Seminar

We discuss the monodromy symplectomorphism associated to a second order equation on a Riemann surface when the potential is either holomorphic or has poles of order up to 2. This problem can be naturally studies using homological Darboux coordinates on moduli spaces of quadratic differentials; we show that the canonical Poisson structure on the space of potentials implies the Goldman Poisson structure on the  monodromy manifold. For the case of potentials with second order poles we study the WKB limit of the generating function of the monodromy  symplectomorphism  (the "Yang-Yang" function) and explicitly compute its leading order in terms of the Bergman tau-function on the moduli spaces of quadratic differentials.