  We study the effective Batalin-Vilkovisky quantization theory for chiral
deformation of two dimensional conformal field theories. We establish an exact
correspondence between renormalized quantum master equations for effective
functionals and Maurer-Cartan equations for chiral vertex operators. The
generating functions are proven to have modular property with mild holomorphic
anomaly. As an application, we construct an exact solution of quantum B-model
(BCOV theory) in complex one dimension that solves the higher genus mirror
symmetry conjecture on elliptic curves.
