Communications in Number Theory and Physics最新文献_第2页

Laplace transform of the $x-y$ symplectic transformation formula in Topological Recursion 拓扑递推中 x-y$ 交映变换公式的拉普拉斯变换

IF 1.9 3区数学

Communications in Number Theory and Physics Pub Date : 2024-01-24 DOI: 10.4310/cntp.2023.v17.n4.a1

Alexander Hock

引用次数: 0

Cohomological Hall algebras and perverse coherent sheaves on toric Calabi–Yau $3$-folds 环卡拉比约 3$ 折叠上的同调霍尔代数和反相干剪切

IF 1.9 3区数学

Communications in Number Theory and Physics Pub Date : 2024-01-24 DOI: 10.4310/cntp.2023.v17.n4.a2

Miroslav Rapčák, Yan Soibelman, Yaping Yang, Gufang Zhao

{"title":"Cohomological Hall algebras and perverse coherent sheaves on toric Calabi–Yau $3$-folds","authors":"Miroslav Rapčák, Yan Soibelman, Yaping Yang, Gufang Zhao","doi":"10.4310/cntp.2023.v17.n4.a2","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n4.a2","url":null,"abstract":"We study the Drinfeld double of the (equivariant spherical) Cohomological Hall algebra in the sense of Kontsevich and Soibelman, associated to a smooth toric Calabi–Yau $3$-fold $X$. By general reasons, the COHA acts on the cohomology of the moduli spaces of certain perverse coherent systems on $X$ via “raising operators”. Conjecturally the COHA action extends to an action of the Drinfeld double by adding the “lowering operators”. In this paper, we show that the Drinfeld double is a generalization of the notion of the Cartan doubled Yangian defined earlier by Finkelberg and others. We extend this “$3d$ Calabi–Yau perspective” on the Lie theory furthermore by associating a root system to certain families of $X$. We formulate a conjecture that the above-mentioned action of the Drinfeld double factors through a shifted Yangian of the root system. The shift is explicitly determined by the moduli problem and the choice of stability conditions, and is expressed explicitly in terms of an intersection number in $X$. We check the conjectures in several examples, including a special case of an earlier conjecture of Costello.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139550665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Numerical experiments on coefficients of instanton partition functions 瞬子分割函数系数的数值实验

IF 1.9 3区数学

Communications in Number Theory and Physics Pub Date : 2024-01-24 DOI: 10.4310/cntp.2023.v17.n4.a3

Aradhita Chattopadhyaya, Jan Manschot

{"title":"Numerical experiments on coefficients of instanton partition functions","authors":"Aradhita Chattopadhyaya, Jan Manschot","doi":"10.4310/cntp.2023.v17.n4.a3","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n4.a3","url":null,"abstract":"We analyze the coefficients of partition functions of Vafa–Witten (VW) theory on a four-manifold. These partition functions factorize into a product of a function enumerating pointlike instantons and a function enumerating smooth instantons. For gauge groups $SU(2)$ and $SU(3)$ and four-manifold the complex projective plane $mathbb{CP}^2$, we experimentally study the latter functions, which are examples of mock modular forms of depth $1$, weight $3/2$, and depth $2$, weight $3$ respectively. We also introduce the notion of “mock cusp form”, and study an example of weight $3$ related to the $SU(3)$ partition function. Numerical experiments on the first 200 coefficients of these mock modular forms suggest that the coefficients of these functions grow as $O(n^{k-1})$ for the respective weights $k = 3/2$ and $3$. This growth is similar to that of a modular form of weight $k$. On the other hand the coefficients of the mock cusp form of weight $3$ appear to grow as $O(n^{3/2})$, which exceeds the growth of classical cusp forms of weight $3$. We provide bounds using saddle point analysis, which however largely exceed the experimental observation.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139550920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Enumeration of hypermaps and Hirota equations for extended rationally constrained KP 扩展合理约束KP的超映射和Hirota方程的计数

IF 1.9 3区数学

Communications in Number Theory and Physics Pub Date : 2023-11-07 DOI: 10.4310/cntp.2023.v17.n3.a3

G. Carlet, J. van de Leur, H. Posthuma, S. Shadrin

引用次数: 2

Weyl invariant $E_8$ Jacobi forms and $E$-strings Weyl不变量$E_8$Jacobi形式和$E$字符串

IF 1.9 3区数学

Communications in Number Theory and Physics Pub Date : 2023-11-07 DOI: 10.4310/cntp.2023.v17.n3.a1

Kaiwen Sun, Haowu Wang

{"title":"Weyl invariant $E_8$ Jacobi forms and $E$-strings","authors":"Kaiwen Sun, Haowu Wang","doi":"10.4310/cntp.2023.v17.n3.a1","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n3.a1","url":null,"abstract":"In 1992 Wirthmüller showed that for any irreducible root system not of type $E_8$ the ring of weak Jacobi forms invariant under Weyl group is a polynomial algebra. However, it has recently been proved that for $E_8$ the ring is not a polynomial algebra. Weyl invariant $E_8$ Jacobi forms have many applications in string theory and it is an open problem to describe such forms. The scaled refined free energies of $E$-strings with certain $eta$-function factors are conjectured to be Weyl invariant $E_8$ quasi-holomorphic Jacobi forms. It is further observed that the scaled refined free energies up to some powers of $E_4$ can be written as polynomials in nine Sakai’s $E_8$ Jacobi forms and Eisenstein series $E_2, E_4, E_6$. Motivated by the physical conjectures, we prove that for any Weyl invariant $E_8$ Jacobi form $phi_t$ of index $t$ the function $E^{[t/5]}_4 Delta^{[5t/6]} phi_t$ can be expressed uniquely as a polynomial in $E_4$, $E_6$ and Sakai’s forms, where $[x]$ is the integer part of $x$. This means that a Weyl invariant $E_8$ Jacobi form is completely determined by a solution of some linear equations. By solving the linear systems, we determine the generators of the free module of Weyl invariant $E_8$ weak (resp. holomorphic) Jacobi forms of given index $t$ when $t leq 13$ (resp. $t leq 11$).","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71516744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Whittaker Fourier type solutions to differential equations arising from string theory 弦理论微分方程的Whittaker-Fourier型解

IF 1.9 3区数学

Communications in Number Theory and Physics Pub Date : 2023-11-07 DOI: 10.4310/cntp.2023.v17.n3.a2

Ksenia Fedosova, Kim Klinger-Logan

引用次数: 6

Resurgence, Stokes constants, and arithmetic functions in topological string theory 拓扑弦理论中的复活、Stokes常数和算术函数

IF 1.9 3区数学

Communications in Number Theory and Physics Pub Date : 2023-11-07 DOI: 10.4310/cntp.2023.v17.n3.a4

Claudia Rella

{"title":"Resurgence, Stokes constants, and arithmetic functions in topological string theory","authors":"Claudia Rella","doi":"10.4310/cntp.2023.v17.n3.a4","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n3.a4","url":null,"abstract":"The quantization of the mirror curve to a toric Calabi–Yau threefold gives rise to quantum-mechanical operators, whose fermionic spectral traces produce factorially divergent power series in the Planck constant. These asymptotic expansions can be promoted to resurgent trans-series. They show infinite towers of periodic singularities in their Borel plane and infinitely many rational Stokes constants, which are encoded in generating functions expressed in closed form in terms of $q$-series. We provide an exact solution to the resurgent structure of the first fermionic spectral trace of the local $mathbb{P}^2$ geometry in the semiclassical limit of the spectral theory, corresponding to the strongly-coupled regime of topological string theory on the same background in the conjectural TS/ST correspondence. Our approach straightforwardly applies to the dual weakly-coupled limit of the topological string. We present and prove closed formulae for the Stokes constants as explicit arithmetic functions and for the perturbative coefficients as special values of known $L$-functions, while the duality between the two scaling regimes of strong and weak string coupling constant appears in number-theoretic form. A preliminary numerical investigation of the local $mathbb{F}_0$ geometry unveils a more complicated resurgent structure with logarithmic sub-leading asymptotics. Finally, we obtain a new analytic prediction on the asymptotic behavior of the fermionic spectral traces in an appropriate WKB double-scaling regime, which is captured by the refined topological string in the Nekrasov–Shatashvili limit.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71517145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

Completing the $c_2$ completion conjecture for $p=2$ 完成$p=2的$c_2$完成猜想$

IF 1.9 3区数学

Communications in Number Theory and Physics Pub Date : 2022-06-15 DOI: 10.4310/cntp.2023.v17.n2.a4

Simone Hu, K. Yeats

引用次数: 2

Equivariant derived equivalence and rational points on K3 surfaces K3曲面上等变导数等价与有理点

IF 1.9 3区数学

Communications in Number Theory and Physics Pub Date : 2022-05-28 DOI: 10.4310/cntp.2023.v17.n2.a2

B. Hassett, Y. Tschinkel

引用次数: 1

Diophantine equations with sum of cubes and cube of sum 丢番图方程的和立方和立方的和

IF 1.9 3区数学

Communications in Number Theory and Physics Pub Date : 2022-04-27 DOI: 10.4310/cntp.2022.v16.n2.a4

Bogdan A. Dobrescu, Patrick J. Fox

引用次数: 0