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

We study symplectic properties of the monodromy map second-order equations on a Riemann surface whose potential is meromorphic with double poles. show that Poisson bracket defined in terms periods quadratic differential implies Goldman structure manifold. apply these results to WKB analysis this equation and leading term expansion generating function symplectomorphism (the Yang–Yang introduced by Nekrasov, Rosly, Shatashvili) determined Bergman tau moduli space differentials.