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\title{The non-commutative geometry of higher-rank Gibbons-Hermsen systems}
\author{Alberto Tacchella}
\institute{ICMC - Universidade de S\~ao Paulo, S\~ao Carlos}
\usetheme{Default}
\begin{document}
\maketitle[width=0.85\textwidth]
\begin{columns}
% FIRST column
\column{0.5}
\block{Gibbons-Hermsen manifolds}{
For every \(n,r\in \bb{N}\) consider the vector space
\[ \begin{aligned}
V_{n,r} &= \Mat_{n,n}(\bb{C})\oplus \Mat_{n,n}(\bb{C})\oplus
\Mat_{n,r}(\bb{C})\oplus \Mat_{r,n}(\bb{C})\\
&= T^{*}(\Mat_{n,n}(\bb{C})\oplus \Mat_{r,n}(\bb{C}))
\end{aligned} \]
The group \(\GL_{n}(\bb{C})\) acts on \(V_{n,r}\) by
\[ G.(X,Y,v,w) = (GXG^{-1}, GYG^{-1}, Gv, wG^{-1}) \]
This action is Hamiltonian with moment map \(V_{n,r}\to
\mathfrak{gl}_{n}(\bb{C})\) given by
\[ \mu(X,Y,v,w) = [X,Y] - vw \]
For every \(\tau\in \bb{C}^{*}\) the action of \(\GL_{n}(\bb{C})\) on
\(\mu^{-1}(\tau I)\) is free. Then we can perform a Marsden-Weinstein
reduction to obtain a smooth symplectic manifold of dimension \(2nr\)
\[ \CM_{n,r} = \mu^{-1}(\tau I)/\GL_{n}(\bb{C}) \]
}
\block{The Gibbons-Hermsen integrable system}{
Denote by \(\CM_{n,r}'\) (resp. \(\CM_{n,r}''\)) the space of quadruples
\([X,Y,v,w]\) in \(\CM_{n,r}\) such that \(X\) (resp. \(Y\)) is regular
semisimple (= diagonalizable with distinct eigenvalues). Define
\[ A_{r}\deq \set{(f,e)\in \bb{C}^{r}\times \bb{C}^{r} | \pair{f}{e} =
\tau}/\bb{C}^{*} \]
and denote by \(f_{1}\dots f_{n}\) the rows of the matrix \(v\) and by
\(e_{1}\dots e_{n}\) the columns of the matrix \(w\).
\textbf{Theorem.} There is a bijection \(\CM_{n,r}' \to
(\bb{C}^{n}_{\mathrm{reg}}\times \bb{C}^{n}\times A_{r}^{n})/S_{n}\) defined
by sending \([X,Y,v,w]\) to the spectrum of \(X\), the \(n\)-tuple of
diagonal elements of \(Y\) and the pairs \((f_{i},e_{i})\) for each
\(i=1\dots n\).
The restriction to \(\CM_{n,r}'\) of the reduced symplectic form is
\[ \omega = \sum_{i=1}^{n} (\de p_{i}\wedge \de x_{i} + \de f_{i}\wedge \de
e_{i}) \]
The Gibbons-Hermsen Hamiltonians [GH84] are given by \(J_{k,\alpha}\deq \tr
Y^{k}v\alpha w\), where \(k\in \bb{N}\) and \(\alpha\in
\Mat_{r,r}(\bb{C})\). Complete integrability follows from the relations
\[ \{J_{k,\alpha},J_{\ell,\beta}\} = J_{k+\ell,[\alpha,\beta]} \]
In particular (taking \(\tau=1\))
\[ J_{2,I} = \frac{1}{2}\sum_{i=1}^{n} p_{i}^{2} - \frac{1}{2} \sum_{i\neq j=1}^{n}
\frac{\pair{f_{i}}{e_{j}}\pair{f_{j}}{e_{i}}}{(q_{i}-q_{j})^{2}} \]
If \(r=1\) there are no additional degrees of freedom and we recover the
well-known (rational) \emph{Calogero-Moser system}.
}
\block{The case $r=1$}{
The space \(\CM\deq \bigsqcup_{n\in \bb{N}} \CM_{n,1}\) parametrizes
isomorphism classes of right ideals in the first Weyl algebra over
\(\bb{C}\),
\[ A\deq \bb{C}\langle a,a^{*}\rangle / (aa^{*}-a^{*}a-1) \]
Moreover, the natural action of the group \(\Aut A\) on each of the
\(\CM_{n,1}\) is \emph{transitive}.
The group \(\Aut A\) is generated by the family of automorphisms
\(\Phi_{p}\) defined by \((x,y)\mapsto (x-p'(y),y)\)
together with the ``formal Fourier transform'' \(\mathcal{F}\) defined by
\((x,y)\mapsto (-y,x)\).
These generators act on \(\CM_{n,1}\) as follows:
\begin{equation}
\label{eq:0}
\Phi_{p}.(X,Y) = (X - p'(Y), Y) \quad \mathcal{F}.(X,Y) = (-Y,X)
\end{equation}
It turns out that the correct way to generalize these results is to pass
through \emph{noncommutative symplectic geometry} (Kontsevich, Ginzburg,
Bocklandt-Le Bruyn).
The group \(\Aut A\) is isomorphic to the group of automorphisms of the
free algebra \(F_{2}\deq \bb{C}\langle a,a^{*}\rangle\) preserving the
element \([a,a^{*}]\) (``noncommutative symplectomorphisms'' of \(F_{2}\)).
Also, \(F_{2}\) is the path algebra of the quiver
\[ \ol{Q}_{0} = \xymatrix{\bullet \ar@(ul,dl)[]_{a} \ar@(ur,dr)[]^{a^{*}}} \]
and \(\CM_{n,1}\) can be seen as a submanifold in the moduli space of
representations of \(\ol{Q}_{0}\) with dimension vector \((n)\). The action
\eqref{eq:0} then becomes simply the natural action of \(\Aut
(\bb{C}\ol{Q}_{0};[a,a^{*}])\) on this space.
}
\block{The case $r=2$}{
The idea is now to find a suitable family of quivers such that the same
construction works for every \(r>1\). When \(r=2\), Bielawski and
Pidstrygach [BP11] considered the quiver
\[ Q_{2} = {} \xymatrix{\bullet \ar@(ul,dl)[]_{a} \ar@<0.6ex>[rr]^{y} &&
\bullet \ar@<0.6ex>[ll]^{x}} \]
Then \(\CM_{n,2}\) can be seen as a submanifold in the moduli space of
linear representations of \(\ol{Q}_{2}\) with dimension vector \((n,1)\).
\textbf{Theorem (Bielawski, Pidstrygach 2011).} The group
\(\TAut(\bb{C}\ol{Q}_{2};c)\) of \emph{tame} symplectic automorphisms of the
path algebra of \(\ol{Q}_{2}\) acts transitively on \(\CM_{n,2}\).
Tame automorphisms are defined as follows. Given \(\psi\in \Aut
\bb{C}\ol{Q}_{2}\), we say that \(\psi\) is:
\begin{compactitem}
\item \emph{triangular} if \(\psi\) fixes the arrows \(a\), \(x\) and \(y\);
\item \emph{affine} if \(\psi\) sends each arrow in \(\ol{Q}_{2}\) to a
polynomial which is at most linear in each arrow.
\end{compactitem}
An automorphism of \(\bb{C}\ol{Q}_{2}\) is called \emph{tame} if it is
generated by triangular and affine automorphisms.
Wilson [W09] proposed another group for a (possibly transitive) action on
\(\CM_{n,2}\):
\[ \Gamma^{\alg}\deq \Gamma^{\alg}_{\mathrm{sc}} \times \GL_{2}(\bb{C}[z]) \]
where \(\Gamma^{\alg}_{\mathrm{sc}}\) is the group of matrix-valued
functions of the form \(\e^{p} I_{2}\) for some \(p\in z\bb{C}[z]\). In
[MT13] we proved that these two very different approaches are, in fact,
related:
\textbf{Theorem (Mencattini, T. 2013).} The group \(\Gamma^{\alg}\) can be
embedded in \(\TAut(\bb{C}\ol{Q}_{2};c)\) in such a manner that, denoting by
\(\mathcal{P}\) the subgroup generated by the image of the embedding and the
symplectic automorphism
\[ \mathcal{F}_{2}(a,a^{*},x,x^{*},y,y^{*}) = (-a^{*},a,-y^{*},y,-x^{*},x) \]
the induced action of \(\mathcal{P}\) on \(\CM_{n,2}\) is transitive (at
least) on the open subset \(\CM_{n,2}'\cup \CM_{n,2}''\) of \(\CM_{n,2}\).
}
% SECOND column
\column{0.5}
\block{The higher rank case}{
Our work [T] generalizes Bielawski and Pidstrygach's approach to every
\(r>2\). The idea is to consider the family of ``zigzag'' quivers
\[ Z_{r} = \xymatrix{
\bullet \ar@(ul,dl)[]_{a} \ar@/^2.4ex/[rrrr]|{y_{1}}
\ar@/_4ex/[rrrr]_{y_{r}}="b" &&&&
\bullet \ar@/_4ex/[llll]_{x_{1}} \ar[llll]|{x_{2}}="a"
\ar@{.}"a";"b"} \]
with \(\ceil{r/2}\) arrows \(x_{i}\colon \bullet\leftarrow \bullet\) and
\(\floor{r/2}\) arrows \(y_{j}\colon \bullet\rightarrow \bullet\). Then,
denoting by \(\Q_{r}\deq \ol{Z}_{r}\) their doubles, we can again embed the
manifold \(\CM_{n,r}\) in the moduli space of representations of \(\Q_{r}\)
with dimension vector \((n,1)\). This automatically gives for every \(r\in
\bb{N}\) an action of the group of symplectic automorphisms of the path
algebra \(\bb{C}\Q_{r}\) on each \(\CM_{n,r}\).
}
\block{Structure of the automorphism group}{
Let us write
\begin{equation}
\label{eq:1}
\bb{C}\Q_{r} = \mathcal{A}_{11}\oplus \mathcal{A}_{12}\oplus
\mathcal{A}_{21}\oplus \mathcal{A}_{22}
\end{equation}
where \(\mathcal{A}_{ij}\) denotes the linear subspace spanned by the paths
\(j\to i\). We have the following easy results:
\begin{compactitem}
\item \(\mathcal{A}_{11}\) is a free algebra on \(r^{2}+2\) generators;
\item \(\mathcal{A}_{12}\) is a free left \(\mathcal{A}_{11}\)-module;
\item \(\mathcal{A}_{21}\) is a free right \(\mathcal{A}_{11}\)-module.
\end{compactitem}
Now let \(\psi\) be an automorphism of \(\bb{C}\Q_{r}\) fixing the trivial
paths at the two vertices. Then \(\psi\) respects the decomposition
\eqref{eq:1}, hence it induces three bijections
\[ \app{\psi_{11}}{\mathcal{A}_{11}}{\mathcal{A}_{11}}, \quad
\app{\psi_{12}}{\mathcal{A}_{12}}{\mathcal{A}_{12}} \quad\text{ and }\quad
\app{\psi_{21}}{\mathcal{A}_{21}}{\mathcal{A}_{21}} \]
The correspondence \(\psi\mapsto \psi_{11}\) defines a morphism of groups
\(\Aut \bb{C}\Q_{r}\to \Aut \mathcal{A}_{11}\); as a consequence, the group
\(\Aut \bb{C}\Q_{r}\) acts on \(\GL(\mathcal{A}_{11})\) by
\begin{equation}
\label{eq:2}
\psi(M)\deq (\psi_{11}(M_{\alpha\beta}))_{\alpha,\beta=1\dots r} \quad\text{
for all } M\in \GL_{r}(\mathcal{A}_{11}).
\end{equation}
Also, as \(\mathcal{A}_{21}\) is a free \(\mathcal{A}_{11}\)-module of rank
\(r\) the bijection \(\psi_{21}\) is uniquely determined by a matrix in
\(\GL_{r}(\mathcal{A}_{11})\).
\textbf{Theorem.} The map that sends an automorphism \(\psi\) to the
(unique) matrix \(N^{\psi}\) associated to \(\psi_{12}\) is a crossed
morphism \(\Aut \bb{C}\Q_{r}\to \GL_{r}(\mathcal{A}_{11})\) with respect to
the action \eqref{eq:2}, in the sense that
\[ N^{\sigma\circ \psi} = N^{\sigma} \sigma(N^{\psi}) \]
}
\block{Symplectic automorphisms}{
Let \(c_{r}\deq [a,a^{*}] + \sum_{i} [x_{i},x_{i}^{*}] + \sum_{j}
[y_{j},y_{j}^{*}]\). An automorphism of \(\bb{C}\Q_{r}\) is
\emph{symplectic} if \(\psi(c_{r})=c_{r}\). Let us call again an
automorphism \(\psi\) \emph{triangular} if it fixes the unstarred arrows
(i.e. \(a\), \(x_{1}, \dots, x_{\ceil{r/2}}\) and \(y_{1}, \dots,
y_{\floor{r/2}}\)) and \emph{affine} if it depends at most linearly on the
arrows in \(\Q_{r}\). Then we have the following classification:
\begin{compactitem}
\item Triangular symplectomorphisms are of the following form:
\[ \begin{cases}
a^{*}\mapsto a^{*} + \frac{\partial f}{\partial a}\\
x_{i}^{*}\mapsto x_{i}^{*} + \sum_{j=1}^{\floor{r/2}} y_{j}
\frac{\partial f}{\partial b_{ij}}\\
y_{j}^{*}\mapsto y_{j}^{*} + \sum_{i=1}^{\ceil{r/2}}
\frac{\partial f}{\partial b_{ij}} x_{i}
\end{cases} \]
where \(f\) is a \emph{necklace word} (i.e., a word modulo cyclic
permutations) in the free subalgebra of \(\mathcal{A}_{11}\) generated by
the elements \(a\) and \(b_{ij}\deq x_{i}y_{j}\);
\item Affine symplectomorphisms form a group isomorphic to
\(\ASL_{2}(\bb{C})\times \GL_{r}(\bb{C})\), where the first factor acts on
the linear subspace of \(\bb{C}\Q_{r}\) spanned by \(a\) and \(a^{*}\) and
the second factor acts on the subspace spanned by the other arrows.
\end{compactitem}
The group \(\TAut(\bb{C}\Q_{r};c_{r})\) is again defined as the subgroup
generated by these two classes of automorphisms.
}
\block{The action on Gibbons-Hermsen manifolds}{
For every \(r\in \bb{N}\), the group \(\TAut(\bb{C}\Q_{r};c_{r})\) acts on
\[ \CM_{r}\deq \bigsqcup_{n\in \bb{N}} \CM_{n,r} \]
The results of [MT13] generalize as follows. Denote by \(P_{r}\) the
subgroup of \(\TAut(\bb{C}\Q_{r};c_{r})\) generated by those triangular
symplectomorphisms for which the necklace word \(f\) is of the form \(f =
p(a)b_{11}\) for some \(p\in \bb{C}[a]\), and by affine symplectomorphisms
fixing \(a\) and \(a^{*}\).
\textbf{Theorem.} The map \(\psi\mapsto N^{\psi}\) restricts to an
isomorphism \(P_{r}\to \GL(\bb{C}[a])\).
Also, denote by \(\mathcal{P}_{r}\) the semidirect product \(P_{r}\rtimes
\Aut (\bb{C}\Q_{0};[a,a^{*}])\), where the factor \(\Aut
(\bb{C}\Q_{0};[a,a^{*}])\) acts only on \(a\) and \(a^{*}\). Then we have
again the following ``partial'' transitivity result.
\textbf{Theorem.} For every pair of points \(p,p'\in \CM_{n,r}'\cup
\CM_{n,r}''\) there exists \(\psi\in \mathcal{P}_{r}\) such that \(p.\psi =
p'\).
}
\block{References}{
\begin{description}
\item[BP11] Roger Bielawski and Victor Pidstrygach, \emph{On the symplectic
structure of instanton moduli spaces}, Adv. in Math. \textbf{226}
(2011), 2796--2824, \texttt{arXiv:0812.4918}.
\item[GH84] John Gibbons and Theo Hermsen, \emph{A generalization of the
Calogero-Moser system}, Physica \textbf{11D} (1984), 337--348.
\item[MT13] Igor Mencattini and Alberto Tacchella, \emph{A note on the
automorphism group of the Bielawski-Pidstrygach quiver}, SIGMA
\textbf{9} (2013), no.~037, {\tt arXiv:1208.3613}.
\item[T] Alberto Tacchella, \emph{On a family of quivers related to the
Gibbons-Hermsen system}, {\tt arXiv:1311.4403}.
\item[W09] George Wilson, \emph{Notes on the vector adelic Grassmannian},
Unpublished, 31/12/2009.
\end{description}
}
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\end{document}