Correlators in the theory of Integral Discriminants

TL;DR

This work advances the study of integral discriminants by focusing on invariant correlators under $SL(n)$ and introducing a differential-operator framework to compute them in terms of fundamental polynomial invariants. Across the cases $(n|r)=(2|3),(2|4),(3|3)$, the authors demonstrate striking polynomiality: certain correlators become polynomials in invariants such as $I_4$, $I_2$, or $I_6$, revealing a superintegrability-like structure. The approach combines operator action on the partition function with invariant-based recursions, yielding explicit results for correlators like $\langle Q^m\rangle$ and highlighting deep connections to hypergeometric (GKZ) functions of the invariants. These findings suggest a path toward simplifying the non-Gaussian integral theory and motivate exploration of larger $(n,r)$ and GKZ frameworks in integral discriminants.

Abstract

Integral discriminants provide a simple and fundamental model for non-Gaussian integrals, associated with homogeneous polynomials of degree r in n variables. We argue that, in this context, the study of correlators is equally if not more important. In this paper, we study a natural class of correlators in this model -- the invariant correlators. We suggest a general method to compute invariant correlators using differential operators that act on the partition function. This method allows to compute general invariant correlators in terms of the fundamental invariants. Moreover, in some cases the correlators appear to be simply polynomials in the invariants. This could be an interesting manifestation of superintegrability phenomenon in the theory of integral discriminants.