Abstract
Modular graph functions are SL(2, ℤ)-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus τ. For one-loop graphs they reduce to real analytic Eisenstein series. We obtain the Fourier series, including the constant and non-constant Fourier modes, of all two-loop modular graph functions, as well as their Poincaré series with respect to Γ∞\PSL(2, ℤ). The Fourier and Poincaré series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Rankin-Selberg-Zagier methods. Modular graph functions which are odd under τ → −\( \overline{\tau} \) are cuspidal functions, with exponential decay near the cusp, and exist starting at two loops. Holomorphic subgraph reduction and the sieve algorithm, developed in earlier work, are used to give a lower bound on the dimension of the space \( \mathfrak{A} \)w of odd two-loop modular graph functions of weight w. For w ≤ 11 the bound is saturated and we exhibit a basis for \( \mathfrak{A} \)w.
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ArXiv ePrint: 1902.04180
This research is supported in part by the National Science Foundation under grant PHY-16-19926.
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D’Hoker, E., Kaidi, J. Modular graph functions and odd cuspidal functions. Fourier and Poincaré series. J. High Energ. Phys. 2019, 136 (2019). https://doi.org/10.1007/JHEP04(2019)136
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DOI: https://doi.org/10.1007/JHEP04(2019)136