Geodesic
WKB expansion for a~Yang--Yang generating function and the~Bergman tau function
Teoretičeskaâ i matematičeskaâ fizika, Tome 206 (2021) no. 3, pp. 295-338

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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, symplectic map generating function, tau function, Goldman bracket.
Mots-clés : monodromy map 
@article{TMF_2021_206_3_a1,
     author = {M. Bertola and D. A. Korotkin},
     title = {WKB expansion for {a~Yang--Yang} generating function and {the~Bergman} tau function},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {295--338},
     publisher = {mathdoc},
     volume = {206},
     number = {3},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2021_206_3_a1/}
}
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M. Bertola; D. A. Korotkin. WKB expansion for a~Yang--Yang generating function and the~Bergman tau function. Teoretičeskaâ i matematičeskaâ fizika, Tome 206 (2021) no. 3, pp. 295-338. http://geodesic.mathdoc.fr/item/TMF_2021_206_3_a1/