Abstract
We obtain a second order differential equation on moduli space satisfied by certain modular graph functions at genus two, each of which has two links. This eigenvalue equation is obtained by analyzing the variations of these graphs under the variation of the Beltrami differentials. This equation involves seven distinct graphs, three of which appear in the integrand of the D8 mathrm{mathcal{R}} 4 term in the low momentum expansion of the four graviton amplitude at genus two in type II string theory.
Highlights
Multiloop string amplitudes provide useful insight into the structure of terms in the effective action of string theory
We obtain a second order differential equation on moduli space satisfied by certain modular graph functions at genus two, each of which has two links
This equation involves seven distinct graphs, three of which appear in the integrand of the D8R4 term in the low momentum expansion of the four graviton amplitude at genus two in type II string theory
Summary
Multiloop string amplitudes provide useful insight into the structure of terms in the effective action of string theory. In order to calculate the analytic terms in the low momentum expansion, it is very useful to obtain eigenvalue equations which the modular invariant integrand satisfies. This helps us to have an understanding of the detailed structure of the integrand, and to calculate the integral over moduli space. We would like to mention that the structure of the eigenvalue equation we obtain seems natural, as they all involve modular graphs having the same set of skeleton graphs with different dressing factors.
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