Selberg integral and multiple zeta values

A Selberg Integral Type Formula for an sl_2 One-Dimensional Space of Conformal Blocks

For distinct complex numbers $z_1,...,z_{2N}$, we give a polynomial $P(y_1,...,y_{2N})$ in the variables $y_1,...,y_{2N}$, which is homogeneous of degree $N$, linear with respect to each variable, $sl_2$-invariant with respect to a natural $sl_2$-action, and is of order $N-1$ at $(y_1,...,y_{2N})=(z_1,...,z_{2N})$. We give also a Selberg integral type formula for the associated one-dimensional space of conformal blocks.

The $\mathbb F_p$-Selberg Integral

We prove an $\mathbb F_p$-Selberg integral formula, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$.

Dimensional interpolation and the Selberg integral

We show that a version of dimensional interpolation for the Riemann--Roch--Hirzebruch formalism in the case of a grassmannian leads to an expression for the Euler characteristic of line bundles in terms of a Selberg integral. We propose a way to interpolate higher Bessel equations, their wedge powers, and monodromies thereof to non--integer orders, and link the result with the dimensional interpolation of the RRH formalism in the spirit of the gamma conjectures.

Regularizable cycles associated with a Selberg type integral under some resonance condition

We study the twisted homology group attached to a Selberg type integral under some resonance condition, which naturally appears in the su_2-conformal field theory and the representation of the Iwahori-Hecke algebra. We determine the dimension of the space of the regularizable cycles. The dimension-formula is given in terms of the generalized hypergeometric series.

The $\mathbb F_p$-Selberg integral of type $A_n$

We prove an $\mathbb F_p$-Selberg integral formula of type $A_n$, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$. For the type $A_1$ the formula was proved in a previous paper by the authors.

关于二维 $\mathbb F_p$-Selberg 积分的注记

Notes on $2D$ $\mathbb F_p$-Selberg integrals

本研究证明了依赖于多组正整数参数、取值于奇素数阶有限域的二维 $\mathbb F_p$-Selberg 积分公式，其灵感源于KZ方程的多维超几何解与模p约化后多项式解的相似性，丰富了有限域上积分公式的研究成果。