text:
Disney's Aladdin on Broadway: Great Family Fun!
Review By Stella Schour for Kababayan Media
The Broadway adaptation of the 1992 Disney film, "Aladdin" is nothing short of spectacular. Having debuted in 2014, this play is still capturing the hearts of theatergoers all over the world.
This show stays true to the film with a few adaptations for the theater such as Iago the parrot of Jafar with a human character of the same name, and Abu, Aladdin's monkey friend with three friends; Babkak, Omar and Kassim.
Flowing and vibrantly hued costumes transport the viewers to a whole new world in its opening scene and does not fail to impress from start to finish. The very impressive set decor with 84 illusions and effects magically takes you to a Middle Eastern marketplace, the palace, the cave of wonders and of course, a magic carpet ride! No detail was spared and all the 337 costumes were a sight to behold. More impressive were the costume changes that happen right before your eyes!
Aladdin is a lighthearted Broadway musical peppered with jokes with modern references. The performances of Adam Jacobs (Aladdin) and Courtney Reed (Jasmine) were stellar, but what really made the play very enjoyable for me was the way Alfie Parker, Jr played Genie with his impeccable timing on the jokes and his charming flamboyance.
Aladdin's non-stop action is definitely a treat for everyone in the family.
(Photos: Deen van Meer)
Tickets are available for purchase through AladdinTheMusical.com, by calling Ticketmaster's Disney on Broadway hotline at 866-870-2717, or in person at the New Amsterdam Theatre box office (214 W. 42nd Street). The New Amsterdam Theatre box office is open Monday through Friday 9:00am – 8:00pm, Saturday 10:00am – 8:00pm and Sunday 10:00am – 6:30pm.
Premium and group tickets are also available. For information about tickets for groups of 20 or more, CLICK HERE.
For more information on Aladdin, visit AladdinTheMusical.com. Connect with Aladdin on Facebook.com/Aladdin and Twitter.com/Aladdin.


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text:
\section{Introduction}

This lecture reviews a recent progress in studying the large N limit
of simple supersymmetric quantum mechanical systems which result
from the dimensional reduction of field theories with gauge
symmetry. Models of this type have been studied for a long time
\cite{CH,dWLN,JW1} and have many different applications, depending
on the space-time dimension, D, of the unreduced theory
\cite{CW1,CW2} (cf. Table \ref{roadmap}). For D=2 they
 can be often solved analytically \cite{CH,CW1,S,T}, providing quantitative
 realization of supersymmetry. In four space-time dimensions one of them
 is nothing but the small
volume limit of the Yang-Mills gluodynamics revealing the spectrum
of zero volume glueballas as a special case \cite{L,LM,vB,K}.
Finally, for D=10 they make contact with the M-theory via the BFSS
hypothesis \cite{BFSS}.

\begin{table}[h]
\begin{center}
\begin{tabular}{ccccc}
 \hline\hline
  $D$        &  $ 2 $ & $ 4 $ & $ \dots $ & $ 10 $  \\  \hline
  $N$        &        &       &           &         \\
   \hline
  2          &   $\bullet $    &    $\bullet $       &          &   $\circ$       \\
  3          &   $\bullet $       &        &          &          \\
  .          &   $\circ$     &        &          &          \\
  .          &   $\circ$      &        &          &          \\
  .          &        &        &          &          \\
  $\infty$   &   $\checkmark$     &        &          &  M     \\
   \hline\hline
\end{tabular}
\end{center}
\caption{A "road map" of a parameter space of supersymmetric
Yang-Mills quantum mechanics. Black dots denote models well
understood/solved, while open circles mark ones under study. A
checkmark labels parameters of a system discussed in this lecture.}
\label{roadmap}
\end{table}

Although we shall be mainly concerned with the large N limit, we
would like to give one example of the N=2 model.

\subsection{Four-dimensional supersymmetric Yang-Mills quantum mechanics at
finite N }
The system has three bosons and two fermions, both in the adjoint representation
of SU(2). The Hamiltonian reads \cite{CH}
\begin{eqnarray} H &=&  {1\over 2} p_a^i p_a^i + {g^2\over
4}\epsilon_{abc} \epsilon_{ade}x_b^i x_c^j x_d^i x_e^j + {i g
\over 2} \epsilon_{abc}\psi_a^{\dagger}\Gamma^k\psi_b x_c^k,
\label{eq:Hamiltonian} \nonumber \\
i=1,2,3&& a=1,2,3. \nonumber
\end{eqnarray}
The spectrum is obtained numerically by diagonalizing  $H$ in the gauge
 invariant eigen-basis of the occupation numbers of all degrees of freedom
 $n_a^i={a^{\dagger}}_a^i a_a^i, \eta_a^i={f^{\dagger}}_a^i f_a^i$
 \cite{JW1}
\begin{eqnarray}
|\{n_a^i,\eta_c^j\}>=\sum_{contractions}{a^{\dagger}}^i_c {{a^{\dagger}}}^j_d {a^{\dagger}}^k_e {f^{\dagger}}_b^m {f^{\dagger}}_a^n \dots |0>. \nonumber
\end{eqnarray}
The Hilbert space was cut by restricting the gauge invariant total number
 of bosonic quanta
\begin{eqnarray}
B=\sum_{b,i} {a^{\dagger}}_b^i a_b^i < B_{max}. \nonumber
\end{eqnarray}

\begin{figure}[ht]
\centering
 \psfig{figure=fig3D.eps,height=11cm}
 \caption{ The spectrum, and its supersymmetry structure, of the
D=4 supersymmetric Yang-Mills quantum mechanics.
\label{fig:3D}}
\end{figure}
It turned out that the physical (i.e. convergent with the cutoff
eigenenergies) could be computed for sizes of bases well within a
reach of a reasonably fast PC.
   The spectrum obtained in this way is shown in Fig.~\ref{fig:3D} \cite{CW2}.
It reveals a series of dynamical supermultiplets with degenerate
eigenenergies. The eigenstates within a supermultiplet are indeed
the supersymmetric images of one another. These states are localized
and they form a discrete spectrum. In addition there is also a
continuum. It occupies  central sectors of the figure, i.e. ones
with the conserved fermion number $F=2,3,4$. Supermultiplets of
non-localized states occur at every second value of the angular
momentum. Moreover, in these sectors the discrete and continuous
spectra coexists at the same energies. This unusual feature of gauge
interactions has never been observed so directly before. There are
two supersymmetric vacua with fermion numbers $F=2$ and $4$. This is
nothing but the zero-volume manifestation of the existence of the
$\lambda\lambda$ condensate in the space extended theories
\cite{JW2}. The condensate assumes two different (in fact opposite
for SU(2)) values, again in agreement with the non-trivial
predictions of the unreduced theory \cite{VY,NSVZ} .

\section{The large N limit and the planar calculus}
Above approach, even though quite successful, is naturally limited to the
low number of colours. For higher N the Hilbert spaces become too large.
Fortunately {\em at infinite N} Fock bases simplify enormously allowing
again to reach quantitative results \cite{VW1}-\cite{VW3}. This simplification is usually phrased
in terms of the Feynman diagrams and topological expansion \cite{tH}. However it can be
also formulated in the way suitable for the Hamiltonian formalism \cite{VW1,Adriano}.
 To this end
introduce matrix creation and annihilation (c/a) operators.
\begin{eqnarray}
a_{ik}=\sqrt{2} a^a T^a_{ik},\;\;\;f_{ik}= \sqrt{2} f^a
T^a_{ik},\;\;\; i,k=1,...,N. \nonumber \end{eqnarray}
The physical basis can be
conveniently generated by acting with the gauge invariant building
blocks (bricks)
\begin{eqnarray} ({a^{\dagger}} {a^{\dagger}}),({a^{\dagger}}\ad{a^{\dagger}}),({a^{\dagger}}\ad{a^{\dagger}}\ad),\dots,
({a^{\dagger}}^{N}),\;\;\;(.)\equiv Tr[.] \nonumber \end{eqnarray}
and their products, on the
empty Fock state $|0\rangle$. Similarly for the fermionic sectors one
employs bricks with mixed fermionic and bosonic operators. It turns
out that in the large N limit only the single trace operators are
relevant. All products of traces are either non-leading or do not
provide new information (see \cite{PMO} for recent
results on that point). This observation culminates in the set
of simple rules to calculate explicitly matrix elements of various
Hamiltonians. One basically applies the Wick theorem employing the
commutation rules \begin{eqnarray} [a_{ik},{a^{\dagger}}_{jl}]=\delta_{il}\delta_{kj}\nonumber
\end{eqnarray} and identifying the leading contributions. Details of such
"planar calculus" have been presented in \cite{VW1,Adriano,VW2}.
Here we summarize only two examples

A normalized state with n gluons in the F=0 sector reads
\begin{eqnarray}
|n>= \frac{1}{{\cal N}_n}Tr[({a^{\dagger}})^n]|0>.  \label{state}
\end{eqnarray}
The normalization factor
\begin{eqnarray}
{\cal N}_n^2&=&<0|Tr[a^n] Tr[({a^{\dagger}})^n]|0> \nonumber \\
&=&<0|(12)(23)...(n1)[1'2'][2'3']\dots[n'1']|0>,  \label{norm}\nonumber \\
(12)\equiv a_{i_1i_2},&\;\;\;&[12]\equiv {a^{\dagger}}_{i_1i_2}.\nonumber
\end{eqnarray}
receives the maximal contribution only when the adjacent creation and annihilation
operators are contracted. This gives
\begin{eqnarray}
{\cal N}_n^2=n N^{n}.\nonumber
\end{eqnarray}
Similarly a matrix element of a typical term in a generic Hamiltonian
\begin{eqnarray}
 H_{n+2,n}=g^2 <n+2|Tr[{a^{\dagger}}\ad{a^{\dagger}} a]|n>  \label{hmn}\nonumber
\end{eqnarray}
can be explicitly calculated
\begin{eqnarray}
H_{n+2,n} =  g^2 N \sqrt{n(n+2)},&& \nonumber
\end{eqnarray}
and shown to depend only on the 't Hooft coupling $\lambda=g^2 N$.

   The rest of this lecture will be devoted to a rather simple system
which turns out to be surprisingly rich. The model may be considered
as a distant cousin of the $D=1+1$ supersymmetric
Yang-Mills theory reduced to the one point in space.

\section{A simple supersymmetric system }
Consider  one fermion and one boson with the following
supersymmerty generators and the Hamiltonian \cite{VW1}.
\begin{eqnarray}
Q= \sqrt{2} Tr [f {a^{\dagger}}(1+g{a^{\dagger}})],&&
 {Q^{\dagger}} = \sqrt{2} Tr [{f^{\dagger}} (1+g a) a],\nonumber\\
H=&\{Q,{Q^{\dagger}}\}&=H_B+H_F.\label{ham}
\end{eqnarray}
Explicitly
\begin{eqnarray}
H_B&=&{a^{\dagger}} a + g({a^{\dagger}}^2 a + {a^{\dagger}} a^2) + g^2 {a^{\dagger}}^2 a^2,\nonumber \\
H_F&=& {f^{\dagger}} f + g ( {f^{\dagger}} f ({a^{\dagger}}+a) + {f^{\dagger}} ({a^{\dagger}}+a) f) \nonumber \\
& + & g^2 ( {f^{\dagger}} a f {a^{\dagger}} + {f^{\dagger}} a {a^{\dagger}} f + {f^{\dagger}} f {a^{\dagger}} a + {f^{\dagger}} {a^{\dagger}} f a).\nonumber
\end{eqnarray}
This Hamiltonian conserves the gauge invariant fermion number
$F=Tr[{f^{\dagger}} f]$ and can be diagonalized in sectors with well defined $F$.
For any finite N calculation of matrix elements of H quickly becomes cumbersome.
At infinite N, however, the planar rules illustrated above give simple and
compact expressions.
\subsection{Hamiltonian matrix at large N }
The gauge invariant basis in the $F=0$ sector can be chosen as (\ref{state}) with
 $n=0,1,2,3, ...$. Only $H_B$ contributes in this case. The matrix elements
in the planar approximation read
\begin{eqnarray}
<0,n|H|0,n>&=&(1+\lambda(1-\delta_{n1}))n, \nonumber \\
<0,n+1|H|0,n>=&<0,n|H|0,n+1>=&\sqrt{\lambda}\sqrt{n(n+1)}. \label{hmf1}
\end{eqnarray}
In the sector with one fermion the basis is
\begin{eqnarray}
|n>= \frac{1}{{\cal N}_n}Tr[({f^{\dagger}}{a^{\dagger}})^n]|0>,\;\;\;n=0,1,2,\dots . \nonumber
\end{eqnarray}
Now, and for higher $F$,
both $H_B$ and $H_F$ contribute. The Hamiltonian matrix is again simple
\begin{eqnarray}
<1,n|H|1,n>&=&(1 + \lambda)(n+1) +  \lambda,\nonumber \\
<1,n+1|H|1,n>=&<1,n|H_2|1,n+1>=&\sqrt{\lambda}(2+n). \label{hmf2}
\end{eqnarray}
The system reveals many interesting features already in these two lower sectors
\cite{VW1},
therefore we postpone the discussion of higher F's and turn to the physics of
ensembles with at most one fermion.

\subsection{Numerical results}

The Hamiltonian matrix is sparse but infinite with rows and columns labeled by
the gauge invariant number of bosons $B=Tr[{a^{\dagger}} a]$. To obtain,
the spectrum numerically we introduce the cutoff, $B_{max}$, limiting
the number of
bosonic quanta
 \begin{eqnarray}
  B < B_{max}, \nonumber
 \end{eqnarray}
and increase the cutoff until the spectrum converges. Results are shown in
Fig.\ref{fig:h2} for few values of the 't Hooft coupling. The convergence
with $B_{max}$ is satisfactory for $\lambda\ne 1$ and is faster for lower
eigenvalues. There, the limiting, "infinite volume", results can be easily
recovered. The less trivial way so see this is to check for the supersymmetry
which is broken by the cutoff and should be recovered only at infinite $B_{max}$.
Figure \ref{fig:susyf01} shows first few energy levels in four lower fermionic
sectors. The degeneracies between bosonic and fermionic partners of
the $F=(0,1)$ supermultiplets
are excellent. Moreover, there is also an unbalanced supersymmetric vacuum state
with zero eigenenergy \footnote{An apparent lack of the degeneracy between some
states with higher $F$ will be explained later.}. These results provide also the
non-trivial test of our planar rules: the planar approximation does not break
supersymmetry.

The slowing of the cutoff dependence around $\lambda=1$ is a characteristic
signature of a phase transition. At this point the system looses the energy gap
and the spectrum becomes continuous. Any finite number of low-lying levels
collapses to zero at infinite cutoff, but the cutoff dependence becomes
characteristic of that for the continuous spectrum.

\begin{figure}[t]
\begin{center}
\epsfig{width=14cm,file=newh2.eps}
\end{center}
\vskip-4mm
\caption{ The cutoff dependence of the spectra of $H$ in the F=0 sector, and
for a range of $\lambda$'s}
\label{fig:h2}
\end{figure}

Another interesting feature of this phase transition is the rearrangement
of members of supermultiplets. This is shown in Fig.\ref{fig:rearr} where
the $\lambda$ dependence of the first four eigenenergies from both sectors
is displayed for few values of $B_{max}$.
Away from the critical region supersymmetry is quickly restored and the F=0
and F=1 levels are undistinguishable even for low cutoffs. Close to the criticality
 partners do not have the same energies since SUSY is broken at finite
cutoff. Interestingly however, they {\em rearrange} while the 't Hooft coupling
passes its critical value. In particular, the {\em new} vacuum state appears
when $\lambda$ moves from the low to the large coupling phases. This also
implies (and was readily found) that the Witten index
(restricted to the $F=0,1$ sectors) jumps by one unit across the
phase transition point.

\begin{figure}[tbp]
\psfrag{yyy}{${\scriptstyle E}$ }\psfrag{ttt}{${\scriptstyle\lambda=2.0}$}
\psfrag{xx1}{${\scriptstyle F=0\hspace*{2.5cm}F=1}$}\psfrag{tt1}{${\scriptstyle\lambda=0.5}$}
\psfrag{fff}{${\scriptstyle F=2\hspace*{2.5cm}F=3}$}
\epsfig{width=8cm,file=levels01.eps}
\epsfig{width=8cm,file=levels23.eps}
\vskip-4mm \caption{Low lying bosonic and fermionic levels in the
first four fermionic sectors.} \label{fig:susyf01}
\end{figure}

Finally, the system has the exact strong-weak duality
\begin{eqnarray}
b \left(E^{(F=0)}_n(1/b) -\frac{1}{b^2}\right)  &=&
\frac{1}{b} \left( E^{(F=0)}_{n+1}(b) -b^2 \right), \nonumber \\
b \left(E^{(F=1)}_n(1/b) -\frac{1}{b^2}\right)   &=&
 \frac{1}{b} \left( E^{(F=1)}_n(b)   -b^2 \right) . \nonumber
\end{eqnarray}
It follows directly from the matrix representation (\ref{hmf2}) for $F=1$ .
However it is not obvious in the $F=0$ sector, cf. Eq. (\ref{hmf1}),
but it is a direct consequence of supersymmetry.

\subsection{Analytic solution}
All above results have been subsequently derived analytically \cite{VW1}. For example
the second vacuum state has the form $b \equiv \sqrt{\lambda}$
\begin{eqnarray}
|0\rangle_2 = \sum_{n=1}^{\infty} \left(\frac{-1}{b}\right)^{n} \frac{1}{\sqrt{n} }|0,n\rangle \ .
\label{vac2}
\end{eqnarray}
It is indeed annihilated by ${Q^{\dagger}}$,
and exists only for $ b > 1 $.

Surprisingly, one can find analytically the complete spectrum and construct
all eigenstates in the $F=0$ sector. To this end it is convenient to
introduce another, not orthonormal basis,
\begin{eqnarray}
  |B_n\rangle &=&\sqrt{n}|n\rangle+b\sqrt{n+1}|n+1\rangle \, . \nonumber
\end{eqnarray}

\begin{figure}[t]
\begin{center}
\epsfig{width=8cm,file=rearrange.eps}
\end{center}
\vskip-4mm
\caption{Lowest bosonic and fermionic levels, as functions of $\lambda$, for different cutoffs}
\label{fig:rearr}
\end{figure}

This basis is convenient since the action of the Hamiltonian on $|B_n\rangle$
is so simple that the generating function for the expansion of the eigenstates
in that basis
\begin{eqnarray}
f(x)=\sum_{n=0}^{\infty} c_n x^n \, \;\;\;\;\;\;\leftrightarrow\;\;\;\;\;\;
|\psi\rangle = \sum_{n=0}^{\infty} c_n |B_n\rangle \nonumber
 \label{spsi}
\end{eqnarray}
can be constructed. The solution reads
\begin{eqnarray}
f(x)&=& \frac{1}{\alpha}\;\frac{1}{x+1/b}\;  F(1,\alpha;1+\alpha;\frac{x+b}{x+1/b}),\;\;\; b < 1, \nonumber\\
f(x)&=& \frac{1}{1-\alpha}\;\frac{1}{x+b}\; F(1,1-\alpha;2-\alpha;\frac{x+1/b}{x+b}),\;\;\; b > 1, \nonumber\\
E &=& \alpha(b^2-1), \nonumber
\end{eqnarray}
and together with the quantization condition
\begin{eqnarray}
f(0)=0,
\end{eqnarray}
reproduces numerical eigenvalues obtained earlier in the infinite cutoff limit.

 As a one check, set $\alpha=0$ in the generating function, for $b > 1$, to obtain
\begin{eqnarray}
f_0(x)=\frac{1}{1+b x} \log\frac{b+x}{b-1/b}, \;\;\;  b>1,
\end{eqnarray}
which indeed reproduces the second ground state (\ref{vac2}).

\section{Higher fermionic sectors: F=2,3}

Single trace states with two fermions
\begin{eqnarray}
|n_1,n_2\rangle & = & \frac{1}{{\cal N}_{n_1 n_2}}
Tr[{a^{\dagger}}^{n_1}{f^{\dagger}}{a^{\dagger}}^{n_2}{f^{\dagger}} ]|0\rangle,\nonumber
\end{eqnarray}
are labeled by two integers whose ordering
is important modulo a cyclic permutation. Pauli principle eliminates
states with $n_1=n_2$, hence we can always take $n_2 < n_1$. The planar rules
give for the Hamiltonian matrix in this sector \cite{VW2}
\begin{eqnarray}
\langle n_1,n_2|H|n_1,n_2\rangle& = &(n_1+n_2+2)(1+b^2) -
b^2 (2 - \delta_{n_1,0}) - 2 b^2 \delta_{n_2,n_1+1},\nonumber \\
\langle n_1+1,n_2|H|n_1,n_2\rangle = &b (n_1+2)& = \langle n_1,n_2|H|n_1+1,n_2\rangle,\nonumber\\
\langle n_1,n_2+1|H|n_1,n_2\rangle = &b (n_2+2)& = \langle n_1,n_2|H|n_1,n_2+1\rangle.\nonumber\\
\langle n_1+1,n_2-1|H|n_1,n_2\rangle  &=& \langle n_1,n_2|H|n_1+1,n_2-1\rangle
\nonumber\\&=& 2 b^2 (1- \delta_{n_2,n_1+1}).\nonumber
\end{eqnarray}
 Planar  states in the three fermion sector are labeled by three integers
modulo cyclic translations
\begin{eqnarray}
|n_1,n_2,n_3\rangle & = & \frac{1}{{\cal N}_{n_1 n_2 n_3}}Tr[{a^{\dagger}}^{n_1}{f^{\dagger}}{a^{\dagger}}^{n_2}{f^{\dagger}} {a^{\dagger}}^{n_3}{f^{\dagger}} ]|0\rangle,\nonumber\\
 0\leq n_1 ,\;\;\;\;&& n_1 \leq n_2,\;\;\;\;\ n_1 \leq n_3.  \nonumber
\end{eqnarray}
Again the Hamiltonian matrix was explicitly calculated
\begin{eqnarray}
\langle n_1,n_2, n_3|H|n_1,n_2,n_3\rangle = (n_1+n_2+n_3 +3)(1+b^2) \nonumber \\
- b^2 (3 - \delta_{n_1,0} -
\delta_{n_2,0}- \delta_{n_3,0})\nonumber
\label{diag3} ,
\end{eqnarray}
\begin{eqnarray}
\langle n_1+1,n_2,n_3|H|n_1,n_2,n_3\rangle &=b (n_1+2) \Delta=& \langle n_1,n_2,n_3|H|n_1+1,n_2,n_3\rangle
,   \nonumber\\
{\rm plus ~ cyclic} \nonumber
\label{off31}
\end{eqnarray}
\begin{eqnarray}
\langle n_1+1,n_2-1,n_3|H|n_1,n_2, n_3\rangle  &= b^2 \Delta=& \langle n_1,n_2,n_3|H|n_1+1,n_2-1, n_3\rangle ,
\nonumber\\{\rm plus ~ cyclic} \nonumber
\label{off32}.
\end{eqnarray}
where $\Delta=1/\sqrt{3}$ if $n_1=n_2=n_3$, and $\Delta=\sqrt{3}$ if the final state is of this form,
otherwise $\Delta=1$.

    Numerical computation of the spectrum proceeds as before.
There is again a phase transition causing the critical
slowing down around $\lambda_c=1$. Away from it, the eigenenergies converge
satisfactorily, with the infinite volume results shown in Fig.\ref{fig:susyf01}.
   The spectrum in higher fermionic sectors is essentially different from
the $F=0,1$ case. We again find dynamical $F=(2,3)$ supermultiplets, however
not all states with three fermions have their superpartners in the $F=2$ sector.
Instead, they are degenerate with ones from the $F=4$ sector. This pattern continues
now for all F as follows from counting the number of states: it is an increasing
function of F. Another novel feature is that the spectrum is rather irregular,
while the eigenenergies are almost equidistant for $F=0,1$.
    There is again the rearrangement of members of supermultiplets across
the phase transition. Interestingly however, the {\em two} new SUSY vacua
with $F=2$ appear
in the strong coupling phase, while there is none at weak coupling.

    Similarly to the $F=0$ case, the two vacua can be constructed analytically.
To this end consider the "extreme" strong coupling limit of the Hamiltonian
(\ref{ham})\cite{VW2}
\begin{eqnarray}
 H_{SC}=\lim_{\lambda\rightarrow\infty} \frac{1}{\lambda} H  =  Tr ({f^{\dagger}} f) +
 \frac{1}{N}[Tr({a^{\dagger}}^2 a^2) + Tr({a^{\dagger}} {f^{\dagger}} a f) + Tr({f^{\dagger}}{a^{\dagger}} f a)].
 \label{hstr}
\end{eqnarray}
Surprisingly, $H_{SC}$ conserves the number of bosonic quanta as well, and proves
very useful in mapping the structure of the model in {\em all} fermionic sectors
(cf. the following Section). Coming back to $F=2$, one can identify the
finite dimensional
$(F=2,B)$ sectors with zero eigenvalues, and construct the corresponding eigenstates.
They read
\begin{eqnarray}
|F=2,v\rangle_1^{\infty} &=& |0,1\rangle, \\
|F=2,v\rangle_2^{\infty} &=& |0,3\rangle - 2|1,2\rangle,
\end{eqnarray}
the superscript referring to the infinite value of the 't Hooft coupling.
These states can then be used to construct the two vacua at finite $\lambda > 1$
\begin{eqnarray}
|F=2,v\rangle_1^{\lambda} &=& \sum_{n=1}^{\infty} (-\sqrt{\lambda})^{-n}
|0,n\rangle ,\ \nonumber \\
|F=2,v\rangle_2^{\lambda} &=& (1+Q_s H_s^{-1} {Q^{\dagger}}_w)^{-1}|F=2,v\rangle_2^{\infty},
\end{eqnarray}
with $Q_s=g Tr[f {a^{\dagger}}^2],\; H_s=\{Q_s,{Q^{\dagger}}_s\},\; {Q^{\dagger}}_w = Tr[{f^{\dagger}} a]$ \cite{VW2}.

The restoration of supersymmetry can also bee seen on the level of eigenstates.
Members of supermultiplets transform among themselves by SUSY charges
$Q$ and ${Q^{\dagger}}$, which become the well defined matrices in the planar bases. To see
SUSY in this way define the following supersymmetry fractions
\begin{eqnarray}
q_{m n}\equiv \sqrt{\frac{2}{E_m+E_n}} < F+1,E_m|Q^{\dagger}|F,E_n > ,
\end{eqnarray}
which are the coordinates of the supersymmetric images of eigenstates
in the corresponding fermionic sectors. Fig.\ref{fig:sfrac} shows that
SUSY fractions indeed quickly stabilize with incrasing $B_{max}$.

\begin{figure}[tbp]
\psfrag{yyy}{$q_n+n-1$}\psfrag{xxx}{$B_{max}$}
\begin{center}
\epsfig{width=8cm,file=sfrac.eps}
\end{center}
\vskip-4mm \caption{First five supersymmetry fractions.} \label{fig:sfrac}
\end{figure}

Supersymmtery fractions are also useful in defining the restricted Witten index
which smoothly interpolates, at finite cutoff, between the two phases. The straightforward
restriction of the sum
\begin{eqnarray}
W(T,\lambda)= \sum_i (-1)^{F_i} e^{-T E_i} , \nonumber
\end{eqnarray}
to the $F=2,3$ eigenstates does not suffice since some of the states with three
fermions remain unbalanced. Instead, one can define
\begin{eqnarray}
W_R(T,\lambda)= \sum_i  \left( e^{-T E_i} -  e^{-T \bar{E}_i} \right),\;\;\;
\bar{E}_i=\frac{\sum_f E_f |q_{fi}|^2}{\sum_f |q_{fi}|^2},  \nonumber
\end{eqnarray}
i.e. we take as the energy of the supersymmetric partner the energy weighted
by supersymmetric fractions. This definition enforces summation only over
complete supermultiplets away from the critical region, while provides
the smooth smearing among possible candidates around $\lambda_c$.
The index defined this way changes smoothly at finite cutoff, and varies
 by two units
as expected (cf. Fig.\ref{fig:wi}) \cite{VW2}.

\begin{figure}[tbp]
\psfrag{yyy}{${\scriptstyle I_W(6)}$}\psfrag{xxx}{${\scriptstyle \sqrt{\lambda}}$}
\begin{center}
\epsfig{width=8cm,file=wi.eps}
\end{center}
\vskip-4mm \caption{Behaviour of the restricted Witten index,
at $T=6$, around the phase transition.} \label{fig:wi}
\end{figure}

\section{Arbitrary F}

A simple generalization of the $F=2,3$ cases for arbitrary number of fermions
reads \cite{OVW1}
\begin{eqnarray}
|n\rangle=|n_1,n_2,\dots,n_F\rangle =
\frac{1}{{\cal N}_{\{n\}}}
Tr({a^{\dagger}}^{n_1}{f^{\dagger}} {a^{\dagger}}^{n_2}{f^{\dagger}}\dots{a^{\dagger}}^{n_F}{f^{\dagger}})|0\rangle   \label{anyF}
\end{eqnarray}
So the states of the basis may be labeled by $F$ bosonic occupation numbers
$\{n_1,n_2,\dots n_F\}$ (configurations) modulo cyclic shifts. Some states
are excluded by the Pauli principle. For example
\begin{eqnarray}
\{3,3\},\;\;\;\;\;{\rm or}\;\;\;\;\;\{2,1,1,1,2,1,1,1\}  \nonumber
\end{eqnarray}
are not allowed, since they change sign while retaining their identity after
the suitable number of cyclic shifts. When calculating normalizations and
matrix elements one should keep track of symmetry factors
defined as the number of cyclic shifts which bring a state back to itself.
For example the state $|2,1,2,1,2,1\rangle$ has the symmetry factor $d=3$.
Alternatively one can label planar states with the periodic binary strings
(necklaces) with, e.g., zeros/ones corresponding to the bosonic/fermionic
creation operators (for a more complete discussion see \cite{OVW1}).

    Detailed calculation of spectra for arbitrary fermion number
is now in progress. However many properties, e.g. the supersymmetry structure,
has been inferred from the strong coupling limit of the model.

As already mentioned, the strong coupling Hamltonian (\ref{hstr}) conserves
both fermionic and bosonic numbers of quanta. Hence the Hilbert space
splits into the finite dimensional sectors, cf. Table \ref{bastab},
and the Hamiltonian in each sector becomes a finite matrix. Still the
$H_{SC}$ has the exact supersymmetry with the strong-coupling charges
\begin{eqnarray}
Q_{SC}=\frac{1}{\sqrt{N}} Tr[f{a^{\dagger}}^2],\;\;\;{Q^{\dagger}}_{SC}=\frac{1}{\sqrt{N}} Tr[f{a^{\dagger}}^2],
\end{eqnarray}
acting along the "diagonals" $2F+B=const.$ of Table \ref{bastab}.
Diagonalizing few of these matrices we have indeed found expected
degeneracies\footnote{Obviously no cutoff was required this time.}. Interestingly,
we have also found experimentally \cite{OVW1,VW3} that the zero energy eigenstates
are located {\em only} in the sectors with
\begin{eqnarray}
B=F \pm 1,\;\;\;F\; even.  \label{magic}
\end{eqnarray}
which form the regular "magic" staircase, shown in the boldface, in Table \ref{bastab}.

\begin{table}[tbp]
\begin{center}

\begin{tabular}{||c|ccccccccccc}
$11\;\;\;$ & 1  &  1 & 6 &  26  & 91 &  273 & 728 & 1768 & 3978 & 8398  & {\bf 16796} \\
$10\;\;\;$ & 1  &  1 & 5 &  22  & 73 &  201 & 497 & 1144 & 2438 & 4862  & 9226 \\
$9\;\;\;$ & 1  &  1 & 5 &  19  & 55 &  143 & 335 & 715 & {\bf 1430} & 2704  & {\bf 4862} \\
$8\;\;\;$ & 1  &  1 & 4 &  15  & 42 &  99 & 212 & 429 & 809 & 1430  & 2424 \\
$7\;\;\;$ & 1  &  1 & 4 &  12  & 30 &  66 & {\bf 132} & 247 & {\bf 429} & 715  & 1144 \\
$6\;\;\;$ & 1  &  1 & 3 &  10  & 22 &  42 & 76 & 132 & 217 & 335  & 497 \\
$5\;\;\;$ & 1  &  1 & 3 &  7  & {\bf 14} &  26 & {\bf 42} & 66 & 99 & 143  & 201 \\
$4\;\;\;$ & 1  &  1 & 2 &  5  & 9 &  14 & 20 & 30 & 43 & 55  & 70 \\
$3\;\;\;$ & 1  &  1 & {\bf 2} &  4  & {\bf 5} &  7 & 10 & 12 & 15 & 19  & 22 \\
$2\;\;\;$ & 1  &  1 & 1 &  2  & 3 &  3 & 3 & 4 & 5 & 5  & 5 \\
$1\;\;\;$ & {\bf 1}  &  1 & {\bf 1} &  1  & 1 &  1 & 1 & 1 & 1 & 1  & 1 \\
$0\;\;\;$ & 1  &  1 & 0 &  1  & 0 &  1 & 0 & 1 & 0 & 1  & 0 \\
\hline
$B\;\;\;$        &  $  $ & $  $ & $  $
  & $  $ &  $  $ & $  $ & $  $  & $  $ &  $  $ & $  $ & $  $ \\

  $\;\;\;F$        &  $ 0 $ & $ 1 $ & $ 2 $
  & $ 3 $ &  $ 4 $ & $ 5 $ & $ 6 $  & $ 7 $ &  $ 8 $ & $ 9 $ & $ 10 $ \\

   \hline\hline
\end{tabular}
}
\end{center}
\caption
Sizes of gauge invariant bases in the $(F,B)$ sectors. }
\label{bastab}
\end{table}

This observation, when combined with the weak coupling (harmonic oscillator)
limit, explains the structure of SUSY vacua for all $F$
and at any value of the 't Hooft coupling. Namely, for any even $F$ there
are two supersymmetric vacua for any $\lambda$ in the strong coupling phase.
On the other hand, there is only one SUSY vacuum in the weak coupling
 phase and it has $F=0$.

The question {\em why} the supersymmetric vacua are located
in the magic sectors (\ref{magic}) will be answered in the next Section.

\section{Two equivalencies with statistical systems}
Interestingly, our strong-coupling model is equivalent to the two
well known and nontrivial statistical systems \cite{VW3}.
\subsection{A gas of q-bosons}
Consider a one dimensional, periodic lattice with size $F$.
At each lattice site put a bosonic degree of freedom described by its
c/a operators ${a^{\dagger}}_i,a_i$.
The strong coupling Hamiltonian (\ref{hstr}) is equivalent to the following
one expressed solely in terms of bosonic variables
\begin{eqnarray}
H=B + \sum_{i=1}^F \delta_{N_i,0} + \sum_{i=1}^F b_i {b^{\dagger}}_{i+1} + b_i {b^{\dagger}}_{i-1},
\label{hlat}
\end{eqnarray}
where $N_i={a^{\dagger}}_i a_i$,  $B=N_1+N_2+\dots +N_F$, and the action of $b_i, {b^{\dagger}}_i$
is
\begin{eqnarray}
{b^{\dagger}}|n\rangle = |n+1\rangle,\;\;\; & b|n\rangle = |n-1\rangle,&\;\;\; b|0\rangle\equiv 0, \label{qbos}
\end{eqnarray}
that is, they create and annihilate one quantum {\em without} the usual
$\sqrt{n}$ factors. They satisfy the following commutation rules
\begin{eqnarray}
 [b,{b^{\dagger}}]=\delta_{N,0}
\end{eqnarray}
with the same non-linear (in b's) $\delta$ operator as in the Hamiltonian.
Using the planar rules discussed earlier, one can show that the action
of the first two terms of (\ref{hstr}) on the planar basis is exactly the same
as the action of the first two terms of (\ref{hlat}) in the eigenbasis of $N_i$ \cite{VW3}.
The last two terms of (\ref{hstr}) are the same as the hopping terms of
(\ref{hlat}).  Since the planar states acquire a phase $(-1)^{F-1}$ upon
 cyclic shifts, the planar system has the same spectrum as (\ref{hlat})
 in the eigen-sector of the lattice shifts, $U$, with $\lambda_U=(-1)^{F-1}$.
This equivalence was cross-checked numerically for $3\le F, B \le 7$.

The c/a operators are known in the literature \cite{QB1,QB2}. Their algebra is the special case
of the q-deformed harmonic oscillator algebra
\begin{eqnarray}
 [b,{b^{\dagger}}]=q^{-2N}
\end{eqnarray}
with $q\rightarrow\infty$. Transitions (\ref{qbos}) without the square roots
are often referred to as {\em assisted} (i.e. independent of the occupancy) transitions.
The system of q-bosons described by the Hamiltonian (\ref{hlat}) is,
 to our knowledge, regarded as non-soluble. In view of the second equivalence,
 discussed below, it turns out to be in fact soluble.

\subsection{The XXZ Heisegberg chain}
The second equivalence has been proved employing another representation
of the planar states (\ref{anyF}) \cite{VW3}
\begin{eqnarray}
 \frac{1}{{\cal N}_n}|
Tr[{a^{\dagger}}^{m_1}({f^{\dagger}})^{n_1}{a^{\dagger}}^{m_2}({f^{\dagger}})^{n_2} \dots ({f^{\dagger}})^{n_k}]|0\rangle \equiv
\left| (0)^{m_1}(1)^{n_1}\dots (0)^{m_r}(1)^{n_r} \right\rangle  \,
r\ge 1,\;m_i, n_i >0) \, ,
\label{plstates}
\end{eqnarray}
Where the states are labeled by the binary strings with, e.g. 0/1 corresponding
to
the bosonic/fermionic creation operators. In this basis, the action of
the strong coupling Hamiltonian (\ref{hstr}) is equivalent to that of the XXZ
chain
\begin{eqnarray}
H_{\rm XXZ}^{(\Delta)} &=& -\frac12 \sum_{i=1}^L \left( \sigma_i^x \sigma_{i+1}^x
+ \sigma_i^y \sigma_{i+1}^y + \Delta~    \sigma_i^z \sigma_{i+1}^z    \right) \nonumber
\end{eqnarray}
for the particular values of the anisotropy parameter $\Delta$.
The detailed correspondence reads
\begin{eqnarray}
H_{{\rm SC}}(F,B)=\left\{\begin{array}{ccc}
             -H_{\rm XXZ}^{(+ 1/2)}+\frac{3}{4}L \ ,\   & F ~{\rm  odd}, &\\
             +H_{\rm XXZ}^{(- 1/2)}+\frac{3}{4}L \ , \  & F ~{\rm  even}, B~ {\rm odd} \, ,
            \end{array}
            \right.              \label{xxzeq}
\end{eqnarray}
where the spin Hamiltonians are restricted to the translationally invariant
sectors with the lattice size and the total spin fixed by $(F,B)$:
\begin{eqnarray}
L=F+B,\;\;\;\;S^z=\sum_{i=1}^L s_i^z=\frac{1}{2}(F-B),\;\;\;\label{xxzpar}
\end{eqnarray}
Remarkably, this equivalence implies the existence
of the magic staircase of the supersymmetric vacua discussed above.
More than thirty years ago Baxter has found that, for $\Delta=-1/2$,
the ground states with $S_z=\pm 1/2$ have particularly simple eigenenergy
$E_0=-\frac{3}{4}L$ for infinite L \cite{Ba1}. Recently his findings
have been extended by Riazumov and Stroganov to any finite, odd L \cite{RS}.
 In view of (\ref{xxzeq},\ref{xxzpar})
the Riazumov-Stroganov states are nothing but our strong coupling vacua \cite{VW3}.
 Moreover, supersymmetry
requires a host of degeneracies among the, seemingly unrelated,
excitations of the XXZ chain. Amusingly, since the supersymmetry
transformations change the lattice size, these excitations live on
different lattices! \footnote{Supersymmetry of the, $\Delta=-1/2$,
XXZ chain and other statistical systems, has been discussed in the
literature, albeit in slightly different contexts (see, e.g.
\cite{FNS} and references therein). We thank Jan de Gier for
bringing this to our attention. }

Finally, since the XXZ model is exactly soluble (e.g. by the Bethe Ansatz \cite{Fad}),
our supersymmetric system is also soluble at infinite coupling.
As one application we give here the algebraic determination of the Bethe
phase factors for the first three steps of the magic staircase.
The eigenenergies of $H_{XXZ}(\Delta)$ are \cite{XXZ}
\begin{eqnarray}
E_{XXZ}(\Delta)=-L\frac{\Delta}{2} + 2 m \Delta -2 \sum_{j=1}^m \cos{p_j}, \label{baen}
\end{eqnarray}
where the momenta $ -\pi < p_j < \pi $ satisfy the following set of Bethe equations
\begin{eqnarray}
e^{i L p_j} = (-1)^{m-1} \prod_{l=1}^m e^{i(p_j-p_l)}
\frac{e^{i p_l}+e^{-i p_j} - 2\Delta}{e^{i p_j}+e^{-i p_l} - 2\Delta},\;\;\;
j=1,\dots,m. \label{beqs}
\end{eqnarray}
With $m$ denoting the number of down spins in a chain.
For the supersymmetric model $m=B$ and (\ref{baen}) translates into
\begin{eqnarray}
E_{SC}(F,B)=&F+2\sum_{j=1}^B \cos{p_j},& \;\;\;{\rm for}\; F\;\;{\rm odd}~,~
\Delta = + \frac12 ,\\
E_{SC}(F,B)=&F-2\sum_{j=1}^B \cos{p_j},& \;\;\;{\rm for}\; F\;\;{\rm even},
\;\;{\rm and}\;\;\ B\;\; {\rm odd}
~,~ \Delta = -  \frac12 .
\label{SBAeven}
\end{eqnarray}
Solving numerically Eqs.(\ref{beqs}) we have found that the supersymmetric
vacua occur at F even and B odd, and are always given by the yet simpler sub-ansatz of the Bethe Ansatz
\begin{eqnarray}
p_1=0,\;\;\; p_{2k+1}=-p_{2k}, \;\;\; k=1,...,(B-1)/2.    \label{an2}
\end{eqnarray}
which reduces the number of independent variables roughly by a factor of 2.
Still, the Bethe equations can only be solved numerically in general. However,
for the first three sectors the problem can be managed algebraically.
We shall discuss it separately for each sector.
\subsubsection{F=4, B=3}
This corresponds to $L=7$, $m=3$, and Eq.(\ref{an2}) implies
\begin{eqnarray}
p_1=0,\;\; p_2=p,\;\;p_3=-p,
\end{eqnarray}
 and (\ref{beqs}) reduces to ($z=e^{ip}$)
\begin{eqnarray}
z^6=1.
\end{eqnarray}
The admissible solution, which gives $E_{SC}=0$, is
\begin{eqnarray}
z=e^{i\frac{\pi}{3}}.
\end{eqnarray}
\subsubsection{F=4, B=5}
Now $L=9$, $m=5$, and with the aid of (\ref{an2}) Bethe equations reduce to
($x=e^{i p_2},\;\;y=e^{i p_4}$)
\begin{eqnarray}
x^8 & = & \frac{x y + x + y}{x+y+1}\;\frac{x y + x + 1}{x y + y + 1}, \nonumber \\
y^8 & = & \frac{x y + x + y}{x+y+1}\;\frac{x y + y + 1}{x y + x + 1}.  \label{beq45}
\end{eqnarray}
Solving (\ref{beq45}) is difficult in general, however one can easily find solutions with
vanishing energy (\ref{SBAeven}).
\begin{eqnarray}
E_{SC}(4,5)=2-2\left(x+y+\frac{1}{x}+\frac{1}{y}\right) = 0. \label{ben}
\end{eqnarray}
 Introducing two symmetric variables
\begin{eqnarray}
s=x+y,\;\;\;\;p=x y,
\end{eqnarray}
one obtains from (\ref{beq45}), (\ref{ben}) the following  two equations
for $s$ and $p$
\begin{eqnarray}
 \frac{s+p}{s+1}&=& \pm p^4, \\
s+\frac{s}{p}&=&1.
\end{eqnarray}
The admissible solution is on the negative branch and reads
\begin{eqnarray}
x=\frac{1}{64}\left(16+i \sqrt{2}\sqrt{15+\sqrt{33}}(7-\sqrt{33})
-4\sqrt{-16(3+\sqrt{33})-i 2\sqrt{2}\sqrt{15+\sqrt{33}}(9+\sqrt{33})}\right),\nonumber \\
y=\frac{1}{64}\left(16+i \sqrt{2}\sqrt{15+\sqrt{33}}(7-\sqrt{33})
+4\sqrt{-16(3+\sqrt{33})-i 2\sqrt{2}\sqrt{15+\sqrt{33}}(9+\sqrt{33})}\right). \nonumber \\  \label{xy45}
\end{eqnarray}
This pair indeed satisfies both Bethe equations, together with (\ref{ben}),
and therefore corresponds to our supersymmetric vacuum.
\subsubsection{F=6, B=5}
This sector corresponds to $L=11$, $m=5$.
Now the reduced Bethe equations are
\begin{eqnarray}
x^{10} & = & \frac{x y + x + y}{x+y+1}\;\frac{x y + x + 1}{x y + y + 1}, \nonumber \\
y^{10} & = & \frac{x y + x + y}{x+y+1}\;\frac{x y + y + 1}{x y + x + 1}.  \label{beq65}
\end{eqnarray}
As previously, we look for the solutions which satisfy $E_{SC}(6,5)=0$, that is
\begin{eqnarray}
x+y+\frac{1}{x}+\frac{1}{y} = 2,\;\;\;{\rm or}\;\;\;\frac{s}{p}=\frac{2}{1+p}.
\end{eqnarray}
The second equation follows from the product of the Bethe equations
\begin{eqnarray}
p^5=\pm \frac{p+s}{1+s},\;\;\;{\rm or}\;\;\; \frac{s}{p}=-\frac{1 \mp p^4}{1 \mp p^5}.
\end{eqnarray}
These equations reduce to the fourth order polynomial equation for p.
Again there is only one admissible solution and it lies on the negative branch
\begin{eqnarray}
x=\frac{1}{72}\left(36+i \sqrt{2}\sqrt{11+\sqrt{13}}(7+\sqrt{13})
-6\sqrt{2}\sqrt{6(-3+\sqrt{13})+i \sqrt{2}\sqrt{11+\sqrt{13}}(-5+\sqrt{13})}\right),\nonumber \\
y=\frac{1}{72}\left(36+i \sqrt{2}\sqrt{11+\sqrt{13}}(7+\sqrt{13})
+6\sqrt{2}\sqrt{6(-3+\sqrt{13})+i \sqrt{2}\sqrt{11+\sqrt{13}}(-5+\sqrt{13})}\right). \nonumber \\  \label{xy65}
\end{eqnarray}

As the last exercise let us show that these numbers are indeed unimodular which
is not obvious at first sight. However it is easy to calculate
\begin{eqnarray}
|x+y|^2=\frac{1}{3}(7+\sqrt{13}),\;\;|x-y|^2=\frac{1}{3}(5-\sqrt{13}),\;\;
{\rm therefore} \;\;|x|^2+|y|^2=2.  \label{unimod}
\end{eqnarray}
Further, since $x=a+b$ and $y=a-b$, one can observe that
\begin{eqnarray}
(a b^*)^2=\frac{1}{72}(-11+\sqrt{13}),\;\;\Rightarrow\;\; |x|^2=|y|^2 ,
\end{eqnarray}
which together with (\ref{unimod}) implies that $x$ and $y$ are indeed
pure phase factors. As a byproduct one obtains
\begin{eqnarray}
Re(x y^*)\equiv \cos{(p_2-p_4)}=\frac{1}{6}(1+\sqrt{13}).
\end{eqnarray}
\section{Discussion}
The direct diagonalization of a Hamiltonian matrix is usually
considered a viable tool, for finding a spectrum, only for finite
matrices. It turns out however, that the approach works in many
cases with the infinite dimensional Hilbert space as well. Although
such Hilbert spaces appear already in many quantum mechanical
systems with finite number of degrees of freedom, the true challenge
for the Hamiltonian formalizm is posed by the field theoretical
problems. An interesting family of systems results from the
dimensional reduction of field theoretical models to a one
point in space. As such, they again have the finite number of degrees
of freedom, however they inherit many advanced features of
parent field theories, for example their
 symmetries including
supersymmetry. The straightforward diagonalization proved quite successful in uncovering
quite rich and nontrivial spectra of supersymmetric Yang-Mills quantum mechanics in various dimensions.

This lecture reviews the recent study of a system with an infinite
number of degrees of freedom, namely a particular supersymmetric gauge quantum mechanics
(also referred to as a matrix model) with the infinite number of colours.
The model was conceived as the illustration of the planar calculus in the Fock
space. However it was subsequently found that the system has an interesting
physics which connects to many recently discussed issues. For example, the system
undergoes the discontinuous phase transition in the 't Hooft coupling which is
accompanied
by the remarkable rearrangement of dynamical supermultiplets. It enjoys the strong-weak
duality in the lowest (and simplest) fermionic sectors where the complete,
exact spectrum can be found analytically. The full structure of intervening
supermultiplets begins with two fermions and goes on ad infinitum.
In each bosonic  sector two new supersymmetric vacua appear in the strong
coupling phase while there is only one in the weak coupling region.
The behaviour in the strong coupling phase has been found by studying the system
at the infinite value of the 't Hooft coupling and then extending it to the whole
strong coupling regime.

At infinite coupling the model reveals also its connections
with the statistical physics, which proves, among other things, that
a quantum mechanics at infinite $N$ becomes a {\em bona fide} field theory.
The supersymmetric planar model  is exactly equivalent
to the one-dimensional quantum
XXZ Heisenberg chain, and at the same time, to the one-dimensional lattice
gas of q-bosons.
The strong coupling vacua found in the SUSY matrix
model turn out to be nothing but simple ground states of
the XXZ chain which were found more than thirty years ago.
In addition, the XXZ chain appears to have the hidden
supersymmetry which results in many degeneracies among various
energy eigenstates. An unusual feature of these supersymmetry transformations
is that they connect states propagating on different lattices.

Second equivalence is with a gas of q-bosons, with the infinite deformation
parameter. The latter mapping holds exactly in all fermionic and bosonic
sectors while the former does not work for even F and B.

The chain of
correspondences discovered in \cite{VW3} implies also that the specific,
nonlinear $\infty$-bosonic Hamiltonian is in fact soluble via solubility
of the XXZ chain.
{\em Vice versa}: the same solubility implies that the supersymmetric
matrix model is exactly soluble at the infinite value of the 't Hooft coupling.
It is also quite conceivable that this solubility holds in the
whole strong coupling phase \cite{VW2}.

Summarizing: planar calculus applied directly in the Fock space turned
out be rather promising tool in studying some simple, but
non-trivial, models. It allowed to find phenomena which bear a
tantalizing similarity to ones found in the much more advanced
systems  \cite{MZ}-\cite{Ro}. It remains to be seen if this approach
can be extended to the full, i.e. space extended field theories.

\section*{Acknowledgements}
Most of the results reviewed here have been obtained in collaboration with
Gabriele Veneziano.
I thank him for pointing out the practical
advantages of planar calculus in the Hilbert space, and for numerous,
enlightening and stimulating discussions on specific issues.
This work is partially supported by the grant of Polish Ministry of
Science and Education P03B 024 27 (2004)--(2007).

\vspace*{.5cm} A special session during this School is dedicated to
Andrzej Bia\l as in honour of his 70-th birthday. I would like to
thank him for "being there" and for the incredible passion and drive
for physics he is distributing among all of us who have a chance,
and a privilege, to interact with him.

redpajama_set_name: RedPajamaArXiv


text:
The Sex Habits of American Women
Cover graphic by Evern Designs
Julie Marie Myatt
Trade Edition$15.95
In the early 1950s, a psychologist sets out to write the definitive book on THE SEX HABITS OF AMERICAN WOMEN. Meanwhile, his wife is having an affair with his former student, his daughter can't find love, and all his theories refuse to go unchallenged. Juxtaposed to the action in the 1950s is a present-day videographer making a documentary on precisely the same subject.
Categories: The Plays Tags: Family, Non-naturalistic, 1950s
"Funny, gently satiric, at times almost poignant … Myatt, an up-and-coming young playwright, excels at acute, unexpected punch lines. She lightly blends in multimedia elements and deftly plants seeds that pay off in later scenes." —Robert Hurwitt, San Francisco Chronicle
"An intriguing serio-comedy about the stultifying sexual morality and gender roles in place around the time Alfred Kinsey released his roof-rattling academic studies … This play is a rare intelligent new one with direct appeal to older audiences, suggesting a possible shelf life amongst regional companies." —Dennis Harvey, Variety
"Julie Marie Myatt … teases and cajoles us with an intelligent, intriguing and funny play that manages to ask questions without knocking us over the head with them … Myatt's humor freely runs the spectrum from adolescent to coy to poignant, and Myatt strikes a nice balance in treating the material with a mix of matter-of-factness and awe." —Dominic Papatola, Saint Paul Pioneer Press
"An engaging new play built on a sturdy foundation of irony … Julie Marie Myatt is a playwright whose work bears noticing. Based on THE SEX HABITS OF AMERICAN WOMEN, her plays could prove to be habit forming." —Chad Jones, Oakland Tribune
Julie Marie Myatt's play SOMEDAY premiered as part of Cornerstone Theater Company's Justice Cycle in June 2008. Her play WELCOME HOME, JENNY SUTTER recently premiered at Oregon Shakespeare Festival and will tour to the Kennedy Center in July 2008. Her play BOATS ON A RIVER premiered at the Guthrie Theater in May 2007 and was a finalist for the Susan Smith Blackburn Prize. Her play MY WANDERING BOY premiered at South Coast Rep and was featured in the 2007 Summer Play Festival in New York City. Her ten-minute play, MR AND MRS premiered at the 2007 Humana Festival. Her play THE SEX HABITS OF AMERICAN WOMEN was produced by the Guthrie Theater, Signature Theater in Arlington, VA, Synchronicity Performance Group, and premiered at the Magic Theater in San Francisco, and was recently published by Broadway Play Publishing Inc. She received a Walt Disney Studios Screenwriting Fellowship, a Jerome Fellowship at the Playwrights' Center, and a McKnight Advancement Grant. Her other plays include ALICE IN THE BADLANDS, COWBIRD, 49 DAYS TO THE SUN, and THE PINK FACTOR (also published by BPPI). She is a member of New Dramatists.
If original stage producers credits appear in bold below, all licensees are required to include them in the following form on the title page in all programs distributed in connection with performances of the Play and in all advertising in which the full cast appears in size of type not less than ten percent (10%) of the size of the title of the Play:
Originally produced the Magic Theater, San Francisco
The Sex Habits of American Women is produced
by special arrangement with Broadway Play Publishing Inc, NYC
www.broadwayplaypub.com
Anthony Giardina
At an elite women's college in Massachusetts, two minor crises emerge during the fall semester: the philandering head of the English Department decides to callously end his affair with a colleague, and an aging professor is caught making unwanted sexual advances toward a student. But in the academic world of the early 21st century, there is no such thing as a minor crisis. For Jacob Freundlich, an English professor approaching middle age, the sudden necessity of making moral choices, rather than just teaching books about them, leads him into a devastating midlife crisis of his own. A comedy about the necessity, and the exquisite pain, of making choices.
Cast: 10 total (5 female, 5 male)
Them Within Us
Todd David Gross
A romantic comedy with a Sci-fi twist. Troubled lovers, Roger and Susan, visit rural Vermont to work on their relationship. Possession by aliens was not on their list of problems … until they got there.
Mia McCullough
TAKING CARE chronicles the last years of an elderly woman and the mentally ill adult son who has lived with her for almost three decades. Stuck in a cramped, one-bedroom apartment, Ma and Benny have spent much of the past thirty years pretending the other wasn't there. Now that the old woman is more or less house-bound, she foists conversation upon her unwilling son and threatens to institutionalize him if he doesn't clean up his act and start behaving more responsibly. When she breaks her hip, Ma convinces her alienated daughters that their brother is well enough to look after her in her "recovery." Realizing that failure to do so could result in institutionalization for both them, Benny tries to rise to the occasion and care for his mother, even as she begins a quick decent into dementia. His new responsibilities test his limited capabilities and force mother and son to forge new bonds in the waning days of her life.
Two American couples get caught in a chronological palindrome on a trip to Italy. Stigmata, alchemy, adultery, and St. Catherine of Siena's severed head plague a nun and a minister in this circular love story that is a comedy if performed forward and a tragedy if performed backward.
Sybille Pearson
An adoptee searches for her natural mother, seeking to discover who she is from the person from whom she came.
The Distance from Here
Neil LaBute
A riveting, intense look at the dark side of American suburbia. With little to occupy their time other than finding a decent place to hang out — the zoo, the mall, the school parking lot — Darrell and Tim are two American teenagers who lack direction and purpose in their lives. When Darrell's suspicion about the infidelity of his girlfriend is confirmed, and Tim comes to her defense, there is nothing to brake their momentum as all three speed toward disaster.
What if Blanche Dubois didn't go crazy? Or the Three Sisters actually made it to Moscow? When he discovers he's fictitious, a never-seen character in an Oscar Wilde play, Bunbury joins forces with Rosaline, Romeo's never-seen obsession from ROMEO AND JULIET. Together they infiltrate and alter classic literature, including giving ROMEO AND JULIET a happy ending.
About Spontaneous Combustion
Human spontaneous combustion is not something that most people worry about, but Amalia Parker does, and that's why she won't sleep with Rob. Sex with someone she doesn't care about might not be so dangerous, but with the man she truly loves? A definite possibility. This is the kind of logic that only makes sense in the Parker household, where Mom cooks obsessively, Dad has turned the house into an 18-hole golf course, and ancient maiden Aunt Emily does laundry and practices the family art of selective sight. A sweet, quirky comedy.
Mahida's Extra Key to Heaven
Thomas is visiting his mother's island home when he encounters Mahida, an Iranian college student stranded after an argument with her brother. The two strangers begin a beautifully awkward search for a common country, but discover that sometimes the smaller the world the bigger the invasions.
Kenny Finkle
In this unabashedly sentimental comedy, a cat named Samantha has just started living with a guy named Shuman who she believes is her true love. At first everything seems to be going great. Samantha can't get enough of Shuman, and she can't get enough of him. But soon Samantha realizes that Shuman doesn't always seem to understand her. Worse yet, Shuman doesn't always seem all that interested in trying to understand her. Things become truly problematic when one night Oscar, a sexy alley cat, shows up at the door. Suddenly Samantha begins to wonder whether she's an indoor or an outdoor cat. To make matters more complicated, a quirky aspiring cat therapist named Matilda inserts herself into the mix, thinking she can solve everyone's problems but only seems to make things worse. Eventually Samantha finds herself on a journey, both physical and emotional, that leads her to discover what having a home and being loved really mean to her. INDOOR/OUTDOOR is an allegory about the difficulties and joys of all manner of relationships.

redpajama_set_name: RedPajamaCommonCrawl


text:
Alzheimer's Community Corp
Alzheimer's Community CorpAlzheimer's Community CorpAlzheimer's Community Corp
Information about us and how we work
Brian Reynolds, PhD, BCMAS
Brian earned his doctorate in microbiology and immunology from the Indiana University School of Medicine. He is also a Board Certified Medical Affairs Specialist. He spent 3 years at Alma College as a professor of biology and biochemstry before moving to Cary, NC in 1998. He served as the Director of Medical Information at Voyager Pharma, a small biotechnology company focused on finding a therapy for Alzheimer's disease, then as a Senior Director of Clinical Operations in GSK's Alzheimer's program. He left GSK to become a Senior Clinical Project Leader at the Duke University School of Medicine where he worked for 10 years. Brian is passionate about sharing information that might help people live more fully in the midst of Alzheimer's disease or possibly prevent or delay the onset of AD by lifestyle modifications. He has particular interest in finding creative ways to help caregivers live happier lives.
About Alzheimer's Community Corp
Our objective is to provide a place for people to come to get current and sound information on how they might proceed when faced with a diagnosis of dementia either personally or in a loved one. To be a place where they can ask questions and interact with other folks; an alcove for learning about how to possibly prevent or delay the onset of dementia and ideas about how to better cope with circumstances if dementia is already present.
We've been writing and posting information about this and other topics on venues like Facebook for a while now and have decided to make that information more accessible to the public. We have products listed on our site. EVERY ONE of those products has been assessed for its value to a caregiver or patient through 1,000's of hours spent in chat rooms and caregiver support FB pages - asking questions and listening - finding out what they really want and need and what's worked for them. If you click on a product, it will take you to another site - like Amazon for example. You'll be able to buy the product there for exactly the same price that you would going through your Amazon account - in fact you will be going through your Amazon account - but if you click on it through our link, we get a small referral fee that help us funding our work.
All of the products are those that we believe in and many have been referred to us directly by caregivers. If you don't want to buy them through our link, that's fine, but I hope you'll look at them as they might be of value to you.
We are exclusively focused on Alzheimer's disease and other dementias
Our focus is the sharing of peer-reviewed and accurate scientific literature regarding Alzheimer's disease, updates on current thinking regarding possible prevention and community for discussion among caregivers about ways to best interact with their loved ones while supporting their own health and well-being.
Copyright © 2022 Alzheimer's Community Corp - All Rights Reserved.


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David J. Clark is senior vice president of Operations for Broward Health. David is an experienced healthcare operations manager with extensive experience in successfully managing portfolios, programs, corporate non-clinical service departments and projects in both the public and private sector.
Prior to Broward Health, Mr. Clark worked as the associate vice president, facilities construction and design at Jackson Health System. In this role, he oversaw all project initiatives throughout the system, and directed design and construction of all expansions, new construction, interior and exterior renovations and infrastructure expansion and replacement projects.
He has over 18 years of professional healthcare experience in the design and construction industry in both the public and private sector.
He is currently a member of the American College of Healthcare Executives and Kappa Alpha Psi Fraternity. He is a licensed certified general contractor, a LEED accredited professional and a real estate associate.
Mr. Clark earned his undergraduate degree in architectural and facilities engineering from Tennessee State University in Nashville, TN, and his Masters in Business Administration from NOVA Southeastern University in Davie, FL.


We curated this special family of beautiful dark gray sleep sack newborn cocoons just for you! These artistically stitched baby sacks come with two adorned sides for a dual purpose prop. One side will be designed as the photo shows and the opposite side is a tier of 3 wooden buttons for a classic look.
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Social media management for Inception, a creative consultancy focused on sustainable living.
As advocates for sustainable living, Inception's co-founders regularly travel to many different cities across continents. Having set up their creative consultancy, they wanted to use social media to connect with forward-thinking individuals and communities around the world.
The studio's Facebook and Instagram channels enable them to share their thoughts on sustainability and projects by local groups.
Need a social media strategy for your brand?


text:
RMOs to improve Spiritual Combat Power of Navy : CNS
The 109th Midshipmen and 18th Short Service Commission (SSC) Course Commissioning Parade was held at Pakistan Naval Academy PNS RAHBAR. The Commissioning Parade comprised 45 Pakistani and 43 Midshipmen from friendly countries as well as 54 Cadets of Short Service Commission Course including 32 Religious & Motivation Officers.
Chief of the Naval Staff, Admiral Zafar Mahmood Abbasi graced the occasion as Chief Guest. Upon his arrival, the Chief Guest was received by Commander Karachi, Rear Admiral Asif Khaliq.
In his address, Chief of the Naval Staff congratulated and extended felicitations to graduating officers on achieving this milestone and highlighted that in order to enhance Spiritual Combat Power of our men, for the first time, Religious & Motivation Officers (RMOs) are being commissioned as part of the SSC Course to provide religious guidance to the personnel and their families on social and moral aspects.
While taking stock of prevailing geo-strategic situation, Admiral Zafar Mahmood Abbasi said that Pakistan stands for peace and amity in the region and seeks harmonious and peaceful co-existence in the comity of nations. However, it will be a mistake if our desire for peace is construed as our weakness. We are fully capable of giving a befitting response, in case of any aggression against our country.
The Naval Chief reiterated the firm resolve of Armed Forces and underscored that Pakistan Armed Forces are at the forefront of the national resolve to cleanse Pakistan from the scourge of terrorism; and Pakistan Navy on its part is maintaining a robust security posture along the Pakistani Coast and in the regional seas to deter the nefarious elements from any illicit activities in the maritime domain.
The Naval Chief also mentioned the strategic shift from erstwhile participation of Pakistan Navy in the Combined Task Forces 150 and 151 under the auspices of Combined Maritime Forces to Regional Maritime Security Patrols. Admiral Zafar Mahmood Abbasi said that in order to pursue our national interests with strategic autonomy, we have now decided to institute Regional Maritime Security Patrols in the Indian Ocean Region in line with the UN Resolutions and UN Convention on Law of the Sea. The initiative is aimed at maintaining robust security posture in the critical sea areas and choke points in the Indian Ocean Region for protection of national and international shipping against the threats of maritime terrorism, piracy, narco-arms smuggling and human trafficking.
While advising the graduating officers, the Naval Chief urged them to hold fast to the qualities of Faith, professional excellence, loyalty, courage and character in their personality which are the essential ingredients of a true leader. He also expressed his satisfaction for extending support in training future officers of friendly nations who would prove worthy of their mettle in their respective Navies.
Earlier in his welcome address, Commandant Pakistan Naval Academy, Cdre Syed Faisal Hameed highlighted the salient features of the officers' training. The Commandant mentioned that cadets from Bahrain, Maldives, Qatar, Saudi Arabia and Yemen are also undergoing training at Pakistan Naval Academy.
Later, the Chief Guest gave away prizes to the winners. The prestigious Quaid-e-Azam Gold Medal was conferred upon Lt Roheel Shahzad PN. Midshipman Kashif Abdul Quyyom was awarded the coveted Sword of Honour for his overall best performance. Whereas, Midshipman Dayyan Ahmed won the Academy's Dirk; Officer Cadet Badar Ali was given Chairman Joint Chiefs of Staff Committee Gold Medal and Officer Cadet Mohammed Lahim S Aldawsri from Kingdom of Saudi Arabia was awarded Chief of the Naval Staff Gold Medal and Officer Cadet Muhammad Arshad from SSC Course clinched the Commandant Gold Medal. Proficiency Banner was re-claimed by Forecastle Squadron.
The ceremony was attended by a large number of senior military officers, including Commander Royal Bahrain Coast Guards and Commander Royal Bahrain Naval Force both of whom are alumni of Pakistan Naval Academy; Ambassadors, Defence Attaches of various countries, civil dignitaries and parents of passing out midshipmen and cadets.


Ragin' Cajuns Softball strengthened its group of outfielders with the addition of Julie Rawls, as head coach Gerry Glasco announced she is joining the UL roster in 2019.
Rawls, a rising junior, spent the previous two seasons at Northwestern State.
She made significant strides in her second season with the Demons, as she raised her average to .296, increased her home run total to 10 and more than tripled her run production from her freshman season, finishing with a team-high-tying 36 RBIs.
The Hallsville, Texas, product started 70-plus games in the outfield for NSU. She tracked down 113 fly ball outs and had 17 outfield assists (12 in sophomore season) during the timeframe.
Rawls was the MaxPreps Class 5A Player of the Year in Texas as a junior. She was also a three-time All-Atate selection and two-time District MVP as an outfielder.


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asus eah5770 Being asus eah5770, the effective memory speed is a mouth watering 4. Looks like it will run cool but The design of the Asus EAH CuCore videocard's cooler doesn't push air out of the back your case, so it's a good idea to have an internal fan set up to push air out of the back of the case.
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With age or significant weight loss, our skin begins to lose its ability to snap back into place. As a result it starts to sag. One place that this is especially noticeable is on the inside of the thighs.
In addition to the unattractive appearance in the mirror, this sagging of the skin can cause women to feel extremely uncomfortable wearing bathing suits, shorts, skirts or shorter dresses. This creates a real problem in the sun and heat of the Dallas and Fort Worth area.
Fortunately, sagging skin on the thigh can be removed by performing a thigh lift. At North Texas Plastic Surgery, our board-certified plastic surgeon will be happy to answer all of your questions in the comfort and privacy of our offices in Dallas, Plano, and Southlake. Call 817-416-8080 today for a free consultation in the Dallas area.
I Don't Want Anyone to Know I Had a Thigh Lift Done. Will the Scars Be Visible?
Our plastic surgeon performs a thigh lift through an incision that is made in a patient's groin crease. This hides the scar extremely well. The scar will never be visible when a patient wears clothing and will only be seen with a very revealing bathing suit.
Your procedure will be performed at a surgery center in Southlake. The night after surgery, many women are a little sore in the thighs making it difficult to get around much. For this reason, we will keep you overnight so that a nurse may help you with all that you need.
By the next morning, most women are able to get up and around well and are able to go home. You will be able to shower the day after your surgery.
Dr. Obaid will see you back in his office one week after your surgery. At that time he will remove two small drains. The stitches he uses will all dissolve and there will be no difficult or painful bandages to change. At that time you will already see the difference and be feeling great.
Our plastic surgeon recommends that you take it easy for the first two weeks after your surgery. You should be able to resume full normal activities within two weeks.
I Had a Gastric Bypass and Have Lost a Lot of Weight. Now the Skin on My Thighs Sag. Is There Anything That Can Be Done?
Sagging skin both on the inner and outer thighs is one of the most common complaints after weight loss surgery. It can be demoralizing to have had the surgery and successfully shed the weight only to find out that there is now this unattractive sagging skin left behind in the thighs. This sagging skin forces many people to cover up their new slimmer figure.
Fortunately, it can be corrected. With a weight loss thigh lift, Dr. Obaid can remove the extra skin from the thighs allowing for a more attractive contour which will give you the ability and confidence to wear the clothes that will show off your new body!
Discuss your options for toned legs with the team at North Texas Plastic Surgery by calling 817-416-8080 for a free consultation. We serve patients in Dallas and nearby communities from convenient offices in Southlake and Plano.


text:
using std::cout;
using std::endl;

using namespace MathTL;

int main()
{
  cout << "Testing finite difference operators ..." << endl;

  Polynomial<double> p;
  p.set_coefficient(2, 1.0); // x^2

  InfiniteVector<double, int> values;
  for (int i = -4; i <= 4; i++)
    values[i] = p.value(i);

  cout << "- a sampled polynomial on a 1D grid:" << endl;
  cout << values;

  cout << "- result of forward_difference<0>:" << endl;
  cout << forward_difference<0>(values);

  cout << "- result of forward_difference<1>:" << endl;
  cout << forward_difference<1>(values);

  cout << "- result of forward_difference<2>:" << endl;
  cout << forward_difference<2>(values);

  cout << "- result of forward_difference<3>:" << endl;
  cout << forward_difference<3>(values);

  cout << "- result of backward_difference<0>:" << endl;
  cout << backward_difference<0>(values);

  cout << "- result of backward_difference<1>:" << endl;
  cout << backward_difference<1>(values);

  cout << "- result of backward_difference<2>:" << endl;
  cout << backward_difference<2>(values);

  Polynomial<double> q;
  q.set_coefficient(3, -1.0); // x^3

  InfiniteVector<double, MultiIndex<int, 2> > values2;
  for (int x = -2; x <= 3; x++)
    for (int y = -1; y <= 2; y++)
      values2.set_coefficient(MultiIndex<int, 2>(x, y), p.value(x)*q.value(y));

  cout << "- a sampled multivariate polynomial:" << endl;
  cout << values2;

  cout << "- result of multivariate_forward_difference<2, 0>:" << endl;
  cout << multivariate_forward_difference<2, 0>(values2);

  cout << "- result of multivariate_forward_difference<2, 1>:" << endl;
  cout << multivariate_forward_difference<2, 1>(values2);

  cout << "- result of multivariate_forward_difference<2, 0>:" << endl;
  cout << multivariate_backward_difference<2, 0>(values2);

  cout << "- result of multivariate_forward_difference<2, 1>:" << endl;
  cout << multivariate_backward_difference<2, 1>(values2);

  return 0;
}

redpajama_set_name: RedPajamaGithub


text:
Q: JSF Number Validation Is there any inbuilt number validator tag in JSF that checks whether an input entered in h:inputext field is a number?
The first question was answered. Edited to explain the next problem:
<h:inputText id="maxrecs" value="#{simpleBean.numRecords}" required="false" maxlength="4" >
    <f:convertNumber longOnly="true"/>
</h:inputText>

Backing Bean
private long numRecords = null;

If I use String or Integer object in the backing bean , value is not being set. Now when I use primitive int, 0 is being printed on the screen. I would like the screen to be empty.

A: JSF Number validation for inputtext
mention f:converterNumber component in between h inputText component and mention the attributes integerOnly and type.
<h:inputText id="textMobileId" label="Mobile" styleClass="controlfont" value="#{UserRegistrationBean.textMobile}">
    <f:convertNumber integerOnly="true" type="number" />
</h:inputText>

If you enter abcd in Mobile textbox at the time when you click on commandbutton it automatically shows an error like
Mobile: 'abcd' is not a number.

A: You can use f:convertNumber (use the integerOnly attribute).
You can get more information here.

A: i8taken solution converts number into long without validation message (at least in my case: JSF2 / global messages per page). For proper validation message you can
1. check value in action method in bean;
or
2. use converter attribute for inputText:

<h:inputText id="maxrecs" value="#{simpleBean.numRecords}" maxlength="4" converter="javax.faces.Integer" />

A: You can use:
 <f:validateLongRange minimum="20" maximum="1000" />

Where minimum is the smallest number allowed and maximum is the largest.
Look here for more details

A: You can simply use the passthrough, so first add this library
    xmlns:pt="http://xmlns.jcp.org/jsf/passthrough"

and after use this
   <h:inputText id="numberId" pt:type="number" />

redpajama_set_name: RedPajamaStackExchange