Classical theta constants vs. lattice theta series, and super string partition functions

TL;DR

This paper compares classical theta constants and lattice theta series in superstring theory, demonstrating their equivalence up to genus 4 and analyzing their relation at genus 5, with implications for superstring amplitudes.

Contribution

It provides a detailed comparison and establishes the equivalence of classical theta constants and lattice theta series for superstring amplitudes up to genus 4, and explores their relation at genus 5.

Findings

Proves equivalence of the two tools up to genus 4.

Shows solutions are equivalent modulo the Schottky form at genus 5.

Demonstrates coincidence when the cosmological constant vanishes.

Abstract

Recently, various possible expressions for the vacuum-to-vacuum superstring amplitudes has been proposed at genus $g=3,4,5$. To compare the different proposals, here we will present a careful analysis of the comparison between the two main technical tools adopted to realize the proposals: the classical theta constants and the lattice theta series. We compute the relevant Fourier coefficients in order to relate the two spaces. We will prove the equivalence up to genus 4. In genus five we will show that the solutions are equivalent modulo the Schottky form and coincide if we impose the vanishing of the cosmological constant.