Maximal $U(1)_Y$-violating $n$-point correlators in $\mathcal{N}=4$ super-Yang-Mills theory

TL;DR

This paper develops recursion relations for maximal $U(1)_Y$-violating $n$-point correlators in $ =4$ super-Yang-Mills, linking them to string theory amplitudes and modular forms, advancing understanding of their structure and coupling dependence.

Contribution

It introduces $SL(2,b Z)$-covariant recursion relations for these correlators, connecting them to string theory and modular forms, and explores their large-$N$ expansion and low-energy limits.

Findings

Recursion relations relate $n$-point to $(n-1)$-point correlators.

Correlators' Mellin amplitudes are polynomials in Mellin variables.

Coupling dependence is governed by non-holomorphic modular forms.

Abstract

This paper concerns a special class of $n$-point correlation functions of operators in the stress tensor supermultiplet of $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory. These are "maximal $U(1)_Y$-violating" correlators that violate the bonus $U(1)_Y$ charge by a maximum of $2(n-4)$ units. We will demonstrate that such correlators satisfy $SL(2,\mathbb{Z})$-covariant recursion relations that relate $n$-point correlators to $(n-1)$-point correlators in a manner analogous to the soft dilaton relations that relate the corresponding amplitudes in flat-space type IIB superstring theory. These recursion relations are used to determine terms in the large-$N$ expansion of $n$-point maximal $U(1)_Y$-violating correlators in the chiral sector, including correlators with four superconformal stress tensor primaries and $(n-4)$ chiral Lagrangian operators, starting from known properties of the $n=4$ case. We concentrate on the first three orders in $1/N$ beyond the supergravity limit. The Mellin representations of the correlators are polynomials in Mellin variables, which correspond to higher derivative contact terms in the low-energy expansion of type IIB superstring theory in $AdS_5 \times S^5$ at the same orders as $R^4, d^4R^4$ and $d^6R^4$. The coupling constant dependence of these terms is found to be described by non-holomorphic modular forms with holomorphic and anti-holomorphic weights $(n-4,4-n)$ that are $SL(2, \mathbb{Z})$-covariant derivatives of Eisenstein series and certain generalisations. This determines a number of non-leading contributions to $U(1)_Y$-violating $n$-particle interactions ($n>4$) in the low-energy expansion of type IIB superstring amplitudes in $AdS_5\times S^5$.