Abstract
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 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.
Highlights
The low energy expansion of the Type II superstring effective action generates an infinite set of higher derivative interactions that generalize the Einstein-Hilbert action of classical general relativity
We argue that the set of modular functions that contribute to the coefficients of interactions up to order D10R4 are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values
The simplest nontrivial example is provided by uncompactified Type IIB theory, which has maximal supersymmetry and for which the S-duality group is SL(2, Z), and the higher derivative interactions depend upon the complex axion-dilaton through non-holomorphic modular forms
Summary
The low energy expansion of the Type II superstring effective action generates an infinite set of higher derivative interactions that generalize the Einstein-Hilbert action of classical general relativity The dependence of these interactions on the vacuum expectation values of the scalar fields is highly constrained by space-time supersymmetry and duality. The simplest nontrivial example is provided by uncompactified Type IIB theory, which has maximal supersymmetry and for which the S-duality group is SL(2, Z), and the higher derivative interactions depend upon the complex axion-dilaton through non-holomorphic modular forms In this case the low order terms in the low energy expansion preserve a fraction of the supersymmetry. We note that there has been some recent progress in understanding the mathematical structure of N -particle genus-one open string amplitudes [36] based on elliptic multiple-polylogarithms constructed in [37]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
