arXiv 2201.118202022-01-27math-phmath.AGmath.MPmath.NT

关于KZ方程的$p$ -超几何解的个数

On the number of $p$-hypergeometric solutions of KZ equations

Alexander Varchenko

备注：: Latex, 19 pages

摘要：

已知KZ方程的解可以写成多维超几何积分的形式。2017年，在作者与V. Schechtman的联合论文中，修改了超几何解的构造，并构造了KZ方程模素数$p$的解。这些解模$p$，称为$p$ -超几何解，是具有整数系数的多项式。一个普遍的问题是确定独立的$p$ -超几何解的个数并理解这个个数的意义。在本文中，我们考虑的KZ方程与空间的奇异向量的重量$n-2r$在张量功率$W^{\otimes n}$的向量表示$\frak{sl}_2$。在这种情况下，KZ方程的超几何解由$r$维超几何积分给出。我们考虑相应的$p$ -超几何解的模，确定其秩，并证明其秩等于合适的平方可积微分$r$ -形式空间的维数.

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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