Feynman Graph Integrals on Kähler Manifolds

Abstract

In this paper, we establish the convergence of Feynman graph integrals on closed real-analytic Kähler manifolds and uncover the structural mechanism underlying this convergence. The key insight is that, using Getzler's rescaling technique, the graph integrands extend canonically to the Fulton-MacPherson compactification of configuration spaces as forms with divisorial-type singularities. This allows the Feynman graph integrals to be rigorously defined as Cauchy principal value integrals. As an application, these integrals provide a mathematically rigorous construction of the higher-genus B-model invariants on Calabi-Yau threefolds in the sense of Bershadsky-Cecotti-Ooguri-Vafa (BCOV).

55 pages, no figures. We merged previous Proposition 4.8 and Corollary 4.10 to a single Proposition 4.8, and rewritten its proof. The proofs in Appendix B have been greatly simplified