Introduction to Integral Discriminants

TL;DR

This paper explores integral discriminants associated with symmetric forms, demonstrating their calculation in non-Gaussian cases using Ward identities, revealing their expression as generalized hypergeometric functions dependent on SL(n) invariants.

Contribution

It introduces a method to compute integral discriminants for non-Gaussian forms using Ward identities, expanding understanding beyond Gaussian cases.

Findings

Integral discriminants are expressed as generalized hypergeometric functions.

Explicit calculations for J_{2|3}, J_{2|4}, J_{2|5}, J_{3|3} are provided.

Discriminants depend on SL(n) invariants with singularities linked to algebraic discriminants.

Abstract

The simplest partition function, associated with homogeneous symmetric forms S of degree r in n variables, is integral discriminant J_{n|r}(S) = \int e^{-S(x_1 ... x_n)} dx_1 ... dx_n. Actually, S-dependence remains the same if e^{-S} in the integrand is substituted by arbitrary function f(S), i.e. integral discriminant is a characteristic of the form S itself, and not of the averaging procedure. The aim of the present paper is to calculate J_{n|r} in a number of non-Gaussian cases. Using Ward identities -- linear differential equations, satisfied by integral discriminants -- we calculate J_{2|3}, J_{2|4}, J_{2|5} and J_{3|3}. In all these examples, integral discriminant appears to be a generalized hypergeometric function. It depends on several SL(n) invariants of S, with essential singularities controlled by the ordinary algebraic discriminant of S.