Exact expressions for n-point maximal U(1)_Y-violating integrated correlators in SU(N)\mathcal{N}=4 SYM

Abstract

The exact expressions for integrated maximal U(1)Y violating (MUV) n-point correlators in SU(N) N = 4 supersymmetric Yang–Mills theory are determined. The analysis generalises previous results on the integrated correlator of four superconformal primaries and is based on supersymmetric localisation. The integrated correlators are functions of N and τ = θ/(2π) + 4πi/g2 Y M , and are expressed as two-dimensional lattice sums that are modular forms with holomorphic and anti-holomorphic weights (w, −w) where w = n − 4. The correlators satisfy Laplace-difference equations that relate the SU(N + 1), SU(N) and SU(N −1) expressions and generalise the equations previously found in the w = 0 case. The correlators can be expressed as infinite sums of Eisenstein modular forms of weight (w, −w). For any fixed value of N the perturbation expansion of this correlator is found to start at order (g 2 Y M N) w. The contributions of Yang–Mills instantons of charge k > 0 are of the form q k f(gY M ), where q = e 2πiτ and f(gY M ) = O(g −2w Y M ) when g 2 Y M 1. Anti-instanton contributions have charge k < 0 and are of the form ¯q |k| ˆf(gY M ), where ˆf(gY M ) = O(g 2w Y M ) when g 2 Y M 1. Properties of the large-N expansion are in agreement with expectations based on the low energy expansion of flat-space type IIB superstring amplitudes. We also comment on the identification of n-point free-field MUV correlators with the integrands of (n − 4)-loop perturbative contributions to the four-point correlator. In particular, we emphasise the important rˆole of SL(2, Z)-covariance in the construction.