{"title":"When the Fourier transform is one loop exact?","authors":"Maxim Kontsevich, Alexander Odesskii","doi":"10.1007/s00029-024-00920-y","DOIUrl":null,"url":null,"abstract":"<p>We investigate the question: for which functions <span>\\(f(x_1,\\ldots ,x_n),~g(x_1,\\ldots ,x_n)\\)</span> the asymptotic expansion of the integral <span>\\(\\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\\)</span> 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 <span>\\(\\{(1:x_1:\\ldots :x_n:f)\\}\\)</span>. We also construct various examples, in particular we prove that Kummer surface in <span>\\({\\mathbb {P}}^3\\)</span> gives a solution to our problem.\n</p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"67 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Selecta Mathematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00029-024-00920-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.