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International conference "Geometrical Methods in Mathematical Physics"
December 14, 2011 14:45–15:30, Moscow, Lomonosov Moscow State University

On birational nature of isomonodromic deformation equations

M. V. Babich

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
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M. V. Babich

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Abstract: It is known that the famous Painlevé VI equation has surprisingly rich group of birational symmetries. The equation describe isomonodromic deformations of $2\times 2$ Fuchsian system with four poles.
The generic case of $N\times N$ matrices with different eigenvalues has the similar birational symmetries. What about a general case? Are there the same birational symmetries in the degenerated case of low-dimensional orbits with multiple eigenvalues, or the situation is similar to the difference between the twisted and the plane cubics, where the first one is rational and the second one is not?
I will show that the phase spaces of the Isomonodromic Deformation equations have the same structure of the birational symplectic manifold in the degenerated cases too, at least if there are enough number of one-dimensional eigenspaces. The possibility to define the rational canonical variables on the same system in several ways, like the permutations of the basic vectors or renumbering the poles is the source of the birational symmetries in question.
Let us consider the deformation of the Fuchs equation
\begin{equation} \frac{d}{dz}\Psi =\underset{k=1}{\overset{M}{\sum }}\frac{A^{k}}{z-z^{k}} \Psi ; A^{k}\in sl(N,C); z,z^{k}\in C. \label{1} \end{equation}

It is known that the isomonodromic deformation of this equation may be associated with some Hamiltonian system defined on the space that we denote by $O_{J^{1}}\times O_{J^{2}}\times ...\times O_{J^{M}}//$SL($N,$C). This space is the quotient of he product of the several (co)adjoint orbits $ O_{J^{k}}:=\cup _{g\in SL(N,C)}gJ^{k}g^{-1}\ni A^{k}$ over the diagonal (co)adjoint action of SL($N$; C) intersected by the momentum level $ \Sigma :=\underset{k=1}{\overset{M}{\sum }}A^{k}=0$.
Let us built a set of the canonical coordinates on an orbit first. The construction is based on the possibility to project a linear transformation $ A\in $End$V$ along its eigenspace ker($A-\lambda _{1}I)\neq 0$ to End$V/$ker( $A-\lambda _{1}I)$. The Jordan structure of the projection is defined by the Jordan structure of $A$, all the Jordan chains corresponding to $\lambda _{1} $ become one unit shorter. The fiber of the projection is the linear symplectic space, so after the introducing a basis in $V$ we get the symplectic fibration. of the orbit. The iteration of the construction gives the birational symplectomorphism between the orbit $O_{J}$ and the linear symplectic space with the natural Darboux coordinates.
To parameterize the Isomonodromic Deformation phase space $O_{J^{1}}\times O_{J^{2}}\times ...\times O_{J^{M}}//$SL($N,$C) it is possible to construct a basis $\mathbf{e:=}e^{1},...,e^{N}$ rigidly connected with the set of $ A^{1},...,A^{N}:=\bar{A}:$
\begin{equation*} \mathbf{e}(g^{-1}\bar{A}g)=\mathbf{e}(\bar{A})g. \end{equation*}
It is equivalent to the factorization with respect to the diagonal adjoint action of SL($N$; C). The problem is to control the momentum map $\Sigma := \underset{k=1}{\overset{M}{\sum }}A^{k}=0$.
I will present the iteration procedure for the construction of the basis $ \mathbf{e}$ with the necessary properties. The construction is based on the following observation.
Let we project each of the transformations $A^{(k)}\in $End$V$ along its own fixed in someway one-dimensional subspace $K_{1}^{(k)}$ of the eigenspace ker($A^{(k)}-\lambda _{1}^{(k)})\supset K_{1}^{(k)}$ on one hyper-subspace $V_{1}\subset V,\dim V_{1}=\dim V-1$. Denote such a projections by $A_{1}^{(k)}$. Consider the difference $\sigma _{1}$ between two transformations of $V_{1}$. The first one is the projection back to $ V_{1}$ along any fixed direction $e_{1}^{1}$ of the constriction $\Sigma |_{V_{1}}$. The second one is the sum of the projections $\underset{k=1}{ \overset{M}{\sum }}A_{1}^{(k)}=\Sigma _{1}$. The observation is: the transformation $\sigma _{1}\in $End$V_{1}$ depends on the directions ker($A^{(k)}-\lambda _{1}^{(1)}I)$, $im$($A^{(k)}-\lambda _{1}^{(1)}I)$ and $e_{1}^{1}$ only.

Materials: gmmp2011_mbabich.pdf (196.2 Kb)

Language: English

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