On new theta identities of fermion correlation functions on genus g Riemann surfaces

TL;DR

This paper introduces new theta identities on genus g Riemann surfaces that simplify fermion correlation functions and could advance calculations in superstring theory and Kac-Moody currents, especially for higher genus surfaces.

Contribution

It proposes a generalization of genus 1 theta identities to higher genus surfaces, providing a new approach to simplify fermion correlation functions and superstring amplitude calculations.

Findings

New theta identities for genus g surfaces are formulated.

These identities enable simpler summation over spin structures.

Potential applications include closed-form expressions for Kac-Moody current correlations.

Abstract

Theta identities on genus g Riemann surfaces which decompose simple products of fermion correlation functions with a constraint on their variables are considered. This type of theta identities is, in a sense, dual to Fay s formula, by which it is possible to sum over spin structures of certain part of superstring amplitudes in NSR formalism without using Fay s formula nor Riemann s theta formula in much simpler, more transparent way. Also, such identities will help to cast correlation functions among arbitrary numbers of Kac-Moody currents in a closed form. As for genus 1, the identities are reported before in ref[1] [2]. Based on some notes on genus 1 case which were not reported in ref[1] [2] and relating those to the results of the Dolan Goddard method ref[3] on describing Kac-Moody currents in a closed form, we propose an idea of generalizing genus 1 identities to the case of genus g surfaces. This is not a complete derivation of the higher genus formula due to difficulties of investigating singular part of derivatives of genus g Weierstrass Pe functions. Mathematical issues remained unsolved for genus g >1 are described in the text.