Stokes phenomenon and reflection equations

Abstract

In this paper, we study the Stokes phenomenon of the generalized cyclotomic Knizhnik-Zamolodchikov equation, and prove that its two types of Stokes matrices satisfy the Yang-Baxter and reflection equations respectively. We then discuss its isomonodormy deformation, and its relations with cyclotomic associators, twists, and quantum symmetric pairs. In the end, we explain our main goal, i.e., to set up a framework to study the quantization of Frobenius manifolds.

Representations of quantum groups arising from the Stokes phenomenon and applications

“…The Stokes phenomenon of this equation was first studied by Toledano Laredo in [27]. Then motivated by [27], we introduce and study the quantum Stokes matrices in [28,31,33,35]. In particular, similar to the classical case, this equation has two canonical solutions F ± (z) with prescribed asymptotics at z = ∞ within Sect ± = {z ∈ C | ± Re(z) > 0}, see e.g., [33], and Definition 2.8.…”

“…The following theorem can be found in [28,33], and see [31,35] for a proof in the categorical setting. Theorem 2.9.…”

“…Besides, the (quantum) Stokes matrices are compatible with involutions on Lie algebras. See [5] and [35] for some particular classical and quantum cases respectively. In particular, we showed that the quantum Stokes matrices of cyclotomic KZ equations give rise to universal K-matrices for quantum symmetric pairs of type AI [35].…”

“…See [5] and [35] for some particular classical and quantum cases respectively. In particular, we showed that the quantum Stokes matrices of cyclotomic KZ equations give rise to universal K-matrices for quantum symmetric pairs of type AI [35]. Thus the Drinfeld isomorphism ν (u) in Theorem 2.12 restricts to an algebra isomorphism for the classical and quantum symmetric pairs of type AI.…”

“…It has Stokes matrices S ± (u, hT ) ∈ End(C n ) ⊗ End(W ). One can study the isomonodromy deformation of such a system with respect to u, while keeping the multiplicity of each eigenvalue u i , see [35] and see e.g., [7] for a more general setting. Similarly the closure of Stokes matrices in [34] can be generalized to this setting.…”