When the Fourier transform is one loop exact?

Maxim Kontsevich; Alexander Odesskii

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arXiv · 2306.02178

When the Fourier transform is one loop exact?

Maxim Kontsevich·Alexander Odesskii

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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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Maxim Kontsevich, Alexander Odesskii. 2023-06-03. When the Fourier transform is one loop exact?. https://arxiv.org/abs/2306.02178

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