </mml:math> is a superconformal primary with charge (or dimension) p and the superscript i represents possible degeneracy. These integrated correlators are defined by integrating out spacetime dependence with a certain integration measure, and they can be computed via supersymmetric localisation. They are modular functions of complexified Yang-Mills coupling τ . We show that the localisation computation is systematised by appropriately reorganising the operators. After this reorganisation of the operators, we prove that all the integrated correlators for any N , with some crucial normalisation factor, satisfy a universal Laplace-difference equation (with the laplacian defined on the τ -plane) that relates integrated correlators of operators with different charges. This Laplace-difference equation is a recursion relation that completely determines all the integrated correlators, once the initial conditions are given.