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) $$ \mathcal{N} $$ 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/$$ {g}_{YM}^2 $$ g YM 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 ($$ {g}_{YM}^2 $$ g YM 2 N)w. The contributions of Yang-Mills instantons of charge k > 0 are of the form qkf(gYM), where q = e2πiτ and f(gYM) = O($$ {g}_{YM}^{-2w} $$ g YM − 2 w ) when $$ {g}_{YM}^2 $$ g YM 2 ≪ 1. Anti-instanton contributions have charge k < 0 and are of the form $$ {\overline{q}}^{\left|k\right|}\hat{f}\left({g}_{YM}\right) $$ q ¯ k f ̂ g YM , where $$ \hat{f}\left({g}_{YM}\right)=O\left({g}_{YM}^{2w}\right) $$ f ̂ g YM = O g YM 2 w when $$ {g}_{YM}^2 $$ g YM 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.

Harnessing S-duality in $$ \mathcal{N} $$ = 4 SYM & supergravity as SL(2, ℤ)-averaged strings

“…More N = 4 SYM observables: as an aid to future progress we mention some observables one could analyze using SL(2, Z) spectral methods, in order of increasing ambitiousness. There are some nearby extensions, such as the generalization of G N (τ ) to other SYM gauge groups or to the SL(2, Z)-covariant cousin in [159] and the integrated 22pp correlator written in (5.58), for which we made a tentative prediction of its spectral overlap. A more complicated extension would be to the extremal correlators O p O p in 4d N = 2 SQCD: upon multiplying by an appropriate power of y, these become SL(2, Z) invariant [147].…”

Exact results for duality-covariant integrated correlators in $\mathcal{N}=4$ SYM with general classical gauge groups

“…Another challenge is to construct expressions for integrated n-point correlators with n > 4. Although the general problem is daunting, following the methods of this paper and the previous results of [34,35], it should be possible to obtain exact expressions for integrated maximal U (1) Y -violating n-point correlators with n > 4 for all classical gauge groups, which transform covariantly under GNO S-duality (where U (1) Y is the bonus symmetry [36]).…”

Selected Topics in Analytic Conformal Bootstrap: A Guided Journey

“…By studying correlators in this limit, one can access SL(2, Z) properties of superstring amplitudes and perform nonperturbative precision tests of AdS/CFT. For works in this direction, see [224,[236][237][238][239][240][241].…”