The Fay relations satisfied by the elliptic associator

Let $A_\tau$ denote the elliptic associator constructed by Enriquez, a power series in two non-commutative variables $a,b$ defined as an iterated integral of the Kronecker function $F_\tau$. We study a family of {\it Fay relations} satisfied by $A_\tau$, derived from the original Fay relation satisfied by the $F_\tau$. The Fay relations of $A_\tau$ were studied by Broedel, Matthes and Schlotterer, and determined up to non-explicit correction terms that arise from the necessity of regularizing the non-convergent integral. Here we study a reduced version $\bar{A}_\tau$ mod $2\pi i$. We recall a different construction of $\bar{A}_\tau$ in three steps, due to Matthes, Lochak and the author: first one defines the reduced {\it elliptic generating series} $\bar{E}_\tau$ which comes from the reduced Drinfeld associator $\overline{\Phi}_{KZ}$ and whose coefficients generate the same ring $\bar{R}$ as those of $\bar{A}_\tau$; then one defines $\Psi$ to be the automorphism of the free associative ring $\bar{R}\langle\langle a,b\rangle\rangle$ defined by $\Psi(a)=\bar{E}_\tau$ and $\Psi([a,b])=[a,b]$; finally one shows that the reduced elliptic associator $\bar{A}_\tau$ is equal to $\Psi\bigl({{ad(b)}\over{e^{ad(b)}-1}}(a)\bigr)$. Using this construction and mould theory and working with Lie-like versions of the elliptic generating series and associator, we prove the following results: (1) a mould satisfies the Fay relations if and only if a closely related mould satisfies the "swap circ-neutrality" relations defining the elliptic Kashiwara-Vergne Lie algebra $krv_{ell}$, (2) the reduced elliptic generating series satisfies a family of Fay relations with extremely simple correction terms coming directly from those of the Drinfeld associator, and (3) the correction terms for the Fay relations satisfied by the reduced elliptic associator can be deduced explicitly from these.