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Feynman Graph Integrals on $\mathbb{C}^d$

Minghao Wang

TL;DR

The paper develops a rigorous holomorphic analog of configuration-space Feynman integrals on complex space $C^d$ by introducing heat-kernel regularization and a compactified Schwinger-space framework. It proves ultraviolet finiteness for all decorated graphs and dimensions, and identifies boundary-induced anomalies that obstruct holomorphicity, localized to Laman subgraphs. It then derives quadratic relations among anomaly terms, connecting to BV formalism and L-infinity type structures in holomorphic field theories. The framework generalizes prior finiteness results beyond Laman graphs and provides a robust method for studying holomorphic factorization algebras and their correlation functions.

Abstract

We introduce a type of graph integrals which are holomorphic analogs of configuration space integrals. We prove their (ultraviolet) finiteness by considering a compactification of the moduli space of graphs with metrics, and study their failure to be holomorphic.

Feynman Graph Integrals on $\mathbb{C}^d$

TL;DR

The paper develops a rigorous holomorphic analog of configuration-space Feynman integrals on complex space by introducing heat-kernel regularization and a compactified Schwinger-space framework. It proves ultraviolet finiteness for all decorated graphs and dimensions, and identifies boundary-induced anomalies that obstruct holomorphicity, localized to Laman subgraphs. It then derives quadratic relations among anomaly terms, connecting to BV formalism and L-infinity type structures in holomorphic field theories. The framework generalizes prior finiteness results beyond Laman graphs and provides a robust method for studying holomorphic factorization algebras and their correlation functions.

Abstract

We introduce a type of graph integrals which are holomorphic analogs of configuration space integrals. We prove their (ultraviolet) finiteness by considering a compactification of the moduli space of graphs with metrics, and study their failure to be holomorphic.
Paper Structure (13 sections, 30 theorems, 125 equations)

This paper contains 13 sections, 30 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Feynman graph integrals on $\mathbb{C}^d$.
  3. Feynman graph integrals.
  4. Propagator in Schwinger spaces.
  5. Feynman graph integrals in Schwinger spaces.
  6. Compactification of Schwinger spaces and finiteness of the Feynman graph integrals.
  7. Compactification of Schwinger spaces.
  8. Two technical lemmas
  9. Finiteness of Feynman graph integrals.
  10. Holomorphicity of Feynman graph integrals.
  11. Holomorphicity and integrals over boundaries of compactified Schwinger spaces.
  12. Anomaly integrals and Laman graphs.
  13. Quadratic relations.

Key Result

Theorem 1

Theorems & Definitions (86)

  • Theorem 1: See Theorem \ref{['extension theorem1']}, Corollary \ref{['distribution theorem']}
  • Theorem 2: See Theorem \ref{['laman anomaly']}, Corollary \ref{['holomorphicity\nfailure']}
  • Theorem 3: K.Budzik, D.Gaiotto, J.Kulp, J.Wu, M.Yu, See budzik2023feynman and Theorem \ref{['Quadratic relations']}
  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Definition 4
  • Remark 2
  • Example 1
  • ...and 76 more