When the Fourier transform is one loop exact?

Author:

Maxim Kontsevich, Alexander Odesskii

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), High Energy Physics - Theory (hep-th), Mathematical Physics (math-ph)

journal:

date:

2023-06-02 16:00:00

Abstract

We investigate the question: for which functions $f(x_1,...,x_n),~g(x_1,...,x_n)$ the asymptotic expansion of the integral $\int g(x_1,...,x_n) e^{\frac{f(x_1,...,x_n)+x_1y_1+...+x_ny_n}{\hbar}}dx_1...dx_n$ 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.

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