Exact expressions for n-point maximal U(1)(Y)-violating integrated correlators in SU(N) N=4 SYM

Abstract The exact expressions for integrated maximal U (1) Y violating (MUV) n-point correlators in SU (N) 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/ g YM 2 gₘ₌², 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 YM 2 gₘ₌² N) w. The contributions of Yang-Mills instantons of charge k > 0 are of the form q k f (g YM), where q = e 2πiτ and f (g YM) = O (g YM − 2 w gₘ₌^-2w) when g YM 2 gₘ₌² ≪ 1. Anti-instanton contributions have charge k < 0 and are of the form q ¯ k f ̂ g YM q^|k|f (gₘ₌), where f ̂ g YM = O g YM 2 w f (gₘ₌) =O (gₘ₌^2w) when g YM 2 gₘ₌² ≪ 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 rôle of SL (2, ℤ) -covariance in the construction.