# The tt*-Toda equations of $A_{n}$ type

Martin A. Guest, Alexander R. Its, and Chang-Shou Lin

###### Abstract.

In previous articles we have studied the $A_{n}$ tt*-Toda equations of Cecotti
and Vafa, giving details mainly for $n=3$. Here we give a proof of the
existence and uniqueness of global solutions for any $n$, and a new treatment
of their asymptotic data, monodromy data, and Stokes data.

###### 2000 Mathematics Subject Classification:

Primary 81T40; Secondary 53D45, 35Q15, 34M40

##  1\. The equations

The tt*-Toda equations of $A_{n}$ type are:

(1.1) $2(w_{i})_{{t\bar{t}}}=-e^{2(w_{i+1}-w_{i})}+e^{2(w_{i}-w_{i-1})},\
w_{i}:\mathbb{C}^{\ast}\to\mathbb{R},\ i\in\mathbb{Z}$

where the functions $w_{i}$ are subject to the conditions $w_{i}=w_{i+n+1}$
(periodicity), $w_{i}=w_{i}(|t|)$ (radial condition), $w_{i}+w_{n-i}=0$
(“anti-symmetry”). These equations are a version of the 2D periodic Toda
equations, and they first appeared as formula (7.4) in the article [3] of
Cecotti-Vafa. We refer to that article (and our previous articles listed in
the references) for background information and for some of the surpisingly
deep relations with physics and geometry. An important feature of (1.1) is
that it is the Toda system associated to a certain Lie algebra
$\mathfrak{s}\mathfrak{l}^{\Delta}_{n+1}\mathbb{R}$ (isomorphic to
$\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{R}$), which will be defined in the next
section.

Results on the globally smooth solutions were given, mainly in the case $n=3$,
in our previous work [12],[9],[10],[11]. Results for the case $n=1$ were given
by McCoy-Tracy-Wu (see [5]), and for the case $n=2$ by Kitaev [16], and our
method gives alternative proofs of these.

The methods of [12],[9],[10],[11] extend in principle to the case of general
$n$. Nevertheless it would be laborious to use exactly the same arguments. In
the current article we give a new proof of the existence and uniqueness of
global solutions. We also give a more systematic description of the monodromy
data and a more efficient application of the Riemann-Hilbert method. Many of
these improvements are due to our (implicit) use of the Lie-theoretic point of
view from [7],[8], although the proofs presented here can be understood
without knowledge of Lie theory.

The existence of a family of solutions parametrized by points of

$\mathcal{A}=\\{m\in\mathbb{R}^{n+1}\ |\
m_{i-1}-m_{i}\geq-1,m_{i}+m_{n-i}=0\\},$

can be established by p.d.e. methods (cf. our previous articles [12],[9], the
Higgs bundle method of Mochizuki in [19],[20], and the proof given here in
section 10). However p.d.e. methods provide only minimal information on the
properties of the solutions. By using the Riemann-Hilbert method we shall be
able to give more detailed information.

We consider first the solutions parametrized by points of the interior

$\mathring{\mathcal{A}}=\\{m\in\mathbb{R}^{n+1}\ |\
m_{i-1}-m_{i}>-1,m_{i}+m_{n-i}=0\\}$

of this region; we refer to these as “generic solutions”. For such solutions
we have:

(I) The asymptotics of $w_{0},\dots,w_{n}$ at $t=0$: for $0\leq i\leq n$,

$w_{i}(x)=-m_{i}\log x-\tfrac{1}{2}\log l_{i}+o(1),\quad x\to 0$

where

$l_{i}=\textstyle{(n+1)}^{m_{n-i}-m_{i}}\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+1}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+1}}{n+1}\right)}\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+2}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+2}}{n+1}\right)}\cdots\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+n}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+n}}{n+1}\right)},$

$m^{\prime}_{i}=m_{i}-i$, and $x=|t|$; for $i\in\mathbb{Z}$ we put
$m^{\prime}_{i}=m^{\prime}_{i+n+1}+n+1$. (For a subfamily of solutions this
formula was already known from [23].)

(II) The asymptotics of $w_{0},\dots,w_{n}$ at $t=\infty$: for
$j=1,2,\dots,d$,

$\displaystyle w_{0}(x)\sin j\tfrac{\pi}{n+1}$ $\displaystyle+w_{1}(x)\sin
3j\tfrac{\pi}{n+1}+\cdots+w_{d-1}(x)\sin(2d-1)j\tfrac{\pi}{n+1}$
$\displaystyle=-\tfrac{n+1}{8}\ s_{j}\ (\pi
L_{j}x)^{-\frac{1}{2}}e^{-2L_{j}x}+O(x^{-\frac{3}{2}}e^{-2L_{j}x}),\quad
x\to\infty$

where $L_{j}=2\sin\tfrac{j}{n+1}\pi$ and we write $n+1=2d$ or $n+1=2d+1$
according to whether $n+1$ is even or odd. The real number $s_{j}$ is the
$j$-th symmetric function of
$\omega^{m_{0}+\frac{n}{2}},\omega^{m_{1}-1+\frac{n}{2}},\dots,\omega^{m_{n}-n+\frac{n}{2}}$,
where $\omega=e^{{2\pi{\scriptscriptstyle\sqrt{-1}}\,}/{(n+1)}}$. Thus we have
$d$ linear equations which determine $w_{0},\dots,w_{d-1}$; the significance
of these linear equations will be explained later.

The explicit relation between the asymptotic data at $t=0$ (the numbers
$m_{i}$) and the asymptotic data at $t=\infty$ (the numbers $s_{i}$)
constitutes a solution of the “connection problem” for solutions of (1.1).
These results are proved in section 8 (Corollary 8.14).

(III) Explicit expressions for the monodromy data (Stokes matrices and
connection matrices) of the isomonodromic o.d.e. (3.1) associated to $w$. This
is a linear o.d.e. in a complex variable $\lambda$ whose coefficients depend
on $w$ (and hence on $t,\bar{t}$). It has poles of order $2$ at
$\lambda=0,\infty$.

The meaning of “isomonodromic” is that the monodromy data at these poles is
independent of $t,\bar{t}$. Thus, to each solution $w_{0},\dots,w_{n}$ of
(1.1), we can assign a collection of monodromy data. We shall show in section
8 that the monodromy data associated to $w_{0},\dots,w_{n}$ is equivalent to
the “Stokes numbers” $s_{i}$ which appear in the above formula for the
asymptotics at $t=\infty$.

In order to obtain these results on the global solutions, we make a thorough
study of a wider class of solutions, namely those defined on regions of the
form $0<|t|<\epsilon$. We refer to such solutions as “smooth near zero”. They
are constructed from a “subsidiary o.d.e.” which has poles of order $2,1$ at
$\lambda=0,\infty$. For these solutions, the Stokes matrices are given in
Theorem 6.12 and the connection matrices are given in Theorem 6.21, Corollary
6.22.

We show (Corollary 8.13) that the global solutions can be characterized by a
condition on the connection matrices of the subsidiary o.d.e., which can be
expressed most intrinsically as follows:

######  Corollary 1.1.

A solution $w_{0},\dots,w_{n}$ which is smooth near zero is a global solution
if and only if all Stokes matrices and connection matrices for the subsidiary
o.d.e. lie in the group $\textrm{SL}^{\Delta}_{n+1}\mathbb{R}$.

A more precise statement is given in Corollary 8.13. Our proof is indirect; it
relies on explicit computations of Stokes matrices and connection matrices.

We conclude with some comments on the “non-generic global solutions”, i.e.
those which correspond to the boundary points of the region $\mathcal{A}$. It
turns out that most of our methods apply equally well to this case. We have
restricted attention to the generic solutions in this article only because the
construction of solutions near zero is easier in this case. Our argument can
be summarized as follows. First we construct solutions which are smooth near
zero. Then we use the Riemann-Hilbert method to extend the range of solutions
towards infinity, and make contact with the global solutions. To complete the
argument we use the p.d.e. results of section 10.1. The results of section
10.2 are not actually needed at this point.

In fact it is possible to proceed in the other direction. Namely, we could
start with the p.d.e. results of section 10, which apply to all global
solutions, generic or non-generic. The Riemann-Hilbert method of section 7
applies without change, taking the monodromy data (Stokes matrices and
connection matrices) as an Ansatz, and it produces solutions which are smooth
near infinity. The results of section 10.2 are now essential, in order to
connect the Stokes data with the asymptotic data at zero.

Arguing in “reverse order” in this way, we see that the above statements (I),
(II), (III) hold also the non-generic case, except that (I) must be replaced
by the following weaker statement:

$\lim_{x\to 0}\frac{w_{i}(x)}{\log x}=-m_{i}.$

Thus this argument alone would not produce the second term in the asymptotics
of $w$ at zero. Nevertheless, it shows that the Stokes matrices and connection
matrices for the global solutions in the non-generic case are given by exactly
the same formulae as in the generic case.

To produce the second term in the non-generic case, the “forward argument”
(starting with solutions near zero) has to be modified, and then the second
term will be quite different from that in (I) above. In the case $n=3$, these
modifications can be found in [11].

Acknowledgements: The first author was partially supported by JSPS grant
18H03668. The second author was partially supported by NSF grant DMS:1955265,
by RSF grant No. 22-11-00070, and by a Visiting Wolfson Research Fellowship
from the Royal Society. The authors thank Nan-Kuo Ho for comments on a draft
version of the paper.

##  2\. The zero curvature formulation

In this section we review the zero curvature formulation of (1.1), and a
construction of local solutions near $t=0$.

###  2.1. The main connection form $\alpha$

Equation (1.1) is the zero curvature condition $d\alpha+\alpha\wedge\alpha=0$
for the following connection form:

######  Definition 2.1.

$\alpha=(w_{t}+\tfrac{1}{\lambda}W^{T})dt+(-w_{{\bar{t}}}+\lambda
W)d{\bar{t}},$ where

$w=\operatorname{diag}(w_{0},\dots,w_{n}),\ \
W=\left(\begin{array}[]{c|c|c|c}\vphantom{(w_{0})_{(w_{0})}^{(w_{0})}}&\\!e^{w_{1}\\!-\\!w_{0}}\\!&&\\\
\hline\cr&&\ \ddots&\\\
\hline\cr\vphantom{\ddots}&&&e^{w_{n}\\!-\\!w_{n\\!-\\!1}}\\!\\!\\!\\\
\hline\cr\vphantom{(w_{0})_{(w_{0})}^{(w_{0})}}\\!\\!e^{w_{0}\\!-\\!w_{n}}\\!\\!&&&\\!\end{array}\right).$

Here $W^{T}$ denotes the transpose of $W$, and $\lambda$ is a complex
parameter.

At this point we just take $w=w(t,\bar{t})$ to be smooth on some non-empty
open subset of $\mathbb{C}^{\ast}$. Thus, $\alpha$ is an
$\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C}$-valued connection form on that open
set.

It will sometimes be convenient to write
$\alpha=\alpha^{\prime}dt+\alpha^{\prime\prime}d{\bar{t}}$ to indicate the
$(1,0)$ and $(0,1)$ parts, and to write
$\alpha^{T}=A^{\prime}dt+A^{\prime\prime}d{\bar{t}}$ for the transpose. Using
this notation we can say that (1.1) is the compatibility condition for the
linear system

(2.1) $\displaystyle\begin{cases}\Psi_{t}&=A^{\prime}\Psi\\\
\Psi_{\bar{t}}&=A^{\prime\prime}\Psi\end{cases}$

where $\Psi$ takes values in $\textrm{SL}_{n+1}\mathbb{C}$.

We shall make use of the following automorphisms. For
$X\in\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C}$,
$V\in\textrm{SL}_{n+1}\mathbb{C}$, we define

$\displaystyle\tau(X)$ $\displaystyle=d_{n+1}^{-1}Xd_{n+1}$
$\displaystyle\tau(V)$ $\displaystyle=d_{n+1}^{-1}Vd_{n+1}$
$\displaystyle\sigma(X)$ $\displaystyle=-\Delta\,X^{T}\Delta$
$\displaystyle\sigma(V)$ $\displaystyle=\Delta\,V^{-T}\,\Delta$ $\displaystyle
c(X)$ $\displaystyle=\Delta\bar{X}\Delta$ $\displaystyle c(V)$
$\displaystyle=\Delta\bar{V}\Delta$

where $d_{n+1}=\operatorname{diag}(1,\omega,\dots,\omega^{n})$,
$\omega=e^{{2\pi{\scriptscriptstyle\sqrt{-1}}\,}/{(n+1)}}$, and $\Delta$ is
the anti-diagonal matrix with $1$ in positions $(i,n-i)$, $0\leq i\leq n$, and
$0$ elsewhere.

By inspection we see that the connection form $\alpha$ satisfies:

Cyclic symmetry:
$\tau(\alpha(\lambda))=\alpha(e^{{2\pi{\scriptscriptstyle\sqrt{-1}}\,}/{(n+1)}}\lambda)$

Anti-symmetry:  $\sigma(\alpha(\lambda))=\alpha(-\lambda)$

$c$-reality:
$c(\alpha^{\prime}(\lambda))=\alpha^{\prime\prime}(1/\bar{\lambda})$.

The $c$-reality property implies that

(2.2) $\alpha|_{|\lambda|=1}\ \text{is an
$\mathfrak{s}\mathfrak{l}^{\Delta}_{n+1}\mathbb{R}$-valued connection form,}$

where

$\mathfrak{s}\mathfrak{l}^{\Delta}_{n+1}\mathbb{R}=\\{X\in\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C}\
|\ \Delta\bar{X}\Delta=X\\}$

is the fixed point set of $c$. We note that
$\mathfrak{s}\mathfrak{l}^{\Delta}_{n+1}\mathbb{R}=P\,\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{R}\,P^{-1}\cong\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{R}$
and
$\textrm{SL}^{\Delta}_{n+1}\mathbb{R}=P\,\textrm{SL}_{n+1}\mathbb{R}\,P^{-1}\cong\textrm{SL}_{n+1}\mathbb{R}$,
where

$\displaystyle P=\tfrac{1-i}{2}\begin{pmatrix}1\hskip 2.27626pt&\hskip
2.27626pt\hskip 2.27626pt&\hskip 2.27626pt\hskip 2.27626pt&\hskip
2.27626pti\\\ \hskip 2.27626pt&\hskip 2.27626pt\hskip 2.27626pt\ddots\hskip
2.27626pt\hskip 2.27626pt&\hskip 2.27626pt\hskip 2.27626pt\iddots\hskip
2.27626pt\hskip 2.27626pt&\hskip 2.27626pt\\\ \hskip 2.27626pt&\hskip
2.27626pt\iddots\hskip 2.27626pt&\hskip 2.27626pt\ddots\hskip 2.27626pt&\hskip
2.27626pt\\\ i\hskip 2.27626pt&\hskip 2.27626pt\hskip 2.27626pt&\hskip
2.27626pt\hskip 2.27626pt&\hskip 2.27626pt1\end{pmatrix}.$

Here the diagonal entries are all $1$ and the anti-diagonal entries are all
$i={\scriptstyle\sqrt{-1}}\,$, except that the “middle” entry is $1+i$ if
$n+1$ is odd.

######  Remark 2.2.

In the language of the theory of integrable systems, the linear system (2.1)
forms the Lax pair for the nonlinear p.d.e. (1.1). This Lax pair was first
found by Mikhailov [18]. More precisely, in [18], a hyperbolic version of the
2D Toda equations was considered, i.e., (1.1) with the replacement,
$(w_{i})_{t{\bar{t}}}\to(w_{i})_{\xi\eta},$ $\xi,\eta\in\mathbb{R}$, and
neither anti-symmetry nor radial conditions were assumed. ∎

######  Remark 2.3.

Using the Riemann-Hilbert method, Novokshenov [21] constructed the long time
asymptotics of the solution of the Cauchy problem for the hyperbolic version
of equation (1.1), considered in the laboratory coordinates
$x=\xi+\eta,\,\,t=\xi-\eta$. From the physical and analytical points of view
our problem is very different from the one considered in [21], but there are
parallels. ∎

###  2.2. The subsidiary connection form $\omega$

Solving (1.1) is equivalent to constructing connection forms $\alpha$ of the
above type. We are going to construct $\alpha$ from a simpler connection form
$\omega$.

######  Definition 2.4.

Let111The connection form $\omega$ should not be confused with the root of
unity $\omega=e^{{2\pi{\scriptscriptstyle\sqrt{-1}}\,}/{(n+1)}}$ of course;
the context will make the intended meaning clear.
$\omega=\tfrac{1}{\lambda}\eta dz$, where

$\eta=\begin{pmatrix}&&&p_{0}\\\ p_{1}&&&\\\ &\ddots&&\\\
&&p_{n}&\end{pmatrix}$

and $p_{i}(z)=c_{i}z^{k_{i}}$ with $c_{i}>0,k_{i}\geq-1$ and
$p_{i}=p_{n-i+1}$.

Writing $\omega^{T}=Bdz$, we have an associated linear o.d.e.

(2.3) $\Phi_{z}=B\Phi$

where $\Phi$ takes values in $\textrm{SL}_{n+1}\mathbb{C}$.

By inspection we see that the connection form $\omega$ satisfies:

Cyclic symmetry:
$\tau(\omega(\lambda))=\omega(e^{{2\pi{\scriptscriptstyle\sqrt{-1}}\,}/{(n+1)}}\lambda)$

Anti-symmetry:  $\sigma(\omega(\lambda))=\omega(-\lambda)$.

###  2.3. A family of local solutions near $t=0$

We shall now construct some specific local solutions $w$ of (1.1).

If all $k_{i}>-1$ we have a unique local holomorphic222When
$k_{i}\notin\mathbb{Z}$, “holomorphic” should be understood in the multi-
valued sense. solution $\Phi=\Phi(z,\lambda)$ of 2.3 with $\Phi(0,\lambda)=I$.
We refer to this case (i.e. where all $k_{i}>-1$, as opposed to $k_{i}\geq-1$)
as the generic case. From now on — up to and including section 9 — we focus on
the generic case.

In this section we write

$L=\Phi^{T},$

so that $L$ is the unique local solution of the o.d.e.

(2.4) $L^{-1}L_{z}=\tfrac{1}{\lambda}\eta,\quad L|_{z=0}=I.$

Restricting to $S^{1}=\\{\lambda\in\mathbb{C}\ |\ |\lambda|=1\\}$, we may
regard $L$ as a map (from a neighbourhood of $z=0$ in $\mathbb{C}$) to the
loop group $\Lambda\textrm{SL}_{n+1}\mathbb{C}$. For the next step we use the
Iwasawa factorization for $\Lambda SL_{n+1}\mathbb{C}$ with respect to its
real form $\Lambda SL^{\Delta}_{n+1}\mathbb{R}$, which can be described as
follows.

First, we have an (elementary) Lie algebra decomposition

(2.5)
$\Lambda\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C}=\Lambda\mathfrak{s}\mathfrak{l}^{\Delta}_{n+1}\mathbb{R}\
+\ \Lambda_{+}\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C},$

where $\Lambda_{+}\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C}$ denotes the space
of loops $f(\lambda)=\sum_{i\in\mathbb{Z}}f_{i}\lambda^{i}$ in
$\Lambda\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C}$ such that $f_{i}=0$ when
$i<0$, and

$\Lambda\mathfrak{s}\mathfrak{l}^{\Delta}_{n+1}\mathbb{R}=\\{f\in\Lambda\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C}\
|\ c(f(\lambda))=f(1/\bar{\lambda})\\}.$

Concretely, let us write $f=f_{<0}+f_{0}+f_{>0}$ $(=f_{\leq 0}+f_{>0})$ for
the decomposition of $f$ according to terms $f_{i}\lambda^{i}$ with $i<0$,
$i=0$, $i>0$ (respectively). Then the decomposition (2.5) follows from writing
$f=(f_{\leq 0}+\hat{c}(f_{\leq 0})+(f_{>0}-\hat{c}(f_{\leq 0}))$, where
$\hat{c}(f)(\lambda)=c(f(1/\bar{\lambda}))$. We note that the intersection
$\Lambda\mathfrak{s}\mathfrak{l}^{\Delta}_{n+1}\mathbb{R}\ \cap\
\Lambda_{+}\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C}$ is
$\mathfrak{s}\mathfrak{l}^{\Delta}_{n+1}\mathbb{R}$, i.e. the space of
constant loops, so the decomposition (2.5) is not a direct sum.

Next, the symmetries $\tau,\sigma$ may be imposed on (2.5), in the sense that
(2.5) remains valid if we assume that all loops satisfy the “twisting
conditions”
$\tau(f(\lambda))=f(e^{{2\pi{\scriptscriptstyle\sqrt{-1}}\,}/{(n+1)}}\lambda)$,
$\sigma(f(\lambda))=f(-\lambda)$. The intersection of the twisted loop
algebras in (2.5) is the set of matrices
$\operatorname{diag}(z_{0},\dots,z_{n})$ with $z_{i}+z_{n-i}=0$ (from the
twisting conditions), such that $z_{i}=\bar{z}_{n-i}$ (from the definition of
$\mathfrak{s}\mathfrak{l}^{\Delta}_{n+1}\mathbb{R}$). Thus all $z_{i}$ are
pure imaginary.

It follows that any (twisted) $f$ has a unique splitting

(2.6) $f=f_{\mathbb{R}}+f_{+}$

with
$f_{\mathbb{R}}\in\Lambda\mathfrak{s}\mathfrak{l}^{\Delta}_{n+1}\mathbb{R}$,
$f_{+}\in\Lambda_{+}\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C}$ if we insist
that the constant term of $f_{+}$ is real. Thus, after imposing these
conditions, we see that (2.5) becomes a direct sum decomposition.

Finally, the inverse function theorem guarantees — locally, near the identity
loop — the existence of a corresponding multiplicative splitting of Banach Lie
groups.

This particular Iwasawa factorization is all that we need. We refer to
[22],[6],[1],[15] for the general theory of Iwasawa factorizations for loop
groups.

It follows that we have (locally, near $z=0$) a factorization

(2.7) $L=L_{\mathbb{R}}L_{+}$

of the $\Lambda\textrm{SL}_{n+1}\mathbb{C}$-valued function $L$ with
$L_{\mathbb{R}}|_{z=0}=I=L_{+}|_{z=0}$. The normalization of $L_{+}$ means
that $L_{+}(z,\bar{z},\lambda)=b(z,\bar{z})+O(\lambda)$, where
$b=\operatorname{diag}(b_{0},\dots,b_{n})$ and all $b_{i}>0$. We have
$b(0)=I$.

As explained in section 2 of [11], from each such $L_{\mathbb{R}}$ we can
obtain a connection form $\alpha$, and hence a solution of (1.1) near $t=0$.
We review this next, in order to introduce some essential notation. The
required connection form $\alpha$ will be of the form

(2.8)
$\alpha=(L_{\mathbb{R}}G)^{-1}(L_{\mathbb{R}}G)_{t}dt+(L_{\mathbb{R}}G)^{-1}(L_{\mathbb{R}}G)_{\bar{t}}d{\bar{t}},$

where $G$ is a certain diagonal matrix (depending on $t,\bar{t}$) and $t$ is a
new variable. To define these we introduce the following notation.

######  Definition 2.5.

Let $N=\sum_{i=0}^{n}(k_{i}+1)$, $c=\prod_{i=0}^{n}c_{i}$. Matrices
$m=\operatorname{diag}(m_{0},\dots,m_{n}),\hat{c}=\operatorname{diag}(\hat{c}_{0},\dots,\hat{c}_{n})$
are defined by:

(2.9) $m_{i-1}-m_{i}=-1+\tfrac{n+1}{N}(k_{i}+1)$ (2.10)
$\hat{c}_{i-1}/\hat{c}_{i}=\left(\tfrac{n+1}{N}\right)^{m_{i}-m_{i-1}}c_{i}c^{-(k_{i}+1)/N}.$

It is easy to see that $m,\hat{c}$ are well-defined and unique. They satisfy
$m_{i}+m_{n-i}=0,\hat{c}_{i}\hat{c}_{n-i}=1$ (because $p_{i}=p_{n-i+1}$).

######  Definition 2.6.

Let

(2.11) $t=\tfrac{n+1}{N}\ c^{\frac{1}{n+1}}\ z^{\frac{N}{n+1}}$

and

(2.12)
$h=\operatorname{diag}(\hat{c}_{0},\dots,\hat{c}_{n})\,t^{\operatorname{diag}(m_{0},\dots,m_{n})}=\hat{c}\,t^{m}.$

Let

(2.13) $G=\operatorname{diag}(|h_{0}|/h_{0},\dots,|h_{n}|/h_{n})=|h|/h.$

The definitions of $m,\hat{c},t,h$ are made in order to have the simple
formula

(2.14)
$h^{-1}\,\tfrac{n+1}{N}z\eta^{T}\,h=t\Pi,\quad\Pi=(\delta_{i,i+1})_{0\leq
i\leq n}=\begin{pmatrix}&\\!1\\!&&\\\ &&\ \ddots\ &\\\ &&&\\!1\\\
1\\!&&&\end{pmatrix}.$

Computation now shows that
$(L_{\mathbb{R}}G)^{-1}(L_{\mathbb{R}}G)_{t}dt+(L_{\mathbb{R}}G)^{-1}(L_{\mathbb{R}}G)_{\bar{t}}d\bar{t}$
has the form of $\alpha$ in Definition 2.1, if we introduce

(2.15) $w_{i}=\log(b_{i}/|h_{i}|).$

######  Proposition 2.7.

Let $\omega$ be as in Definition 2.4. Assume $k_{i}>-1$ for all $i$. Then the
above construction produces a solution $w$ of (1.1) on some punctured disk of
the form $0<|t|<\epsilon$, with

$w_{i}=-m_{i}\log|t|-\log\hat{c}_{i}+o(1)$

as $t\to 0$.

###### Proof.

We have $b=I+o(1)$ in a neighbourhood of $t=0$, and
$\log|h_{i}|=m_{i}\log|t|+\log\hat{c}_{i}$. ∎

From Definition 2.5 we see that the condition $k_{i}>-1$ corresponds to the
condition $m_{i-1}-m_{i}>-1$.

It is easy to show that $m_{i-1}-m_{i}\geq-1$ is a necessary and sufficient
condition for the existence of local (radial) solutions near $t=0$ satisfying
$w_{i}\sim-m_{i}\log|t|$. These $m$ constitute a compact convex region
$\mathcal{A}$, and the generic case corresponds to the interior
$\mathring{\mathcal{A}}$ of this region.

######  Remark 2.8.

The arguments we have used in passing from the holomorphic equation (2.4) to
the zero curvature form (2.8) are very similar to the technique used by
Krichever in [17] where he introduced a nonlinear analogue of d’Alembert’s
formula for the equations of the principal chiral field and the Sine-Gordon
equation. In our construction, a key role is played by the Iwasawa
factorization (2.7), while, in Krichever’s paper, a Riemann-Hilbert problem
posed on two circles (around zero and infinity) is used. In fact, properly
modified for the case of (1.1), Krichever’s technique can be used to produce a
proof of the factorization (2.7) which is independent from loop group theory.
Although this relation between the group-theoretical Iwasawa factorization
theory and the analytic Riemann-Hilbert theory is well known, we have found
that Krichever’s paper [17] adds interesting new aspects, which we plan to
address in a forthcoming publication. ∎

##  3\. Isomonodromy formulation

In this section we review the isomonodromy formulation of (1.1). This will
eventually (in section 7) be used in a construction of local solutions near
$t=\infty$.

###  3.1. The main isomonodromic connection form $\hat{\alpha}$

As explained in section 1 of [9] (and section 2 of [11]), the radial property
$w=w(|t|)$ leads to a homogeneity property of $\alpha$, and hence to another
connection form $\hat{\alpha}$.

######  Definition 3.1.

$\hat{\alpha}=\left[-\tfrac{t}{\lambda^{2}}\
W^{T}-\tfrac{1}{\lambda}xw_{x}+{\bar{t}}\,W\right]d\lambda,$ where $x=|t|$.

Writing $\hat{\alpha}^{T}=\hat{A}d\lambda$, we have an associated linear
o.d.e.

(3.1) $\Psi_{\lambda}=\hat{A}\Psi.$

This o.d.e. is meromorphic in $\lambda$ with poles of order $2$ at $\lambda=0$
and $\lambda=\infty$.

The argument of section 2.4 of [11] shows that

$\hat{\alpha}=(gL_{\mathbb{R}}G)^{-1}(gL_{\mathbb{R}}G)_{\lambda}d\lambda,$

where

(3.2) $g=\lambda^{m}.$

From the previous section we have

$\displaystyle\alpha$
$\displaystyle=(L_{\mathbb{R}}G)^{-1}(L_{\mathbb{R}}G)_{t}dt+(L_{\mathbb{R}}G)^{-1}(L_{\mathbb{R}}G)_{\bar{t}}d{\bar{t}}$
$\displaystyle=(gL_{\mathbb{R}}G)^{-1}(gL_{\mathbb{R}}G)_{t}dt+(gL_{\mathbb{R}}G)^{-1}(gL_{\mathbb{R}}G)_{\bar{t}}d{\bar{t}}.$

In other words, $gL_{\mathbb{R}}G$ is a fundamental solution matrix for the
(linear system corresponding to the) combined connection
$d+\alpha+\hat{\alpha}$. Hence this connection is flat.

Here we are assuming (as in section 2.1) that we have a solution $w=w(|t|)$
for $t$ in some open set. Then $d+\alpha+\hat{\alpha}$ is defined for such $t$
and any $\lambda\in\mathbb{C}^{\ast}$.

Flatness implies that the $\lambda$-monodromy of $gL_{\mathbb{R}}G$ is
independent of $t$. Namely, if analytic continuation of $gL_{\mathbb{R}}G$
around the origin in the $\lambda$-plane produces $CgL_{\mathbb{R}}G$ for some
$C=C(t,\bar{t})$, then, after substituting into (2.1),(3.1), we deduce that
$C_{t}=C_{\bar{t}}=0$. In fact ([14]; see chapter 4 of [5]), flatness implies
that the system (3.1) is isomonodromic in the sense that all Stokes matrices
and connection matrices of (3.1) (as well as the monodromy matrices) are
independent of $t$.

For future calculations it will be convenient to put $\zeta=\lambda/t$. This
converts $\hat{\alpha}$ to
$\hat{\alpha}(\zeta)=\left[-\tfrac{1}{\zeta^{2}}W^{T}-\tfrac{1}{\zeta}xw_{x}+x^{2}W\right]d\zeta$
and (3.1) to

(3.3)
$\Psi_{\zeta}=\left[-\tfrac{1}{\zeta^{2}}W-\tfrac{1}{\zeta}xw_{x}+x^{2}W^{T}\right]\Psi$

(recall that $x=|t|$). We shall investigate the monodromy data of this
meromorphic o.d.e.

By inspection we see that the connection form $\hat{\alpha}$ satisfies:

Cyclic symmetry:
$\tau(\hat{\alpha}(\zeta))=\hat{\alpha}(e^{{2\pi{\scriptscriptstyle\sqrt{-1}}\,}/{(n+1)}}\zeta)$

Anti-symmetry:  $\sigma(\hat{\alpha}(\zeta))=\hat{\alpha}(-\zeta)$

$c$-reality:
$c(\hat{\alpha}(\zeta))=\hat{\alpha}(1/(x^{2}\bar{\zeta}\vphantom{\tfrac{a}{a}}))$

$\theta$-reality:  $\theta(\hat{\alpha}(\zeta))=\hat{\alpha}(\bar{\zeta})$.

Here $\tau,\sigma,c$ are as in section 2 and the new involution $\theta$ is
defined by:

$\theta(X)=\bar{X}\quad\quad\quad\theta(V)=\bar{V}$

for $X\in\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C}$,
$V\in\textrm{SL}_{n+1}\mathbb{C}$.

###  3.2. The subsidiary connection form $\hat{\omega}$

Just as $\hat{\alpha}$ was constructed from $\alpha$, we can construct (see
section 2 of [11]) a meromorphic connection form
$\hat{\omega}=(gL)^{-1}(gL)_{\lambda}d\lambda$ from $\omega$:

######  Definition 3.2.

$\hat{\omega}=\left[-\tfrac{n+1}{N}\tfrac{z}{\lambda^{2}}\
\eta+\tfrac{1}{\lambda}\ m\right]d\lambda$.

Writing $\hat{\omega}^{T}=\hat{B}d\lambda$, we have an associated linear
o.d.e.

(3.4) $\Phi_{\lambda}=\hat{B}\Phi.$

Again, the extended connection $d+\omega+\hat{\omega}$ is flat.

Putting $\zeta=\lambda/t$ converts $\hat{\omega}$ to
$\hat{\omega}(\zeta)=\left[-\tfrac{1}{\zeta^{2}}h^{-1}\Pi^{T}h+\tfrac{1}{\zeta}m\right]d\zeta$
and equation (3.4) to

(3.5) $\Phi_{\zeta}=\left[-\tfrac{1}{\zeta^{2}}h\Pi
h^{-1}+\tfrac{1}{\zeta}m\right]\Phi.$

Here we are making use of (2.14).

By inspection we see that the connection form $\hat{\omega}$ satisfies:

Cyclic symmetry:
$\tau(\hat{\omega}(\zeta))=\hat{\omega}(e^{{2\pi{\scriptscriptstyle\sqrt{-1}}\,}/{(n+1)}}\zeta)$

Anti-symmetry:  $\sigma(\hat{\omega}(\zeta))=\hat{\omega}(-\zeta)$

$\theta$-reality:  $\theta(\hat{\omega}(\zeta))=\hat{\omega}(\bar{\zeta})$.

###  3.3. How the isomonodromic connection forms will be used.

In section 2.3 we gave a construction of local solutions (near $t=0$) of the
tt*-Toda equations (1.1). This was done by constructing connection forms
$\alpha$ — whose coefficients contain solutions $w$ of (1.1) — from connection
forms $\omega$ — whose coefficients are given explicitly in terms of the data
$c_{i}>0$, $k_{i}>-1$. The isomonodromic connection forms
$\hat{\alpha},\hat{\omega}$ will give another way of constructing solutions.

To exploit this, we shall first (in sections 4-6) compute the monodromy data
of the known local solutions which are smooth near zero. Motivated by this, we
shall (in section 7) formulate and solve a Riemann-Hilbert problem which
produces solutions which are smooth near infinity.

By “monodromy data” we mean Stokes matrices and monodromy matrices associated
to solutions of the linear system at each irregular pole (just monodromy
matrices, in the case of simple poles), together with connection matrices
which relate solutions at different poles. In section 4 we define the relevant
Stokes matrices, and in section 5 the connection matrices. In section 6 we
compute all this data corresponding to the local solutions near $t=0$ of
(1.1), first for $\hat{\omega}$ then for $\hat{\alpha}$. This information will
be used in sections 7 and 8 to identify which of these local solutions are
actually global solutions.

##  4\. Definition of Stokes matrices

In this section we define the Stokes matrices (and the monodromy matrices) for
both $\hat{\alpha}$ and $\hat{\omega}$ when $n+1$ is even. Their computation
will be carried out in section 6. The modifications needed when $n+1$ is odd
will be given in an appendix (section 9).

At first we assume $n+1\geq 3$; special features of the case $n+1=2$ will be
explained at the ends of the relevant subsections.

###  4.1. Stokes matrices for $\hat{\alpha}$ at $\zeta=0$

These depend on a choice of formal solution at each pole. Following
[9],[10],[11] we diagonalize the leading term of equation (3.3) by writing

$W=e^{-w\,}\Pi\,e^{w},\quad\Pi=\Omega\,d_{n+1}\,\Omega^{-1}$

where $\Pi$ is as in formula (2.14) and $\Omega=(\omega^{ij})_{0\leq i,j\leq
n}$. Then there is a (unique) formal solution of equation (3.3) of the form

$\textstyle\Psi^{(0)}_{f}(\zeta)=e^{-w}\,\Omega\left(I+\sum_{k\geq
1}\psi^{(0)}_{k}\zeta^{k}\right)e^{\frac{1}{\zeta}d_{n+1}}.$

The formal monodromy is trivial, i.e. there are no logarithmic terms in the
exponential factor (see section 2 of [9] and Lemma 4.4 of [8]). For reasons to
be given later, we shall also use the formal solution
$\tilde{\Psi}^{(0)}_{f}(\zeta)=\Psi^{(0)}_{f}(\zeta)\,d_{n+1}^{\frac{1}{2}}$
where
$d_{n+1}^{\frac{1}{2}}=\operatorname{diag}(1,\omega^{\frac{1}{2}},\omega^{1},\dots,\omega^{\frac{n}{2}})$,
$\omega=e^{{2\pi{\scriptscriptstyle\sqrt{-1}}\,}/{(n+1)}}$.

Each Stokes sector $\Omega^{(0)}_{k}$ at $\zeta=0$ in the $\zeta$-plane
supports holomorphic solutions $\Psi^{(0)}_{k},\tilde{\Psi}^{(0)}_{k}$ which
are uniquely characterized by the properties
$\Psi^{(0)}_{k}(\zeta)\sim\Psi^{(0)}_{f}(\zeta)$,
$\tilde{\Psi}^{(0)}_{k}(\zeta)\sim\tilde{\Psi}^{(0)}_{f}(\zeta)$ as $\zeta\to
0$ in $\Omega^{(0)}_{k}$. We use the following Stokes sectors which were
defined in [7] (see section 4 of [9] for a more detailed explanation).

As initial Stokes sector we take

$\Omega^{(0)}_{1}=\\{\zeta\in\mathbb{C}^{\ast}\ |\
-\tfrac{\pi}{2}-\tfrac{\pi}{n+1}<\text{arg}\zeta<\tfrac{\pi}{2}\\}.$

We regard this as a subset of the universal covering
$\tilde{\mathbb{C}}^{\ast}$, and define further Stokes sectors by

$\Omega^{(0)}_{k+\scriptstyle\frac{1}{n+1}}=e^{-\frac{\pi}{n+1}{\scriptscriptstyle\sqrt{-1}}\,}\Omega^{(0)}_{k},\quad
k\in\tfrac{1}{n+1}\mathbb{Z}.$

The solutions $\Psi^{(0)}_{k},\tilde{\Psi}^{(0)}_{k}$ extend by analytic
continuation to the whole of $\tilde{\mathbb{C}}^{\ast}$.

It is convenient to write

$\Omega^{(0)}_{k}=\\{\zeta\in\tilde{\mathbb{C}}^{\ast}\ |\
\theta^{(0)}_{k}-\tfrac{\pi}{2}<\text{arg}\zeta<\theta^{(0)}_{k-\scriptstyle\frac{1}{n+1}}+\tfrac{\pi}{2}\\}$

where

$\theta^{(0)}_{k}=-\tfrac{1}{n+1}\pi-(k-1)\pi,\quad
k\in\tfrac{1}{n+1}\mathbb{Z}.$

Figure 1. Singular directions $\theta^{(0)}_{k}=-\tfrac{\pi}{n+1}-(k-1)\pi$.

Then the intersection of successive Stokes sectors

$\Omega^{(0)}_{k}\cap\Omega^{(0)}_{k+\scriptstyle\frac{1}{n+1}}=\\{\zeta\in\tilde{\mathbb{C}}^{\ast}\
|\
\theta^{(0)}_{k}-\tfrac{\pi}{2}<\text{arg}\zeta<\theta^{(0)}_{k}+\tfrac{\pi}{2}\\}$

is seen to be the sector of width $\pi$ bisected by the ray with angle
$\theta^{(0)}_{k}$. On this intersection, successive solutions must differ by
(constant) matrices, i.e.

$\Psi^{(0)}_{k+\scriptstyle\frac{1}{n+1}}=\Psi^{(0)}_{k}Q^{(0)}_{k},$

and these “Stokes factors” $Q^{(0)}_{k}$ are thus indexed by the “singular
directions” $\theta^{(0)}_{k}$.

It is conventional (cf. [5], section 1.4) to define Stokes matrices
$S^{(0)}_{k}$ by

$\Psi^{(0)}_{k+1}=\Psi^{(0)}_{k}S^{(0)}_{k},$

for $k\in\mathbb{Z}$. Thus each Stokes matrix is a product of $n+1$ Stokes
factors:

$S^{(0)}_{k}=Q^{(0)}_{k}Q^{(0)}_{k+\scriptstyle\frac{1}{n+1}}Q^{(0)}_{k+\scriptstyle\frac{2}{n+1}}\dots
Q^{(0)}_{k+\scriptstyle\frac{n}{n+1}}.$

Figure 2. Stokes sectors $\Omega^{(0)}_{k}$. The ray $\theta^{(0)}_{k}$
bisects $\Omega^{(0)}_{k}\cap\Omega^{(0)}_{k+\scriptstyle\frac{1}{n+1}}$.

The monodromy of the solution $\Psi^{(0)}_{k}$ is given by

(4.1)
$\Psi^{(0)}_{k}(e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,}\zeta)=\Psi^{(0)}_{k}(\zeta)S^{(0)}_{k}S^{(0)}_{k+1}.$

In this formula, the left hand side indicates analytic continuation of
$\Psi^{(0)}_{k}(\zeta)$ around the pole $\zeta=0$ in the positive direction.
The formula is an immediate consequence of

(4.2)
$\Psi^{(0)}_{k+2}(e^{-2\pi{\scriptscriptstyle\sqrt{-1}}\,}\zeta)=\Psi^{(0)}_{k}(\zeta),$

which holds because both sides have the same asymptotic expansion
$\Psi^{(0)}_{f}(\zeta)$ as $\zeta\to 0$ in $\Omega^{(0)}_{k}$.

Thus the Stokes factors $Q^{(0)}_{k}$ are the fundamental data at $\zeta=0$.
From (4.2) we obtain $Q^{(0)}_{k+2}=Q^{(0)}_{k}$ and
$S^{(0)}_{k+2}=S^{(0)}_{k}$, so there are $2n+2$ independent Stokes factors,
or $2$ independent Stokes matrices.

From the symmetries of $\hat{\alpha}$ at the end of section 3 we obtain
corresponding symmetries of $\Psi^{(0)}_{k},\tilde{\Psi}^{(0)}_{k}$ and
$Q^{(0)}_{k},\tilde{Q}^{(0)}_{k}$ (cf. section 2 of [10] and section 3 of
[7]). The cyclic symmetry formula

(4.3) $Q^{(0)}_{k+\scriptstyle\frac{2}{n+1}}=\Pi\ Q^{(0)}_{k}\ \Pi^{-1}$

shows that all Stokes factors are determined by any two consecutive Stokes
factors, and that the monodromy matrix of $\Psi^{(0)}_{1}$ can be written

$S^{(0)}_{1}S^{(0)}_{2}=(Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi)^{n+1}.$

######  Definition 4.1.

$M^{(0)}=Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi$.

Thus the matrix $M^{(0)}$ determines the monodromy. We shall see later (see
Proposition 6.10) that it also determines the individual $Q^{(0)}_{1}$,
$Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}$, and hence all Stokes factors. It is,
therefore, the key object in describing the Stokes data.

Using $\tilde{\Psi}^{(0)}_{k}$ instead of $\Psi^{(0)}_{k}$ we can define
$\tilde{Q}^{(0)}_{k},\tilde{S}^{(0)}_{k}$ in a similar way. From this we have

(4.4)
$\tilde{Q}^{(0)}_{k}=d_{n+1}^{-\frac{1}{2}}Q^{(0)}_{k}d_{n+1}^{\frac{1}{2}}.$

A corresponding definition of $\tilde{M}^{(0)}$ will be given later, in
formula (6.6).

###  4.2. Stokes matrices for $\hat{\alpha}$ at $\zeta=\infty$

The definitions at $\zeta=\infty$ are very similar to the definitions at
$\zeta=0$. We diagonalize $W^{T}$ by writing

$W^{T}=e^{w}\Omega^{-1}d_{n+1}\Omega e^{-w}.$

Then we have formal solutions

$\Psi^{(\infty)}_{f}(\zeta)=\textstyle e^{w}\Omega^{-1}\left(I+\sum_{k\geq
1}\psi^{(\infty)}_{k}\zeta^{-k}\right)e^{x^{2}\zeta d_{n+1}}$

and
$\tilde{\Psi}^{(\infty)}_{f}(\zeta)=\Psi^{(\infty)}_{f}(\zeta)\,d_{n+1}^{-\frac{1}{2}}$.
As Stokes sectors at $\zeta=\infty$ we take initial sector

$\Omega^{(\infty)}_{1}=\\{\zeta\in\mathbb{C}^{\ast}\ |\
-\tfrac{\pi}{2}<\text{arg}\zeta<\tfrac{\pi}{2}+\tfrac{\pi}{n+1}\\}=\overline{\Omega^{(0)}_{1}}$

and then define
$\Omega^{(\infty)}_{k+\scriptstyle\frac{1}{n+1}}=e^{\frac{\pi}{n+1}{\scriptscriptstyle\sqrt{-1}}\,}\Omega^{(\infty)}_{k}$.
We can write

$\Omega^{(\infty)}_{k}=\\{\zeta\in\tilde{\mathbb{C}}^{\ast}\ |\
\theta^{(\infty)}_{k-\scriptstyle\frac{1}{n+1}}-\tfrac{\pi}{2}<\text{arg}\zeta<\theta^{(\infty)}_{k}+\tfrac{\pi}{2}\\}=\overline{\Omega^{(0)}_{k}}$

where $\theta^{(\infty)}_{k}=-\theta^{(0)}_{k}$,
$k\in\tfrac{1}{n+1}\mathbb{Z}$. We also have
$\theta^{(\infty)}_{k}=\theta^{(0)}_{\frac{2n}{n+1}-k}$.

On $\Omega^{(\infty)}_{k}$ we have canonical holomorphic solutions
$\Psi^{(\infty)}_{k},\tilde{\Psi}^{(\infty)}_{k}$. We define Stokes factors
$Q^{(\infty)}_{k}$ by
$\Psi^{(\infty)}_{k+\scriptstyle\frac{1}{n+1}}=\Psi^{(\infty)}_{k}Q^{(\infty)}_{k}$,
and Stokes matrices $S^{(\infty)}_{k}$ by
$\Psi^{(\infty)}_{k+1}=\Psi^{(\infty)}_{k}S^{(\infty)}_{k}$. The singular
directions $\theta^{(\infty)}_{k}$ bisect successive intersections
$\Omega^{(\infty)}_{k}\cap\Omega^{(\infty)}_{k+\scriptstyle\frac{1}{n+1}}$,
and index the Stokes factors $Q^{(\infty)}_{k}$. We have

$S^{(\infty)}_{k}=Q^{(\infty)}_{k}Q^{(\infty)}_{k+\scriptstyle\frac{1}{n+1}}Q^{(\infty)}_{k+\scriptstyle\frac{2}{n+1}}\dots
Q^{(\infty)}_{k+\scriptstyle\frac{n}{n+1}}.$

The monodromy of the solution $\Psi^{(\infty)}_{k}$ at $\zeta=\infty$ is given
by

(4.5)
$\Psi^{(\infty)}_{k}(e^{-2\pi{\scriptscriptstyle\sqrt{-1}}\,}\zeta)=\Psi^{(\infty)}_{k}(\zeta)S^{(\infty)}_{k}S^{(\infty)}_{k+1}$

(analytic continuation in the negative direction in the $\zeta$-plane). This
follows from the identity

(4.6)
$\Psi^{(\infty)}_{k+2}(e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,}\zeta)=\Psi^{(\infty)}_{k}(\zeta),$

cf. (4.2). For topological reasons, $S^{(\infty)}_{k}S^{(\infty)}_{k+1}$ must
be conjugate to the inverse of $S^{(0)}_{k}S^{(0)}_{k+1}$; we shall give the
precise relation in the next section.

Finally, the $c$-reality and $\theta$-reality conditions of section 2.1 lead
to

(4.7) $Q^{(\infty)}_{k}=d_{n+1}^{-1}Q^{(0)}_{k}d_{n+1}$

(Lemma 2.4 of [10]). This means that the Stokes data at $\zeta=\infty$ is
equivalent to the Stokes data at $\zeta=0$. The version of (4.4) at
$\zeta=\infty$ is

(4.8)
$\tilde{Q}^{(\infty)}_{k}=d_{n+1}^{\frac{1}{2}}Q^{(\infty)}_{k}d_{n+1}^{-\frac{1}{2}}.$

Using (4.4) we obtain the simple relation

(4.9) $\tilde{Q}^{(\infty)}_{k}=\tilde{Q}^{(0)}_{k},$

i.e. the tilde versions of the Stokes factors at zero and infinity are not
only equivalent but actually coincide.

###  4.3. Stokes matrices for $\hat{\omega}$ at $\zeta=0$

The leading term of equation (3.5) at $\zeta=0$ is diagonalized by $h\Omega$,
as $\Pi=\Omega\,d_{n+1}\,\Omega^{-1}$. Hence there is a formal solution

$\textstyle\Phi^{(0)}_{f}(\zeta)=h\Omega\left(I+\sum_{k\geq
1}\phi^{(0)}_{k}\zeta^{k}\right)e^{\frac{1}{\zeta}d_{n+1}}.$

We shall also make use of the formal solution
$\textstyle\tilde{\Phi}^{(0)}_{f}(\zeta)=\Phi^{(0)}_{f}(\zeta)\,d_{n+1}^{\frac{1}{2}}.$
Using the same Stokes sectors and singular directions as for
$\Psi^{(0)}_{k},\tilde{\Psi}^{(0)}_{k}$, we obtain canonical holomorphic
solutions $\Phi^{(0)}_{k},\tilde{\Phi}^{(0)}_{k}$.

For the Stokes factors we shall not need new notation, as (in the situation of
interest to us) it turns out that these are exactly the same as the Stokes
factors in section 4.1:

######  Proposition 4.2.

The connection form $\hat{\omega}$ has the same Stokes data as $\hat{\alpha}$
at $\zeta=0$, i.e. the solutions $\Phi^{(0)}_{k}$ satisfy
$\Phi^{(0)}_{k+\scriptstyle\frac{1}{n+1}}=\Phi^{(0)}_{k}Q^{(0)}_{k}$ and
$\Phi^{(0)}_{k+1}=\Phi^{(0)}_{k}S^{(0)}_{k}$.

It follows in particular that $\Phi^{(0)}_{k}$ has the same monodromy as
$\Psi^{(0)}_{k}$:

(4.10)
$\Phi^{(0)}_{k}(e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,}\zeta)=\Phi^{(0)}_{k}(\zeta)S^{(0)}_{k}S^{(0)}_{k+1}.$

The proof of the proposition will be given at the end of section 5.

###  4.4. Monodromy of $\hat{\omega}$ at $\zeta=\infty$

At the regular singularity $\zeta=\infty$ it is easier to find a canonical
solution:

######  Proposition 4.3.

Equation (3.5) has a (unique) solution of the form

$\textstyle\Phi^{(\infty)}(\zeta)=\left(I+\sum_{k\geq
1}\phi^{(\infty)}_{k}\zeta^{-k}\right)\zeta^{m}.$

The series in parentheses converges on a non-empty open disk at
$\zeta=\infty$.

This solution extends by analytic continuation to a solution which is
holomorphic on $\tilde{\mathbb{C}}^{\ast}$ but multivalued on
$\mathbb{C}^{\ast}$ because of the factor $\zeta^{m}=e^{m\log\zeta}$. We take
the standard branch of $\log\zeta$ which is positive on the positive real
axis.

###### Proof.

As $\zeta=\infty$ is a simple pole for equation (3.5), o.d.e. theory says that
there is a solution near $\zeta=\infty$ of the form

$\textstyle\Phi^{(\infty)}(\zeta)=\left(I+\sum_{k\geq
1}\phi^{(\infty)}_{k}\zeta^{-k}\right)\zeta^{m}\zeta^{M}$

where $M$ is nilpotent. We shall show later — see Remark 6.15 — that $M=0$.
This depends on our assumption that $m_{i-1}-m_{i}+1>0$. We note that, if
$m_{i-1}-m_{i}+1=0$, it is possible to have $M\neq 0$ (see [11] for examples
of this). ∎

Evidently we have

(4.11)
$\Phi^{(\infty)}(e^{-2\pi{\scriptscriptstyle\sqrt{-1}}\,}\zeta)=\Phi^{(\infty)}(\zeta)e^{-2\pi{\scriptscriptstyle\sqrt{-1}}\,m}$

so the monodromy of this solution is
$e^{-2\pi{\scriptscriptstyle\sqrt{-1}}\,m}$ (analytic continuation in the
negative direction in the $\zeta$-plane).

######  Remark 4.4.

The discussion in sections 4.1, 4.2, 4.3, applies when $n+1\geq 4$. Here the
Stokes sectors $\Omega^{(0)}_{k},\Omega^{(\infty)}_{k}$ are maximal. When
$n+1=2$ they are still Stokes sectors (in the sense of [5], section 1.4), but
they are not maximal. The rays $\theta^{(0)}_{k},\theta^{(\infty)}_{k}$ are
singular directions when $k\in\frac{1}{2}+\mathbb{Z}$, but not when
$k\in\mathbb{Z}$. This has the effect of making all Stokes factors equal to
the identity matrix when $k\in\mathbb{Z}$. With this caveat, the above
discussion applies also in the case $n+1=2$. We have

$S^{(0)}_{1}=Q^{(0)}_{1}Q^{(0)}_{\scriptstyle\frac{3}{2}}=Q^{(0)}_{\scriptstyle\frac{3}{2}},\quad
S^{(0)}_{2}=Q^{(0)}_{2}Q^{(0)}_{\scriptstyle\frac{5}{2}}=Q^{(0)}_{\scriptstyle\frac{5}{2}},$

and
$M^{(0)}=Q^{(0)}_{1}Q^{(0)}_{\scriptstyle\frac{3}{2}}\Pi=Q^{(0)}_{\scriptstyle\frac{3}{2}}\Pi$.
∎

######  Remark 4.5.

From (4.9) we know that $\tilde{Q}^{(0)}_{k}=\tilde{Q}^{(\infty)}_{k}$.
Furthermore, as we shall see in section 6.2, it will turn out that these
matrices are real. Thus the $\tilde{Q}^{(0)}_{k}$ are apparently superior to
the $Q^{(0)}_{k}$. On the other hand, the $Q^{(0)}_{k}$ are easier to define,
as for them the awkward factor $d_{n+1}^{\frac{1}{2}}$ is not needed in the
formal solution. And the apparent advantages of the $\tilde{Q}^{(0)}_{k}$ do
not carry over to the Lie-theoretic context of [8], where neither
$\tilde{Q}^{(0)}_{k}$ nor $Q^{(0)}_{k}$ are optimal; they are a useful but
special feature of $\textrm{SL}_{n+1}\mathbb{C}$. For these reasons we use
both $\tilde{Q}^{(0)}_{k}$ and $Q^{(0)}_{k}$ pragmatically, choosing whichever
is convenient for the task at hand. ∎

##  5\. Definition of connection matrices

In the previous section we have defined monodromy data of
$\hat{\alpha},\hat{\omega}$ at each pole (Stokes factors/monodromy matrices).
In this section we define the remaining part of the monodromy data, namely the
connection matrices. As in section 4 we assume $n+1$ is even.

###  5.1. Connection matrices for $\hat{\alpha}$

The solutions $\Psi^{(0)}_{k},\Psi^{(\infty)}_{k}$ (regarded as holomorphic
functions on $\tilde{\mathbb{C}}^{\ast}$) must be related by “connection
matrices” $E_{k}$, i.e.

$\Psi^{(\infty)}_{k}=\Psi^{(0)}_{k}E_{k},$

and similarly
$\tilde{\Psi}^{(\infty)}_{k}=\tilde{\Psi}^{(0)}_{k}\tilde{E}_{k}$. We have
$\tilde{E}_{k}=d_{n+1}^{-\frac{1}{2}}E_{k}d_{n+1}^{-\frac{1}{2}}$.

From the definition we obtain immediately

(5.1)
$E_{k}=\left(Q^{(0)}_{k-\scriptstyle\frac{1}{n+1}}\right)^{-1}E_{k-\scriptstyle\frac{1}{n+1}}\
Q^{(\infty)}_{k-\scriptstyle\frac{1}{n+1}}.$

Thus it suffices to consider $E_{1}$.

Using the monodromy formulae (4.1), (4.5), we obtain the “cyclic relation”

$E_{1}=S^{(0)}_{1}S^{(0)}_{2}\,E_{1}\ S^{(\infty)}_{1}S^{(\infty)}_{2}.$

This shows that $E_{1}$ conjugates $S^{(\infty)}_{1}S^{(\infty)}_{2}$ to
$(S^{(0)}_{1}S^{(0)}_{2})^{-1}$. In fact (see Lemma 2.5 of [10]) the cyclic
symmetry leads to the stronger relation

(5.2) $E_{1}=\left(Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi\right)\
E_{1}\
\left(Q^{(\infty)}_{1}Q^{(\infty)}_{1+\scriptstyle\frac{1}{n+1}}\Pi\right)$

(stronger in the sense that $(n+1)$-fold iteration produces the cyclic
relation).

###  5.2. Connection matrices for $\hat{\omega}$

With respect to the solutions $\Phi^{(\infty)},\Phi^{(0)}_{k}$ we define
connection matrices $D_{k}$ by

$\Phi^{(\infty)}=\Phi^{(0)}_{k}D_{k},$

and similarly $\Phi^{(\infty)}=\tilde{\Phi}^{(0)}_{k}\tilde{D}_{k}$. We have
$\tilde{D}_{k}=d_{n+1}^{-\frac{1}{2}}D_{k}$.

It follows that

(5.3) $D_{k}=Q^{(0)}_{k}D_{k+\scriptstyle\frac{1}{n+1}}.$

The monodromy formulae (4.10) and (4.11) give the cyclic relation

$D_{k}=S^{(0)}_{k}S^{(0)}_{k+1}\,D_{k}\,e^{-2\pi{\scriptscriptstyle\sqrt{-1}}\,m},$

which shows that $D_{k}$ conjugates
$e^{-2\pi{\scriptscriptstyle\sqrt{-1}}\,m}$ to
$(S^{(0)}_{k}S^{(0)}_{k+1})^{-1}$. The cyclic symmetry gives the stronger
relation

(5.4)
$D_{k}=\left(Q^{(0)}_{k}Q^{(0)}_{k+\scriptstyle\frac{1}{n+1}}\Pi\right)\,D_{k}\,e^{-2\pi{\scriptscriptstyle\sqrt{-1}}\,m/(n+1)}d_{n+1}.$

######  Remark 5.1.

With the conventions of [11], indicated here by GIL, where we used the
$\lambda$-o.d.e. (3.4), the canonical solution at $\zeta=\infty$ was chosen to
be of the form
$\Phi^{(\infty)}_{\text{GIL}}(\lambda)=(I+O(\lambda^{-1})\lambda^{m}$. Our
current convention is $\Phi^{(\infty)}(\zeta)=(I+O(\zeta^{-1}))\zeta^{m}$.
Thus $\Phi^{(\infty)}_{\text{GIL}}(\lambda)=\Phi^{(\infty)}(\zeta)t^{m}$. On
the other hand, the canonical solutions at $\zeta=0$ satisfy
$\Phi^{(0)}_{\text{GIL}}(\lambda)=\Phi^{(0)}(\zeta)$. Thus the connection
matrix $D_{k}$ is related to the corresponding connection matrix
$D^{\text{GIL}}_{k}$ in [11] by $D^{\text{GIL}}_{k}=D_{k}t^{m}$. ∎

###  5.3. Relation between $E_{k}$ and $D_{k}$

The construction of $\alpha$ from $\omega$, and hence the construction of
$\hat{\alpha}$ from $\hat{\omega}$, leads to a relation between their
connection matrices:

######  Theorem 5.2.

$E_{k}=\tfrac{1}{n+1}D_{k}t^{m}\,\Delta\,\bar{t}^{-m}\bar{D}_{\frac{2n+1}{n+1}-k}^{-1}\,d_{n+1}^{-1}\,C,$
where

(5.5)
$C=\Omega\bar{\Omega}^{-1}=\tfrac{1}{n+1}\Omega^{2}={\scriptsize\left(\begin{array}[]{c|ccc}\\!\\!1&&&\\\
\hline\cr&&&1\\!\\\ &&\iddots&\\\ &\\!\\!1&&\end{array}\right)}.$

In particular we obtain

(5.6)
$E_{1}=\tfrac{1}{n+1}D_{1}t^{m}\,\Delta\,(\overline{D_{1}t^{m}})^{-1}(\bar{Q}^{(0)}_{\frac{n}{n+1}})^{-1}d_{n+1}^{-1}\,C.$

###### Proof.

From section 3.1 (and section 2 of [11]) we know that
$(gL_{\mathbb{R}}G)^{T}(\lambda)$ is a solution of equation (3.1), hence
$(gL_{\mathbb{R}}G)^{T}(\zeta t)$ is a solution of equation (3.3). We remind
the reader that $G,g$ were defined in (2.13), (3.2). Thus, we must have

$\Psi^{(0)}_{k}(\zeta)=(gL_{\mathbb{R}}G)^{T}(\zeta
t)Y_{k},\quad\Psi^{(\infty)}_{k}(\zeta)=(gL_{\mathbb{R}}G)^{T}(\zeta t)Z_{k},$

for some constant matrices $Y_{k},Z_{k}$ (independent of $\zeta$).

We shall show that

(1) $Y_{k}=(D_{k}t^{m})^{-1}$

(2)
$Z_{k}=\tfrac{1}{n+1}\Delta\,\bar{Y}_{\frac{2n+1}{n+1}-k}\,d_{n+1}^{-1}\,C$,

from which the above formula for $E_{k}=Y_{k}^{-1}Z_{k}$ follows immediately.

Proof of (1). By definition of the canonical solutions
$\Phi^{(0)}_{k},\Psi^{(0)}_{k}$, we have

$\displaystyle\Phi^{(0)}_{k}(\zeta)$
$\displaystyle\sim\Phi^{(0)}_{f}(\zeta)=h\Omega(I+O(\zeta))e^{\frac{1}{\zeta}d_{n+1}}$
$\displaystyle\Psi^{(0)}_{k}(\zeta)$
$\displaystyle\sim\Psi^{(0)}_{f}(\zeta)=e^{-w}\Omega(I+O(\zeta))e^{\frac{1}{\zeta}d_{n+1}}$

as $\zeta\to 0$ in the sector $\Omega^{(0)}_{k}$. Let us examine

$\displaystyle\Psi^{(0)}_{k}$ $\displaystyle=(gL_{\mathbb{R}}G)^{T}Y_{k}$
$\displaystyle=(gLL_{+}^{-1}G)^{T}Y_{k}$
$\displaystyle=G(L_{+}^{-1})^{T}(gL)^{T}Y_{k}$

more closely. From the Iwasawa factorization we have

$(L_{+}^{-1})^{T}=b^{-1}+O(\lambda),$

where $b$ is real and diagonal. From section 3.2 (and section 2 of [11]) we
know that $(gL)^{T}(\lambda)$ is a solution of equation (3.4), hence
$(gL)^{T}(\zeta t)$ is a solution of equation (3.5). From the form of the
respective series expansions we must have333 With the conventions of [11] we
had $(gL)^{T}(\lambda)=\Phi^{(\infty)}_{\text{GIL}}(\lambda)$. With our
current convention — see Remark 5.1 — we have
$(gL)^{T}(\lambda)=\Phi^{(\infty)}(\zeta)t^{m}$. This affects the formula for
$Y_{k}$: here we have $D_{k}t^{m}=Y_{k}^{-1}$, while [11] had
$D^{\text{GIL}}_{k}=Y_{k}^{-1}$.  $(gL)^{T}(\zeta
t)=\Phi^{(\infty)}(\zeta)t^{m}$. Hence

$(gL)^{T}(\zeta
t)=\Phi^{(\infty)}(\zeta)t^{m}=\Phi^{(0)}_{k}(\zeta)D_{k}t^{m}\sim
h\Omega(I+O(\zeta))e^{\frac{1}{\zeta}d_{n+1}}D_{k}t^{m}.$

Substituting these expressions for $(L_{+}^{-1})^{T}$ and $(gL)^{T}$, we
obtain

$\displaystyle\Psi^{(0)}_{k}(\zeta)$ $\displaystyle\sim
G\,b^{-1}h\Omega(I+O(\zeta))e^{\frac{1}{\zeta}d_{n+1}}D_{k}t^{m}Y_{k}$
$\displaystyle=e^{-w}\Omega(I+O(\zeta))e^{\frac{1}{\zeta}d_{n+1}}D_{k}t^{m}Y_{k},$

as $Gb^{-1}h=|h|h^{-1}b^{-1}h=e^{-w}$, from the definition of $w$ in (2.15).
By the uniqueness of the formal solution $\Psi^{(0)}_{f}$, it follows that
$D_{k}t^{m}Y_{k}=I$ This gives formula (1) for $Y_{k}$.

Proof of (2). If $\Psi(\zeta)$ is a solution of (3.3), then
$\Delta\overline{\Psi(\frac{1}{x^{2}\bar{\zeta}\vphantom{\tfrac{a}{a}}})}$ is
also a solution of (3.3), hence the latter must be equal to $\Psi(\zeta)$
times a constant matrix. For the formal solution
$\Psi(\zeta)=\Psi^{(0)}_{f}(\zeta)$ at $\zeta=0$,
$\Delta\overline{\Psi^{(0)}_{f}(\frac{1}{x^{2}\bar{\zeta}\vphantom{\tfrac{a}{a}}})}$
is then a formal solution at $\zeta=\infty$, and the constant matrix is
$(n+1)Cd_{n+1}$. This is easily verified, using the identities
$\Omega\Delta=(n+1)d_{n+1}^{-1}\Omega^{-1}$ and $d_{n+1}Cd_{n+1}=C$ (cf.
Appendix A of [10]). We obtain

$\Delta\overline{\Psi^{(0)}_{f}(\tfrac{1}{x^{2}\bar{\zeta}\vphantom{\tfrac{a}{a}}})}=(n+1)\Psi^{(\infty)}_{f}(\zeta)Cd_{n+1},$

from which it follows that

(5.7)
$\Delta\overline{\Psi^{(0)}_{k}(\tfrac{1}{x^{2}\bar{\zeta}\vphantom{\tfrac{a}{a}}})}=(n+1)\Psi^{(\infty)}_{\frac{2n+1}{n+1}-k}(\zeta)Cd_{n+1}.$

Let us substitute

$\Psi^{(0)}_{k}(\zeta)=(gL_{\mathbb{R}}G)^{T}(\zeta
t)Y_{k},\quad\Psi^{(\infty)}_{k}(\zeta)=(gL_{\mathbb{R}}G)^{T}(\zeta t)Z_{k}$

into (5.7), using $\overline{(gL_{\mathbb{R}}G)(1/\bar{\lambda})}=g^{-1}\Delta
L_{\mathbb{R}}(\lambda)\Delta\bar{G}$ (which follows from (5.8) and the
definitions of $g,G$). We obtain

$\Delta\bar{G}\Delta L_{\mathbb{R}}(\lambda)^{T}\Delta
g^{-1}\bar{Y}_{k}=(n+1)GL_{\mathbb{R}}(\lambda)^{T}gZ_{\frac{2n+1}{n+1}-k}Cd_{n+1}$

As $G^{-1}\Delta\bar{G}\Delta=I$ and $g^{-1}\Delta g^{-1}=\Delta$, this
reduces to

$\Delta\bar{Y}_{k}=(n+1)Z_{\frac{2n+1}{n+1}-k}Cd_{n+1},$

which gives formula (2) for $Z_{k}$. This completes the proof of the formula
for $E_{k}$ in Theorem 5.2. Formula (5.6) for $E_{1}$ follows from this and
(5.3). ∎

######  Remark 5.3.

More conceptually, the key formula (5.7) in the proof arises from the fact
(section 2.3) that the Iwasawa factorization $L=L_{\mathbb{R}}L_{+}$ is taken
with respect to the real form $\Lambda SL^{\Delta}_{n+1}\mathbb{R}$; thus, by
definition, $L_{\mathbb{R}}$ satisfies

(5.8)
$\Delta\overline{L_{\mathbb{R}}(1/\bar{\lambda})}\Delta=L_{\mathbb{R}}(\lambda),$

i.e. $c(L_{\mathbb{R}}(1/\bar{\lambda}))=L_{\mathbb{R}}(\lambda)$. This gives
rise to the $c$-reality property of $\alpha$ (section 2.1), and the
corresponding $c$-reality property of $\hat{\alpha}$ (section 3.1). The latter
predicts that
$\Delta\overline{\Psi^{(0)}_{f}(\tfrac{1}{x^{2}\bar{\zeta}\vphantom{\tfrac{a}{a}}})}\Delta$
is equal to $\Psi_{f}(\zeta)$ times a constant matrix. However, this constant
matrix depends on the chosen normalizations of
$\Psi^{(0)}_{f},\Psi^{(\infty)}_{f}$, so a direct calculation of this matrix
is unavoidable. ∎

Formula (1) of the proof allows us to show that the Stokes factors associated
to the solutions $\Phi^{(0)}_{k}$ of (3.5) agree with the Stokes factors
$Q^{(0)}_{k}$ associated to the solutions $\Psi^{(0)}_{k}$ of (3.3), as stated
in Proposition 4.2. We give the omitted proof here.

Proof of Proposition 4.2. Using the notation of the proof of Theorem 5.2, we
have

$\displaystyle Q^{(0)}_{k}$
$\displaystyle=(\Psi^{(0)}_{k})^{-1}\Psi^{(0)}_{k+\scriptstyle\frac{1}{n+1}}$
$\displaystyle=((gL_{\mathbb{R}}G)^{T}Y_{k})^{-1}(gL_{\mathbb{R}}G)^{T}Y_{k+\scriptstyle\frac{1}{n+1}}$
$\displaystyle=Y_{k}^{-1}Y_{k+\scriptstyle\frac{1}{n+1}}$
$\displaystyle=D_{k}t^{m}(D_{k+\scriptstyle\frac{1}{n+1}}t^{m})^{-1}=D_{k}(D_{k+\scriptstyle\frac{1}{n+1}})^{-1}$
$\displaystyle=(\Phi^{(\infty)}D_{k}^{-1})^{-1}\Phi^{(\infty)}(D_{k+\scriptstyle\frac{1}{n+1}})^{-1}$
$\displaystyle=(\Phi^{(0)}_{k})^{-1}\Phi^{(0)}_{k+\scriptstyle\frac{1}{n+1}},$

which is the Stokes factor for $\Phi^{(0)}_{k}$, as required. ∎

##  6\. Computation of some Stokes matrices and connection matrices

The monodromy data corresponding to the solutions of (1.1) which arise from
Proposition 2.7 will be calculated explicitly in this section. Our target is
the monodromy data of $\hat{\alpha}$, which will be calculated in terms of the
monodromy data of $\hat{\omega}$, and this will be found explicitly in terms
of $c_{0}z^{k_{0}},\dots,c_{n}z^{k_{n}}$. The calculation will be simplified
by normalizing $\hat{\omega}$ suitably.

As in sections 4 and 5 we assume that $n+1$ is even. The modifications needed
when $n+1$ is odd will be given in an appendix (section 9). Special features
of the case $n+1=2$ will be mentioned when appropriate, in remarks at the end
of each subsection.

###  6.1. The normalized o.d.e.

Recall the $\zeta$-equation (3.5) associated to the connection form
$\hat{\omega}$:

$\Phi_{\zeta}=\left[-\tfrac{1}{\zeta^{2}}h\Pi
h^{-1}+\tfrac{1}{\zeta}m\right]\Phi.$

The matrices $m,h,\Pi$ were specified in Definitions 2.5 and 2.6.

Introducing $\mathcal{X}=h^{-1}\Phi$, we have

(6.1)
$\mathcal{X}_{\zeta}=\left[-\tfrac{1}{\zeta^{2}}\Pi+\tfrac{1}{\zeta}m\right]\mathcal{X}.$

We regard (6.1) as the fundamental (normalized) equation. We shall use it for
all calculations in this section.

First we give the relation between the monodromy data for $\mathcal{X}$ and
that for $\Phi$. At $\zeta=0$, we have the formal solution

$\textstyle\mathcal{X}^{(0)}_{f}(\zeta)=\Omega\left(I+\sum_{k\geq
1}x^{(0)}_{k}\zeta^{k}\right)e^{\frac{1}{\zeta}d_{n+1}}.$

The sector $\Omega^{(0)}_{k}$ then determines a canonical holomorphic solution
$\mathcal{X}^{(0)}_{k}$. We define Stokes factors $\mathcal{Q}^{(0)}_{k}$ by

$\mathcal{X}^{(0)}_{k+\scriptstyle\frac{1}{n+1}}=\mathcal{X}^{(0)}_{k}\mathcal{Q}^{(0)}_{k}.$

At $\zeta=\infty$ we have the canonical solution

$\textstyle\mathcal{X}^{(\infty)}(\zeta)=\left(I+\sum_{k\geq
1}x^{(\infty)}_{k}\zeta^{-k}\right)\zeta^{m}.$

We define connection matrices $\mathcal{D}_{k}$ by

$\mathcal{X}^{(\infty)}=\mathcal{X}^{(0)}_{k}\mathcal{D}_{k},$

from which it follows that

(6.2)
$\mathcal{D}_{k}=\mathcal{Q}^{(0)}_{k}\mathcal{D}_{k+\scriptstyle\frac{1}{n+1}}.$

######  Proposition 6.1.

The monodromy data for (6.1) is related to the monodromy data for (3.5) by:
(1) $\mathcal{Q}^{(0)}_{k}=Q^{(0)}_{k}$, (2) $\mathcal{D}_{k}=D_{k}h$.

###### Proof.

(1) As $\Phi^{(0)}_{f}=h\mathcal{X}^{(0)}_{f}$, we have
$\Phi^{(0)}_{k}=h\mathcal{X}^{(0)}_{k}$. It follows that
$\mathcal{Q}^{(0)}_{k}=Q^{(0)}_{k}$. (2) As $h^{-1}\Phi^{(\infty)}$ is a
solution of (6.1), it must be $\mathcal{X}^{(\infty)}$ times a constant
matrix. Comparison of the definitions of
$\Phi^{(\infty)},\mathcal{X}^{(\infty)}$ gives
$h^{-1}\Phi^{(\infty)}=\mathcal{X}^{(\infty)}h^{-1}$, i.e.
$\Phi^{(\infty)}=h\mathcal{X}^{(\infty)}h^{-1}$. It follows that
$\mathcal{D}_{k}=D_{k}h$. ∎

###  6.2. Computation of the Stokes data

The singular directions $\theta^{(0)}_{k}\in\mathbb{R}$ (regarded as angles of
rays in the universal covering of $\mathbb{C}^{\ast}$) index the Stokes
factors $Q^{(0)}_{k}=\mathcal{Q}^{(0)}_{k}$. We shall now compute these
matrices.

######  Proposition 6.2.

All diagonal entries of $Q^{(0)}_{k}$ are $1$. The $(i,j)$-entry $(0\leq i\neq
j\leq n)$ can be nonzero only when $(i,j)$ satisfies the condition

(6.3) $\arg(\omega^{j}-\omega^{i})=\theta^{(0)}_{k}\quad\text{mod}\ 2\pi.$

###### Proof.

This is the classical criterion, based on the eigenvalues of the leading term
$-\Pi=\Omega^{-1}(-d_{n+1})\Omega$ of (6.1); cf. Lemma 3.1 of [7]. ∎

######  Definition 6.3.

For $k\in\tfrac{1}{n+1}\mathbb{Z}$, let $\mathcal{R}^{(0)}_{k}=\\{(i,j)\ |\
(i,j)\ \text{satisfies (\ref{anglecriterion})}\\}$.

By the cyclic symmetry formula (4.3), the matrices $Q^{(0)}_{1}$,
$Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}$ determine all Stokes factors. The
corresponding $\mathcal{R}^{(0)}_{1}$,
$\mathcal{R}^{(0)}_{1+\scriptstyle\frac{1}{n+1}}$ are as follows (Proposition
3.4 of [7]):

######  Proposition 6.4.

(1) If $n+1=4c$:

$\mathcal{R}^{(0)}_{1}=\\{(2c-1,0),(2c-2,1),\dots,(c,c-1)\\}\newline
\quad\quad\quad\cup\\{(2c,4c-1),(2c+1,4c-2),\dots,(3c-1,3c)\\}$

$\mathcal{R}^{(0)}_{1+\scriptstyle\frac{1}{n+1}}=\\{(2c-1,4c-1),(2c,4c-2),\dots,(3c-2,3c)\\}\newline
\quad\quad\quad\quad\cup\\{(2c-2,0),(2c-3,1),\dots,(c,c-2)\\}$

(2) If $n+1=4c+2$:

$\mathcal{R}^{(0)}_{1}=\\{(2c+1,4c+1),(2c+2,4c),\dots,(3c,3c+2)\\}\newline
\quad\quad\quad\cup\\{(2c,0),(2c-1,1),\dots,(c+1,c-1)\\}$

$\mathcal{R}^{(0)}_{1+\scriptstyle\frac{1}{n+1}}=\\{(2c-1,0),(2c-2,1),\dots,(c,c-1)\\}\newline
\quad\quad\quad\quad\cup\\{(2c,4c+1),(2c+1,4c),\dots,(3c,3c+1)\\}$ ∎

######  Remark 6.5.

The set $\mathcal{R}^{(0)}_{k}$ may be described more conceptually as a subset
of the roots of the Lie algebra $\mathfrak{s}\mathfrak{l}_{n+1}\mathbb{C}$. As
explained in [7],[8], the significance of
$\mathcal{R}^{(0)}_{1}\cup\mathcal{R}^{(0)}_{1+\scriptstyle\frac{1}{n+1}}$ is
that it is a set of representatives for the orbits of the action of the
Coxeter element $(n\ n-1\ \dots\ 1\ 0)$ on the roots. We shall not exploit
this Lie-theoretic point of view, but we note the consequence that
$\mathcal{R}^{(0)}_{k+\scriptstyle\frac{2}{n+1}}$ is obtained from
$\mathcal{R}^{(0)}_{k}$ by applying $(n\ n-1\ \dots\ 1\ 0)$. Thus the
proposition determines all $\mathcal{R}^{(0)}_{k}$. ∎

Propositions 6.2 and 6.4 specify the “shape” of $Q^{(0)}_{k}$, i.e. the values
of the diagonal entries and which off-diagonal entries can be nonzero. The
tilde version
$\tilde{Q}^{(0)}_{k}=d_{n+1}^{-\frac{1}{2}}Q^{(0)}_{k}d_{n+1}^{\frac{1}{2}}$
has the same shape, as $d_{n+1}^{-\frac{1}{2}}$ is diagonal. It will be
convenient to use $\tilde{Q}^{(0)}_{k}$ instead $Q^{(0)}_{k}$ from now on in
this section, because all the entries of the $\tilde{Q}^{(0)}_{k}$ turn out to
be real (Proposition 6.7 below).

From the discussion above, we may write

$\tilde{Q}^{(0)}_{k}=I+\sum_{(i,j)\in\mathcal{R}^{(0)}_{k}}\tilde{q}_{i,j}E_{i,j}$

for some (a priori, complex) scalars $\tilde{q}_{i,j}$, where $E_{i,j}$ is the
matrix which has $1$ in the $(i,j)$-entry and $0$ elsewhere.

The symmetries of the $\tilde{Q}^{(0)}_{k}$ imply the following symmetries of
the $\tilde{q}_{i,j}$:

######  Proposition 6.6.

Interpreting $i,j\in\\{0,1,\dots,n\\}$ mod $n+1$, we have:

(1) $\tilde{q}_{i-1,j-1}=\begin{cases}\ \ \tilde{q}_{i,j}\ \text{if $i,j\geq
1$}\\\ -\tilde{q}_{i,j}\ \text{if $i=0$ or $j=0$}\end{cases}$

(2) $\tilde{q}_{j,i}=-\tilde{q}_{i,j}$

(3) $\overline{\tilde{q}_{i,j}}=\begin{cases}-\tilde{q}_{n+1-i,n+1-j}\
\text{if $i,j\geq 1$}\\\ \ \ \tilde{q}_{n+1-i,n+1-j}\ \text{if $i=0$ or
$j=0$}\end{cases}$

###### Proof.

These correspond to the symmetries of the $\tilde{Q}^{(0)}_{k}$ given by the
automorphisms $\tau,\sigma,c$. ∎

In view of (1) and (3) let us modify the signs of the $E_{i,j}$ as follows:

$e_{i,j}=\begin{cases}\ \ E_{i,j}\ \text{if $0\leq i<j\leq n$}\\\ -E_{i,j}\
\text{if $0\leq i>j\leq n$}\end{cases}s_{i,j}=\begin{cases}\ \
\tilde{q}_{i,j}\ \text{if $0\leq i<j\leq n$}\\\ -\tilde{q}_{i,j}\ \text{if
$0\leq i>j\leq n$}\end{cases}$

Then we may write

(6.4)
$\tilde{Q}^{(0)}_{k}=I+\sum_{(i,j)\in\mathcal{R}^{(0)}_{k}}s_{i,j}\,e_{i,j},$

where the scalars $s_{i,j}$ satisfy the following simpler relations:

######  Proposition 6.7.

Interpreting $i,j\in\\{0,1,\dots,n\\}$ mod $n+1$, we have:

(1) $s_{i-1,j-1}=s_{i,j}$

(2) $s_{j,i}=s_{i,j}$

(3) $\bar{s}_{i,j}=s_{n+1-i,n+1-j}$

These relations imply that all $s_{i,j}\in\mathbb{R}$.

###### Proof.

(1),(2),(3) are direct translations from Proposition 6.6. Using these, for
$i<j$, we have $\bar{s}_{i,j}=s_{n+1-i,n+1-j}=s_{j-i,0}=s_{0,j-i}=s_{i,j}$, so
$s_{i,j}$ is real. ∎

######  Definition 6.8.

For $i,j\in\\{0,1,\dots,n\\}$, let $[i,j]$ denote the equivalence class of
$(i,j)$ under $(i,j)\sim(i-1,j-i)$ (interpreted mod $n+1$).

By (1) of Proposition 6.7, $s_{i,j}$ depends only on the equivalence class
$[i,j]$, so we can denote it by $s_{[i,j]}$.

######  Definition 6.9.

$s_{i}=s_{[0,i]}$.

By (1) and (2) of Proposition 6.7 we have $s_{i}=s_{n+1-i}$. Thus the
essential Stokes parameters444In terms of the notation
$s_{1}^{\mathbb{R}},s_{2}^{\mathbb{R}}$ of [10],[11],[7] (the case $n=3$), we
have $s_{1}=s_{1}^{\mathbb{R}},s_{2}=-s_{2}^{\mathbb{R}}$. are the real
numbers $s_{1},\dots,s_{\frac{1}{2}(n+1)}$.

This completes our description of the shape of the Stokes factor
$\tilde{Q}^{(0)}_{k}$. We should now recall that $\tilde{Q}^{(0)}_{k}$ was
associated to a certain solution of (1.1). In section 2.3, this solution was
constructed from the data $c_{0}z^{k_{0}},\dots c_{n}z^{k_{n}}$ where
$c_{i}>0$ and $k_{i}>-1$. To compute $\tilde{Q}^{(0)}_{k}$ it remains to
express $s_{1},\dots,s_{\frac{1}{2}(n+1)}$ in terms of this data. In order to
do this, we need one more result on the structure of $\tilde{Q}^{(0)}_{k}$.

Recall (Definition 4.1) that we have introduced
$M^{(0)}=Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi$. From formula
(4.4) we have
$Q^{(0)}_{k}=d_{n+1}^{\frac{1}{2}}\tilde{Q}^{(0)}_{k}d_{n+1}^{-\frac{1}{2}}$,
so

$M^{(0)}=d_{n+1}^{\frac{1}{2}}\tilde{Q}^{(0)}_{1}\tilde{Q}^{(0)}_{1+\scriptstyle\frac{1}{n+1}}d_{n+1}^{-\frac{1}{2}}\Pi=\omega^{\frac{1}{2}}d_{n+1}^{\frac{1}{2}}\tilde{Q}^{(0)}_{1}\tilde{Q}^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\hat{\Pi}d_{n+1}^{-\frac{1}{2}}$

where

(6.5) $\hat{\Pi}=\omega^{-\frac{1}{2}}d_{n+1}^{-\frac{1}{2}}\Pi
d_{n+1}^{\frac{1}{2}}=\left(\begin{smallmatrix}&1&&&\\\ &&\ \ddots&&\\\
&&&&1\\\ -1&&&\end{smallmatrix}\right).$

In view of this we introduce

(6.6)
$\tilde{M}^{(0)}=\tilde{Q}^{(0)}_{1}\tilde{Q}^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\hat{\Pi}=\omega^{-\frac{1}{2}}d_{n+1}^{-\frac{1}{2}}M^{(0)}d_{n+1}^{\frac{1}{2}}.$

This satisfies
$(\tilde{M}^{(0)})^{n+1}=-\tilde{S}^{(0)}_{1}\tilde{S}^{(0)}_{2}$. Then the
result we need is:

######  Proposition 6.10.

The characteristic polynomial of
$\tilde{M}^{(0)}=\tilde{Q}^{(0)}_{1}\tilde{Q}^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\hat{\Pi}$
is $\sum_{i=0}^{n+1}s_{i}\mu^{n+1-i}$. Here we put $s_{0}=s_{n+1}=1$.

This shows that $\tilde{M}^{(0)}$ (in fact, just the conjugacy class of
$\tilde{M}^{(0)}$) determines the $s_{i}$ and hence all Stokes factors.

###### Proof.

A proof by direct computation is given in Proposition 3.4 [13], where $s_{i}$
is denoted by $p_{i}$. ∎

######  Remark 6.11.

A Lie-theoretic proof could be given by observing, as in [7], that
$\tilde{M}^{(0)}$ is a Steinberg cross-section of the set of regular conjugacy
classes. ∎

Using this, we obtain explicit expressions for the $s_{i}$:

######  Theorem 6.12.

The Stokes factor $\tilde{Q}^{(0)}_{k}$ associated to the data
$c_{0}z^{k_{0}},\dots c_{n}z^{k_{n}}$ (or to the corresponding local solution
of (1.1)) is

$\tilde{Q}^{(0)}_{k}=I+\sum_{(i,j)\in\mathcal{R}^{(0)}_{k}}s_{|i-j|}\,e_{i,j}$

where $s_{i}$ is the $i$-th symmetric function of
$\omega^{m_{0}+\frac{n}{2}},\omega^{m_{1}-1+\frac{n}{2}},\dots,\omega^{m_{n}-n+\frac{n}{2}}$.

###### Proof.

The cyclic symmetry (5.4)

$Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi=D_{k}\,d_{n+1}^{-1}\omega^{m}D_{k}^{-1}$

shows that the eigenvalues of
$M^{(0)}=Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi$ are those of
$d_{n+1}^{-1}\omega^{m}$. Hence the eigenvalues of $\tilde{M}^{(0)}$ are those
of
$\omega^{-\frac{1}{2}}d_{n+1}^{-1}\omega^{m}=\omega^{\frac{n}{2}}d_{n+1}^{-1}\omega^{m}$.

Let $\sigma_{i}$ be the $i$-th symmetric function of the $n+1$ complex numbers
$\omega^{m_{0}+\frac{n}{2}},\omega^{m_{1}-1+\frac{n}{2}},\dots,\omega^{m_{n}-n+\frac{n}{2}}$.
These are the eigenvalues of
$\omega^{\frac{n}{2}}d_{n+1}^{-1}\omega^{m}=-\omega^{-\frac{1}{2}}d_{n+1}^{-1}\omega^{m}$
(we are assuming that $n+1$ is even). Thus, the characteristic polynomial of
$-\tilde{M}^{(0)}$ is $\sum_{i=0}^{n+1}(-1)^{i}\sigma_{i}\mu^{n+1-i}$, and the
characteristic polynomial of $\tilde{M}^{(0)}$ is
$\sum_{i=0}^{n+1}\sigma_{i}\mu^{n+1-i}$. By Proposition 6.10, we must have
$s_{i}=\sigma_{i}$. ∎

The theorem shows that the $s_{i}$ depend only on the $k_{i}$ (and not on the
$c_{i}$).

######  Example 6.13.

In the case $n+1=2$ we have $\mathcal{R}^{(0)}_{k}=\emptyset$ and
$\tilde{Q}^{(0)}_{k}=I$ for $k\in\mathbb{Z}$. There is only one Stokes
parameter $s=s_{1}$, and

$\tilde{Q}^{(0)}_{\scriptstyle\frac{3}{2}}=\begin{pmatrix}1&s\\\
0&1\end{pmatrix},\quad\tilde{Q}^{(0)}_{\scriptstyle\frac{5}{2}}=\begin{pmatrix}1&0\\\
-s&1\end{pmatrix}$

(and $\tilde{Q}^{(0)}_{k+2}=\tilde{Q}^{(0)}_{k}$). We have

$\tilde{M}^{(0)}=\tilde{Q}^{(0)}_{1}\tilde{Q}^{(0)}_{\scriptstyle\frac{3}{2}}\hat{\Pi}=\begin{pmatrix}-s&1\\\
-1&0\end{pmatrix},$

whose characteristic polynomial is $\mu^{2}+s\mu+1$. Theorem 6.12 gives
$s=-2\sin\pi m_{0}$. We have $m_{0}=\frac{k_{1}-k_{0}}{2k_{0}+2k_{1}+4}$ (from
Definition 2.5). ∎

###  6.3. Computation of the connection matrices $D_{k}$

To compute $D_{k}$, we need some particular solutions $\mathcal{X}$ of
equation (6.1) whose behaviour at both $\zeta=0$ and $\zeta=\infty$ can be
computed. First, we write the fundamental solution matrix as

$\mathcal{X}=\begin{pmatrix}|&&|\\\ X^{(0)}&\cdots&X^{(n)}\\\
|&&|\end{pmatrix},$

and then study the system

(6.7) $\partial_{\zeta}X=[-\tfrac{1}{\zeta}\Pi+m]X,\quad
X=\begin{pmatrix}x_{0}\\\ \vdots\\\ \vdots\\\ x_{n}\\\ \end{pmatrix}.$

In fact, we shall focus on the scalar o.d.e. for $x_{0}$:

(6.8)
$\zeta(\partial_{\zeta}-m_{n})\cdots\zeta(\partial_{\zeta}-m_{0})x_{0}=(-1)^{n+1}x_{0}.$

The function $X$ may be recovered from $x_{0}$ and the formula
$x_{k+1}=-\zeta(\partial_{\zeta}-m_{k})x_{k}$.

Note that (6.8) can be written

$\left[(\partial_{\zeta}-m_{n}^{\prime})(\partial_{\zeta}-m_{n-1}^{\prime})\cdots(\partial_{\zeta}-m_{0}^{\prime})-(-1)^{n+1}\zeta^{-(n+1)}\right]x_{0}=0$

where

(6.9) $m_{i}^{\prime}=m_{i}-i.$

Thus the indicial roots are $m_{0}^{\prime},\dots,m_{n}^{\prime}$.

It will be convenient to allow $i\in\mathbb{Z}$ in (6.9). (Note that
$m_{i}=m_{i+n+1}$ but $m^{\prime}_{i}=m^{\prime}_{i+n+1}+n+1$.) Then the
condition $k_{i}>-1$ in Definition 2.5 is equivalent to

(6.10) $m^{\prime}_{i}<m^{\prime}_{i-1},\quad i\in\mathbb{Z}.$

######  Lemma 6.14.

Let $c$ and $b_{0},\dots,b_{n}$ be real numbers such that
$\frac{b_{i}+i}{n+1}>c$ for all $i$. Let
$g(\zeta)=g^{b_{0},\dots,b_{n}}(\zeta)=$

$\int_{c-{\scriptscriptstyle\sqrt{-1}}\,\infty}^{c+{\scriptscriptstyle\sqrt{-1}}\,\infty}\textstyle\Gamma(\frac{b_{0}+0}{n+1}-t)\Gamma(\frac{b_{1}+1}{n+1}-t)\cdots\Gamma(\frac{b_{n}+n}{n+1}-t)(n+1)^{-(n+1)t}\zeta^{-(n+1)t}dt.$

This defines a holomorphic function of $\zeta$ on the sector
$-\tfrac{\pi}{2}<\arg\zeta<\tfrac{\pi}{2}$ (taking the standard branch of
$\log\zeta$ in $\zeta^{-(n+1)t}$). Furthermore:

(1) $g$ satisfies the equation
$\zeta(\partial_{\zeta}+b_{n})\cdots\zeta(\partial_{\zeta}+b_{0})g(\zeta)=g(\zeta)$,
i.e.
$\left[(\partial_{\zeta}+(b_{0}+0))(\partial_{\zeta}+(b_{1}+1))\cdots(\partial_{\zeta}+(b_{n}+n))-\zeta^{-(n+1)}\right]g(\zeta)=0.$

(2) We have
$\zeta(\partial_{\zeta}+b_{0})g^{b_{0},\dots,b_{n}}(\zeta)=g^{b_{1},\dots,b_{n},b_{0}}(\zeta)$;
let us call this $g^{[1]}(\zeta)$. For $i=1,2,\dots$ we define
$g^{[i+1]}=(g^{[i]})^{[1]}=g^{b_{i+1},\dots,b_{n},b_{0},\dots,b_{i}}$.

(3)
$g(\zeta)\sim{\scriptstyle\sqrt{-1}}\,(2\pi)^{\frac{1}{2}(n+2)}(n+1)^{-\frac{1}{2}-\frac{1}{n+1}\sum_{0}^{n}b_{i}}\,\zeta^{-\frac{1}{n+1}\sum_{0}^{n}b_{i}}\,e^{-\frac{1}{\zeta}}$
as $\zeta\to 0$.

(4) Assume that $0<|(b_{i}+i)-(b_{j}+j)|<n+1$ for all $i,j$. Then
$g(\zeta)=\sum_{i=0}^{n}C_{i}\zeta^{-(b_{i}+i)}(1+O(\zeta^{-(n+1)}))$ for some
$C_{0},\dots,C_{n}$ as $\zeta\to\infty$. Only $C_{0}$ will be needed later. It
is given by

$C_{0}=2\pi{\scriptstyle\sqrt{-1}}\,(n+1)^{-b_{0}}\textstyle\Gamma(\frac{b_{1}-b_{0}+1}{n+1})\Gamma(\frac{b_{2}-b_{0}+2}{n+1})\cdots\Gamma(\frac{b_{n}-b_{0}+n}{n+1}).$

###### Proof.

Stirling’s Formula shows that the integral converges whenever
$-\tfrac{\pi}{2}<\arg\zeta<\tfrac{\pi}{2}$. Note that the poles of
$\Gamma(\frac{b_{i}+i}{n+1}-t)$ are $\frac{b_{i}+i}{n+1}+k$, $k=0,1,2,\dots$,
all of which lie to the right of the contour of integration. We sketch the
proofs of (1)-(4) below.

(1) Application of $\partial_{\zeta}+b_{i}+i$ to $\zeta^{-(n+1)t}$ produces
the factor $b_{i}+i-(n+1)t=(n+1)\left[\frac{b_{i}+i}{n+1}-t\right]$. The
factor $\frac{b_{i}+i}{n+1}-t$ combines with $\Gamma(\frac{b_{i}+i}{n+1}-t)$
to produce $\Gamma(\frac{b_{i}+i}{n+1}-t+1)$. Application of
$(\partial_{\zeta}+(b_{0}+0))(\partial_{\zeta}+(b_{1}+1))\cdots(\partial_{\zeta}+(b_{n}+n))$
to $g(\zeta)$ thus produces, after the change of variable $t^{\prime}=t-1$,
$\zeta^{-(n+1)}g(\zeta)$, as required.

(2) Similar to the proof of (1).

(3) Substituting the definition
$\Gamma(k)=\int_{0}^{\infty}e^{-\tau}\tau^{k-1}d\tau$ of the gamma function
into the integral, and computing555 In Proposition 3.11 of [11], the initial
factor $2$ was omitted. The authors thank Yudai Hateruma for this correction.
as in Proposition 3.11 of [11], we obtain $g(\zeta)=$

$\tfrac{2\pi\sqrt{-1}}{(\zeta(n+1))^{\sum_{0}^{n}B_{i}}}\int_{0}^{\infty}\\!\\!\\!\\!\\!\\!\cdots\\!\\!\int_{0}^{\infty}\\!\overset{\scriptscriptstyle
n}{\underset{\scriptstyle
i=1}{\Pi}}x_{i}^{B_{i}-1-B_{0}}e^{-\frac{1}{\zeta}\frac{1}{n+1}\left(x_{1}+\cdots+x_{n}+\frac{1}{x_{1}\cdots
x_{n}}\right)}dx_{1}\\!\dots\\!dx_{n},$

where $B_{i}=\frac{b_{i}+i}{n+1}$. We shall apply Laplace’s Method to obtain
the asymptotics as $\zeta\to 0$ from this formula.

In general, for a (real) function $\phi$ on $(0,\infty)$ with a nondegenerate
minimum at $c$, Laplace’s Method says that
$\int_{0}^{\infty}f(x)e^{-\frac{1}{\zeta}\phi(x)}dx\sim
f(c)e^{-\frac{1}{\zeta}\phi(c)}(2\pi)^{\frac{1}{2}}\zeta^{\frac{1}{2}}\phi^{\prime\prime}(c)^{-\frac{1}{2}}$
as $\zeta\to 0$ with $-\tfrac{\pi}{2}<\arg\zeta<\tfrac{\pi}{2}$.

The function

$\phi(x_{1},\dots,x_{n})=\tfrac{1}{n+1}(x_{1}+\cdots+x_{n}+\tfrac{1}{x_{1}\cdots
x_{n}})$

has a critical point at $(x_{1},\dots,x_{n})=(1,\dots,1)$, and the eigenvalues
of the Hessian at this point are $1$ (with multiplicity $1$), $\frac{1}{n+1}$
(with multiplicity $n-1$). We obtain

$g(\zeta)\sim\tfrac{2\pi\sqrt{-1}}{(\zeta(n+1))^{\sum_{0}^{n}B_{i}}}\
e^{-\frac{1}{\zeta}}\ (2\pi)^{\frac{1}{2}n}\ \zeta^{\frac{1}{2}n}\
(n+1)^{\frac{1}{2}(n-1)}.$

Now,
$\sum_{0}^{n}B_{i}=\tfrac{1}{n+1}(\tfrac{1}{2}n(n+1)+\sum_{i=0}^{n}b_{i})=\tfrac{1}{2}n+\tfrac{1}{n+1}\sum_{i=0}^{n}b_{i}$.
This gives the stated formula.

(4) As stated earlier, all poles of the integrand lie in the half-plane
$\text{Re\,}t>c$. By the assumption of (4), these poles are distinct.

Consider a semicircle of radius $R$ in the half-plane $\text{Re\,}t>c$ which,
together with the segment $\\{c+iT\ |\ -R\leq T\leq R\\}$, forms a closed
contour. It can be shown that the integral over the semicircle tends to zero
as $R\to\infty$ — this argument is summarized in Lemma A.6 of [2], for
example.

Cauchy’s Integral Theorem then shows that
$g(\zeta)/(-2\pi{\scriptstyle\sqrt{-1}}\,)$ is equal to the sum of the
residues of the integrand. The residue of $\Gamma(u-t)$ is $-(-1)^{k}/k!$ at
its simple pole $u+k$. The stated formula follows from these facts (as in the
proof of Corollary 3.10 of [11]). ∎

######  Remark 6.15.

The residue expansion in the proof of (4) shows that, near $\zeta=\infty$,
$g(\zeta)$ is the solution of the scalar equation predicted by the Frobenius
Method. Note that there are no “polynomial in $\log\zeta$” terms, even though
the differences of indicial roots may be integers. This justifies our
assertion in the proof of Proposition 4.3 that $M=0$ there. ∎

We shall use Lemma 6.14 with $b_{i}=-m_{i}$, so we begin by verifying the
assumptions on $b_{0},\dots,b_{n}$. For convenience let us write
$b^{\prime}_{i}=b_{i}+i$ and extend this to $i\in\mathbb{Z}$ by defining
$b_{i+n+1}=b_{i}$, $b^{\prime}_{i+n+1}=b^{\prime}_{i}+n+1$ (so that
$b_{i}=-m_{i}$ and $b^{\prime}_{i}=-m^{\prime}_{i}$ for all $i\in\mathbb{Z}$).
Then (6.10) gives

(6.11)
$b^{\prime}_{n}-(n+1)=b^{\prime}_{-1}<b^{\prime}_{0}<\cdots<b^{\prime}_{n}<b^{\prime}_{n+1}=b^{\prime}_{0}+n+1.$

From $m_{0}+m_{n}=0$ we have $b^{\prime}_{0}+b^{\prime}_{n}=n$, so the left-
most inequality gives $-b^{\prime}_{0}-1<b^{\prime}_{0}$, hence
$b^{\prime}_{0}>-\tfrac{1}{2}$. It follows that $b^{\prime}_{i}>-\tfrac{1}{2}$
for $i\geq 0$, hence $\frac{b_{i}+i}{n+1}>-\tfrac{1}{2n+2}$. Thus the initial
assumption in Lemma 6.14 holds if we take $c\leq-\tfrac{1}{2n+2}$. The
validity of the assumption in (4) of Lemma 6.14 follows directly from (6.11),
as $b^{\prime}_{n+1}-b^{\prime}_{0}=n+1$.

To construct a solution $\mathcal{X}$ of (6.1), we begin with the scalar
equation (6.8). As we are assuming that $n+1$ is even, the scalar equation of
(1) coincides with (6.8) if we take $x_{0}=g$. As
$x_{k+1}=-\zeta(\partial_{\zeta}-m_{k})x_{k}$, (2) implies that
$x_{k+1}=(-1)^{k+1}g^{[k+1]}$. Thus we have a solution

$X=\begin{pmatrix}x_{0}\\\ \vdots\\\ \vdots\\\ x_{n}\\\
\end{pmatrix}=\begin{pmatrix}\ \ \ \ \ \,g\\\ (-1)^{1}g^{[1]}\\\ \vdots\\\
(-1)^{n}g^{[n]}\end{pmatrix}$

of (6.7).

Next we need a basis of solutions $x_{0}^{(0)},\dots,x_{0}^{(n)}$ of the
scalar equation, in order to construct $\mathcal{X}$ from $X$. We shall take
$n+1$ functions of the form $g(\omega^{i}\zeta)$. These are solutions of the
scalar equation as only $\zeta^{n+1}$ occurs in the coefficients of the
equation, and $(\omega^{i}\zeta)^{n+1}=\zeta^{n+1}$. It follows from (4) that
any consecutive $n+1$ such functions are linearly independent. We shall take

$\mathcal{X}=\begin{pmatrix}&\dots&g(\omega\zeta)&g(\zeta)&g(\omega^{-1}\zeta)&\dots&\\\
\ \\\ &&\vdots&\vdots&\vdots&&\\\ \ \end{pmatrix}$

where row $i+1$ is given by applying $-\zeta(\partial_{\zeta}-m_{i})$ to row
$i$ ($0\leq i\leq n-1$), and where $g(\zeta)$ is in column $\tfrac{1}{2}(n+1)$
(out of columns $0,1,\dots,n$).

Thus we obtain the particular solution

$\mathcal{X}=\begin{pmatrix}\vphantom{\boxed{{A^{B}_{C}}}}&\dots&g(\omega\zeta)&g(\zeta)&g(\omega^{-1}\zeta)&\dots&\\\
\vphantom{\boxed{{A^{B}_{C}}}}&\dots&-\omega^{-1}g^{[1]}(\omega\zeta)&-g^{[1]}(\zeta)&-\omega
g^{[1]}(\omega^{-1}\zeta)&\dots&\\\
\vphantom{\boxed{{A^{B}_{C}}}}&\dots&\omega^{-2}g^{[2]}(\omega\zeta)&g^{[2]}(\zeta)&\omega^{2}g^{[2]}(\omega^{-1}\zeta)&\dots&\\\
&&\vdots&\vdots&\vdots&&\\\ \end{pmatrix}$

of (6.1). We shall use this to compute $\mathcal{D}_{k}$, by relating
$\mathcal{X}$ to the canonical solutions
$\mathcal{X}^{(\infty)},\mathcal{X}^{(0)}_{k}$.

The relation between $\mathcal{X}$ and $\mathcal{X}^{(\infty)}$. Let us write

$\mathcal{X}^{(\infty)}(\zeta)=\mathcal{X}(\zeta)B.$

Then

$B^{-1}=\lim_{\zeta\to\infty}\mathcal{X}^{(\infty)}(\zeta)^{-1}\mathcal{X}(\zeta)=\lim_{\zeta\to\infty}\zeta^{-m}\mathcal{X}(\zeta).$

Applying (4) of Lemma 6.14, we obtain

$\textstyle
g^{-m_{0},\dots,-m_{n}}(\zeta)=\sum_{i=0}^{n}C_{i}\,\zeta^{m_{i}^{\prime}}(1+O(\zeta^{-(n+1)}))$

where

$C_{0}=2\pi{\scriptstyle\sqrt{-1}}\,(n+1)^{m_{0}}\textstyle\Gamma(\frac{-m_{1}^{\prime}+m_{0}^{\prime}}{n+1})\Gamma(\frac{-m_{2}^{\prime}+m_{0}^{\prime}}{n+1})\cdots\Gamma(\frac{-m_{n}^{\prime}+m_{0}^{\prime}}{n+1}).$

By (6.10), $m^{\prime}_{i}<m^{\prime}_{0}=m_{0}$ for $1\leq i\leq n$. Hence

$\textstyle\lim_{\zeta\to\infty}\zeta^{-m_{0}}g(\zeta)=\lim_{\zeta\to\infty}\sum_{i=0}^{n}C_{i}\,\zeta^{m_{i}^{\prime}-m_{0}}(1+O(\zeta^{-(n+1)}))=C_{0}.$

It follows that the first row of $B^{-1}$ is

$\dots\quad C_{0}\omega^{2m_{0}}\quad C_{0}\omega^{m_{0}}\quad C_{0}\quad
C_{0}\omega^{-m_{0}}\quad\dots$

For the second row, we use the fact that
$-\zeta(\partial_{\zeta}+m_{0})g(\zeta)=-g^{[1]}(\zeta)$. A similar argument
gives the second row of $B^{-1}$ as

$\dots\quad-C_{0}^{[1]}\omega^{-2}\omega^{2m_{1}}\quad-
C_{0}^{[1]}\omega^{-1}\omega^{m_{1}}\quad-C_{0}^{[1]}\quad-
C_{0}^{[1]}\omega^{1}\omega^{-m_{1}}\quad\dots$

where

$C_{0}^{[1]}=2\pi{\scriptstyle\sqrt{-1}}\,(n+1)^{m_{1}}\textstyle\Gamma(\frac{-m_{2}^{\prime}+m_{1}^{\prime}}{n+1})\cdots\Gamma(\frac{-m_{n}^{\prime}+m_{1}^{\prime}}{n+1})\Gamma(\frac{-m_{n+1}^{\prime}+m_{1}^{\prime}}{n+1}).$

Continuing in this way, with $C_{0}^{[i+1]}=(C_{0}^{[i]})^{[1]}$, we obtain

$B^{-1}=\Gamma_{m}\ V_{m}$

where $\Gamma_{m}$, $V_{m}$ are (respectively)

$\begin{pmatrix}C_{0}&&&\\\ &-C_{0}^{[1]}&&\\\ &&\ddots&\\\
&&&(-1)^{n}C_{0}^{[n]}\end{pmatrix},\begin{pmatrix}\dots&\omega^{2m_{0}^{\prime}}&\omega^{m_{0}^{\prime}}&1&\omega^{-m_{0}^{\prime}}&\dots\\\
\dots&\omega^{2m_{1}^{\prime}}&\omega^{m_{1}^{\prime}}&1&\omega^{-m_{1}^{\prime}}&\dots\\\
&\vdots&\vdots&\vdots&\vdots&\\\
\dots&\omega^{2m_{n}^{\prime}}&\omega^{m_{n}^{\prime}}&1&\omega^{-m_{n}^{\prime}}&\dots\end{pmatrix}.$

In the matrix $V_{m}$ (on the right), column $\tfrac{1}{2}(n+1)$ (out of
columns $0,1,\dots,n$) is the column whose entries are all equal to $1$.

The relation between $\mathcal{X}$ and $\mathcal{X}_{1}$. Let us write

$\mathcal{X}_{1}(\zeta)=\mathcal{X}(\zeta)B_{1}.$

To calculate $B_{1}$, we begin by writing the first row of $\mathcal{X}_{1}$
as

$(l_{0}(\zeta),\dots,l_{n}(\zeta)).$

Since $\mathcal{X}_{1}\sim\mathcal{X}_{f}$ as $\zeta\to 0$ on $\Omega_{1}$, we
have

$(l_{0}(\zeta),\dots,l_{n}(\zeta))\sim(e^{\frac{1}{\zeta}},e^{\frac{\omega}{\zeta}},\dots,e^{\frac{\omega^{n}}{\zeta}}).$

In particular this holds when $\zeta\to 0$ on the positive real axis
$\zeta>0$. Since

— $g(\zeta)$ must be a linear combination of $l_{0}(\zeta),\dots,l_{n}(\zeta)$

— $g(\zeta)\to 0$ by (3) of Lemma 6.14

— amongst $l_{0}(\zeta),\dots,l_{n}(\zeta)$ only
$l_{\frac{1}{2}(n+1)}(\zeta)\sim e^{-\frac{1}{\zeta}}\to 0$

we conclude that $g(\zeta)$ must be a scalar multiple of
$l_{\frac{1}{2}(n+1)}(\zeta)$. By (3) of Lemma 6.14 again, this scalar is just
${\scriptstyle\sqrt{-1}}\,(2\pi)^{\frac{1}{2}(n+2)}(n+1)^{-\frac{1}{2}}$,
because $\sum_{0}^{n}b_{i}=0$ when $b_{i}=-m_{i}$. We have proved that

(6.12)
$g(\zeta)={\scriptstyle\sqrt{-1}}\,(2\pi)^{\frac{1}{2}(n+2)}(n+1)^{-\frac{1}{2}}l(\zeta),\quad
l(\zeta)=l_{\frac{1}{2}(n+1)}(\zeta).$

Next we shall express $(l_{0}(\zeta),\dots,l_{n}(\zeta))$ in terms of
$l(\zeta)$:

######  Proposition 6.16.

We have $(l_{0}(\zeta),\dots,l_{n}(\zeta))=$

$\left(l(\omega^{\frac{1}{2}(n+1)}\zeta),\ \dots,\
l(\omega\zeta),l(\zeta),l(\omega^{-1}\zeta),\ \dots,\
l(\omega^{-\frac{1}{2}(n-1)}\zeta)\right)P_{m}^{T}$

where $P_{m}^{T}=d_{n+1}^{\frac{1}{2}}\,F\,d_{n+1}^{-\frac{1}{2}}$, and $F$ is
the matrix in formula (3.7) of [13].

###### Proof.

Proposition 3.4 of [13] gives

(6.13) $F\tilde{M}^{(0)}F^{-1}=\begin{pmatrix}-s_{n}&1&0&\cdots&0\\\
-s_{n-1}&0&1&\cdots&0\\\ \vdots&\vdots&\vdots&&\vdots\\\
-s_{1}&0&0&\cdots&1\\\ -1&0&0&\cdots&0\end{pmatrix}.$

Using (6.6), it follows from this that $P_{m}^{T}M^{(0)}P_{m}^{-T}$ has the
same form, but with $s_{1},\dots,s_{n}$ modified. Thus the last $n$ columns of
$P_{m}^{T}M^{(0)}P_{m}^{-T}$ are exactly as in (6.13).

The cyclic symmetry for the o.d.e. (6.1) says that

$d_{n+1}\mathcal{X}^{(0)}_{k}(\omega^{-1}\zeta)\Pi=\mathcal{X}^{(0)}_{k-\scriptstyle\frac{2}{n+1}}(\zeta)$

(as $\mathcal{X}^{(0)}_{k}=h^{-1}\Phi^{(0)}_{k}$ this is equivalent to the
cyclic symmetry for $\Phi^{(0)}_{k}$; the latter is given in section 3 of
[10]). Hence we obtain

$d_{n+1}\mathcal{X}^{(0)}_{1}(\omega^{-1}\zeta)Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi=\mathcal{X}^{(0)}_{1}(\zeta).$

For the first rows of these matrices, this means that

$(l_{0}(\omega^{-1}\zeta),\dots,l_{n}(\omega^{-1}\zeta))M^{(0)}=(l_{0}(\zeta),\dots,l_{n}(\zeta)),$

and therefore

$(l_{0}(\omega^{-1}\zeta),\dots,l_{n}(\omega^{-1}\zeta))P_{m}^{-T}\left(P_{m}^{T}M^{(0)}P_{m}^{-T}\right)=(l_{0}(\zeta),\dots,l_{n}(\zeta))P_{m}^{-T}.$

Let us introduce

$(u_{0},\dots,u_{n})=(l_{0}(\omega^{-1}),\dots,l_{n}(\omega^{-1}))P_{m}^{-T}.$

Then the version of formula (6.13) for $P_{m}^{T}M^{(0)}P_{m}^{-T}$ gives

$u_{i}(\omega^{-1}\zeta)=u_{i+1}(\zeta),\quad 0\leq i\leq n-1.$

We obtain $(u_{0}(\zeta),\dots,u_{n}(\zeta))=$

$(u_{\frac{1}{2}(n+1)}(\omega^{\frac{1}{2}(n+1)}\zeta),\ \dots,\
u_{\frac{1}{2}(n+1)}(\zeta),\ \dots,\
u_{\frac{1}{2}(n+1)}(\omega^{-\frac{1}{2}(n-1)}\zeta))P_{m}^{T}.$

To complete the proof it suffices to show that $u_{\frac{1}{2}(n+1)}$ is
$l_{\frac{1}{2}(n+1)}$ (i.e. $l$). But this is immediate from the definition
of $F$ (formula (3.7) of [13]), as $F=\left(\begin{smallmatrix}L&0\\\
0&U\end{smallmatrix}\right)$ where $L$ and $U$ are, respectively, lower and
upper triangular $\frac{1}{2}(n+1)\times\frac{1}{2}(n+1)$ matrices with all
diagonal entries equal to $1$. ∎

From Proposition 6.16 we obtain

$B_{1}^{-1}={\scriptstyle\sqrt{-1}}\,(2\pi)^{\frac{1}{2}(n+2)}(n+1)^{-\frac{1}{2}}P_{m}^{-T},$

and we arrive at the main result of this section:

######  Theorem 6.17.

$\mathcal{D}_{1}=B_{1}^{-1}B={\scriptstyle\sqrt{-1}}\,(2\pi)^{\frac{1}{2}(n+2)}(n+1)^{-\frac{1}{2}}P_{m}^{-T}V_{m}^{-1}\Gamma_{m}^{-1}$,
where $P_{m},V_{m},\Gamma_{m}$ are as defined above.

###### Proof.

By definition, $\mathcal{X}^{(\infty)}=\mathcal{X}_{1}\mathcal{D}_{1}$. We
have defined $B,B_{1}$ by $\mathcal{X}^{(\infty)}=\mathcal{X}B$ and
$\mathcal{X}_{1}=\mathcal{X}B_{1}$. Thus $\mathcal{D}_{1}=B_{1}^{-1}B$.
Substituting the values of $B,B_{1}$ obtained earlier, we obtain the result. ∎

The connection matrix $D_{1}$ for $\hat{\omega}$ is given by
$D_{1}=\mathcal{D}_{1}h^{-1}$ where $h=\hat{c}t^{m}$ (Proposition 6.1).

######  Remark 6.18.

In the case $n+1=2$ we have $P_{m}=I$. ∎

###  6.4. Computation of the connection matrices $E_{k}$

Theorem 5.2 expresses $E_{1}$ in terms of $D_{1}$, and we have just calculated
$D_{1}$. We shall deduce:

######  Theorem 6.19.

$E_{1}=-(V_{m}P_{m}^{T})^{-1}(\hat{c}\Gamma_{m})^{-1}\Delta\,\hat{c}\Gamma_{m}\,\Delta(V_{m}P_{m}^{T})E_{1}^{\operatorname{id}},$
where $E_{1}^{\operatorname{id}}=\tfrac{1}{n+1}CQ^{(\infty)}_{\frac{n}{n+1}}$.

The significance of $E_{1}^{\operatorname{id}}$ will become clear in the next
section.

###### Proof.

Formula (5.6) gives
$E_{1}=\tfrac{1}{n+1}D_{1}t^{m}\,\Delta\,(\overline{D_{1}t^{m}})^{-1}(\bar{Q}^{(0)}_{\frac{n}{n+1}})^{-1}d_{n+1}^{-1}\,C$.
Let us substitute the value of
$D_{1}t^{m}=\mathcal{D}_{1}h^{-1}t^{m}=\mathcal{D}_{1}\hat{c}^{-1}$ from
Theorem 6.17, which is

$D_{1}t^{m}=\kappa
P_{m}^{-T}V_{m}^{-1}\Gamma_{m}^{-1}\hat{c}^{-1},\quad\kappa={\scriptstyle\sqrt{-1}}\,\,(2\pi)^{\frac{1}{2}(n+2)}(n+1)^{-\frac{1}{2}}.$

Noting that $\bar{\kappa}=-\kappa$ and $\bar{\Gamma}_{m}=-\Gamma_{m}$, we
obtain

$E_{1}=\tfrac{1}{n+1}(V_{m}P_{m}^{T})^{-1}\Gamma_{m}^{-1}\hat{c}^{-1}\Delta\hat{c}\Gamma_{m}\bar{V}_{m}\bar{P}_{m}^{T}(\bar{Q}^{(0)}_{\frac{n}{n+1}})^{-1}d_{n+1}^{-1}C$

By direct calculation we find that

$\bar{V}_{m}=-\Delta V_{m}d_{n+1}.$

We also have

$\bar{P}_{m}^{T}=d_{n+1}^{-1}P_{m}^{T}d_{n+1},$

as this is equivalent to the fact that the matrix
$F=d_{n+1}^{-\frac{1}{2}}P_{m}^{T}d_{n+1}^{\frac{1}{2}}$ (defined in
Proposition 6.16) is real. We obtain

$E_{1}=-(V_{m}P_{m}^{T})^{-1}\Gamma_{m}^{-1}\hat{c}^{-1}\Delta\hat{c}\Gamma_{m}\Delta(V_{m}P_{m}^{T})\tfrac{1}{n+1}d_{n+1}(\bar{Q}^{(0)}_{\frac{n}{n+1}})^{-1}d_{n+1}^{-1}C.$

Now, the $\theta$-reality condition (the general formula is stated later on as
(7.6) in section 7) gives

$(\bar{Q}^{(0)}_{\frac{n}{n+1}})^{-1}=CQ^{(0)}_{\frac{n}{n+1}}C.$

Using the fact that $d_{n+1}C=Cd_{n+1}^{-1}$, we obtain

$d_{n+1}(\bar{Q}^{(0)}_{\frac{n}{n+1}})^{-1}d_{n+1}^{-1}C=d_{n+1}CQ^{(0)}_{\frac{n}{n+1}}Cd_{n+1}^{-1}C=Cd_{n+1}^{-1}Q^{(0)}_{\frac{n}{n+1}}d_{n+1}=CQ^{(\infty)}_{\frac{n}{n+1}}.$

The stated result follows. ∎

Observe that

$(\hat{c}\Gamma_{m})^{-1}\Delta\,\hat{c}\Gamma_{m}\,\Delta=\begin{pmatrix}\frac{\hat{c}_{n}\Gamma^{(n)}_{m}}{\hat{c}_{0}\Gamma^{(0)}_{m}}&&&\\\
&\frac{\hat{c}_{n-1}\Gamma^{(n-1)}_{m}}{\hat{c}_{1}\Gamma^{(1)}_{m}}&&\\\
\phantom{\frac{\hat{c}_{0}\Gamma^{(0)}_{m}}{\hat{c}_{n}\Gamma^{(n)}_{m}}}&&\ddots&\\\
&&&\frac{\hat{c}_{0}\Gamma^{(0)}_{m}}{\hat{c}_{n}\Gamma^{(n)}_{m}}\end{pmatrix}$

where we have written
$\Gamma_{m}=\operatorname{diag}(\Gamma^{(0)}_{m},\dots,\Gamma^{(n)}_{m})$. In
view of this, we make the following definition:

######  Definition 6.20.

For $i=0,1,\dots,n$, let
$e_{i}=-(\hat{c}_{n-i}\Gamma^{(n-i)}_{m})/(\hat{c}_{i}\Gamma^{(i)}_{m})$.

With this notation, Theorem 6.19 becomes:

######  Theorem 6.21.

$E_{1}(E_{1}^{\operatorname{id}})^{-1}=(V_{m}P_{m}^{T})^{-1}\\!\begin{pmatrix}e_{0}&&&\\\
&e_{1}&&\\\ &&\ddots&\\\ &&&e_{n}\end{pmatrix}\\!(V_{m}P_{m}^{T}).$ ∎

Using the values of the $\Gamma^{(i)}_{m}=(-1)^{i}C_{0}^{[i]}$ which were
computed in section 6.3, we find explicit expressions for the $e_{i}$:

######  Corollary 6.22.

For $i=0,1,\dots,n$, we have $e_{i}=$

$\textstyle\frac{\hat{c}_{n-i}}{\hat{c}_{i}}\frac{(n+1)^{m_{n-i}}}{(n+1)^{m_{i}}}\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+1}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+1}}{n+1}\right)}\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+2}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+2}}{n+1}\right)}\cdots\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+n}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+n}}{n+1}\right)}.\qed$

Note that the minus sign in Definition 6.20 is cancelled by the minus signs in
the $\Gamma^{(n-i)}_{m}/\Gamma^{(i)}_{m}$. We recall that $m_{i}+m_{n-i}=0$
and $\hat{c}_{i}\hat{c}_{n-i}=1$ here. As in section 6.3, we extend to
$i\in\mathbb{Z}$ the definitions of $m_{i}$ and $m^{\prime}_{i}=m_{i}-i$ (that
is, we put $m_{i+n+1}=m_{i}$ and $m^{\prime}_{i+n+1}=m^{\prime}_{i}-(n+1)$).

######  Remark 6.23.

The (positive) constants $c,N$ in Definition 2.5 may be chosen freely. Let us
normalize them both to be $1$. Then for $i=0,1,\dots,\tfrac{1}{2}(n+1)$ the
formula for $e_{i}$ can be written very simply in terms of the original data
$c_{i},k_{i}$ of the connection form $\hat{\omega}$ as

$\textstyle e_{i}=\frac{1}{c_{i+1}c_{i+2}\cdots c_{n-i}}\
\frac{\Gamma(\alpha_{n-i+1})}{\Gamma(\alpha_{i+1})}\frac{\Gamma(\alpha_{n-i+1}+\alpha_{n-i+2})}{\Gamma(\alpha_{i+1}+\alpha_{i+2})}\cdots\frac{\Gamma(\alpha_{n-i+1}+\cdots+\alpha_{n-i+n})}{\Gamma(\alpha_{i+1}+\cdots+\alpha_{i+n})}$

where $\alpha_{i}=k_{i}+1$ (and we extend this to $i\in\mathbb{Z}$ by
$\alpha_{i+n+1}=\alpha_{i}$). ∎

######  Remark 6.24.

Substituting $D_{1}t^{m}=\kappa
P_{m}^{-T}V_{m}^{-1}\Gamma_{m}^{-1}\hat{c}^{-1}$ into (5.4) gives a
diagonalization of $Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi$:

$Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi=D_{1}\omega^{m^{\prime}}D_{1}^{-1}\\\
=(V_{m}P_{m}^{T})^{-1}\omega^{m^{\prime}}V_{m}P_{m}^{T}.$

Thus, both $E_{1}(E_{1}^{\operatorname{id}})^{-1}$ and
$Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi$ are diagonalized by
$V_{m}P_{m}^{T}$. This is a fact of independent interest, so it may be
worthwhile making some further comments here.

We could have deduced the diagonalization of
$Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi$ (without calculating
$D_{1}$) from formula (6.13) in the proof of Proposition 6.16, because that
formula says that
$P_{m}^{T}Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi P_{m}^{-T}$ is
in rational normal form, and it is well known that the rational normal form is
diagonalized by a Vandermonde matrix ($V_{m}$ in the present situation).

The simultaneous diagonalizability could also have been deduced a priori, from
the cyclic symmetry (5.2): we have

$\displaystyle(a)\ E_{1}=Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi\
E_{1}\ Q^{(\infty)}_{1}Q^{(\infty)}_{1+\scriptstyle\frac{1}{n+1}}\Pi$
$\displaystyle(b)\
E_{1}^{\operatorname{id}}=Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi\
E_{1}^{\operatorname{id}}\
Q^{(\infty)}_{1}Q^{(\infty)}_{1+\scriptstyle\frac{1}{n+1}}\Pi$

then multiplication of (a) by the inverse of (b) gives

$E_{1}(E_{1}^{\operatorname{id}})^{-1}=Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi\
E_{1}(E_{1}^{\operatorname{id}})^{-1}(Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi)^{-1},$

i.e. $E_{1}(E_{1}^{\operatorname{id}})^{-1}$ commutes with
$Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi$.

Nevertheless, the explicit formula for the eigenvalues of
$E_{1}(E_{1}^{\operatorname{id}})^{-1}$ in Corollary 6.22 depends on the
calculation of $D_{1}$ in this section. ∎

To conclude, we can say that the eigenvalues $e_{0},\dots,e_{n}$ of
$E_{1}(E_{1}^{\operatorname{id}})^{-1}$ and the eigenvalues
$\omega^{m^{\prime}_{0}},\dots,\omega^{m^{\prime}_{n}}$ of
$M^{(0)}=Q^{(0)}_{1}Q^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\Pi$ represent the
monodromy data of the o.d.e. (3.3) corresponding to the solutions of the
tt*-Toda equations (1.1) which were constructed in section 2.3. Similarly, the
tilde versions $\tilde{E}_{1}(\tilde{E}_{1}^{\operatorname{id}})^{-1}$ and
$\tilde{M}^{(0)}=\tilde{Q}^{(0)}_{1}\tilde{Q}^{(0)}_{1+\scriptstyle\frac{1}{n+1}}\hat{\Pi}$
commute. Their eigenvalues are $e_{0},\dots,e_{n}$ and
$\omega^{m^{\prime}_{0}-\frac{1}{2}},\dots,\omega^{m^{\prime}_{n}-\frac{1}{2}}$.

##  7\. Riemann-Hilbert problem

In sections 6.2 and 6.3 we computed the monodromy data associated to certain
local solutions of (1.1) near $t=0$, namely those arising as in Proposition
2.7 from the data $c_{0}z^{k_{0}},\dots,c_{n}z^{k_{n}}$ (with
$c_{i}>0,k_{i}>-1$). The method used the Iwasawa factorization, which can be
interpreted as solving a Riemann-Hilbert problem.

Our aim in this section is to go in the opposite direction: to reconstruct
solutions from monodromy data. This is a different type of Riemann-Hilbert
problem, which we shall solve under certain conditions; in particular we shall
produce local solutions of (1.1) near $t=\infty$, some of which will be global
solutions. How these are related to the local solutions in Proposition 2.7
will be discussed in section 8.

###  7.1. Preparation

As monodromy data we take hypothetical Stokes factors
$Q^{(0)}_{k},Q^{(\infty)}_{k}$ and hypothetical connection matrices $E_{k}$,
motivated by the fact that any local solution of (1.1) (not necessarily at
$t=0$) must give rise to such data as in sections 4.1, 4.2, 5.1. Moreover, the
shape of the Stokes factors must be as described in section 6.2, excluding
only the final statement of Theorem 6.12 which expresses
$s_{1},\dots,s_{\frac{1}{2}(n+1)}$ in terms of $m_{0},\dots,m_{n}$.

Let us choose hypothetical “Stokes parameters”
$s_{1},\dots,s_{\frac{1}{2}(n+1)}$ as in Definition 6.9, hence hypothetical
Stokes factors $Q^{(0)}_{k},Q^{(\infty)}_{k}$. We do not assume that
$s_{1},\dots,s_{\frac{1}{2}(n+1)}$ arise from the solutions in Proposition
2.7; we just take $s_{1},\dots,s_{\frac{1}{2}(n+1)}$ to be arbitrary real
numbers. We choose also hypothetical connection matrices $E_{k}$. We shall
specify $E_{k}$ more precisely in a moment.

From this data, our aim is to construct functions
$\Psi^{(0)}_{k},\Psi^{(\infty)}_{k}$ which are holomorphic on the appropriate
sectors and which satisfy the “jump conditions”

$\Psi^{(0)}_{k+\scriptstyle\frac{1}{n+1}}=\Psi^{(0)}_{k}Q^{(0)}_{k},\quad\Psi^{(\infty)}_{k+\scriptstyle\frac{1}{n+1}}=\Psi^{(\infty)}_{k}Q^{(\infty)}_{k},\quad\Psi^{(\infty)}_{k}=\Psi^{(0)}_{k}E_{k}.$

It will be convenient to replace the connection matrices $E_{k}$ by new
matrices $Z_{k}$, defined by

(7.1)
$\Psi^{(\infty)}_{k}=\Psi^{(0)}_{{\scriptstyle\frac{2n+1}{n+1}-k}}Z_{{\scriptstyle\frac{2n+1}{n+1}-k}},$

as these matrices relate “opposite” sectors (cf. Figure 4 below). From the
definition it follows that

(7.2)
$Z_{k+\scriptstyle\frac{1}{n+1}}=\left(Q^{(0)}_{k}\right)^{-1}Z_{k}\left(Q^{(\infty)}_{{\scriptstyle\frac{2n}{n+1}-k}}\right)^{-1}.$

We recall the analogous relation (5.1) for the $E_{k}$:

(7.3) $E_{k+\scriptstyle\frac{1}{n+1}}=\left(Q^{(0)}_{k}\right)^{-1}E_{k}\
Q^{(\infty)}_{k}.$

For $k=1$ we have the relation

(7.4) $E_{1}=Z_{1}Q^{(\infty)}_{\scriptstyle\frac{n}{n+1}},$

as
$\Psi^{(\infty)}_{1}(\zeta)=\Psi^{(0)}_{1}(\zeta)E_{1}=\Psi^{(\infty)}_{\scriptstyle\frac{n}{n+1}}(\zeta)Z_{1}^{-1}E_{1}=\Psi^{(\infty)}_{1}(\zeta)\left(Q^{(\infty)}_{\scriptstyle\frac{n}{n+1}}\right)^{-1}Z_{1}^{-1}E_{1}.$
Formulae (7.2), (7.3), (7.4) show that the $Z_{k}$ are equivalent to the
$E_{k}$.

We aim to choose specific $E_{k}$ in such a way that all the $Z_{k}$ are the
same, independent of $k$. One such choice of $E_{k}$ is given by:

######  Lemma 7.1.

Let $E_{1}=\tfrac{1}{n+1}CQ^{(\infty)}_{\scriptstyle\frac{n}{n+1}}$. Then
$Z_{k}=\tfrac{1}{n+1}C$ for all $k$.

###### Proof.

Using the symmetries, we shall prove first that

(7.5)
$Q^{(\infty)}_{k}=\left(\tfrac{1}{n+1}C\right)^{-1}\left(Q^{(0)}_{\scriptstyle\frac{2n}{n+1}-k}\right)^{-1}\tfrac{1}{n+1}C$

for all $k$.

To begin, we note that the $\theta$-reality condition (section 3 of [7]) gives

(7.6) $Q^{(0)}_{k}=C\
\left(\,\overline{Q^{(0)}_{{\scriptstyle\frac{2n}{n+1}-k}}}\,\right)^{-1}\ C.$

On the other hand, (4.7) gives:

$\text{(a)}\quad Q^{(\infty)}_{k}=d_{n+1}^{-1}Q^{(0)}_{k}d_{n+1}.$

From (6.4) and Proposition 6.7 we know that all $\tilde{Q}^{(0)}_{k}$ are
real. By (4.4), this means:

$\text{(b)}\quad\overline{Q^{(0)}_{k}}=d_{n+1}^{-1}Q^{(0)}_{k}d_{n+1}.$

Substituting (a) and (b) into (7.6) we obtain (7.5).

To prove the lemma, let us put
$E_{1}=\tfrac{1}{n+1}CQ^{(\infty)}_{\scriptstyle\frac{n}{n+1}}$. By (7.4),
this means $Z_{1}=\tfrac{1}{n+1}C$. Proceeding by induction, if we assume that
$Z_{k}=\tfrac{1}{n+1}C$, then (7.2) and (7.5) give
$Z_{k+\scriptstyle\frac{1}{n+1}}=\tfrac{1}{n+1}C$. This completes the proof. ∎

######  Remark 7.2.

Evidently the scalar $\tfrac{1}{n+1}$ does not play any role in the above
proof; if we set $E_{1}=a\,CQ^{(\infty)}_{\scriptstyle\frac{n}{n+1}}$, then we
obtain $Z_{k}=a\,C$ for all $k$. The choice $a=\tfrac{1}{n+1}$ comes from the
fact (from $\Psi^{(\infty)}_{k}=\Psi^{(0)}_{k}E_{k}$) that $\det
E_{1}=\det\Omega^{-2}=\det\frac{1}{n+1}C$, when $E_{1}$ arises from a solution
of (1.1). (The choice $a=-\tfrac{1}{n+1}$ also gives the same determinant, but
it turns out not to give a (real) solution of (1.1) — cf. section 4 of [10].)
∎

###  7.2. The contour

To formulate a Riemann-Hilbert problem (in the manner of [5], chapter 3) it is
necessary to specify an oriented contour with jump matrices in the punctured
$\zeta$-plane $\mathbb{C}^{\ast}$ (not $\tilde{\mathbb{C}}^{\ast}$). For this
purpose we choose to parametrize angles in the $\zeta$-plane using the
interval $[-\frac{\pi}{n+1},2\pi-\frac{\pi}{n+1})$, i.e.
$[\theta^{(0)}_{1},2\pi+\theta^{(0)}_{1})$.

Figure 3. Contour.

For the contour we take the union of the $2n+2$ singular directions in our
“reference sector” $[-\frac{\pi}{n+1},2\pi-\frac{\pi}{n+1})$, together with
the circle $|\zeta|=x^{2}$ (where $x\in(0,\infty)$ is fixed), as in Figure 3.
These singular directions are:

$\displaystyle-\tfrac{\pi}{n+1}=\theta^{(0)}_{1},\theta^{(0)}_{\scriptstyle\frac{n}{n+1}},\dots,\theta^{(0)}_{-\scriptstyle\frac{n}{n+1}}=2\pi-2\tfrac{\pi}{n+1}\left(<2\pi-\tfrac{\pi}{n+1}\right)$
$\displaystyle-\tfrac{\pi}{n+1}=\theta^{(\infty)}_{\scriptstyle\frac{n-1}{n+1}},\theta^{(\infty)}_{\scriptstyle\frac{n}{n+1}},\dots,\theta^{(\infty)}_{2+\scriptstyle\frac{n-2}{n+1}}=2\pi-2\tfrac{\pi}{n+1}\left(<2\pi-\tfrac{\pi}{n+1}\right)$

To emphasize that we are working in $\mathbb{C}^{\ast}$, let us write
$\,\underline{\theta\\!}\,^{(0)}_{k}$, for these particular values of $k$. For
arbitrary $k$ we define
$\,\underline{\theta\\!}\,^{(0)}_{k}=\,\underline{\theta\\!}\,^{(0)}_{k^{\prime}}$
when $k^{\prime}\cong k$ mod $2\mathbb{Z}$ and $-\frac{n}{n+1}\leq
k^{\prime}\leq 1$. It follows that
$\,\underline{\theta\\!}\,^{(0)}_{k+2}=\,\underline{\theta\\!}\,^{(0)}_{k}$,
in contrast to the relation $\theta^{(0)}_{k+2}=\theta^{(0)}_{k}-2\pi$ which
holds in $\tilde{\mathbb{C}}^{\ast}$. For
$\,\underline{\theta\\!}\,^{(\infty)}_{k}$ we make similar definitions. On the
closed sector of $\mathbb{C}^{\ast}$ bounded by
$\,\underline{\theta\\!}\,^{(0)}_{k},\,\underline{\theta\\!}\,^{(0)}_{k-\scriptstyle\frac{1}{n+1}}$
(inside the circle) we define a holomorphic function
$\,\underline{\\!\Psi\\!}\,^{(0)}_{k}$ by

(7.7)
$\,\underline{\\!\Psi\\!}\,^{(0)}_{k}(\zeta)=\Psi^{(0)}_{k}(\zeta^{\prime})$

where $\zeta^{\prime}\in\Omega^{(0)}_{k}$ covers $\zeta\in\mathbb{C}^{\ast}$.
Similarly we define $\,\underline{\\!\Psi\\!}\,^{(\infty)}_{k}$ on the closed
sector bounded by
$\,\underline{\theta\\!}\,^{(\infty)}_{k-\scriptstyle\frac{1}{n+1}},\,\underline{\theta\\!}\,^{(\infty)}_{k}$
(outside the circle). Whereas the functions
$\Psi^{(0)}_{k},\Psi^{(\infty)}_{k}$ are defined on the universal cover
$\tilde{\mathbb{C}}^{\ast}$, the functions
$\,\underline{\\!\Psi\\!}\,^{(0)}_{k},\,\underline{\\!\Psi\\!}\,^{(\infty)}_{k}$
are defined only on the sectors specified. These are illustrated in Figure 4.
We refer to this diagram as the “Riemann-Hilbert diagram”. The jumps on the
circle are given by the connection matrices $Z_{\ast}$ and the jumps on the
rays are given by the Stokes factors $Q_{\ast}$.

Figure 4. Riemann-Hilbert diagram.

The contours are oriented so that

(7.8)
$\,\underline{\\!\Psi\\!}\,_{\text{left}}=\,\underline{\\!\Psi\\!}\,_{\text{right}}\
\times\ \text{jump matrix}.$

The problem of recovering the functions from the jumps on the contour will be
referred to as “the Riemann-Hilbert problem”. The Riemann-Hilbert diagram in
Figure 4 constructed from the functions
$\,\underline{\\!\Psi\\!}\,^{(0)}_{k},\,\underline{\\!\Psi\\!}\,^{(\infty)}_{k}$
is the model for this. Tautologically, it is solvable (i.e. the functions
$\,\underline{\\!\Psi\\!}\,^{(0)}_{k},\,\underline{\\!\Psi\\!}\,^{(\infty)}_{k}$
exist) if the monodromy data is exactly the data that we have calculated in
section 6. We shall now consider other monodromy data and attempt to find
corresponding functions
$\,\underline{\\!\Psi\\!}\,^{(0)}_{k},\,\underline{\\!\Psi\\!}\,^{(\infty)}_{k}$.
For this purpose it is convenient to modify the Riemann-Hilbert diagram.

If we replace $\Psi^{(0)}_{k}$ by $\Psi^{(0)}_{k}\tfrac{1}{n+1}C$ then all
jumps on the circle will be the identity. The Riemann-Hilbert diagram in
Figure 4 can then be replaced by the simplified Riemann-Hilbert diagram in
Figure 5.

Figure 5. Simplified Riemann-Hilbert diagram.

A further modification will be useful. Bearing in mind the asymptotics

$\displaystyle\Psi^{(0)}_{k}(\zeta)$ $\displaystyle\sim
e^{-w}\,\Omega\left(I+O(\zeta)\right)e^{\frac{1}{\zeta}d_{n+1}},\quad\zeta\to
0$ $\displaystyle\Psi^{(\infty)}_{k}(\zeta)$ $\displaystyle\sim
e^{w}\Omega^{-1}\left(I+O(1/\zeta)\right)e^{x^{2}\zeta
d_{n+1}},\quad\zeta\to\infty$

(from sections 4.1 and 4.2), we shall replace
$\,\underline{\\!\Psi\\!}\,^{(\infty)}_{k}$ by $Y_{k}$, defined as follows:

######  Definition 7.3.

Let
$Y_{k}(\zeta)=(e^{w}\Omega^{-1})^{-1}\,\underline{\\!\Psi\\!}\,^{(\infty)}_{k}(\zeta)e(\zeta)^{-1}$
where $e(\zeta)$ is defined by
$e(\zeta)=e^{\frac{1}{\zeta}d_{n+1}^{-1}}e^{x^{2}\zeta d_{n+1}}$.

The form of $e(\zeta)$ is motivated by:

######  Proposition 7.4.

If $Y_{k}$ arises from any (local) solution $w$ of (1.1), then

(1) $\lim_{\zeta\to 0}Y_{k}(\zeta)=\Omega e^{-2w}\Omega^{-1}$

(2) $\lim_{\zeta\to\infty}Y_{k}(\zeta)=I$

with exponential convergence in both cases.

###### Proof.

As $\zeta\to\infty$, we have
$Y_{k}(\zeta)=(e^{w}\Omega^{-1})^{-1}\,\underline{\\!\Psi\\!}\,^{(\infty)}_{k}(\zeta)e(\zeta)^{-1}=(I+O(1/\zeta))e^{x^{2}\zeta
d_{n+1}}e(\zeta)^{-1}$, from which (2) is immediate.

Since
$\Psi^{(\infty)}_{k}(\zeta)=\Psi^{(0)}_{{\scriptstyle\frac{2n+1}{n+1}-k}}(\zeta)\tfrac{1}{n+1}C$
by (7.1), and $\Psi^{(0)}_{k}(\zeta)\sim
e^{-w}\,\Omega(I+O(\zeta))e^{\frac{1}{\zeta}d_{n+1}}$ as $\zeta\to 0$, we
obtain $\lim_{\zeta\to 0}Y_{k}(\zeta)=\Omega e^{-2w}\Omega\tfrac{1}{n+1}C$.
Here we make use of the fact that $d_{n+1}Cd_{n+1}=C$. As
$\frac{1}{n+1}C=\Omega^{-2}$, we obtain (1). ∎

The modified Riemann-Hilbert diagram (with $Y_{k}$ instead of
$\,\underline{\\!\Psi\\!}\,^{(\infty)}_{k}$) will be the basis of our Riemann-
Hilbert problem: to reconstruct functions $Y_{k}$ starting from the monodromy
data. By (1) of Proposition 7.4, this will produce a solution $w$ of the
tt*-Toda equation. Instead of the jumps $Q^{(\infty)}_{k}$ in Figure 5,
however, the modification gives new jumps $G_{k}(\zeta)$:

######  Definition 7.5.

For $\zeta$ in the ray $\,\underline{\theta\\!}\,^{(\infty)}_{k}$,
$G_{k}(\zeta)=e(\zeta)\,Q^{(\infty)}_{k}e(\zeta)^{-1}$.

The tilde version of $Y_{k}$ is
$\tilde{Y}_{k}(\zeta)=(e^{w}\Omega^{-1}d_{n+1}^{-\frac{1}{2}})^{-1}\,\underline{\\!\tilde{\Psi}\\!}\,^{(\infty)}_{k}(\zeta)e(\zeta)^{-1}.$
We obtain

(7.9) $\tilde{G}_{k}(\zeta)=e(\zeta)\,\tilde{Q}^{(\infty)}_{k}e(\zeta)^{-1}.$

The tilde versions of (1) and (2) in Proposition 7.4 are:

(7.10) $\displaystyle\lim_{\zeta\to
0}\tilde{Y}_{k}(\zeta)=d_{n+1}^{\frac{1}{2}}\Omega
e^{-2w}\Omega^{-1}d_{n+1}^{-\frac{1}{2}}$ (7.11)
$\displaystyle\lim_{\zeta\to\infty}\tilde{Y}_{k}(\zeta)=I.$

###  7.3. The Riemann-Hilbert problem

Figure 6. The Riemann-Hilbert problem.

In this section we take as our starting point the functions $G_{k}$ (or
$\tilde{G}_{k}$) on the contour, as in Figure 6, and seek corresponding
functions $Y_{k}$ (or $\tilde{Y}_{k}$). This is the Riemann-Hilbert problem.
The function $G_{k}$ was defined in terms of the hypothetical Stokes factors
$Q^{(\infty)}_{k}$. We emphasize that $G_{k}$ depends on the variable $x=|t|$
of (1.1) as well as on $\zeta$; thus we have a Riemann-Hilbert problem for
each $x$.

We shall show that the Riemann-Hilbert problem is solvable for sufficiently
large $x$. For this we use the formulation of [5] (pages 102/3).

Our contour is

$\Gamma=\,\underline{\theta\\!}\,^{(\infty)}_{\scriptstyle\frac{n-1}{n+1}}\cup\,\underline{\theta\\!}\,^{(\infty)}_{\scriptstyle\frac{n}{n+1}}\cup\cdots\cup\,\underline{\theta\\!}\,^{(\infty)}_{2+\scriptstyle\frac{n-2}{n+1}}.$

We shall denote by $G,Q^{(\infty)}$ the matrix-valued functions on $\Gamma$
defined respectively by

$\Gamma|_{\,\underline{\theta\\!}\,^{(\infty)}_{k}}=G_{k},\quad
Q^{(\infty)}|_{\,\underline{\theta\\!}\,^{(\infty)}_{k}}=Q^{(\infty)}_{k},$

and similarly for $\tilde{G},\tilde{Q}^{(\infty)}$.

The formulation of [5] requires that (for fixed $x$) $G(\zeta)$ approaches $I$
rapidly (both in $L^{2}$ norm and $L^{\infty}$ norm) as $\zeta\to\infty$ along
any infinite component of $\Gamma$, and we shall verify that $G(\zeta)$ has
this property this next. From the formula (6.4) for
$\tilde{Q}^{(0)}_{k}(=\tilde{Q}^{(\infty)}_{k})$ we have

$\tilde{Q}^{(\infty)}=I+\sum_{i\neq j}s_{i,j}\,e_{i,j},$

so let us work with
$\tilde{G}=d_{n+1}^{\frac{1}{2}}Gd_{n+1}^{-\frac{1}{2}}=e(\zeta)\tilde{Q}^{(\infty)}e(\zeta)^{-1}$
(instead of $G$). To calculate this explicitly, we need to calculate
$e(\zeta)\,e_{i,j}\,e(\zeta)^{-1}$:

######  Lemma 7.6.

For
$\zeta=re^{{\scriptscriptstyle\sqrt{-1}}\,\,\underline{\theta\\!}\,^{(\infty)}_{k}}$,
i.e. for $\zeta$ in the $\,\underline{\theta\\!}\,^{(\infty)}_{k}$ ray, we
have
$e(\zeta)\,e_{i,j}\,e(\zeta)^{-1}=e^{(-rx^{2}-\frac{1}{r})|\omega^{j}-\omega^{i}|}\,e_{i,j}$.

###### Proof.

We have $e^{x^{2}\zeta d_{n+1}}\ e_{i,j}\ e^{-x^{2}\zeta
d_{n+1}}=e^{x^{2}\zeta(\omega^{i}-\omega^{j})}e_{i,j}$. On the ray
$\zeta=re^{{\scriptscriptstyle\sqrt{-1}}\,\,\underline{\theta\\!}\,^{(\infty)}_{k}}$
(with $r>0$) we have

$\displaystyle x^{2}\zeta(\omega^{i}-\omega^{j})$
$\displaystyle=-rx^{2}(\omega^{j}-\omega^{i})e^{{\scriptscriptstyle\sqrt{-1}}\,\,\underline{\theta\\!}\,^{(\infty)}_{k}}$
$\displaystyle=-rx^{2}(\omega^{j}-\omega^{i})e^{-{\scriptscriptstyle\sqrt{-1}}\,\arg(\omega^{j}-\omega^{i})}$
$\displaystyle=-rx^{2}|\omega^{j}-\omega^{i}|,$

as
$\,\underline{\theta\\!}\,^{(\infty)}_{k}=-\theta^{(0)}_{k}=-\arg(\omega^{j}-\omega^{i})$.
Thus, $e^{x^{2}\zeta d_{n+1}}\ e_{i,j}\ e^{-x^{2}\zeta
d_{n+1}}=e^{-rx^{2}|\omega^{j}-\omega^{i}|}\,e_{i,j}$.

Similarly, $e^{\frac{1}{\zeta}d_{n+1}^{-1}}\ e_{i,j}\
e^{-\frac{1}{\zeta}d_{n+1}^{-1}}=e^{\frac{1}{\zeta}(\omega^{-i}-\omega^{-j})}e_{i,j}$,
and

$\displaystyle\overline{\tfrac{1}{\zeta}(\omega^{-i}-\omega^{-j})}$
$\displaystyle=\tfrac{1}{r}(\omega^{i}-\omega^{j})e^{{\scriptscriptstyle\sqrt{-1}}\,\,\underline{\theta\\!}\,^{(\infty)}_{k}}$
$\displaystyle=-\tfrac{1}{r}(\omega^{j}-\omega^{i})e^{-{\scriptscriptstyle\sqrt{-1}}\,\arg(\omega^{j}-\omega^{i})}$
$\displaystyle=-\tfrac{1}{r}|\omega^{j}-\omega^{i}|.$

Thus $e^{\frac{1}{\zeta}d_{n+1}^{-1}}\ e_{i,j}\
e^{-\frac{1}{\zeta}d_{n+1}^{-1}}=e^{-\frac{1}{r}|\omega^{j}-\omega^{i}|}\,e_{i,j}$.
The stated result follows. ∎

The following notation will be useful.

######  Definition 7.7.

For $p=1,\dots,n$, let $L_{p}=|\omega^{p}-\omega^{0}|=2\sin\tfrac{p}{n+1}\pi$.

Using this, our explicit formula for $\tilde{G}$ is:

$\tilde{G}(\zeta)=I+\sum_{i\neq
j}e^{(-rx^{2}-\frac{1}{r})L_{|j-i|}}\,s_{i,j}\,e_{i,j}.$

We can re-parametrize the rays by putting $l=rx$; this gives the more
symmetrical expression

(7.12) $\tilde{G}(\zeta)=I+\sum_{i\neq
j}e^{-x(l+\frac{1}{l})L_{|j-i|}}\,s_{i,j}\,e_{i,j}.$

Evidently $\tilde{G}(\zeta)$ approaches $I$ exponentially as $r\to\infty$ (or
$l\to\infty$). In fact we have

(7.13) $|(\tilde{G}(\zeta)-I)_{ij}|\leq A_{|j-i|}e^{-2L_{|j-i|}x}$

for some constants $A_{|j-i|}$ when $x$ is sufficiently large.

This shows that $G(\zeta)$ approaches $I$ (both in $L^{2}$ norm and
$L^{\infty}$ norm) sufficiently rapidly as to satisfy the solvability
criterion of [5] (Theorem 8.1):

######  Theorem 7.8.

The Riemann-Hilbert problem of Figure 6 is uniquely solvable when $x$ is
sufficiently large. Concretely this means that there is some $R>0$, depending
on the given monodromy data, such that the function $\tilde{Y}$ is defined for
$x\in(R,\infty)$ and has jumps on the contour $\Gamma$ given by $\tilde{G}$.

Furthermore the solution $\tilde{Y}$ satisfies

(7.14)
$\displaystyle\tilde{Y}(0)=I+\tfrac{1}{2\pi{\scriptstyle\sqrt{-1}}\,}\int_{\Gamma}\frac{\tilde{G}(\zeta)-I}{\zeta}\
d\zeta+(O_{i,j})$

as $x\to\infty$, where $(O_{i,j})$ is the matrix with $(i,j)$ entry
$O(e^{-4L_{|j-i|}x})$. ∎

The solution $\tilde{Y}$ (i.e. the collection of piecewise holomorphic
functions $\tilde{Y}_{k}$, with jumps $\tilde{G}_{k}$) produces functions
$\,\underline{\\!\tilde{\Psi}\\!}\,^{(\infty)}_{k}=(e^{w}\Omega^{-1}d_{n+1}^{-\frac{1}{2}})\tilde{Y}_{k}(\zeta)e(\zeta)$
with jumps $\tilde{Q}^{(\infty)}_{k}$.

Moreover, it can be shown these functions satisfy the system (3.1), hence the
corresponding functions $w_{i}:(R,\infty)\to\mathbb{R}$ (defined by formula
(1) of Proposition 7.4) satisfy the tt*-Toda equations (1.1). This argument is
the same as that given in section 3.4 of [10].

We have now achieved our goal of producing local solutions of (1.1) near
$t=\infty$:

######  Theorem 7.9.

Let $s_{1},\dots,s_{\frac{1}{2}(n+1)}$ be real numbers, and let the matrices
$\tilde{Q}^{(\infty)}_{k}$ be defined in terms of
$s_{1},\dots,s_{\frac{1}{2}(n+1)}$ as in section 6. Then there is a unique
solution $w$ of (1.1) on an interval $(R,\infty)$, where $R$ depends on
$s_{1},\dots,s_{\frac{1}{2}(n+1)}$, such that the associated monodromy data is
given by the $Q^{(\infty)}_{k}$ and
$E_{1}^{\operatorname{id}}=\tfrac{1}{n+1}CQ^{(\infty)}_{\scriptstyle\frac{n}{n+1}}$.
∎

In the next section we shall investigate the relation between these solutions
and the local solutions of near $t=0$ which were constructed in section 2.3.

To end this section, we note the important consequence that Theorem 7.8 gives
information on the asymptotics of the solutions at $t=\infty$. Before stating
this, we recall that $w_{i}+w_{n-i}=0$, and $n+1$ is even, so it suffices to
specify the behaviour of $w_{0},\dots,w_{\frac{n-1}{2}}$. In the course of the
proof, certain linear combinations of the $w_{i}$ arise naturally, and we
state the result using these.

######  Theorem 7.10.

Let $w$ be the solution of (1.1) constructed above from
$s_{1},\dots,s_{\frac{1}{2}(n+1)}$. Then, for $1\leq p\leq\frac{n+1}{2}$, we
have

$\displaystyle w_{0}\sin p\tfrac{\pi}{n+1}+w_{1}$ $\displaystyle\sin
3p\tfrac{\pi}{n+1}+\cdots+w_{\frac{n-1}{2}}\sin np\tfrac{\pi}{n+1}$
$\displaystyle=-\tfrac{n+1}{8}\ s_{p}\ (\pi
L_{p}x)^{-\frac{1}{2}}e^{-2L_{p}x}+O(x^{-\frac{3}{2}}e^{-2L_{p}x})$

as $x\to\infty$.

###### Proof.

Using (7.12), the right hand side of formula (7.14) becomes

$I+\tfrac{1}{2\pi{\scriptstyle\sqrt{-1}}\,}\sum_{i\neq
j}s_{i,j}\int_{0}^{\infty}e^{-x(l+\frac{1}{l})L_{|j-i|}}\tfrac{dl}{l}\
e_{i,j}+(O_{i,j}).$

With the notation of section 6 this can be written more compactly as

$I+\tfrac{1}{2\pi{\scriptstyle\sqrt{-1}}\,}\sum_{p=1}^{n}s_{p}\,\int_{0}^{\infty}e^{-x(l+\frac{1}{l})L_{p}}\tfrac{dl}{l}\
\hat{\Pi}\,\vphantom{\hat{\Pi}}^{p}+\sum_{p=1}^{n}\,O(e^{-4L_{p}x})\hat{\Pi}\,\vphantom{\hat{\Pi}}^{p},$

where, as in formula (6.5),

$\hat{\Pi}=\left(\begin{smallmatrix}&1&&&\\\ &&\ \ddots&&\\\ &&&&1\\\
-1&&&\end{smallmatrix}\right)=\omega^{-\frac{1}{2}}d_{n+1}^{-\frac{1}{2}}\Pi\,d_{n+1}^{\frac{1}{2}}.$

From formula (7.10), the left hand side of (7.14) can be replaced by
$d_{n+1}^{\frac{1}{2}}\Omega e^{-2w}\Omega^{-1}d_{n+1}^{-\frac{1}{2}}.$ We
obtain

$d_{n+1}^{\frac{1}{2}}\Omega
e^{-2w}\Omega^{-1}d_{n+1}^{-\frac{1}{2}}=I+\tfrac{1}{2\pi{\scriptstyle\sqrt{-1}}\,}\sum_{p=1}^{n}s_{p}\,\int_{0}^{\infty}e^{-x(l+\frac{1}{l})L_{p}}\tfrac{dl}{l}\
\hat{\Pi}\,\vphantom{\hat{\Pi}}^{p}+\sum_{p=1}^{n}\,O_{p}\hat{\Pi}\,\vphantom{\hat{\Pi}}^{p},$

where $O_{p}=O(e^{-4L_{p}x})$. Now, Laplace’s method gives
$\frac{1}{2\pi}\int_{0}^{\infty}e^{-x(\frac{1}{l}+l)}\frac{dl}{l}=\tfrac{1}{2}(\pi
x)^{-\frac{1}{2}}\,e^{-2x}+O(x^{-\frac{3}{2}}e^{-2x})$ as $x\to\infty$. Taking
logarithms, we obtain

$d_{n+1}^{\frac{1}{2}}\Omega(-2w)\Omega^{-1}d_{n+1}^{-\frac{1}{2}}=\tfrac{1}{{\scriptstyle\sqrt{-1}}\,}\sum_{p=1}^{n}s_{p}\,\tfrac{1}{2}(\pi
L_{p}x)^{-\frac{1}{2}}e^{-2L_{p}x}\
\hat{\Pi}\,\vphantom{\hat{\Pi}}^{p}+\sum_{p=1}^{n}\,O_{p}^{\text{Lap}}\hat{\Pi}\,\vphantom{\hat{\Pi}}^{p},$

where $O_{p}^{\text{Lap}}=O(x^{-\frac{3}{2}}e^{-2L_{p}x})$. We claim next that

(7.15)
$d_{n+1}^{\frac{1}{2}}\Omega(-2w)\Omega^{-1}d_{n+1}^{-\frac{1}{2}}=\hat{w}_{0}+\hat{w}_{1}\hat{\Pi}+\cdots+\hat{w}_{n}\hat{\Pi}\,\vphantom{\hat{\Pi}}^{n},$

where the $\hat{w}_{p}$ are given by

$\hat{w}_{p}=\tfrac{4}{n+1}{\scriptstyle\sqrt{-1}}\,\left[w_{0}\sin
p\tfrac{\pi}{n+1}+w_{1}\sin 3p\tfrac{\pi}{n+1}+\cdots+w_{\frac{n-1}{2}}\sin
np\tfrac{\pi}{n+1}\right].$

Comparison with the previous formula will then give

(7.16) $\hat{w}_{p}=-{\scriptstyle\sqrt{-1}}\,s_{p}\,\tfrac{1}{2}(\pi
L_{p}x)^{-\frac{1}{2}}e^{-2L_{p}x}+O_{p}^{\text{Lap}},$

which is the statement of the theorem.

To prove (7.15), we note first that $\Omega(-2w)\Omega^{-1}$ must be a linear
combination of $I,\Pi,\dots,\Pi^{n}$, because $\Omega d_{n+1}\Omega^{-1}=\Pi$
and any diagonal matrix must be a linear combination of
$I,d_{n+1},\dots,d_{n+1}^{n}$.

Let us write $-2w=v_{0}+v_{1}d_{n+1}+\cdots+v_{n}d_{n+1}^{n}$. As
$I,d_{n+1},\dots,d_{n+1}^{n}$ are the columns of $\Omega$, this is equivalent
to

$\begin{pmatrix}-2w_{0}\\\ -2w_{1}\\\ \vdots\\\
-2w_{n}\end{pmatrix}=\Omega\begin{pmatrix}v_{0}\\\ v_{1}\\\ \vdots\\\
v_{n}\end{pmatrix}.$

The complex conjugate of the formula
$\hat{\Pi}=\omega^{-\frac{1}{2}}d_{n+1}^{-\frac{1}{2}}\Pi
d_{n+1}^{\frac{1}{2}}$ is
$\hat{\Pi}=\omega^{\frac{1}{2}}d_{n+1}^{\frac{1}{2}}\Pi
d_{n+1}^{-\frac{1}{2}}$, and we know that $\Pi=\Omega d_{n+1}\Omega^{-1}$, so
$d_{n+1}^{\frac{1}{2}}\Omega
d_{n+1}\Omega^{-1}d_{n+1}^{-\frac{1}{2}}=\omega^{-\frac{1}{2}}\hat{\Pi}$. From
$-2w=v_{0}+v_{1}d_{n+1}+\cdots+v_{n}d_{n+1}^{n}$ we deduce that

$\displaystyle
d_{n+1}^{\frac{1}{2}}\Omega(-2w)\Omega^{-1}d_{n+1}^{-\frac{1}{2}}$
$\displaystyle=v_{0}+\omega^{-\frac{1}{2}}v_{1}\hat{\Pi}+\cdots+(\omega^{-\frac{1}{2}})^{n}v_{n}\hat{\Pi}\,\vphantom{\hat{\Pi}}^{n}$
$\displaystyle=\hat{w}_{0}+\hat{w}_{1}\hat{\Pi}+\cdots+\hat{w}_{n}\hat{\Pi}\,\vphantom{\hat{\Pi}}^{n},$

where

(7.17) $\begin{pmatrix}\hat{w}_{0}\\\ \hat{w}_{1}\\\ \vdots\\\
\hat{w}_{n}\end{pmatrix}=d_{n+1}^{-\frac{1}{2}}\begin{pmatrix}v_{0}\\\
v_{1}\\\ \vdots\\\
v_{n}\end{pmatrix}=d_{n+1}^{-\frac{1}{2}}\Omega^{-1}\begin{pmatrix}-2w_{0}\\\
-2w_{1}\\\ \vdots\\\ -2w_{n}\end{pmatrix}.$

Let us compute this more explicitly. From (5.5) we have
$\Omega^{-1}=\frac{1}{n+1}\bar{\Omega}$. Using this in (7.17), we see that
$\hat{w}_{0}=0$ and, for $1\leq p\leq\frac{n+1}{2}$, $\hat{w}_{p}=$

$-\tfrac{2}{n+1}\left[(\omega^{-\frac{1}{2}p}-\omega^{\frac{1}{2}p})w_{0}+(\omega^{-\frac{3}{2}p}-\omega^{\frac{3}{2}p})w_{1}+\cdots+(\omega^{-\frac{n}{2}p}-\omega^{\frac{n}{2}p})w_{\frac{n-1}{2}}\right],$

and this is $\frac{4}{n+1}{\scriptstyle\sqrt{-1}}\,(w_{0}\sin
p\tfrac{\pi}{n+1}+w_{1}\sin 3p\tfrac{\pi}{n+1}+\cdots+w_{\frac{n-1}{2}}\sin
np\tfrac{\pi}{n+1})$, as required. ∎

##  8\. Global solutions

Now we shall relate the local solutions at $t=0$ (section 2) to the local
solutions at $t=\infty$ (section 7). We begin by summarizing both
constructions.

###  8.1. Monodromy data for local solutions at $t=0$: summary

In section 2.3, solutions $w=w(|t|)$ of (1.1) on intervals of the form
$(0,\epsilon)$ were constructed from data $(k_{0},\dots,k_{n})$,
$(c_{0},\dots,c_{n})$ with $c_{i}>0$ and $k_{i}>-1$. These solutions have the
asymptotics

$w_{i}\sim-m_{i}\log|t|-\log\hat{c}_{i}$

as $t\to 0$, where

$\displaystyle m$ $\displaystyle=(m_{0},\dots,m_{n})$ $\displaystyle\hat{c}$
$\displaystyle=(\hat{c}_{0},\dots,\hat{c}_{n})$

are equivalent to $(k_{0},\dots,k_{n})$, $(c_{0},\dots,c_{n})$ through the
formulae in Definition 2.5. We have $\hat{c}_{i}>0$ and $m_{i-1}-m_{i}>-1$.

######  Definition 8.1.

We denote by $w^{0}_{m,\hat{c}}$ the solution corresponding to the pair
$(m,\hat{c})$. The domain $(0,\epsilon)$ of $w^{0}_{m,\hat{c}}$ depends on $m$
and $\hat{c}$.

Let

$\mathcal{A}=\\{m\in\mathbb{R}^{n+1}\ |\
m_{i-1}-m_{i}\geq-1,m_{i}+m_{n-i}=0\\}.$

This is a compact convex region of $\mathbb{R}^{n+1}$, of dimension
$\tfrac{1}{2}(n+1)$. (We are continuing to assume that $n+1$ is even in this
section.) As noted at the end of section 2, the condition $m\in\mathcal{A}$ is
a necessary and sufficient condition for the existence of local (radial)
solutions near $t=0$ satisfying $w_{i}\sim-m_{i}\log|t|$. Our solutions
$w^{0}_{m,\hat{c}}$ are of this type, and in fact $m\in\mathring{\mathcal{A}}$
(the interior of $\mathcal{A}$, given by $m_{i-1}-m_{i}>-1$) as we are
considering the generic case.

The monodromy data of $w^{0}_{m,\hat{c}}$ consists of Stokes factors and
connection matrices. The ingredients of this monodromy data are real numbers
$s_{i},e_{i}$ (see Definition 6.9 and Definition 6.20). Writing

$\displaystyle s$ $\displaystyle=(s_{1},\dots,s_{n})$ $\displaystyle e$
$\displaystyle=(e_{0},\dots,e_{n})$

we can say that the “monodromy map” is the map

$\mu:(m,\hat{c})\mapsto(s,e)=(\mu_{1}(m),\mu_{2}(m,\hat{c}))$

because the Stokes factors are given in Theorem 6.12 in terms of $s$, and the
connection matrices are given in Theorem 6.21 and Corollary 6.22 in terms of
$s$ and $e$.

The map $\mu$ is injective. This follows from the injectivity of its first
component $\mu_{1}$:

######  Lemma 8.2.

The map $\mu_{1}:\mathring{\mathcal{A}}\to\mathbb{R}^{n+1}$, $m\mapsto s$, is
injective.

###### Proof.

We have to prove that $s$ determines $m$. First, $s$ determines the unordered
set
$\\{\omega^{m_{0}+\frac{n}{2}},\omega^{m_{1}-1+\frac{n}{2}},\dots,\omega^{m_{n}-n+\frac{n}{2}}\\}$
by Proposition 6.10 and Theorem 6.12. Multiplying by $\omega^{-\frac{n}{2}}$
we obtain the unordered set
$\\{\omega^{m_{0}^{\prime}},\omega^{m_{1}^{\prime}},\dots,\omega^{m_{n}^{\prime}}\\}$.
By (6.10) we have $m^{\prime}_{i}<m^{\prime}_{i-1}$ (for $i\in\mathbb{Z}$),
and by (6.11) we have $|m_{i}^{\prime}-m_{j}^{\prime}|<n+1$ (for $0\leq
i,j\leq n$). As $\omega=e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,/(n+1)}$, these
facts show that $s$ determines $m_{n}^{\prime}<\cdots<m_{0}^{\prime}$, hence
$m$. ∎

Hence $\mu$ is bijective to its image. We conclude that, for the solutions
$w^{0}_{m,\hat{c}}$, the monodromy data $s,e$ is equivalent to the data
$m,\hat{c}$.

###  8.2. Monodromy data for local solutions at $t=\infty$: summary

In section 7.3, solutions $w=w(|t|)$ of (1.1) on intervals of the form
$(R,\infty)$ were constructed from monodromy data $s\in\mathbb{R}^{n}$,
$e=(1,\dots,1)$. (The real number $R$ depends on $s$.) The corresponding
Stokes factors have the same shape as in Theorem 6.12 (without assuming that
$s_{i}$ is the $i$-th symmetric function of powers of $\omega$). The
connection matrix is
$E_{1}^{\operatorname{id}}=\tfrac{1}{n+1}CQ^{(\infty)}_{\frac{n}{n+1}}$.

######  Definition 8.3.

We denote by $w^{\infty}_{s}$ the solution corresponding to
$s\in\mathbb{R}^{n}$.

######  Lemma 8.4.

Let $m\in\mathring{\mathcal{A}}$. Let $s=\mu_{1}(m)$. Then there exists
exactly one $\hat{c}$ with the property $\mu(m,\hat{c})=(s,(1,\dots,1))$. We
denote this by $\hat{c}^{\operatorname{id}}$.

###### Proof.

By Corollary 6.22 the condition $\mu_{2}(m,\hat{c})=(1,\dots,1)$ means

$\textstyle(\hat{c}_{i})^{2}=\scriptstyle{(n+1)}^{m_{n-i}-m_{i}}\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+1}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+1}}{n+1}\right)}\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+2}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+2}}{n+1}\right)}\cdots\frac{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{n-i}-m^{\prime}_{n-i+n}}{n+1}\right)}{{\displaystyle\Gamma}\left(\frac{m^{\prime}_{i}-m^{\prime}_{i+n}}{n+1}\right)}.$

This determines $\hat{c}_{i}$ as all $\hat{c}_{i}$ are positive. ∎

It follows that the monodromy data of $w^{\infty}_{s}$ coincides with the
monodromy data of $w^{0}_{m,\hat{c}^{\operatorname{id}}}$ if $m$ corresponds
to $s$, i.e. $s=\mu_{1}(m)$. We are going to prove that, when
$m\in\mathring{\mathcal{A}}$, there exist global solutions which coincide near
$t=0$ with $w^{0}_{m,\hat{c}^{\operatorname{id}}}$ and near $t=\infty$ with
$w^{\infty}_{s}$.

###  8.3. Global solutions and their monodromy data

When $m=0$ we have $s=0$, and this data corresponds to the trivial solution
$w=0$ of (1.1), which (obviously) is a globally smooth solution. The following
result, which makes use of most of the calculations in this article, extends
this to an open neighbourhood of the trivial solution:

######  Theorem 8.5.

There exists an open neighbourhood $V$ of $0$ in
$\mu_{1}(\mathring{\mathcal{A}})$ such that, if $s\in V$, then the local
solution $w^{\infty}_{s}$ at $t=\infty$ is in fact smooth on the interval
$0<|t|<\infty$, i.e. is a globally smooth solution.

###### Proof.

The strategy of the proof is the same as that in section 5 of [10], using the
“Vanishing Lemma”. Namely, we shall show that the “homogeneous” Riemann-
Hilbert problem, in which the condition $Y|_{\zeta=\infty}=I$ is replaced by
the condition $Y|_{\zeta=\infty}=0$, has only the trivial solution $Y\equiv
0$. It follows from this (Corollary 3.2 of [5]) that the original problem is
solvable.

We shall work with a modified (but equivalent) Riemann-Hilbert problem on the
contour consisting of the two rays with arguments
$\scriptstyle\frac{\pi}{2(n+1)}$, $\pi+\scriptstyle\frac{\pi}{2(n+1)}$ (Figure
7). This contour divides $\mathbb{C}$ into two half-planes. We denote the
upper region by $\mathbb{C}_{+}$ and the lower region by $\mathbb{C}_{-}$.
Explicitly,

$\mathbb{C}_{+}=\\{\zeta\in\mathbb{C}^{\ast}\ |\
\tfrac{\pi}{2(n+1)}<\text{arg}\zeta<\pi+\tfrac{\pi}{2(n+1)}\\}.$

Our Riemann-Hilbert problem is motivated (in analogy with section 7.2) by
considering first the holomorphic functions
$\,\underline{\\!\Psi\\!}\,^{(\infty)}_{\scriptstyle 3/2}$ on $\mathbb{C}_{+}$
and $\,\underline{\\!\Psi\\!}\,^{(\infty)}_{5/2}$ on $\mathbb{C}_{-}$. By
$\,\underline{\\!\Psi\\!}\,^{(\infty)}_{3/2},\,\underline{\\!\Psi\\!}\,^{(\infty)}_{5/2}$
we mean the functions defined on $\bar{\mathbb{C}}_{+},\bar{\mathbb{C}}_{-}$
by the procedure of section 7.2, i.e. they are obtained from
$\Psi^{(\infty)}_{3/2}$, $\Psi^{(\infty)}_{5/2}$ by projecting to the
reference sector $[-\frac{\pi}{n+1},2\pi-\frac{\pi}{n+1})$.

Figure 7. $\mathbb{C}_{+}$ (above heavy line) and $\mathbb{C}_{-}$ (below
heavy line).

Let us recall that the sectors $\Omega^{(\infty)}_{3/2}$,
$\Omega^{(\infty)}_{5/2}$ in $\tilde{\mathbb{C}}^{\ast}$ (which were used to
define $\Psi^{(\infty)}_{3/2}$, $\Psi^{(\infty)}_{5/2}$) were defined,
respectively, by the conditions
$0<\text{arg}\zeta<\pi+\scriptstyle\frac{\pi}{n+1}$,
$\pi<\text{arg}\zeta<2\pi+\scriptstyle\frac{\pi}{n+1}$. These are indicated by
the heavy circular segments in Figure 7. It follows that
$\,\underline{\\!\Psi\\!}\,^{(\infty)}_{3/2}(\zeta)=\Psi^{(\infty)}_{3/2}(\zeta)$
for $\zeta\in\bar{\mathbb{C}}_{+}$. On the other hand
$\,\underline{\\!\Psi\\!}\,^{(\infty)}_{5/2}(\zeta)=\Psi^{(\infty)}_{5/2}(\zeta)$
only when
$\pi+{\scriptstyle\frac{\pi}{2(n+1)}}\leq\arg\zeta<2\pi-\scriptstyle\frac{\pi}{n+1})$;
for
${-\scriptstyle\frac{\pi}{n+1}}\leq\arg\zeta\leq\scriptstyle\frac{\pi}{2(n+1)}$
we have
$\,\underline{\\!\Psi\\!}\,^{(\infty)}_{5/2}(\zeta)=\Psi^{(\infty)}_{5/2}(e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,}\zeta)$.

We need the jump matrices on the rays which separate $\mathbb{C}_{+}$ from
$\mathbb{C}_{-}$. For $\zeta$ on the $\pi+\scriptstyle\frac{\pi}{2(n+1)}$ ray
we have
$\,\underline{\\!\Psi\\!}\,^{(\infty)}_{\scriptstyle\frac{3}{2}}(\zeta)=\Psi^{(\infty)}_{\scriptstyle\frac{3}{2}}(\zeta)$,
$\,\underline{\\!\Psi\\!}\,^{(\infty)}_{\scriptstyle\frac{5}{2}}(\zeta)=\Psi^{(\infty)}_{\scriptstyle\frac{5}{2}}(\zeta)$,
so the jump matrix is $S^{(\infty)}_{3/2}$. On the $\frac{\pi}{2(n+1)}$ ray we
have
$\,\underline{\\!\Psi\\!}\,^{(\infty)}_{3/2}(\zeta)=\Psi^{(\infty)}_{3/2}(\zeta)$,
but
$\,\underline{\\!\Psi\\!}\,^{(\infty)}_{5/2}(\zeta)=\Psi^{(\infty)}_{5/2}(e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,}\zeta)$,
which is equal to $\Psi^{(\infty)}_{1/2}(\zeta)$ by formula (4.6). It follows
that the jump matrix here is $S^{(\infty)}_{1/2}$.

Next, we replace the $\,\underline{\\!\Psi\\!}\,^{(\infty)}_{k}$ by
$Y_{k}(\zeta)=(e^{w}\Omega^{-1})^{-1}\,\underline{\\!\Psi\\!}\,^{(\infty)}_{k}(\zeta)e(\zeta)^{-1}$
as in Definition 7.3, where
$e(\zeta)=e^{\frac{1}{\zeta}d_{n+1}^{-1}}e^{x^{2}\zeta d_{n+1}}$. Then the
jump matrices become $eS^{(\infty)}_{3/2}e^{-1}$, $eS^{(\infty)}_{5/2}e^{-1}$.

Motivated by this, we can now formulate our Riemann-Hilbert problem: to
reconstruct (holomorphic) functions $Y_{3/2},Y_{5/2}$ on
$\bar{\mathbb{C}}_{+},\bar{\mathbb{C}}_{-}$ such that the jump matrix on the
$\scriptstyle\frac{\pi}{2(n+1)}$ ray is $eS^{(\infty)}_{3/2}e^{-1}$, and the
jump matrix on the $\pi+\scriptstyle\frac{\pi}{2(n+1)}$ ray is
$eS^{(\infty)}_{5/2}e^{-1}$.

In order to apply the theory of [5] we must show that these jump matrices
approach $I$ exponentially as $\zeta\to\infty$ along either ray. For this we
need the following modification of Lemma 7.6, which can be proved in exactly
the same way:

######  Lemma 8.6.

For
$\zeta=re^{{\scriptscriptstyle\sqrt{-1}}\,(\,\underline{\theta\\!}\,^{(\infty)}_{k}+\alpha)}$,
i.e. for $\zeta$ in the $\,\underline{\theta\\!}\,^{(\infty)}_{k}+\alpha$ ray
($\alpha\in\mathbb{R}$),
$e(\zeta)\,e_{i,j}\,e(\zeta)^{-1}=e^{(-rx^{2}-\frac{1}{r})|\omega^{j}-\omega^{i}|\cos\alpha+{\scriptscriptstyle\sqrt{-1}}\,(-rx^{2}+\frac{1}{r})|\omega^{j}-\omega^{i}|\sin\alpha}\,e_{i,j}$.
∎

Let us consider now the behaviour of $eS^{(\infty)}_{3/2}e^{-1}$ on the
$\pi+\scriptstyle\frac{\pi}{2(n+1)}$ ray. We have

$S^{(\infty)}_{\scriptstyle\frac{3}{2}}=Q^{(\infty)}_{\scriptstyle\frac{3}{2}}\dots
Q^{(\infty)}_{1+\scriptstyle\frac{n}{n+1}}Q^{(\infty)}_{2}\dots
Q^{(\infty)}_{{\scriptstyle\frac{3}{2}}+\scriptstyle\frac{n}{n+1}}$

so it suffices show that $eQ^{(\infty)}_{k}e^{-1}$ approaches $I$
exponentially on the $\pi+\scriptstyle\frac{\pi}{2(n+1)}$ ray as
$\zeta\to\infty$, for each
$k\in\\{\frac{3}{2},\frac{3}{2}+\frac{1}{n+1},\dots,\frac{3}{2}+\frac{n}{n+1}\\}$.

Adjacent $\,\underline{\theta\\!}\,^{(\infty)}_{k}$ rays are separated by
$\scriptstyle\frac{\pi}{n+1}$, and the $\pi+\scriptstyle\frac{\pi}{2(n+1)}$
ray lies strictly between the
$\,\underline{\theta\\!}\,^{(\infty)}_{1+\scriptstyle\frac{n}{n+1}}$ and
$\,\underline{\theta\\!}\,^{(\infty)}_{2}$ rays. It follows that, if we write
$\pi+{\scriptstyle\frac{\pi}{2(n+1)}}=\,\underline{\theta\\!}\,^{(\infty)}_{k}+\alpha_{k}$,
then we have $-\frac{\pi}{2}<\alpha_{k}<\frac{\pi}{2}$ for each
$k\in\\{\frac{3}{2},\frac{3}{2}+\frac{1}{n+1},\dots,\frac{3}{2}+\frac{n}{n+1}\\}$.
As all $\cos\alpha_{k}>0$ here, the lemma shows that all
$eQ^{(\infty)}_{k}e^{-1}$ approach $I$ exponentially as $\zeta\to\infty$.

A similar argument applies to the jump matrix on the
$\scriptstyle\frac{\pi}{2(n+1)}$ ray.

As in the case of the Riemann-Hilbert problem in section 7.3, it follows that
the Riemann-Hilbert problem (with $Y|_{\zeta=\infty}=I$) is solvable when $x$
is sufficiently large. The Vanishing Lemma argument will allow us to establish
solvability, not just for large $x$, but for all $x\in(0,\infty)$.

For this we consider the homogeneous Riemann-Hilbert problem, in which we
require $Y|_{\zeta=\infty}=0$ (in contrast to the Riemann-Hilbert problem
based on the functions $\,\underline{\\!\Psi\\!}\,^{(\infty)}$, for which we
had $Y|_{\zeta=\infty}=I$).

For brevity let us denote the $\scriptstyle\frac{\pi}{2(n+1)}$ ray by
$\Gamma_{1/2}$ and the $\pi+\scriptstyle\frac{\pi}{2(n+1)}$ ray by
$\Gamma_{3/2}$, and the jump matrices on these rays by $G_{1/2}$, $G_{3/2}$.
If $Y^{0}_{5/2}$, $Y^{0}_{3/2}$ solve the homogeneous Riemann-Hilbert problem,
we aim to find sufficient conditions which ensure that $Y^{0}_{5/2}=0$,
$Y^{0}_{3/2}=0$.

As in Proposition 5.1 of [10], Cauchy’s Theorem implies that

(a) $\displaystyle\int_{\Gamma}Y^{0}_{5/2}(\zeta)\
\overline{Y^{0}_{3/2}(\bar{\zeta}e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,/2n+2})}^{\,T}\
d\zeta=0$

(b) $\displaystyle\int_{\Gamma}Y^{0}_{3/2}(\zeta)\
\overline{Y^{0}_{5/2}(\bar{\zeta}e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,/2n+2})}^{\,T}\
d\zeta=0$

where $\Gamma=\Gamma_{3/2}\cup\Gamma_{1/2}$. Let us add (a) and (b) together,
and use the fact that $Y^{0}_{5/2}=Y^{0}_{3/2}G_{3/2}$ on $\Gamma_{3/2}$, and
$Y^{0}_{3/2}=Y^{0}_{5/2}G_{1/2}$ on $\Gamma_{1/2}$. From the resulting
equation we deduce that, if
$G_{3/2}(\zeta)+\overline{G_{3/2}(\bar{\zeta}e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,/2n+2})}^{\,T}>0$
on $\Gamma_{3/2}$, and
$G_{1/2}(\zeta)+\overline{G_{1/2}(\bar{\zeta}e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,/2n+2})}^{\,T}>0$
on $\Gamma_{1/2}$, then $Y^{0}_{5/2}$ and $Y^{0}_{3/2}$ must both be
identically zero.

For any $s$, and for $x$ sufficiently large this criterion is satisfied, as
$G_{1/2}$ and $G_{3/2}$ are then close to the identity, so we recover the fact
stated earlier that the original Riemann-Hilbert problem is solvable near
$x=\infty$. For $s=0$ the criterion is satisfied (for all $x$), as then
$G_{1/2}=I=G_{3/2}$. Let us now consider the behaviour of
$H_{3/2}=G_{3/2}(\zeta)+\overline{G_{3/2}(\bar{\zeta}e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,/2n+2})}^{\,T}$
and
$H_{1/2}=G_{1/2}(\zeta)+\overline{G_{1/2}(\bar{\zeta}e^{2\pi{\scriptscriptstyle\sqrt{-1}}\,/2n+2})}^{\,T}$
when $s$ is close to zero. A Hermitian matrix is positive definite if and only
if its minors are positive, and (in our situation, by the formula for $G$ and
the lemma above) the minors are polynomials in variables
$\lambda_{i,j}s_{i,j}$ where $|\lambda_{i,j}|<1$ for all $i,j$.

If all $\lambda_{i,j}s_{i,j}$ are replaced by $1$ (i.e. $e$ is replaced by
$I$), then we obtain the matrices