Nonperturbative approach to finite-dimensional non-Gaussian integrals

We study the homogeneous non-Gaussian integral $J_{n|r}(S)=\int e^{-S(x_1,\dots,x_n)}\,d^nx$, where $S(x_1,\dots,x_n)$ 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.