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.
Categories: math-ph hep-th math.MP