Cyclic products of higher-genus Szegö kernels, modular tensors and polylogarithms

Abstract

A wealth of information on multiloop string amplitudes is encoded in fermionic two-point functions known as Szegö kernels. In this paper we show that cyclic products of any number of Szegö kernels on a Riemann surface of arbitrary genus may be decomposed into linear combinations of modular tensors on moduli space that carry all the dependence on the spin structure $\delta$. The $\delta$-independent coefficients in these combinations carry all the dependence on the marked points and are composed of the integration kernels of higher-genus polylogarithms. We determine the antiholomorphic moduli derivatives of the $\delta$-dependent modular tensors.

5.5 + 1.5 pages; v2: version to be published in Physics Review Letters, merged with the supplemental material as appendices; v3: corrections in and below equations (57), (58) of appendix D relative to v2