Asyzygies, modular forms, and the superstring measure II

TL;DR

This paper investigates the superstring measure at three loops, exploring polynomial Ansätze in theta constants, and proposes a new approach involving the square of the measure density that satisfies key modular and degeneration criteria.

Contribution

It introduces a novel Ansatz for the superstring measure's square, satisfying modular covariance and degeneration limits, suggesting the measure's square may be polynomial in theta constants.

Findings

An Ansatz for the square of the measure density satisfies key criteria.

The proposed Ansatz implies a vanishing cosmological constant.

None of the initial polynomial Ansätze for the measure density meet all conditions.

Abstract

Precise factorization constraints are formulated for the three-loop superstring chiral measure, in the separating degeneration limit. Several natural Ans\"atze in terms of polynomials in theta constants for the density of the measure are examined. None of these Ans\"atze turns out to satisfy the dual criteria of modular covariance of weight 6, and of tending to the desired degeneration limit. However, an Ansatz is found which does satisfy these criteria for the square of the density of the measure, raising the possibility that it is not the density of the measure, but its square which is a polynomial in theta constants. A key notion is that of totally asyzygous sextets of spin structures. It is argued that the Ansatz produces a vanishing cosmological constant.