Abstract
We study symplectic properties of the monodromy map of second-order equations on a Riemann surface whose potential is meromorphic with double poles. We show that the Poisson bracket defined in terms of periods of the meromorphic quadratic differential implies the Goldman Poisson structure on the monodromy manifold. We apply these results to a WKB analysis of this equation and show that the leading term in the WKB expansion of the generating function of the monodromy symplectomorphism (the Yang–Yang function introduced by Nekrasov, Rosly, and Shatashvili) is determined by the Bergman tau function on the moduli space of meromorphic quadratic differentials.
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Acknowledgments
The authors thank D. Allegretti, T. Bridgeland, A. Nietzke, and C. Norton for the illuminating discussions.
Funding
This research was supported by the National Science Foundation (Grant No. DMS-1440140) while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2019 semester Holomorphic Differentials in Mathematics and Physics. The research that led to the present paper was supported in part by a grant of the Gruppo Nazionale per la Fisica Matematica (GNFM), INdAM.
The research of M. Bertola was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Grant No. RGPIN-2016-06660).
The research of D. A. Korotkin was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Grant No. RGPIN/3827-2015).
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Dedicated to the memory of Boris Dubrovin
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Bertola, M., Korotkin, D.A. WKB expansion for a Yang–Yang generating function and the Bergman tau function. Theor Math Phys 206, 258–295 (2021). https://doi.org/10.1134/S0040577921030028
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DOI: https://doi.org/10.1134/S0040577921030028