  This is a brief summary of our works [arXiv:1112.4063, arXiv:1201.4501] on
constructing higher genus B-model from perturbative quantization of BCOV
theory. We analyze Givental's symplectic loop space formalism in the context of
B-model geometry on Calabi-Yau manifolds, and explain the Fock space
construction via the renormalization techniques of gauge theory. We also give a
physics interpretation of the Virasoro constraints as the symmetry of the
classical BCOV action functional, and discuss the Virasoro constraints in the
quantum theory.
