Universal KZB equations I: the elliptic case

By Damien Calaque, B. Enriquez, Pavel Etingof. 2007

130 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/06e216f74d988b52c35740a5315833314924837e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. F. Brown
2. Stephen Griffeth
3. B. Enriquez
4. Emanuel Stoica
5. Pavel Etingof
6. I. Gordon
7. Erik Panzer
8. I. Losev

In Mathematical Physics

1. F. Brown
2. Stephen Griffeth
3. B. Enriquez
4. Pavel Etingof
5. I. Gordon
6. Erik Panzer
7. I. Losev
8. V. Ginzburg

Citations in the corpus

• Unitary representations of rational Cherednik algebras (2009, background): “5.2, since if τ∗ contains a part equal 1 then N(τ) = ℓ(τ). Proposition 5.4. Let τ= (p,p,...,p), where pis a divisor of n. Then L 1/p(τ) is unitary. Proof. This is shown in the proof of Theorem 8.8 in [CEE]. The main result of this subsection is the following theorem. Theorem 5.5. For any τ 6= ( n),(”
• A CLASS OF NONHOLOMORPHIC MODULAR FORMS II: EQUIVARIANT ITERATED EISENSTEIN INTEGRALS (2017, methodology): “The universal elliptic KZB equation [8, 9, 16, 19, 25] that we shall use is d J (z) = ωJ (z), where ω is the formal 1-form:”
• Analogues elliptiques des nombres multiz\'etas (2013, background): “On a A(τ) = Φ(ỹ, t)e i Φ(ỹ, t) +O(e i τ ), (8) B(τ) = eΦ(−ỹ − t, t)e i e i Φ(ỹ, t) +O(e i(1−ǫ)τ ) (9) pour tout ǫ > 0, lorsque τ → i∞ ([CEE], démonstration de Proposition 4.”