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

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.

arXiv comment15 pages. Typos corrected, refs. and comments added

References in corpus (31)

• Super-Poincare Covariant Quantization of the Superstring
• Two-Loop Superstrings VI: Non-Renormalization Theorems and the 4-Point Function
• Two-Loop Superstrings I, Main Formulas
• Two-Loop Superstrings II, The Chiral Measure on Moduli Space
• Two-loop superstrings and S-duality
• Two-Loop Superstrings IV, The Cosmological Constant and Modular Forms
• Two-Loop Superstrings III, Slice Independence and Absence of Ambiguities
• Two-Loop Superstrings V: Gauge Slice Independence of the N-Point Function
• The Superembedding Origin of the Berkovits Pure Spinor Covariant Quantization of Superstrings
• Modular Forms and Three Loop Superstring Amplitudes
• Asyzygies, modular forms, and the superstring measure II
• Superstring scattering amplitudes in higher genus
• Uniformization theory and 2D gravity I. Liouville action and intersection numbers
• Higher genus superstring amplitudes from the geometry of moduli spaces
• Remarks on Superstring amplitudes in higher genus
• Asyzygies, modular forms, and the superstring measure I
• Two-Loop Superstrings VII, Cohomology of Chiral Amplitudes
• Getting superstring amplitudes by degenerating Riemann surfaces
• Superstring measure and non-renormalization of the three-point amplitude
• The vanishing of two-point functions for three-loop superstring scattering amplitudes
• Two loop superstring amplitudes and S_6 representations
• Genus four superstring measures
• Lattice Theta Constants vs Riemann Theta Constants and NSR Superstring Measures
• NSR measures on hyperelliptic locus and non-renormalization of 1,2,3-point functions
• More on superstring chiral measures
• Classical theta constants vs. lattice theta series, and super string partition functions
• Linear relations among holomorphic quadratic differentials and induced Siegel's metric on M_g
• NSR superstring measures in genus 5
• The superstring cosmological constant and the Schottky form in genus 5
• The Singular Locus of the Theta Divisor and Quadrics through a Canonical Curve
• Binary Forms and the Hyperelliptic Superstring Ansatz

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