Quantization of open-closed BCOV theory, I

TL;DR

This paper establishes a unique quantization of BCOV theory coupled with holomorphic Chern-Simons theory on flat space, and explores conjectural links to supersymmetric theories and supergravity in various dimensions.

Contribution

It provides the first rigorous quantization of open-closed BCOV theory on flat space and proposes connections to physical theories in higher dimensions.

Findings

Existence of a unique quantization on $\, ext{C}^d$ for odd d.

Canonical quantization of BCOV theory alone.

Conjectural links to supersymmetric theories and supergravity in dimensions 3 and 5.

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.