Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model

Abstract: Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed that the B-model of mirror symmetry should be described by a quantum field theory on a Calabi-Yau variety, which they called the Kodaira-Spenser theory (we call it the BCOV theory). This is the first of three papers in which we construct and analyze the quantum BCOV theory. In this paper, we construct the classical field theory on a Calabi-Yau variety of arbitrary dimension; define what it means to give a quantization; analyze the relation Givental's symplectic formalism for Gromov-Witten theory; prove uniqueness of the quantization on an elliptic curve; and prove the Virasoro constraints on an elliptic curve. The second paper (arXiv:1112.4063) proves that the partition function of the quantum BCOV theory on the elliptic curve is equivalent to the Gromov-Witten theory of the mirror elliptic curve. The third paper, in progress, constructs the quantum BCOV theory on a general Calabi-Yau.