论文标题
霍奇和动机相关器的洗牌关系
Shuffle relations for Hodge and motivic correlators
论文作者
论文摘要
Hodge相关器$ {\ rm cor} _ {\ Mathcal H}（Z_0，Z_1，\ dots，Z_n）$是由Goncharov（Arxiv：0803.0297）定义的几个复杂变量的函数。他们满足了一些线性关系：二面体对称性关系，分布关系和洗牌关系。我们发现了新的第二次洗牌关系。当$ z_i \ in0 \cupμ_n$时，$μ_n$是$ n $ th的统一根源，预计它们几乎会提供所有关系。当$ z_i $通过有限的子集$ s的$ \ mathbb c $运行时，hodge相关器描述了基本组$π_1^{\ rm nil nil}的真正混合杂物结构，这混合$ \ Mathbb Q $ -HODGE-TATE结构的类别。 hodge相关器被提升为规范元素$ {\ rm cor_ {hod}}（z_0，\ dots，z_n）$在此类别的Tannakian Lie Calgebra中。我们证明这些元素满足了第二个洗牌关系。令$ s \ subset \ overline {\ mathbb q} $。 Pronilpotent基本组是动机基本组的Betti实现，这是$ \ overline {\ mathbb Q} $的混合泰特动机类别中的一个代数。将hodge相关器提升为元素$ {\ rm cor_ {mot}}（z_0，\ dots，z_n）$中的tannakian lie calgebra $ \ rm lim lie_ {mt}^\ vee $。我们证明了这些动机元素的第二种混乱关系。 Goncharov通过动机多个Polyrogarithms描述了$ \ rm lie_ {mt}^\ vee $的通用包围代数，该章节遵守了一套类似但不同的双重混合关系。动机相关因素具有几个优势：它们在各个时刻服从二面体对称关系，不仅是团结的根源；它们是针对任何曲线定义的，并且双层式关系承认对椭圆曲线的概括。他们描述了动机谎言煤炭的元素，而不是其普遍包围的代数。
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 $z_i\in0\cupμ_N$, where $μ_N$ 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 $π_1^{\rm nil}(\mathbb{CP}^1-(S\cup\infty),v_\infty)$, 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.
