Stokes Phenomenon and Yang–Baxter Equations

Abstract

We study the Stokes phenomenon of the generalized Knizhnik–Zamolodchikov (gKZ) equations, and prove that their Stokes matrices satisfy the Yang–Baxter equations. In particular, the monodromy of the gKZ equations defines a family of braid groups representations via the Stokes matrices.

Appendix

1.1 A. Canonical solutions and Stokes matrices

Let us consider the meromorphic linear system

$$\begin{aligned} \frac{dF}{dz}=(\lambda +\frac{A}{z})\cdot F, \end{aligned}$$

(12)

where $F(z)\in \mathbb {C}^m$, $\lambda =\mathrm{diag}(\lambda _1,\ldots ,\lambda _m)$ is a diagonal matrix, and $A\in \mathrm{gl}_m$ an arbitrary matrix. Thus $z=\infty$ is an irregular singularity, and the equation in general has only a formal solution around $\infty$ taking the form

$$\begin{aligned} {\hat{F}}(z)={\hat{H}}(z)z^{[A]} e^{z \lambda }, \ \ \ { for} \ \ {\hat{H}}(z)=1+H_1z^{-1}+H_2z^{-2}+\cdot \cdot \cdot . \end{aligned}$$

(13)

Here [A] takes the projection of A to the centralizer of $\lambda$ in $\mathrm{gl}_m$. In particular, if $\lambda$ has distinct diagonal elements, [A] takes the diagonal part of A.

Such a formal power series is, in fact, asymptotic to canonical holomorphic solutions $F_i(z)$ (given by resummation methods, see e.g., [2], Chapter 4–6) in certain different sectors $\mathrm{Sect}_i$ in the complex plane. The discontinuity of asymptotic expansions is known as the Stokes phenomenon, and is measured by the so called Stokes matrices which connect the different solutions $F_i$ with the fixed asymptotic expansion ${\hat{F}}$ in the various sectors. We give more details in the following.

Definition 2.6

The anti-Stokes rays of the Eq. (12) are the directions along which $e^{(\lambda _i-\lambda _j)z}$ decays most rapidly as $z\mapsto \infty$ for some $i\ne j$. The Stokes sectors are the open regions of $\mathbb {C}$ bounded by two adjacent anti-Stokes rays.

On each Stokes sector $\mathrm{Sect}_i$ bounded by two Stokes rays $d_i$ and $d_{i+1}$, there is a canonical solution $F_i$ of (12) with prescribed asymptotics on the supersector $\widehat{\mathrm{Sect}_i}=(d_i-\frac{\pi }{2},d_{i+1}+\frac{\pi }{2})$. In particular, the following result can be found in e.g., [2] Chapter 8 or [3, 18].

Theorem 2.7

On $\mathrm{Sect}_i$, there is a unique (therefore canonical) holomorphic function $H_i:\mathrm{Sect}_i\rightarrow \mathrm{GL}_m$ such that the function

$$\begin{aligned} F_i=H_i e^{z\lambda } z^{[A]} \end{aligned}$$

satisfies Eq. (12), and $H_i$ can be analytically continued to $\widehat{\mathrm{Sect}_i}$ and then $H_i$ is asymptotic to ${\hat{H}}$ at $z\mapsto \infty$ within $\widehat{\mathrm{Sect}_i}$.

Suppose we are given a Stokes sector $\mathrm{Sect}_0$ (with a chosen branch of $\mathrm{log}(z)$ on it) and the opposite sector $\mathrm{Sect}_l$.

Definition 2.8

The Stokes matrices of the Eq. (12) (with respect to to $\mathrm{Sect}_0$) are the matrices $S_\pm$ determined by

$$\begin{aligned}F_l=F_{0}\cdot S_+, \ \ \ \ \ F_{0}=F_l\cdot e^{2\pi i [A]} S_- \end{aligned}$$

where the first (resp. second) identity is understood to hold in $\mathrm{Sect}_l$ (resp. $\mathrm{Sect}_0$) after $F_0$ (resp. $F_{l}$) has been analytically continued counterclockwise.

We remark that the Stokes matrices $S_\pm$ will in general depend on the irregular data $\lambda$ in (12). Such dependence was studied by many authors, see e.g., [5, 14].

1.2 B. Confluence of $\mathrm{KZ}$ equations

First note that the results in the paper can be generalized to the gKZ equations associated to any complex simple Lie algebra. Let $\mathfrak {g}$ be such a Lie algebra with a Cartan subalgebra $\mathfrak {h}$ and a root system $\Delta =\Delta _+\sqcup \Delta _-$. For $\alpha \in \Delta$, we choose generators $e_\alpha$ of the root subspaces $\mathfrak {g}_\alpha$ such that $(e_\alpha ,e_{-\alpha })=1$, and choose $\{x_i\}$ an orthogonal basis of $\mathbb {h}$. We set

$$\begin{aligned}{}[\Omega ]=\sum x_i\otimes x_i, \ \ \ \ \Omega ^+=\frac{1}{2}[\Omega ]+\sum _{\alpha \in \Delta _+}e_\alpha \otimes e_{-\alpha }, \ \ \ \Omega ^-=\frac{1}{2}[\Omega ]+\sum _{\alpha \in \Delta _+}e_{-\alpha } \otimes e_\alpha . \end{aligned}$$

Define the Casimir element and the trigonometric r-matrix (see e.g., [11], Section 3.8 or [10]) by

$$\begin{aligned} \Omega =\Omega ^++\Omega ^-, \ \ \ \ \ \ r(z)=\frac{\Omega ^+z+\Omega ^-}{z-1}. \end{aligned}$$

Then the trigonometric $\mathrm{KZ}_n$ equation associated to a $\mathfrak {g}$-module V and a regular element $u\in \mathfrak {h}_\mathrm{reg}$ is

$$\begin{aligned} \kappa z_i \frac{\partial F}{\partial z_i}=(t u^{(i)}+\sum _{j\ne i} r(z_i/z_j)^{ij})\cdot F, \ \ \ { for} \ i=1,\ldots ,n, \end{aligned}$$

where $F(z_1,\ldots ,z_n)$ is valued in $V^{\otimes n}$, $r^{ij}$ and $u^{(i)}$ denote r acting in the ith and jth factors of the tensor product and u acting in the ith factor, and t is a complex parameter.

In terms of $s_i=t (1+z_i)$, the above $\mathrm{KZ}_n$ equation becomes

$$\begin{aligned} \kappa \frac{\partial F}{\partial s_i}=\left( \frac{t u^{(i)}}{s_i-t}+\sum _{j\ne i} (\frac{\Omega _{ij}}{s_i-s_j}-\frac{\Omega _{ij}^-}{s_i-t})\right) \cdot F, \ \ \ { for} \ i=1,\ldots ,n. \end{aligned}$$

When $t\rightarrow \infty$, it approaches to the $\mathrm{gKZ}_n$ equation

$$\begin{aligned} \kappa \frac{\partial F}{\partial s_i}=(-u^{(i)}+\sum _{j\ne i}\frac{\Omega _{ij}}{s_i-s_j})\cdot F, \ \ \ { for} \ i=1,\ldots ,n. \end{aligned}$$

Therefore, at the level of equations, the $\mathrm{gKZ}_n$ (4) is a limit of the $\mathrm{KZ}_n$ (14). At the level of solutions, the limit is related to the fact that the Kummer’s function (or confluent hypergeometric function) is a limit of the hypergeometric function. Thus we expect that many theories related to the $\mathrm{KZ}$ equations should have a degeneration, which is coupled with irregular singularities and Stokes phenomenon.