# Eigenvalue equation for the modular graph $C_{a,b,c,d}$

## TL;DR

This paper derives an eigenvalue equation for the modular graph $C_{a,b,c,d}$ on the torus, involving auxiliary graphs and Eisenstein series, advancing understanding of modular graph functions in string theory.

## Contribution

It introduces a novel eigenvalue equation for the modular graph $C_{a,b,c,d}$, incorporating auxiliary graphs and Eisenstein series, obtained via variation with respect to the Beltrami differential.

## Key findings

- Eigenvalue equation for $C_{a,b,c,d}$ derived
- Involves sums of squares of Eisenstein series
- Eigenfunction includes specific subtractions

## Abstract

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 $d$.