Knizhnik-Zamolodchikov equations in Deligne categories

TL;DR

This work extends Knizhnik-Zamolodchikov (KZ) theory to Deligne categories $\underline{Rep}(GL_t)$ and $\underline{Rep}(O_t)$, focusing on dualities with $(\mathfrak{gl}_m,\mathfrak{gl}_n)$ and $(\mathfrak{so}_m,\mathfrak{so}_{2n})$. It establishes a duality that maps KZ equations to a corresponding dynamical differential system on $\mathfrak{gl}_{m+n}$, and provides explicit integral formulas for the dynamical solutions together with a Bethe-ansatz interpretation. For the orthogonal case, it describes the KZ-type connection in $\underline{Rep}(O_t)$ and discusses reductions to a dual dynamical structure. A generalized Drinfeld-Kohno theorem is proved in the Deligne/skein category setting, relating KZ monodromy to braiding data in the quantum/interpolated framework. Overall, the paper builds a coherent bridge between KZ theory, dynamical systems, and braided/skein quantizations in interpolated representation categories, with implications for Gaudin models and monodromy computations in nonclassical settings.

Abstract

We consider the Knizhnik-Zamolodchikov equations in Deligne Categories in the context of $(\mathfrak{gl}_m,\mathfrak{gl}_{n})$ and $(\mathfrak{so}_m,\mathfrak{so}_{2n})$ dualities. We derive integral formulas for the solutions in the first case and compute monodromy in both cases.

Theorems & Definitions (30)

• Example 3.1.1: EV
• Lemma 3.2.1
• proof
• Theorem 3.2.2
• proof
• Lemma 3.2.3
• proof
• Theorem 3.2.4
• proof
• Theorem 4.1.1: FMTVSV