On the number of p-hypergeometric solutions of KZ equations

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 $$\mathfrak {sl}_2$$ . 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 is equal to the dimension of the space of suitable square integrable differential r-forms.