Yang-Yang generating function and Bergman tau-function

In this paper we consider the symplectic properties of the monodromy map of second order equations on a Riemann surface whose potential is meromorphic with second order poles. We show that the Poisson bracket defined by the periods of meromorphic quadratic differential implies the Goldman Poisson structure on character variety. These results generalize the previous works of the authors where the case of holomorphic potentials was considered and another paper of the second author where the symplectomorphism was proved for potentials with first order poles. We show that the leading term of the generating function of the monodromy symplectomorphism (the "Yang-Yang" function) is proportional to the Bergman tau-function on moduli spaces of meromorphic quadratic differentials with second order poles.