In this dissertation, we consider moduli spaces of meromorphic quadratic differentials in homological coordinates and applications of underlying deformation theory of Ahlfors-Rauch type. 

At first, we derive variational formulas for objects associated with generalized $SL(2)$ Hitchin’s spectral covers: Prym matrix, Prym bidifferential, Bergman tau-function. The resulting formulas are antisymmetric versions of Donagi-Markman residue formula. Then we adapt the framework of topological recursion to the case of double covers to compute higher-order variations. 

Another application of the deformation theory lies within the symplectic geometry of the monodromy map of the Schrödinger equation on a Riemann surface with a meromorphic potential having second order poles. We discuss the conditions for the base projective connection, which induces its own set of Darboux homological coordinates, to imply the Goldman Poisson structure on the character variety. Using this result, we perform generalized WKB expansion of the generating function of monodromy symplectomorphism (the Yang-Yang function) and compute its leading asymptotics. 

Finally, we relate these two studies by showing how the variational analysis on Hitchin’s spectral covers could be applied towards the computation of higher asymptotics of the WKB expansion.