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Regularized limits of Stokes matrices, isomonodromy deformation and crystal basis

Xiaomeng Xu

Abstract

In the first part of the paper, we solve the boundary and monodromy problems for the isomonodromy equation of the $n\times n$ meromorphic linear system of ordinary differential equations with Poncaré rank $1$. In particular, we derive an explicit expression of the Stokes matrices of the linear system, via the boundary value of the solutions of the isomonodromy equation at a critical point. Motivated by this result, we then describe the regularized limits of Stokes matrices as the irregular data $u={\rm diag}(u_1,...,u_n)$ in the linear system degenerates, i.e., as some $u_i, u_j,...,u_k$ collapse. The prescription of the regularized limit is controlled by the geometry of the De Concini-Procesi wonderful compactification space. As applications, many analysis problems about higher rank Painlevé transcendents can be solved. In the second part of the paper, we show some important applications of the above analysis results in representation theory and Poisson geometry: we obtain the first transcendental realization of crystals in representations of $\frak{gl}_n$ via the Stokes phenomenon in the WKB approximation; we develop a wall-crossing formula that characterizes the discontinuous jump of the regularized limits of Stokes matrices as crossing walls in the compactification space, and interpret the known cactus group actions on crystals arising from representation theory as a wall-crossing phenomenon; and we find the first explicit linearization of the standard dual Poisson Lie group for $U(n)$.

Regularized limits of Stokes matrices, isomonodromy deformation and crystal basis

Abstract

In the first part of the paper, we solve the boundary and monodromy problems for the isomonodromy equation of the meromorphic linear system of ordinary differential equations with Poncaré rank . In particular, we derive an explicit expression of the Stokes matrices of the linear system, via the boundary value of the solutions of the isomonodromy equation at a critical point. Motivated by this result, we then describe the regularized limits of Stokes matrices as the irregular data in the linear system degenerates, i.e., as some collapse. The prescription of the regularized limit is controlled by the geometry of the De Concini-Procesi wonderful compactification space. As applications, many analysis problems about higher rank Painlevé transcendents can be solved. In the second part of the paper, we show some important applications of the above analysis results in representation theory and Poisson geometry: we obtain the first transcendental realization of crystals in representations of via the Stokes phenomenon in the WKB approximation; we develop a wall-crossing formula that characterizes the discontinuous jump of the regularized limits of Stokes matrices as crossing walls in the compactification space, and interpret the known cactus group actions on crystals arising from representation theory as a wall-crossing phenomenon; and we find the first explicit linearization of the standard dual Poisson Lie group for .

Paper Structure

This paper contains 62 sections, 72 theorems, 414 equations, 5 figures.

Table of Contents

  1. Introduction and main results
  2. Explicit expression of Stokes matrices via the boundary value
  3. Regularized limits of Stokes matrices at a caterpillar point
  4. Regularized limit of Stokes matrices in the De Concini-Procesi space and wall-crossing formula
  5. Application I: the first explicit Ginzburg-Weinstein diffeomorphism
  6. The quantum case: representations of quantum groups arising from the quantum Stokes matrices
  7. The WKB approximation in the Stokes matrices is the crystal limit in the quantum groups
  8. Leading asymptotics of quantum Stokes matrices in terms of Gelfand-Tsetlin basis
  9. Application II: a realization of $\frak{gl}_n$-crystals via the WKB approximation in Stokes phenomenon
  10. Application III: cactus group actions on $\frak{gl}_n$-crystals arising from the wall-crossing phenomenon
  11. Witham dynamics and Halacheva-Kamnitzer-Rybnikov-Weekes covers
  12. A continuous path connecting the canonical basis and the Gelfand-Tsetlin basis
  13. Extension to $\frak{gl}_n(\mathbb{C})$ valued solutions and higher rank Painlevé transcendents
  14. Stokes phenomenon and monodromy data
  15. Canonical solutions
  16. ...and 47 more sections

Key Result

Theorem 1.1

For any solution $\Phi(u)$ of the isomonodromy equation introisoeq on the connected component $U_{\rm id}:=\{u\in \mathfrak h_{\rm reg}(\mathbb{R})~|~u_1<\cdots <u_n\}$, there exists a unique constant $\Phi_0\in{\rm Herm}(n)$ such that as the real numbers $\frac{u_{k+1}-u_{k}}{u_{k}-u_{k-1}}\rightar where ${\rm Ad}(g)X=gXg^{-1}$ for any $g\in U(n)$ and $X\in {\rm Herm}(n)$, the product $\overright

Figures (5)

  • Figure 1: A planar binary rooted tree with $6$ leaves colored by $u_1,...,u_6$.
  • Figure 2: A planar rooted tree with coloring
  • Figure 3: A planar rooted tree with coloring that represents the caterpillar point $u_{\rm cat}$
  • Figure 4: The planar rooted tree for Example \ref{['ex3']}
  • Figure 5: A caterpillar point with a planar embedding given by $\tau_i$

Theorems & Definitions (150)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Proposition 1.8: Wall-crossing formula at $u_{\rm cat}$
  • Remark 1.9
  • Remark 1.10
  • ...and 140 more