Solutions of the $sl_2$ qKZ equations modulo an integer

Abstract

We study the qKZ difference equations with values in the $n$-th tensor power of the vector $sl_2$ representation $V$, variables $z_1,\dots,z_n$ and integer step $\kappa$. For any integer $N$ relatively prime to the step $\kappa$, we construct a family of polynomials $f_r(z)$ in variables $z_1,\dots,z_n$ with values in $V^{\otimes n}$ such that the coordinates of these polynomials with respect to the standard basis of $V^{\otimes n}$ are polynomials with integer coefficients. We show that the polynomials $f_r(z)$ satisfy the qKZ equations modulo $N$. Polynomials $f_r(z)$ are modulo $N$ analogs of the hypergeometric solutions of the \qKZ/ equations given in the form of multidimensional Barnes integrals.