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Feynman Amplitudes and Cosmic Galois group

Francis Brown

TL;DR

The work constructs a comprehensive motivic framework for Feynman amplitudes by associating to each graph a graph motive and a motivic amplitude, organized by a conjectural cosmic Galois group C_{Q,M}. It develops a motic Hopf algebra and a geometric theory of linear blow-ups to model UV/IR factorization and to define affine models, enabling precise coaction and face-relations that tie amplitudes across graphs. It proves key structural results like finite weight-boundedness (the finiteness theorem) and establishes a robust program to classify and constrain Feynman periods via Galois-theoretic invariants, with substantial connections to BEK motives, symbol calculus, and single-valued constructions. The framework aims to unify perturbative physics across QFT, string theory, and bootstrap approaches by revealing a stable, algebraic action of a cosmic Galois group on motivic amplitudes, and by proposing concrete conjectures and finite-dimensional constraints driven by graph topology (the small-graphs principle).

Abstract

The first part of a set of notes based on lectures given at the IHES in May 2015 on Feynman amplitudes and motivic periods.

Feynman Amplitudes and Cosmic Galois group

TL;DR

The work constructs a comprehensive motivic framework for Feynman amplitudes by associating to each graph a graph motive and a motivic amplitude, organized by a conjectural cosmic Galois group C_{Q,M}. It develops a motic Hopf algebra and a geometric theory of linear blow-ups to model UV/IR factorization and to define affine models, enabling precise coaction and face-relations that tie amplitudes across graphs. It proves key structural results like finite weight-boundedness (the finiteness theorem) and establishes a robust program to classify and constrain Feynman periods via Galois-theoretic invariants, with substantial connections to BEK motives, symbol calculus, and single-valued constructions. The framework aims to unify perturbative physics across QFT, string theory, and bootstrap approaches by revealing a stable, algebraic action of a cosmic Galois group on motivic amplitudes, and by proposing concrete conjectures and finite-dimensional constraints driven by graph topology (the small-graphs principle).

Abstract

The first part of a set of notes based on lectures given at the IHES in May 2015 on Feynman amplitudes and motivic periods.

Paper Structure

This paper contains 77 sections, 44 theorems, 296 equations, 4 figures.

Table of Contents

  1. Some motivation for physicists
  2. Statement of results
  3. Product structure and stability
  4. Contents
  5. Relation to other work
  6. Acknowledgements
  7. Scalar Feynman graphs and Symanzik polynomials
  8. Feynman graphs
  9. Graph polynomials
  10. Feynman integral in projective space
  11. Edge subgraphs and their quotients
  12. Contraction-deletion
  13. Generic kinematics and non-vanishing of graph polynomials
  14. Space of generic kinematics
  15. Partial factorization theorems
  16. ...and 62 more sections

Key Result

Theorem 1

For any Feynman graph $G$ with generic kinematics $q,m$, there is a canonical way to associate to a convergent integral $(introIG)$: (i). an object $\mathrm{mot}_G$ in $\mathcal{H} (S)$, where $S$ is a Zariski open in a space of kinematics, (ii). a de Rham class $[\omega_G]$ in the generic fiber of

Figures (4)

  • Figure 1: A decomposition of the coordinate simplex in $\mathbb P^3$ (here $B=\emptyset$), defined by hyperplanes $z_i \alpha_i= z_{j} \alpha_j$ for $0<z_{i}<\infty$ and $i=1,2,3$. The affine open sets $U_i: \alpha_i \neq 0$ for $i=1,2,3$ are depicted schematically by grey arcs. On the right, $z_2,z_3 \rightarrow 0$.
  • Figure 2: Integration of a period. Left: the non-trivial contributions to the period integral. Right: taking the limit as $t \rightarrow \infty$ and replacing the paths with tangential base points.
  • Figure :
  • Figure :

Theorems & Definitions (133)

  • Theorem 1
  • Conjecture 1
  • Theorem 2
  • Example 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Example 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 123 more