Generating functions and large-charge expansion of integrated correlators in 𝒩 = 4 supersymmetric Yang-Mills theory

Abstract

Abstract We recently proved that, when integrating out the spacetime dependence with a certain integration measure, four-point correlators $$ \left\langle {\mathcal{O}}_2{\mathcal{O}}_2{\mathcal{O}}_p^{(i)}{\mathcal{O}}_p^{(i)}\right\rangle $$ O 2 O 2 O p i O p i in 𝒩 = 4 supersymmetric Yang-Mills theory with SU(N) gauge group are governed by a universal Laplace-difference equation. Here $$ {\mathcal{O}}_p^{(i)} $$ O p i is a superconformal primary with charge p and degeneracy i. These physical observables, called integrated correlators, are modular-invariant functions of Yang-Mills coupling τ. The Laplace-difference equation is a recursion relation that relates integrated correlators of operators with different charges. In this paper, we introduce the generating functions for these integrated correlators that sum over the charge. By utilising the Laplace-difference equation, we determine the generating functions for all the integrated correlators, in terms of the initial data of the recursion relation. We show that the transseries of the integrated correlators in the large-p (i.e. large-charge) expansion for a fixed N consists of three parts: 1) is independent of τ, which behaves as a power series in 1/p, plus an additional log(p) term when i = j; 2) is a power series in 1/p, with coefficients given by a sum of the non-holomorphic Eisenstein series; 3) is a sum of exponentially decayed modular functions in the large-p limit, which can be viewed as a generalisation of the non-holomorphic Eisenstein series. When i = j, there is an additional modular function of τ that is independent of p and is fully determined in terms of the integrated correlator with p = 2. The Laplace-difference equation was obtained with a reorganisation of the operators that means the large-charge limit is taken in a particular way here. From these SL(2, ℤ)-invariant results, we also determine the generalised ’t Hooft genus expansion and the associated large-p non-perturbative corrections of the integrated correlators by introducing λ = p$$ {g}_{YM}^2 $$ g YM 2 . The generating functions have subtle differences between even and odd N, which have important consequences in the large-charge expansion and resurgence analysis. We also consider the generating functions of the integrated correlators for some fixed p by summing over N, and we study their large-N behaviour, as well as comment on the similarities and differences between the large-p expansion and the large-N expansion.