An invariant subbundle of the KZ connection mod $p$ and reducibility of ${\widehat{sl}}_2$ Verma modules mod $p$

Abstract

We consider the KZ differential equations over $\mathbb{C}$ in the case, when its multidimensional hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field ${\mathbb{F}}_p$. We study the space of polynomial solutions of these differential equations over ${\mathbb{F}}_p$, constructed in a previous work by V. Schechtman and the author. The module of these polynomial solutions defines an invariant subbundle of the associated KZ connection modulo $p$. We describe the algebraic equations for that subbundle and argue that the equations correspond to highest weight vectors of the associated ${\widehat{sl}}_2$ Verma modules over the field ${\mathbb{F}}_p$.

Latex, 14 pages, v2: a misprint in formula (4.11) is corrected

References in corpus (1)

• Solutions modulo $p$ of Gauss-Manin differential equations for multidimensional hypergeometric integrals and associated Bethe ansatz

Cited by in corpus (3)

• The ${\mathbb{F}}_p$-Selberg integral of type $A_n$
• Notes on solutions of KZ equations modulo $p^s$ and $p$-adic limit $s\to \infty$
• Determinant of ${\mathbb{F}}_p$-hypergeometric solutions under ample reduction