亏格为1的弦论三圈梯形图的Poisson方程
Poisson equation for the three loop ladder diagram in string theory at genus one
摘要：
三环梯形图是一个有六个链环和四个立方顶点的图，它对II型弦论亏格1处的D^{12} R^4振幅有贡献。顶点表示顶点算子在环形世界表上的插入点，链接表示连接它们的标量格林函数。利用格林函数的性质，通过对各种表达式的处理，得到了该图所满足的模不变Poisson方程，其源项包含一个、两个和三个循环图.不同于泊松方程中的源项，对于动量展开或梅赛德斯图中的低阶图，一个特定的源项涉及一个五点函数，该函数包含一个全纯和一个反全纯世界面导数，作用于不同的格林函数。我们也得到简单的拓扑不同的图之间的等式，并考虑一些基本的例子。
英文摘要：
The three loop ladder diagram is a graph with six links and four cubic vertices that contributes to the D^{12} R^4 amplitude at genus one in type II string theory. The vertices represent the insertion points of vertex operators on the toroidal worldsheet and the links represent scalar Green functions connecting them. By using the properties of the Green function and manipulating the various expressions, we obtain a modular invariant Poisson equation satisfied by this diagram, with source terms involving one, two and three loop diagrams. Unlike the source terms in the Poisson equations for diagrams at lower orders in the momentum expansion or the Mercedes diagram, a particular source term involves a five point function containing a holomorphic and a antiholomorphic worldsheet derivative acting on different Green functions. We also obtain simple equalities between topologically distinct diagrams, and consider some elementary examples.
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