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

TL;DR

This paper investigates the number of independent solutions of KZ equations modulo a prime p, showing that their count matches the dimension of a space of square integrable differential forms, thus linking algebraic solutions to geometric structures.

Contribution

It determines the rank of the module of p-hypergeometric solutions for specific KZ equations and establishes its equality with a geometric space's dimension.

Findings

The rank of the p-hypergeometric solutions module is explicitly determined.

The rank equals the dimension of a space of square integrable differential forms.

The results connect algebraic solutions of KZ equations to geometric structures.

Abstract

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.