The superstring cosmological constant and the Schottky form in genus 5

Abstract

Combining certain identities for modular forms due to Igusa with Schottky-Jung relations, we study the cosmological constant for the recently proposed ansatz for the chiral superstring measure in genus 5. The vanishing of this cosmological constant turns out to be equivalent to the long-conjectured vanishing of a certain explicit modular form of genus 5 on the moduli of curves M_5, and we disprove this conjecture, thus showing that the cosmological constant for the proposed ansatz does not vanish identically. We exhibit an easy modification of the genus 5 ansatz satisfying factorization constraints and yielding a vanishing cosmological constant. We also give an expression for the cosmological constant for the proposed ansatz that should hold for any genus if certain generalized Schottky-Jung identities hold.

v2: A modified genus 5 ansatz with a vanishing cosmological constant proposed. v3: TeXnical corrections; published version, Amer. J. Math. v4: incorporates an erratum, correcting the mistake in the proof of theorem 15

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• Modular Forms and Three Loop Superstring Amplitudes
• Superstring scattering amplitudes in higher genus
• Higher genus superstring amplitudes from the geometry of moduli spaces
• Remarks on Superstring amplitudes in higher genus
• The vanishing of two-point functions for three-loop superstring scattering amplitudes
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• Getting superstring amplitudes by degenerating Riemann surfaces
• Lattice Theta Constants vs Riemann Theta Constants and NSR Superstring Measures
• Higher genus partition functions of meromorphic conformal field theories
• Classical theta constants vs. lattice theta series, and super string partition functions
• Extending the Belavin-Knizhnik "wonderful formula" by the characterization of the Jacobian
• Uniformization, Unipotent Flows and the Riemann Hypothesis