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Elliptic associators

B. Enriquez

TL;DR

This work constructs a genus-one analogue of the Grothendieck-Teichmüller program by defining elliptic structures and the elliptic GT group GT_{ell}, along with its pro-p, pro-l, and prounipotent variants. It introduces elliptic associators Ell, proves their nonemptiness, and establishes a torsor structure under GT_{ell}(-); it also connects these to a graded Lie algebra grt_{ell} and reveals a semidirect product decomposition with explicit generators. A concrete family of elliptic associators e(τ) is produced via the universal KZB connection, yielding relations between multiple zeta values and iterated Eisenstein integrals, and enabling a precise description of Zariski closures of genus-one mapping class actions. The paper also develops a rich interplay between categorical formalisms (elliptic BMCs), algebraic groups, and Galois actions, extending the genus-zero GT/GRT framework to genus one and illuminating deep links to modular and arithmetic structures. Overall, it provides foundational structures for elliptic GT theory, explicit morphisms and actions, and bridges between algebraic topology, number theory, and mathematical physics.

Abstract

We construct a genus one analogue of the theory of associators and the Grothendieck-Teichmueller group. The analogue of the Galois action on the profinite braid groups is an action of the arithmetic fundamental group of a moduli space of elliptic curves on the profinite braid groups in genus one. This action factors through an explicit profinite group hat GT_ell, which admits an interpretation in terms of decorations of braided monoidal categories. We relate this group to its prounipotent group scheme version GT_ell(-). We construct a torsor over the latter group, the scheme of elliptic associators. An explicit family of elliptic associators is constructed, based on our earlier work with Calaque and Etingof on the universal KZB connexion. The existence of elliptic associators enables one to show that the Lie algebra of GT_ell(-) is isomorphic to a graded Lie algebra, on which we obtain several results: semidirect product structure; explicit generators. This existence also allows one to compute the Zariski closure of the mapping class group in genus one (isomorphic to the braid group B_3) in the automorphism groups of the prounipotent completions of braid groups in genus one. The analytic study of the family of elliptic associators produces relations between MZVs and iterated integrals of Eisenstein series.

Elliptic associators

TL;DR

This work constructs a genus-one analogue of the Grothendieck-Teichmüller program by defining elliptic structures and the elliptic GT group GT_{ell}, along with its pro-p, pro-l, and prounipotent variants. It introduces elliptic associators Ell, proves their nonemptiness, and establishes a torsor structure under GT_{ell}(-); it also connects these to a graded Lie algebra grt_{ell} and reveals a semidirect product decomposition with explicit generators. A concrete family of elliptic associators e(τ) is produced via the universal KZB connection, yielding relations between multiple zeta values and iterated Eisenstein integrals, and enabling a precise description of Zariski closures of genus-one mapping class actions. The paper also develops a rich interplay between categorical formalisms (elliptic BMCs), algebraic groups, and Galois actions, extending the genus-zero GT/GRT framework to genus one and illuminating deep links to modular and arithmetic structures. Overall, it provides foundational structures for elliptic GT theory, explicit morphisms and actions, and bridges between algebraic topology, number theory, and mathematical physics.

Abstract

We construct a genus one analogue of the theory of associators and the Grothendieck-Teichmueller group. The analogue of the Galois action on the profinite braid groups is an action of the arithmetic fundamental group of a moduli space of elliptic curves on the profinite braid groups in genus one. This action factors through an explicit profinite group hat GT_ell, which admits an interpretation in terms of decorations of braided monoidal categories. We relate this group to its prounipotent group scheme version GT_ell(-). We construct a torsor over the latter group, the scheme of elliptic associators. An explicit family of elliptic associators is constructed, based on our earlier work with Calaque and Etingof on the universal KZB connexion. The existence of elliptic associators enables one to show that the Lie algebra of GT_ell(-) is isomorphic to a graded Lie algebra, on which we obtain several results: semidirect product structure; explicit generators. This existence also allows one to compute the Zariski closure of the mapping class group in genus one (isomorphic to the braid group B_3) in the automorphism groups of the prounipotent completions of braid groups in genus one. The analytic study of the family of elliptic associators produces relations between MZVs and iterated integrals of Eisenstein series.

Paper Structure

This paper contains 64 sections, 67 theorems, 382 equations.

Table of Contents

  1. Elliptic structures over braided monoidal categories
  2. Definition
  3. Relation with elliptic braid groups
  4. Universal elliptic structures
  5. The elliptic Grothendieck-Teichmüller group
  6. Reminders about ${\ul{\operatorname{GT}}}$ and its variants
  7. The semigroup ${\ul{\operatorname{GT}}}_{ell}$ and its variants
  8. Computation of ${\ul{\operatorname{GT}}}_{ell}$
  9. The semigroup scheme ${\ul{\operatorname{GT}}}_{ell}(-)$
  10. The Zariski closure $\langle B_3\rangle \subset R_{ell}(-)$
  11. A morphism ${\ul{\operatorname{GT}}}\to{\ul{\operatorname{GT}}}_{ell}$ and its variants
  12. Elliptic associators
  13. Lie algebras ${\mathfrak{t}}_n$ and ${\mathfrak{t}}_{1,n}$
  14. Elliptic associators
  15. Categorical interpretations
  16. ...and 49 more sections

Key Result

Proposition 1.3

If $(\tilde{{\mathcal{C}}},F,A^\pm_{\ldots})$ is an elliptic structure over ${\mathcal{C}}$ and $X\in\operatorname{Ob}{\mathcal{C}}$, then there is a unique group morphism $B_{1,n}\to \operatorname{Aut}_{\tilde{{\mathcal{C}}}}(F(X^{\otimes n}))$, such that

Theorems & Definitions (93)

  • Definition 1.1
  • Remark 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • ...and 83 more