More on superstring chiral measures

TL;DR

This paper investigates superstring chiral measures for genera up to 5, providing new expressions involving theta constants and exploring their properties, especially for genus 5 where a candidate measure is proposed.

Contribution

It introduces new expressions for superstring measures in terms of higher powers of theta constants and analyzes their properties across different genera, including a candidate for genus five.

Findings

For g=3, measures are expressed via fourth powers of theta constants.

For g=4, measures are expressed via squares of theta constants.

A genus five measure candidate is proposed with specific constraints.

Abstract

In this paper we study the expressions of the superstring chiral measures for $g\leq 5$. We obtain certain new expressions which are functions of higher powers of theta constants. For $g=3$ we show that the measures can be written in terms of fourth power of theta constants and for $g=4$ in terms of squares of theta constants. In both cases the forms $\Xi_8^{(g)}[0^{(g)}]$ appearing in the expression of the measures are defined on the whole Siegel upper half space. Instead, for $g=5$ we find a form $\Xi_8^{(5)}[0^{(5)}]$ which is a polynomial in the classical theta constants, well defined on the Siegel upper half space and satisfying some suitable constraints on the moduli space of curves (and not on the whole Siegel upper half space) that could be a candidate for the genus five superstring measure. Moreover, we discuss the problem of the uniqueness of this form in genus five. We also determine the dimension of certain spaces of modular forms and reinterpret the vanishing of the cosmological constant in terms of group representations.