Introduction to integral 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) = ∫e-S(x 1,x n) dx 1dx 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. © 2009 SISSA.