Shift operators of the Dotsenko-Fateev equation and its higher order versions

TL;DR

This work develops a comprehensive shift-operator framework for the Dotsenko–Fateev equation and its higher-order relatives, tying the codimension-2 specializations $S\!E_3$, $S\!E_4$, $S\!E_5$, and $S\!E_6$ to families $E_3$, $E_4$, $E_5$, and $E_6$ via addition and middle convolution. It provides explicit shift operators, S-values, and reducibility criteria, often deriving operators directly rather than solely via Osh’s recipe, and shows how reducibility manifests through factorizations into lower-order hypergeometric-type equations or ST$_4$-type components. A key theme is the interplay between integral representations (middle convolution), spectral data (Riemann schemes), and monodromy-invariant subspaces that govern when shifts yield reducible systems. The results deepen understanding of the structure of rigid and non-rigid Fuchsian systems in this lineage and offer computational recipes for constructing shift operators and identifying reducible cases with potential applications to related integrable models. Overall, the paper advances the theory of shift operators and reducibility in a broad hierarchy of connected Fuchsian equations, linking Dotsenko–Fateev to a spectrum of higher-order analogues.

Abstract

We find shift operators for the Dotsenko-Fateev equation, which is a differential equation of order 3, and for the three Fuchsian differential equations of order 4, 5 and 6, respectively, which are connected with the Dotsenko-Fateev equation via addition and middle convolution. These shift operators are used to study reducible cases.

Theorems & Definitions (76)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• Proposition 2.6
• Proposition 2.7
• Remark 2.8
• Definition 2.9
• Proposition 2.10