Elliptic KZB connections via universal vector extensions

Using the formalism of bar complexes and their relative versions, we give a new, purely algebraic, construction of the so-called universal elliptic KZB connection in arbitrary level. We compute explicit analytic formulae, and we compare our results with previous approaches to elliptic KZB equations and multiple elliptic polylogarithms in the literature. Our approach is based on a number of results concerning logarithmic differential forms on universal vector extensions of elliptic curves. Let $S$ be a scheme of characteristic zero, $E\to S$ be an elliptic curve, $f:E^{\mathrm{♮}}\to S$ be its universal vector extension, and $\pi :E^{\mathrm{♮}}\to E$ be the natural projection. Given a finite subset of torsion sections $Z\subset E(S)$, we study the dg-algebra over ${\mathcal{O}}_S$ of relative logarithmic differentials $\mathcal{A}=f_{\ast }{\mathrm{\Omega }}_{E^{\mathrm{♮}}\mathrm{/}S}^{\bullet }(\log {\pi }^{-1}Z)$. In particular, we prove that the residue exact sequence in degree one splits canonically, and we derive the formality of $\mathcal{A}$. When $S$ is smooth over a field $k$ of characteristic zero, we also prove that sections of ${\mathcal{A}}^1$ admit canonical lifts to absolute logarithmic differentials in $f_{\ast }{\mathrm{\Omega }}_{E^{\mathrm{♮}}\mathrm{/}k}^1(\log {\pi }^{-1}Z)$, which extends a well known property for regular differentials given by the `crystalline nature’ of universal vector extensions.