Notes on the Universal Elliptic KZB Equation

Abstract

The universal elliptic KZB equation is the integrable connection on the pro-vector bundle over M_{1,2} whose fiber over the point corresponding to the elliptic curve E and a non-zero point x of E is the unipotent completion of π_1(E-{0},x). This was written down independently by Calaque, Enriquez and Etingof (arXiv:math/0702670), and by Levin and Racinet (arXiv:math/0703237). It generalizes the KZ-equation in genus 0. These notes are in four parts. The first two parts provide a detailed exposition of this connection (following Levin-Racinet); the third is a leisurely exploration of the connection in which, for example, we compute the limit mixed Hodge structure on the unipotent fundamental group of the Tate curve minus its identity. In the fourth part we elaborate on ideas of Levin and Racinet and explicitly compute the connection over the moduli space of elliptic curves with a non-zero abelian differential, showing that it is defined over Q.

Equation on line 25 of page 39 corrected; 60 pages

References in corpus (3)

• Towards multiple elliptic polylogarithms
• Lectures on Moduli Spaces of Elliptic Curves
• Relative Weight Filtrations on Completions of Mapping Class Groups

Cited by in corpus (22)

• Elliptic multiple zeta values and one-loop superstring amplitudes
• Multiple Modular Values and the relative completion of the fundamental group of $M_{1,1}$
• Elliptic modular graph forms I: Identities and generating series
• Zeta elements in depth 3 and the fundamental Lie algebra of a punctured elliptic curve
• Elliptic Quantum Curves of 6d SO(N) theories
• Algebraic de Rham theory for weakly holomorphic modular forms of level one
• Higher-genus Fay-like identities from meromorphic generating functions
• Two dialects for KZB equations: generating one-loop open-string integrals
• On the universal ellipsitomic KZB connection
• The elliptic KZB connection and algebraic de Rham theory for unipotent fundamental groups of elliptic curves
• Multiple zeta values and iterated Eisenstein integrals
• Deligne-Beilinson Cohomology of Affine Groups
• Moduli of framed formal curves
• Algebraic de Rham theory for relative completion of ${\mathrm{SL}}_2(\mathbb{Z})$
• Towards algebraic iterated integrals on elliptic curves via the universal vectorial extension
• Periods of generalized Tate curves
• Elliptic KZB connections via universal vector extensions
• Notes on mixed Teichmüller motives
• The meta-abelian elliptic KZB associator and periods of Eisenstein series
• Iterated integrals and unipotent periods on families of marked elliptic curves
• 1-minimal models for $C_{\infty }$-algebras and flat connections
• Homotopy transfer theorem and KZB connections