Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, Sergey Shadrin
{"title":"超几何型卡多姆采夫-彼得维亚什维利陶函数的拓扑递归","authors":"Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, Sergey Shadrin","doi":"10.1112/jlms.12946","DOIUrl":null,"url":null,"abstract":"<p>We study the <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>-point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their <span></span><math>\n <semantics>\n <msup>\n <mi>ℏ</mi>\n <mn>2</mn>\n </msup>\n <annotation>$\\hbar ^2$</annotation>\n </semantics></math>-deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so-called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of <span></span><math>\n <semantics>\n <msup>\n <mi>ℏ</mi>\n <mn>2</mn>\n </msup>\n <annotation>$\\hbar ^2$</annotation>\n </semantics></math>-deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)-type formulas discussed in the literature.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12946","citationCount":"0","resultStr":"{\"title\":\"Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type\",\"authors\":\"Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, Sergey Shadrin\",\"doi\":\"10.1112/jlms.12946\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study the <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math>-point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their <span></span><math>\\n <semantics>\\n <msup>\\n <mi>ℏ</mi>\\n <mn>2</mn>\\n </msup>\\n <annotation>$\\\\hbar ^2$</annotation>\\n </semantics></math>-deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so-called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of <span></span><math>\\n <semantics>\\n <msup>\\n <mi>ℏ</mi>\\n <mn>2</mn>\\n </msup>\\n <annotation>$\\\\hbar ^2$</annotation>\\n </semantics></math>-deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. 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Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type
We study the -point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their -deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so-called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of -deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)-type formulas discussed in the literature.
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