When the Fourier transform is one loop exact?

Abstract

We investigate the question: for which functions $f(x_1,...,x_n),g(x_1,...,x_n)$ the asymptotic expansion of the integral $\int g(x_1,...,x_n)e^{\frac{f(x_1,...,x_n)+x_1{y}_1+...+x_n{y}_n}{\hslash }}d{x}_1...d{x}_n$ consists only of the first term. We reveal a hidden projective invariance of the problem which establishes its relation with geometry of projective hypersurfaces of the form $\{(1:x_1:...:x_n:f)\}$. We also construct various examples, in particular we prove that Kummer surface in ${\mathbb{P}}^3$ gives a solution to our problem.

53 pages