Quantization of open-closed BCOV theory, I

Abstract

This is the first in a series of papers which analyze the problem of quantizing the theory coupling Kodaira-Spencer gravity (or BCOV theory) on Calabi-Yau manifolds using the formalism for perturbative QFT developed by the first author. In this paper, we focus on flat space \(\mathbb{C}^d\) for \(d\) odd. We prove that there exists a unique quantization of the theory coupling BCOV theory and holomorphic Chern-Simons theory with gauge group the supergroup \(GL(N \mid N)\). We deduce a canonically defined quantization of BCOV theory on its own. We also discuss some conjectural links between BCOV theory in various dimensions and twists of physical theories: in complex dimension \(3\) we conjecture a relationship to twists of \((1,0)\) supersymmetric theories and in complex dimension \(5\) to a twist of type IIB supergravity.