Lattice Theta Constants vs Riemann Theta Constants and NSR Superstring Measures

TL;DR

This paper explores the mathematical relationships between lattice and Riemann theta constants in the context of NSR superstring measures, revealing elegant connections and highlighting challenges in higher-genus constructions.

Contribution

It establishes explicit formulas linking lattice theta constants to Riemann theta constants and discusses the limitations of modular-form approaches at higher genera.

Findings

Explicit relations between lattice and Riemann theta constants at genus 5

Challenges in constructing NSR measures beyond genus 5

Potential failure of modular-form methods at genus 6

Abstract

We discuss relations between two different representations of hypothetical holomorphic NSR measures, based on two different ways of constructing the semi-modular forms of weight 8. One of these ways is to build forms from the ordinary Riemann theta constants and another -- from the lattice theta constants. We discuss unexpectedly elegant relations between lattice theta constants, corresponding to 16-dimensional self-dual lattices, and Riemann theta constants and present explicit formulae expressing the former ones through the latter. Starting from genus 5 the modular-form approach to construction of NSR measures runs into serious problems and there is a risk that it fails completely already at genus 6.