Correlators in the theory of Integral Discriminants

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.

💡 Research Summary

The paper investigates a class of invariant correlators within the framework of integral discriminants, which are non‑Gaussian integrals defined by homogeneous polynomials S(x₁,…,xₙ) of degree r. The partition function Zₙ|ᵣ(S)=∫_γ dⁿx e^{−S(x)} generalizes the Gaussian case (r=2) to genuinely non‑perturbative models for r>2, where the Gaussian term is absent. A key property is SL(n) invariance: Zₙ|ᵣ(e S)=Zₙ|ᵣ(S) for any linear transformation e∈SL(n). Consequently, Zₙ|ᵣ(S) can be expressed solely through a finite set of fundamental invariants I₁(S), I₂(S), … of the tensor S.

The authors define correlators ⟨F⟩ = (∫_γ dⁿx F e^{−S})/(∫_γ dⁿx e^{−S}) and focus on “invariant correlators” where the observable F transforms covariantly under SL(n) in the same way as the integration measure, ensuring that ⟨F⟩ depends only on the invariants of S. The simplest such observable is Q(x)=det(∂²S/∂x_i∂x_j), a homogeneous polynomial in the coordinates built from the second‑derivative tensor of S.

The central methodological contribution is the introduction of a differential operator
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