All-order differential equations for one-loop closed-string integrals and modular graph forms
All-order differential equations for one-loop closed-string integrals and modular graph forms
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一环闭弦积分和模图形式的全阶微分方程
DOI：
10.1007/jhep01(2020)064
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发表时间：
2020
影响因子：
5.4
通讯作者：
O. Schlotterer
中科院分区：
文献类型：
--
作者：
Jan E. Gerken;A. Kleinschmidt;O. Schlotterer
我们研究了玻色子、异质和 II 类理论的闭弦单环振幅中出现的世界片环面积分的生成函数。这些闭弦积分在环面模参数中服从齐次线性微分方程。我们阐明了任意数量外部状态的生成函数的一阶柯西-黎曼方程和二阶拉普拉斯方程。这种环面积分的低能展开引入了无限族的非全纯模形式，称为模图形式。我们的结果为任意此类模图形式生成齐次一阶和二阶微分方程，并且可以被视为迈向闭弦积分的全阶低能展开的一步。
We investigate generating functions for the integrals over world-sheet tori appearing in closed-string one-loop amplitudes of bosonic, heterotic and type-II theories. These closed-string integrals are shown to obey homogeneous and linear differential equations in the modular parameter of the torus. We spell out the first-order Cauchy-Riemann and second-order Laplace equations for the generating functions for any number of external states. The low-energy expansion of such torus integrals introduces infinite families of non-holomorphic modular forms known as modular graph forms. Our results generate homogeneous first- and second-order differential equations for arbitrary such modular graph forms and can be viewed as a step towards all-order low-energy expansions of closed-string integrals.