Feynman graph integrals of topological field theories have been proved to be ultraviolet finite by Axelrod and Singer, and Kontsevich independently. This result leads to many applications including universal finite type knot invariants and the formality of $E_n$ operads. In this talk, I will extend the finiteness results (and some anomaly cancellation results) to Feynman graph integrals of topological-holomorphic theories on flat spaces. The main technique for the proof is compactification of the moduli space of metric graphs. As a result, we can construct many factorization algebras from quantum topological-holomorphic theories. In the special case of 4d Chern–Simons theory, the factorization algebra structure encodes the Yang–Baxter equation. If time permits, I will sketch how to extend these results to Feynman graph integrals on Kähler manifolds. Part of this work is joint with Brian Williams.

Reference: https://arxiv.org/abs/2401.08113