模图$C_{a,b,c,d}$的特征值方程
Eigenvalue equation for the modular graph $C_{a,b,c,d}$
摘要：
环面上的模图$C_{a,b,c,d}$是一个三圈平面图，其中两个顶点的坐标数为4，而其他顶点的坐标数为2。我们得到了$C_{a,b,c,d}$对$a,b,c$和$d$的一般值所满足的特征值方程，其中源项涉及各种模图.这是通过改变关于环形世界面上的贝尔特拉米微分的图来获得的。在分析的各个中间阶段使用几个辅助图对于获得方程是至关重要的。事实上，本征函数不是简单的$C_{a,b,c,d}$，而是涉及到从它减去特定的平方和的非全纯爱森斯坦系列的特点是$a,b,c$和$d$。
英文摘要：
The modular graph $C_{a,b,c,d}$ on the torus is a three loop planar graph in which two of the vertices have coordination number four, while the others have coordination number two. We obtain an eigenvalue equation satisfied by $C_{a,b,c,d}$ for generic values of $a,b,c$ and $d$, where the source terms involve various modular graphs. This is obtained by varying the graph with respect to the Beltrami differential on the toroidal worldsheet. Use of several auxiliary graphs at various intermediate stages of the analysis is crucial in obtaining the equation. In fact, the eigenfunction is not simply $C_{a,b,c,d}$ but involves subtracting from it specific sums of squares of non--holomorphic Eisenstein series characterized by $a,b,c$ and $d$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
