Isomonodromy for the Degenerate Fifth Painlevé Equation

Isomonodromy for the Degenerate
Fifth Painlevé Equation

Primitivo B. ACOSTA-HUMÁNEZ †, Marius VAN DER PUT ‡ and Jaap TOP ‡

P.B. Acosta-Humánez, M. van der Put and J. Top

† Universidad Simón Bolívar, Barranquilla, Colombia
<EMAIL_ADDRESS>
${\ddagger}$ University of Groningen, Groningen, The Netherlands
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received December 12, 2016, in final form May 01, 2017; Published online May
09, 2017

This is a sequel to papers by the last two authors making the Riemann–Hilbert
correspondence and isomonodromy explicit. For the degenerate fifth Painlevé
equation, the moduli spaces for connections and for monodromy are explicitly
computed. It is proven that the extended Riemann–Hilbert morphism is an
isomorphism. As a consequence these equations have the Painlevé property and
the Okamoto–Painlevé space is identified with a moduli space of connections.
Using MAPLE computations, one obtains formulas for the degenerate fifth
Painlevé equation, for the Bäcklund transformations.

moduli space for linear connections; irregular singularities; Stokes matrices;
monodromy spaces; isomonodromic deformations; Painlevé equations

33E17; 14D20; 14D22; 34M55

##  1 Introduction

In the series of papers [17, 18, 19, 21, 22] on isomonodromy families for
Painlevé equations the cases ${\rm P}_{\rm I}$–${\rm P}_{\rm IV}$ are treated.
Here we apply our methods to ${\rm degP}_{\rm V}$, the degenerate fifth
Painlevé equation. We hope to extend this in a later paper to ${\rm P}_{\rm
V}$. We now describe the method of the Riemann–Hilbert correspondence for
${\rm degP}_{\rm V}$, following closely [21, 22].

The degenerate fifth Painlevé equation ${\rm degP}_{\rm
V}(\theta_{0},\theta_{1})$ depends on two parameters $\theta_{0}$,
$\theta_{1}$ and the corresponding isomonodromy family is given, to begin
with, by the set ${\bf S}(\theta_{0},\theta_{1})$ of differential modules $M$
over $\mathbb{C}(z)$ defined by: $\dim M=2$; the exterior product
$\Lambda^{2}M$ is trivial; the points $0$, $1$ are regular singular with local
exponents $\pm\frac{\theta_{0}}{2}$ and $\pm\frac{\theta_{1}}{2}$. Finally
$z=\infty$ is irregular singular with Katz invariant $\frac{1}{2}$, which
means that the ‘generalized eigenvalues’ at $z=\infty$ are $\pm t\cdot
z^{1/2}$ with $t\in\mathbb{C}^{*}$.

This set is made into an algebraic variety
$\mathcal{M}(\theta_{0},\theta_{1})$ which is a moduli space for connections
on a fixed bundle on $\mathbb{P}^{1}$ of rank two and degree $-1$ with
prescribed data (see Section 2.1) at $z=0,1,\infty$. If $\theta_{0}\neq 0$,
$\theta_{1}\neq 0$, then $\mathcal{M}(\theta_{0},\theta_{1})$ is a fine modul
space and is smooth. For $\theta_{0}=0$ and/or $\theta_{1}=0$, the moduli
problem will be changed by adding “an invariant line”. This is called a
parabolic structure in the literature, see [4, 5, 6, 7]. It leads to a fine
moduli space $\mathcal{M}^{+}(\theta_{0},\theta_{1})$ which is a
desingularisation of $\mathcal{M}(\theta_{0},\theta_{1})$.

The analytic data attached to modules in ${\bf S}(\theta_{0},\theta_{1})$ are
two monodromy matrices and one Stokes matrix. They produce a ‘monodromy space’
$\mathcal{R}(s_{0},s_{1})$ depending on $s_{0}=e^{\pi i\theta_{0}}+e^{-\pi
i\theta_{0}}$ and $s_{1}=e^{\pi i\theta_{1}}+e^{-\pi i\theta_{1}}$. For
$s_{0}\neq\pm 2$ and $s_{1}\neq\pm 2$ the monodromy space is a fine moduli
space and is smooth. It is in fact a smooth affine cubic surface with three
lines at infinity. For the other cases one changes the moduli problem by
adding an ‘invariant line’. The fine moduli space
$\mathcal{R}^{+}(s_{0},s_{1})$ for these new data is a minimal resolution of
$\mathcal{R}(s_{0},s_{1})$.

The Stokes matrix attached to a module in ${\bf S}(\theta_{0},\theta_{1})$
depends on the choice of a summation direction at $z=\infty$. This direction
has to be different from the singular direction at $z=\infty$ which turns
around $\infty$ for $t$ varying in $T:=\mathbb{C}^{*}$. Therefore there is a
locally, with respect to $t\in\mathbb{C}^{*}$, defined analytic
Riemann–Hilbert map ${\rm
RH}\colon\mathcal{M}^{+}(\theta_{0},\theta_{1})\rightarrow\mathcal{R}^{+}(s_{0},s_{1})$
with $s_{0}=e^{\pi i\theta_{0}}+e^{-\pi i\theta_{0}}$ and $s_{1}=e^{\pi
i\theta_{1}}+e^{-\pi i\theta_{1}}$. After replacing $T$ by its universal
covering $\tilde{T}=\mathbb{C}$ the morphism ${\rm RH}$ is well defined. The
main result is that the extended Riemann–Hilbert map

$\displaystyle{\rm RH}^{+}\colon\
\mathcal{M}^{+}(\theta_{0},\theta_{1})\times_{T}\tilde{T}\rightarrow\mathcal{R}^{+}(s_{0},s_{1})\times\tilde{T}$

is an analytic isomorphism. It follows from this that ${\rm degP}_{\rm V}$ has
the Painlevé property and that $\mathcal{M}^{+}(\theta_{0},\theta_{1})$
coincides with the Okamoto–Painlevé space (see [23, 24, 25] for this subject).
The explicit computations of the spaces
$\mathcal{M}^{+}(\theta_{0},\theta_{1})$ and $\mathcal{R}^{+}(s_{0},s_{1})$
lead, using MAPLE, to formulas for ${\rm degP}_{\rm V}$ and for the Bäcklund
transformations.

We note that our definition of ${\rm degP}_{\rm V}$ is not quite the same as
the ‘classical’ degenerate fifth Painlevé equation. The close relation between
the two is given in Section 3. We were informed by Y. Ohyama about the
‘equivalence’ between ${\rm degP}_{\rm V}$ and ${\rm P}_{\rm III}(D_{6})$
found by V.I. Gromak [3]. A Hamiltonian for ${\rm P}_{\rm III}(D_{6})$ is

$\displaystyle\frac{1}{t}\big{(}q^{2}p^{2}-\big{(}q^{2}-(\alpha+\beta)q-t\big{)}p-\alpha
q\big{)}.$

The degrees in $p$ and $q$ are $\leq 2$. Therefore the Hamilton equations
allow to eliminate $p$ (in terms of $q$, $q^{\prime}$) and also to eliminate
$q$ (in terms of $p$, $p^{\prime}$). In the first case one obtains ${\rm
P}_{\rm III}(D_{6})$ and in the second case the classical degenerate fifth
Painlevé equation.

This equivalence does not seem to produce a relation between the isomonodromy
families for ${\rm degP}_{\rm V}$ and for ${\rm P}_{\rm III}(D_{6})$.

Apart from the references given at the beginning of this introduction, several
sources discuss geometric aspects of Painlevé equations. Many of these can be
found in the references of our papers [17, 18, 19, 21, 22]. Classical’ papers
on the subject are [8, 9, 12, 13, 14, 15, 16]. Especially relevant for the
present text are the paper by Ohyama and Okumura [11], Witte’s paper [26]. The
book [2] by Fokas, Its, Kapaev, and Novokshenov discusses more analytic
aspects of the Riemann–Hilbert correspondence, but it does not discuss the
degenerate fifth Painlevé equation. Finally, we mention the recent paper [1]
by Chekhov, Mazzocco, and Rubtsov which also provides an interesting geometric
appoach and overview.

##  2 The moduli space $\boldsymbol{\mathcal{M}(\theta_{0},\theta_{1})}$ of
connections

###  2.1 Definition of $\boldsymbol{{\bf S}(\theta_{0},\theta_{1})}$

An element of the set ${\bf S}(\theta_{0},\theta_{1})$ is (the isomorphy class
of) a tuple $(M,\theta_{0},\theta_{1},t)$ where $M$ is a differential module
over $\mathbb{C}(z)$ such that $\dim M=2$; $\det M:=\Lambda^{2}M$ is the
trivial module; $M$ has three singular points $0$, $1$, $\infty$ and their
Katz invariants are $r(0)=0$, $r(1)=0$, $r(\infty)=1/2$. Further, the
singularities are represented:

  * •

at $z=0$ by $\frac{d}{dz}+\frac{1}{z}\left(\begin{smallmatrix}\omega_{0}&*\\\
0&-\omega_{0}\end{smallmatrix}\right)$ with $\omega_{0}=\frac{\theta_{0}}{2}$
and $\theta_{0},*\in\mathbb{C}$;

  * •

at $z=1$ by
$\frac{d}{dz}+\frac{1}{z-1}\left(\begin{smallmatrix}\omega_{1}&*\\\
0&-\omega_{1}\end{smallmatrix}\right)$ with $\omega_{1}=\frac{\theta_{1}}{2}$
and $\theta_{1},*\in\mathbb{C}$;

  * •

at $z=\infty$ by $z\frac{d}{dz}+\left(\begin{smallmatrix}\omega_{\infty}&0\\\
0&-\omega_{\infty}\end{smallmatrix}\right)$ with $\omega_{\infty}=tz^{1/2}$
and $t\in\mathbb{C}^{*}$.

Let $\partial$ denote the given differential operator on $M$, corresponding
with the derivation $\frac{d}{dz}$ on $\mathbb{C}(z)$, i.e.,
$\partial(fm)=\frac{df}{dz}m+f\partial(m)$ for $f\in\mathbb{C}(z)$ and $m\in
M$.

The condition at $z=0$ means that $\mathbb{C}((z))\otimes M$ has a
$\mathbb{C}[[z]]$-lattice with a basis such that the matrix of $z\partial$ is
$\left(\begin{smallmatrix}\omega_{0}&*\\\
0&-\omega_{0}\end{smallmatrix}\right)$. For $\theta_{0}\neq 0$ one can always
take $*=0$. For $\theta_{0}=0$, any $*$ is admissible. At $z=1$ the condition
is similar for the matrix of $(z-1)\partial$.

The condition at $z=\infty$ means that $\mathbb{C}((z^{-1/2}))\otimes M$ has a
basis for which $z\partial$ has the matrix
$\left(\begin{smallmatrix}\omega_{\infty}&0\\\
0&-\omega_{\infty}\end{smallmatrix}\right)$. In the construction of the moduli
space $\mathcal{M}(\theta_{0},\theta_{1})$ this matrix will be changed in a
matrix defined over $\mathbb{C}((z^{-1}))$.

There is an obvious bijection ${\bf S}(\theta_{0},\theta_{1})\rightarrow{\bf
S}(\theta_{0}+a,\theta_{1}+b)$ for any $a,b\in 2\mathbb{Z}$, obtained by
changing the lattices for $z=0$ and $z=1$.

###  2.2 Choosing connections

We represent $M\in{\bf S}(\theta_{0},\theta_{1})$ by a connection
$\nabla\colon\mathcal{V}\rightarrow\Omega([0]+[1]+2[\infty])\otimes\mathcal{V}$,
where $\mathcal{V}$ is a vector bundle on $\mathbb{P}^{1}$ of rank two. Thus
the generic fiber of $\nabla$ is the map $M\rightarrow
M\otimes_{\mathbb{C}(z)}\mathbb{C}(z)dz$, given by $m\mapsto\partial(m)dz$.

The vector bundle $\mathcal{V}$ is determined by the choice of the lattices at
every point of $\mathbb{P}^{1}$. For the points $z=a\neq 0,1,\infty$ the
lattice in $\mathbb{C}((z-a))\otimes M$ is the $\mathbb{C}[[z-a]]$-module
generated by $\ker(\partial,\mathbb{C}((z-a))\otimes M)$. For the points $z=0$
and $z=1$ we choose the lattices corresponding to the given $\theta_{0}$,
$\theta_{1}$. In other words, the characteristic polynomials of the matrices
of $z\partial$ and $(z-1)\partial$ are prescribed by
$X^{2}-\frac{\theta_{0}^{2}}{4}$ and $X^{2}-\frac{\theta_{1}^{2}}{4}$.

At $z=\infty$ the situation is more complicated. An invariant lattice at
$\infty$ is represented over the differential field $\mathbb{C}((z^{-1/2}))$
by $z\frac{d}{dz}+\left(\begin{smallmatrix}tz^{1/2}&0\\\
0&-tz^{1/2}\end{smallmatrix}\right).$ We need an expression over the field
$\mathbb{C}((z^{-1}))$ or “invariant lattices” for
$\mathbb{C}((z^{-1}))\otimes M$. For Katz invariant $1/2$ or $1$ at $\infty$,
a lattice $\Lambda$ is called invariant if $z\partial\Lambda\subset\Lambda$.

We adopt here the terminology [20] for the classification of differential
modules over $\mathbb{C}((z^{-1}))$.

The formal solution space $V$ at $z=\infty$ is described as $V=V_{q}\oplus
V_{-q}=\mathbb{C}e_{1}\oplus\mathbb{C}e_{2}$ with $q=tz^{1/2}$ and the formal
monodromy $\gamma$ is given by $\gamma(e_{1})=e_{2}$ and
$\gamma(e_{2})=-e_{1}$. Indeed, the determinant of $M$ is trivial and hence
$\det(\gamma)=1$. The (formal local) differential module
$\mathbb{C}((z^{-1}))\otimes M$ and its invariant lattices are now obtained by
considering the invariants of $\mathcal{U}\otimes_{\mathbb{C}}V$ (here
$\mathcal{U}$ is the universal Picard–Vessiot ring for $\mathbb{C}((z^{-1}))$)
under the actions of the differential automorphisms of $\mathcal{U}$ over
$\mathbb{C}((z^{-1}))$. A computation yields invariant lattices $\Lambda_{1}$
and $\Lambda_{2}$ represented by

$\displaystyle z\frac{d}{dz}+\left(\begin{matrix}-\frac{1}{4}&tz\\\
t&\frac{1}{4}\end{matrix}\right)\qquad\text{and}\qquad
z\frac{d}{dz}+\left(\begin{matrix}-\frac{3}{4}&tz\\\
t&-\frac{1}{4}\end{matrix}\right).$

All lattices are given by $z^{n}\Lambda_{1}$ and $z^{n}\Lambda_{2}$ with
$n\in\mathbb{Z}$.

######  Remark 2.1.

By conjugation with the constant matrix $\left(\begin{smallmatrix}t&0\\\
0&1\end{smallmatrix}\right)$ one can change these formulas into

$\displaystyle z\frac{d}{dz}+\left(\begin{matrix}-\frac{1}{4}&t^{2}z\\\
1&\frac{1}{4}\end{matrix}\right)\qquad\text{and}\qquad
z\frac{d}{dz}+\left(\begin{matrix}-\frac{3}{4}&t^{2}z\\\
1&-\frac{1}{4}\end{matrix}\right).$

We want that $\det\mathcal{V}:=\Lambda^{2}\mathcal{V}$ has degree $-1$. The
reason is that, in general $\mathcal{V}=O(d_{1})\oplus O(d_{2})$ with
$d_{1}\leq d_{2}$. The differential module $\mathbb{C}((z^{-1}))\otimes M$ is
irreducible because of the ramification at $z=\infty$. Hence $M$ and also the
connection on $\mathcal{V}$ is irreducible. This implies (in this special
case) that $d_{2}-d_{1}\leq 2$. If we make the natural assumption that
$\deg\mathcal{V}=0$, then there are two possibilities for $\mathcal{V}$,
namely $O\oplus O$ and $O(-1)\oplus O(+1)$. The construction of a family of
connection would involve a construction of a family of vector bundles of rank
two and degree 0 on $\mathbb{P}^{1}$. However, we will avoid this by imposing
$\deg\mathcal{V}=-1$. Then there is only one possibility for $\mathcal{V}$,
namely $O\oplus O(-1)$.

Finally, the degree of $\mathcal{V}$ is $-1$ precisely for the choice of the
lattice $\Lambda_{2}$ corresponding to
$z\frac{d}{dz}+\left(\begin{smallmatrix}-\frac{3}{4}&tz\\\
t&-\frac{1}{4}\end{smallmatrix}\right)$.

###  2.3 Computation of the connection

We identify $\mathcal{V}$ with $Oe_{1}\oplus O(-[\infty])e_{2}$ (a subvector
bundle of the free (i.e., trivial) vector bundle $Oe_{1}\oplus Oe_{2}$). Let
$D=\nabla_{\frac{d}{dz}}$ and $\tilde{D}:=z(z-1)D$. Then
$\tilde{D}e_{1}\in\langle 1,z,z^{2}\rangle e_{1}+\langle 1,z\rangle e_{2}$;
$\tilde{D}e_{2}\in\langle 1,z,z^{2},z^{3}\rangle e_{1}+\langle
1,z,z^{2}\rangle e_{2}$. Here $\langle*,*,*\rangle$ denotes the
$\mathbb{C}$-vector spaces generated by these $*$’s. The vector bundle
$\mathcal{V}$ has an automorphism group $G$ and the moduli space
$\mathcal{M}(\theta_{0},\theta_{1})$ (note that we fix the parameters
$\theta_{0}$ and $\theta_{1}$ in this construction), that we are constructing,
is obtained by dividing the space of all matrices for the connections by the
group $G$. This group consists of the elements $e_{1}\mapsto\lambda e_{1}$,
$e_{2}\mapsto\mu e_{2}+(x_{0}+x_{1}z)e_{1}$ (with
$\lambda,\mu\in\mathbb{C}^{*}$, $x_{0},x_{1}\in\mathbb{C}$). Further the
multiples of the identity act trivially and we have only to consider the
automorphisms $e_{1}\mapsto\lambda e_{1}$, $e_{2}\mapsto
e_{2}+(x_{0}+x_{1}z)e_{1}$.

Since the connection is irreducible we have that $\tilde{D}e_{1}=\langle
1,z,z^{2}\rangle e_{1}+(b_{1}z+b_{0})e_{2}$ with $(b_{1}z+b_{0})\neq 0$. We
consider now two affine parts: $b_{1}\neq 0$ and $b_{0}\neq 0$. The two affine
parts are divided out by the action of $G$. These quotients are geometric
quotients and they are obtained by normalization of some of the entries of the
matrices. One obtains two affine varieties
$\mathcal{M}_{1}(\theta_{0},\theta_{1})$ and
$\mathcal{M}_{2}(\theta_{0},\theta_{1})$ which are glued to the moduli space
$\mathcal{M}(\theta_{0},\theta_{1})$.

####  2.3.1 The first affine part
$\boldsymbol{\mathcal{M}_{1}(\theta_{0},\theta_{1})}$

This is obtained by dividing the open subspace $b_{1}\neq 0$ by the action of
the group $G$. First one normalizes $b_{1}$ by the automorphism
$e_{1}\mapsto\lambda e_{1}$ to $b_{1}=1$. Using $e_{2}\mapsto
e_{2}+(a_{0}+a_{1}z)e_{1}$ one normalizes further to
$\tilde{D}e_{1}=a_{0}e_{1}+(z+b_{0})e_{2}$ and
$\tilde{D}e_{2}=(c_{0}+c_{1}z+c_{2}z^{2}+c_{3}z^{3})e_{1}+(d_{0}+d_{1}z+d_{2}z^{2})e_{2}$.
Now we have to compute the equations between the variables due to the
prescription of the three invariant lattices.

For $z=0$. Now $\frac{1}{z-1}\tilde{D}$ has modulo $z=0$ the eigenvalues
$\pm\frac{\theta_{0}}{2}$. This yields $a_{0}+d_{0}=0$ and
$a_{0}d_{0}-b_{0}c_{0}=-\frac{\theta_{0}^{2}}{4}$.

For $z=1$. Now $\frac{1}{z}\tilde{D}$ has modulo $z=1$ the eigenvalues
$\pm\frac{\theta_{1}}{2}$. This yields $a_{0}+d_{0}+d_{1}+d_{2}=0$ and
$a_{0}(d_{0}+d_{1}+d_{2})-(1+b_{0})(c_{0}+c_{1}+c_{2}+c_{3})=-\frac{\theta_{1}^{2}}{4}.$

For $z=\infty$. The local basis of $\mathcal{V}$ at $z=\infty$ is $e_{1}$,
$z^{-1}e_{2}$. The operator $\frac{1}{z-1}\tilde{D}$ w.r.t. this local basis
is
$z\frac{d}{dz}+\left(\begin{smallmatrix}\frac{a_{0}}{z-1}&\frac{c_{0}+c_{1}z+c_{2}z^{2}+c_{3}z^{3}}{z(z-1)}\\\
z\frac{z+b_{0}}{z-1}&-1+\frac{d_{0}+d_{1}z+d_{2}z^{2}}{z-1}\end{smallmatrix}\right)$.
This should be equivalent to
$z\frac{d}{dz}+\left(\begin{smallmatrix}\frac{-3}{4}&tz\\\
t&\frac{-1}{4}\end{smallmatrix}\right)$. The characteristic polynomials of the
two matrices should be equal modulo $\mathbb{C}[[z^{-1}]]$.

######  Remark 2.2.

A more precise computation is needed to verify the correctness of the last
statement. On another basis of the lattice, the operator
$z\frac{d}{dz}+\left(\begin{smallmatrix}\frac{-3}{4}&tz\\\
t&\frac{-1}{4}\end{smallmatrix}\right)$ reads

$\displaystyle\big{(}1+A_{1}z^{-1}+\cdots\big{)}^{-1}A_{0}^{-1}\left(z\frac{d}{dz}+\left(\begin{matrix}\frac{-3}{4}&tz\\\
t&\frac{-1}{4}\end{matrix}\right)\right)A_{0}\big{(}1+A_{1}z^{-1}+\cdots\big{)},$

leading to

$\displaystyle
z\frac{d}{dz}+A_{0}^{-1}\left(\left(\begin{matrix}\frac{-3}{4}&tz\\\
t&\frac{-1}{4}\end{matrix}\right)+t\left(\begin{matrix}x_{3}&x_{4}-x_{1}\\\
0&-x_{3}\end{matrix}\right)\right)A_{0},$

where $A_{0}\in{\rm GL}_{2}(\mathbb{C})$, $x_{1},x_{3},x_{4}\in\mathbb{C}$.
Thus $d_{2}=0$, $d_{1}=0$, $c_{3}=0$ and $c_{2}=t^{2}$.

Finally the connection as matrix differential operator w.r.t. $e_{1}$, $e_{2}$
reads

$\displaystyle\frac{d}{dz}+\frac{1}{z(z-1)}\left(\begin{matrix}a_{0}&c_{0}+c_{1}z+t^{2}z^{2}\\\
z+b_{0}&-a_{0}\end{matrix}\right)$

and there are two equations

$\displaystyle a_{0}^{2}+b_{0}c_{0}=\frac{\theta_{0}^{2}}{4},\qquad
a_{0}^{2}+(1+b_{0})\big{(}c_{0}+c_{1}+t^{2}\big{)}=\frac{\theta_{1}^{2}}{4}.$

The last equation can be changed into

$\displaystyle
c_{0}=-\frac{\theta_{0}^{2}}{4}+\frac{\theta_{1}^{2}}{4}-c_{1}-t^{2}-b_{0}c_{1}-b_{0}t^{2}$

and this is used to eliminate $c_{0}$. This leaves four variables $a_{0}$,
$b_{0}$, $c_{1}$, $t$ and one equation

$\displaystyle
a_{0}^{2}+b_{0}\left(-\frac{\theta_{0}^{2}}{4}+\frac{\theta_{1}^{2}}{4}-c_{1}-t^{2}-b_{0}c_{1}-b_{0}t^{2}\right)-\frac{\theta_{0}^{2}}{4}=0.$

The above describes the space $\mathcal{M}_{1}(\theta_{0},\theta_{1})$. We
note that this space depends only on $\theta_{0}^{2}$ and $\theta_{1}^{2}$.
The space $\mathcal{M}_{1}(\theta_{0},\theta_{1})$ is smooth if
$\theta_{0}\neq 0$ and $\theta_{1}\neq 0$. For $\theta_{0}=0$, $\theta_{1}\neq
0$, the singular locus is given by $a_{0}=0$, $b_{0}=0$,
$c_{1}=-t^{2}+\frac{\theta_{1}^{2}}{4}$. For $\theta_{0}\neq 0$,
$\theta_{1}=0$, the singular locus is given by $a_{0}=0$, $b_{0}=-1$,
$c_{1}=-t^{2}+\frac{\theta_{0}^{2}}{4}$. For $\theta_{0}=\theta_{1}=0$, the
singular locus is given by $a_{0}=0$, $b_{0}(b_{0}+1)=0$, $c_{1}=-t^{2}$.

Moreover $\mathcal{M}_{1}(\theta_{0},\theta_{1})$ seen as a two dimensional
space over the field $\mathbb{C}(t)$ has the same singular locus but now seen
as a set of at most two points.

######  Observation 2.3.

For every $a\in\mathbb{C}^{*}$ the closed subspace of
$\mathcal{M}_{1}(\theta_{0},\theta_{1})$, defined by $t=a$, is simply
connected.

Indeed, the equation

$\displaystyle
a_{0}^{2}+b_{0}\left(-\frac{\theta_{0}^{2}}{4}+\frac{\theta_{1}^{2}}{4}-c_{1}-a^{2}-b_{0}c_{1}-b_{0}a^{2}\right)-\frac{\theta_{0}^{2}}{4}=0$

defines this two-dimensional space. It is mapped to $\mathbb{C}^{2}$ by
$(a_{0},b_{0},c_{1})\mapsto(a_{0},b_{0})$. The fibre is either empty, or a
point, or $\mathbb{C}$. Hence it suffices to show that the image
$B\subset\mathbb{C}^{2}$ is simply connected. Now $B$ is the union of
$X:=\mathbb{C}\times(\mathbb{C}\setminus\\{0,-1\\})$ and the points
$(\pm\frac{\theta_{0}}{2},0)$ and $(\pm\frac{\theta_{1}}{2},-1)$. The
canonical map $\pi_{1}(X,*)\rightarrow\pi_{1}(B,*)$ is surjective. Consider
one of the two generators,
$s\in[0,1]\mapsto\big{(}\frac{\theta_{0}}{2},e^{2\pi is}\big{)}$ of
$\pi_{1}(X,*)$. In $B$ this closed path is homotopic to the constant closed
path by the homotopy
$s,\lambda\in[0,1]\mapsto\big{(}\frac{\theta_{0}}{2},\lambda e^{2\pi
is}\big{)}$. The same observation can be made for the other generator of
$\pi_{1}(X,*)$. Hence $B$ is simply connected.

####  2.3.2 The second affine part
$\boldsymbol{\mathcal{M}_{2}(\theta_{0},\theta_{1})}$

This space is obtained by dividing the open subset $b_{0}\neq 0$ by the action
of $G$. Now $b_{0}\neq 0$ is normalized by the automorphism
$e_{1}\mapsto\lambda e_{1}$ to $b_{0}=1$.

Using $e_{2}\mapsto e_{2}+(x_{0}+x_{1}z)e_{2}$ one normalizes further to
$\tilde{D}e_{1}=a_{2}z^{2}e_{1}+(b_{1}z+1)e_{2}$ and
$\tilde{D}e_{2}=(c_{0}+c_{1}z+c_{2}z^{2}+c_{3}z^{3})e_{1}+(d_{0}+d_{1}z+d_{2}z^{2})e_{2}$.
The equations derived from the prescribed invariant lattices are

For $z=0$: $d_{0}=0$ and $c_{0}=\frac{\theta_{0}^{2}}{4}$.

For $z=1$: $a_{2}+d_{1}+d_{2}=0$ and
$a_{2}(d_{1}+d_{2})-(1+b_{1})(c_{0}+c_{1}+c_{2}+c_{3})=-\frac{\theta_{1}^{2}}{4}$.

For $z=\infty$: the operator $\frac{1}{z-1}\tilde{D}$ w.r.t. $e_{1}$,
$z^{-1}e_{2}$ is

$\displaystyle
z\frac{d}{dz}+\left(\begin{matrix}\frac{a_{2}z^{2}}{z-1}&\frac{c_{0}+c_{1}z+c_{2}z^{2}+c_{3}z^{3}}{z(z-1)}\\\
(1+b_{1}z)\frac{z}{z-1}&-1+\frac{d_{0}+d_{1}z+d_{2}z^{2}}{z-1}\end{matrix}\right)$

and is equivalent to

$\displaystyle z\frac{d}{dz}+\left(\begin{matrix}\frac{-3}{4}&tz\\\
t&\frac{-1}{4}\end{matrix}\right).$

This yields the equations $d_{2}=-a_{2}$ (and from before $d_{1}+d_{2}=-a_{2}$
and thus $d_{1}=0$). Further $a_{2}^{2}+b_{1}c_{3}=0$ and
$c_{3}=t^{2}-b_{1}c_{2}-a_{2}$ is used to eliminate $c_{3}$. One obtains the
equation

$\displaystyle
c_{1}+c_{2}+t^{2}+b_{1}\frac{\theta_{0}^{2}}{4}+b_{1}c_{1}-a_{2}+\frac{\theta_{0}^{2}}{4}-\frac{\theta_{1}^{2}}{4}=0$

which eliminates $c_{2}$. The matrix differential operator w.r.t. the basis
$e_{1}$, $e_{2}$ is now

$\displaystyle\frac{d}{dz}+\frac{1}{z(z-1)}\left(\begin{matrix}a_{2}z^{2}&\frac{\theta_{0}^{2}}{4}+c_{1}z+c_{2}z^{2}+(t^{2}-b_{1}c_{2}-a_{2})z^{3}\\\
1+b_{1}z&-a_{2}z^{2}\end{matrix}\right)$

in the variables $a_{2}$, $b_{1}$, $c_{1}$, $t$ and one equation
$a_{2}^{2}+b_{1}(-a_{2}-b_{1}c_{2}+t^{2})=0$ with $c_{2}$ eliminated as above.
The above describes the space $\mathcal{M}_{2}(\theta_{0},\theta_{1})$.

The singular locus of this three-dimensional variety is given by
$\theta_{1}=0$, $a_{2}=0$, $b_{1}=-1$, $c_{1}=t^{2}-\frac{\theta_{0}^{2}}{4}$
(note that $t\neq 0$). As a variety over $\mathbb{C}(t)$ the singularity
occurs only for $\theta_{1}=0$ and consists of one point.

The two parts glue to the required space $\mathcal{M}(\theta_{0},\theta_{1})$.

######  Observation 2.4.

For any $a\in\mathbb{C}^{*}$ the closed subspace of
$\mathcal{M}(\theta_{0},\theta_{1})$, defined by $t=a$, is simply connected.

Indeed, by Observation 2.3, this holds for the first open affine part of this
space. For the second open affine part the same reasoning proves the
statement. Van Kampen’s theorem finishes the proof.

####  2.3.3 Resolving the singularities of
$\boldsymbol{\mathcal{M}(\theta_{0},\theta_{1})}$

For $\theta_{0}\neq 0$, $\theta_{1}\neq 0$, the space
$\mathcal{M}(\theta_{0},\theta_{1})$ is the fine moduli space for the
connections on the fixed vector bundle $\mathcal{V}$ of rank 2 and degree $-1$
with the prescribed singularities. The space
$\mathcal{M}(\theta_{0},\theta_{1})$ is smooth and simply connected for any
nonzero fixed value of $t$. Its set of closed points
$\mathcal{M}(\theta_{0},\theta_{1})(\mathbb{C})$ coincides by construction
with ${\bf S}(\theta_{0},\theta_{1})$.

For $\theta_{0}=0$ and/or for $\theta_{1}=0$ the space
$\mathcal{M}(\theta_{0},\theta_{1})$ has singularities and is no longer a fine
moduli space. Consider the case $\theta_{0}=0$ and $\theta_{1}\neq 0$.

A tuple $(M,\theta_{0}=0,\theta_{1},t)$ is represented at $z=0$ by a lattice
$\Lambda\subset\mathbb{C}((z))\otimes M$ such that
$\delta(\Lambda)\subset\Lambda$ and the action of $\delta:=zD$ on
$\Lambda/z\Lambda$ has trace zero and determinant zero. There exists a basis
of $\Lambda$ over $\mathbb{C}[[z]]$ such that $\delta$ has the matrix
$\left(\begin{smallmatrix}0&0\\\ 0&0\end{smallmatrix}\right)$ or
$\left(\begin{smallmatrix}0&1\\\ 0&0\end{smallmatrix}\right)$. The space
$\mathcal{M}(\theta_{0}=0,\theta_{1})$ does not distinguish between these
cases. Therefore it is not a fine moduli space and moreover it has a
singularity.

There is a geometric way to treat these problems. One adds to the data
$(M,\theta_{0}=0,\theta_{1},t)$ a ‘line’. This means the following. The
assumption $\theta_{0}=0$ defines a lattice
$\Lambda\subset\mathbb{C}((z))\otimes M$ invariant under $\delta$. The ‘line’
is a 1-dimensional summand of $\Lambda$, invariant under $\delta$. We note
that there are two cases:

  1. (1)

$\delta$ has matrix $\left(\begin{smallmatrix}0&0\\\
0&0\end{smallmatrix}\right)$ on a basis $e_{1}$, $e_{2}$ of $\Lambda$. Then
the possible lines are $\mathbb{C}[[z]]e$ with
$e\in\mathbb{C}e_{1}+\mathbb{C}e_{2}$, $e\neq 0$. The possibilities form a
$\mathbb{P}^{1}$ over $\mathbb{C}$.

  2. (2)

$\delta$ has matrix $\left(\begin{smallmatrix}0&1\\\
0&0\end{smallmatrix}\right)$ on a basis $e_{1}$, $e_{2}$ of $\Lambda$. Then
$\mathbb{C}[[z]]e_{1}$ is the only possible ‘line’.

This additional ‘line’ is called a “parabolic structure” (see [4, 6, 7]) or a
“level structure”. This defines a new set ${\bf
S}^{+}(\theta_{0}=0,\theta_{1})$ and a new moduli problem consisting of
connections on the above $\mathcal{V}$ and an invariant line in
$\mathcal{V}_{0}\otimes\mathbb{C}[[z]]$ (here $\mathcal{V}_{0}$ denotes the
stalk of $\mathcal{V}$ at $z=0$). There is a fine moduli space which we call
$\mathcal{M}^{+}(\theta_{0}=0,\theta_{1})$. The natural morphism
$\mathcal{M}^{+}(\theta_{0}=0,\theta_{1})\rightarrow\mathcal{M}(\theta_{0}=0,\theta_{1})$
turns out to be the resolution of the latter space, seen as a surface over
$\mathbb{C}(t)$ or as a surface after fixing a value for $t$. The preimage of
the singular point is the projective line over $\mathbb{C}$. This construction
is also present in the papers [21, 22] and we will not make it explicit here.

Something similar has to be done for the case $\theta_{0}\neq 0$,
$\theta_{1}=0$ and the case $\theta_{0}=\theta_{1}=0$. In all cases we write
$\mathcal{M}^{+}(\theta_{0},\theta_{1})$ for the moduli space obtained in this
way and, for notational convenience we write
$\mathcal{M}^{+}(\theta_{0},\theta_{1})=\mathcal{M}(\theta_{0},\theta_{1})$
also in the case $\theta_{0}\neq 0$, $\theta_{1}\neq 0$.

######  Observation 2.5.

Consider a lattice $\Lambda$ over $\mathbb{C}[[z]]$ of rank two with a
differential operator $\delta$ which has operator form
$z\frac{d}{dz}+\left(\begin{smallmatrix}a&c\\\ b&d\end{smallmatrix}\right)$
with $a,b,c,d\in\mathbb{C}[[z]]$ and $\left(\begin{smallmatrix}a&c\\\
b&d\end{smallmatrix}\right)\equiv\left(\begin{smallmatrix}\alpha&0\\\
0&\beta\end{smallmatrix}\right)\mod(z)$. Suppose that
$\beta-\alpha\not\in\mathbb{Z}_{<0}$. Then $\Lambda$ has a unique direct
summand $\mathbb{C}[[z]]e$ such that $\delta e=\alpha e$.

The proof is obtained by conjugating the operator
$z\frac{d}{dz}+\left(\begin{smallmatrix}a&c\\\ b&d\end{smallmatrix}\right)$
with a suitable invertible matrix $\left(\begin{smallmatrix}t_{1}&0\\\
t_{2}&t_{3}\end{smallmatrix}\right)\in{\rm GL}_{2}(\mathbb{C}[[z]])$. After
conjugation with a diagonal matrix in ${\rm GL}_{2}(\mathbb{C}[[z]])$ one can
suppose that the differential operator is
$z\frac{d}{dz}+\left(\begin{smallmatrix}\alpha&c\\\
b&\beta\end{smallmatrix}\right)$ and $b,c\in z\mathbb{C}[[z]]$. There exists
$x\in\mathbb{C}[[z]]$ such that $\left(\begin{smallmatrix}1&0\\\
-x&1\end{smallmatrix}\right)\big{\\{}z\frac{d}{dz}+\left(\begin{smallmatrix}\alpha&c\\\
b&\beta\end{smallmatrix}\right)\big{\\}}\left(\begin{smallmatrix}1&0\\\
x&1\end{smallmatrix}\right)$ equals
$z\frac{d}{dz}+\left(\begin{smallmatrix}\alpha&*\\\
0&\beta\end{smallmatrix}\right)$. Indeed, this condition is equivalent to the
equation $z\frac{dx}{dz}+(\beta-\alpha)x+b-cx^{2}=0$. Write
$x=\sum\limits_{n\geq 1}a_{n}z^{n}$. The equation reads

$\displaystyle\sum_{n\geq 1}(n+\beta-\alpha)a_{n}z^{n}+\sum_{n\geq
1}b_{n}z^{n}-\sum_{n\geq 1}c_{n}z^{n}\cdot\bigg{(}\sum_{n\geq
1}a_{n}z^{n}\bigg{)}^{2}=0$

and there is a unique solution.

Now we consider the moduli space $\mathcal{M}(\theta_{0},\theta_{1})$ for some
$\theta_{0}\in\mathbb{Z}_{>0}$. From Observation 2.5 one concludes that the
invariant lattice $\Lambda$ over $\mathbb{C}[[z]]$ has a direct summand
$\mathbb{C}[[z]]e$ such that $\delta e=-\frac{\theta_{0}}{2}e$. In other
words, there is a unique ‘line’ present in this situation and the ‘level
structure’ at $z=0$ is already present. This explains why
$\mathcal{M}(\theta_{0},\theta_{1})$ has no singularities for non zero integer
values of $\theta_{0}$ and $\theta_{1}$.

Another interesting way to produce the correct moduli space for the cases
$\theta_{0}=0$ and/or $\theta_{1}=0$ is the following. A differential module
$M$ with the required data is represented by a connection on a vector bundle
$\mathcal{V}$ of rank 2, with degree $-3$ instead of $-1$. Then $\mathcal{V}$
is identified with $O(-[\infty])e_{1}\oplus O(-2[\infty])e_{2}$. The
connection
$\nabla\colon\mathcal{V}\rightarrow\mathcal{V}\otimes\Omega([0]+[1]+2[\infty])$
is prescribed by the local differential operators

$\displaystyle z\frac{d}{dz}+\left(\begin{matrix}\frac{\theta_{0}}{2}&0\\\
0&-\frac{\theta_{0}}{2}-1\end{matrix}\right),\qquad(z-1)\frac{d}{dz}+\left(\begin{matrix}\frac{\theta_{1}}{2}&0\\\
0&-\frac{\theta_{1}}{2}-1\end{matrix}\right),\qquad
z\frac{d}{dz}+\left(\begin{matrix}-\frac{3}{4}&t^{2}z\\\
1&-\frac{1}{4}\end{matrix}\right).$

As in Section 2.3 one has to consider two affine parts. On the first part the
operator reads

$\displaystyle\frac{d}{dz}+\frac{1}{z(z-1)}\left(\begin{matrix}a_{0}&c_{0}+c_{1}z+c_{2}z^{2}+c_{3}z^{3}\\\
z+b_{0}&d_{0}+d_{1}z+d_{2}z^{2}\end{matrix}\right)$

and on the second part it is

$\displaystyle\frac{d}{dz}+\frac{1}{z(z-1)}\left(\begin{matrix}a_{2}z^{2}&c_{0}+c_{1}z+c_{2}z^{2}+c_{3}z^{3}\\\
1+b_{1}z&d_{0}+d_{1}z+d_{2}z^{2}\end{matrix}\right).$

The equations for the entries in these matrices are similar to those of
Sections 2.3.1 and 2.3.2. After a computation one finds that singular points
only occur for the cases $\theta_{0}=-1$ and/or $\theta_{1}=-1$. In
particular, this produces a smooth moduli space for, say, $\theta_{0}=0$ and
$\theta_{1}\neq-1$. Other choices for $\mathcal{V}$ with negative odd degree
$-2d-1$ and local equations at $z=0$ and $z=1$ where the above matrices have
traces $-d$, $-d$ can be used to construct smooth moduli spaces for all
combinations of $\theta_{0}$ and $\theta_{1}$.

In the sequel we write $\mathcal{M}^{+}(\theta_{0},\theta_{1})$ for the
resolution of the space $\mathcal{M}(\theta_{0},\theta_{1})$. We note that
$\mathcal{M}^{+}(\theta_{0},\theta_{1})$ is in general not yet the
Okamoto–Painlevé space for the following reason. For any $a\in\mathbb{C}^{*}$,
the closed subspace of $\mathcal{M}^{+}(\theta_{0},\theta_{1})$, given by
$t=a$, is simply connected. This follows from Observation 2.4 and the fact
that a fibre of
$\mathcal{M}^{+}(\theta_{0},\theta_{1})\rightarrow\mathcal{M}(\theta_{0},\theta_{1})$
is either a point or a projective line over $\mathbb{C}$.

The ‘$t$-part’ of $\mathcal{M}^{+}(\theta_{0},\theta_{1})$ runs in
$\mathbb{C}^{*}$, which is not simply connected. In the earlier definition of
Okamoto’s space of initial values ‘simply connected’ was required. However, in
one of the later papers of Okamoto et al. [10] the condition ‘simply
connected’ for ‘the space of initial values’ is removed.

The other reason to replace $t$ by $e^{2\pi iu}$ with $u\in\mathbb{C}$ is the
following. The map from a tuple $(M,\theta_{0},\theta_{1},t)$ to the monodromy
data at $z=\infty$ depends on the choice of a direction for multisummation.
This direction has to be different from the singular direction and the latter
moves with $t$.

As we will see in Section 4, the monodromy space $\mathcal{R}(s_{0},s_{1})$
has for the values $s_{0}=\pm 2$ and $s_{1}=\pm 2$ singular points. Further
$s_{0}=e^{\pi i\theta_{0}}+e^{-\pi i\theta_{0}}$ and $s_{1}=e^{\pi
i\theta_{1}}+e^{-\pi i\theta_{1}}$. Also in this case one has to add a similar
level structure as a method to obtain a desingularisation
$\mathcal{R}^{+}(s_{0},s_{1})$ of $\mathcal{R}(s_{0},s_{1})$. Further we will
show that $\mathcal{R}^{+}(s_{0},s_{1})$ is simply connected.

##  3 Computation of the Painlevé equation $\boldsymbol{{\rm degP}_{\rm V}}$

This calculation is done on the first affine part of the space
$\mathcal{M}(\theta_{0},\theta_{1})$. There an isomonodromic family
$\frac{d}{dz}+A$ with $A=A(z,t)$ is given and computed by the assumption that
this operator commutes with an unknown operator $\frac{d}{dt}+B$ with
$B=B(z,t)$. This is equivalent to the formula $\frac{\partial A}{\partial
t}=\frac{\partial B}{\partial z}+[A,B]$.

From the singularities of $A$ one derives that $B$ is w.r.t. $z$ a polynomial
matrix of degree at most 1. Further, both $A$ and $B$ are $2\times 2$-matrices
with trace 0. This we use to make the computations smoother.

Write $H=\left(\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\right)$,
$E_{1}=\left(\begin{smallmatrix}0&1\\\ 0&0\end{smallmatrix}\right)$,
$E_{2}=\left(\begin{smallmatrix}0&0\\\ 1&0\end{smallmatrix}\right)$. Observe
$[H,E_{1}]=2E_{1}$, $[H,E_{2}]=-2E_{2}$, $[E_{1},E_{2}]=H$. In the sequel we
write $f^{\prime}$ for $\frac{df}{dt}$. Write
$A=\frac{a_{0}}{z(z-1)}H+\frac{c_{0}+c_{1}z+t^{2}z^{2}}{z(z-1)}E_{1}+\frac{z+b_{0}}{z(z-1)}E_{2}$.
Write $B=B_{H}H+B_{1}E_{1}+B_{2}E_{2}$ and $B_{H}=B_{H,0}+B_{H,1}z$,
$B_{1}=B_{1,0}+B_{1,1}z$, $B_{2}=B_{2,0}+B_{2,1}z$ where the $B_{*,*}$ only
depend on $t$. The equation $\frac{\partial A}{\partial t}=\frac{\partial
B}{\partial z}+[A,B]$, multiplied by $z(z-1)$, has coefficients with respect
to the basis $H$, $E_{1}$, $E_{2}$ which read:

$\displaystyle(H)\quad$ $\displaystyle
a_{0}^{\prime}=z(z-1)B_{H,1}+\big{(}c_{0}+c_{1}z+t^{2}z^{2}\big{)}(B_{2,0}+B_{2,1}z)-(z+b_{0})(B_{1,0}+zB_{1,1}),$
$\displaystyle(E_{1})\quad$ $\displaystyle
c_{0}^{\prime}+c_{1}^{\prime}z+2tz^{2}=z(z-1)B_{1,1}+2a_{0}(B_{1,0}+B_{1,1}z)$
$\displaystyle\qquad{}-2(B_{H,0}+B_{H,1}z)\big{(}c_{0}+c_{1}z+t^{2}z^{2}\big{)},$
$\displaystyle(E_{2})\quad$ $\displaystyle
b_{0}^{\prime}=z(z-1)B_{2,1}-2a_{0}(B_{2,0}+B_{2,1}z)+2(B_{H,0}+B_{H,1}z)(z+b_{0}).$

Each of these three equations is considered with respect to the degrees in
$z$. A sequence of solving equations (in a suitable order!) yields the
following:

  1. $(H)$

degree 3 implies $B_{2,1}=0$; degree 2 implies
$0=B_{H,1}+t^{2}B_{2,0}-B_{1,1}$ and so $B_{2,0}=2t^{-1}$; degree 1 implies
$0=c_{1}B_{2,0}-B_{1,0}-b_{0}B_{1,1}$ and so $B_{1,0}=c_{1}\cdot
2t^{-1}-b_{0}\cdot 2t$; degree 0 implies
$a_{0}^{\prime}=c_{0}B_{2,0}-b_{0}B_{1,0}$ and so
$a_{0}^{\prime}=2t^{-1}c_{0}-b_{0}(2t^{-1}c_{1}-2tb_{0})$;

  2. $(E_{1})$

degree 3 implies $B_{H,1}=0$; degree 2 implies $2t=B_{1,1}-2B_{H,0}t^{2}$ and
so $B_{1,1}=2t$; degree 1 implies $c_{1}^{\prime}=-B_{1,1}+2a_{0}B_{1,1}$ and
so $c_{1}^{\prime}=2t(2a_{0}-1)$; degree 0 implies
$c_{0}^{\prime}=2a_{0}B_{1,0}$ and so $c_{0}^{\prime}=2a_{0}(c_{1}\cdot
2t^{-1}-b_{0}\cdot 2t)$;

  3. $(E_{2})$

$b_{0}^{\prime}=-2a_{0}B_{2,0}+2B_{H,0}(z+b_{0})$ and thus $B_{H,0}=0$ and
$b_{0}^{\prime}=-2a_{0}B_{2,0}$ and so $b_{0}^{\prime}=-2a_{0}\cdot 2t^{-1}$.

This leads to the following set of equations:

$\displaystyle(1)\quad$ $\displaystyle
a_{0}^{2}+b_{0}c_{0}=\frac{\theta_{0}^{2}}{4},$ $\displaystyle(2)\quad$
$\displaystyle
c_{0}=-\frac{\theta_{0}^{2}}{4}+\frac{\theta_{1}^{2}}{4}-c_{1}-t^{2}-b_{0}c_{1}-b_{0}t^{2},$
$\displaystyle(3)\quad$ $\displaystyle
a_{0}^{\prime}=2t^{-1}c_{0}-b_{0}\big{(}2t^{-1}c_{1}-2tb_{0}\big{)},$
$\displaystyle(4)\quad$ $\displaystyle c_{1}^{\prime}=-2t+4a_{0}t,$
$\displaystyle(5)\quad$ $\displaystyle
c_{0}^{\prime}=2a_{0}\big{(}2t^{-1}c_{1}-2tb_{0}\big{)},$
$\displaystyle(6)\quad$ $\displaystyle b_{0}^{\prime}=-4a_{0}t^{-1}.$

This is solved, using MAPLE, by the following steps: Eliminate $c_{0}$ by (2)
and $a_{0}$ by using (6). In the new set of equations one can eliminate
$c_{1}$ in a linear way. Then MAPLE yields a second-order equation for
$b_{0}$. Write the matrix differential operator as
$\frac{d}{dz}+\left(\begin{smallmatrix}a&c\\\ b&-a\end{smallmatrix}\right)$.
Then taking the first vector as cyclic vector one obtains a scalar equation
$\big{(}\frac{d}{dz}\big{)}^{2}-\frac{c^{\prime}}{c}\frac{d}{dz}-a^{\prime}-a^{2}-bc+a\frac{c^{\prime}}{c}$.
The $q$ for the Painlevé equation is the pole $\neq 0,1,\infty$ of this scalar
equation. Thus $q=-b_{0}$. The Hamiltonian system for the Painlevé equation
has variables $q$ and $p$ where $p$ is the residue of the term
$-a^{\prime}-a^{2}-bc+a\frac{c^{\prime}}{c}$ at $z=q$. One finds in this way
for ${\rm degP}_{\rm V}$ and the Hamiltonian function $\mathcal{H}$ the
formulas

$\displaystyle
q^{\prime\prime}=\frac{1}{2}\left(\frac{1}{q}+\frac{1}{q-1}\right)(q^{\prime})^{2}-\frac{q^{\prime}}{t}+\frac{2(q-1)\theta_{0}^{2}}{qt^{2}}-\frac{2q\theta_{1}^{2}}{(q-1)t^{2}}+8q(q-1),$
$\displaystyle
p=\frac{t}{4}q^{\prime},\qquad\mathcal{H}=\frac{2(p^{2}-\frac{\theta_{0}^{2}}{4})}{tq}-\frac{2(p^{2}-\frac{\theta_{1}^{2}}{4})}{t(q-1)}+2qt,$

with here, exceptionally,
$q^{\prime}=q(1-q)\frac{\partial\mathcal{H}}{\partial p}$ and
$p^{\prime}=-q(1-q)\frac{\partial\mathcal{H}}{\partial q}$.

We note that the formula for ${\rm degP}_{\rm V}$ coincides with the one in
[19].

The relation between this ${\rm degP}_{\rm V}$ and the classical degenerate
${\rm P}_{\rm V}$ is the following. Consider solutions $q(t)$ of ${\rm
degP}_{\rm V}$ which are even. Then these are written as $q(t)=Q(t^{2})$ for
some function $Q(s)$. One easily computes that the second-order differential
equation for $Q$ is

$\displaystyle
Q^{\prime\prime}=\frac{1}{2}\left(\frac{1}{Q}+\frac{1}{Q-1}\right)(Q^{\prime})^{2}-\frac{Q^{\prime}}{s}+\frac{(Q-1)\frac{\theta_{o}^{2}}{2}}{Qs^{2}}-\frac{Q\frac{\theta_{1}^{2}}{2}}{(Q-1)s^{2}}+\frac{2Q(Q-1)}{s}.$

One substitutes $Q=\frac{y}{y-1}$ and finds for $y$ the second-order
differential equation

$\displaystyle
y^{\prime\prime}=\frac{1}{2}\frac{3y-1}{y(y-1)}(y^{\prime})^{2}-\frac{y^{\prime}}{s}+\frac{y(y-1)^{2}\theta_{1}^{2}}{2s^{2}}-\frac{(y-1)^{2}\theta_{0}^{2}}{2s^{2}y}-\frac{2y}{s}.$

This is the classical degenerate ${\rm P}_{\rm
V}(\alpha,\beta,\gamma,\delta)$, normalized as the following special case
${\rm P}_{\rm V}\big{(}\frac{\theta_{1}^{2}}{2}$,
$-\frac{\theta_{0}^{2}}{2},-2,0\big{)}$ of ${\rm P}_{\rm V}$.

######  Remark 3.1.

The formula for ${\rm P}_{\rm V}$ in [19] reduces to the classical formula for
${\rm P}_{\rm V}$ as well, using the same substitution $q=\frac{y}{y-1}$.

##  4 The moduli space for the analytic data
$\boldsymbol{\mathcal{R}(s_{0},s_{1})}$

We reproduce here, with a slightly different choice of signs, the paper [19].
The solution space $V$ at $z=\infty$ has a basis $e_{1}$, $e_{2}$ such that
the formal monodromy is $\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right)$ and the only Stokes matrix has the matrix
$\left(\begin{smallmatrix}1&0\\\ e&1\end{smallmatrix}\right)$. The topological
monodromy at $z=\infty$ is the product
$M_{\infty}=\left(\begin{smallmatrix}-e&-1\\\ 1&0\end{smallmatrix}\right)$. By
multisummation in a suitable direction we combine this with the monodromy
matrices $M_{0}$ and $M_{1}$ for loops around $z=0$ and $z=1$. Put
$s_{0}=e^{\pi i\theta_{0}}+e^{-\pi i\theta_{0}}$, $s_{1}=e^{\pi
i\theta_{1}}+e^{-\pi i\theta_{1}}$. Then $M_{0}$, $M_{1}$ have determinants 1
and traces $s_{0}$, $s_{1}$. One has the relation $M_{0}M_{1}M_{\infty}=1$.

The basis $e_{1}$, $e_{2}$ is unique up to $e_{1},e_{2}\mapsto\lambda
e_{1},\lambda e_{2}$. This transformation acts trivially on the matrices.
Write $M_{1}=\left(\begin{smallmatrix}a_{1}&b_{1}\\\
c_{1}&d_{1}\end{smallmatrix}\right)$. Then $M_{0}$ is the inverse of
$M_{1}M_{\infty}$. Hence $\mathcal{R}(s_{0},s_{1})$ is the affine space with
coordinate ring $\mathbb{C}[a_{1},b_{1},c_{1},d_{1},e]$ with the relations

$\displaystyle a_{1}d_{1}-b_{1}c_{1}=1,\qquad a_{1}+d_{1}=s_{1},\qquad-
a_{1}e+b_{1}-c_{1}=s_{0}.$

Elimination of $c_{1}$, $d_{1}$ and $x_{1}=-b_{1}$, $x_{2}=-a_{1}$, $x_{3}=-e$
leads to the ring
$\mathbb{C}[x_{1},x_{2},x_{3}]/(x_{1}x_{2}x_{3}+x_{1}^{2}+x_{2}^{2}+s_{0}x_{1}+s_{1}x_{2}+1)$.

Only for $s_{1}=\pm 2$ and for $s_{0}=\pm 2$ the corresponding affine cubic
surface $\mathcal{R}(s_{0},s_{1})$ has singularities:

$\displaystyle s_{1}=\pm 2,\qquad x_{1}=0,\qquad x_{2}=\mp 1,\qquad x_{3}=\pm
s_{0},\qquad\text{and}$ $\displaystyle s_{0}=\pm 2,\qquad x_{1}=\mp 1,\qquad
x_{2}=0,\qquad x_{3}=\pm s_{1}.$

In particular for $s_{0}\neq\pm 2$, $s_{1}\neq\pm 2$ the space
$\mathcal{R}(s_{0},s_{1})$ has no singularities.

######  Observation 4.1.

$\mathcal{R}(s_{0},s_{1})$ is simply connected for all $s_{0}$, $s_{1}$.

###### Proof.

Consider the projection of
$\mathcal{R}(s_{0},s_{1})\rightarrow\mathbb{C}^{2}$, by
$(x_{1},x_{2},x_{3})\mapsto(x_{1},x_{2})$. The fibres of this map are either a
point or $\mathbb{C}$ or empty. The image $B$ is the union of
$(\mathbb{C}^{*})^{2}$ with the points
$\\{(0,x_{2})\,|\,x_{2}^{2}+s_{1}x_{2}+1=0\\}$ and
$\\{(x_{1},0)\,|\,x_{1}^{2}+s_{0}x_{1}+1=0\\}$. It suffices to show that $B$
is simply connected. The inclusion $(\mathbb{C}^{*})^{2}\subset B$ induces a
surjection $\pi_{1}((\mathbb{C}^{*})^{2},*)\rightarrow\pi_{1}(B,*)$. Consider
a generator of $\pi_{1}((\mathbb{C}^{*})^{2},*)$, represented by the loop
$s\in[0,1]\mapsto(\tilde{x}_{1},e^{2\pi is})$, where $\tilde{x}_{1}$ is chosen
such that $\tilde{x}_{1}^{2}+s_{0}\tilde{x}_{1}+1=0$. In $B$ this loop is
homotopic to the constant loop $s\mapsto(\tilde{x}_{1},0)$ by the homotopy
$(s,\lambda)\in[0,1]^{2}\mapsto(\tilde{x}_{1},\lambda e^{2\pi is})$. One
concludes that the two generators of $\pi_{1}((\mathbb{C}^{*})^{2},*)$ have
trivial image in $\pi_{1}(B,*)$ and that $\pi_{1}(B,*)=1$. ∎

As in Section 2.3.3, the geometric way to resolve the singularities of
$\mathcal{R}(s_{0},s_{1})$ for $s_{0}=\pm 2$ and/or $s_{1}=\pm 2$ is to add a
level structure consisting of a line (or two lines if both $s_{0}=\pm 2$ and
$s_{1}=\pm 2$). The resulting space is denoted by
$\mathcal{R}^{+}(s_{0},s_{1})$. We will work out the details for $s_{0}\neq\pm
2$ and $s_{1}=2$.

The fibre of the surjective morphism
$\mathcal{R}^{+}(s_{0},s_{1})\rightarrow\mathcal{R}(s_{0},s_{1})$ is in
general a point and there are at most two fibres isomorphic to
$\mathbb{P}^{1}$. It follows that $\mathcal{R}^{+}(s_{0},s_{1})$ is simply
connected as well.

For the formulation of the (extended) Riemann–Hilbert morphism we need to
replace the space $T=\mathbb{C}^{*}$ of the variable $t$ by its universal
covering $\tilde{T}=\mathbb{C}$. The reason is that the singular direction at
infinity varies with $t$. Then

$\displaystyle{\rm RH}^{+}\colon\
\mathcal{M}^{+}(\theta_{0},\theta_{1})\times_{T}\tilde{T}\rightarrow\mathcal{R}^{+}(s_{0},s_{1})\times\tilde{T}$

is a well defined analytic map. This (extended) Riemann–Hilbert map ${\rm
RH}^{+}$ is bijective on points. Indeed, the points on the left hand side
correspond to the tuples $(M,\theta_{0},\theta_{1},t)$ with additionally
$u\in\mathbb{C}$ with $t=e^{2\pi iu}$ and level structure(s) if needed. The
points on the right hand side correspond to the analytic data with level
structure (if needed), $u\in\mathbb{C}$ and the formal structure at
$z=\infty$. By [19, Theorem 1.7], these two sets coincide. As in [18, Theorem
1.5] we conclude that ${\rm RH}^{+}$ is an analytic isomorphism between two
algebraic varieties over $\mathbb{C}$. Moreover, as in the proof of loc. sit.,
from the isomorphism one obtains (compare [10])

######  Theorem 4.2.

The Painlevé property for ${\rm degP}_{\rm V}(\theta_{0},\theta_{1})$ holds.
Moreover $\mathcal{M}^{+}(\theta_{0},\theta_{1})\times_{T}\tilde{T}$ is the
Okamoto–Painlevé space.

The resolution $\mathcal{R}^{+}(s_{0},2)\rightarrow\mathcal{R}(s_{0},2)$ for
$s_{0}\neq\pm 2$. As before, any differential module $M$ (with the given data)
determines a basis $e_{1}$, $e_{2}$ of the solution space $V$ at $z=\infty$
such that the formal monodromy $\gamma$ has matrix
$\left(\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix}\right)$. This basis is
unique up to multiplication of $e_{1}$, $e_{2}$ by the same constant. All maps
are written as matrices with respect to this basis.

A point of $\mathcal{R}^{+}(s_{0},2)$ corresponds to a tuple
$(M_{0},M_{1},M_{\infty},\mathbb{C}v)$ where $M_{0}$, $M_{1}$, $M_{\infty}$
are the matrices for the topological monodromies for the points $0$, $1$,
$\infty$. Then $M_{0}M_{1}M_{\infty}=1$ and $v$ is a non zero eigenvector of
$M_{1}$ (for the eigenvalue 1). Write
$M_{1}=\left(\begin{smallmatrix}a_{1}&b_{1}\\\
c_{1}&d_{1}\end{smallmatrix}\right)$,
$M_{\infty}=\left(\begin{smallmatrix}-e&-1\\\ 1&0\end{smallmatrix}\right)$,
$v=\left(\begin{smallmatrix}y_{0}\\\ y_{1}\end{smallmatrix}\right)$. There are
two affine parts given by the cases $y_{0}\neq 0$ and $y_{1}\neq 0$. We
consider here the first case and normalize $y_{0}=1$.

The matrix $M_{0}$ is determined by $M_{0}M_{1}M_{\infty}=1$. The equations in
the variables $a_{1}$, $b_{1}$, $c_{1}$, $d_{1}$, $e$, $y_{1}$ are

$\displaystyle a_{1}+d_{1}=2,\qquad a_{1}d_{1}-b_{1}c_{1}=1,\qquad
a_{1}-1+b_{1}y_{1}=0,\qquad c_{1}+(d_{1}-1)y_{1}=0,$

where the last two equations come from: $v$ is eigenvector for $M_{1}$. One
eliminates $c_{1}$, $d_{1}$ by $a_{1}+d_{1}=2$ and
$-a_{1}e+b_{1}-c_{1}=s_{0}$. One writes $x_{1}=-b_{1}$, $x_{2}=-a_{1}$,
$x_{3}=-e$. Thus the ring of regular functions on the affine part that we are
looking at has the form $\mathbb{C}[x_{1},x_{2},x_{3},y_{1}]/{\rm relations}$
and the relations are

$\displaystyle
x_{1}x_{2}x_{3}+x_{1}^{2}+x_{2}^{2}+s_{0}x_{1}+2x_{2}+1=0,\qquad(x_{2}+1)+x_{1}y_{1}=0,$
$\displaystyle(1+x_{2})y_{1}=x_{2}x_{3}+x_{1}+s_{0}.$

The morphism $\mathcal{R}^{+}(s_{0},2)\rightarrow\mathcal{R}(s_{0},2)$,
restricted to this affine part, is given by the obvious homomorphism

$\displaystyle\mathbb{C}[x_{1},x_{2},x_{3}]/\big{(}x_{1}x_{2}x_{3}+x_{1}^{2}+x_{2}^{2}+s_{0}x_{1}+2x_{2}+1\big{)}\rightarrow\mathbb{C}[x_{1},x_{2},x_{3},y_{1}]/{\rm
relations}.$

For $x_{1}\neq 0$ one can eliminate $y_{1}$ by using the equation
$(x_{2}+1)+x_{1}y_{1}=0$. In fact, the above map is an isomorphism after
inverting the element $x_{1}$. For $x_{1}=0$ one finds that
$(x_{1},x_{2},x_{3})=(0,-1,s_{0})$ and this is the unique singular point of
$\mathcal{R}(s_{0},2)$. The points lying above this singular point are
$(x_{1},x_{2},x_{3},y_{1})=(0,-1,s_{0},a)$ for all $a\in\mathbb{C}$. One
easily verifies that these points of $\mathcal{R}^{+}(s_{0},2)$ are smooth.

The computation of the affine part $y_{1}\neq 0$ is similar. One concludes
that $\mathcal{R}^{+}(s_{0},2)\rightarrow\mathcal{R}(s_{0},2)$ is a resolution
of singularities and that the fibre above the singular point is
$\mathbb{P}^{1}$.

##  5 Computing the Bäcklund transformations

Write $\bf S$ for the union of the sets ${\bf S}(\theta_{0},\theta_{1})$ taken
over all $\theta_{0}$, $\theta_{1}$. We recall that $\theta_{0}$,
$\theta_{1}$, $t$ determine the invariant lattices at $z=0,1,\infty$. A
‘natural’ automorphism of $\bf S$ may change a given tuple
$(M,\theta_{0},\theta_{1},t)$ into the same module $M$ but with different
lattices. For example, the lattice at $z=0$ will be changed by replacing
$\theta_{0}$ by $\theta_{0}+2$. Further the module $M$ can be changed and one
can consider the automorphism of $\mathbb{P}^{1}$ which interchanges $z=0,1$
and has $z=\infty$ as fixed point.

Some ‘natural’ automorphisms of $\bf S$ are given by the tuple
$(M,\theta_{0},\theta_{1},t)\mapsto$

  1. (1)

$(M,\theta_{0},\theta_{1},-t)$,

  2. (2)

$(M,-\theta_{0},\theta_{1},t)$,

  3. (3)

$(M,\theta_{0},-\theta_{1},t)$,

  4. (4)

$(\tilde{M},\theta_{1},\theta_{0})$ where $\tilde{M}$ is obtained from $M$ by
the automorphism $z\mapsto 1-z$,

  5. (5)

$(N\otimes M,\theta_{0}+1,\theta_{1},t)$ where $N$ is the 1-dimensional module
represented by $\frac{d}{dz}+\frac{1}{2z}$.

These special automorphisms will be lifted to isomorphisms between various
moduli spaces $\mathcal{M}^{+}(\theta_{0},\theta_{1})$. The isomorphisms
preserve the analytic data and map therefore solutions of one ${\rm degP}_{\rm
V}$ equation to solutions of another. They are Bäcklund transformations for
the ${\rm degP}_{\rm V}$ and we hope that we found all of them in this way.

Discussion of the transformations: (1) only changes the variable. It induces
the Bäcklund transformation which sends a solution $q(t)$ of ${\rm degP}_{\rm
V}(\theta_{0},\theta_{1})$ to another solution $q(-t)$ of that equation. (2)
and (3) induce the identity on the solutions of all ${\rm degP}_{\rm V}$.

(4) Consider a point $M$ of $\mathcal{M}(\theta_{0},\theta_{1})$ belonging to
the first affine chart of this variety. It is represented by
$\frac{d}{dz}+\frac{1}{z(z-1)}\left(\begin{smallmatrix}a_{0}&c_{0}+c_{1}z+t^{2}z^{2}\\\
z-q&-a_{0}\end{smallmatrix}\right)$. The change $z\mapsto 1-z$ applied to this
operator yields

$\displaystyle\frac{d}{dz}+\frac{1}{z(z-1)}\left(\begin{matrix}-a_{0}&-c_{0}-c_{1}(1-z)-t^{2}(1-z)^{2}\\\
z+q-1&a_{0}\end{matrix}\right).$

A small computation shows that after changing $t$ into $it$ this becomes a
differential operator belonging to $\mathcal{M}(\theta_{1},\theta_{0})$. This
produces the Bäcklund transformation $q(t)\mapsto-q(it)+1$ which sends a
solution of ${\rm degP}_{\rm V}(\theta_{0},\theta_{1})$ to a solution of ${\rm
degP}_{\rm V}(\theta_{1},\theta_{0})$.

(5) $M$ is locally represented by
$z\frac{d}{dz}+\left(\begin{smallmatrix}\omega_{0}&0\\\
0&-\omega_{0}\end{smallmatrix}\right)$ with $\omega_{0}=\frac{\theta_{0}}{2}$
and by $z\frac{d}{dz}+\left(\begin{smallmatrix}\frac{-3}{4}&t^{2}z\\\
1&-\frac{1}{4}\end{smallmatrix}\right)$. After taking the tensor product with
$N$ these operators become
$z\frac{d}{dz}+\left(\begin{smallmatrix}\omega_{0}+\frac{1}{2}&0\\\
0&-\omega_{0}+\frac{1}{2}\end{smallmatrix}\right)$ and
$z\frac{d}{dz}+\left(\begin{smallmatrix}\frac{-3}{4}+\frac{1}{2}&t^{2}z\\\
1&-\frac{1}{4}+\frac{1}{2}\end{smallmatrix}\right)$. By multiplying one basis
vector by $z^{-1}$, the first operator becomes
$z\frac{d}{dz}+\left(\begin{smallmatrix}\omega_{0}+\frac{1}{2}&0\\\
0&-\omega_{0}-\frac{1}{2}\end{smallmatrix}\right)$ and the second becomes
$z\frac{d}{dz}+\left(\begin{smallmatrix}\frac{-3}{4}-\frac{1}{2}&t^{2}z\\\
1&-\frac{1}{4}+\frac{1}{2}\end{smallmatrix}\right)$. This shows the validity
of (5).

Using MAPLE one can compute the actual isomorphism between the two moduli
spaces. In terms of matrix differential operators, this works as follows. Let
the differential operator $\frac{d}{dz}+A(a,q)$ represent an open part of the
first affine chart $\mathcal{M}_{1}(\theta_{0},\theta_{1})$. Here $q$ stands
for $-b_{0}$ (as before) and $a$ denotes $a_{0}$. Further $c_{1}$ is written
as a rational function in $a$ and $q$.

Similarly, let $\frac{d}{dz}+\tilde{A}(\tilde{a},\tilde{q})$ denote the
differential operator on an open part of
$\mathcal{M}(\theta_{0}+1,\theta_{1})$. Then there exists $U\in{\rm
GL}_{2}(\mathbb{C}(z))$ such that

$\displaystyle
U\left(\frac{d}{dz}+A(a,q)+\left(\begin{matrix}\frac{1}{2z}&0\\\
0&\frac{1}{2z}\end{matrix}\right)\right)U^{-1}=\frac{d}{dz}+\tilde{A}(\tilde{a},\tilde{q}).$

Further $U$, $U^{-1}$ are seen to have poles of order $\leq 1$ at $z=0$ and
$z=\infty$ and no further poles. This information suffices for the computation
of the solution $\tilde{q}$ of ${\rm degP}_{\rm V}(\theta_{0}+1,\theta_{1})$
in terms of the solution $q$ of ${\rm degP}_{\rm V}(\theta_{0},\theta_{1})$
and its first derivative (and $t$, $\theta_{0}$, $\theta_{1}$)

$\displaystyle\tilde{q}=1-\frac{\theta_{0}^{2}(q-1)}{4q^{2}t^{2}}+\frac{a\theta_{0}}{q^{2}t^{2}}+\frac{\theta_{1}^{2}}{4t^{2}(q-1)}$

and a longer formula for $\tilde{a}$, namely

$\displaystyle\frac{q-1}{8q^{3}t^{2}}\theta_{0}^{3}+\frac{q^{2}+2aq-q-6a}{8q^{3}t^{2}}\theta_{0}^{2}+\frac{q-1}{2q}\theta_{0}-\frac{a(q^{2}+2aq-q-3a)}{2(q-1)q^{3}t^{2}}\theta_{0}-\frac{\theta_{0}\theta_{1}^{2}}{8qt^{2}(q-1)}$
$\displaystyle\qquad{}-\frac{a}{q}-\frac{q+2a}{8q(q-1)t^{2}}\theta_{1}^{2}+\frac{a^{2}(2a+q)}{q^{3}t^{2}(q-1)}.$

Now we compare the group of the Bäcklund transformations for ${\rm degP}_{\rm
V}$ with the work of N.S. Witte [26]. We restrict our transformations to the
case of even solutions of ${\rm degP}_{\rm V}$ and find the classical ${\rm
P}_{\rm
V}\big{(}\frac{\theta_{1}^{2}}{2},-\frac{\theta_{0}^{2}}{2},-2,0\big{)}$. The
formulas (21), (22), (23) of [26] lead to $v_{1}=\theta_{0}+\theta_{1}$ and
$v_{2}=\theta_{0}-\theta_{1}$. Our group of transformations for the even
solutions of ${\rm degP}_{\rm V}$ is generated by
$(\theta_{0},\theta_{1})\mapsto(\theta_{0}+1,\theta_{1})$ and
$(\theta_{0},\theta_{1})\mapsto(\theta_{1},\theta_{0})$ and the ‘trivial’
transformations
$(\theta_{0},\theta_{1})\rightarrow(\pm\theta_{0},\pm\theta_{1})$. One easily
verifies that this coincides with the group of transformations in [26].

We remark that the transformations $\theta_{0}\mapsto-\theta_{0}$ and/or
$\theta_{1}\mapsto-\theta_{1}$ act trivially on solutions of ${\rm degP}_{\rm
V}$. However (compare [26]), they do not act as the identity on a suitable
Hamiltonian system for ${\rm degP}_{\rm V}$.

### Acknowledgments

The authors thank Yousuke Ohyama for his many helpful answers to our
questions, and the referees for their careful reading and useful suggestions.
The first author thanks the Johann Bernoulli Institute of the University of
Groningen and the Universidad Simón Bolívar for financial support to
participate in this project.

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# Timbre Transfer using Image-to-Image Denoising Diffusion Implicit Models

###### Abstract

Timbre transfer techniques aim at converting the sound of a musical piece
generated by one instrument into the same one as if it was played by another
instrument, while maintaining as much as possible the content in terms of
musical characteristics such as melody and dynamics. Following their recent
breakthroughs in deep learning-based generation, we apply Denoising Diffusion
Models (DDMs) to perform timbre transfer. Specifically, we apply the recently
proposed Denoising Diffusion Implicit Models (DDIMs) that enable to accelerate
the sampling procedure. Inspired by the recent application of DDMs to image
translation problems we formulate the timbre transfer task similarly, by first
converting the audio tracks into log mel spectrograms and by conditioning the
generation of the desired timbre spectrogram through the input timbre
spectrogram. We perform both one-to-one and many-to-many timbre transfer, by
converting audio waveforms containing only single instruments and multiple
instruments, respectively. We compare the proposed technique with existing
state-of-the-art methods both through listening tests and objective measures
in order to demonstrate the effectiveness of the proposed model.

##  1 Introduction

Figure 1: Training scheme of the proposed DiffTransfer technique. The target
instrument spectrogram is summed with noise following a simplified cosine
schedule. The decoder, conditioned on the conditioning instrument spectrogram
and on the sinusoidal embedding representing the current time instant
estimates the added noise. The decoder parameters are estimated by computing
the L1 loss between the ground truth and the estimated diffusion noise.

Timbre is an extremely important perceptual aspect of music, yet it is hard to
both model and define. The concept of musical timbre can be defined as the
perceived characteristics of a musical sound that are different from pitch and
amplitude contours [1].

Timbre Transfer concerns the task of converting a musical piece from one
timbre to another while preserving the other music-related characteristics.
While this operation is not trivial, it is of extreme interest for several
applications, from the development of plugins to be used in Digital Audio
Workstations (DAW) to enabling the possibility of playing sounds corresponding
to not widely available musical instruments.

In this paper, we present DiffTransfer, a technique for timbre transfer which
is tested both between single and multiple instruments and is based on a
continuous Denoising Diffusion Implicit Model (DDIM) with deterministic
sampling [2], a modified version of Denoising Diffusion Probabilistic Models
(DDPMs) that are trained using the same procedure, but allow for faster
sampling times. Specifically, in [2] it was empirically shown that DDIMs allow
for $10\times-50\times$ faster wall-clock time performances with respect to
DDPMs.

In order to be able to convert one timbre into another, we use a procedure
similar to the recently proposed image-to-image technique Palette [3].
Specifically, we use as input to the diffusion model the noise and condition
it with the chosen input timbre spectrogram, then, through the denoising
procedure, the model learns to reconstruct spectrograms of the desired timbre.
We consider the scenario where the timbre-transfer task is paired, which means
that the desired and input spectrograms have the same melodic/harmonic
content, but differ in terms of timbre.

We experiment both with the possibility of converting between tracks
containing only single instruments and also mixtures of instruments, with no
prior separation step, while making no modifications to the model in order to
take into account both configurations.

In order to demonstrate the effectiveness of the proposed model, we compare
DiffTransfer with state-of-the-art techniques, both through objective measures
and by performing a user-based listening test. The source code and audio
excerpts can be found at https://lucacoma.github.io/DiffTransfer/.

##  2 Related Work

Several types of timbre Transfer techniques have been proposed in the
literature. In [4] a CycleGAN [5] is applied in order to perform an unpaired
transfer using the Constant-Q transform and the audio is then recovered
through a WaveNet [6] model. In [7] an attention-based architecture is applied
in order to convert mel spectrograms, which are then inverted through a MelGAN
architecture [8]. Gaussian mixture-based variational autoencoders are applied
[9] in order to learn a latent space where pitch and timbre representations
are disentangled.

Another class of methods, instead, extracts musical parameters such as pitch
and loudness from the input audio tracks and performs the transfer by
resynthesizing sound through a network that has learned to generate tracks
with the desired timbre. The most known example of these techniques is the
Differentiable Digital Signal Processing (DDSP) [10] model. Other similar
techniques were proposed such as [11], where a hierarchical model is used in
order to reconstruct the signal at increasing resolutions. Recently there have
been proposed also models that directly work on the audio waveform such as
[12], where music pieces are translated to specific timbre domains. The only
model that, to the best of our knowledge and except for the one proposed in
this paper, is tested on multi-instrument timbre transfer without any source
separation pre-processing is the Music-STAR network, presented in [13]. In
Music-STAR a WaveNet autoencoder [14] is trained by applying teacher-forcing
[15] to the decoders in order to recover the desired timbre.

Denoising Diffusion Probabilistic Models (DDPMs) [16] have recently become the
latest state-of-the-art for what concerns deep learning-based generation
fastly replacing Generative Adversarial Networks (GANs) [17] and Variational
Autoencoders [18], due to their easier training procedure and increased
quality of the produced results.

DDPMs have been successfully applied to a wide variety of image-related tasks
such as generation [19] and translation [3].

More recently, DDPMs have been also used for audio-related tasks. In [20] a
diffusion model is applied in order to convert midi tracks to spectrograms,
while in [21] a text-to-music diffusion model is proposed. DDPMs have also
been applied to symbolic music generation [22], speech synthesis [23] and
singing voice extraction [24].

While DDPMs have extremely powerful generation capabilities they suffer from
slow sampling times. To ameliorate this issue, recently Denoising Diffusion
Implicit Models (DDIMs) [2], which allow for faster sampling times and were
recently applied to image inpainting [25].

##  3 Proposed Model

In this section, we describe the proposed DiffTransfer technique for timbre
transfer. Instead of working directly with raw audio signals, we convert them
into log mel-scaled spectrograms, due to their easier handling by deep
learning models. We then propose a model that, given as input the spectrogram
corresponding to the conditioning instrument, generates the corresponding
target spectrogram that would have been obtained by playing the same piece of
music with the target instrument. Operatively we achieve this through a
conditional continuous-time DDIM, which learns to denoise the target
instrument spectrogram, while conditioned on the input instrument spectrogram,
as depicted in Fig. 1. At inference time, the model is fed with the input
conditioning instrument concatenated with Gaussian noise and generates the
corresponding target spectrogram. We retrieve the audio signal by applying to
the log mel spectrograms the
SoundStream111https://tfhub.dev/google/soundstream/mel/decoder/music/1 model
[26], provided by [20] where it was trained on a custom music dataset.

In the following, we’ll provide a brief overview of the DDIM framework and
notation used in this paper, in order to make the tractation as compact as
possible, for additional and more thorough formulations, we refer the reader
to [2] and [3]. We aim at giving a general overview of the process and we’ll
use a slight abuse of notation to describe the diffusion process using the
continuous time framework, in order to make it more similar to the more common
literature regarding DDPMs and DDIMs.

###  3.1 Diffusion Decoder

Figure 2: Deployment scheme of the proposed DiffTransfer technique. The
decoder is fed with Gaussian noise and with the conditioning instrument
spectrogram. The noise estimate provided by the decoder is then subtracted
from the input noise in order to provide an estimate of the desired target
spectrogram, from which the audio is estimated via the SoundStream model [26,
20].

We adopt a procedure similar to the Palette [3] image-to-image translation
technique in order to train the timbre transfer decoder as a Denoising
Diffusion Implicit Model (DDIM) [2]. Broadly speaking, DDIMs work by learning
how to generate data from noise in a two-part procedure. The first part is
denoted as the forward process, where Gaussian noise
$\gamma\sim\mathcal{N}(0,1)$ is subsequently added to the input until it is
indistinguishable from the former. The second part consists of the reverse
process where a decoder learns how to invert the forward process, effectively
reconstructing data from the noise. DDIMs can be seen as a generalization of
DDPMs that shares the same training procedure, however, they differ in the
modeling of the reverse process, by using a non-markovian diffusion process,
which allows for faster generation times.

####  3.1.1 Forward Process

Let us define $\mathbf{X}$ and $\mathbf{Y}$ as the log mel spectrograms
corresponding to the conditioning and target instruments, respectively. We
choose a continuous diffusion time [27, 28, 29]in order to be able to change
the number of desired sampling steps. If we consider $T$ steps, then the
diffusion time can be defined as $t\in\\{0,1\\}$, where consecutive times are
separated by $\Delta_{t}=1/T$. Then, the forward process is defined similarly
to the case of DDPMs by subsequently adding noise to the target spectrogram
for $T$ steps

$\begin{split}&q(\mathbf{Y}_{t}|\mathbf{Y}_{t-\Delta_{t}})=\mathcal{N}(\mathbf{Y}_{t},\sqrt{(\alpha_{t})}\mathbf{Y}_{t-\Delta_{t}},\beta_{t}\mathbf{I}),\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}q(\mathbf{Y}_{1:T}|\mathbf{Y}_{0})=\prod_{t=1}^{T}q(\mathbf{Y}_{t-\Delta_{t}})\end{split}$
(1)

where $\alpha$ and $\beta$ are parameters defined by a simplified cosine
schedule [30].

####  3.1.2 Reverse Process

In the case of DDIMs, the reverse diffusion process is operated by introducing
an additional distribution $p_{\theta}$, where a sample $\mathbf{Y}_{t-\Delta
t}$ can be generated from a sample $\mathbf{Y}_{t}$ as

$\begin{split}\mathbf{Y}_{t-\Delta t}=&\sqrt{\beta_{t-\Delta
t}}\left(\frac{c-\sqrt{\beta_{t}}\gamma_{\theta}^{(t)}(\mathbf{Y}_{t},\mathbf{X})}{\sqrt{(}\alpha_{t})}\right)+\\\
&\sqrt{1-\alpha_{t-\Delta_{t}}}\cdot\gamma_{\theta}^{(t)}(\mathbf{Y}_{t},\mathbf{X}),\end{split}$
(2)

, where $\gamma$ is the noise estimated by a network with parameters $\theta$.
The noise at time $t$ $\gamma_{\theta}^{(t)}$ is estimated by a network that
is conditioned also on the input timbre spectrogram $\mathbf{X}$, similarly to
the formulation proposed in Palette [3].

####  3.1.3 Training Procedure

The denoising process is operated through a U-Net architecture which is
conditioned on $\mathbf{X}$ and trained to predict the added noise in order to
minimize the L1 loss

$\mathbb{E}=||\gamma_{\theta}^{(t)}(\mathbf{Y}_{t},\mathbf{X})-\gamma||_{1}^{1},$
(3)

where $\gamma$ is the true perturbation, while
$\gamma_{\theta}^{(t)}(\mathbf{Y}_{t},\mathbf{X})$ is the estimate of the
noise added to the target spectrogram at time $t$, conditioned on the input
spectrogram $\mathbf{X}$.

###  3.2 Architecture

The decoder architecture is based on a U-Net model. The building element is
made of residual blocks, in each of these the input is processed by (i) a 2D
convolutional layer with swish activation, followed by batch normalization and
by (ii) a convolutional layer with no activation. Both convolutional layers
have kernel size $3$. The output of this procedure is then summed with the
residual, which is obtained by processing the input with a convolutional layer
with kernel size $1$.

The encoder part of the network consists of $3$ downsampling blocks, each
consisting of $4$ residual blocks having filter sizes $64,128,256$. The output
of each downsampling block is followed by average pooling, with pool size $2$
in order to compress the dimension of the spectrograms. The last block of the
encoder is followed a self-attention block.

The bottleneck obtained through the encoder is processed by a residual block
with $512$ filters and is then processed by the decoder, which is a specular
version of the encoder. The only difference lies in the use of transposed
convolutions in order to create upsampling layers needed to increase the
dimension of the features.

The last downsampling layer of the encoder, the bottleneck and the first
upsampling layer of the decoder are followed by self-attention.

###  3.3 Deployment

The proposed model takes as input spectrograms of a fixed size, therefore
audio tracks longer than the ones used for training need to be sliced
accordingly.

The decoder takes as input the conditioning spectrogram $\mathbf{X}$ and the
diffusion noise and retrieves an estimate of the latter, which can then be
subtracted in order to obtain an estimate of the desired output timbre
spectrogram $\hat{\mathbf{Y}}$. The output waveform $y$ can then be obtained
by feeding the pre-trained SoundStream model with $\hat{\mathbf{Y}}$.

##  4 EXPERIMENTS

In this section, we describe experiments performed with the aim of
demonstrating the capabilities of the proposed DiffTransfer technique both in
the single-instrument and multi-instrument application scenarios.

In Fig. 3 we show an example of input, generated and ground-truth
spectrograms, obtained via the DiffTransfer model when converting from a
Clarinet to Strings.

(a)

(b)

(c)

Figure 3: Example of Timbre Conversion log mel Spectrograms using the
DiffTransfer architecture, obtained when converting Clarinet
LABEL:sub@subfig:cond to Strings LABEL:sub@subfig:est. The ground truth
Strings spectrogram is shown in LABEL:sub@subfig:gt.

###  4.1 Dataset

In order to train the model we considered the StarNet dataset [31], which
contains a set of tracks that are played with two timbre-domains, namely
strings-piano and vibraphone-clarinet. The dataset consists of roughly 22
hours of audio. We used the reduced version of the dataset, where tracks are
resampled to $16000~{}\mathrm{Hz}$ and converted them to mono. In order to
perform the evaluation, we use the same ten tracks considered in [13], in
order to ease the comparison with their model.

###  4.2 Techniques Under Comparison

We consider two baselines in order to compare the performances of the proposed
DiffTransfer architecture. For what concerns the single-instrument timbre
transfer task, we consider the Universal Network [12] fine-tuned on the
StarNet dataset as done in [13]. For what concerns the multi-timbre task, we
consider the mixture-supervised version of the Music-STAR network proposed in
[13]. We perform three different types of timbre transfer tasks: single, where
only single instruments are converted, single/mixed where the separate
conversions of single instruments are mixed in order to create the desired
mixture track and mixture, where the mixture is directly converted. These
nomenclatures are used just to ease the presentation of the results, we would
like to point out that, for what concerns the DiffTransfer architecture, no
specific changes are required for the various types of applications, except
for the choice of desired input data.

###  4.3 Experiment Setup

The Universal Network and Music-STAR architectures are trained with the
procedure described in [13]. The DiffTransfer network is trained for $5000$
epochs using a batch size of $16$, with the AdamW optimizer [32] with learning
rate $2e-5$ and weight decay $1e-4$. The epoch that minimizes the $L1$ noise
prediction loss is chosen in order to retain the model used to compute the
results. We train a total of six models, performing the following timbre
transfer conversions: vibraphone to piano, piano to vibraphone, clarinet to
strings, strings to clarinet vibraphone/clarinet to piano/strings and
piano/strings to vibraphone/clarinet.

The network input features are computed by first applying the Short-Time
Fourier Transform (STFT) with a Hann window of size $0.020~{}\mathrm{s}$ and
$50\%$ overlap to normalized audio tracks. Then the log mel spectrogram is
computed over $128$ bins corresponding to the range of
$0-16000~{}\mathrm{Hz}$. We do not feed the entire audio tracks as input to
the network, instead, during each epoch we extract $128$ frames from the log
mel spectrogram, corresponding to $\approx~{}2~{}\mathrm{s}$. Each spectrogram
slice is normalized between $-1$ and $1$ before being given as input to the
network and the output spectrograms are denormalized before being fed to the
SoundStream model in order to recover the audio waveform. Since the tracks
considered for the test are of length $10~{}\mathrm{s}$ and the model gets as
input a fixed $128$ frames spectrogram we slice the conditioning spectrogram
before feeding into the model and we keep the input noise fixed for all
slices, in order to ensure consistency in the generation. All spectrogram
slices are normalized in the range $[-1,1]$ and denormalized before being fed
to the SoundStream decoder.

###  4.4 Objective Evaluation

We evaluate the model objectively in order to analyze the perceptual
similarity and content preservation capabilities of the generated tracks with
respect to the ground truth audio.

Objective Evaluation
---
Method |  FAD $\downarrow$ |  JD $\downarrow$
Universal Network (single) | 7.09 | 0.53
DiffTransfer (single) | 2.58 | 0.28
Universal Network (single/mixed) | 10.47 | 0.64
DiffTransfer (single/mixed) | 4.73 | 0.46
Music-STAR (mixture) | 8.93 | 0.57
DiffTransfer (mixture) | 4.37 | 0.38
Table 1: Objective Evaluation of the proposed DiffTransfer Method compared to
the baselines, in terms of Fréchet Audio Distance (FAD) and Jaccard Distance
(JD). Results are averaged over all participants and over all the tracks
considered for each part of the test.

In order to evaluate the perceptual similarity, we compute the Fréchet Audio
Distance (FAD) [33] using the VGGish embeddings [34], through a PyTorch
implementation222https://pypi.org/project/frechet-audio-distance/. FAD is a
reference-free metric for music enhancement algorithms, which views the
embeddings as a continous multivariate Gaussian and is computed between the
real and generated data as

$\mathrm{FAD}=||\mu_{r}-\mu_{g}||^{2}+\mathrm{tr}(\Sigma_{r}+\mu_{g}-2\sqrt{\Sigma_{r}\Sigma_{g}}),$
(4)

where $(\mu_{r},\Sigma_{r})$ and $(\mu_{g},\Sigma_{g})$ are the mean and
covariances of the embeddings corresponding to the real and generated data,
respectively. Similarly to [20], we compute FAD in order to analyze the
perceptual similarity between the generated audios with respect to the ground
truth one, corresponding to the original StarNet dataset.

To understand the content-preservation capabilities of the model, following
[35], we compute how the pitch contours of generated ground truth audio tracks
are dissimilar, by calculating the mismatch between two sets of pitches $A$
and $B$ through the Jaccard Distance

$JD(A,B)=1-\frac{|A\cap B|}{|A\cup B|},$ (5)

where a lower value corresponds to a lower mismatch and thus to a higher
degree of similarity between the generated pitch contours. Pitch contours are
computed using a multi-pitch version of the MELODIA [36] as implemented in the
Essentia library [37], rounding pitches to the nearest semitone. We report the
values obtained by computing the metrics on the test dataset in Table 1.

###  4.5 Subjective Evaluation

In order to evaluate subjectively the timbre transfer capabilities, we perform
a listening test with 18 human participants. The web page of the test is
available at 333https://listening-test-ismir-ttd.000webhostapp.com/. The test
was split into two parts corresponding to the single and multiple instrument
application scenarios, respectively.

During the single instrument part of the test, the users listened to four
tracks, corresponding to the four types of conversions performed, namely:
clarinet to strings, strings to clarinet, piano to vibraphone, vibraphone to
piano. Each example consisted of two conditions, one obtained via the
DiffTransfer model and the other through the Universal Network.

In the second part of the test, concerning multiple instrument timbre
transfer, a total of four tracks were considered, two for the conversion from
vibraphone/strings to piano/strings waveforms and two for the reverse
conversion. Each example consisted of four conditions, namely DiffStar
(single/mix), Universal Network (single/mix), DiffStar (mixture) and Music-
STAR (mixture).

Both the order of conditions and the order of examples in each separate part
of the test were randomized.

The participants were asked to rate the conditions in terms of similarity with
respect to the reference track on a 5 elements Likert scale where $1$
corresponds to bad and $5$ to excellent. We report the results obtained
through the listening test in Table 2.

###  4.6 Discussion

Subjective Evaluation
---
Method | Similarity
Universal Network (single) | 1.82
DiffTransfer (single) | 3.68
Universal Network (single/mixed) | 1.69
DiffTransfer (single/mixed) | 3.78
Music-STAR (mixture) | 2.89
DiffTransfer (mixture) | 3.80
Table 2: Objective Evaluation of the proposed DiffTransfer Method compared to
the baselines, in terms of perceived similarity with respect to the ground
truth on a Likert scale from 1 (Bad) to 5 (Excellent). Results are averaged
over all test tracks.

By briefly inspecting both the objective and subjective results, reported in
Table 1 and 2, respectively, it is clear how the proposed DiffTransfer model
outperforms the Universal Network and Music-STAR baselines both for what
concerns the single and multiple timbre transfer tasks.

When considering single timbre results, DiffTransfer is able to achieve
significantly better performances in terms of FAD, Jaccard Distance and
Perceived Similarity, with respect to the Universal network. The gap between
the two methods becomes even more evident when considering the single/mixed
case, i.e. when single timbre transfer tracks are mixed in order to form the
desired mixture audio.

For what concerns the Music-STAR method, the gap with respect to DiffTransfer
remains high in terms of FAD, but becomes less noticeable when considering JD
and the perceived subjective similarity.

##  5 Conclusion

In this paper, we have presented DiffTransfer a technique for both single- and
multi-instrument timbre transfer using Denoising Diffusion Implicit models.
The novelty of the proposed approach lies in the fact that in addition to
being, to the best of our knowledge, the first application of diffusion models
to timbre transfer, it is the first model to be tested in order to perform
single and multi-timbre transfer, without varying the architecture depending
on which application is chosen. We compared the proposed model with state-of-
the-art Universal Network and Music-STAR baselines through both objective
evaluation measures and a listening test, demonstrating the better
capabilities of the proposed DiffTransfer approach.

Future works will involve increasing the audio quality of the generated audio,
by taking into account the consistency of subsequent generated spectrograms.
Furthermore, we plan on modifying the model in order to be able to perform
unpaired timbre transfer, which greatly eases the dataset requirements and
applicability of the technique.

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