The Sum Formula of Multiple Zeta Values and Connection Problem of the Formal Knizhnik-Zamolodchikov Equation

Connection Problem of Knizhnik-Zamolodchikov Equation on Moduli Space ${\mathcal M}_{0,5}$

In this article, we consider the connection problem of the KZ (Knizhnik-Zamolodchikov) equation on the moduli space $\cM_{0,5}$, and show that the connection matrices are expressed in terms of the Drinfel'd associator. As the compatibility condition on the connection problem, we have the pentagon relation for the Drinfeld associators. As an application of the connection problem, we derive the five term relation for dilogarithms.

Emptiness Formation Probability and Quantum Knizhnik-Zamolodchikov Equation

We consider the one-dimensional XXX spin 1/2 Heisenberg antiferromagnet at zero temperature and zero magnetic field. We are interested in a probability of formation of a ferromagnetic string in the antiferromagnetic ground-state. We call it emptiness formation probability [EFP]. We suggest a new technique for computation of EFP in the inhomogeneous case. It is based on quantum Knizhnik-Zamolodchikov equation. We evalauted EFP for strings of the length six in the inhomogeneous case. The homogeneous limit confirms our hypothesis about the relation of quantum correlations to number theory. We als…

Fundamental solutions of the Knizhnik-Zamolodchikov equation of one variable and the Riemann-Hilbert problem

In this article, we derive multiple polylogarithms from multiple zeta values by using a recursive Riemann-Hilbert problem of additive type. Furthermore we show that this Riemann-Hilbert problem is regarded as an inverse problem for the connection problem of the KZ equation of one variable, so that the fundamental solutions to the equation are derived from the Drinfel'd associator by using a Riemann-Hilbert problem of multiplicative type. These results say that the duality relation for the Drinfel'd associator can be interpreted as the solvability condition for this inverse problem.

On an algebraic version of the Knizhnik-Zamolodchikov equation

A difference equation analogue of the Knizhnik-Zamolodchikov equation is exhibited by developing a theory of the generating function $H(z)$ of Hurwitz polyzeta functions to parallel that of the polylogarithms. By emulating the role of the KZ equation as a connection on a suitable bundle, a difference equation version of the notion of connection is developed for which $H(z)$ is a flat section. Solving a family of difference equations satisfied by the Hurwitz polyzetas leads to the normalized multiple Bernoulli polynomials (NMBPs) as the counterpart to the Hurwitz polyzeta functions, at tuples o…

多重zeta值与Drinfeld结合子级数的delta值之间的关系

Relations between multiple zeta values and delta values from Drinfeld's associator series

该研究通过比较Drinfeld结合子的两个级数公式，推导得出多重zeta值与delta值的新关系，并从五阶项中发现两个新关系。