Nonperturbative approach to finite-dimensional non-Gaussian integrals

We study the homogeneous non-Gaussian integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ J_{n|r} (S) = \int {e^{ - S(x_1 , \ldots ,x_n )} } d^n x $$\end{document}, where S(x1,…,xn) is a symmetric form of degree r in n variables. This integral is naturally invariant under SL(n) transformations and therefore depends only on the invariants of the form. For example, in the case of quadratic forms, it is equal to the (−1/2)th power of the determinant of the form. For higher-degree forms, the integral can be calculated in some cases using the so-called Ward identities, which are second-order linear differential equations. We describe the method for calculating the integral and present detailed calculations in the case where n = 2 and r = 5. It is interesting that the answer is a hypergeometric function of the invariants of the form.