Vertex operator algebras and the zeta function

λ-环中的多重ζ值

Multiple zeta values in $λ$-rings

本文提出λ-环中多重ζ值的统一框架，证明其代数同态性质与多重对数公式，应用中明确正则多重q-ζ值的代数、计算庞加莱级数并给出生成集。

多重Eisenstein级数与多重q-zeta值的李代数

Lie algebras for multiple Eisenstein series and multiple q-zeta values

本文证明了多重Eisenstein级数和多重q-zeta值的两个类似李代数猜想，并建立了它们之间的同构关系。

Selberg Integrals, Multiple Zeta Values and Feynman Diagrams

We prove that there is an isomorphism between the Hopf Algebra of Feynman diagrams and the Hopf algebra corresponding to the Homogenous Multiple Zeta Value ring H in C<<X,Y>> . In other words, Feynman diagrams evaluate to Multiple Zeta Values in all cases. This proves a recent conjecture of Connes-Kreimer, and others including Broadhurst and Kontsevich. The key step of our theorem is to present the Selberg integral as discussed in Terasoma [22] as a Functional from the Rooted Trees Operad to the Hopf algebra of Multiple Zeta Values. This is a new construction which provides illumination to the…

一个代数框架内多重zeta值变形生成级数的恒等式

An identity for generating series of deformations of multiple zeta values within an algebraic framework

本文在代数框架内推广Bachmann恒等式，证明多重zeta值变形生成级数的一般恒等式，并得到含Lewis-Zagier周期类似函数的新公式。

Relations of multiple zeta values and their algebraic expression

We establish a new class of relations among the multiple zeta values ζ(k_1,k_2,...,k_n), which we call the cyclic sum identities. These identities have an elementary proof, and imply the "sum theorem" for multiple zeta values. They also have a succinct statement in terms of "cyclic derivations" as introduced by Rota, Sagan and Stein. In addition, we discuss the expression of other relations of multiple zeta values via the shuffle and "harmonic" products on the underlying vector space H of the noncommutative polynomial ring Q<x,y>, and also using an action on Q<x,y> of the Hopf algebra of quasi…

Formal weights in Kontsevich's formality construction and multiple zeta values

We construct a functor that associates to any dg cooperad of dg commutative algebras (satisfying some conditions) an augmented commutative algebra. When applied to the cohomology operad of Francis Brown's moduli spaces it produces an algebra that formally models the algebra of multiple zeta values. We prove that there is an injection from the graded dual of the Grothendieck-Teichmueller Lie algebra into the indecomposables of the algebra associated to the Gerstenhaber cooperad, and that there is a morphism from the algebra associated to Brown's moduli spaces to the algebra associated to the Ge…

与多重 Eisenstein 级数和多重 q-zeta 值相关的形式多重 zeta 值的一种推广

A generalization of formal multiple zeta values related to multiple Eisenstein series and multiple q-zeta values

本文构造 $τ$-不变平衡拟洗牌代数 $\mathcal{G}^{\operatorname{f}}$ 以统一形式化多重 Eisenstein 级数与多重 q-zeta 值，并通过仿射概形 BM 推广 Racinet 的 DM，建立向形式多重 zeta 值的投影。

Quasi-Symmetric Functions, Multiple Zeta Values, and Rooted Trees

We review the relation between the Hopf algebra QSym of quasi-symmetric functions and the multiple zeta values, and then discuss a commutative diagram involving the Hopf algebra Sym of symmetric functions, the Hopf algebra dual NSym of QSym, and the Hopf algebras of rooted trees and planar rooted trees as defined by Kreimer and Foissy respectively.

Analysis of Zeta Functions, Multiple Zeta Values, and Related Integrals

In this work, we begin to uncover the architecture of the general family of zeta functions and multiple zeta values as they appear in the theory of integrable systems and conformal field theory. One of the key steps in this process is to recognize the roles that zeta functions play in various arenas using transform methods. Other logical connections are provided by the the appearance of the Drinfeld associator, Hopf algebras, and techniques of conformal field theory and braid groups. These recurring themes are subtly linked in a vast scheme of a logically woven tapestry. An immediate applicati…

A universal A-infinity structure on Batalin-Vilkovisky algebras with multiple zeta value coefficients

We explicitly construct a universal A-infinity deformation of Batalin-Vilkovisky algebras, with all coefficients expressed as rational sums of multiple zeta values. If the Batalin-Vilkovisky algebra that we start with is cyclic, then so is the A-infinity deformation. Moreover, the adjoint action of the odd Poisson bracket acts by derivations of the A-infinity structure. The construction conjecturally defines a new presentation of the Grothendieck-Teichmueller Lie algebra.

Quasi-shuffle products revisited

Quasi-shuffle products, introduced by the first author, have been useful in studying multiple zeta values and some of their analogues and generalizations. The second author, together with Kajikawa, Ohno, and Okuda, significantly extended the definition of quasi-shuffle algebras so it could be applied to multiple zeta q-values. This article extends some of the algebraic machinery of the first author's original paper to the more general definition, and uses this extension to obtain various algebraic formulas in the quasi-shuffle algebra in a transparent way. Some applications to multiple zeta va…

On relations among multiple zeta values obtained in knot theory

This paper focuses linear and algebraic relations among multiple zeta values which were obtained in knot theory. It is shown that they can be derived from the associator relations, i.e. the pentagon equation and the shuffle relation.

Renormalised conical zeta values

Conical zeta values associated with rational convex polyhedral cones generalise multiple zeta values. We renormalise conical zeta values at poles by means of a generalisation of Connes and Kreimer's Algebraic Birkhoff Factorisation. This paper serves as a motivation for and an application of this generalised renormalisation scheme. The latter also yields an Euler-Maclaurin formula on rational convex polyhedral lattice cones which relates exponential sums to exponential integrals. When restricted to Chen cones, it reduces to Connes and Kreimer's Algebraic Birkhoff Decomposition for maps with va…

The Algebra of a q-Analogue of Multiple Harmonic Series

We introduce an algebra which describes the multiplication structure of a family of q-series containing a q-analogue of multiple zeta values. The double shuffle relations are formulated in our framework. They contain a q-analogue of Hoffman's identity for multiple zeta values. We also discuss the dimension of the space spanned by the linear relations realized in our algebra.

Supercongruences of multiple harmonic $q$-sums and generalized finite/symmetric multiple zeta values

The Kaneko--Zagier conjecture describes a correspondence between finite multiple zeta values and symmetric multiple zeta values. Its refined version has been established by Jarossay, Rosen and Ono--Seki--Yamamoto. In this paper, we explicate these conjectures through studies of multiple harmonic $q$-sums. We show that the (generalized) finite/symmetric multiple zeta value are obtained by taking an algebraic/analytic limit of multiple harmonic $q$-sums. As applications, new proofs of reversal, duality and cyclic sum formulas for the generalized finite/symmetric multiple zeta values are given.

Hurwitz 空间、Nichols 代数与 Igusa zeta 函数

Hurwitz spaces, Nichols algebras, and Igusa zeta functions

本文构造次数 4、5 的 Hurwitz 空间拟映射紧化，建立算术统计、Nichols 代数与 Igusa zeta 函数的精确联系，并回答相关计数与拓扑起源问题。

Local Index Formulae on Noncommutative Orbifolds and Equivariant Zeta Functions for the Affine Metaplectic Group

We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations. The algebra $A$ includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple $(A, H, D)$ with $H=L^2(\mathbb R^n, Λ(\mathbb R^n))$ and the Euler operator $D$, a first order differential operator of index $1$. We show that this spectral triple has simple dimension spectrum: For every operator $B$ in the algebra $Ψ(A,H,D)$ generated by the Shubin type pseudodifferential operators and the elements of $A$, the zeta f…

Vertex Lie algebras, vertex Poisson algebras and vertex algebras

The notions of vertex Lie algebra and vertex Poisson algebra are presented and connections among vertex Lie algebras, vertex Poisson algebras and vertex algebras are discussed.