Calabi-Yau Geometry and Higher Genus Mirror Symmetry

We study closed string mirror symmetry on compact Calabi-Yau manifolds at higher genus. String theory predicts the existence of two sets of geometric invariants, from the A-model and the B-model on Calabi-Yau manifolds, each indexed by a non-negative integer called genus. The A-model has been mathematically established at all genera by the Gromov-Witten theory, but little is known in mathematics for B-model beyond genus zero.;We develop a mathematical theory of higher genus B-model from perturbative quantization techniques of gauge theory. The relevant gauge theory is the Kodaira-Spencer gauge theory, which is originally discovered by Bershadsky-Cecotti-Ooguri-Vafa as the closed string field theory of B-twisted topological string on Calabi-Yau three-folds. We generalize this to Calabi-Yau manifolds of arbitrary dimensions including also gravitational descendants, which we call BCOV theory. We give the geometric description of the perturbative quantization of BCOV theory in terms of deformation-obstruction theory. The vanishing of the relevant obstruction classes will enable us to construct the higher genus B-model. We carry out this construction on the elliptic curve and establish the corresponding higher genus B-model. Furthermore, we show that the B-model invariants constructed from BCOV theory on the elliptic curve can be identified with descendant Gromov-Witten invariants on the mirror elliptic curve. This gives the first compact Calabi-Yau example where mirror symmetry can be established at all genera.