Journal of High Energy Physics

, 2016:51 | Cite as

Hierarchy of modular graph identities

  • Eric D’HokerEmail author
  • Justin Kaidi
Open Access
Regular Article - Theoretical Physics


The low energy expansion of Type II superstring amplitudes at genus one is organized in terms of modular graph functions associated with Feynman graphs of a conformal scalar field on the torus. In earlier work, surprising identities between two-loop graphs at all weights, and between higher-loop graphs of weights four and five were constructed. In the present paper, these results are generalized in two complementary directions. First, all identities at weight six and all dihedral identities at weight seven are obtained and proven. Whenever the Laurent polynomial at the cusp is available, the form of these identities confirms the pattern by which the vanishing of the Laurent polynomial governs the full modular identity. Second, the family of modular graph functions is extended to include all graphs with derivative couplings and worldsheet fermions. These extended families of modular graph functions are shown to obey a hierarchy of inhomogeneous Laplace eigenvalue equations. The eigenvalues are calculated analytically for the simplest infinite sub-families and obtained by Maple for successively more complicated sub-families. The spectrum is shown to consist solely of eigenvalues s(s − 1) for positive integers s bounded by the weight, with multiplicities which exhibit rich representation-theoretic patterns.


Discrete Symmetries Integrable Hierarchies Superstrings and Heterotic Strings 


Open Access

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Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and AstronomyUniversity of CaliforniaLos AngelesU.S.A.

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