  We study the 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
meromorphic quadratic differential implies the Goldman Poisson structure on the
monodromy manifoldThese results are applied to the WKB analysis of the
equation. It is shown that the leading term in the WKB expansion of the
generating function of the monodromy symplectomorphism (the "Yang-Yang
function" of Nekrasov, Rosly and Shatashvili) is determined by the Bergman
tau-function on the moduli space of meromorphic quadratic differentials.
