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 $SU(N)$ $\mathcal{N}=4$ 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 $\tau =\theta /(2\pi )+4\pi i/g_{{}_{YM}}^2$, 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 $(g_{{}_{YM}}^{2}N{)}^w$. The contributions of Yang--Mills instantons of charge $k>0$ are of the form $q^{k}f(g_{{}_{YM}})$, where $q=e^{2\pi i\tau }$ and $f(g_{{}_{YM}})=O(g_{{}_{YM}}^{-2w})$ when $g_{{}_{YM}}^2\ll 1$. Anti-instanton contributions have charge $k<0$ and are of the form ${\bar{q}}^{|k|}\hat{f}(g_{{}_{YM}})$, where $\hat{f}(g_{{}_{YM}})=O(g_{{}_{YM}}^{2w})$ when $g_{{}_{YM}}^2\ll 1$. 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 $SL(2,\mathbb{Z})$-covariance in the construction.

40 pages. v2 minor typos corrected

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