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.
Categories: math-ph hep-th math.DG math.MP