When the Fourier transform is one loop exact?

Maxim Kontsevich, Alexander Odesskii
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Abstract

We investigate the question: for which functions \(f(x_1,\ldots ,x_n),~g(x_1,\ldots ,x_n)\) the asymptotic expansion of the integral \(\int g(x_1,\ldots ,x_n) e^{\frac{f(x_1,\ldots ,x_n)+x_1y_1+\dots +x_ny_n}{\hbar }}dx_1\ldots dx_n\) 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 \(\{(1:x_1:\ldots :x_n:f)\}\). 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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当傅立叶变换为单圈精确变换时?
我们研究的问题是对于哪些函数(f(x_1,\ldots ,x_n),~g(x_1,\ldots ,x_n)),积分(int g(x_1、\e^{frac{f(x_1,\ldots ,x_n)+x_1y_1+\dots +x_ny_n}\{hbar }}dx_1\ldots dx_n\) 只包含第一项。我们揭示了问题的一个隐藏的投影不变性,它建立了问题与投影超曲面几何的关系,其形式为\(\{(1:x_1:\ldots :x_n:f)\}\).我们还构造了各种例子,特别是我们证明了库默曲面在 \({\mathbb {P}}^3\) 中给出了问题的解。
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