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Finding All Solutions of qKZ Equations in Characteristic $p$

Evgeny Mukhin, Alexander Varchenko

TL;DR

This work studies the difference qKZ equations for ${\mathfrak{sl}_2}$ in characteristic $p$, focusing on the first nontrivial family of solutions in mod-$p$ and showing that all qKZ solutions in this setting arise from the $p$-hypergeometric family. It introduces the $p$-curvature framework, constructs a canonical orthogonal (Lagrangian) subbundle generated by $p$-hypergeometric sections, and analyzes duality, limits to the differential KZ system, and the role of Pochhammer-based master polynomials in building these solutions. The authors prove linear independence of the $p$-hypergeometric solutions, establish explicit leading-term asymptotics, and derive orthogonality relations that tightly constrain the solution space. They also characterize invariant subbundles and reduced $p$-curvature operators, providing a complete description of all qKZ solutions for $\kappa\in{\mathbb F_p^\times}$ and clarifying the nonexistence of flat solutions when $\kappa\notin{\mathbb F_p}$, thus extending differential KZ mod-$p$ results to the difference setting. Overall, the paper furnishes a thorough mod-$p$ account of qKZ solutions, linking hypergeometric, curvature, and duality structures to yield a robust classification and a bridge to the KZ differential theory.

Abstract

In [J. Lond. Math. Soc. 109 (2024), e12884, 22 pages, arXiv:2208.09721], the difference qKZ equations were considered modulo a prime number $p$ and a family of polynomial solutions of the qKZ equations modulo $p$ was constructed by an elementary procedure as suitable $p$-approximations of the hypergeometric integrals. In this paper, we study in detail the first family of nontrivial examples of the qKZ equations in characteristic $p$. We describe all solutions of these qKZ equations in characteristic $p$ by demonstrating that they all stem from the $p$-hypergeometric solutions. We also prove a Lagrangian property (called the orthogonality property) of the subbundle of the qKZ bundle spanned by the $p$-hypergeometric sections. This paper extends the results of [arXiv:2405.05159] on the differential KZ equations to the difference qKZ equations.

Finding All Solutions of qKZ Equations in Characteristic $p$

TL;DR

This work studies the difference qKZ equations for in characteristic , focusing on the first nontrivial family of solutions in mod- and showing that all qKZ solutions in this setting arise from the -hypergeometric family. It introduces the -curvature framework, constructs a canonical orthogonal (Lagrangian) subbundle generated by -hypergeometric sections, and analyzes duality, limits to the differential KZ system, and the role of Pochhammer-based master polynomials in building these solutions. The authors prove linear independence of the -hypergeometric solutions, establish explicit leading-term asymptotics, and derive orthogonality relations that tightly constrain the solution space. They also characterize invariant subbundles and reduced -curvature operators, providing a complete description of all qKZ solutions for and clarifying the nonexistence of flat solutions when , thus extending differential KZ mod- results to the difference setting. Overall, the paper furnishes a thorough mod- account of qKZ solutions, linking hypergeometric, curvature, and duality structures to yield a robust classification and a bridge to the KZ differential theory.

Abstract

In [J. Lond. Math. Soc. 109 (2024), e12884, 22 pages, arXiv:2208.09721], the difference qKZ equations were considered modulo a prime number and a family of polynomial solutions of the qKZ equations modulo was constructed by an elementary procedure as suitable -approximations of the hypergeometric integrals. In this paper, we study in detail the first family of nontrivial examples of the qKZ equations in characteristic . We describe all solutions of these qKZ equations in characteristic by demonstrating that they all stem from the -hypergeometric solutions. We also prove a Lagrangian property (called the orthogonality property) of the subbundle of the qKZ bundle spanned by the -hypergeometric sections. This paper extends the results of [arXiv:2405.05159] on the differential KZ equations to the difference qKZ equations.

Paper Structure

This paper contains 29 sections, 34 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Difference qKZ equations
  3. Notations
  4. qKZ equations
  5. $\boldsymbol{ p}$-curvature of the qKZ connection
  6. Differential KZ equations
  7. KZ equations as a limit of qKZ equations
  8. Limit of solutions
  9. Dual qKZ equations
  10. Pochhammer polynomials
  11. $\boldsymbol{ p}$-hypergeometric solutions for $\boldsymbol{\kappa\in{\mathbb F}_p^\times \subset {\mathbb K}}$
  12. Solutions in $\boldsymbol{\operatorname{Sing} L^{\otimes n}[n-2]}$
  13. Master polynomial and weight functions
  14. Step $\boldsymbol{-\kappa}$
  15. $\boldsymbol{ p}$-hypergeometric solutions of KZ equations
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $p$ be a prime number that does not divide $n$. For $\kappa\in{\mathbb K}\setminus {\mathbb F}_p$, there does not exist a nonzero rational $V$-valued function $f(z_1,\dots,z_n)$ which is a solution of the qKZ equations with parameter $\kappa$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 4.1
  • Example 4.2
  • ...and 31 more