On the renormalization and quantization of topological-holomorphic field theories

Topological field theories and holomorphic field theories naturally appear in both mathematics and physics. However, there exist intriguing hybrid theories that are 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 rigorously prove the ultraviolet (UV) finiteness for such hybrid theories on the model manifold ${\mathbb{R}}^{d^{\mathrm{\prime }}}\times {\mathbb{C}}^d$, and present two significant vanishing results regarding anomalies: in the case $d^{\mathrm{\prime }}=1$, the odd-loop obstructions to quantization on ${\mathbb{R}}^{d^{\mathrm{\prime }}}\times {\mathbb{C}}^d$ vanish; in the case $d^{\mathrm{\prime }}>1$, all obstructions disappear, allowing us to define a factorization algebra structure for quantum observables. Previous versions circulated under the title “Factorization algebras from topological-holomorphic field theories”.