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Universal KZB equations I: the elliptic case

D. Calaque, B. Enriquez, P. Etingof

Abstract

We define a universal version of the Knizhnik-Zamolodchikov-Bernard (KZB) connection in genus 1. This is a flat connection over a principal bundle on the moduli space of elliptic curves with marked points. It restricts to a flat connection on configuration spaces of points on elliptic curves, which can be used for proving the formality of the pure braid groups on genus 1 surfaces. We study the monodromy of this connection and show that it gives rise to a relation between the KZ associator and a generating series for iterated integrals of Eisenstein forms. We show that the universal KZB connection realizes as the usual KZB connection for simple Lie algebras, and that in the sl_n case this realization factors through the Cherednik algebras. This leads us to define a functor from the category of equivariant D-modules on sl_n to that of modules over the Cherednik algebra, and to compute the character of irreducible equivariant D-modules over sl_n which are supported on the nilpotent cone.

Universal KZB equations I: the elliptic case

Abstract

We define a universal version of the Knizhnik-Zamolodchikov-Bernard (KZB) connection in genus 1. This is a flat connection over a principal bundle on the moduli space of elliptic curves with marked points. It restricts to a flat connection on configuration spaces of points on elliptic curves, which can be used for proving the formality of the pure braid groups on genus 1 surfaces. We study the monodromy of this connection and show that it gives rise to a relation between the KZ associator and a generating series for iterated integrals of Eisenstein forms. We show that the universal KZB connection realizes as the usual KZB connection for simple Lie algebras, and that in the sl_n case this realization factors through the Cherednik algebras. This leads us to define a functor from the category of equivariant D-modules on sl_n to that of modules over the Cherednik algebra, and to compute the character of irreducible equivariant D-modules over sl_n which are supported on the nilpotent cone.

Paper Structure

This paper contains 59 sections, 65 theorems, 353 equations.

Table of Contents

  1. Bundles with flat connections on (reduced) configuration spaces
  2. The Lie algebras ${\mathfrak{t}}_{1,n}$ and $\bar{{\mathfrak{t}}}_{1,n}$
  3. Bundles with flat connections over $C(E,n)$ and $\bar{C}(E,n)$
  4. Bundles with flat connections on $C(E,n)/S_{n}$ and $\bar{C}(E,n)/S_{n}$
  5. Formality of pure braid groups on the torus
  6. Reminders on Malcev Lie algebras
  7. Presentation of $\operatorname{PB}_{1,n}$
  8. Alternative presentations of ${\mathfrak{t}}_{1,n}$
  9. The formality of $\operatorname{PB}_{1,n}$
  10. The formality of $\overline{\operatorname{PB}}_{1,n}$
  11. The isomorphisms $\operatorname{B}_{1,n}({\mathbb C})\simeq \operatorname{exp}(\hat{{\mathfrak{t}}}_{1,n}) \rtimes S_{n}$, $\overline{\operatorname{B}}_{1,n}({\mathbb C}) \simeq \operatorname{exp}(\hat{\bar{{\mathfrak{t}}}}_{1,n})\rtimes S_{n}$
  12. Bundles with flat connection on ${\cal M}_{1,n}$ and ${\cal M}_{1,[n]}$
  13. Derivations of the Lie algebras ${\mathfrak{t}}_{1,n}$ and $\bar{{\mathfrak{t}}}_{1,n}$ and associated groups
  14. Bundle with flat connection on ${\cal M}_{1,n}$
  15. Bundle with flat connection over ${\cal M}_{1,[n]}$
  16. ...and 44 more sections

Key Result

Lemma 1.1

$K_{i}({\bf z}+\delta_{j}|\tau) = K_{i}({\bf z}|\tau)$ and $K_{i}({\bf z}+\tau\delta_{j}|\tau) = e^{-2\pi\operatorname{i} \operatorname{ad}x_{j}}(K_{i}({\bf z}|\tau))$, i.e., the $K_{i}({\bf z}|\tau)$ satisfy condition (a).

Theorems & Definitions (84)

  • Lemma 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • ...and 74 more