Abstract
The integral of an arbitrary two-loop modular graph function over the fundamental domain for SL(2, ℤ) in the upper half plane is evaluated using recent results on the Poincaré series for these functions.
Highlights
A generalization of the Rankin-Selberg method, which is applicable to modular functions with polynomial growth at the cusp, was introduced in [25] and evaluates integrals over ML of expressions linear, bilinear, and trilinear in Eisenstein series
The integrals over moduli space required for superstring amplitudes are unique and free of divergences, but their construction requires subtle analytic continuation in the kinematic variables of the amplitude
The components of the connectivity matrix Γ are denoted by Γv r where v = 1, · · ·, V labels the vertices and r = 1, · · ·, R labels the edges
Summary
We shall give a brief review of the definition and basic properties of modular graph functions needed in the sequel of the paper. To a connected decorated graph (Γ, A, B) we associate a complex-valued function on H, defined by a multiple KroneckerEisenstein sum over p1, · · · , pR in Λ = Λ \ {0} where Λ = Z + τ Z for τ ∈ H, A. We shall assume that Γ remains connected upon removing any single edge or vertex and that the equality of the sums of A and B exponents in (2.3) holds. It was shown in [2] that the behavior near the cusp of an arbitrary modular graph function of weight w ≥ 2, whose exponents satisfy (2.3), is governed by a Laurent polynomial in τ2 of degree (w, 1 − w) up to exponentially suppressed terms,. We shall provide more explicit formulas in the cases of one-loop and two-loop modular graph functions
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