论文标题
椭圆属和模块化微分方程
Elliptic genus and modular differential equations
论文作者
论文摘要
我们研究了一个Abelian变量中基本弱的雅各比形式的模块化微分方程，并应用于Calabi-Yau品种的椭圆属。我们表明，任何$ cy_3 $的椭圆属都满足了与热运算符相对于一学位的微分方程。对于$ k3 $的表面或任何$ cy_5 $，微分方程的度数为$ 3 $。我们证明，对于一般$ cy_4 $，其椭圆形属满足$ 5 $的模块化微分方程。我们给出了相对于热算子类似于一个变量的模块化形式的kaneko-zagier方程的二级程度差分方程的示例。我们发现Kaneko的模块化微分方程 - Zagier类型的学位$ 2 $或$ 3 $的第二，第三和第四势力的雅各比Theta系列。
We study modular differential equations for the basic weak Jacobi forms in one abelian variable with applications to the elliptic genus of Calabi--Yau varieties. We show that the elliptic genus of any $CY_3$ satisfies a differential equation of degree one with respect to the heat operator. For a $K3$ surface or any $CY_5$ the degree of the differential equation is $3$. We prove that for a general $CY_4$ its elliptic genus satisfies a modular differential equation of degree $5$. We give examples of differential equations of degree two with respect to the heat operator similar to the Kaneko--Zagier equation for modular forms in one variable. We find modular differential equations of Kaneko--Zagier type of degree $2$ or $3$ for the second, third and fourth powers of the Jacobi theta-series.
