Math @ Duke

Publications [#324840] of Richard Hain
search www.ams.org.Papers Published
 Hain, R, Notes on the Universal Elliptic KZB Equation,
Pure and Applied Mathematics Quarterly, vol. 12 no. 2
(July, 2016), International Press [arXiv:1309.0580], [1309.0580v3]
(last updated on 2019/04/19)
Abstract: The universal elliptic KZB equation is the integrable connection on the
provector bundle over M_{1,2} whose fiber over the point corresponding to the
elliptic curve E and a nonzero point x of E is the unipotent completion of
\pi_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 KZequation in genus 0. These notes are in four parts. The
first two parts provide a detailed exposition of this connection (following
LevinRacinet); 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 nonzero abelian differential,
showing that it is defined over Q.


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