On the modular structure of the genus-one Type II superstring low energy expansion

  • Eric D’Hoker
  • Michael B. Green
  • Pierre Vanhove
Open Access
Regular Article - Theoretical Physics


The analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions up to order \( {D}^{10}{\mathrm{\mathcal{R}}}^4 \) are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values.


Superstrings and Heterotic Strings String Field Theory 


Open Access

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Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Eric D’Hoker
    • 1
  • Michael B. Green
    • 2
  • Pierre Vanhove
    • 3
    • 4
  1. 1.Department of Physics and AstronomyUniversity of CaliforniaLos AngelesUnited States
  2. 2.Department of Applied Mathematics and Theoretical PhysicsCambridgeUnited Kingdom
  3. 3.Institut des Hautes Études Scientifiques, Le Bois-MarieBures-sur-YvetteFrance
  4. 4.Institut de physique théorique, Université Paris Saclay, CEA, CNRSGif-sur-YvetteFrance

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