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Stokes Phenomenon and Yangians

Qian Tang, Xiaomeng Xu

TL;DR

This work establishes a deep link between the Stokes phenomenon for a quantum confluent hypergeometric system and Yangian algebras. By analyzing formal solutions, applying Borel–Laplace resummation, and employing a path algebroid framework, the authors show that Stokes matrices can be realized as infinite products of representations of the Yangian $Y_{\hbar}(\mathfrak{gl}_{\nu-1})$, with explicit Olshanski-type deformations $O(u)_k$. They further connect these structures to quantum groups via RLL relations, a morphism from the path algebroid to $U_q(\mathfrak{gl}_\nu)$, and braid-group actions on Stokes data, providing a unifying algebraic picture of irregular singularities and monodromy. The paper also outlines extensions to classical Lie algebras and twisted Yangians, and proposes a rich interplay between differential/difference systems, resurgence, and quantum algebras with potential implications for isomonodromy and quantum integrable systems.

Abstract

In this paper, we first establish a connection between Yangians and the unique formal solution of the quantum hypergeometric differential equations at irregular singularities. We then realize the Stokes matrices of the hypergeometric equations as infinite matrix products of representations of Yangains, with the help of the theory of difference systems. Along the way, we also investigate the algebroid structure associated with the Stokes matrices.

Stokes Phenomenon and Yangians

TL;DR

This work establishes a deep link between the Stokes phenomenon for a quantum confluent hypergeometric system and Yangian algebras. By analyzing formal solutions, applying Borel–Laplace resummation, and employing a path algebroid framework, the authors show that Stokes matrices can be realized as infinite products of representations of the Yangian , with explicit Olshanski-type deformations . They further connect these structures to quantum groups via RLL relations, a morphism from the path algebroid to , and braid-group actions on Stokes data, providing a unifying algebraic picture of irregular singularities and monodromy. The paper also outlines extensions to classical Lie algebras and twisted Yangians, and proposes a rich interplay between differential/difference systems, resurgence, and quantum algebras with potential implications for isomonodromy and quantum integrable systems.

Abstract

In this paper, we first establish a connection between Yangians and the unique formal solution of the quantum hypergeometric differential equations at irregular singularities. We then realize the Stokes matrices of the hypergeometric equations as infinite matrix products of representations of Yangains, with the help of the theory of difference systems. Along the way, we also investigate the algebroid structure associated with the Stokes matrices.

Paper Structure

This paper contains 19 sections, 31 theorems, 141 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Formal Solutions and Yangian Y(gl(ν-1))
  3. Stokes matrices as Infinite Product of Representations of Yangian
  4. Outlook: Classical Lie Algebra Cases and Twisted Yangians
  5. Monodromy Data of the Confluent Hypergeometric System
  6. Notations and Basic Properties
  7. Borel-Laplace Transform
  8. Darboux's Method and Path Class with Arguments
  9. The Associated Difference Systems
  10. Infinite Matrix Product as Solutions of Difference Systems
  11. Bilateral Infinite Matrix Product and Connection Matrices of Difference Systems
  12. Path Algebroid
  13. Path Algebroid and its Representation
  14. Stokes Matrices and Quantum Groups
  15. A Morphism from the Algebroid S to the Quantum Group U
  16. ...and 4 more sections

Key Result

Theorem 1.2

For $\hbar\notin \mathbb{Q}$, the system introqeq has a unique formal solution where $I$ is the identity matrix, and the coefficients $H_p\in\operatorname{Mat}_{n} (U(\mathfrak{gl}_\nu{}))$ are recursively determined by the relations Here for any $\lambda\in \mathbb{C}\setminus\{0\}$, $T_k(\lambda)$ is an $(\nu-1)\times(\nu-1)$-matrix with entries Furthermore, $T_k(\lambda)$ satisfies the Yangi

Figures (3)

  • Figure 1: Examples of $\gamma_{st}^{(d)}$ defined in Proposition \ref{['Pro:gammajk']} for $\nu=3$
  • Figure 2: The composition \ref{['Composition']} for $m\neq\infty$ and $m=\infty$
  • Figure 3: Example of the action of the element $\sigma_1$ in the braid group $B_4$

Theorems & Definitions (47)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: Formal solutions at irregular singularity
  • Definition 2.5
  • ...and 37 more