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Feynman Graph Integrals on Kähler Manifolds

Minghao Wang, Junrong Yan

TL;DR

This work delivers a rigorous convergence theory for holomorphic Feynman graph integrals on closed real-analytic Kähler manifolds by showing the graph integrands extend canonically to the Fulton–MacPherson configuration spaces and define convergent Cauchy principal value integrals. The key technical innovation is combining Getzler’s rescaling with a heat-kernel analysis to establish divisorial-type singularities for propagators, which underpins the PV integration over configuration spaces. The framework yields a mathematically robust construction of higher-genus BCOV B-model invariants for Calabi–Yau threefolds and situates mirror symmetry within a precise analytic setting, including the expected independence from the Kähler class and the holomorphic limit correspondence to Gromov–Witten invariants when M and M^∨ are mirrors. Overall, the paper provides a canonical, intrinsic approach to BCOV-type invariants via convergent Feynman graph integrals on real-analytic Kähler manifolds, with potential connections to index theory and factorization algebras.

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).

Feynman Graph Integrals on Kähler Manifolds

TL;DR

This work delivers a rigorous convergence theory for holomorphic Feynman graph integrals on closed real-analytic Kähler manifolds by showing the graph integrands extend canonically to the Fulton–MacPherson configuration spaces and define convergent Cauchy principal value integrals. The key technical innovation is combining Getzler’s rescaling with a heat-kernel analysis to establish divisorial-type singularities for propagators, which underpins the PV integration over configuration spaces. The framework yields a mathematically robust construction of higher-genus BCOV B-model invariants for Calabi–Yau threefolds and situates mirror symmetry within a precise analytic setting, including the expected independence from the Kähler class and the holomorphic limit correspondence to Gromov–Witten invariants when M and M^∨ are mirrors. Overall, the paper provides a canonical, intrinsic approach to BCOV-type invariants via convergent Feynman graph integrals on real-analytic Kähler manifolds, with potential connections to index theory and factorization algebras.

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).

Paper Structure

This paper contains 18 sections, 53 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. BCOV's formulation of Mirror symmetry conjecture
  4. Organizations
  5. Feynman Graph Integrals on Kähler Manifolds
  6. Propagators
  7. Feynman graph integrals
  8. Regular Expressions
  9. Regular expressions and its filtration
  10. Regular expressions and divisorial type singularities
  11. Regularities of Propagator in Schwinger Space
  12. Heat kernel expansion
  13. Getzler's rescaling
  14. Proof of \ref{['main0']} and \ref{['propagator has divisorial type singularities']}
  15. Proof of \ref{['lem36']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\vec{\Gamma}$ be a directed graph, whose vertex set $\vec{\Gamma}_{0}$ and edge set $\vec{\Gamma}_{1}$ are ordered sets. Then exists, where $t,h:\vec{\Gamma}_{1}\rightarrow\vec{\Gamma}_{0}$ denotes the tail and head of a given directed edge, and $|\vec{\Gamma}_{0}|$ is the number of vertices. Here $\chi_{\epsilon}$ is the function on $M^{|\vec{\Gamma}_{0}|}$ defined by and for a finite set

Theorems & Definitions (131)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 1.5
  • Conjecture 1.6: Bershadsky–Cecotti–Ooguri–Vafa
  • Remark 1.7
  • Remark 2.1
  • Lemma 2.2
  • proof
  • ...and 121 more