  In both mathematics and physics, topological field theories and holomorphic
field theories appear naturally, but there are interesting theories that are
hybrids -- looking topological in some directions and holomorphic in others --
such as twists of supersymmetric field theories or Costello's 4-dimensional
Chern--Simons theory. In this paper we construct perturbative, one-loop
quantizations rigorously on the model manifold $\mathbb{R}^m \times
\mathbb{C}^n$, and find a remarkable vanishing result about anomalies: the
one-loop obstruction to quantization on $\mathbb{R}^m\times \mathbb{C}^n$
vanishes when $m\geq 1$. A concrete consequence of our results is the existence
of exact and finite quantizations at one-loop for twists of pure $\mathcal{N}
=2$ four-dimensional supersymmetric Yang--Mills theory.
