Introduction to integral discriminants

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(x₁,...,x_n)}dx₁...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.