Shuffle relations for Hodge and motivic correlators

Abstract

The Hodge correlators ${\rm Cor}_{\mathcal H}(z_0,z_1,\dots,z_n)$ are functions of several complex variables, defined by Goncharov (arXiv:0803.0297) by an explicit integral formula. They satisfy some linear relations: dihedral symmetry relations, distribution relations, and shuffle relations. We found new second shuffle relations. When $z_i\in0\cup\mu_N$, where $\mu_N$ are the $N$-th roots of unity, they are expected to give almost all relations. When $z_i$ run through a finite subset $S$ of $\mathbb C$, the Hodge correlators describe the real mixed Hodge-Tate structure on the pronilpotent completion of the fundamental group $\pi_1^{\rm nil}(\mathbb{CP}^1-(S\cup\infty),v_\infty)$, a Lie algebra in the category of mixed $\mathbb Q$-Hodge-Tate structures. The Hodge correlators are lifted to canonical elements ${\rm Cor_{Hod}}(z_0,\dots,z_n)$ in the Tannakian Lie coalgebra of this category. We prove that these elements satisfy the second shuffle relations. Let $S\subset\overline{\mathbb Q}$. The pronilpotent fundamental group is the Betti realization of the motivic fundamental group, a Lie algebra in the category of mixed Tate motives over $\overline{\mathbb Q}$. The Hodge correlators are lifted to elements ${\rm Cor_{Mot}}(z_0,\dots,z_n)$ in its Tannakian Lie coalgebra $\rm Lie_{MT}^\vee$. We prove the second shuffle relations for these motivic elements. The universal enveloping algebra of $\rm Lie_{MT}^\vee$ was described by Goncharov via motivic multiple polylogarithms, which obey a similar yet different set of double shuffle relations. Motivic correlators have several advantages: they obey dihedral symmetry relations at all points, not only at roots of unity; they are defined for any curve, and the double shuffle relations admit a generalization to elliptic curves; and they describe elements of the motivic Lie coalgebra rather than its universal enveloping algebra.