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

TL;DR

This paper constructs and analyzes the classical and quantum BCOV theory on Calabi-Yau manifolds, establishing foundational aspects and constraints, with specific results on elliptic curves and connections to Gromov-Witten theory.

Contribution

It introduces the classical BCOV theory on arbitrary Calabi-Yau varieties, defines quantization, and proves key properties like uniqueness and Virasoro constraints on elliptic curves.

Findings

Constructed classical BCOV theory on Calabi-Yau varieties.

Defined quantization and analyzed its properties.

Proved Virasoro constraints for elliptic curves.

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.