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We study the symplectic properties of the monodromy map of second orderequations on a Riemann surface whose potential is meromorphic with doublepoles. We show that the Poisson bracket defined in terms of periods ofmeromorphic quadratic differential implies the Goldman Poisson structure on themonodromy manifoldThese results are applied to the WKB analysis of theequation. It is shown that the leading term in the WKB expansion of thegenerating function of the monodromy symplectomorphism (the "Yang-Yangfunction" of Nekrasov, Rosly and Shatashvili) is determined by the Bergmantau-function on the moduli space of meromorphic quadratic differentials.

WKB expansion of Yang-Yang generating function and Bergman tau-function