Exact properties of an integrated correlator in N=4 SU(N) SYM

We present a novel expression for an integrated correlation function of four superconformal primaries in SU(N) N = 4 supersymmetric Yang-Mills (N = 4 SYM) theory. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. In this paper the correlator is re-expressed as a sum over a two dimensional lattice that is valid for all N and all values of the complex Yang-Mills coupling tau=theta/2 pi+4 pi i/g(YM)(2). In this form it is manifestly invariant under SL(2, Z) Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the SU(N) correlator to the SU(N + 1) and SU(N - 1) correlators. For any fixed value of N the correlator can be expressed as an infinite series of non-holomorphic Eisenstein series, E(s tau(tau) over bar with s is an element of Z, and rational coefficients that depend on the values of N and s. The perturbative expansion of the integrated correlator is an asymptotic but Borel summable series, in which the n-loop coefficient of order (g(YM)/pi)(2n) is a rational multiple of zeta(2n + 1). The n = 1 and n = 2 terms agree precisely with results determined directly by integrating the expressions in one-loop and two-loop perturbative N = 4 SYM field theory. Likewise, the charge-k instanton contributions (|k| = 1, 2, . . .) have an asymptotic, but Borel summable, series of perturbative corrections. The large-N expansion of the correlator with fixed tau is a series in powers of N1/2-l (l is an element of Z) with coefficients that are rational sums of Es;tau(tau) over bar with s is an element of Z + 1/2. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider the 't Hooft topological expansion of large-N Yang-Mills theory in which lambda=g(YM)(2) N is fixed. The coefficient of each order in the 1/N expansion can be expanded as a series of powers of lambda that converges for |lambda| < pi(2). For large lambda this becomes an asymptotic series when expanded in powers of 1/root lambda with coefficients that are again rational multiples of odd zeta values, in agreement with earlier results and providing new ones. We demonstrate that the large-lambda series is not Borel summable, and determine its resurgent non-perturbative completion, which is O(exp(-2 root lambda)).

Automated summary

This paper presents a novel expression for an integrated correlator in SU(N) N=4 supersymmetric Yang-Mills theory, applicable to all N and values of the complex Yang-Mills coupling. The correlator can be expressed as an infinite series of non-holomorphic Eisenstein series and rational coefficients, and is invariant under SL(2,Z) Montonen-Olive duality. Furthermore, the large-N expansion of the correlator with fixed tau is a series in powers of N1/2-l with coefficients that are rational sums of Es;tau(tau) over bar, giving an all orders derivation of the conjectured expansion.