A single theta-lift construction generates both two-loop modular graph functions and S-dual modular functions as rational combinations of generalised Eisenstein series with all cusp-form L-values cancelling.
Modular local polynomials
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abstract
In this paper, we consider modular local polynomials. These functions satisfy modularity while they are locally defined as polynomials outside of an exceptional set. We prove an inequality for the dimension of the space of such forms when the exceptional set is given by certain natural geodesics related to binary quadratic forms of (positive) discriminant $D$. We furthermore show that the dimension is the largest possible if and only if $D$ is an even square. Following this, we describe how to use the methods developped in this paper to establish an algorithm which explicitly determines the space of modular local polynomials for each $D$.
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Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts
A single theta-lift construction generates both two-loop modular graph functions and S-dual modular functions as rational combinations of generalised Eisenstein series with all cusp-form L-values cancelling.