Pub Date : 2024-01-24DOI: 10.4310/cntp.2023.v17.n4.a1
Alexander Hock
The functional relation coming from the $x-y$ symplectic transformation of Topological Recursion has a lot of applications; for instance it is the higher order moment-cumulant relation in free probability or can be used to compute intersection numbers on the moduli space of complex curves. We derive the Laplace transform of this functional relation, which has a very nice and compact form as a formal power series in $hbar$. We apply the Laplace transformed formula to the Airy curve and the Lambert curve which provides simple formulas for $psi$-class intersections numbers and Hodge integrals on $overline{mathcal{M}}_{g,n}$.
{"title":"Laplace transform of the $x-y$ symplectic transformation formula in Topological Recursion","authors":"Alexander Hock","doi":"10.4310/cntp.2023.v17.n4.a1","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n4.a1","url":null,"abstract":"The functional relation coming from the $x-y$ symplectic transformation of Topological Recursion has a lot of applications; for instance it is the higher order moment-cumulant relation in free probability or can be used to compute intersection numbers on the moduli space of complex curves. We derive the Laplace transform of this functional relation, which has a very nice and compact form as a formal power series in $hbar$. We apply the Laplace transformed formula to the Airy curve and the Lambert curve which provides simple formulas for $psi$-class intersections numbers and Hodge integrals on $overline{mathcal{M}}_{g,n}$.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"107 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139550677","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}
Pub Date : 2024-01-24DOI: 10.4310/cntp.2023.v17.n4.a2
Miroslav Rapčák, Yan Soibelman, Yaping Yang, Gufang Zhao
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.
{"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":"33 1","pages":""},"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}
Pub Date : 2024-01-24DOI: 10.4310/cntp.2023.v17.n4.a3
Aradhita Chattopadhyaya, Jan Manschot
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.
{"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":"27 1","pages":""},"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}
Pub Date : 2023-11-07DOI: 10.4310/cntp.2023.v17.n3.a3
G. Carlet, J. van de Leur, H. Posthuma, S. Shadrin
We consider the Hurwitz Dubrovin–Frobenius manifold structure on the space of meromorphic functions on the Riemann sphere with exactly two poles, one simple and one of arbitrary order. We prove that the all genera partition function (also known as the total descendant potential) associated with this Dubrovin–Frobenius manifold is a tau function of a rational reduction of the Kadomtsev–Petviashvili hierarchy. This statement was conjectured by Liu, Zhang, and Zhou. We also provide a partial enumerative meaning for this partition function associating one particular set of times with enumeration of rooted hypermaps.
{"title":"Enumeration of hypermaps and Hirota equations for extended rationally constrained KP","authors":"G. Carlet, J. van de Leur, H. Posthuma, S. Shadrin","doi":"10.4310/cntp.2023.v17.n3.a3","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n3.a3","url":null,"abstract":"We consider the Hurwitz Dubrovin–Frobenius manifold structure on the space of meromorphic functions on the Riemann sphere with exactly two poles, one simple and one of arbitrary order. We prove that the all genera partition function (also known as the total descendant potential) associated with this Dubrovin–Frobenius manifold is a tau function of a rational reduction of the Kadomtsev–Petviashvili hierarchy. This statement was conjectured by Liu, Zhang, and Zhou. We also provide a partial enumerative meaning for this partition function associating one particular set of times with enumeration of rooted hypermaps.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"56 10","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71517146","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}
Pub Date : 2023-11-07DOI: 10.4310/cntp.2023.v17.n3.a1
Kaiwen Sun, Haowu Wang
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. $t leq 11$).
{"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":"51 8","pages":""},"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}
Pub Date : 2023-11-07DOI: 10.4310/cntp.2023.v17.n3.a2
Ksenia Fedosova, Kim Klinger-Logan
In this article, we find the full Fourier expansion for solutions of $(Delta-lambda)f(z) = -E_k (z) E_ell (z)$ for $z = x + i y in mathfrak{H}$ for certain values of parameters $k$, $ell$ and $lambda$. When such an $f$ is fully automorphic these functions are referred to as generalized non-holomorphic Eisenstein series. We give a connection of the boundary condition on such Fourier series with convolution formulas on the divisor functions. Additionally, we discuss a possible relation with the differential Galois theory.
{"title":"Whittaker Fourier type solutions to differential equations arising from string theory","authors":"Ksenia Fedosova, Kim Klinger-Logan","doi":"10.4310/cntp.2023.v17.n3.a2","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n3.a2","url":null,"abstract":"In this article, we find the full Fourier expansion for solutions of $(Delta-lambda)f(z) = -E_k (z) E_ell (z)$ for $z = x + i y in mathfrak{H}$ for certain values of parameters $k$, $ell$ and $lambda$. When such an $f$ is fully automorphic these functions are referred to as generalized non-holomorphic Eisenstein series. We give a connection of the boundary condition on such Fourier series with convolution formulas on the divisor functions. Additionally, we discuss a possible relation with the differential Galois theory.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"51 9","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71516743","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}
Pub Date : 2023-11-07DOI: 10.4310/cntp.2023.v17.n3.a4
Claudia Rella
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.
{"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":"56 11","pages":""},"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}
Pub Date : 2022-06-15DOI: 10.4310/cntp.2023.v17.n2.a4
Simone Hu, K. Yeats
The $c_2$-invariant is an arithmetic graph invariant useful for understanding Feynman periods. Brown and Schnetz conjectured that the $c_2$-invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of the $c_2$-invariant in the $p=2$ case, extending previous work of one of us. The methods are combinatorial and enumerative involving counting certain partitions of the edges of the graph.
{"title":"Completing the $c_2$ completion conjecture for $p=2$","authors":"Simone Hu, K. Yeats","doi":"10.4310/cntp.2023.v17.n2.a4","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n2.a4","url":null,"abstract":"The $c_2$-invariant is an arithmetic graph invariant useful for understanding Feynman periods. Brown and Schnetz conjectured that the $c_2$-invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of the $c_2$-invariant in the $p=2$ case, extending previous work of one of us. The methods are combinatorial and enumerative involving counting certain partitions of the edges of the graph.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43597831","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}
Pub Date : 2022-05-28DOI: 10.4310/cntp.2023.v17.n2.a2
B. Hassett, Y. Tschinkel
We study arithmetic properties of derived equivalent K3 surfaces over the field of Laurent power series, using the equivariant geometry of K3 surfaces with cyclic groups actions.
{"title":"Equivariant derived equivalence and rational points on K3 surfaces","authors":"B. Hassett, Y. Tschinkel","doi":"10.4310/cntp.2023.v17.n2.a2","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n2.a2","url":null,"abstract":"We study arithmetic properties of derived equivalent K3 surfaces over the field of Laurent power series, using the equivariant geometry of K3 surfaces with cyclic groups actions.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43033009","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}
Pub Date : 2022-04-27DOI: 10.4310/cntp.2022.v16.n2.a4
Bogdan A. Dobrescu, Patrick J. Fox
We solve Diophantine equations of the type $a(x^3+y^3+z^3)=(x+y+z)^3$, where $x$, $y$, $z$ are integer variables, and the coefficient $a neq 0$ is rational. We show that there are infinite families of such equations, including those where $a$ is any cube or certain rational fractions, that have nontrivial solutions. There are also infinite families of equations that do not have any nontrivial solution, including those where $1/a=1-24/m$ with restrictions on the integer $m$. The equations can be represented by elliptic curves unless $a=9$ or $1$, and any elliptic curve of nonzero $j$-invariant and torsion group $mathbb{Z}/3kmathbb{Z}$ for $k=2,3,4$, or $mathbb{Z}/2mathbb{Z} times mathbb{Z}/6mathbb{Z}$ corresponds to a particular $a$. We prove that for any $a$ the number of nontrivial solutions is at most $3$ or is infinite, and for integer $a$ it is either $0$ or $infty$. For $a=9$, we find the general solution, which depends on two integer parameters. These cubic equations are important in particle physics, because they determine the fermion charges under the $U(1)$ gauge group.
我们求解类型为$a(x^3+y^3+z^3)=(x+y+z)^3$的丢芬图方程，其中$x$, $y$, $z$为整数变量，系数$a neq 0$为有理数。我们证明有无限的这样的方程族，包括$a$是任意立方体或某些有理数的方程族，它们具有非平凡解。也有无限的方程族没有任何非平凡解，包括那些$1/a=1-24/m$对整数$m$有限制的方程族。方程可以用椭圆曲线表示，除非$a=9$或$1$，对于$k=2,3,4$或$mathbb{Z}/2mathbb{Z} times mathbb{Z}/6mathbb{Z}$，任何非零的$j$ -不变量和扭转群$mathbb{Z}/3kmathbb{Z}$的椭圆曲线对应于一个特定的$a$。证明了对于任意$a$，非平凡解的个数不超过$3$或无穷大，对于整数$a$，非平凡解的个数不超过$0$或$infty$。对于$a=9$，我们找到了通解，它依赖于两个整数参数。这些三次方程在粒子物理学中很重要，因为它们决定了$U(1)$规范群下的费米子电荷。
{"title":"Diophantine equations with sum of cubes and cube of sum","authors":"Bogdan A. Dobrescu, Patrick J. Fox","doi":"10.4310/cntp.2022.v16.n2.a4","DOIUrl":"https://doi.org/10.4310/cntp.2022.v16.n2.a4","url":null,"abstract":"We solve Diophantine equations of the type $a(x^3+y^3+z^3)=(x+y+z)^3$, where $x$, $y$, $z$ are integer variables, and the coefficient $a neq 0$ is rational. We show that there are infinite families of such equations, including those where $a$ is any cube or certain rational fractions, that have nontrivial solutions. There are also infinite families of equations that do not have any nontrivial solution, including those where $1/a=1-24/m$ with restrictions on the integer $m$. The equations can be represented by elliptic curves unless $a=9$ or $1$, and any elliptic curve of nonzero $j$-invariant and torsion group $mathbb{Z}/3kmathbb{Z}$ for $k=2,3,4$, or $mathbb{Z}/2mathbb{Z} times mathbb{Z}/6mathbb{Z}$ corresponds to a particular $a$. We prove that for any $a$ the number of nontrivial solutions is at most $3$ or is infinite, and for integer $a$ it is either $0$ or $infty$. For $a=9$, we find the general solution, which depends on two integer parameters. These cubic equations are important in particle physics, because they determine the fermion charges under the $U(1)$ gauge group.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"30 3","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520810","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：