椭圆结合子

The Fay relations satisfied by the elliptic associator

Let $A_τ$ 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 $F_τ$. We study a family of {\it Fay relations} satisfied by $A_τ$, derived from the original Fay relation satisfied by the $F_τ$. The Fay relations of $A_τ$ 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 $\bar{A}_τ$ mod $2πi$. We recall a different constructio…

椭球体结合子

Ellipsitomic Associators

本文通过operad理论提出椭球体结合子概念，利用通用椭球体KZB联络的单值性证明其存在性，并将其与同余子群的Eisenstein级数及扭曲椭圆多重zeta值建立联系。

Anatomy of an associator

We study some Lie algebras defined by solutions to the double shuffle equations with poles and construct families of explicit solutions to these equations in all weights and depths. These provide universal coordinates in which to write down `zeta elements': the images of generators of the Lie algebra of the motivic Galois group of mixed Tate motives over the integers. We expect that a similar statement holds for associators. In particular, these coordinates encode algebraic relations between multiple zeta values, and enable one to compress the currently used tables for relations between multip…

Stable cohomology of graph complexes

We study three graph complexes related to the higher genus Grothendieck-Teichmüller Lie algebra and diffeomorphism groups of manifolds. We show how the cohomology of these graph complexes is related, and we compute the cohomology as the genus $g$ tends to $\infty$. As a byproduct, we find that the Malcev completion of the genus $g$ mapping class group relative to the symplectic group is Koszul in the stable limit (partially answering a question of Hain). Moreover, we obtain that any elliptic associator gives a solution to the elliptic Kashiwara-Vergne problem.

通用 KZB 方程 I：椭圆情形

Universal KZB equations I: the elliptic case

本文定义亏格 1 上的通用 KZB 联络，分析其单值性及在单李代数与 sl_n/Cherednik 代数中的实现，并建立相关函子与不可约等变 D-模特征标计算。

规范化ζ生成元：零亏格与一亏格情形

Canonicalizing zeta generators: genus zero and genus one

本文针对零亏格与一亏格的ζ生成元开展规范化研究，通过多重ζ值对偶空间确定规范多项式，构造对应同构，解决一亏格生成元非几何部分的定义歧义，建立了两类亏格下ζ生成元的关联，连通了两类数学对象的构造。