[Paper Review] Universal KZB equations I: the elliptic case

D. Calaque, B. Enriquez+1 more|arXiv (Cornell University)|Feb 22, 2007

Advanced Algebra and Geometry4 citations

TL;DR

This paper introduces a universal Knizhnik-Zamolodchikov-Bernard (KZB) connection on the moduli space of elliptic curves with marked points, constructing a flat connection on a principal bundle that restricts to configuration spaces of points on tori. The monodromy of this connection yields a link between the KZ associator and iterated integrals of Eisenstein forms, and it realizes the standard KZB connection for simple Lie algebras, factoring through Cherednik algebras and enabling a functor from equivariant D-modules on sl_n to modules over Cherednik algebras with explicit character computations.

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.

Motivation & Objective

• To define a universal version of the KZB connection in genus 1, extending the KZ connection to elliptic curves.
• To prove the formality of pure braid groups on the torus using the monodromy of this flat connection.
• To establish a realization of the universal KZB system via representations of Cherednik algebras and equivariant D-modules on the nilpotent cone.
• To compute the characters of irreducible equivariant D-modules supported on the nilpotent cone for sl_n.

Proposed method

• Construct a flat connection on configuration spaces of points on elliptic curves, lifting to the moduli space M_{1,[n]} of elliptic curves with marked points.
• Define the Lie algebras t_{1,n} and tbar_{1,n} as holonomy Lie algebras of the fundamental group of the configuration space.
• Use the compatibility of derivations and group actions to extend the connection to the moduli space, ensuring flatness via curvature vanishing.
• Derive monodromy representations via solutions to differential systems involving logarithmic derivatives and associators.
• Realize the universal KZB system as a representation of the rational Cherednik algebra of type A_{n-1}, using a homomorphism from tbar_{1,n} to the algebra.
• Construct a functor from equivariant D-modules on the nilpotent cone of sl_n to modules over the Cherednik algebra, computing characters via symmetric and spherical parts.

Experimental results

Research questions

• RQ1 How can a universal KZB connection be defined on the moduli space of elliptic curves with marked points in genus 1?
• RQ2 What is the monodromy of the universal KZB connection, and how does it relate to the KZ associator and iterated integrals of Eisenstein forms?
• RQ3 How does the universal KZB connection realize the standard KZB connection for simple Lie algebras, particularly sl_n?
• RQ4 Can the universal KZB system be realized through representations of Cherednik algebras, and what is the structure of this realization?
• RQ5 What is the character of irreducible equivariant D-modules supported on the nilpotent cone for sl_n, and how is it computed via the Cherednik algebra functor?

Key findings

• The universal KZB connection is a flat connection on a principal bundle over the moduli space M_{1,[n]}, restricting to configuration spaces of points on elliptic curves.
• The monodromy of the connection gives a morphism from the fundamental group Γ_{1,[n]} to G_n ⋊ S_n, relating the KZ associator to iterated integrals of Eisenstein forms.
• The universal KZB system realizes the standard KZB connection for sl_n, factoring through the rational Cherednik algebra of type A_{n-1}.
• A functor is defined from the category of equivariant D-modules on the nilpotent cone of sl_n to modules over the Cherednik algebra, with explicit character computations provided.
• The character of irreducible equivariant D-modules supported on the nilpotent cone is computed via the spherical and symmetric parts of the Cherednik algebra representation.
• The irreducibility and character formula for the representation V_N of the Cherednik algebra are established, with a precise formula derived for the character.