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 TMF, 2021, Volume 206, Number 3, Pages 295–338 (Mi tmf9834)

WKB expansion for a Yang–Yang generating function and the Bergman tau function

M. Bertolaab, D. A. Korotkina

a Concordia University, Department of Mathematics and Statistics, Montréal, Québec, Canada
b International School for Advanced Studies (SISSA), Area of Mathematics, Trieste, Italy

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.

Keywords: Riemann surface, monodromy map, symplectic map generating function, tau function, Goldman bracket.

 Funding Agency Grant Number Natural Sciences and Engineering Research Council of Canada (NSERC) RGPIN-2016-06660RGPIN/3827-2015 National Science Foundation DMS-1440140 Istituto Nazionale di Alta Matematica "Francesco Severi" 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).

DOI: https://doi.org/10.4213/tmf9834

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English version:
Theoretical and Mathematical Physics, 2021, 206:3, 258–295

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Revised: 02.12.2020

Citation: M. Bertola, D. A. Korotkin, “WKB expansion for a Yang–Yang generating function and the Bergman tau function”, TMF, 206:3 (2021), 295–338; Theoret. and Math. Phys., 206:3 (2021), 258–295

Citation in format AMSBIB
\Bibitem{BerKor21} \by M.~Bertola, D.~A.~Korotkin \paper WKB expansion for a~Yang--Yang generating function and the~Bergman tau function \jour TMF \yr 2021 \vol 206 \issue 3 \pages 295--338 \mathnet{http://mi.mathnet.ru/tmf9834} \crossref{https://doi.org/10.4213/tmf9834} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4224012} \transl \jour Theoret. and Math. Phys. \yr 2021 \vol 206 \issue 3 \pages 258--295 \crossref{https://doi.org/10.1134/S0040577921030028} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000664262900002}