Finding all solutions to the KZ equations in characteristic $p$
Alexander Varchenko, Vadim Vologodsky
TL;DR
This paper analyzes the Knizhnik–Zamolodchikov equations modulo a prime $p$ by developing a geometric and cohomological framework: reinterpretation of KZ as a Gauss–Manin connection for the master-function local system $\mathcal{P}^h$ on $\mathbb{P}^1_S$, and a study of the resulting de Rham cohomology $E$ in both generic and special rational levels. For irrational levels $h\notin\mathbb{F}_p$, the KZ connection is shown to be irreducible with no formal solutions, via a steepest-descent–type analysis linked to the critical locus of the master function. When $h\in\mathbb{F}_p$, all solutions are $p$-hypergeometric and there exists a flat subbundle $\mathcal{U}$ of rank $[n\tilde h/p]$, with a dual, orthogonal quotient, providing a Lagrangian structure and a complete basis of flat sections in terms of $p$-hypergeometric data. The results are underpinned by a detailed study of the $p$-curvature (Katz theory), a construction of a family of curves $X$ with a $\mu_q$-action whose isotypic components reproduce the $\mathcal{P}^h$-picture, and explicit Cartie–Ogus–Katz computations linking to the master function’s geometry. Overall, the work gives a sharp, representation-theoretic and algebro-geometric classification of KZ solutions in characteristic $p$, and exposes deep connections between Gauss–Manin theory, $p$-curvature, and hypergeometric-type structures in positive characteristic.
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.