SciPost Phys. 13, 092 (2022) ·
published 11 October 2022

· pdf
We present exact expressions for certain integrated correlators of four superconformal primary operators in the stress tensor multiplet of $\cN=4$ supersymmetric YangMills (SYM) theory with classical gauge group, $G_N$ $= SO(2N)$, $SO(2N+1)$, $USp(2N)$. These integrated correlators are expressed as twodimensional lattice sums by considering derivatives of the localised partition functions, generalising the expression obtained for $SU(N)$ \mbg{gauge group} in our previous works. These expressions are manifestly covariant under GoddardNuytsOlive duality. The integrated correlators can also be formally written as infinite sums of nonholomorphic Eisenstein series with integer indices and rational coefficients. Furthermore, the action of the hyperbolic Laplace operator with respect to the complex coupling $\tau=\theta/(2\pi) + 4\pi i /g^2_{_{YM}}$ on any integrated correlator for gauge group $G_N$ relates it to a linear combination of correlators with gauge groups $G_{N+1}$, $G_N$ and $G_{N1}$. These ``Laplacedifference equations'' determine the expressions of integrated correlators for all classical gauge groups for any value of $N$ in terms of the correlator for the gauge group $SU(2)$. The perturbation expansions of these integrated correlators for any finite value of $N$ agree with properties obtained from perturbative YangMills quantum field theory, together with various multiinstanton calculations which are also shown to agree with those determined by supersymmetric localisation. The coefficients of terms in the large$N$ expansion are sums of nonholomorphic Eisenstein series with halfinteger indices, which extend recent results and make contact with low order terms in the low energy expansion of type IIB superstring theory in an $AdS_5\times S^5/\mathbb{Z}_2$ background.