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Notes on Motivic Periods

Francis Brown

TL;DR

The notes develop a comprehensive, algebraic-geometry–flavored framework for periods by embedding them into a Tannakian setting via the category $\mathcal{H}$ of Betti/de Rham realizations. Motives give rise to a universal ring of periods with a Galois action, enabling a decomposed view into semi-simple and unipotent parts through a coradical filtration and a coaction formalism. Concrete constructions include motivic algebraic numbers, Lefschetz periods, motivic logarithms, and motivic multiple zeta values, with explicit decomposition maps and single-valued variants that relate complex and de Rham data. The framework unifies symbols, families of periods, and geometric examples, offering a path toward a motivic classification and a Galois-theoretic understanding of period relations with potential implications for physics, number theory, and transcendence theory.

Abstract

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

Notes on Motivic Periods

TL;DR

The notes develop a comprehensive, algebraic-geometry–flavored framework for periods by embedding them into a Tannakian setting via the category of Betti/de Rham realizations. Motives give rise to a universal ring of periods with a Galois action, enabling a decomposed view into semi-simple and unipotent parts through a coradical filtration and a coaction formalism. Concrete constructions include motivic algebraic numbers, Lefschetz periods, motivic logarithms, and motivic multiple zeta values, with explicit decomposition maps and single-valued variants that relate complex and de Rham data. The framework unifies symbols, families of periods, and geometric examples, offering a path toward a motivic classification and a Galois-theoretic understanding of period relations with potential implications for physics, number theory, and transcendence theory.

Abstract

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

Paper Structure

This paper contains 69 sections, 24 theorems, 356 equations.

Table of Contents

  1. Introduction
  2. Contents
  3. Generalities
  4. Recap on Tannakian categories
  5. Matrix coefficients.
  6. Tannakian case
  7. Minimal objects
  8. Coradical filtration and decomposition
  9. Multiplicative structure
  10. Cohomological interpretation
  11. Motivic periods over $\mathbb Q$
  12. Category $\mathcal{H}$ of Betti and de Rham realisations
  13. The ring of $\mathcal{H}$-periods
  14. Some variants
  15. * Weight filtrations on $\mathcal{P}^{\mathfrak{dr}}_{\mathcal{H} }$
  16. ...and 54 more sections

Key Result

Theorem 2.1

(Saavedra, corrected in Tannaka). Let $\mathcal{T}$ be a Tannakian category with a fiber functor to $S$, a non-empty scheme over $k$. Then the groupoid of tensor automorphisms $Aut^{\otimes}_{\mathcal{T} }(\omega)$ is faithfully flat on $S\times S$, and $\omega$ defines an equivalence of categories

Theorems & Definitions (81)

  • Example 1.1
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 71 more