Abstract
The supermeasure whose integral is the genus g vacuum amplitude of superstring theory is potentially singular on the locus in the moduli space of supercurves where the corresponding even theta-characteristic has nontrivial sections. We show that the supermeasure is actually regular for \(g\le 11\). The result relies on the study of the superperiod map. We also show that the minimal power of the classical Schottky ideal that annihilates the image of the superperiod map is equal to g if g is odd and is equal to g or \(g-1\) if g is even.
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Acknowledgements
G.F. is partially supported by the National Centre of Competence in Research “SwissMAP — The Mathematics of Physics” of the Swiss National Science Foundation. He thanks the Hebrew University of Jerusalem, where part of this work was done, for hospitality. D.K. is partially supported by the ERC under Grant Agreement 669655. A.P. is partially supported by the NSF Grant DMS-2001224, by the National Center of Competence in Research “SwissMAP — The Mathematics of Physics” of the Swiss National Science Foundation, and by the Russian Academic Excellence Project ‘5-100’ within the framework of the HSE University Basic Research Program. While working on this project, A.P. was visiting ETH Zurich and Hebrew University of Jerusalem. He would like to thank these institutions for hospitality and excellent working conditions.
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Felder, G., Kazhdan, D. & Polishchuk, A. Regularity of the superstring supermeasure and the superperiod map. Sel. Math. New Ser. 28, 17 (2022). https://doi.org/10.1007/s00029-021-00727-1
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DOI: https://doi.org/10.1007/s00029-021-00727-1