Finding all solutions to the KZ equations in characteristic $p$

Abstract: The KZ equations are differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the $n$-point functions of affine primary fields. In [SV1] the KZ equations were identified with equations for flat sections of suitable Gauss-Manin connections, and solutions of the KZ equations were constructed in the form of multidimensional hypergeometric integrals. In [SV2] the KZ equations were considered modulo a prime number $p$, and, for rational levels, polynomial solutions of the KZ equations modulo $p$ were constructed by an elementary procedure as suitable $p$-approximations of the hypergeometric integrals. In this paper we study in detail the first nontrivial example of the KZ equations in characteristic $p$. In particular, if the level is irrational, we prove a version of the steepest descent result that relates the KZ local system to the space of functions on the critical locus of the master function. We use this result to prove the generic irreducibility of the KZ local system at any irrational level. If the level is rational, we describe all solutions of the KZ equations in characteristic $p$ by demonstrating that they all stem from the $p$-hypergeometric solutions. Finally, we prove a Lagrangian property of the subbundle of the KZ bundle spanned by the $p$-hypergeometric sections.