When the Fourier transform is one loop exact?

2023/06/03 by Maxim Kontsevich, Alexander Odesskii, Kontsevich, Maxim +1
Mathematics · #Meromorphic and Entire Functions #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2306.02178

Abstract

We investigate the question: for which functions f(x₁,...,x_n),~g(x₁,...,x_n) the asymptotic expansion of the integral ∫ g(x₁,...,x_n) e^{(f(x₁,...,x_n)+x₁y₁+...+x_ny_n)/(ℏ)}dx₁...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:x₁:...:x_n:f)\. We also construct various examples, in particular we prove that Kummer surface in ℙ³ gives a solution to our problem.

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