| Summary: | Abstract We present a novel expression for an integrated correlation function of four superconformal primaries in SU(N) N $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills ( N $$ \mathcal{N} $$ = 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 τ = θ / 2 π + 4 πi / g YM 2 $$ \tau =\theta /2\pi +4\pi i/{g}_{\mathrm{YM}}^2 $$ . 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 τ τ ¯ $$ E\left(s;\tau, \overline{\tau}\right) $$ with s ∈ 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/π)2n is a rational multiple of ζ(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 $$ \mathcal{N} $$ = 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 τ is a series in powers of N 1 2 − ℓ $$ {N}^{\frac{1}{2}-\mathrm{\ell}} $$ (ℓ ∈ ℤ) with coefficients that are rational sums of E s τ τ ¯ $$ E\left(s;\tau, \overline{\tau}\right) $$ with s ∈ 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 λ = g YM 2 N $$ \lambda ={g}_{\mathrm{YM}}^2N $$ is fixed. The coefficient of each order in the 1/N expansion can be expanded as a series of powers of λ that converges for |λ| < π 2. For large λ this becomes an asymptotic series when expanded in powers of 1 / λ $$ 1/\sqrt{\lambda } $$ 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-λ series is not Borel summable, and determine its resurgent non-perturbative completion, which is O exp − 2 λ $$ O\left(\exp \left(-2\sqrt{\lambda}\right)\right) $$ .
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