Abstract
For certain finite groups \(G\) of Bäcklund transformations, we show that a dynamics of \(G\)-invariant configurations of \(n|G|\) Calogero–Painlevé particles is equivalent to a certain \(n\)-particle Calogero–Painlevé system. We also show that the reduction of a dynamics on \(G\)-invariant subset of \(n|G|\times n|G|\) matrix Painlevé system is equivalent to a certain \(n\times n \) matrix Painlevé system. The groups \(G\) correspond to folding transformations of Painlevé equations. The proofs are based on Hamiltonian reductions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In the paper we construct a relation between matrix Painlevé equations of different size and Calogero–Painlevé systems of different number of particles. Such relations are in a correspondence with folding transformations in the Painlevé theory. We recall these notions and illustrate our results by simple instructive examples.
Folding transformations This notion was introduced in [13], but actual examples of such transformations were known for many years. By definition, the folding transformation is an algebraic (of degree greater than 1) map between solutions of Painlevé equations. Moreover, this map should go through a quotient of Okamoto-Sakai space of initial conditions (see e.g. [8] for the review). Probably, the simplest example is the folding transformation of Painlevé II to itself.
Example 1.1
The Painlevé \(\textrm{II}\) equation is a second-order differential equation with parameter \(\theta \)
This equation is equivalent to a Hamiltonian system with Hamiltonian
This equation (system) has a natural symmetry \(r\) which transforms parameter \(\theta \mapsto -\theta \) and maps \((p,q) \mapsto (-p,-q)\). Such symmetries are called Bäcklund transformations. For the special value of the parameter \(\theta =0\) the equation is preserved by \(r\) so one can ask for an equation on functions invariant under this transformation. If we introduce new invariant coordinates \(P,Q\) and new time \(s\)
we get Painlevé \(\textrm{II}\) equation on Q with parameter \(\theta =-1/2\)
This is a transformation of degree 2 between the spaces of initial conditions.
Note that in this example we start from the Painlevé equation with the special value of the parameter (namely \(\theta =0\)) and come to the Painlevé equation with the special value of the parameter (namely \(\theta =-1/2\)). But, it appears that we can come to the equation with an arbitrary value of parameter if we start from Calogero–Painlevé system.
Calogero–Painlevé systems These systems can be viewed as a certain \(N\)-particle generalization of the Painlevé equations. Let \(q_1,\dots ,q_{N}\) be coordinates of these particles and \(p_1,\dots ,p_{N}\) be the corresponding momenta with the standard Poisson bracket between them. The dynamics of a Calogero–Painlevé system is defined by the Hamiltonian of the form [12]
Here \(V^{(1)}(q)\) is a Painlevé potential (i.e. for \(N=1\) we get a Hamiltonian of the Painlevé equation) and \(V^{(2)}(q_1,q_2)\) is a Calogero-type interaction. Note that the Hamiltonian (1.5) is symmetric under permutations. By definition the phase space of a Calogero–Painlevé system is a quotient by the action of permutation group \(S_{N}\).
Note that the Hamiltonian (1.5) is non-autonomous, since Painlevé potential \(V^{(1)}\) depends on time \(t\). In the autonomous limit, in which \(t\) is a coupling constant, the Hamiltonian (1.5) belongs to Inozemtsev extension of Calogero integrable system [4, 5].
Another important feature of Calogero–Painlevé systems is that they describe isomonodromic deformations of certain natural \(2N\times 2N\) systems [2, 6, 12].
In the paper we construct a natural analog of the folding transformations for the Calogero–Painlevé systems. Let us give an example of Calogero–Painlevé II for \(N=2\).
Example 1.2
Hamiltonian (1.5) in this case has the form
This system has a symmetry \(r_{\textrm{CP}}: \{(p_1, q_{1}), (p_{2}, q_{2})\} \mapsto \{(-p_1, -q_{1}), (-p_{2}, -q_{2})\}\) with \(\theta \mapsto -\theta \). For \(\theta =0\) the subset of \(r_{\textrm{CP}}\)–invariant points is defined by the equations (recall that we take the quotient by permutation group \(S_2\))
It is easy to check that these equations are preserved by the dynamics.
For \(g = 0\) Calogero–Painlevé system is equivalent to the system of two non-interacting Painlevé particles \(q_1,q_2\) up to the permutation. Hence, by Example 1.1, the dynamics on \(r_{\textrm{CP}}\)-invariant subset is equivalent to the Painlevé \(\textrm{II}\) equation with \(\theta =-1/2\).
For \(g \ne 0\) let us take the following coordinates P, Q on \(r_{\textrm{CP}}\)-invariant subset and rescale the time
Note that these formulas are g-deformed version of formulas (1.3). It is straightforward to compute the dynamics in terms of \(p_1,q_1\)
and then get
which is the Painlevé \(\textrm{II}\) equation with parameter \(\theta =-\frac{1}{2} -\text {i}g\).
It appears that this example can be generalized to the Calogero–Painlevé system with more than 2 particles. Namely, for \(2n\)-particle Calogero–Painlevé II with \(\theta =0\) the dynamics on (dense open subset of) \(r_{\textrm{CP}}\)-invariant subset is equivalent to the dynamics of \(n\)-particle Calogero Painlevé II system with \(\theta =-\frac{1}{2} -\text {i}g\). Moreover, similar statements hold for other folding transformations, see Theorem 4.1 and remarks after it. This is one of the main results of the paper.
Remark 1.1
Part of our motivation comes from the papers by Rumanov [10, 11], where certain solutions of \(N\)-particle Calogero–Painlevé II systems are related to a spectrum of \(\beta \)-ensemble with \(\beta =2N\). It was also observed in [11] that this solution for \(N=2\) is described by the scalar Painlevé II equation. It appears that this solution is invariant with respect to \(r_{\textrm{CP}}\), hence this observation is a corollary of Example 1.2 above. It would be interesting to study subsets corresponding to Rumanov’s solutions.
Matrix Painlevé systems Calogero–Painlevé systems can be obtained by a Hamiltonian reduction à la Kazhdan–Kostant–Sternberg [7] from matrix Painlevé systems [2].Footnote 1 In the Calogero–Painlevé case we consider the Bäcklund invariant subset of the phase space. It appears that in the matrix case it is natural to consider a subset invariant under the Bäcklund transformation twisted with conjugation by a certain permutation matrix. In the matrix case we also perform an additional reduction.
Example 1.3
The phase space of matrix Painlevé \(\textrm{II}\) consists of pairs of \(N\times N\) matrices \(p,q\) with symplectic form \(\textrm{Tr}(\textrm{d}p\wedge \textrm{d}q)\). The Hamiltonian has the form [cf. Hamiltonian in scalar case (1.2)]
where parameter \(\theta \) remains a scalar variable.
As before we have Bäcklund transformation \(r: (p,q)\mapsto (-p,-q)\) with \(\theta \mapsto -\theta \). Let us take \(2n\times 2n\) matrix Painlevé \(\textrm{II}\) with \(\theta =0\) and consider the subset of the phase space invariant under \(\textrm{Ad}_{S_2} \circ r\), where \(S_2=\mathrm (\textbf{1}_{n\times n}, -\textbf{1}_{n\times n})\). This invariant subset is given by block matrices with \(n\times n\) blocks
It is easy to see that the dynamics preserves this subset. It is given by
One can check that \(m_2={\mathfrak {p}}_{21}{\mathfrak {q}}_{12} - {\mathfrak {q}}_{21}{\mathfrak {p}}_{12}\) is an integral of motion. Let us fix its value as \(m_2=\text {i}g \textbf{1}_{n\times n}\). Then, taking \(n\times n\) matrices (P, Q) and rescaling the time
we obtain matrix Painlevé \(\textrm{II}\) with \(\theta =-\text {i}g-1/2\)
Note that the parameter value for the resulting matrix Painlevé system coincides with one for the scalar Painlevé \(\textrm{II}\) obtained in Example 1.2 [cf. (1.15) with (1.10)].
Actually, the last step in the example is a Hamiltonian reduction with respect to the conjugation by a certain \(\textrm{GL}_n \subset \textrm{GL}_{2n}\). In Theorem 3.1 we construct such Hamiltonian reductions of matrix Painlevé systems, corresponding to all folding transformations of the Painlevé equations. This is one of the main results of the paper. Moreover, our proof of Theorem 4.1 mentioned above is based on Theorem 3.1.
Plan of the paper In Sect. 2 we recall matrix Painlevé equations in their Hamiltonian forms. We also lift all Bäcklund transformations known in scalar case to the matrix case. This lift is not completely straightforward due to noncommutativity of variables, see e.g. Tables 1 and 2 below.
In Sect. 3 we construct block reductions for the matrix Painlevé systems. The construction works for Bäcklund transformations which preserve parameters of certain Painlevé systems but act on \(p,q\) nontrivially (similarly to the folding transformations). Let \(w\) be such a transformation, \(d\) denotes order of \(w\) and \({\bar{w}}\) denotes \(w\) twisted by adjoint action of a certain permutation matrix of order \(d\). In Theorem 3.1 we consider a subset in the phase space of \(nd \times nd\) matrix Painlevé system that is invariant under the action of \({\bar{w}}\), and show that the dynamics on its Hamiltonian reduction is equivalent to \(n\times n\) matrix Painlevé system. The proof of this theorem is based on case by case considerations, which occupy the bulk of Sect. 3. In Sect. 3.4 we prove a similar statement for the non-cyclic subgroups.
In Sect. 4.1 we recall the definition of Calogero–Painlevé systems. In the following Sect. 4.2 we prove Theorem 4.1. Roughly speaking, this theorem states that the dynamics on \(w\)-invariant set of \(nd\) Calogero–Painlevé particles is equivalent to the dynamics of \(n\) Calogero–Painlevé particles. As we mentioned above, the proof is based on Theorem 3.1 and the construction of Calogero–Painlevé systems via the Hamiltonian reduction of matrix Painlevé systems. In particular, we do not need a case by case analysis here.
It appears that there are more relations between matrix Painlevé systems and Calogero–Painlevé systems, similar to ones found in Theorems 3.1 and 4.1. We do not intend to classify them and just give several examples in Sect. 5. In particular, in Sect. 5.1 we study another block reductions of matrix Painlevé equations. In Sect. 5.2 we study \(w\)-invariant configurations of Calogero–Painlevé particles in which several particles evolve by algebraic solutions of a Painlevé equation. Finally, in Sect. 5.3 we discuss spin generalization of the Calogero–Painlevé systems.
2 Matrix Painlevé systems and their Bäcklund transformations
We follow [2] in conventions on matrix Painlevé systems.
2.1 Painlevé II
Scalar case We will consider a parametrized family PII\((\theta )\) of ordinary differential equations
where by dot we denote \(\frac{\text {d}}{\text {d}t}\). These equations are Hamiltonian. Namely, one can take \({\mathbb {C}}^2\) with standard symplectic structure \(\omega = \textrm{d}p\wedge \textrm{d}q\), then the Hamiltonian (1.2)
leads to dynamics (2.1).
This family of ordinary differential equations has discrete symmetries, called Bäcklund transformations. For example, it is easy to see that if q(t) is a solution of PII\((\theta )\), then \(-q(t)\) is a solution of PII\((-\theta )\).
Let us define Bäcklund transformations in general case. Let \({\mathbb {D}}\left( \alpha \right) \) be a system of ordinary differential equations with extended phase space \(\textrm{M}\), which depends on the set of parameters \(\alpha \in {\mathbb {C}}^k\).
Definition 2.1
A pair of maps \(\left( \pi , {\tilde{\pi }}\right) : \textrm{M}\times {\mathbb {C}}^k \rightarrow \textrm{M}\times {\mathbb {C}}^k\) is called Bäcklund transformation if \(\pi \) maps solutions of \({\mathbb {D}}\left( \alpha \right) \) to solutions of \({\mathbb {D}}\left( {\tilde{\pi }}\left( \alpha \right) \right) \).
Let us consider a hamiltonian dynamics of a system of ordinary differential equations \(\mathbb {D}\left( \alpha \right) \) on a symplectic manifold \(\left( M, \omega \right) \) with a time-dependent Hamiltonian \(H_{\alpha }(x; t),\, (x,t)\in \textrm{M} = M \times \mathbb {C}\). The equations of motion can be identified with the one dimensional distribution on \( \textrm{TM}\) defined as \( \textrm{Ker}\left( \omega - \textrm{d}H_{\alpha }\wedge \textrm{d}t\right) \). Then the sufficient condition for \( \left( \pi , \tilde{\pi }\right) \) to be a Bäcklund transformation is
for a certain function \(h_{\alpha }(x;t)\). Note that it is important to consider extended phase space since we work with non-autonomous systems and non-autonomous symmetries.
Below we follow [13] in the description of Bäcklund transformations.
Matrix generalisation Hamiltonian system corresponding to PII\((\theta )\) can be generalized to the matrix case. Let us consider a phase space \(\textrm{Mat}_{N}\left( {\mathbb {C}}\right) \times \textrm{Mat}_{N}\left( {\mathbb {C}}\right) \) with coordinates (p, q) and symplectic structure \(\omega = \textrm{Tr}\left( \textrm{d}p\wedge \textrm{d}q\right) \). Matrix PII\((\theta )\) can be defined by the Hamiltonian (1.11)
Note that the parameter \(\theta \) remains scalar. In all cases below we will consider matrix Painlevé equations with scalar parameters only. Hamiltonian (2.4) leads to equations of motion
In case \(N = 1\) this system is equivalent to the ordinary differential equation PII\((\theta )\).
Now we generalize formulas for the Bäcklund transformations of scalar PII\((\theta )\) to the matrix case. Here and below we will use notation \(C_n\) for the cyclic group of order n.
Proposition 2.1
Transformations in the table below are Bäcklund transformations of matrix \(\textrm{PII}\). These transformations generate group \(C_2\ltimes W\left( A_1^{(1)}\right) \).
Let us explain the notation. Here \(f = p + q^2 + \frac{t}{2},\;\;\alpha _1 = \theta + \frac{1}{2},\;\;\alpha _0 = 1 - \alpha _1\). Parameters \(\alpha _0\) and \(\alpha _1\) are called root variables. Action of \(s_1\) on them is given by \(s_{1}(\alpha _1) = -\alpha _1,\;\;s_{1}(\alpha _0) = \alpha _0 + \alpha _1 \). The action of r on root variables is indicated on the diagram.
For any Painlevé equation group of Bäcklund transformations is isomorphic to an extended affine Weyl group. Thus it is convenient to encode generators and relations of these groups through diagrams. Solid lines and nodes define an affine Dynkin diagram. On the i-th node we write the corresponding root variable \(\alpha _i\). These root variables parametrize the family of equations. By \(s_i\) we denote the reflection corresponding to i’th simple root. A solid line between i’th and j’th nodes corresponds to the relation \(s_is_js_i = s_js_is_j\). Absence of a solid line between i’th and j’th nodes corresponds to the relation \(s_is_j = s_js_i\). The action of reflections on the root variables is given by
where \(\{a_{ij}\}\) is the Cartan matrix corresponding to a given Dynkin diagram. Then we extend Coxeter group by a finite group, acting by automorphisms of the diagram. Dashed arrows show action of automorphisms on parameters and define adjoint action on \(\{s_i\}\)
The generators coming from the automorphisms of diagrams will be very important in Sect. 3.
Proof of proposition 2.1
Let us check that transformations from Table 1 are symmetries of matrix PII\((\alpha _0,\alpha _1)\) using (2.3).
For transformation r it is easy to see that it maps Hamiltonian with parameter \(\theta \) to the Hamiltonian with parameter \(-\theta \). Also it preserves \(\omega \) and t. Thus r satisfies equation (2.3) with \(h_{\theta }(x;t) = 1\).
For transformation \(s_1\) let us consider matrix coordinates \(\left( f, q\right) \). Then
So we have
It is easy to see that \(s_1\) acts as \( f \mapsto f, \alpha _1 \mapsto -\alpha _1\). Hence we get
Then we have
Hence \(s_1\) is a symmetry of the matrix PII\(\left( -\theta + \frac{1}{2}, \theta + \frac{1}{2}\right) \).
Now it remains to check group relations between generators. In the case considered we have to check that \(s_1\), r are involutions, which can be done by a straightforward computation. The group obtained is not smaller than \(C_2 \ltimes W\left( A_{1}^{(1)}\right) \) since the action on parameters is the same as in the case of scalar PII\(\left( -\theta + \frac{1}{2}, \theta + \frac{1}{2}\right) \) and action on parameters determines an element of \(C_2 \ltimes W\left( A_{1}^{(1)}\right) \) uniquely. \(\square \)
2.2 Painlevé VI
The definition, Hamiltonian and Bäcklund transformations of scalar PVI\(\left( \theta \right) \) can be found in [13]. Let us start from the definition of the matrix PVI. Here and below to define a matrix Painlevé equation we will consider a symplectic structure \(\omega = \textrm{Tr}\left( \textrm{d}p \wedge \textrm{d}q\right) \) on the space of pairs of matrices \(\textrm{Mat}_{n}\left( {\mathbb {C}}\right) \times \textrm{Mat}_{n}\left( {\mathbb {C}}\right) \ni (p, q)\). Then the dynamics can be defined by the Hamiltonian, which for matrix PVI is given by
Proposition 2.2
Transformations in the table below are Bäcklund transformations of the matrix \(\textrm{PVI}\). These transformations generate group \(S_4 \ltimes W\left( D_4^{(1)}\right) \).
Automorphisms \(\textrm{Aut}\left( D_4^{(1)}\right) =S_4\) are generated by permutations \(\sigma _{34}, \sigma _{14}, \sigma _{03}\). In general this group changes the time variable t, the time preserving subgroup is \(C_2^2\), generated by \(\pi _1=\sigma _{14}\sigma _{03}, \, \pi _2=(\sigma _{34}\sigma _{03}\sigma _{14})^2\). This is just a group of automorphisms of the affine Weyl group.
Proof of Proposition 2.2 can be done by a tedious but straightforward calculation, similar to the above PII case.
Note that \(s_1, s_2\) and \(\sigma \)’s generate the rest of the transformations. Therefore it is sufficient to check the Bäcklund symmetry condition only for them. The following remarks simplify the proofs.
Remark 2.1
Let \(m_1, m_2\) be monomials in p, q with coefficient 1 such that
If \(\deg _{p}(m_1) + \deg _{q}(m_1) \le 3\), then
If \(\deg _{p}(m_1) + \deg _{q}(m_1) = 4\), then the same is true except for the case \(\deg _{p}(m_1) = \deg _{q}(m_1) = 2\). In this case either \(\textrm{Tr}\left( m_1\right) = \textrm{Tr}\left( p^2q^2\right) \) or \(\textrm{Tr}\left( m_1\right) = \textrm{Tr}\left( pqpq\right) \).
Remark 2.2
Let \(H_1\) and \(H_2\) be two Hamiltonians such that \(H_1 - H_2 = f(t)\left( \textrm{Tr}\left( pqpq\right) \right. \left. - \textrm{Tr}\left( p^2q^2\right) \right) \). Then
where
In other words, the dynamics generated by \(H_1\) can be mapped to the dynamics generated by \(H_2\) by the certain non-autonomous change of variables.
One can see with the help of Remark 2.1 that for transformations \(s_0, s_2\) form \(\omega - \textrm{d}H \wedge \textrm{d}t\) changes in the same way as in the commutative case.
There is an additional conjugation in \(\sigma _{14}, \sigma _{03}\) which disappears in the commutative case. Let us explain its appearance. Consider \({\tilde{\sigma }}_{03}\) defined by the formula
In commutative case \({\tilde{\sigma }}_{03}\) is a Bäcklund transformation. In the matrix case almost all terms in the Hamiltonian transform in the same way as in the commutative case. The only difference appears in the transformation of the first term in the Hamiltonian
Hence we have
We see that \({\tilde{\sigma }}_{03}\) fails to satisfy Eq. (2.3). But it follows from Remark 2.2 that for \(\sigma _{03}\) we have
Transformations \(\sigma _{14}, \sigma _{34}\) can be treated similarly.
To finish the proof it remains to check group relations encoded in Table 2. It can be done directly.
2.3 Answers for the other Painlevé equations
In this part we present groups of Bäcklund transformations which generalize the Bäcklund groups of the scalar Painlevé equations for the matrix case. Namely, for each equation we list the following data:
-
1.
A Hamiltonian which defines the equation.
-
2.
A table and a diagram which encode action of Bäcklund transformations of the equation.
-
3.
Answer for the Bäcklund group.
Note that all transformations we list are symplectic. Also, they do preserve [p, q].
Painlevé V The system is defined by Hamiltonian
These transformations generate group \(\left( C_2\ltimes C_4 \right) \ltimes W(A_3^{(1)})\).
Painlevé III \(\mathrm {\big (D_6^{(1)}\big )}\)
The system is defined by Hamiltonian
These transformations generate group \(\left( C_2 \ltimes \left( C_2 \times C_2\right) \right) \ltimes W\left( A_1^{(1)}\right) ^2\).
Painlevé III \(\mathrm {\big (D_7^{(1)}\big )}\)
The system is defined by Hamiltonian
These transformations generate group \(C_2\ltimes W(A_1^{(1)})\).
Painlevé III \(\mathrm {\big (D_8^{(1)}\big )}\)
The system is defined by Hamiltonian
This transformation generates group \(C_2\).
Painlevé IV
Painlevé IV The system is defined by Hamiltonian
These transformations generate group \(\left( C_4 \ltimes C_3\right) \ltimes W(A_2^{(1)})\). Structure of the semidirect product in \(C_4 \ltimes C_3\) is defined by the relations
Remark 2.3
There is central subgroup \(\{\sigma _1^2, e\} \subset C_4 \ltimes C_3\) which acts trivially on the parameters. Quotient \(C_4 \ltimes C_3 / \{\sigma _1^2, e\}\) is isomorphic to the group of automorphisms of the diagram. This is only the case when the finite group we extend affine Weyl group by is not isomorphic to the group of automorphisms of the diagram.
Painlevé I The system is defined by Hamiltonian
Here \(\varpi _5\) is a fifth root of unity: \(\varpi _5^5 = 1,\, \varpi _5\ne 1\). This transformation generates group \(C_5\).
3 Block reduction of Matrix Painlevé systems
3.1 General construction
In this section we provide a construction which connects two matrix Painlevé systems of different sizes. In this construction input is a matrix Painlevé system and a Bäcklund transformation w of this system. Output is the matrix Painlevé system which we call the image. We specify input and output in the Table 9 below. Let us denote the extended phase space by \(M = \textrm{Mat}_{N}\left( {\mathbb {C}}\right) \times \textrm{Mat}_{N}\left( {\mathbb {C}}\right) \times {\mathbb {C}} \times {\mathbb {C}}^k = \{(p, q, t, \alpha )\}\), where k is the number of parameters of the equation. For the phase space we will use notations \(M_{\alpha } = \textrm{Mat}_{N}\left( {\mathbb {C}}\right) \times \textrm{Mat}_{N}\left( {\mathbb {C}}\right) \times {\mathbb {C}} = \{(p, q, t)\},\;\; M_{\alpha , t} = \textrm{Mat}_{N}\left( {\mathbb {C}}\right) \times \textrm{Mat}_{N}\left( {\mathbb {C}}\right) = \{(p,q)\}\). By d we denote the order of the Bäcklund transformation w.
Remark 3.1
Transformations from Table 9 are specified by following conditions
-
There exists \(\alpha \in {\mathbb {C}}^k\) such that \(M_{\alpha , t}\) is invariant under the action of w.
-
w acts on \(M_{\alpha , t}\) nontrivially.
Note that all of these transformations come from automorphisms of the corresponding Dynkin diagrams and preserve the time. They are known to be related to folding transformations and are classified in [13].
For each case from Table 9 we consider a matrix Painlevé of size \(nd\times nd\). There is the adjoint action of \(\textrm{GL}_{nd}\left( {\mathbb {C}}\right) \) on \(M_{\alpha , t}\), namely \(S:(p,q)\mapsto (SpS^{-1}, SqS^{-1})\). Consider a twisted Bäcklund transformation \({\bar{w}} = \textrm{Ad}_{S_{d}} \circ w\), where \(S_d = \textrm{diag}\left( \textbf{1}_{n\times n}, e^{\frac{2\pi \text {i}}{d}}\textbf{1}_{n\times n},..., e^{\frac{2\pi (d-1) }{d}}\textbf{1}_{n\times n}\right) \). Transformation \({\bar{w}}\) is an order d symmetry of the equation. Then for \(M^{{\bar{w}}} = \{x \in M | {\bar{w}}(x) = x\}\) standard arguments imply
-
\(M_{\alpha , t}^{{\bar{w}}}\) is a symplectic submanifold of \(M_{\alpha , t}\).
-
\(M_{\alpha }^{{\bar{w}}}\) is preserved by the dynamics of the equation.
Lemma 3.1
Let w be a transformation from Table 9. Let \(\alpha \in {\mathbb {C}}^k\) be preserved by w. Let M be the extended phase space of the corresponding matrix \( n d\times nd\) Painlevé system and H be the corresponding Hamiltonian. Then restriction of the dynamics, generated by H to \(M^{{\bar{w}}}_{\alpha }\) is Hamiltonian.
Let us denote the corresponding Hamiltonian by \({\tilde{H}}\). Note that adjoint action of \(\textrm{GL}_{nd}\left( {\mathbb {C}}\right) \) restricted to \(\textrm{GL}_{n}^d\left( {\mathbb {C}}\right) = \{\textrm{diag} \left( h_1, h_2, \ldots , h_d\right) | h_{i}\in \textrm{GL}_{n}\left( {\mathbb {C}}\right) \}\) do commute with \({\bar{w}}\). Thus this action preserves \(M^{{\bar{w}}}_{\alpha }\). Consider a Hamiltonian reduction on \(M^{{\bar{w}}}_{\alpha }\) by a subgroup \(\textrm{GL}^{d-1}_{n}\left( {\mathbb {C}}\right) = \{\textrm{diag}\left( 1, h_2, \ldots , h_d\right) |\) \(h_i\in \textrm{GL}_{n}\left( {\mathbb {C}}\right) \} \subset \textrm{GL}^{d}_{n}\left( {\mathbb {C}}\right) \) with the moment map value fixed as \(\textbf{g} = (\text {i}g_2\textbf{1}_{n\times n}, \ldots , \text {i}g_{d}\textbf{1}_{n\times n})\). Let us denote the reduced space by \({\mathbb {M}}_{\alpha }:= M^{{\bar{w}}}_{\alpha }{//}_{\textbf{g}}\textrm{GL}_{n}^{d-1}\left( {\mathbb {C}}\right) \).
Theorem 3.1
Under the conditions from Lemma 3.1 the dynamics on the manifold \({\mathbb {M}}_{\alpha }\) corresponding to \({\tilde{H}}\) is equivalent to the dynamics of a matrix \(n\times n\) Painlevé system written as Image in Table 9.
Proof of Theorem 3.1 and Lemma 3.1 is given below case by case. We perform the following steps for all cases.
- Step 1.:
-
Solve fixed-point equations and obtain Darboux coordinates on \(M^{{\bar{w}}}_{\alpha }\).
- Step 2.:
-
Compute the action of \(\textrm{GL}^{d}_{n}\left( {\mathbb {C}}\right) \) on \(M^{{\bar{w}}}\) and the moment map of this action in these coordinates.
- Step 3.:
-
Obtain Darboux coordinates on \({\mathbb {M}}_{\alpha ,t}\).
- Step 4.:
-
Calculate the Hamiltonian for the dynamics on \({\mathbb {M}}_{\alpha }\) and find coordinates on \({\mathbb {M}}_{\alpha }\), in which the Hamiltonian is a Painlevé’s standard one.
Step 1, Step 2, Step 3 can be performed simultaneously for cases 1, 2, 3 and for cases 6, 7, 8 in Table 9. We call cases 1 – 5 linear, since for them to obtain a parametrization of \(M^{{\bar{w}}}\) one has to solve only linear equations. Remaining cases 6, 7, 8 are called non-linear.
Remark 3.2
We use the following notation.
-
Standard small letters (for example, (p, q)) to denote canonical coordinates on \(M_{\alpha , t}\).
-
Gothic letters (for example, \(({\mathfrak {p}}, {\mathfrak {q}})\)) to denote coordinates on \(M_{\alpha , t}^{{\bar{w}}}\).
-
Standard capital letters (for example, \((P,Q)\)) to denote canonical coordinates on \({\mathbb {M}}_{\alpha , t}\).
3.2 Linear cases
Step 1. Equations that determine the symplectic submanifold \(M_{\alpha , t}^{{\bar{w}}}\) for cases 1, 2, 3
Here \(\zeta , \xi \in {\mathbb {C}}\), \(\eta \in {\mathbb {C}}[t]\) are specified for each case. For all cases which we consider \(\zeta \xi = 0\) and for them we get
Hence we get a symplectic submanifold of dimension \(4n^2\) with symplectic form \({\text {Tr}}(\textrm{d}{\mathfrak {p}}_{12}\wedge \textrm{d}{\mathfrak {q}}_{21}) + {\text {Tr}}(\textrm{d}{\mathfrak {p}}_{21}\wedge \textrm{d}{\mathfrak {q}}_{12})\).
Recall that H is the Hamiltonian which defines a matrix Painlevé dynamics on \(M_{\alpha }\). Then the equations of motion can be identified with the one dimensional distribution on \(M_{\alpha }\), namely
Let us denote the embedding of the set of \({\bar{w}}\)–invariant points by \(\iota : M_{\alpha }^{{\bar{w}}} \rightarrow M_{\alpha }\). Since \(\iota ^*(\omega ) = {\text {Tr}}(\textrm{d}{\mathfrak {p}}_{12}\wedge \textrm{d}{\mathfrak {q}}_{21}) + {\text {Tr}}(\textrm{d}{\mathfrak {p}}_{21}\wedge \textrm{d}{\mathfrak {q}}_{12})\) and the dynamics on \(M_{\alpha }^{{\bar{w}}}\) is defined as \(\textrm{Ker}\left( \iota ^*\left( \omega - \textrm{d}H\wedge \textrm{d}t\right) \right) \) it follows that the dynamics on \(M_{\alpha }^{{\bar{w}}}\) is also Hamiltonian and defined by Hamiltonian \(\iota ^*\left( H\right) \) and \(\left( {\mathfrak {p}}_{12},{\mathfrak {p}}_{21}, {\mathfrak {q}}_{21}, {\mathfrak {q}}_{12} \right) \) are Darboux coordinates on \(M_{\alpha }^{{\bar{w}}}\).
Step 2. The remaining gauge freedom consists of block diagonal matrices \(h=\textrm{diag}(h_1, h_2),\, h_1, h_2 \in \textrm{GL}_n(\mathbb {C})\) and the moment map is also block diagonal
Remark 3.3
Block structure of [p, q] appears not accidentally. Consider \((p, q)\in M_{\alpha , t}^{{\bar{w}}}\) with \({\bar{w}} = \textrm{Ad}_{S_2} \circ w\). As we mentioned above [p, q] is preserved by Bäcklund transformation w. Then we have
which implies
From the last equation it follows that [p, q] is block diagonal.
Step 3. Let us perform Hamiltonian reduction with respect to \(\textrm{GL}_{n}\left( {\mathbb {C}}\right) = \{\textrm{diag}(\textbf{1}_{n\times n}, h_2) | h_2 \in \textrm{GL}_{n}\left( {\mathbb {C}}\right) \}\). We fix the moment map value as \(m_2 = {\mathfrak {p}}_{21}{\mathfrak {q}}_{12} - {\mathfrak {q}}_{21}{\mathfrak {p}}_{12} = \text {i}g_2\textbf{1}_{n\times n}\). Hence we get \({\mathfrak {p}}_{21}=({\mathfrak {q}}_{21}{\mathfrak {p}}_{12}+\text {i}g_2){\mathfrak {q}}_{12}^{-1}\). The functions \({\tilde{P}} = {\mathfrak {p}}_{12} {\mathfrak {q}}_{12}^{-1}\) and \({\tilde{Q}} = {\mathfrak {q}}_{12}{\mathfrak {q}}_{21}\) are invariant under the action of \(\textrm{diag}\left( 1, h_2\right) \) and thus define functions on \({\mathbb {M}}_{\alpha }\). Moreover \(\big ({\tilde{P}}, {\tilde{Q}}\big )\) are Darboux coordinates on \({\mathbb {M}}_{\alpha , t}\).
3.2.1 PII to PII
Steps 1, 2, 3. We consider matrix \(2n \times 2n\) PII with \(\alpha _1 = \frac{1}{2}\). In this case \(\xi = 0,\; \eta (t) = 0,\; \zeta = 0\).
Step 4. Substituting Darboux coordinates \(\tilde{P}, \tilde{Q}\) into the Hamiltonian of matrix PII we get
After the change of coordinates
we obtain a system with matrix Darboux coordinates \((\breve{P}, \breve{Q})\) and Hamiltonian
Then after the last change of coordinates
we obtain a system with matrix Darboux coordinates (P, Q) and Hamiltonian
which is the Hamiltonian of matrix \(n\times n\) PII\(\big (1 + \text {i}g_2, - \text {i}g_2\big )\).
3.2.2 PIII\(\mathrm {\big (D_6^{(1)}\big )}\) to PIII\(\mathrm {\big (D_8^{(1)}\big )}\)
Steps 1, 2, 3. We consider matrix \(2n\times 2n\) PIII\(\mathrm {\big (D_{6}^{(1)}\big )}\) with \(\alpha _0 = \alpha _1 = \frac{1}{2},\; \beta _0 = \beta _1 = \frac{1}{2}\). In this case we have \(\xi = 0,\; \eta (t) = 1,\; \zeta = 1\).
Step 4. Substituting Darboux coordinates \({\tilde{P}}, \tilde{Q}\) into the Hamiltonian of matrix PIII\(\mathrm {\big (D_6^{(1)}\big )}\) we get
After the change of variables
we obtain a system with matrix Darboux coordinates (P, Q) and Hamiltonian
which is the Hamiltonian of matrix \(n\times n\) PIII\(\mathrm {\big (D_8^{(1)}\big )}\).
3.2.3 PV to PIII\(\mathrm {\big (D_6^{(1)}\big )}\)
Steps 1, 2, 3. We consider matrix \(2n \times 2n\) PV with \(\alpha _0 = \alpha _2 = \epsilon + \frac{1}{2},\; \alpha _1 = \alpha _3 = -\epsilon \). In this case \(\xi = 1,\; \eta (t) = -t,\; \zeta = 0\).
Step 4. Substituting Darboux coordinates \({\tilde{P}}, {\tilde{Q}}\) into the Hamiltonian of matrix PV we get
After the change of variables
we obtain a system with matrix Darboux coordinates (P, Q) and Hamiltonian
which is the Hamiltonian of matrix \(n\times n\) PIII\(\mathrm {\big (D_6^{(1)}\big )}\big (1+\text {i}g_2, -\text {i}g_2, 1+2\epsilon , -2\epsilon \big )\).
3.2.4 PIV to PIV
Step 1. This is case 4 in Table 9. We consider matrix \(3n\times 3n\) PIV with \(\alpha _0 = \alpha _1 = \alpha _2 = \frac{1}{3}\).
Equations that determine symplectic submanifold \(M^{{\bar{w}}}_{\alpha ,t}\) are
Then
So we get a symplectic manifold of dimension \(6n^2\), with symplectic form \(\sqrt{3}\textrm{i}{\text {Tr}}(\textrm{d}{\mathfrak {q}}_{12}\wedge \textrm{d}{\mathfrak {q}}_{21} + \textrm{d}{\mathfrak {q}}_{31}\wedge \textrm{d}{\mathfrak {q}}_{13} + \textrm{d}{\mathfrak {q}}_{23}\wedge \textrm{d}{\mathfrak {q}}_{32}).\)
Similarly to cases above it follows that the restriction of the dynamics defined by Hamiltonian H on \(M^{{\bar{w}}}_{\alpha }\) is Hamiltonian and defined by restriction of H on \(M^{{\bar{w}}}_{\alpha }\).
Step 2. The remaining gauge freedom consists of block diagonal matrices \(h=\textrm{diag}(h_1, h_2, h_3),\, h_1, h_2, h_3 \in \textrm{GL}_n(\mathbb {C})\) and moment map is also block diagonal
Step 3. Now we can perform Hamiltonian reduction with respect to \(\textrm{GL}_n\left( {\mathbb {C}}\right) ^2 = \{ \textrm{diag}\left( 1_{n \times n}, h_2, h_3\right) | h_2, h_3 \in \textrm{GL}_n\left( {\mathbb {C}}\right) \} \). We fix the moment map value as \(m_2 = \text {i}g_2\textbf{1}_{n\times n},\;\; m_3 = \text {i}g_3\textbf{1}_{n\times n}\).
The functions \({\tilde{P}} = (\varpi - \varpi ^{-1}) {\mathfrak {q}}_{12}{\mathfrak {q}}_{23}{\mathfrak {q}}_{13}^{-1}\) and \({\tilde{Q}} = {\mathfrak {q}}_{13}{\mathfrak {q}}_{23}^{-1}{\mathfrak {q}}_{21}\) are invariant under the action of \(\textrm{GL}_n^2\left( {\mathbb {C}}\right) \), so they define functions on the manifold \({\mathbb {M}}_{\alpha }\). Moreover these functions are Darboux coordinates on \({\mathbb {M}}_{\alpha , t}\).
Step 4. Substitution \({\tilde{P}}, {\tilde{Q}}\) into the Hamiltonian gives
Then after the change of coordinates
we get
which is the Hamiltonian of matrix \(n\times n\) PIV\(\big (1+\frac{\text {i}g_3}{2}, -\frac{\text {i}g_2 + \text {i}g_3}{2}, \frac{\text {i}g_2}{2}\big )\).
3.2.5 PV to PV
Step 1. This is case 5 from Table 9. We consider matrix \(4n\times 4n\) PV with \(\alpha _0 = \alpha _1 = \alpha _2 = \alpha _3 = \frac{1}{4}\).
Equations that determine the symplectic submanifold \(M_{\alpha , t}^{{\bar{w}}}\) in case 5 are
The solution is
So we get symplectic manifold of dimension \(8n^2\), with symplectic form
Similarly to cases 1–3 it follows that the restriction of the dynamics defined by the Hamiltonian H on \(M^{{\bar{w}}}_{\alpha }\) is Hamiltonian and defined by restriction of H on \(M^{{\bar{w}}}_{\alpha }\).
Step 2. The remaining gauge freedom consists of block diagonal matrices \(h=\textrm{diag}(h_1, h_2, h_3, h_4),\, h_1, h_2, h_3, h_4 \in \textrm{GL}_n(\mathbb {C})\) and moment map is also block diagonal
Step 3. Now we can perform reduction with respect to \(\textrm{GL}_n^{3}\left( {\mathbb {C}}\right) = \textrm{diag}\left( 1_{ n \times n }, h_2, h_3, h_4|h_2, h_3, h_4\in \textrm{GL}_n\left( {\mathbb {C}}\right) \right) \). We fix the moment map value as \(m_2 = \text {i}g_2\textbf{1}_{n\times n},\;\; m_3 = \text {i}g_3\textbf{1}_{n\times n},\;\; m_4 = \text {i}g_4\textbf{1}_{n\times n}\).
The functions \({\tilde{P}} = 2\text {i}t {\mathfrak {q}}_{12}{\mathfrak {q}}_{32}^{-1}{\mathfrak {q}}_{34}{\mathfrak {q}}_{41}\) and \({\tilde{Q}} = {\mathfrak {q}}_{14}{\mathfrak {q}}_{34}^{-1}{\mathfrak {q}}_{32}{\mathfrak {q}}_{12}^{-1}\) are invariant under the action of \(\textrm{GL}_n^{3}\left( {\mathbb {C}}\right) \), so define functions on the manifold \({\mathbb {M}}_{\alpha }\). Moreover these functions are Darboux coordinates on \({\mathbb {M}}_{\alpha , t}\).
Step 4. Substituting \({\tilde{P}}, {\tilde{Q}}\) into the Hamiltonian we get
Then after the change of coordinates
we get the Hamiltonian
which is the Hamiltonian of matrix \(n\times n\) PV\(\big (1+\text {i}g_3, \text {i}g_2 + \text {i}g_4,-\text {i}g_4, - \text {i}g_2 - \text {i}g_3\big )\).
3.3 Non-linear cases
Step 1. Equations that determine symplectic submanifold \(M^{{\bar{w}}}_{\alpha ,t}\) for cases 6–8 are
On the open dense subset where the lower left \(n\times n\) block of q is invertible the solution is given by
So, \( \tilde{{\mathfrak {p}}}_{11}, \tilde{{\mathfrak {q}}}_{11}, \tilde{{\mathfrak {p}}}_{21}, \tilde{{\mathfrak {q}}}_{21}\) are local coordinates on \(M_{\alpha , t}^{{\bar{w}}}\). By \(\iota \) denote the embedding \( M_{\alpha }^{{\bar{w}}} \rightarrow M_{\alpha }\).
We can obtain restriction of the canonical 1-form \(\Theta \) on \(M_{\alpha }^{{\bar{w}}}\), namely
where Darboux coordinates on \(M_{\alpha , t}^{{\bar{w}}}\) are
Recall that H is the Hamiltonian which defines a matrix Painlevé dynamics on \(M_{\alpha }\). Then the equations of motion can be identified with the one dimensional distribution on \(M_{\alpha }\), namely \( \textrm{Ker}\left( \omega - \textrm{d}H\wedge \textrm{d}t\right) \). It follows that the dynamics on \(M_{\alpha }^{{\bar{w}}}\) is also Hamiltonian and defined by the Hamiltonian \(\iota ^*\left( H\right) + \textrm{Tr}\left( \tilde{{\mathfrak {p}}}_{21}\tilde{{\mathfrak {q}}}_{21}^{-1}\right) \).
Even though \(({\mathfrak {p}}_{11}, {\mathfrak {q}}_{11}, {\mathfrak {p}}_{12}, {\mathfrak {q}}_{21})\) are Darboux coordinates on \(M_{\alpha , t}^{{\bar{w}}}\), it will be more convenient for us to use \((\tilde{{\mathfrak {p}}}_{11}, \tilde{{\mathfrak {q}}}_{11}, \tilde{{\mathfrak {p}}}_{12}, \tilde{{\mathfrak {q}}}_{21})\).
Step 2. As in the linear cases we have the action of block diagonal matrices
The moment map corresponding to this action is block diagonal, namely \([p,q]=\textrm{diag}(m_1,m_2)\), where
Step 3. Let us perform Hamiltonian reduction with respect to \(\textrm{GL}_{n}\left( {\mathbb {C}}\right) = \{\textrm{diag}(1_{n\times n}, h_2) | h_2 \in \textrm{GL}_{n}\left( {\mathbb {C}}\right) \}\). Then, fixing \(m_2 =\text {i}g_2\textbf{1}_{n\times n}\) and resolving it with respect to \(\tilde{{\mathfrak {p}}}_{11}\) on the open dense subset where \(\tilde{{\mathfrak {q}}}_{11}\) is invertible, we get
We can take the following coordinates on the reduction
Then we have a section from \({\mathbb {M}}_{\alpha }\) to the intersection \(M_{\alpha }^{{\bar{w}}}\cap \{m_2 =\text {i}g_2\textbf{1}_{n\times n}\}\) which maps
Step 4. Hamiltonian on the reduction is
3.3.1 PIII\(\mathrm {\big (D_8^{(1)}\big )}\) to PIII\(\mathrm {\big (D_6^{(1)}\big )}\)
Steps 1, 2, 3. We consider matrix \(2n\times 2n\) PIII\(\mathrm {\big (D_{8}^{(1)}\big )}\). In this case \(\nu = \frac{1}{2}\). Step 4. Here we start from coordinates
in which we get Hamiltonian
Then after change of variables
we get
which is the Hamiltonian of matrix \(n\times n\) PIII\(\mathrm {\big (D_{6}^{(1)}\big )}\big (1-\text {i}g_2, \text {i}g_2, 1-\text {i}g_2, \text {i}g_2\big )\).
3.3.2 PIII\(\mathrm {\big (D_6^{(1)}\big )}\) to PV
Steps 1, 2, 3. We consider matrix \(2n\times 2n\) PIII\(\mathrm {\big (D_{6}^{(1)}\big )}\) with \(\beta _0=\beta _1= \frac{1}{2},\alpha _1=\epsilon \). In this case \(\nu =\epsilon \).
Step 4. Here we start from coordinates
we get a system with Hamiltonian
After change of coordinates
we get
which is the Hamiltonian of matrix \(n\times n\) PV\(\big (1-\text {i}g_2 - \epsilon , \text {i}g_2, \epsilon - \text {i}g_2, \text {i}g_2\big )\).
3.3.3 PVI to PVI
Steps 1, 2, 3. We consider matrix \(2n\times 2n\) PVI with \(\alpha _0=\alpha _3 =\epsilon _0,\; \alpha _1=\alpha _4 = \epsilon _1\). In this case \(\nu = \alpha _2 = \frac{1}{2} - \epsilon _0 - \epsilon _1\).
Step 4. We start from coordinates
in which we get Hamiltonian
After the last substitution
we get
which is the Hamiltonian of matrix \(n\times n\) PVI\(\big (2\epsilon _0, 2\epsilon _1,\frac{1}{2}-\text {i}g_2 - \epsilon _0 - \epsilon _1, \text {i}g_2, \text {i}g_2 \big )\).
3.4 Special \(C_2\times C_2\) cases
Let G be a group consisting of Bäcklund transformations which preserve a certain matrix Painlevé system and act trivially on the time variable. For the scalar Painlevé I-VI it was shown in [13] that such group G consists of transformations coming from automorphisms of diagram (up to overall conjugation of G). This is indeed true also for the described above Bäcklund transformation groups of matrix Painlevé equations. From the description of transformations corresponding to automorphisms in Sect. 2 it follows that possible G’s are either cyclic, or isomorphic to \(C_2\times C_2\).
Twisting generators by \(\textrm{GL}_{n|G|}\left( {\mathbb {C}}\right) \) action we get the group of symmetries \({\bar{G}}\), and the submanifold \(M_{\alpha }^{{\bar{G}}}\) which is preserved by the dynamics. Condition of \({\bar{G}}\)-invariance in case of cyclic G simplifies to invariance under the action of generator \({\bar{w}}\). These cases are described by Theorem 3.1.
There are only two cases of non-cyclic G. For them there is a construction of the reduction similar to one described by Theorem 3.1. The input and output are given in Table 10.
Let us use notation \(w_{+}, w_{-}\) for generators of G. We take twists \(S_{+} = \textrm{diag}\left( \textbf{1}_{n\times n}, \textbf{1}_{n\times n}, -\textbf{1}_{n\times n}, -\textbf{1}_{n\times n}\right) \), \(S_{-} = \textrm{diag}\left( \textbf{1}_{n\times n}, -\textbf{1}_{n\times n}, \textbf{1}_{n\times n}, -\textbf{1}_{n\times n}\right) \). Then twisted generators are \({\bar{w}}_{i} = \textrm{Ad}_{S_i} \circ w_i\).
3.4.1 PIII\(\mathrm {\big (D_6^{(1)}\big )}\) to PIII\(\mathrm {\big (D_6^{(1)}\big )}\)
Step 1. Consider matrix \(4n\times 4n\) PIII\(\mathrm {\big (D_6^{(1)}\big )}\) with parameters \(\alpha _1 = \beta _1 = \frac{1}{2}\). In this case we take \(w_{+} = \pi '\) and \(w_{-} = \pi \circ \pi '\).
Since \(M_{\alpha }^{{\bar{G}}} = M_{\alpha }^{{\bar{w}}_{+}}\cap M_{\alpha }^{{\bar{w}}_{-}}\) let us start from \(M_{\alpha }^{{\bar{w}}_{+}}\). Manifold \(M_{\alpha }^{{\bar{w}}_{+}}\) is defined by equations (3.29) with \(\nu = \frac{1}{2}\). So we have the solution on the dense open subset
Here \(\breve{{\mathfrak {q}}}_{ij}, \breve{{\mathfrak {p}}}_{ij}\) are \(n\times n\) blocks and \(*\)’s are defined in terms of \(\breve{{\mathfrak {q}}}_{ij}\) and \(\breve{{\mathfrak {p}}}_{ij}\) by (3.30). Note that for this case in equations (3.30) blocks have size \(2n\times 2n\).
The Darboux coordinates on \(M_{\alpha }^{{\bar{w}}_{+}}\) are given by (3.32). Note that one half of these coordinates are simply \(\{\breve{{\mathfrak {q}}}_{ij}\}_{1\le i\le 4,\; 1\le j\le 2}\), the other half are coordinates conjugate to them, defined by rather complicated formulas following from (3.32).
Now let us impose invariance under \({\bar{w}}_{-}\) to obtain \(M_{\alpha }^{{\bar{G}}}\) as the submanifold of \(M_{\alpha }^{{\bar{w}}_{+}}\).
These equations are solved by
So matrix coordinates on \(M_{\alpha , t}^{{\bar{G}}}\) are \((\breve{{\mathfrak {q}}}_{12}, \breve{{\mathfrak {q}}}_{21}, \breve{{\mathfrak {q}}}_{32}, \breve{{\mathfrak {q}}}_{41}, \breve{{\mathfrak {p}}}_{12}, \breve{{\mathfrak {p}}}_{21}, \breve{{\mathfrak {p}}}_{32}, \breve{{\mathfrak {p}}}_{41})\).
From the consideration above it follows that Darboux coordinates on \(M_{\alpha , t}^{{\bar{G}}}\) are \(\breve{{\mathfrak {q}}}_{12}, \breve{{\mathfrak {q}}}_{21}, \breve{{\mathfrak {q}}}_{32}, \breve{{\mathfrak {q}}}_{41}\) and conjugated to them which are restrictions of ones conjugated to them on \(M_{\alpha , t}^{{\bar{w}}_{+}}\) to \(M_{\alpha , t}^{{\bar{G}}}\). For example, to obtain matrix coordinate conjugated to \(\breve{{\mathfrak {q}}}_{21}\) one should take upper right \(n\times n\) block of \({\mathfrak {p}}_{11}\) from formula (3.32) and then restrict it to (3.53). Darboux coordinates on \(M_{\alpha , t}^{{\bar{G}}}\) are
Step 2. We have the action of block diagonal matrices on \(M_{\alpha }^{{\bar{G}}}\) which maps
Moment map of this action is defined by blocks of block diagonal matrix [p, q]
Step 3. Let us perform Hamiltonian reduction with respect to \(\textrm{GL}_{n}^3\left( {\mathbb {C}}\right) = \{\textrm{diag}(1_{n\times n}, h_2, h_3, h_4) | h_2, h_3, h_4 \in \textrm{GL}_{n}\left( {\mathbb {C}}\right) \}\). We fix the moment map value as \(m_2 = \text {i}g_2\textbf{1}_{n\times n},\;\; m_3 = \text {i}g_3\textbf{1}_{n\times n},\;\; m_4 = \text {i}g_4\textbf{1}_{n\times n}.\)
Darboux coordinates on the reduction are \(({\tilde{P}}, {\tilde{Q}}) {=} \left( 2(\breve{{\mathfrak {q}}}_{21}^{-1}\breve{{\mathfrak {p}}}_{21} + \breve{{\mathfrak {q}}}_{41}^{-1}\breve{{\mathfrak {p}}}_{41}), \breve{{\mathfrak {q}}}_{12}\breve{{\mathfrak {q}}}_{21} \right) .\)
Step 4. As in the non-linear cases restriction on the set of points invariant under \(\pi '\) shifts the Hamiltonian by \(\textrm{Tr}\left( \tilde{{\mathfrak {p}}}_{21}\tilde{{\mathfrak {q}}}_{21}^{-1}\right) \), which is in our case equal to \(\textrm{Tr}\left( \breve{{\mathfrak {p}}}_{32}\breve{{\mathfrak {q}}}_{32}^{-1} + \breve{{\mathfrak {p}}}_{41}\breve{{\mathfrak {q}}}_{41}^{-1}\right) \). Then it remains to restrict the Hamiltonian obtained to the set of points invariant under \(\pi \circ \pi '\) and satisfying the moment map conditions and substitute the section from the reduction. Then we get
Then after change of variables
we get
which is the Hamiltonian of matrix \(n\times n\) PIII\(\mathrm {\big (D_{6}^{(1)}\big )}\big (1- \text {i}g_2+ \text {i}g_3, \text {i}g_2 + \text {i}g_3, 1 - \text {i}g_3 - \text {i}g_4, \text {i}g_3 + \text {i}g_4\big )\).
3.4.2 PVI to PVI
Step 1. Consider matrix \(4n\times 4n\) PVI with parameters \(\alpha _0 = \alpha _1 = \alpha _3 = \alpha _4 = \epsilon \). In this case we take \(w_{+} = \pi _1\) and \(w_{-} = \pi _2\).
Let us obtain \(M_{\alpha }^{{\bar{w}}_{+}}\). This manifold is defined by equations (3.29) with \(\nu = \alpha _2\). So we have the solution on the dense open subset
Here \(\breve{{\mathfrak {q}}}_{ij}, \breve{{\mathfrak {p}}}_{ij}\) are \(n\times n\) blocks and \(*\)’s are defined in terms of \(\breve{{\mathfrak {q}}}_{ij}\) and \(\breve{{\mathfrak {p}}}_{ij}\) by (3.30). Note that for this case in equations (3.30) blocks \(\tilde{{\mathfrak {p}}}_{ij}, \tilde{{\mathfrak {q}}}_{ij}\) have size \(2n\times 2n\).
Let us obtain \(M_{\alpha }^{{\bar{G}}} = M_{\alpha }^{{\bar{w}}_{+}}\cap M_{\alpha }^{{\bar{w}}_{-}} \subset M_{\alpha }^{{\bar{w}}_{+}}\). Then on \(M_{\alpha }^{{\bar{w}}_{+}}\) we have to solve
On the open dense subset, where \(\breve{{\mathfrak {q}}}_{21}, \breve{{\mathfrak {q}}}_{31}, \breve{{\mathfrak {q}}}_{41}\) are invertible, these equations can be solved by
So matrix coordinates on \(M_{\alpha , t}^{{\bar{G}}}\) are \(\breve{{\mathfrak {q}}}_{11}, \breve{{\mathfrak {q}}}_{21}, \breve{{\mathfrak {q}}}_{31}, \breve{{\mathfrak {q}}}_{41}, \breve{{\mathfrak {p}}}_{21}, \breve{{\mathfrak {p}}}_{41}, \breve{{\mathfrak {p}}}_{32}, \breve{{\mathfrak {p}}}_{42}\). Let us denote embedding by \(\iota :M_{\alpha }^{{\bar{G}}}\rightarrow M_{\alpha }\). Note that on \(M_{\alpha }^{{\bar{G}}}\) blocks of q depend only on \(\breve{{\mathfrak {q}}}_{11}, \breve{{\mathfrak {q}}}_{21}, \breve{{\mathfrak {q}}}_{31}, \breve{{\mathfrak {q}}}_{41}, t\), hence
for certain \(\hat{{\mathfrak {p}}}_{11}, \hat{{\mathfrak {p}}}_{12}, \hat{{\mathfrak {p}}}_{13}, \hat{{\mathfrak {p}}}_{14}, F\). One can calculate all of them, but we will use only \(\hat{{\mathfrak {p}}}_{11}, F\), given by
Step 2. We have a Hamiltonian action of \(\textrm{GL}_{n}^{4}\left( {\mathbb {C}}\right) \) on \(M_{\alpha , t}^{{\bar{G}}}\)
The moment map of this action is given by blocks of block diagonal matrix [p, q]
Step 3. Let us perform Hamiltonian reduction with respect to \(\textrm{GL}_{n}^3\left( {\mathbb {C}}\right) = \{\textrm{diag}(1_{n\times n}, h_2, h_3, h_4) | h_2, h_3, h_4 \in \textrm{GL}_{n}\left( {\mathbb {C}}\right) \}\). We fix the moment map value as \(m_2 = \text {i}g_2\textbf{1}_{n\times n},\;\; m_3 = \text {i}g_3\textbf{1}_{n\times n},\;\; m_4 = \text {i}g_4\textbf{1}_{n\times n}.\)
Darboux coordinates on the reduction are
Step 4. From (3.63) it follows that the dynamics on \(M_{\alpha }^{{\bar{G}}}\) is given by the Hamiltonian \(\iota ^{*}(H) + F\). The Hamiltonian on the reduction in the coordinates
is given by
After change of coordinates
we get
which is the Hamiltonian of matrix \(n\times n\) PVI\(\big (\text {i}g_3{+}\text {i}g_4, 4\epsilon , \frac{1}{2}\left( 1 - 4\epsilon - 2(\text {i}g_2{+}\text {i}g_3 \right. \left. {+}\text {i}g_4) \right) ,\text {i}g_2{+}\text {i}g_4, \text {i}g_2{+}\text {i}g_3\big )\).
4 Application to Calogero–Painlevé systems
4.1 From Matrix Painlevé to Calogero–Painlevé
Every matrix \(N\times N\) Painlevé system defined above corresponds to a Calogero–Painlevé system. We briefly recall its construction following [2].
Consider phase space of matrix Painlevé system, which is \(M_{\alpha , t} = \{(p, q)\in \textrm{Mat}_{N\times N}^{2}\left( {\mathbb {C}}\right) \}\). There is an action of \(\textrm{GL}_{N}\left( {\mathbb {C}}\right) \) by overall conjugation of p and q. This action is Hamiltonian and the moment map equals \(\mu _{N}(p, q) = [p,q]\). Let us define phase space of the corresponding Calogero–Painlevé system as a Hamiltonian reduction
Here \(\textbf{O}_{N,g}\) is a coadjoint orbit \(\textbf{O}_{N,g} = \{\text {i}g(\textbf{1}_{N\times N} - \xi \otimes \eta )| \xi \in \textrm{Mat}_{N\times 1}\left( {\mathbb {C}}\right) , \eta \in \textrm{Mat}_{1\times N}\left( {\mathbb {C}}\right) , \eta \xi = N\}\), \(g\in {\mathbb {C}}\).
Let H be the Hamiltonian of matrix \(N\times N\) Painlevé system. The Hamiltonian H is invariant with respect to the action of \(\textrm{GL}_{N}\left( {\mathbb {C}}\right) \), thus H defines a Hamiltonian dynamics on \({\textsf{M}}_{\alpha }\), which is called dynamics of the corresponding Calogero–Painlevé system.
The open dense subset of \({\textsf{M}}_{\alpha , t}\) can be described as the phase space of system of particles considered up to permutations. This space is \(\textrm{T}^*\left( \left( {\mathbb {C}}^N\backslash \textrm{diags}\right) / \textrm{S}_{N}\right) \), its points are sets \(\{(p_j, q_j)\}_{j = 1,\ldots , N}\) where \(q_{j}\)’s are distinct. Let us consider a map \(\zeta _{N}: \textrm{T}^*\left( \left( {\mathbb {C}}^N\backslash \textrm{diags}\right) / \textrm{S}_{N}\right) \rightarrow M_{\alpha , t}\)
Note that \(\textrm{Im}(\zeta _{N})\subset \mu _{N}^{-1}\left( \{\text {i}g\left( \textbf{1}_{N\times N} - v_{N}\otimes v_{N}^t\right) \}\right) \), where \(v_{N} = \begin{pmatrix} 1\\ \vdots \\ 1\end{pmatrix}\).
Let us denote by [(p, q)] the orbit of (p, q) with respect to the action of \(\textrm{GL}_{N}\left( {\mathbb {C}}\right) \). Consider a map
The map \({\tilde{\zeta }}_{N}\) is injective and the image of \({\tilde{\zeta }}_{N}\) consists of classes [(p, q)] such that q is diagonalisable with different eigenvalues. Let us use the notations \(\textrm{Im}({\tilde{\zeta }}_{N}) = {\textsf{M}}^{\textrm{reg}}_{\alpha , t},\;\; M_{\alpha ,t}^{\textrm{reg}} = \{(p, q)\in \textrm{Mat}_{N\times N}^{2}\left( {\mathbb {C}}\right) : [(p,q)] \in {\textsf{M}}^{\textrm{reg}}_{\alpha , t}\}\). We have the inverse map
Here \(q_{j}\)’s are eigenvalues of q and \(p_{j}\) is the j-th element on the diagonal of p in a basis where \(q = \textrm{diag}\left( q_1, \ldots , q_N\right) \).
Bäcklund transformations of matrix Painlevé systems obtained in Sect. 2 are rational in p, q, t and thus do commute with the action of \(\textrm{GL}_{N}\left( {\mathbb {C}}\right) \). Also, Bäcklund transformations preserve [p, q]. Thus we have
Proposition 4.1
Bäcklund transformation of a matrix Painlevé system defines Bäcklund transformation for the corresponding Calogero–Painlevé system.
Indeed, let w be a Bäcklund transformation of a matrix Painlevé system. Then the corresponding transformation of the Calogero–Painlevé system can be written as \(\pi _N \circ w\circ \zeta _{N} = {\textsf{w}}\). As a result, we have a homomorphism from the described above Bäcklund transformation groups of matrix Painlevé equations to Bäcklund transformations of the corresponding Calogero–Painlevé systems.
Remark 4.1
One can lift the condition of \({\textsf{w}}\)-invariance of [(p, q)] to the level of matrix representatives (p, q), namely
Let us illustrate constructions above by an example (in addition to Example 1.2)
Example 4.1
To obtain the Calogero–Painlevé \(\mathrm {III\big (D_6^{(1)}\big )}\) Hamiltonian one should restrict the corresponding matrix Painlevé \(\mathrm {III\big (D_6^{(1)}\big )}\) Hamiltonian (2.23) to the image of the map (4.2). In this way we obtain
Note that this Hamiltonian is not of the physical form (1.5), to obtain such form, one should make logarithmic change of variables
which gives
So this is a system of trigonometric Calogero type, in difference with rational Calogero–Painlevé \(\textrm{II}\) (1.6). However, Hamiltonian (4.6) is more convenient for us than (4.8) because it is given in terms of rational functions.
4.2 Reduction at Calogero–Painlevé level
Let G be a group of symmetries of a certain matrix Painlevé system and let \({\textsf{G}}\) be the group of corresponding symmetries of the Calogero–Painlevé system. Then \({\textsf{M}}_{\alpha }^{{\textsf{G}}}\) is preserved by the Calogero–Painlevé dynamics. We aim to study the dynamics on this subset.
Consider a certain matrix \(n|G|\times n|G|\) Painlevé system with the finite group G of its symmetries from Table 9 or from Table 10.
Theorem 4.1
Let \(G\cong C_2\) or \(G\cong C_2 \times C_2\). Then there is an open subset \(U \subset \left( {\textsf{M}}^{\textrm{reg}}_{\alpha }\right) ^{{\textsf{G}}}\) such that
-
The dynamics on U is equivalent to the dynamics of the n–particle Calogero–Painlevé system. The type of this Calogero–Painlevé system is given at Image column in Tables 9 and 10. The coupling constant of this system is |G|g.
-
U is open and dense on the connected component of the largest dimension in \(\left( {\textsf{M}}^{\textrm{reg}}_{\alpha }\right) ^{{\textsf{G}}}\).
In general \(\left( {\textsf{M}}^{\textrm{reg}}_{\alpha }\right) ^{{\textsf{G}}}\) is not connected. We will see in the proof that the component of the largest dimension in \(\left( {\textsf{M}}^{\textrm{reg}}_{\alpha }\right) ^{{\textsf{G}}}\) is unique.
Proof
In the proof we combine two different Hamiltonian reductions. First, recall that above we defined reduction (4.1), with the corresponding moment map \(\mu _{N}:M^{\textrm{reg}}_{\alpha }\rightarrow \mathfrak {gl}_{N}\left( {\mathbb {C}}\right) \) and the projection \(\pi _N: \mu _{N}^{-1}\left( \textbf{O}_{N,g}\right) \cap M_{\alpha , t}^{\textrm{reg}}\rightarrow {\textsf{M}}_{\alpha , t}^{\textrm{reg}}\). Second, in the setting of Theorem 3.1 we have the corresponding moment map \(\textbf{m}:M_{\alpha }^{{\bar{G}}}\rightarrow \left( \mathfrak {gl}_{n}\left( {\mathbb {C}}\right) \right) ^{|{\bar{G}}| - 1}\) and the projection \(\textrm{pr}: \textbf{m}^{-1}\left( \textbf{g}\right) \rightarrow {\mathbb {M}}_{\alpha }\). Recall that \(\textbf{m}\) is given just by the diagonal \(n\times n\) blocks of [p, q] from the second to the last one.
The main idea of the proof is to construct a map \(\varphi \) such that diagram on Fig. 1 is commutative.
Description of the map \(\varphi \)
Let us consider the cases \(G \cong C_2\) only (for the cases \(G \cong C_2\times C_2\) the proof is similar). Group \(G\cong C_2\) is generated by the transformation w. Let \({\textsf{w}}\) be the corresponding transformation of the Calogero–Painlevé system.
Step 1. Using explicit formulas for w we see that \({\textsf{w}}\) has special form.
Here \(a_{{\textsf{w}}}, b_{{\textsf{w}}}, c_{{\textsf{w}}}\) are certain rational functions. Invariance condition then can be written as follows.
The permutation \(\sigma \) is unique since \(q_i\)’s are pairwise distinct.
Only conjugacy class of \(\sigma \) is well–defined since coordinates \(((p_1,q_1), (p_2, q_2), \ldots , (p_{2n}, q_{2n}))\) are defined up to permutation. Hence for any point \(x\in \left( {\textsf{M}}_{\alpha }^{\textrm{reg}}\right) ^{{\textsf{w}}}\) we have a conjugacy class which we denote by \([\sigma _{x}]\). In notation \(U_{[\sigma ]} = \{x\in \left( {\textsf{M}}_{\alpha }^{\textrm{reg}}\right) ^{{\textsf{w}}} | [\sigma _{x}] = [\sigma ]\}\), we get \(\left( {\textsf{M}}_{\alpha }^{\textrm{reg}}\right) ^{{\textsf{w}}} = \sqcup _{[\sigma ]}U_{[\sigma ]}\).
Step 2. Let us express \(\sigma \) as the product of independent cycles. Recall that \({\textsf{w}}\) is an involution, thus, taking square of \({\textsf{w}}\), we get \(q_{\sigma ^2(i)} = q_{i}\), hence \(\sigma ^2 = \textrm{Id}\). Then \(\sigma \) is the product of independent transpositions. We denote the cyclic type of \(\sigma \) by \([2^{k}1^{2n-2k}]\). Let us compute \(\textrm{dim}\left( U_{[2^{k}1^{2n-2k}]}\right) \). It is easy to see that each cycle in \(\sigma \) imply system of equations (4.10) and systems for different cycles are independent.
Let \(\sigma (i) = i\), then
Hence \((p_i(t), q_i(t))\) is an algebraic solution of the corresponding Painlevé equation.
If we have cycle \((l_1,l_2)\) in \(\sigma \), then we get
Note that since \({\textsf{w}}\) is an involution (4.12b) follows from (4.12a). Thus the cycle \((l_1,l_2)\) implies two independent equations.
As the result we get (for fixed t)
So \(U_t:= U_{[2^{n}],t}\) has the maximal dimension equal to 2n.
Step 3a. Map \(\pi _{2n}\) should be surjective for the existence of \(\varphi \). In other words we have to check \(\forall x\in U: \pi _{2n}^{-1}(\{x\})\cap \left( M_{\alpha }^{\textrm{reg}}\right) ^{{\bar{w}}}\cap m_2^{-1}\left( \text {i}g \textbf{1}_{n\times n}\right) \ne \varnothing \).
For \(\sigma \in \textrm{S}_{2n}\) let us denote by \(S_{\sigma }\) the matrix corresponding to \(\sigma \). Then for \(x\in U_{[\sigma ]}\) consider \((\breve{p},\breve{q}) = \zeta _{2n}(x)\in \pi _{2n}^{-1}(\{x\})\). Then we have
Since \(x\in U_{[2^n]}\) we get \(\exists A\in \textrm{GL}_{2n}\left( {\mathbb {C}}\right) : AS_{\sigma }A^{-1} = S_{2}\). Recall that \(S_2 = \textrm{diag}\left( \textbf{1}_{n\times n}, -\textbf{1}_{n\times n}\right) \) was defined in Sect. 3.1. Then we get
Therefore \(\exists ({\hat{p}},{\hat{q}}) = (A\breve{p}A^{-1}, A\breve{q}A^{-1})\in \pi _{2n}^{-1}(\{x\}) \cap M_{\alpha , t}^{{\bar{w}}}\).
From Remark 3.3 it follows that \([{\hat{p}}, {\hat{q}}]\) is block diagonal. Let us use the notation \([{\hat{p}}, {\hat{q}}] = \textrm{diag}(m_1, m_2)\).
Since \(({\hat{p}}, {\hat{q}}) \in \mu _{2n}^{-1}\left( \textbf{O}_{2n, g}\right) \) we get either \(m_1 = \text {i}g \textbf{1}_{n\times n}\) or \(m_2 = \text {i}g \textbf{1}_{n\times n}\). In the second case let us take \((p,q):=({\hat{p}}, {\hat{q}}) \in m_{2}^{-1}\left( \text {i}g \textbf{1}_{n\times n}\right) \). In the first case we take \((p, q):= (R{\hat{p}}R^{-1}, R{\hat{q}}R^{-1})\), where \( R = \begin{pmatrix} 0 &{}\quad \textbf{1}_{n\times n}\\ \textbf{1}_{n\times n} &{}\quad 0 \end{pmatrix}\) is a matrix, which preserves \(\mu _{2n}^{-1}\left( \textbf{O}_{2n, g}\right) \cap \left( M_{\alpha }^{\textrm{reg}}\right) ^{{\bar{w}}}\). In both cases we get
Step 3b. Let \((p,q) \in \pi _{2n}^{-1}(\{x\})\cap \left( M_{\alpha ,t}^{\textrm{reg}}\right) ^{{\bar{w}}}\cap m_2^{-1}\left( \text {i}g \textbf{1}_{n\times n}\right) \). Then we have to check that \(\pi ((p,q)) \in \mu _{n}^{-1}\left( \textbf{O}_{n, |G|g}\right) \).
We introduced coordinates \(({\tilde{P}}, {\tilde{Q}})\) on \({\mathbb {M}}_{\alpha ,t}\) on Steps 3 in Sects. 3.2, 3.3 and 3.4. For them we get
For the final coordinates (P, Q) in each case one can check that
Since \([p,q] = \textrm{diag}(m_1, \text {i}g\textbf{1}_{n\times n})\in \mu _{2n}^{-1}\left( \textbf{O}_{2n, g}\right) \) we get
which means \((P,Q) \in \mu _{n}^{-1}\left( \textbf{O}_{n, 2g}\right) \).
Step 3c. Let us define \(\varphi (x) = \pi _n \circ \textrm{pr}\left( \left( p, q\right) \right) \). We have to check that the right side does not depend on the choice of \((p,q)\in \pi _{2n}^{-1}(\{x\})\cap \left( M_{\alpha ,t}^{\textrm{reg}}\right) ^{{\bar{w}}} \cap m_2^{-1}\left( \text {i}g\textbf{1}_{n\times n}\right) \).
Consider \((p', q')\in \pi _{2n}^{-1}(\{x\})\cap \left( M_{\alpha ,t}^{\textrm{reg}}\right) ^{{\bar{w}}} \cap m_2^{-1}\left( \text {i}g\textbf{1}_{n\times n}\right) \), then
Acting on both sides by \({\bar{w}}\) we get
which implies
Since \((p,q)\in \mu _{2n}^{-1}\left( \textbf{O}_{2n, g}\right) \cap M_{\alpha , t}^{\textrm{reg}}\) we can consider these equalities in a gauge, where q is diagonal and eigenvalues of q are different. Then we get
Rewriting this as \(B^{-1}S_2B = \lambda S_2\) and taking square of both sides we get \(\lambda ^2 = 1\). Then
Here \(b_1, b_2\in \textrm{GL}_{n}\left( {\mathbb {C}}\right) \). Note that conjugation R does not preserve \(\mu _{n}^{-1}\left( \textbf{O}_{n, 2\,g}\right) \cap m_2^{-1}\left( \text {i}g\textbf{1}_{n\times n}\right) \), while conjugation by \(\textrm{diag}(b_1, b_2)\) does. Then the only possible case is \(\lambda = 1\).
Conjugation by \(\textrm{diag}(\textbf{1}_{n\times n}, b_2)\) preserves \(\textrm{pr}\) by the definition. Conjugation by \(\textrm{diag}(b_1, b_2)\) descends to the overall conjugation of (P, Q) by \(b_1\). But this conjugation preserves \(\pi _n\) by the definition. Hence \(\pi _n \circ \textrm{pr} ((p,q)) = \pi _n \circ \textrm{pr} ((p',q'))\).
Step 4. One can inverse \(\varphi \) on \(M_{\beta }^{\textrm{reg}}\) taking \(\varphi ^{-1} = \pi _{2n}\circ s \circ \zeta _{n}\). Here \(s:{\mathbb {M}}_{\alpha , t} \rightarrow M_{\alpha , t}^{{\bar{w}}}\) is a section for the reduction from Theorem 3.1 (for an example in non-linear cases see (3.37)).
Step 5. It remains to check that \(\varphi \) maps the dynamics on U to the dynamics of the n–particle Calogero–Painlevé system called by Image in Table 9.
Let \(\gamma \subset U\) be an integral curve of 2n–particle Calogero–Painlevé system. Then for \((x_0, t_0)\in \gamma \) let \(({\tilde{x}}_0, t_0)\in \mu _{2n}^{-1}\left( \textbf{O}_{2n, g}\right) \cap \left( M_{\alpha }^{\textrm{reg}}\right) ^{{\bar{w}}}\cap \textbf{m}^{-1}\left( \textbf{g}\right) \) be a lift of \((x_0, t_0)\) i.e. \(\pi _{2n}({\tilde{x}}_0) = x_0\). Let us denote by \(\Gamma \) the integral curve of the corresponding matrix \(2n\times 2n\) Painlevé system through \(({\tilde{x}}_0, t_0)\). Note that \(\mu _{2n}^{-1}\left( \textbf{O}_{2n, g}\right) \cap \left( M_{\alpha }^{\textrm{reg}}\right) ^{{\bar{w}}}\cap \textbf{m}^{-1}\left( \textbf{g}\right) \) is locally preserved by the dynamics of \(2n\times 2n\) matrix Painlevé system, so \(\Gamma \subset \mu _{2n}^{-1}\left( \textbf{O}_{2n, g}\right) \cap \left( M_{\alpha }^{\textrm{reg}}\right) ^{{\bar{w}}}\cap \textbf{m}^{-1}\left( \textbf{g}\right) \). Then \(\Gamma \) is a lift of \(\gamma \) to \(\mu _{2n}^{-1}\left( \textbf{O}_{2n, g}\right) \cap \left( M_{\alpha }^{\textrm{reg}}\right) ^{{\bar{w}}}\cap \textbf{m}^{-1}\left( \textbf{g}\right) \). By Theorem 3.1, \(\textrm{pr}\) maps \(\Gamma \) to an integral curve of the \(n\times n\) matrix Painlevé system, called by Image in Table 9. Then, by the definition \(\pi _{n}\) maps \(\textrm{pr}(\Gamma )\) to an integral curve of the corresponding n–particle Calogero–Painlevé system.
\(\square \)
Remark 4.2
In cases 4, 5 from Table 9 we have open subset
For this subset one can also construct \(\varphi \) as in Theorem 4.1. Namely surjectivity of \(\pi _{|G|n}\) follows from definition of U, while steps 3b–5 of the proof can be performed with the slight modification. Note that for cases 4, 5 Bäcklund transformations are not of the form (4.9), then steps 1, 2 of the proof have no sense. So for this U we do not have the second statement of Theorem 4.1.
Example 4.2
Let us consider Calogero–Painlevé \(\mathrm {III\big (D_6^{(1)}\big )}\) and corresponding matrix Painlevé \(\mathrm {III\big (D_6^{(1)}\big )}\). Hamiltonian (4.6) gives equations of motion
Let us at first illustrate Theorem 4.1 for G generated by \(\pi \circ \pi '\) (case 2 from Table 9). We take \(N=2n\) particles and consider open subset \(U_{[2^n]}\subset \left( {\textsf{M}}_{\alpha }^{\textrm{reg}}\right) ^{{\textsf{w}}}\) with \({\textsf{w}}=\pi _N \circ (\pi \circ \pi ') \circ \zeta _N\), choosing \(\sigma =(1,n+1) \ldots (n,2n)\). Then \(\alpha _0=\alpha _1=\beta _0=\beta _1=1/2\) and on \(U_{[2^n]}\) we have
Then equations of motion (4.26) reduce to equations on \(\{(p_i,q_i)\}_{i=1,\ldots n}\)
Here for convenience we introduce \({\tilde{p}}_i=p_i-1/2+t/(2q_i^2)\), such that \({\textsf{w}}^*({\tilde{p}}_i)=-{\tilde{p}}_i\). Now we find coordinates in which this dynamics has Calogero–Painlevé type. In other words, we have to find map \(\varphi \) from Theorem 4.1 using the Hamiltonian reduction on the matrix level (with \(g_2=g\)). We take \(\{(p_i,q_i)\}_{i=1,\ldots 2n}\in U_{[2^n]}\), namely under condition (4.27). Choosing the following matrix A from the proof of Theorem 4.1 we have the following \((p,q)\in M_{\alpha }^{{\bar{w}}}\)
After the conjugation by matrix A the moment map condition becomes
So, using coordinates \(({\tilde{P}},{\tilde{Q}})=({\mathfrak {p}}_{12}{\mathfrak {q}}_{12}^{-1},{\mathfrak {q}}_{12}{\mathfrak {q}}_{21})\) from Sect. 3.2 we obtain on \( \textrm{pr}\left( \textrm{Ad}_A\circ \zeta _{2n}(U_{[2^n]})\right) \)
In coordinates (3.11) we obtain following coordinates \(\{(P_i,Q_i)\}_{i=1,\ldots n}\) and time s on \( \pi _n\circ \textrm{pr}\left( \textrm{Ad}_A\circ \zeta _{2n}(U_{[2^n]})\right) \)
Finally, one can check from (4.28) that these \(\{(P_i,Q_i)\}_{i=1,\ldots n}\) satisfy Calogero–Painlevé \(\mathrm {III\big (D_8^{(1)}\big )}\) with coupling constant 2g
Example 4.3
Let us then illustrate Theorem 4.1 also for Calogero–Painlevé \(\mathrm {III\big (D_6^{(1)}\big )}\)system, but for G generated by \(\pi '\) (case 7 from Table 9). Note that this case is non-linear in difference with the previous example. We take \(N=2n\) particles and consider \(U_{[2^n]}\subset \left( {\textsf{M}}_{\alpha }^{\textrm{reg}}\right) ^{{\textsf{w}}}\) with \({\textsf{w}}=\pi _N\circ \pi ' \circ \zeta _N\). Then \(\beta _0=\beta _1=1/2\) and on \(U_{[2^n]}\) we have
Then equations of motion (4.26) reduce to the equations on \(\{(p_i,q_i)\}_{i=1,\ldots n}\), namely
We want to find \(\varphi \) from Theorem 4.1 using the Hamiltonian reduction at matrix level (with \(g_2=g\)). We choose A as in the previous example, and take \(\{(p_i,q_i)\}_{i=1,\ldots 2n}\in U_{[2^n]}\), namely under condition (4.35). Then we have following \((p,q)\in M_{\alpha }^{{\bar{w}}}\)
where we calculate only necessary matrix block for the momentum. So, using formulas from Sect. 3.3 we obtain following \(({\tilde{P}},{\tilde{Q}})\) on \(\textrm{pr}\left( \textrm{Ad}_A\circ \zeta _{2n}(U_{[2^n]})\right) \) (only diagonal elements of \({\tilde{P}}\) are calculated)
In coordinates (3.43) we obtain following coordinates \(\{(P_i,Q_i)\}_{i=1,\ldots n}\) and time s on \(\pi _n\circ \textrm{pr}\left( \textrm{Ad}_A\circ \zeta _{2n}(U_{[2^n]})\right) \)
Finally, these \(\{(P_i,Q_i)\}_{i=1,\ldots n}\) satisfy Calogero–Painlevé \(\textrm{V}(1-\alpha _1-\text {i}g,\text {i}g,\alpha _1-\text {i}g,\text {i}g)\) with coupling constant 2g, which one can check from (4.36)
Example 4.4
Let us now consider Calogero–Painlevé \(\textrm{II}\) system, as in Example 1.2 from Introduction, but with an arbitrary number of particles. Then, instead of Hamiltonian (1.6), we have
The corresponding equations of motion are
Let us illustrate Theorem 4.1 for G generated by r (case 1 from Table 9). We take \(N=2n\) particles and consider \(U_{[2^n]}\subset \left( {\textsf{M}}_{\alpha }^{\textrm{reg}}\right) ^{{\textsf{w}}}\) with \({\textsf{w}}=\pi _N \circ r \circ \zeta _N\). Then \(\alpha _1=1/2\) and on \(U_{[2^n]}\) we have
Then equations of motion (4.42) reduce to equations on \(\{(p_i,q_i)\}_{i=1,\ldots n}\)
We want to find \(\varphi \) from Theorem 4.1 using the Hamiltonian reduction at matrix level (with \(g_2=g\)). The calculations just resemble those from Example 4.2. As a result, we have (4.32) for \(({\tilde{P}}\), \({\tilde{Q}})\) with \(p_i\) instead of \({\tilde{p}}_i\). Then, in coordinates (3.6) we obtain momentum \(\breve{P}\) in a diagonal form
where \(\breve{Q}_i\) and \(\breve{P}_i\) are diagonal elements of \(\breve{Q}\) and \(\breve{P}\) respectively. So the natural projection \(\pi _N\) from \(\textrm{pr}(\textrm{Ad}_A\circ \zeta _{2n}(U_{[2^n]}))\) gives us Calogero–Painlevé system in a gauge, where \(\breve{P}\) is diagonal.
It follows from (4.44) that
This dynamics is Hamiltonian with
It is easy to see that this Hamiltonian can be obtained from the matrix Painlevé Hamiltonian (2.9) in the gauge, where \(\breve{P}\) (f in loc. cit.) is diagonal. So we get Calogero–Painlevé \(\textrm{II} (-\text {i}g-1/2)\) with coupling constant 2g.
In order to obtain a transformation of Calogero–Painlevé system to the standard gauge (as in r.h.s. of (4.2)), one should diagonalize matrix Q. This cannot give an algebraic formula for the general size n.
Remark 4.3
Note that for \(N=2\) and \(g=0\) coordinate transformations from above Examples 4.2, 4.3, 4.4 reproduce formulas [13, 5.10–5.12, 5.5–5.7, 9.5–9.7] for the folding transformations of the corresponding Painlevé equations.
5 Further examples
5.1 Matrix reduction generalization
For the construction in Sect. 3 we choose for a certain Bäcklund transformation the additional twist \(S_d = \textrm{diag}\left( \textbf{1}_{n\times n}, e^{\frac{2\pi \text {i}}{d}}\textbf{1}_{n\times n}, \ldots , e^{\frac{2\pi (d-1) }{d}}\textbf{1}_{n\times n}\right) \). At the Calogero level this twist corresponds to permutation class \([d^n]\). We also choose moment map value \(\textbf{g} = (\text {i}g_2\textbf{1}_{n\times n}, \ldots , \text {i}g_{d}\textbf{1}_{n\times n})\). It is natural to try to weaken such restrictions. Below we present two examples for such generalizations.
Example 5.1
Let us consider \(3n\times 3n\) matrix Painlevé \(\textrm{II}\). We take \(M_{\alpha }^{{\bar{r}}}\) with \({\bar{r}}=\textrm{Ad}_{S} \circ r\), where \(S=\textrm{diag}\left( \textbf{1}_{n\times n},- \textbf{1}_{2n\times 2n}\right) \). Recall that \(r: (p,q)\mapsto (-p,-q)\) and \(\theta \mapsto -\theta \), so we set \(\theta =0\).
Step 1. The matrices \((p,q)\in M_{\alpha }^{{\bar{r}}}\) are given by
with a symplectic form \(\textrm{Tr}(\textrm{d}{\mathfrak {p}}_{12}\wedge \textrm{d}{\mathfrak {q}}_{21}) + \textrm{Tr}(\textrm{d}{\mathfrak {p}}_{21}\wedge \textrm{d}{\mathfrak {q}}_{12})+\textrm{Tr}(\textrm{d}{\mathfrak {p}}_{13}\wedge \textrm{d}{\mathfrak {q}}_{31}) + \textrm{Tr}(\textrm{d}{\mathfrak {p}}_{31}\wedge \textrm{d}{\mathfrak {q}}_{13})\).
Step 2. The remaining gauge freedom is \(\textrm{diag} (\textrm{GL}_{n}({\mathbb {C}})),\textrm{GL}_{2n}({\mathbb {C}}))\) and the moment map is
Step 3. We perform the Hamiltonian reduction with respect to \(\textrm{GL}_{2n}({\mathbb {C}})=\{\textrm{diag}(\textbf{1}_{n\times n},h_2)|h_2\in \textrm{GL}_{2n}({\mathbb {C}}) \}\). We take the moment map value \(m_2=\textrm{diag}\left( \text {i}g_2\textbf{1}_{n\times n},\right. \left. \text {i}g_3\textbf{1}_{n\times n}\right) \), where \(g_2\ne g_3\). Note that its stabilizer is \(\textrm{GL}_n^2({\mathbb {C}})=\{\textrm{diag}(\textbf{1}_{n\times n},h_2,h_3)|h_2, h_3\in \textrm{GL}_n({\mathbb {C}}) \}\). On the reduction \({\mathbb {M}}_{\alpha ,t}\) we can choose the following Darboux coordinates
Step 4. After such Hamiltonian reduction we obtain \(n\times n\) matrix system with Hamiltonian
Omitting \(3t^2/8\) and making standard substitution
we obtain the standard Hamiltonian of \(\textrm{PII}(-\text {i}(g_2-g_3)-1/2)\).
The block sizes and the moment map value in above example, as well as in Sect. 3 are quite special. At least after the Hamiltonian reduction the phase space dimension should correspond to a matrix system, namely it should be equal to a doubled square of an integer. For the standard situation from Sect. 3 we have
where the last two terms correspond to the value of moment map and its stabilizer correspondingly. For the situation from Example 5.1 an analogous calculation gives
However, even if the dimension is not a doubled square of integer, sometimes we can make a certain additional reduction to obtain a matrix Painlevé. We illustrate this by the following example.
Example 5.2
Let us again consider a Hamiltonian reduction of \(M^{{\bar{r}}}_{\alpha }\) for \(3n\times 3n\) matrix Painlevé \(\textrm{II}\), but, in difference with Example 5.1, with respect to \(\mathrm {GL_n}({\mathbb {C}})=\textrm{diag}(h_1,\textbf{1}_{2n\times 2n}|h_1\in \mathrm {GL_n}({\mathbb {C}}))\). We take for the moment map (see (5.2)) the value
so after the reduction we obtain \(6n^2\)-dimensional system
On the reduction we introduce Darboux coordinates
This matrix system has Hamiltonian
On coordinates \(({\tilde{P}}_3,{\tilde{Q}}_3)\) we have almost the matrix Painlevé \(\textrm{II}\) Hamiltonian. It appears that we can perform two successive Hamiltonian reductions to obtain a matrix system only on \(({\tilde{P}}_3,{\tilde{Q}}_3)\). For the first one, with respect to translations of \({\tilde{Q}}_1\), we fix moment map value \({\tilde{P}}_1=0\). Then the Hamiltonian becomes invariant with respect to \(({\tilde{P}}_2,{\tilde{Q}}_2)\mapsto (h{\tilde{P}}_2,{\tilde{Q}}_2h^{-1}), h\in \textrm{GL}_n({\mathbb {C}})\). We fix the value of the corresponding moment map \({\tilde{P}}_2{\tilde{Q}}_2\) to be a scalar matrix, namely \({\tilde{P}}_2{\tilde{Q}}_2=\theta \textbf{1}_{n\times n}={\tilde{Q}}_2{\tilde{P}}_2\). Then after an additional coordinate and time rescaling
we see that Hamiltonian (5.11) becomes the standard Hamiltonian of matrix \(n\times n\) \(\textrm{PII}\big (-\text {i}g-\frac{1}{2}-2\theta \big )\).
5.2 Adding algebraic solutions to Calogero–Painlevé systems
Let us consider a Calogero–Painlevé system together with order 2 Bäcklund transformation \({\textsf{w}}\). Let us suppose that \({\textsf{w}}\) leads to the permutations of cyclic type \([2^n 1^m]\), in difference with Step 2 from the proof of Theorem 4.1. Then we obtain that \({\textsf{w}}((p_i,q_i))=(p_i,q_i)\) for \(2n<i\le 2n+m\), so the last m particles evolve as certain algebraic functions. Below we give two examples for the dynamics on the Calogero–Painlevé invariant subset involving such additional algebraic solutions.
Example 5.3
We modify Example 4.4 for Calogero–Painlevé \(\textrm{II}\). The \( {\textsf{w}}\)-invariant algebraic solution is
We can add only one such particle, due to condition \(q_i\ne q_j\) for \(i\ne j\) on \({\textsf{M}}^{\textrm{reg}}_{\alpha }\). Adding such particle \(q_{2n+1}=p_{2n+1}=0\), for the rest of the particles we have equations of motion on \((p_i,q_i)\), \(1\le i\le n\)
which differ from (4.44) only in coefficient of term \(q_i^{-3}\). Then it is easy to see that we can modify Example 4.4 by hand. Namely, it is enough to modify \(Q_i\) from (4.45) by \(g\rightarrow 3g\)
Finally we obtain Calogero–Painlevé \(\textrm{II}(-3\text {i}g-1/2)\) instead of \((-\text {i}g-1/2)\).
Example 5.4
We modify Example 4.3 for Calogero–Painlevé \(\textrm{III}\big (D_6^{(1)}\big )\). The \({\textsf{w}}\)-invariant algebraic solutions are
We can add one of them or both to our system. So let us add to the Calogero–Painlevé \(\textrm{III}\big (D_6^{(1)}\big )\) from Example 4.3\(n_1=0,1\) particles \(\sqrt{t}\) and \(n_2=0,1\) particles \((-\sqrt{t})\). Modifying momenta \(P_i\) from (4.39) by an additional term as follows
we obtain Calogero–Painlevé \(\textrm{V}(1-\alpha _1-(1+2n_1n_2)\text {i}g,(1-2(n_1-n_2)+2n_1n_2)\text {i}g,\alpha _1-(1+2n_1n_2)\text {i}g,(1-2(n_2-n_1)+2n_1n_2)\text {i}g)\).
Remark 5.1
It would be interesting to obtain such Calogero–Painlevé relations from matrix Painlevé ones. Note that the permutation matrix for \([2^n1^m]\) is conjugated to \(S=\textrm{diag}(\textbf{1}_{(m+n)\times (m+n)},-\textbf{1}_{n\times n})\), cf. Section 5.1.
5.3 Spin Calogero–Painlevé systems
One can consider a more general case of Calogero-type systems taking in (4.1) general orbit \(\textbf{O}\) instead of \(\textbf{O}_{N, g}\). Let us denote the corresponding reduction map by \(\pi _{\textbf{O}}\) instead of \(\pi _{N}\).
The regular part of the phase space of a spin Calogero–Painlevé system can be defined as
It will be more convenient for us to use the identification (for the details see [9], Theorem 3).
Let us recall the construction of the reduction \(\textbf{O}//_{0}\textrm{GL}_{1}^{N}\left( {\mathbb {C}}\right) \) in the right hand side of (5.19).
Let \(X \in \mathfrak {gl}_{N}\left( {\mathbb {C}}\right) \), then the fundamental vector field, corresponding to the coadjoint action of X on \(\textbf{O}\) is \(v_{X}(m) = -\textrm{ad}^*_{X}(m)\). Then Kirillov–Kostant–Souriau form can be written as \(\omega _{\textrm{KKS}}{|_{m}}(v_{X}, v_{Y}) = \langle m, [X, Y]\rangle \). Then we have \(\iota _{v_{X}}\omega _{\textrm{KKS}}{|_{m}}(v_{Y}) = -\left< \text {ad}^*_{Y}(m), X\right> = \langle \text {d}m(v_{Y}), X\rangle \), which means that the coadjoint action on \(\textbf{O}\) is Hamiltonian with the moment map \(\mu _{KKS}: m \mapsto m\). Below we will identify points of \(\textbf{O} \subset \mathfrak {gl}_{N}\left( {\mathbb {C}}\right) ^*\) with matrices using Killing form.
Next, we restrict coadjoint action to the subgroup of diagonal matrices, so the moment map becomes the projection, which maps \((m_{ij})_{i,j=1,\ldots ,N}\mapsto (m_{ii})_{i = 1,\ldots , N}\). Then the element of \(\textbf{O}//_0\textrm{GL}_{1}^{N}\left( {\mathbb {C}}\right) \) is a matrix \(m \in \textbf{O}\) such that \(m_{ii} = 0,\, i = 1,\ldots N\), considered up to the conjugations by diagonal matrices, namely \((m_{ij})_{i,j=1,\ldots ,N} \sim (a_{i}a_{j}^{-1}m_{ij})_{i,j=1,\ldots ,N}\). We will denote this class by [m]. It will be useful to denote the coadjoint orbit of m by \(\textbf{O}\left( m\right) \).
Points of \(\textrm{T}^*\left( {\mathbb {C}}^N\backslash \textrm{diags}\right) \) are ordered sets of pairs \(((p_j, q_j))_{j = 1,\ldots , N}\), where \(q_j\)’s are distinct. The action of \(S_N\) on \(\left( \textrm{T}^*\left( {\mathbb {C}}^N\backslash \textrm{diags}\right) \times \textbf{O}//_{0}\textrm{GL}_{1}^{N}\left( {\mathbb {C}}\right) \right) / \textrm{S}_{N}\) is defined as follows
where \(S_{\sigma }\) is the matrix corresponding to \(\sigma \in S_N\).
Finally the identification 5.19 can be done as follows (cf. \({\tilde{\zeta }}_{N}\) in Sect. 4.1)
The dynamics of a spin Calogero–Painlevé system is defined as descent of the matrix Painlevé dynamics with respect to the reduction (5.18). Let us consider an analogue of Theorem 4.1 for the spin Calogero–Painlevé \(\mathrm {III\big (D_6^{(1)}\big )}\).
Example 5.5
The spin Calogero–Painlevé \(\mathrm {III\big (D_6^{(1)}\big )}\) is defined by the Hamiltonian (cf. (4.6)).
We will consider a spin generalization of Example 4.2. So, we take \(N = 2n\) and \(\alpha _1 = \beta _1 = \frac{1}{2}\).
From the Diagram 1 it can be seen that the natural candidate for U from Theorem 4.1 is \(\pi _{\textbf{O}}\left( {\tilde{U}}\right) \), where \({\tilde{U}} = \left( \left( M_{\alpha , t}^{\textrm{reg}}\right) ^{\overline{\pi \circ \pi '}}\cap \mu _{2n}^{-1}\left( \textbf{O}\right) \cap \textbf{m}^{-1}\left( \text {i}g_2 \textbf{1}_{n\times n}\right) \right) \). We want to obtain coordinates on \(\pi _{\textbf{O}}\left( {\tilde{U}}\right) \) in which the dynamics corresponds to spin Calogero–Painlevé \(\mathrm {III\big (D_8^{(1)}\big )}\) i.e. we want to find an analogue of the map \(\varphi \) from the Diagram 1.
Step 1. Let us obtain coordinates on \(\pi _{\textbf{O}}\left( {\tilde{U}}\right) \). Consider \([(((p_j, q_j))_{j = 1,\ldots , N}, [m])] = \pi _{\textbf{O}}((p,q))\), where \((p,q) \in {\tilde{U}}\). We have that \([p, q] = \textrm{diag}(m_1, \text {i}g_2 \textbf{1}_{n\times n})\). Let A be a matrix such that \({\tilde{q}}:= A^{-1}q A = \textrm{diag}\left( q_1, \ldots , q_{2n}\right) \). Then we get
where \({\tilde{S}}_{2} = A^{-1}S_2 A,\;\; {\tilde{p}} = A^{-1} p A\). Then without loss of generality we get
It can be seen that after additional multiplication of A by a certain diagonal matrix from the right (which preserves \({\tilde{q}}\)) we get \({\tilde{S}}_{2} = \begin{pmatrix} 0 &{}\quad \textbf{1}_{n\times n}\\ \textbf{1}_{n\times n} &{}\quad 0 \end{pmatrix}\). So, for diagonal entries of \({\tilde{p}}\) from (5.23) we get
Now let us obtain conditions on spin variables \(m_{ij}\). Note that since \(A^{-1}S_2A = {\tilde{S}}_2\), we have \( A^{-1} = \frac{1}{\sqrt{2}}\begin{pmatrix} \textbf{1}_{n\times n} &{}\quad -\textbf{1}_{n\times n}\\ \textbf{1}_{n\times n} &{}\quad \textbf{1}_{n\times n} \end{pmatrix} \begin{pmatrix} \alpha _{1} &{}\quad 0\\ 0 &{}\quad \alpha _{2} \end{pmatrix}, \) where \(\alpha _1, \alpha _2 \in \textrm{GL}_{n}\left( {\mathbb {C}}\right) \). Then we get
where \(M = \alpha _{1}m_1\alpha _{1}^{-1} + \text {i}g_2 \textbf{1}_{n\times n}\). Then we have \(M\in \tilde{\textbf{O}}\), where \(\tilde{\textbf{O}} = \textbf{O}\left( m_1\right) + \text {i}g_2 \textbf{1}_{n\times n}\). Note that M has zero diagonal entries and is defined up to conjugations by diagonal matrices, which means that we have \([M] \in \tilde{\textbf{O}}//_{0}\textrm{GL}_{n}\left( {\mathbb {C}}\right) \).
Taking into account remaining symmetry corresponding to permutations of \((p_{i}, q_{i})\)’s we get that \(\pi _{\textbf{O}}\left( {\tilde{U}}\right) \) is parametrized by points \([(((P_i,Q_i))_{i = 1,\ldots , n}, [M])]\in \left( \tilde{\textbf{O}}//_{0}\textrm{GL}_{1}^{n}\left( {\mathbb {C}}\right) \times \textrm{T}^*(\mathbb {C}^n\backslash \textrm{diags})\right) / \textrm{S}_{n}\), where
We will explain meaning of formulas (5.27) below.
Step 2. Let us explain why \([(((P_i,Q_i))_{i = 1,\ldots , n}, [M])]\) are desired coordinates corresponding to the dynamics of spin Calogero–Painlevé \(\mathrm {III\big (D_8^{(1)}\big )}\). Following the arguments similar to Step 5 from the proof of Theorem 4.1 we get that the coordinates given by the map \(\pi _{\textbf{O}}\circ \textrm{pr}\) match the dynamics of spin Calogero–Painlevé \(\mathrm {III\big (D_8^{(1)}\big )}\).
We have \(\textrm{pr}((p,q)) = (P,Q)\), where (P, Q) is given by (3.11). One can check that the pair (P, Q) is conjugated to \(\left( {\hat{P}}, {\hat{Q}}\right) : =\left( 2\left( {\mathfrak {p}}_{12}{\mathfrak {q}}_{12}^{-1} + {\mathfrak {q}}_{21}^{-1}{\mathfrak {p}}_{21}\right) , \frac{1}{4}{\mathfrak {q}}_{12}{\mathfrak {q}}_{21}\right) \), where \({\mathfrak {p}}_{ij}, {\mathfrak {q}}_{ij}\)’s are given by (3.2) properly specialized for case 3.2.2.
To obtain \(\pi _{\textbf{O}}((P, Q)) = \pi _{\textbf{O}}\left( \left( {\hat{P}}, {\hat{Q}}\right) \right) \) following the identification (5.19) one should consider the pair \(\left( {\hat{P}}, {\hat{Q}}\right) \) in the basis, where \({\hat{Q}} = \frac{1}{4}{\mathfrak {q}}_{12}{\mathfrak {q}}_{21}\) is diagonal. Since \(A^{-1}qA = {\tilde{q}}\) it is easy to see that
This proves that the formula \(Q_{i} = \frac{1}{4}q_{i}^2\) gives part of proper coordinates. As well it implies that \(\alpha _{1}\) diagonalizes \(\frac{1}{4}{\mathfrak {q}}_{12}{\mathfrak {q}}_{21}\), so momenta conjugated to \(Q_{i}\)’s are diagonal entries of \(\textrm{Ad}_{\alpha _1}\left( {\hat{P}}\right) \). At first let us compute
Then we obtain
So, we have proved that the formula (5.27) gives part of coordinates corresponding to the dynamics of spin Calogero–Painlevé \(\mathrm {III\big (D_8^{(1)}\big )}\). The rest of coordinates is given by \([\textrm{Ad}_{\alpha _1}\left( [{\hat{P}}, {\hat{Q}}]\right) ]\). By straightforward computation we obtain that \([{\hat{P}}, {\hat{Q}}] = m_1 + \text {i}g_2 \textbf{1}_{n\times n}\), so
Additionally we can check that \([((P_{j},Q_{j}))_{j=1,\ldots ,n},[M]]\) are proper coordinates at the level of dynamics. Since coordinates [M] on \(\pi _{\textbf{O}}({\tilde{U}})\) are tricky it is more convenient to use the analogue of \(\varphi ^{-1}\) which is defined as follows
One can check that this map is symplectic. Let us compute the Hamiltonian with respect to variables \([(((P_{j},Q_{j}))_{j=1,\ldots ,n}, [M])]\). Taking \(s = \frac{t^2}{16}\) as the new time variable we get
Note that on \(\tilde{\textbf{O}}\), the term \(- \frac{1}{2}\sum _{1\le i<j \le n}M_{ij}M_{ji} + \frac{ng^2}{4}\) is just a constant since \(\sum _{1\le i<j \le n}M_{ij}M_{ji}\) is a Casimir function. So we get the system with the Hamiltonian
which is the Hamiltonian of spin Calogero–Painlevé \(\mathrm {III\big (D_8^{(1)}\big )}\).
Notes
In the paper we consider only matrix Painlevé systems that are Hamiltonian, there also exist other interesting matrix Painlevé analogs, see e.g. [1].
References
Adler, V.E., Sokolov, V.V.: Matrix Painlevé II equations. Teoret. Mat. Fiz. 207(2), 188–201 (2021). arXiv:2012.05639
Bertola, M., Cafasso, M., Roubtsov, V.: Noncommutative Painlevé equations and systems of Calogero type. Commun. Math. Phys. 363(2), 503–530 (2018). arXiv:1710.00736
Helton, W.J., de Oliveira, M.C., Miller, B., Stankus, M.: NCAlgebra package. https://mathweb.ucsd.edu/~ncalg/
Inozemtsev, V.I., Meshcheryakov, D.V.: Extension of the class of integrable dynamical systems connected with semisimple Lie algebras. Lett. Math. Phys. 9(1), 13–18 (1985)
Inozemtsev, V.I.: Lax representation with spectral parameter on a torus for integrable particle systems. Lett. Math. Phys. 17(1), 11–17 (1989)
Kawakami, H.: Matrix Painlevé systems. J. Math. Phys. 56(3), 033503–27 (2015)
Kazhdan, D., Kostant, B., Sternberg, S.: Hamiltonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math. 31(4), 481–507 (1978)
Kajiwara, K., Noumi, M., Yamada, Y.: Geometric aspects of Painlevé equations. J. Phys. A: Math. Theor. 50(7), 073001 (2017). arXiv:1509.08186
Reshetikhin, N.: Degenerate integrability of spin Calogero–Moser systems and the duality with the spin Ruijsenaars systems. Lett. Math. Phys. 63, 03 (2002)
Rumanov, I.: Classical integrability for beta-ensembles and general Fokker–Planck equations. J. Math. Phys. 56(1), 013508 (2015). arXiv:1306.2117
Rumanov, I.: Painlevé representation of Tracy–Widom\({_\beta }\) distribution for \({\beta }\) = 6. Commun. Math. Phys. 342(3), 843–868 (2016). arXiv:1408.3779
Takasaki, K.: Painlevé–Calogero correspondence revisited. J. Math. Phys. 42(3), 1443–1473 (2001). arXiv:math/0004118
Tsuda, T., Okamoto, K., Sakai, H.: Folding transformations of the Painlevé equations. Math. Ann. 331(4), 713–738 (2005)
Acknowledgements
We are grateful to V. Adler, I. Sechin, V. Sokolov and V. Roubtsov for useful discussions and paying attention to the literature. Large part of calculations related to Sects. 2 and 3 is performed with the help of the package for noncommutative computations [3]. Authors were partially supported by the HSE University basic research program.
Funding
Open access funding provided by Scuola Internazionale Superiore di Studi Avanzati - SISSA within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bershtein, M., Grigorev, A. & Shchechkin, A. Hamiltonian reductions in matrix Painlevé systems. Lett Math Phys 113, 47 (2023). https://doi.org/10.1007/s11005-023-01651-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11005-023-01651-5