Feynman Graph Integrals on K\"ahler Manifolds

TL;DR

This paper proves the convergence of Feynman graph integrals on K"ahler manifolds, extending integrands to compactified configuration spaces, and uses this to rigorously construct higher-genus B-model invariants on Calabi-Yau threefolds.

Contribution

It establishes the convergence and rigorous definition of Feynman graph integrals on K"ahler manifolds and applies this to construct higher-genus B-model invariants.

Findings

Feynman graph integrals converge on closed real-analytic K"ahler manifolds.

Integrands extend to Fulton-MacPherson compactification as forms with divisorial singularities.

Provides a rigorous mathematical construction of higher-genus B-model invariants on Calabi-Yau threefolds.

Abstract

In this paper, we establish the convergence of Feynman graph integrals on closed real-analytic K\"ahler 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).