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Modular and Holomorphic Graph Functions from Superstring Amplitudes

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Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

We compare two classes of functions arising from genus-one superstring amplitudes: modular and holomorphic graph functions. We focus on their analytic properties, we recall the known asymptotic behaviour of modular graph functions and we refine the formula for the asymptotic behaviour of holomorphic graph functions. Moreover, we give new evidence of a conjecture appeared in [4] which relates these two asymptotic expansions.

IPHT-t18/089.

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Notes

  1. 1.

    The reason for the rational coefficient appearing in Eq. (1) is that we want to follow the notation adopted in the modular graph function literature. Setting \(y=-2\pi \, Im(\tau )\), which mathematically would be a more natural choice, one would get a cleaner statement.

  2. 2.

    After tensoring with \(\mathbb {C}\), more generally considering “relative cohomologies”.

  3. 3.

    More precisely, it should be super-Riemann surfaces, but this does not matter here.

  4. 4.

    It is well known that any punctured genus-one Riemann surface can be realized as a complex torus.

  5. 5.

    Note that we have changed the sign of the propagator w.r.t. [6], following [4].

  6. 6.

    Recent indications suggest that considering only the action of the Laplace operator we lose some information, and it is instead better to consider the action of the Cauchy-Riemann derivative \(\nabla _\tau =2i(Im(\tau ))^2\partial _\tau \) and of its complex conjugate \(\overline{\nabla }_\tau \) [22, 25].

  7. 7.

    It is however believed that this should not always be true.

  8. 8.

    There is a typo in the coefficient of \(y^{-4}\) in the corresponding formula in [42].

  9. 9.

    This identity was first proven by Zagier by a complicated direct computation (private communication).

  10. 10.

    This is only true for graphs with at most four vertices, but there is an obvious n-point version of the integral (27) whose coefficients, i.e. all possible graph functions, must be combinations of A-elliptic MZVs.

  11. 11.

    Here we deviate from [7] and we prefer to exclude the quasi-modular form \(G_2(\tau )\).

  12. 12.

    Since \(\mathbb {H}\) is simply connected, we can choose arbitrary paths from \(\tau ^\prime \) to \(i\infty \) and from 0 to \(\tau ^\prime \).

  13. 13.

    While for A-elliptic MZVs we know that inverting \(2\pi i\) is necessary, we suspect that no inverse powers of \(2\pi i\) should appear in the asymptotic expansion of A-cycle graph functions.

  14. 14.

    This case could only be checked numerically, for about five-hundred digits.

  15. 15.

    In this case, one simply takes the real part and lands on non-holomorphic Eisenstein series [16].

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Acknowledgements

We would like to thank KMPB for the organization of this successful conference. Moreover, we would like to thank C. Dupont, O. Schlotterer and M. Tapus̆ković for useful comments on a first draft and J. Brödel and E. Garcia–Failde for their help with the figures. Our research was supported by a French public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n. PCOFUND-GA-2013-609102, through the PRESTIGE programme coordinated by Campus France.

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  1. Institute de Physique Théorique (IPHT), Orme des Merisiers batiment 774, Point courrier 136, 91191, Gif-sur-Yvette Cedex, France

    Federico Zerbini

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  1. Deutsches Elektronen-Synchrotron, DESY, Zeuthen, Germany

    Johannes Blümlein

  2. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Carsten Schneider

  3. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Peter Paule

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Zerbini, F. (2019). Modular and Holomorphic Graph Functions from Superstring Amplitudes. In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-04480-0_18

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