  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.
