Abstract
Modular graph functions (MGFs) are SL(2, ℤ)-invariant functions on the Poincaré upper half-plane associated with Feynman graphs of a conformal scalar field on a torus. The low-energy expansion of genus-one superstring amplitudes involves suitably regularized integrals of MGFs over the fundamental domain for SL(2, ℤ). In earlier work, these integrals were evaluated for all MGFs up to two loops and for higher loops up to weight six. These results led to the conjectured uniform transcendentality of the genus-one four-graviton amplitude in Type II superstring theory. In this paper, we explicitly evaluate the integrals of several infinite families of three-loop MGFs and investigate their transcendental structure. Up to weight seven, the structure of the integral of each individual MGF is consistent with the uniform transcendentality of string amplitudes. Starting at weight eight, the transcendental weights obtained for the integrals of individual MGFs are no longer consistent with the uniform transcendentality of string amplitudes. However, in all the cases we examine, the violations of uniform transcendentality take on a special form given by the integrals of triple products of non-holomorphic Eisenstein series. If Type II superstring amplitudes do exhibit uniform transcendentality, then the special combinations of MGFs which enter the amplitudes must be such that these integrals of triple products of Eisenstein series precisely cancel one another. Whether this indeed is the case poses a novel challenge to the conjectured uniform transcendentality of genus-one string amplitudes.
Highlights
Modular graph functions (MGFs) are SL(2, Z)-invariant functions on the Poincaré upper half-plane associated with Feynman graphs of a conformal scalar field on a torus
We find that the transcendental structure of the integrals of all two-loop MGFs is consistent with the uniform transcendentality of superstring amplitudes
The main result of this paper is that the violations of uniform transcendentality occurring in the integrals of individual three-loop MGFs are all of the same form as the violations endemic to Zagier’s integrals of triple products of Eisenstein series
Summary
Modular graph functions (MGFs) are SL(2, Z)-invariant functions on the Poincaré upper half-plane associated with Feynman graphs for a conformal scalar field theory on a torus. MGFs may be viewed as generalizations of the non-holomorphic Eisenstein series, which themselves provide a one-dimensional basis for all one-loop MGFs. MGFs may be obtained as special values of elliptic modular graph functions which are closely related to single-valued elliptic polylogarithms [5, 20–22] and iterated modular integrals [23, 24]. These series were used in [30] to evaluate the integrals of two-loop MGFs using the unfolding trick familiar from the RankinSelberg-Zagier method [31–33] These integrals may be expressed in terms of zeta-values and assigned a definite transcendental weight, thereby providing the starting point for a systematic investigation of the transcendentality properties of the genus-one four-graviton amplitude in Type II superstring theory in [34]. Significant partial results, to be explained below, support the validity of uniform transcendentality to arbitrary order in the low-energy expansion of the genus-one four graviton amplitude in Type II superstring theory, as conjectured in [34]. Before turning to the detailed calculations involved, we shall provide a brief overview of the questions pursued and the results obtained in the sequel of the paper
Sign-in/Register to view highlights & summary 
Published Version (
Free)
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 call