伊辛模型：从椭圆曲线到模形式和卡-丘方程
The Ising model: from elliptic curves to modular forms and Calabi-Yau equations
摘要：
我们发现，几乎所有的线性微分算子因子中得到的n粒子贡献的伊辛模型的$\, n \le 6$的磁化率的分析，是运营商“与椭圆曲线”。除了与完全椭圆积分E相关的二阶算子的对称幂同态的最简单因子之外，二阶和三阶微分算子实际上可以被解释为伊辛模型的椭圆曲线的模形式。最后一个四阶整体幂零算子是不能约化到这个椭圆曲线，模形式计划。它实际上对应于这个椭圆曲线的自然推广，模形式方案，随着Calabi-Yau方程的出现，对应于选定的$_4F_3$超几何函数，其也可以被看作是完整椭圆积分K的Hadamard乘积，具有非常简单的代数拉回，相应的Calabi-Yau四阶微分算子具有辛微分Galois群SP（4，C）.相关的镜像映射和高阶Schwarzian常微分方程具有重整化群的生成元的精确（同源）表示，将模群SL（2，Z）扩展到GL（2，Z）对称群。
英文摘要：
We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contribution of the susceptibility of the Ising model for $\, n \le 6$, are operators "associated with elliptic curves". Beyond the simplest factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral E, the second and third order differential operators can actually be interpreted as modular forms of the elliptic curve of the Ising model. A last order-four globally nilpotent operator is not reducible to this elliptic curve, modular forms scheme. It is shown to actually correspond to a natural generalization of this elliptic curve, modular forms scheme, with the emergence of a Calabi-Yau equation, corresponding to a selected $_4F_3$ hypergeometric function which can also be seen as a Hadamard product of the complete elliptic integral K, with a remarkably simple algebraic pull-back, the corresponding Calabi-Yau fourth-order differential operator having a symplectic differential Galois group SP(4,C). The associated mirror maps and higher order Schwarzian ODEs has an exact (isogenies) representation of the generators of the renormalization group, extending the modular group SL(2,Z) to a GL(2, Z) symmetry group.
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