Extending the Belavin-Knizhnik "wonderful formula" by the characterization of the Jacobian

TL;DR

This paper extends the Belavin-Knizhnik formula for the bosonic string measure to arbitrary genus, linking it to the characterization of the Jacobian locus and providing a geometric interpretation involving quadrics in projective space.

Contribution

It introduces a generalized formula for the bosonic measure at higher genus, connecting it to the geometry of Jacobians and modular forms, and proposes a superstring analog.

Findings

Extended the Belavin-Knizhnik formula to arbitrary genus.

Linked the measure to the characterization of the Jacobian locus.

Proposed a geometric interpretation involving quadrics in projective space.

Abstract

A long-standing question in string theory is to find the explicit expression of the bosonic measure, a crucial issue also in determining the superstring measure. Such a measure was known up to genus three. Belavin and Knizhnik conjectured an expression for genus four which has been proved in the framework of the recently introduced vector-valued Teichmueller modular forms. It turns out that for g>3 the bosonic measure is expressed in terms of such forms. In particular, the genus four Belavin-Knizhnik "wonderful formula" has a remarkable extension to arbitrary genus whose structure is deeply related to the characterization of the Jacobian locus. Furthermore, it turns out that the bosonic string measure has an elegant geometrical interpretation as generating the quadrics in P^{g-1} characterizing the Riemann surface. All this leads to identify forms on the Siegel upper half-space that, if certain conditions related to the characterization of the Jacobian are satisfied, express the bosonic measure as a multiresidue in the Siegel upper half-space. We also suggest that it may exist a super analog on the super Siegel half-space.