论文标题
关于Kz方程的$ p $ - 流行地几个地面解决方案的数量
On the number of $p$-hypergeometric solutions of KZ equations
论文作者
论文摘要
众所周知，KZ方程的解可以以多维超测定积分的形式编写。 2017年，在作者与V. Schechtman的联合论文中，修改了超几何解决方案的构建，并构建了KZ方程式Modulo的解决方案。这些解决方案模拟$ p $，称为$ P $ - 缩小地几乎值解决方案，是具有整数系数的多项式。一个一般问题是确定独立的$ p $ - hypheremetric解决方案的数量，并了解该数字的含义。 在本文中，我们考虑了与张量$ w^{\ otimes n} $在$ \ frak {sl} _2 $的向量表示的张量$ w^{\ otimes n} $中相关的kz方程。在这种情况下，Kz方程的超几何解由$ r $二维超几何积分给出。我们考虑相应的$ p $ - hypheremetric解决方案的模块，确定其排名，并表明等级等于合适的平方可集成差异$ r $ r $ forms的空间的尺寸。
It is known that solutions of the KZ equations can be written in the form of multidimensional hypergeometric integrals. In 2017 in a joint paper of the author with V. Schechtman the construction of hypergeometric solutions was modified, and solutions of the KZ equations modulo a prime number $p$ were constructed. These solutions modulo $p$, called the $p$-hypergeometric solutions, are polynomials with integer coefficients. A general problem is to determine the number of independent $p$-hypergeometric solutions and understand the meaning of that number. In this paper we consider the KZ equations associated with the space of singular vectors of weight $n-2r$ in the tensor power $W^{\otimes n}$ of the vector representation of $\frak{sl}_2$. In this case, the hypergeometric solutions of the KZ equations are given by $r$-dimensional hypergeometric integrals. We consider the module of the corresponding $p$-hypergeometric solutions, determine its rank, and show that the rank equals the dimension of the space of suitable square integrable differential $r$-forms.
