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Higher Genus Modular Graph Functions, String Invariants, and their Exact Asymptotics

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Abstract

The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of such functions. A general method is developed for analyzing the behavior of modular graph functions under non-separating degenerations in terms of a natural real parameter t. For arbitrary genus, the Arakelov–Green function and the Kawazumi–Zhang invariant degenerate to a Laurent polynomial in t of degree (1, 1) in the limit \({t\to\infty}\) . For genus two, each coefficient of the low energy expansion of the string amplitude degenerates to a Laurent polynomial of degree (w, w) in t, where w + 2 is the degree of homogeneity in the kinematic invariants. These results are exact to all orders in t, up to exponentially suppressed corrections. The non-separating degeneration of a general class of modular graph functions at arbitrary genus is sketched and similarly results in a Laurent polynomial in t of bounded degree. The coefficients in the Laurent polynomial are generalized modular graph functions for a punctured Riemann surface of lower genus.

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Authors and Affiliations

  1. Department of Physics and Astronomy, Mani L. Bhaumik Institute for Theoretical Physics, University of California, Los Angeles, CA, 90095, USA

    Eric D’Hoker

  2. Department of Applied Mathematics and Theoretical Physics, Cambridge University, Wilberforce Road, Cambridge, CB3 0WA, UK

    Michael B. Green

  3. Centre for Research in String Theory, School of Physics, Queen Mary University of London, Mile End Road, London, E1 4NS, England, UK

    Michael B. Green

  4. Laboratoire de Physique Théorique et Hautes Energies, CNRS and Sorbonne Université UMR7589, Campus Pierre et Marie Curie, 4 place Jussieu, 75252, Paris, France

    Boris Pioline

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  1. Eric D’Hoker
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  2. Michael B. Green
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  3. Boris Pioline
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Correspondence to Boris Pioline.

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Communicated by N. Nekrasov

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D’Hoker, E., Green, M.B. & Pioline, B. Higher Genus Modular Graph Functions, String Invariants, and their Exact Asymptotics. Commun. Math. Phys. 366, 927–979 (2019). https://doi.org/10.1007/s00220-018-3244-3

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