A-cycle integrals at genus one satisfy linear differential equations whose Picard iterations yield all-order alpha-prime expansions in iterated Eisenstein integrals, with cusp values given by disk integrals.
The meta-abelian elliptic KZB associator and periods of Eisenstein series
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abstract
We compute the image of Enriquez' elliptic KZB associator in the (maximal) meta-abelian quotient of the fundamental Lie algebra of a once-punctured elliptic curve. Our main result is an explicit formula for this image in terms of Eichler integrals of Eisenstein series, and is analogous to Deligne's computation of the depth one quotient of the Drinfeld associator. We also show how to retrieve Zagier's extended period polynomials of Eisenstein series, as well as the values at zero of Beilinson--Levin's elliptic polylogarithms from the meta-abelian elliptic KZB associator.
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One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points
A-cycle integrals at genus one satisfy linear differential equations whose Picard iterations yield all-order alpha-prime expansions in iterated Eisenstein integrals, with cusp values given by disk integrals.