Quantization of topological-holomorphic field theories: local aspects

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 $$\mathbb{R}$^m \times $\mathbb{C}$^n$, and find a remarkable vanishing result about anomalies: the one-loop obstruction to quantization on ${\mathbb{R}}^m\times {\mathbb{C}}^n$ vanishes when $m\ge 1$. A concrete consequence of our results is the existence of exact and finite quantizations at one-loop for twists of pure $$\mathcal{N}$ =2$ four-dimensional supersymmetric Yang–Mills theory.