text:
\section{Introduction}
Supersymmetric field theories exhibit many remarkable properties. Chief
among these are the cancellations which occur between boson and
fermion contributions in the perturbative calculation of physical
quantities. These cancellations eliminate many of the divergences
typical of quantum field theory and are at the heart of its use to
solve the gauge hierarchy problem \cite{wein}. Additionally, supersymmetric
versions of Yang Mills theory, while more tractable analytically than their
non-supersymmetric counterparts, exhibit many of the same features such as
confinement and chiral symmetry breaking \cite{prop}. Super Yang-Mills
theories with extended supersymmetry are also conjectured to be dual
to various supergravity theories in the limit of a large number of
colors and have been proposed as a regularization for M-theory \cite{dual,M}.

Since low energy physics is manifestly not
supersymmetric it is necessary that this symmetry be broken at some
energy scale. A set of non-renormalization theorems ensure that if SUSY is
not broken at tree level then it cannot be broken in any finite order of
perturbation theory see eg.~\cite{wein}. Thus the general expectation
is that any breaking
should occur non-perturbatively. The lattice furnishes the only
tool for a systematic investigation of non-perturbative
effects in field theories and so significant effort has gone into
formulating SUSY theories on the lattice -- see \cite{general} and the
recent reviews \cite{feo},\cite{kaplan}.

Unfortunately, there
are several barriers to such lattice formulations.
Firstly, supersymmetry is a spacetime symmetry which is generically
broken by the discretization procedure. In this it resembles Poincare invariance
which is also not preserved in a lattice theory. However, unlike
Poincare invariance there is usually no SUSY analog of
the discrete translation and cubic rotation groups which are left unbroken on
the lattice.
In the latter case the existence of these
remaining discrete symmetries is sufficient to prohibit the appearance of
relevant operators in the long wavelength lattice effective action which violate
the {\it full} symmetry group. This ensures that Poincare
invariance is achieved automatically
{\it without fine tuning} in the continuum limit.
Since generic latticizations of supersymmetric theories do
not have this property their effective actions typically
contain relevant supersymmetry breaking interactions.
To achieve a supersymmetric
continuum limit then requires fine tuning the bare lattice couplings
of all these SUSY violating terms - typically a
very difficult proposition.

Secondly, supersymmetric theories necessarily involve fermionic fields which
generically suffer from doubling problems when we attempt to define them on
the lattice. The presence of extra fermionic modes furnishes yet
another source of supersymmetry breaking since typically they are not
paired with corresponding bosonic states. Furthermore,
most methods of eliminating the extra fermionic modes serve to
break supersymmetry also.

Recently there has been renewed interest in this problem stemming from
the realization that in certain classes of theory it may be possible to
preserve at least some of the supersymmetry {\it exactly} at finite
lattice spacing. It is hoped that this residual supersymmetry may
protect the lattice theory from the
dangerous SUSY-violating radiative corrections that leading to fine
tuning.
In the case of
extended supersymmetry two approaches have been followed\footnote{
For examples of ${\cal N}=1$ models with exact SUSY see the recent
work \cite{feowz} and \cite{moto}}; in the first, pioneered
by Kaplan and collaborators, the lattice theory is
constructed by orbifolding a certain supersymmetric matrix model and
then extracting the lattice theory by expansion around some vacuum
state
\cite{kap,kap4,unsal,others}.
The second approach relies on reformulating the supersymmetric theory in
terms of a new set of variables -- the {\it twisted} fields. In this
procedure a scalar nilpotent supercharge is exposed and it is the algebra
of this charge that one may hope to preserve under discretization
\cite{top},\cite{joel}, \cite{kaw2dtwist}.
This approach was initially used to construct
and study lattice formulations of a variety of
low dimensional theories without gauge symmetry
\cite{qm,qm_joel,wz2,wz2_joel,sigma}. Subsequently it was extended
to case of Yang-Mills theories by Sugino \cite{sugino1,sugino2}
although that construction was confined to low dimensional theories.
In this
review we will confine ourselves to a discussion of a new
geometrical discretization of the twisted theory which
accommodates both two and four dimensional super Yang Mills theories.

The basic idea in this approach is decompose the fermion fields as a set of
antisymmetric tensor fields which are subsequently interpreted as
components of single Dirac-K\"{a}hler  field. This same procedure when applied to
the supercharges exposes a scalar nilpotent supercharge $Q$. Furthermore, the action of
the supersymmetric field theory can then be written as the $Q$-variation
of some function. Provided we can then maintain the nilpotent property
of $Q$ under discretization it is straightforward to write down a
lattice action in terms of these twisted fields which is exactly
invariant under $Q$ at finite lattice spacing.
As a bonus we will find that that this twisted formulation of the
theory, being written entirely in geometrical terms, can be discretized
without encountering spectrum doubling.

We first describe the nature of the twist in two dimensions and explain
how it is related to the use of Dirac-K\"{a}hler  fermions. We then give a detailed
discussion of continuum ${\cal N}=2$ super Yang-Mills in two dimensions which
furnishes a simple example of the twisted approach. Our discretization
prescription is then introduced and a lattice action for this
theory constructed. The latter is local, free of spectrum doubling,
gauge invariant and preserves a single (twisted) supersymmetry at finite
lattice spacing. It is naturally formulated in terms of complex fields
and we discuss how this lattice model may be used to target the
true continuum theory. Preliminary results are then shown coming from
a RHMC simulation of this lattice model which indeed show an exact
$Q$-symmetry emerging at large coupling $\beta$.
The technique can be generalized to provide a lattice action for
${\cal N}=4$ super Yang-Mills in four dimensions and this is then
discussed in some detail.

\section{The 2D Twist}
Consider a theory with ${\cal N}=2$ supersymmetry in two
dimensional Euclidean space. Such a theory contains
two Majorana supercharges $q^I_\alpha$ transforming under the global
symmetry group $SO(2)\times SO_R(2)$ where the subscript
corresponds to two dimensional rotations and the superscript
describes the behavior under the R-symmetry corresponding to
rotating the two Majorana fields into one another.

The basic idea of twisting which goes back to
Witten \cite{tqft} is to introduce a new rotation group
\begin{equation} SO(2)^\prime={\rm diagonal\; subgroup}(SO(2)\times SO(2)_R)\end{equation}
and to decompose all fields now as representations of this new
rotation group. In simple terms what this means is that
whenever I do a rotation in the base space by some angle
I must do an equal rotation in the R-symmetry space. Thus it
is equivalent to treating the two indices $I$ and $\alpha$ as
equivalent. Thus the supercharges are to be interpreted as
{\it matrices} in this twisted picture \cite{tak,kato,kaw2dtwist,lat04}.
\begin{equation} q^{I}_\alpha\to q_{\alpha\beta}\end{equation}
It is now natural to expand such a matrix on a basis of products
of 2D gamma matrices
\begin{equation} q=QI+Q_\mu\gamma_\mu+Q_{12}\gamma_1\gamma_2\end{equation}
The fields $(Q,Q_\mu,Q_{12})$ are called the {\it twisted supercharges}.
The original SUSY algebra then implies a corresponding twisted algebra
which takes the form
\begin{equation}
\{q,q\}_{\alpha\beta}=4\gamma^\mu_{\alpha\beta}p^\mu
\end{equation}
In components this reads
\begin{eqnarray}
\{Q,Q\}&=&\{Q_{12},Q_{12}\}=\{Q,Q_{12}\}=\{Q_\mu,Q_\nu\}=0\\
\{Q,Q_\mu\}&=&P_\mu\\
\{Q_{12},Q_\mu\}&=&-\epsilon_{\mu\nu}P_\nu
\end{eqnarray}
Notice that the scalar component of the supercharge matrix
is nilpotent as previously advertised. It is also
important to realize that the twisted superalgebra implies that
the momentum $P$ is now the $Q$-variation of something -- it is
said to be $Q$-exact. This fact renders it plausible that the
entire energy momentum tensor may be $Q$-exact in such twisted
theories. Since $T_{\mu\nu}=\frac{\delta S}{\delta g_{\mu\nu}}$ this
would imply that the entire action of the theory could
be written in a $Q$-exact form $S=Q\Lambda$ \cite{lat04}.
This turns out to be
true in essentially all the theories that have been studied.
Specifically we will show that the action of ${\cal N}=2$ super Yang-Mills
theory in two dimensions can be written in this way \cite{2dsuper}.
Notice also that to match the four real supercharges of
the original supersymmetric theory it is natural to take the
twisted supercharges to be real also. This in turn implies that
the supercharge matrix obeys a reality condition
\begin{equation} \overline{q}=q^\dagger=q^T\;{\rm in\;2D} \end{equation}

If the presence of
a nilpotent scalar fermionic charge and a corresponding $Q$-exact
action all sound reminiscent of BRST gauge fixing you would be right.
These twisted theories can be derived by gauge fixing an
underlying {\it topological} symmetry \cite{toprev}.
If the gauge fixing function
is chosen carefully the resultant theory can be untwisted to
reveal an associated supersymmetric field theory. However, it must
be stressed that this correspondence is not one to one. In the
topological field theory the set of physical states must be
restricted in the usual way to those annihilated by the
scalar charge. In the language of the supersymmetric field theory
such a constraint would project one to the vacuum state. This
explains why such a theory contains no physically propagating degrees
of freedom -- it is a theory defined only on the moduli space of
classical solutions. Our proposal is not to impose this
physical state condition but merely to treat the twisting
procedure as an exotic change of variables in the supersymmetric
field theory. Certainly in flat space the physical content
of the twisted and untwisted theories is the same.

Let us turn now to a discussion of the fields
in such a twisted theory. It should be clear
that if the supercharges can be written as matrix so can
the fermions
\begin{equation} \Psi=\frac{\eta}{2}I+\psi_\mu\gamma_\mu+\chi_{12}\gamma_1\gamma_2\end{equation}
Thus the twisted theory will not contain spinors but antisymmetric
tensor fields. Indeed it is possible to abstract these p-form
fields and consider them as components of a geometrical object -- a
Dirac-K\"{a}hler  field as first emphasized by Kawamoto and collaborators \cite{kaw2dtwist}.
\begin{equation} \Psi=(\frac{\eta}{2},\psi_\mu,\chi_{12})\end{equation}
It is a remarkable fact that
the original Dirac equation for 2 (Majorana) fermions is
then equivalent to the geometrical Dirac-K\"{a}hler  equation \cite{banks}
\begin{equation} (d-d^\dagger)\Psi=0 \end{equation}
Here $d$ and $d^\dagger$ are the usual exterior derivative and
its adjoint. Their action on a generic Dirac-K\"{a}hler  field $\omega$ is given
by
\begin{equation}
d\omega = \left(0,\partial_\mu f, \partial_\mu f_\nu - \partial_\nu
f_\mu,\ldots\right)\end{equation}
and
\begin{equation}-d^\dagger\omega=\left(f^\nu,f^\nu_\mu,\ldots,0\right)_{;\nu}\end{equation}
The Dirac-K\"{a}hler  equation can be derived from an action
\begin{equation} S_F=\left<\Psi^\dagger .(d-d^\dagger)\Psi\right> \end{equation}
where the dot product of two Dirac-K\"{a}hler  fields $A$ and $B$ is defined by
\begin{equation}<A|B>=\int d^Dx\sqrt{g}\sum_p \frac{1}{p!}A^{\mu_1\ldots\mu_p}B_{\mu_1\ldots\mu_p}\end{equation}
Writing out these components explicitly
and restricting ourselves to flat space we find
\begin{equation} S_F=\frac{1}{2}\psi^\dagger_\mu \partial_\mu \eta+\frac{1}{2}\chi^\dagger_{\mu\nu}
\partial_{\left[\mu\right.}\psi_{\left.\nu\right]}\end{equation}
This equivalence of the Dirac equation to the Dirac-K\"{a}hler  equation holds equally
when ordinary derivatives are replaced by gauge covariant derivatives.

The usual reason why the Dirac equation is preferred over the Dirac-K\"{a}hler
equation is that the latter naturally describes the propagation of
2 degenerate fermions in two dimensions. In a theory like QCD where
the bare quark masses are not usually thought of as equal this
appears problematic. However, a theory with ${\cal N}=2$ supersymmetry
in $D=2$ dimensions naturally requires such degenerate fermionic
states. Thus we have argued that for these special theories the
twisting procedure we have described is actually exactly equivalent
to the decomposition of the fermion fields in terms a single
Dirac-K\"{a}hler  field.
Furthermore the equivalence between the Dirac equation and the Dirac-K\"{a}hler
equation holds also in four dimensions where the latter equations
describes the propagation of four degenerate (Majorana) fermions.
Again, this is precisely the field content of ${\cal N}=4$ super
Yang-Mills theory and we should not be surprised then to find out
that the latter theory when suitably twisted can be expressed
in terms of the components of a single real Dirac-K\"{a}hler  field
\cite{4dsuper},\cite{kaw4dtwist}.

Finally any theory invariant under the $Q$-symmetry
must contain superpartners for these fermionic fields.
In two dimensions we then expect to find the
following bosonic (commuting) fields
\begin{equation} (\ensuremath{\overline{\phi}},A_\mu,B_{12})\end{equation}
The nilpotency of $Q$ then requires that $Q^2=0$ up to gauge
transformations when acting on any of these component fields. If we
interpret $B_{12}$ as a multiplier field we see that in this way
we have generated the bosonic field content of
${\cal N}=2$ SYM in 2D if we interpret each field as
taking values in the adjoint of some gauge group.
We turn now to an explicit construction of the twisted form
of this theory.

\section{Twisted ${\cal N}=2$ SYM in $D=2$}

In this section we render all these ideas concrete by writing down
the twisted formulation of ${\cal N}=2$ super Yang-Mills theory in
two (continuum) dimensions. The construction will expose the scalar
supersymmetric invariance explicitly and the $Q$-exact form of
the action. We will also show that the resulting on-shell action can be
rewritten in terms of the usual spinor degrees of freedom exhibiting
the fact that (in flat space) the twisted theory is nothing
but a change of variables in the original spinor based
super Yang-Mills model. The twisted gauge fermion is written
\cite{2dsuper},\cite{sugino1}
\begin{eqnarray}
S&=&\beta Q{\rm Tr}\int
d^2x\left(
\frac{1}{4}\eta[\phi,\ensuremath{\overline{\phi}}]+2\chi_{12}F_{12}+
\right.\\
&+&\left.\chi_{12}B_{12}+\psi_\mu D_\mu \ensuremath{\overline{\phi}}\right)\end{eqnarray}
where each field is in the adjoint of some gauge $U(N)$ group
$X=\sum_{a=1}^N X^a T^a$ and we employ antihermitian generators $T^a$.
The action of $Q$ on the fields is given by
\begin{eqnarray}
QA_\mu&=&\psi_\mu\nonumber\\
Q\psi_\mu&=&-D_\mu\phi\nonumber\\
Q\phi&=&0\nonumber\\
Q\chi_{12}&=&B_{12}\nonumber\\
QB_{12}&=&[\phi,\chi_{12}]\nonumber\\
Q\ensuremath{\overline{\phi}}&=&\eta\nonumber\\
Q\eta&=&[\phi,\ensuremath{\overline{\phi}}]
\end{eqnarray}
Notice that $Q^2$ generates an (infinitessimal) gauge transformation
parametrized by the field $\phi$. Carrying out the $Q$-variation
and subsequently integrating out the multiplier field $B_{12}$ we find
the on-shell action
\begin{eqnarray}
S&=&\beta {\rm Tr}\int d^2x\left(
\frac{1}{4}[\phi,\ensuremath{\overline{\phi}}]^2-\frac{1}{4}\eta [\phi,\eta]-F_{12}^2\right.\\
&-&D_\mu \phi D_\mu \ensuremath{\overline{\phi}}-\chi_{12}[\phi,\chi_{12}]\\
&-&2\chi_{12}\left(D_1\psi_2-D_2\psi_1\right)-2\psi_\mu D_\mu\eta/2\\
&+&\left.\psi_\mu [\ensuremath{\overline{\phi}},\psi_\mu]\right)
\end{eqnarray}
The gauge and scalar part of this action are easily recognized to be
the usual bosonic sector of ${\cal N}=2$ super Yang-Mills. If I choose
the contour where $\ensuremath{\overline{\phi}}^a=(\phi^a)^*$ it is clear that the bosonic action
is real and positive semi-definite. The fermionic sector looks a little
unconventional but by looking back at the previous discussion it can be
seen that it merely corresponds to a Dirac-K\"{a}hler  decomposition of the
usual spinor action corresponding to two Majorana fermions. To convince
ourselves of the equivalence of this twisted formulation to the usual one
it is convenient to pick a Euclidean chiral
basis for the two dimensional $\gamma$-matrices
\begin{equation} \begin{array}{cc}\gamma_1=
\left(\begin{array}{cc}
0&1\\
1&0
\end{array}\right)&
\gamma_2=\left(\begin{array}{cc}
0&i\\
-i&0
\end{array}\right)
\end{array}\end{equation}
and to subsequently define a Dirac spinor $\lambda$ as
\begin{equation}
\lambda=\left(\begin{array}{c}
\frac{\eta}{2}-i\chi_{12}\\
\psi_1-i\psi_2
\end{array}\right)\end{equation}
It is then a straightforward exercise to show that the Dirac
action $\lambda^\dagger \gamma.D\lambda$ reduces to the
kinetic terms in the twisted action we have previously uncovered.
Indeed the entire twisted action including the Yukawas can be
rewritten in terms of this Dirac spinor and can be
recognized as the usual fermionic sector of two
dimensional ${\cal N}=2$ super
Yang-Mills.
\begin{equation} S=\lambda^\dagger M(\phi) \lambda\end{equation}
where
\begin{equation} M=\gamma .D+\frac{\left(1+\gamma_5\right)}{2}[\ensuremath{\overline{\phi}},\cdots]
             -\frac{\left(1-\gamma_5\right)}{2}[\phi,\cdots]\end{equation}

\section{Discretization}
We will adopt this geometrical formulation as a useful starting off
point for constructing a lattice theory. It is convenient from the
point of view of supersymmetry as the scalar component of the
twisted supersymmetry involves no derivatives and offers the
hope, therefore, of being compatible with a lattice structure.
Secondly, the use of Dirac-K\"{a}hler  fermions is attractive as it is known
how to discretize the Dirac-K\"{a}hler  action without inducing spectrum
doubling. Indeed, Dirac-K\"{a}hler  fermions at the level of free field theory
are known to be equivalent to staggered fermions.  The usual
degeneracy of staggered fermions now becomes a plus -- it allows us to
replicated the degenerate flavors required by extended
supersymmetry. The use of Dirac-K\"{a}hler  fermions to formulate (Hamiltonian) lattice
supersymmetric models was first proposed in \cite{schwim}.

Clearly, continuum scalar fields should map to fields defined on
sites in a (hypercubic) lattice, continuum vector fields to fields
defined on links and rank 2 antisymmetric fields to lattice
fields living on plaquettes. Notice that all lattice p-cubes whose
dimension is greater than zero come in two possible
orientations corresponding to whether the defining vertices are written
down as an even or odd permutation of some standard ordering. In the
case of links this is just the usual statement that the link has two
possible directions. It is then natural (and as we will see later essential)
that this 2-fold orientation of the underlying p-cube can be
associated with the existence of two independent lattice fields for
each continuum field. These two fields can be represented by allowing
the lattice fields $X^a$ to be complex numbers thus promoting the
group from $U(N)$ to $GL(N,C)$.
The lattice fields are also assigned gauge transformation properties
which generalize the usual transformation properties of site fields
and links in lattice gauge theory \cite{adjoint}:
\begin{eqnarray}
f(x)&\to&
G(x)f_(x)G^{-1}(x)\\
f_\mu(x)&\to& G(x)
f_\mu(x)G^{-1}(x+\mu)\\
f_{12}(x)&\to&G(x)f_{12}(x)G^{-1}(x+1+2)
\end{eqnarray}
where $G=e^{\phi}$ is a lattice gauge transformation. Notice in the
naive continuum limit where $G\sim 1+\phi$ these reduce to the usual commutator
structure typical of adjoint fields.
In general this leads to a point split or {\it shifted} commutator. For link
fields this looks like
\begin{equation} [\phi(x)f_\mu(x)-f_\mu(x)\phi(x+\mu)]\end{equation}
Notice that this commutator has a well-defined lattice gauge transformation
property.
Finally, we must replace the continuum gauge connection $A_\mu$ with
a Wilson gauge link $U_\mu(x)=e^{A_\mu}$. Notice now that
$A_\mu$ will be complex which implies that the gauge links
are {\it not} unitary at this stage in the construction.

Of course we have not yet finished. We need also to give a prescription
for replacing continuum covariant derivatives by difference operators.
Furthermore, as we will argue later, we are naturally led to
use covariant versions of forward and backward difference operators.
The results of these difference operators when acting on lattice
fields should also result in new lattice fields with well-defined
gauge transformation properties. Thus hitting a site field with
such a (forward) difference operator should lead to a field which gauge
transforms like a link field etc. It is not hard to show that
the following definition of
a forward difference operator acting on site and link
field satisfy these properties \cite{adjoint}
\begin{eqnarray}
D^+_\mu f(x)&=&U_\mu(x)f(x+\mu)-f(x)U_\mu(x)\\
D^+_\mu f_\nu(x)&=&U_\mu(x)f_\nu(x+\mu)-f_\nu(x)U_\mu(x+\nu)
\end{eqnarray}
They clearly reduce to continuum derivatives as $a\to 0$ and ensure
that the resulting fields gauge transform in the appropriate way.
Rather remarkably it is possible to write a covariant backward
difference which is truly adjoint to the above operator for
gauge invariant quantities. Notice that it will necessarily act
as a divergence operation (for a proof of this statement we refer
the reader to the paper by Aratyn et al \cite{adjoint})
\begin{eqnarray}
D^-_\mu
f_{\mu}(x)&=&f_\mu(x)U^\dagger_\mu(x)-U^\dagger_\mu(x-\mu)f_\mu(x-\mu)\\
D^-_\mu f_{\mu\nu}(x)&=&f_{\mu\nu}(x)U^\dagger_\mu(x+\nu)-U^\dagger_\mu(x-\mu)f_{\mu\nu}(x-\mu)
\end{eqnarray}
We have now almost all the ingredients needed
to construct our lattice super Yang-Mills
theory. The remaining key observation goes back to work by Becher, Joos,
Rabin and others \cite{becher,rabin} concerning discretizing actions written
in terms of p-forms and exterior derivatives. They give a topological
proof using results from homology theory that such actions
can be discretized in such a way that the resulting lattice theories
are completely free of spectrum doubling -- that is, the
resulting lattice theories do
{\it not} contain any modes having no analog in the
continuum theory. However, such theories, like staggered fermion
models, will necessarily describe degenerate flavors of fermion. While this
is a problem for QCD it is a bonus in the supersymmetric theories
studied here as these degenerate flavors can represent the multiple fermions
inherent in extended supersymmetry. While these original papers
showed how to do this for ordinary derivatives, a generalization to
covariant derivatives was first given by Aratyn at al.\cite{adjoint} which
is the one employed here. The discretization prescription is given
explicitly by replacing the usual partial derivatives in the
continuum theory by appropriate difference operators:
\begin{center}
$D_\mu\to D^+$ if it acts like $d$\\
$D_\mu\to D^-$ if it acts like $d^\dagger$
\end{center}
Following this rule we find that the Yang-Mills field strength is
discretized as follows
\begin{equation}
F_{\mu\nu}(x)=D^+_\mu U_\nu(x)\to F^{\rm cont}_{\mu\nu}\;{\rm as}\;a\to 0\end{equation}
Notice it is automatically antisymmetric in its indices.
Using these ingredients we propose the following lattice action
for ${\cal N}=2$ super Yang-Mills
\begin{eqnarray}
S_L&=&\frac{1}{2}\beta Q{\rm Tr}\sum_{x}\left(
-\frac{1}{4}\eta^\dagger[\phi,\ensuremath{\overline{\phi}}]-2\chi^\dagger_{12}F_{12}\right.\\
&-&\left.\chi^\dagger_{12}B_{12}-
\psi^\dagger_\mu D^+_\mu\ensuremath{\overline{\phi}}+{\rm h.c}
\right)
\end{eqnarray}
Notice the necessity of introducing both fields and their hermitian
conjugates in order
to write down gauge invariant expressions. The corresponding lattice
supersymmetry is a simple generalization of the continuum ones \cite{2dsuper}
\begin{eqnarray}
QU_\mu&=&\psi_\mu\nonumber\\
Q\psi_\mu&=&-D^+_\mu\phi\nonumber\\
Q\phi&=&0\nonumber\\
Q\chi_{12}&=&B_{12}\nonumber\\
QB_{12}&=&[\phi,\chi_{12}]^{(12)}\nonumber\\
Q\ensuremath{\overline{\phi}}&=&\eta\nonumber\\
Q\eta&=&[\phi,\ensuremath{\overline{\phi}}]
\end{eqnarray}
where I have replaced $D_\mu\phi$ by $D^+_\mu\phi$ as required by
gauge invariance and the prime reflects the fact that we must
use a {\it shifted} commutator as previously discussed.
Carrying out the $Q$-variation and then integrating out the field
$B$ as in the continuum we obtain the on-shell lattice
action
\begin{eqnarray}
S_L&=&\frac{\beta}{2}{\rm Tr}\sum_x \left(
\frac{1}{4}[\phi,\ensuremath{\overline{\phi}}]^2+F^\dagger_{12} F_{12}\right.\\
&-&\left.\frac{1}{4}\eta^\dagger[\phi,\eta]-\chi^\dagger_{12}[\phi,\chi_{12}]^{(12)}+
\psi^\dagger_\mu[\ensuremath{\overline{\phi}},\psi_\mu]^{(\mu)}\right.\\
&+&\left.(D^+_\mu \phi)^\dagger D^+_\mu \ensuremath{\overline{\phi}}-
2\chi^\dagger_{12}\left(D^+_1\psi_2 -D^+_2\psi_1\right)\right.\\
&-&\left.2\psi^\dagger_\mu D^+_\mu\frac{\eta}{2}+{\rm h.c}\right)
\end{eqnarray}
This action is invariant under $Q$, finite gauge transformations and
an additional $SO(1,1)$ symmetry
\begin{equation} \psi_\mu\to \lambda\psi_\mu,\;\;\eta ,\chi_{12}\to
\frac{1}{\lambda}\eta, \chi_{12}\end{equation}
\begin{equation}
\ensuremath{\overline{\phi}}\to \frac{1}{\lambda^2}\ensuremath{\overline{\phi}},\;\;\phi\to \lambda^2\phi\end{equation}
This symmetry is the analog of the usual exact $U(1)$ chiral symmetry of
staggered quarks. It is important to realize however that our coupling of
gauge fields to lattice p-form fields
does {\it not} correspond to the usual gauging of
staggered quarks.

It is illuminating to examine the gauge action in more detail. Substituting
the definitions of $F_{\mu\nu}$ we obtain
\begin{equation}
\beta {\rm Tr}\sum_x F^\dagger_{12}(x)F_{12}(x)\end{equation}
\begin{equation}
\beta{\rm Tr}\sum_x \left(2I-U_P-U^\dagger_P\right)+
\beta{\rm Tr}\sum_x \left(M_{12}+M_{21}-2I\right)
\end{equation}
where
\begin{equation}
U_P=U_1(x)U_2(x+1)U^\dagger_1(x+2)U^\dagger_2(x)\end{equation}
resembles the usual Wilson plaquette term
and
\begin{equation} M_{12}=U_1(x)U^\dagger_1(x)U_2^\dagger(x+1)U_2(x+1)\end{equation}
is a new zero area Wilson loop term which would vanish if the
link variables were restricted to unitary matrices. Notice the appearance of
the Wilson term depends crucially on the appearance of both
$F_{\mu\nu}$ and $F^\dagger_{\mu\nu}$ which lends some support to
the use of complex variables in the formulation of the theory.
Such a theory does not exhibit doubling of course and we might guess that
the exact supersymmetry will thus prevent doubles from occurring in
the fermionic sector also. We turn to this question now.

We can assemble the various p-form components of the Dirac-K\"{a}hler  fermion
back into a 4 component object in the following way
\begin{equation}
\Psi=\left(\begin{array}{c}
\eta/2\\
\chi_{12}\\
\psi_1\\
\psi_2\end{array}\right)
\end{equation}
The twisted fermion action can then be written in the matrix form
$\Psi^\dagger M\Psi$
where
\begin{equation}
M=\left(\begin{array}{cc}
-[\phi,]^{(p)}&K\\
-K^\dagger&[\ensuremath{\overline{\phi}},]^{(p)}
\end{array}\right)\end{equation}
and the operator
\begin{equation}
K=\left(\begin{array}{cc}
D^+_2&-D^+_1\\
-D^-_1&-D^-_2
\end{array}\right)\end{equation}
is just a discrete kinetic term for the fermions. The Yukawas appear
on the diagonal as indicated.
After integration we expect that the result will be a Pfaffian of
the operator $M$ (the reality conditions in the continuum are
important here).
To examine potential doubling problems in this
operator it suffices to examine $M$ in the free limit
where its determinant reduces to the usual determinant of
a two dimensional {\it double free} scalar laplacian.
\begin{equation} {\rm Pf}(M)={\rm det}(K)={\rm det}(D^+_\mu D^-_\mu)\end{equation}
Thus, as advertised, the use of Dirac-K\"{a}hler  fermions avoids the appearance of
spurious zeroes in the fermion operator.

\section{Continuum Limit}

The lattice formulation we have given requires a doubling of
degrees of freedom -- we have argued that this is quite natural
in any lattice theory and can be associated with the two possible
orientations of the underlying p-cube. We have parametrized this
doubling in terms of complex fields. However, the
target continuum theory that we are hoping to reproduce
in the limit of vanishing lattice spacing corresponds to setting
\begin{eqnarray}
{\rm Im}X_\mu^a&=&0\;\; \mbox{all fields $X$ bar scalars}\\
\ensuremath{\overline{\phi}}&=&-\phi^\dagger
\end{eqnarray}
At this point we must address how this continuum theory can be recovered
from our complexified theory. First, consider the effective
action that results from integrating out the grassmann variables.
It is clearly gauge invariant if we ignore possible phase
problems and replace the Pfaffian by a square root of
the (gauge invariant) determinant. It also clearly has the right
classical continuum limit. The only remaining question is whether
the Ward identities corresponding to the twisted
supersymmetry still hold (or more conservatively, hold in the
limit of vanishing lattice spacing with no additional fine
tuning) We conjecture that this may be so and have followed this
approach so far in our numerical work.

It is possible to make some progress in understanding why this might indeed be
true. First parametrize the general $GL(N,C)$ link field $U_\mu(x)$ in terms
of a unitary component $u_\mu(x)$ and a positive definite hermitian
piece $R_\mu(x)$ in the following way
\begin{equation} U_\mu(x)=R_\mu(x)u_\mu(x)\end{equation}
Now consider some Ward identity corresponding to the expectation
value of the $Q$-variation of
some operator $<QO>$. It is not hard to show that this object can be computed
{\it exactly} in the semi-classical limit $\beta\to\infty$. The argument
follows by recognizing that
\begin{equation} \frac{\partial}{\partial\beta}<QO>=<Q(OS)>=0\end{equation}
where $Q$-exactness of the action has been used. Thus the expectation
value is independent of coupling $\beta$ and hence can be computed in
the semi-classical limit. This is a standard result of topological
field theory \cite{toprev}.
Now consider the second term in the gauge action
\begin{equation} \beta{\rm Tr}\sum_x \left(M_{12}+M_{21}-2I\right)\end{equation}
where
\begin{equation} M_{12}=U_1(x)U^\dagger_1(x)U_2^\dagger(x+1)U_2(x+1)\end{equation}
and insert the general decomposition of the gauge link given above. The result
for $M_{12}$ is
\begin{equation} M_{12}=R^2_1(x)R^2_2(x+1)\end{equation}
with a similar result for $M_{21}$. Notice it is independent
of the unitary piece $u_\mu(x)$. Thus in the limit $\beta\to\infty$
each $R_\mu(x)$ is driven to the identity and the Ward identity can be
computed in the space of purely unitary gauge links. Thus the Ward
identities of the truncated theory should hold at least for large
$\beta$ since they can be viewed as coming from the complexified theory
in the limit of infinite $\beta$.
Of course large $\beta$ also corresponds to the limit of vanishing lattice
spacing and we see that this line of reasoning constitutes an
argument that the
Ward identities in the truncated lattice theory should be realized
without fine tuning in the continuum limit. As we shall show in the next
section our preliminary numerical results are consistent with this.

\section{Simulations}

For our simulations we have taken the gauge links to be
unitary matrices taking values in $U(2)$. As in the continuum
theory the scalars $\phi$ and $\ensuremath{\overline{\phi}}$ are taken to be
complex conjugates of each other\footnote{the antihermitian nature of our basis
$T^a$ actually ensures that $\phi^\dagger=-\ensuremath{\overline{\phi}}$}.
To regulate possible IR divergences associated with the
flat directions where $[\phi,\ensuremath{\overline{\phi}}]=0$ we have
added a scalar mass term $m^2\phi^\dagger\phi$. This will
break supersymmetry softly. At the end of the calculation we should
like to send $m\to 0$ to recover the correct target theory.
This bosonic action is real positive semi-definite,
gauge invariant and clearly has the
correct naive continuum limit. To this gauge and scalar action
we should add the effective action gotten by integrating
over the grassmann valued fields. We will represent this as
\begin{equation} {\rm det}^{\frac{1}{4}} (M(U,\phi)^\dagger M(U,\phi)+m^2)\end{equation}
where $M$ is the lattice Dirac-K\"{a}hler  operator introduced earlier.
Notice that we have added a gluino mass term for the fermions
which helps regulate potential zero modes.
It is chosen to be identical in magnitude to the scalar mass. This ensures
the scalar and fermion determinants will cancel for zero gauge
coupling. In these preliminary simulations we have neglected the
issue of the phase of the fermion determinant which is reflected in
the appearance of just $M^\dagger M$ in the effective action. Finally,
the fourth root reflects the
Majorana nature of the continuum fermions. Notice that the determinant
is also gauge invariant.

To simulate this model we have used the RHMC algorithm
developed by Clark and Kennedy \cite{rhmc}.
The first step of this algorithm replaces the effective action
by an integration over auxiliary {\it commuting}
pseudofermion fields $F$, $F^\dagger$
in the following way
\begin{equation} {\rm det}^{\frac{1}{4}}(M^\dagger M+m^2)=\int DFDF^\dagger e^{-F^\dagger
\left(M^\dagger M+m^2\right)^{-\frac{1}{4}}F}\end{equation}
The key idea of RHMC is to use an optimal (in the minimax sense)
rational approximation to this inverse fractional power.
In practice it is important to use a partial fraction representation
of this rational approximation
\begin{equation} \frac{1}{x^{\frac{1}{4}}}\sim
\alpha_0+\sum_{i=1}^{N}\frac{\alpha_i}{x+\beta_i}\end{equation}
The numbers $\alpha_i$, $\beta_i$ for $i=1\ldots N$
can be computed offline using
the remez algorithm\footnote{many thanks to Mike Clark for providing us
with a copy of his remez code}.
The resulting pseudofermion action becomes
\begin{equation} \alpha_0 F^\dagger F+\sum_{i=1}^NF^\dagger
\frac{\alpha_i}{M^\dagger M+m^2+\beta_i}F\end{equation}
It is thus just a sum of standard 2 flavor pseudofermion actions
with varying amplitudes and mass parameters. In principle
this pseudofermion action can now be used in a conventional
HMC algorithm to yield an {\it exact} simulation of the original
effective action \cite{hmc}. The final
trick needed to render this approach feasible is to utilize a {\it multi-mass
solver} to solve all $N$ sparse linear systems simultaneously and with
a computational cost determined primarily by the smallest shift
$\beta_i$. We use a multi-mass version of the
usual conjugate gradient algorithm \cite{multi}. In practice for the
preliminary simulations shown here we have used $N=12$ and an approximation
that gives an absolute bound on the relative error of $10^{-6}$ over the
entire range of the spectrum (the lower eigenvalue is
given by $m^2$ and the upper can be determined from
trial simulations using a Lanczos method). Further details will
be presented elsewhere \cite{soon}.

The numerical results we present here come from
simulations where the lattice length takes the values
$L=2,4,8$ and the mass parameter $m=0.1,0.05$
over a range of coupling $\beta=0.5\to 4.0$. Notice that these
lattices are roughly equivalent to
$L=4,8,16$ for staggered quark simulations. We are currently extending
these calculations to larger lattices and smaller mass \cite{soon}.

We have devoted our initial runs to a check of the simplest
possible supersymmetric Ward identity -- namely the value of the
total action. If the latter corresponds to a $Q$-exact observable this
should vanish. This fact, together with the quadratic nature
of the fermion action
allows us to compute the bosonic (gauge plus scalar)
action exactly and for {\it all}
values of the coupling constant $\beta$ we find
\begin{equation} \beta<S_B>=\frac{3}{2}RL^2\end{equation}
where $R=4$ is the number of generators of $U(2)$.
The following graph shows a plot of $\frac{\beta}{6L^2}<S_B>$ for a range of
couplings $\beta$ and three different lattice sizes using a mass $m=0.1$.
The bold lines shows the analytic prediction based on supersymmetry (for
clarity we have added multiples of $-0.25$ to the curves and lines to
split up the data from different lattice sizes)
While there are clearly deviations of order 4-5\% at small coupling these
appear to disappear at large $\beta$ in line with our
expectations.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=3.2in]
{baction_c.eps}
\caption{$\frac{\beta}{6L^2}<S_B>$}
\end{center}
\end{figure}

We have also examined the gauge action and the string tension
determined by the smallest creutz ratio
$\sigma=-\ln{W(1,1)W(2,2)/(W(1,2)W(2,1))}$ and plot these as a function of
gauge coupling in the following plots ($m=0.1$ and $L=4$).
These data are consistent with
the existence of a confining phase.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=3.2in]
{gauge_act.eps}
\caption{Gauge action for $U(2)$ model}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=3.2in]
{creutz.eps}
\caption{String tension for $U(2)$ model}
\end{center}
\end{figure}

\section{Twisting ${\cal N}=4$ SYM in D=4}

The ${\cal N}=4$ super Yang-Mills theory in four
dimensions contains four Majorana spinors $\Psi^I_\alpha$
transforming under the Euclidean Lorentz group $SO(4)$ and
an additional $SO_R(4)$ R-symmetry group.
The K\"{a}hler-Dirac twist (see \cite{4dsuper},\cite{kaw4dtwist})
consists of decomposing those fields into representations of
the diagonal subgroup
\begin{equation} SO(4)^\prime={\rm diag}(SO(4)\times SO(4)_R)\end{equation}
As for two dimensions this twisting procedure is
equivalent to regarding the supercharges and fermions as matrices:
\begin{eqnarray}
\Psi&=&\eta
I+\psi_\mu\gamma_\mu+\frac{1}{2!}\chi_{\mu\nu}\gamma_\mu\gamma_\nu+\\
&+&\frac{1}{3!}\theta_{\mu\nu\lambda}\gamma_\mu\gamma_\nu\gamma_\lambda
+\frac{1}{4!}\kappa_{\mu\nu\lambda\rho}\gamma_\mu\gamma_\nu\gamma_\lambda\gamma_\rho
\end{eqnarray}
To match the 16 real supercharges of the
continuum theory it is natural to take these
antisymmetric tensor fields real. They can then be regarded as
the 16 grassmann components of a single real K\"{a}hler-Dirac field just
as for two dimensions. There will now be a scalar nilpotent
supercharge\footnote{the full twisted superalgebra was given by
D'Adda et al in\cite{kaw4dtwist}} and corresponding bosonic
superpartners
$(\ensuremath{\overline{\phi}},A_\mu,B_{\mu\nu},W_{\mu\nu\lambda},C_{\mu\nu\lambda\rho})$
In the end $B$ and $C$ will turn out to be multiplier fields and are
integrated out of the final theory leaving two scalars $\ensuremath{\overline{\phi}}$, $\phi$
(which arises as a gauge parameter as in two dimensions), one vector
$A_mu$ and the 4 independent components of $W_{\mu\nu\lambda}$.
These latter four fields will combine with the two scalars to yield the
usual 6 scalars of ${\cal N}=4$ super Yang-Mills.

The form of the $Q$-variations of these fields under the
scalar supersymmetry can be obtained in analogy to the situation
in two dimensions.
\begin{eqnarray}
Q\ensuremath{\overline{\phi}}&=&\eta\;\;Q\eta=[\phi,\ensuremath{\overline{\phi}}]\nonumber\\
QA_\mu&=&\psi_\mu\;\;Q\psi_\mu=-D_\mu\phi\nonumber\\
QB_{\mu\nu}&=&[\phi,\chi_{\mu\nu}]\;\;Q\chi_{\mu\nu}=B_{\mu\nu}\nonumber\\
QW_{\mu\nu\lambda}&=&\theta_{\mu\nu\lambda}\;\;Q\theta_{\mu\nu\lambda}=
[\phi,W_{\mu\nu\lambda}]\nonumber\\
QC_{\mu\nu\lambda\rho}&=&[\phi,\kappa_{\mu\nu\lambda\rho}]\;\;Q\kappa_{\mu\nu\lambda\rho}=
C_{\mu\nu\lambda\rho}\nonumber\\
Q\phi&=&0
\label{Q}
\end{eqnarray}
In deciding how to write down these transformations one is guided by
the identification of physical fields. If a field is ultimately to
be identified as a propagating field it is necessary to make its
$Q$-variation yield another field. Multiplier fields vary into
commutators. Notice again that the square of the supercharge is
an infinitessimal gauge transformation parametrized by the field $\phi$.

We hypothesize that, following on from our general arguments, it should
be possible to write the action as $Q$-exact function
\begin{equation} S=\beta Q\Lambda\end{equation}
where the
correct gauge fermion of ${\cal N}=4$ super Yang-Mills turns out
to be
\begin{eqnarray}
\Lambda&=&\int d^4x {\rm Tr}\left[
\chi_{\mu\nu}\left(F_{\mu\nu}+\frac{1}{2}B_{\mu\nu}-
\frac{1}{2}[W_{\mu\lambda\rho},W_{\nu\lambda\rho}]\right.\right.\nonumber\\
&+&\left.\left.D_\lambda W_{\lambda\mu\nu}\right)\right.\nonumber\\
&+&\left.\psi_\mu D_\mu\ensuremath{\overline{\phi}}+\frac{1}{4}\eta[\phi,\ensuremath{\overline{\phi}}]+
\frac{1}{3!}\theta_{\mu\nu\lambda}[W_{\mu\nu\lambda},\ensuremath{\overline{\phi}}]\right.\nonumber\\
&+&\left.\frac{1}{4!}\kappa_{\mu\nu\lambda\rho}\left(
\sqrt{2}D_{\left[\mu\right.}W_{\left.\nu\lambda\rho\right]}+
\frac{1}{2}C_{\mu\nu\lambda\rho}\right)
\right]
\label{qact}
\end{eqnarray}\\
Several of these terms are in common with the gauge fermion of ${\cal N}=2$ super
Yang-Mills theory in two dimensions. The new ones involve the $3$ and $4$-form
fields. Of these the terms involving derivatives must be present to generate the
correct Dirac-K\"{a}hler  action for the twisted fermions (and will simultaneously
generate the appropriate kinetic terms for
the W-field).
The commutator term involving $W_{\mu\nu\lambda}$ coupled to
$\chi_{\mu\nu}$ will generate a quartic potential for the W-field analogous to
that generated for the scalars $\phi$ and $\ensuremath{\overline{\phi}}$. This will allow contact
to be made eventually with the supersymmetric theory
where one expects the scalars and W-field to play similar roles.
In the same way the
commutator term involving $\ensuremath{\overline{\phi}}$ and $W$ will also generate the
necessary mixed quartic
couplings between the scalars and the W-field.
Carrying out the $Q$-variation leads to the following action
\begin{equation} S=\beta\left(S_B+S_F+S_Y\right)\end{equation}
where the piece of the action $S_B$ involving the bosonic fields takes the form
\begin{eqnarray}
S_B&=&\int d^4x {\rm Tr}\left[
B_{\mu\nu}\left(F_{\mu\nu}-
\frac{1}{2}[W_{\mu\lambda\rho},W_{\nu\lambda\rho}]+
D_\lambda W_{\lambda\mu\nu}+
\frac{1}{2}B_{\mu\nu}\right)\right.\nonumber\\
&-&\left.D_\mu\phi D_\mu\ensuremath{\overline{\phi}}+\frac{1}{4}[\phi,\ensuremath{\overline{\phi}}]^2-
\frac{1}{3!}[\phi,W_{\mu\nu\lambda}][\ensuremath{\overline{\phi}},W_{\mu\nu\lambda}]\right.\nonumber\\
&+&\left.\frac{1}{4!}C_{\mu\nu\lambda\rho}\left(
\sqrt{2}D_{\left[\mu\right.}W_{\left.\nu\lambda\rho\right]}+
\frac{1}{2}C_{\mu\nu\lambda\rho}\right)
\right]
\end{eqnarray}
and the fermion kinetic terms are given by $S_F$ with
\begin{eqnarray}
S_F&=&\int d^4 x {\rm Tr}\left[
-\chi_{\mu\nu}D_{\left[\mu\right.}\psi_{\left.\nu\right]}
-\chi_{\mu\nu}D_\lambda\theta_{\lambda\mu\nu}
-\eta D_\mu\psi_\mu-
\frac{\sqrt{2}}{4!}\kappa_{\mu\nu\lambda\rho}D_{\left[\mu\right.}
\theta_{\left.\nu\lambda\rho\right]}
\right]
\end{eqnarray}
and $S_Y$ contains the Yukawa couplings
\begin{eqnarray}
S_Y&=&\int d^4x {\rm Tr}\left[
-\frac{1}{4}\eta[\phi,\eta]
-\frac{1}{2}\frac{1}{4!}
\kappa_{\mu\nu\lambda\rho}[\phi,\kappa_{\mu\nu\lambda\rho}]-
\frac{1}{2}\chi_{\mu\nu}[\phi,\chi_{\mu\nu}]\right.\nonumber\\
&+&\left.\psi_\mu[\ensuremath{\overline{\phi}},\psi_\mu]+
\frac{1}{3!}\theta_{\mu\nu\lambda}[\ensuremath{\overline{\phi}},\theta_{\mu\nu\lambda}]
\right.\nonumber\\
&+&\left.\frac{1}{3!}\eta[\theta_{\mu\nu\lambda},W_{\mu\nu\lambda}]-
\frac{\sqrt{2}}{4!}\kappa_{\mu\nu\lambda\rho}
[\psi_{\left[\mu\right.},W_{\left.\nu\lambda\rho\right]}]\right.\nonumber\\
&+&\left.\chi_{\mu\nu}[\theta_{\mu\lambda\rho},W_{\nu\lambda\rho}]-
\chi_{\mu\nu}[\psi_\lambda,W_{\lambda\mu\nu}]
\right]
\end{eqnarray}
Integrating over the multiplier fields $B_{\mu\nu}$ and $C_{\mu\nu\lambda\rho}$
and subsequently
utilizing the Bianchi identity leads to a new bosonic action
of the form
\begin{eqnarray}
S_B&=&\int d^4 x {\rm Tr}\left[
-\frac{1}{2}\left(
\left(F_{\mu\nu}-
\frac{1}{2}[W_{\mu\lambda\rho},W_{\nu\lambda\rho}]\right)^2
+\left(D_\lambda W_{\lambda\mu\nu}\right)^2+
\frac{2}{4!}\left(D_{\left[\mu\right.}
W_{\left.\nu\lambda\rho\right]}\right)^2\right)
\right.\nonumber\\
&-&\left.D_\mu\phi D_\mu\ensuremath{\overline{\phi}}+\frac{1}{4}[\phi,\ensuremath{\overline{\phi}}]^2-
\frac{1}{3!}[\phi,W_{\mu\nu\lambda}][\ensuremath{\overline{\phi}},W_{\mu\nu\lambda}]
\right]
\end{eqnarray}
At this point it is useful to trade the $W$, $\theta$, and $\kappa$ fields
for new variables which will allow contact to be made between this
theory and one of the conventional twists of ${\cal N}=4$ super Yang-Mills.
We write
\begin{eqnarray}
W_{\mu\nu\lambda}&=&\epsilon_{\mu\nu\lambda\rho}V_\rho\nonumber\\
\theta_{\mu\nu\lambda}&=&\epsilon_{\mu\nu\lambda\rho}\ensuremath{\overline{\psi}}_\rho\nonumber\\
\kappa_{\mu\nu\lambda\rho}&=&\epsilon_{\mu\nu\lambda\rho}\ensuremath{\overline{\eta}}
\end{eqnarray}
In terms of these variables and after an additional integration
by parts the bosonic action reads (notice that terms linear in $F_{\mu\nu}$
cancel)
\begin{eqnarray}
S_B&=&\int d^4x {\rm Tr}\left[
-\frac{1}{2}F^2_{\mu\nu}-\frac{1}{2}[V_\mu,V_\nu]^2
-\left(D_\mu V_\nu\right)^2\right.\nonumber\\
&-&\left.D_\mu\phi D_\mu\ensuremath{\overline{\phi}}+\frac{1}{4}[\phi,\ensuremath{\overline{\phi}}]^2-
[\phi,V_{\mu}][\ensuremath{\overline{\phi}},V_{\mu}]
\right]
\end{eqnarray}
Making the additional rescalings $\chi\to 2\chi$ and $\ensuremath{\overline{\eta}}\to \frac{1}{\sqrt{2}}\ensuremath{\overline{\eta}}$ we find the fermion kinetic term takes the form
\begin{equation}
S_F=\int d^4 x {\rm Tr}\left[
-2\chi_{\mu\nu}D_{\left[\mu\right.}\psi_{\left.\nu\right]}
-2\chi^*_{\mu\nu}D_{\left[\mu\right.}\ensuremath{\overline{\psi}}_{\left.\nu\right]}
-2\frac{\eta}{2} D_\mu\psi_\mu
-2\frac{\ensuremath{\overline{\eta}}}{2} D_\mu\ensuremath{\overline{\psi}}_\mu
\right]
\end{equation}
where
$\chi^*_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\lambda\rho}\chi_{\lambda\rho}$
is the dual field.
In these variables the Yukawa's take on the more symmetrical
form
\begin{eqnarray}
S_Y&=&\int d^4x {\rm Tr}\left[
-\frac{\eta}{2}[\phi,\frac{\eta}{2}]
-\frac{\ensuremath{\overline{\eta}}}{2}[\phi,\frac{\ensuremath{\overline{\eta}}}{2}]
-2\chi_{\mu\nu}[\phi,\chi_{\mu\nu}]\right.\nonumber\\
&+&\left.\psi_\mu[\ensuremath{\overline{\phi}},\psi_\mu]+
\ensuremath{\overline{\psi}}_\mu[\ensuremath{\overline{\phi}},\ensuremath{\overline{\psi}}_\mu]
\right.\nonumber\\
&+&\left.2\frac{\eta}{2}[\ensuremath{\overline{\psi}}_{\mu},V_\mu]-2\frac{\ensuremath{\overline{\eta}}}{2}[\psi_\mu,V_\mu]
\right.\nonumber\\
&+&4\left.\chi_{\mu\nu}[\ensuremath{\overline{\psi}}_{\mu},V_{\nu}]-
4\chi^*_{\mu\nu}[\psi_\mu,V_{\nu}]
\right]
\end{eqnarray}
The new action can be recognized as the twist of ${\cal N}=4$ super
Yang-Mills due to Marcus \cite{marcus}.

Finally we will show how this twisted model may be reinterpreted in
terms of the usual formulation of ${\cal N}=4$ super Yang-Mills theory.
First, concentrate on the bosonic action and introduce the new
fields ($\phi=\phi_1+i\phi_2$)
\begin{eqnarray}
X^\mu&=&V_\mu\;\;\mu=0\ldots 3\nonumber\\
X^4&=&\phi_1\nonumber\\
X^5&=&\phi_2
\end{eqnarray}
Then the bosonic action may be trivially rewritten
as
\begin{equation}
S_B=-\frac{1}{2}F^2_{\mu\nu}-\left(D_\mu X^i\right)^2-
\frac{1}{2}\sum_{ij}[X_i,X_j]^2
\end{equation}
Notice this is real, positive semidefinite on account of the
antihermitian basis for the fields. It is also precisely the bosonic
sector of the ${\cal N}=4$ super Yang-Mills action with $X^i$
the usual $6$ real scalars of that theory \cite{4dspinor}.

Next let us turn our attention to the fermion kinetic term.
From our previous discussion
it should be clear that the twisted fermion kinetic term is nothing
more than the component expansion of a Dirac-K\"{a}hler  action:
\begin{equation} S_F=\frac{1}{2}{\rm Tr}\Psi^\dagger \gamma .D \Psi \end{equation}
where $\Psi$ is the Dirac-K\"{a}hler  field defined earlier with $\eta\to\eta/2$,
$\theta_{\mu\nu\lambda}\to\epsilon_{\mu\nu\lambda\rho}\ensuremath{\overline{\psi}}_\rho$
and $\kappa_{\mu\nu\lambda\rho}\to\epsilon_{\mu\nu\lambda\rho}\ensuremath{\overline{\eta}}/2$.
Naively such an action appears to describe a theory with {\it four}
Dirac spinor fields - rather than the two one would have expected
for ${\cal N}=4$ super Yang-Mills. However it is evident that $\Psi$
obeys a reality condition if its associated Dirac-K\"{a}hler  field is real. This
reduces the action to that of two degenerate
Dirac fermions. To see this in detail let us
adopt a Euclidean chiral basis for the $\gamma$-matrices
\begin{equation}
\begin{array}{cc}
\gamma_0=\left(\begin{array}{cc}
0&I\\
I&0\end{array}\right)
&
\gamma_i=\left(\begin{array}{cc}
0&-i\sigma_i\\
i\sigma_i&0\end{array}\right)
\end{array}
\end{equation}
It is straightforward to see that the $\gamma$ matrices (and any
products of them) obey a reality condition
\begin{equation}
\gamma_\mu^*=C\gamma_\mu C^{-1}\end{equation}
where the matrix $C$ is given by
\begin{equation} C=\left(\begin{array}{cc}
\sigma_2&0\\
0&\sigma_2\end{array}\right)\end{equation}
With real p-form coefficients this implies
a reality condition on $\Psi$ itself
\begin{equation} \Psi^*=C\Psi C^{-1}\end{equation}
This in turn implies that $\Psi^\dagger=C\Psi^T C^{-1}$ which makes it
clear that the result of integrating over the
Dirac-K\"{a}hler  field $\Psi$ should be the Pfaffian of the Dirac-K\"{a}hler  operator
(let us neglect the Yukawa couplings for the moment).
This, in turn, will correspond to the product of {\it two}
Dirac determinants.

To see this in more detail one can use the
reality condition to show that successive columns $\Psi^{(n)}$ of $\Psi$
are not independent but are charge conjugates of each other
\begin{equation} \Psi^{(2)}=C\left(\Psi^{(1)}\right)^*\;\;
\Psi^{(4)}=C\left(\Psi^{(3)}\right)^*\end{equation}
These conditions allow us to rewrite
the twisted fermion kinetic term in the conventional form
\begin{equation} \frac{1}{2}\sum_{\alpha=1,2}\lambda^{\dagger}_\alpha \gamma .D \lambda_\alpha\end{equation}
where the spinors are read off as the first and third columns of
the $\Psi$ matrix in this chiral basis:
\begin{equation}
\begin{array}{cc}
\lambda_1=\left(\begin{array}{c}
\frac{\eta}{2}-\frac{\ensuremath{\overline{\eta}}}{2}+2i\chi^+_{03}\\
-2\chi^+_{02}+2i\chi^+_{01}\\
\left(\psi_0+\ensuremath{\overline{\psi}}_0\right)+i\left(\psi_3+\ensuremath{\overline{\psi}}_3\right)\\
-\left(\psi_2+\ensuremath{\overline{\psi}}_2\right)+i\left(\psi_1+\ensuremath{\overline{\psi}}_1\right)
\end{array}\right)
&
\lambda_2=\left(\begin{array}{c}
\left(\psi_0-\ensuremath{\overline{\psi}}_0\right)+i\left(-\psi_3+\ensuremath{\overline{\psi}}_3\right)\\
\left(\psi_2-\ensuremath{\overline{\psi}}_2\right)+i\left(\ensuremath{\overline{\psi}}_1-\psi_1\right)\\
\frac{\eta}{2}+\frac{\ensuremath{\overline{\eta}}}{2}-2i\chi^-_{03}\\
2\chi^-_{02}-2i\chi^-_{01}
\end{array}\right)\\
\end{array}
\end{equation}
Here, $\chi^{\pm}=\frac{1}{2}\left(\chi\pm\chi^*\right)$ are the usual
self-dual and antiself-dual parts of the field.
The Yukawa's can also be put in the general form
\begin{equation} \frac{1}{2}\sum_{\alpha=1,2}\lambda^\dagger_\alpha
C^\alpha_i\Gamma^i[X^i,\lambda_\alpha]\end{equation}
where $\Gamma^i=\left\{I,\gamma_5,\gamma_\mu\gamma_5,\mu=0\ldots 4\right\}$
and the coefficients $C^\alpha_i$ are just numbers. This is just
the structure expected of ${\cal N}=4$ super Yang-Mills.
Notice that both Dirac operators $M^\alpha$ including the Yukawa
interactions possess the symmetry
\begin{equation} \left(D^\alpha\right)^*=C D^\alpha C^{-1}\end{equation}
which shows that their eigenvalues come in complex conjugate
pairs and hence the associated determinants are positive definite.

\section{Lattice ${\cal N}=4$ super Yang-Mills}
The lattice action is obtained by discretization of the
$Q$-exact action given in eqn.~\ref{qact}. Notice it is
important {\it not} to use the same dual variables as Marcus
in discretizing the theory as
the duality transformation is {\it not} compatible with the lattice
gauge transformations we have described. With
that proviso the prescription is essentially the
same as for the two dimensional theory \cite{4dsuper}. We list it now
for a $D$-dimensional theory.
\begin{itemize}
\item A continuum p-form field $f_{\mu_1\ldots \mu_p}(x)$ will be mapped
to a corresponding lattice field associated with
the $p$-dimensional hypercube at lattice site $x$
spanned by the (positively directed) unit vectors $\{\mu_1\ldots\mu_p\}$.
\item Such a lattice
field is taken to transform under gauge transformations in the following way
\begin{equation}
f_{\mu_1\ldots\mu_p}(x)\to G(x)f_{\mu_1\ldots \mu_p}(x)G^{-1}(x+e_{\mu_1\ldots\mu_p})\end{equation}
where the vector $e_{\mu_1\ldots\mu_p}=\sum_{j=1}^p\mu_j$.
\item To construct gauge invariant quantities we will need to introduce both
$f_{\mu_1\ldots\mu_p}$
and its hermitian conjugate $f^\dagger_{\mu_1\ldots\mu_p}(x)$. The
latter transforms as
\begin{equation}
f^\dagger_{\mu_1\ldots\mu_p}(x)\to G(x+e_{\mu_1\ldots\mu_p})
f_{\mu_1\ldots \mu_p}(x)G^{-1}(x)\end{equation}
Since for all fields bar the gauge field we will assume
$f_{\mu_1\ldots\mu_p}=\sum_a f^a_{\mu_1\ldots\mu_p}T^a$ where
$T^a$ are antihermitian generators of $U(N)$, this
necessitates taking the fields $f^a_{\mu_1\ldots\mu_p}$ to be
{\it complex}.
\item For a continuum gauge field we introduce lattice link
fields $U_\mu(x)=e^{A_\mu(x)}=e^{A^a_\mu(x)T^a}$ with $A^a_\mu(x)$
complex together with
its conjugate
$U^\dagger_\mu=e^{A^\dagger_\mu(x)}$.
\item A covariant forward difference operator can be defined which acts on a field
$f_{\mu_1\ldots\mu_p}(x)$ as follows
\begin{equation}
D^+_\mu f_{\mu_1\ldots\mu_p}(x)=
U_\mu(x)f_{\mu_1\ldots\mu_p}(x+\mu)-
f_{\mu_1\ldots\mu_p}(x)U_\mu(x+e_{\mu_1\ldots\mu_p})\end{equation}
This operator acts like a lattice exterior derivative with
respect to gauge transformations in mapping a $p$-form lattice field
to a $(p+1)$-form lattice field.
\item Similarly we can define an adjoint operator $D^-_\mu$
whose action on some field
$f_{\mu_1\ldots\mu_p}$ is given by
\begin{equation}
D^-_\mu f_{\mu_1\ldots\mu_p}(x)=
f_{\mu_1\ldots\mu_p}(x)U^\dagger_\mu(x+e_{\mu_1\ldots\mu_p}-\mu)-
U^\dagger_\mu(x-\mu)f_{\mu_1\ldots\mu_p}(x-\mu)\end{equation}
It thus acts like the adjoint of the exterior derivative.
\item As for two dimensions all instances of $\partial_\mu$ in the
continuum action will be replaced by $D^+_\mu$ if the derivative
acts like $d$ (curl-like operation) and $D^-_\mu$ if the derivative acts
like $d^\dagger$ (divergence-like operation).
\end{itemize}
This prescription preserves the twisted scalar supersymmetry and gauge
invariance and targets a complexified version of the
${\cal N}=4$ super Yang-Mills model.
Again, to obtain ${\cal N}=4$ SYM we conjecture that the
Ward identities corresponding to
the scalar supersymmetry will be satisfied {\it without fine tuning} in the
continuum limit $\beta\to\infty$ for the truncated lattice model
in which the fields are taken real.

\section{Conclusions}
In this paper we have presented a new way to
put supersymmetric theories on the
lattice. The approach only works for theories with a number of supercharges
which is a integer multiple of $2^D$ if $D$ is the dimension of
(Euclidean) spacetime. This includes the important cases of ${\cal N}=2$
and ${\cal N}=4$ super Yang-Mills theory in two and four dimensions
respectively.
The basic idea is to start from a {\it twisted} formulation of
the supersymmetric theory in which the fermions are represented
as grassmann valued antisymmetric tensor fields. The original supersymmetry
algebra is replaced by a twisted algebra which contains a scalar nilpotent
supercharge $Q$. Furthermore the action of the theory can then be written as
the $Q$-variation of some function.
It is straightforward to generalize the action of the scalar
supersymmetry to the lattice while maintaining its nilpotent
property. Invariance of the lattice theory under the scalar supersymmetry
can then be obtained easily. We have described the details of our
discretization prescription in detail. It requires the introduction
of particular covariant forward and backward difference operators and non standard
gauge transformation properties of the fields. The final theory is gauge
invariant, free of spectrum doubling and possesses an exact supersymmetry.
The price one pays for this is that the lattice fields are in general
complexified. We have argued that the $Q$-exactness of the action ensures
nevertheless that the theory truncated to the real line will satisfy the Ward
identities corresponding to the scalar supersymmetry without fine
tuning in the continuum limit. We have also presented preliminary
numerical results stemming from a dynamical fermion simulation of the
$D=2$ $U(2)$ model using
the new RHMC algorithm.

There are several directions for further work.
The most obvious is the need for both perturbative and numerical checks
of both the exact and broken Ward identities
corresponding to the twisted supersymmetry.
Exactly how much residual fine tuning would be
required in the four dimensional theory is currently unclear and could
be revealed by such an analysis.
It should also be possible to use these ideas to
construct lattice regularizations for the ${\cal N}=4,8$ theories
in two dimensions and the ${\cal N}=8$ theory in three dimensions.
This would allow contact to be made with predictions from
string theory and supergravity theories in
lattice models which could be simulated relatively easily and with
high precision.
The geometrical nature of the twisted theory lends one to speculate that it
should be possible to formulate these models on general simplicial
lattices. If this could be realized, then the resulting
models could be coupled to gravity in the usual way by
carrying out a sum over random lattices resulting in a lattice
regulated supergravity theory. It would also be very interesting to
understand the relationship, if any, between twisting and the orbifold constructions
of Kaplan at al.

It appears that the twisted formulation of a theory with extended
supersymmetry affords a more useful starting point for
constructing lattice models over conventional approaches based
on spinors. However,
the precise details of the discretization procedure are not determined
uniquely. While this work was in preparation a new paper by
D'Adda et al \cite{2dfullQ} was posted which claims to maintain
{\it all} the supercharges of the two dimensional twisted theory by
adopting a different mapping of the continuum twisted theory to the
lattice. This appears to be a very intriguing development.

redpajama_set_name: RedPajamaArXiv


text:
The Etihad Tennis Centre In Manchester Will Transform Into A Mass COVID-19 Vaccination Centre
Laura Rogan - Editor • January 6, 2021
It will be one of seven to open in the UK next week.
Pop-up vaccination centres are set to open in the UK next week, with the Etihad Tennis Centre now confirmed to be one of seven to open first. Starting from Monday, the space will be used by NHS staff and volunteers to administer the vaccine – with the government aiming to vaccinate 14 million by February 15th. 1.3 million have so far received the vaccine, according to the Prime Minister.
Credit: Better.org.uk
Covid Vaccine Deployment Minister Nadhim Zahaw told Sky News ahead of the announcement: "I'm confident that as we begin to deploy and get more sites operational – I talked about the hospitals, the GPs, the community pharmacies and the national vaccination centres – so we will be at over 1,000 sites vaccinating.
"I'm confident you will see, certainly from January 4 to January 11, a real step up. So we've done over 1.3 million since December 8, you're going to see a massive acceleration of that."
Here's the full list of locations announced today:
Etihad Tennis Centre, Manchester
Robertson House, Stevenage
Centre for Life, Newcastle
Epsom Race Course, Surrey
Ashton Gate Stadium, Bristol
Millennium Point, Birmingham
Excel Centre, London
The vulnerable will be continued to be prioritised during the vaccine roll-out, with Oldham Council confirming just yesterday that they would be adding the homeless to their priority list. The new mass vaccination centres will open alongside 775 GP-led centres and 207 hospital sites that are also confirmed to open by the end of this week.
Here's the list of who will be vaccinated during 'Phase 1', in order of priority:
Care home residents and their carers
Over 80's and frontline workers
Over 70's and the clinically vulnerable
Anyone over 16 with underlying health issues
Anyone outside of the 'Phase 1' priority list will need to wait until the second phase of vaccinations to receive the jab. A roll-out date for the second phase is currently TBC, and is not expected to begin until prioritised patients are fully vaccinated.
Read more: International Travellers Must Provide Negative Covid-19 Test Before Entering The UK

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: mx record for a mail sender only server For configuring the dns for a mail sender server and go trought the antispam policies,
Is it necessary the mx record for it?

A: No. MX records are only for mail exchangers, which a send-only server is not.

A: With regards to anti-spam policies.
You may also need to look at ensuring you have a PTR DNS entry:
We've come across rejections of our mail service by other users because we migrated to another ISP (new IP Addresses) and didn't get the PTR entry updated fast enough.
Wikipedia

The most common uses of the reverse DNS include:
... One e-mail anti-spam technique: checking the domain names in the rDNS to see if they are likely from dialup users, dynamically assigned addresses, or other inexpensive internet services. Owners of such IP addresses typically assign them generic rDNS names such as "1-2-3-4-dynamic-ip.example.com." Since the vast majority, but by no means all, of e-mail that originates from these computers is spam, many spam filters refuse e-mail with such rDNS names.

Basic DNS: PTR records and why you care

You may have come to this because you've been told that you need a PTR record to avoid rejections from sites like AOL and others.
You may have thought that you can add your own PTR record. You usually cannot.
A PTR record (sometimes called a "host PTR record") is what lets someone do a "reverse" DNS lookup - that is, they have your IP address and want to know what your host/domain is.

A: You need an SPF record, not an MX record.

A: If you're following John's advice of setting up the SPF record, you may want to check out this wizard. It will walk you through the creation of your SPF record.
http://www.unlocktheinbox.com/spfwizard/
Hope that helps some. Good luck.

redpajama_set_name: RedPajamaStackExchange


text:
MOVIE REVIEW- Indomitable <i>Invictus: </i>Eastwood and Freeman make it look easy
By STEVE WARREN | FILM@READTHEHOOK.COM
Published online 9:00am Thursday Dec 10th, 2009
and in print issue #0849 dated Thursday Dec 10th, 2009
PUBLICITY PHOTO
With one of my acting gods, Morgan Freeman, playing one of my political gods, Nelson Mandela, Invictus couldn't help being a spiritual experience for me. Freeman's every line of dialogue– even something as simple as "Good morning"– sounded like a pronouncement from on high. That makes it hard to write an objective review. I know the movie's good, but I'm not sure it's great.
Clint Eastwood makes directing look easy, telling stories in a no-frills style that doesn't call attention to itself and remind you who's behind the camera. Freeman does the same thing as an actor, and the story they're telling is of Mandela finding relatively simple ways to begin the healing between the black and white populations of South Africa.
With everyone making it look easy, it's easy to undervalue the incredible accomplishments of all concerned; but I was certainly moved, on the verge of tears, for most of the two hours-plus.
Mandela's 1990 release after 27 years in the Robben Island prison and subsequent 1994 election as president of South Africa are covered in the first four minutes of the film; then the work begins. A newspaper headline blares, "He can win an election but can he run a country?" "It's a legitimate question," Mandela says (and one most Americans will recall having heard closer to home in the past year).
Arriving at work on the first day of his term and seeing the former administration's all-white staff packing up, Mandela invites them to stay. He can't be president of a "rainbow nation" if all the people aren't represented in his office. This strategy extends to his personal security team, where Jason Tshabalala (Tony Kgoroge) and Linga Moonsamy (Patrick Mofokeng) are surprised to be augmented by Hendrick Booyens (Matt Stern) and Etienne Feyder (Julian Lewis), former members of the Special Branch that used violence against black people.
With one of their own running the country the blacks expect vengeance and retribution for what they were subjected to under more than four decades of apartheid, but Mandela knows the country can't be united by transferring bitterness and rage from one side to the other. When the national rugby team, the Springboks, is about to be stripped of its status, Mandela intervenes.
The team is such a symbol of apartheid that whenever they play the country's blacks root for their opponent. That they're perennial losers doesn't give the whites much to be proud of, but proud they are. South Africa is hosting the 1995 Rugby World Cup Championship, and Mandela sees a chance to unite the nation behind the Springboks.
He starts by inviting team captain Francois Pienaar (Matt Damon) to tea, where they discuss leadership and inspiration and find some common ground. Though he grew up under institutionalized racism in a home where the maid's feelings were ignored, we don't know where Pienaar stood before meeting Mandela; but we see how he's affected by that meeting and by things that happen in the following year.
The team trains harder but it's not clear – unless you want to credit a miracle – how they transform from underdogs to top dogs in that time. Mandela sends them into the townships to conduct workshops with children for public relations purposes, and it works. At first the black kids are only excited to see the one black player, Chester Williams (McNeil Hendricks), but after the play with the whole team the color barriers fall away. The president's support of the team wins him some respect from the white community.
It should be noted that rugby is a lot like American football except the players don't wear helmets and padding. It's a rough game and they make our footballers look like pussies.
Much of the film's second half takes place during the World Cup games, as the Springboks defeat Australia, Western Samoa, France and move on to face the unbeatable (i.e., invictus) New Zealand team. By this time blacks and whites alike are rooting for the home team and small moments, some clichéd and some clever, show the races interacting in ways that would have been unthinkable a year earlier.
The film's title comes from the William Ernest Henley poem that ends, "I am the master of my fate: I am the captain of my soul." Mandela, who found inspiration from it in prison, passes it on to Pienaar.
The screenplay by South African native Anthony Peckham, from the book Playing the Enemy by John Carlin, makes everything seem so smooth that some false alarms have to be invented to provide some tension.
Judging from one white South African I met a year ago, things aren't as kumbaya there as the movie makes them look (it took the U.S. a century to begin to heal from slavery), but Mandela deserves his Nobel Peace Prize and Invictus deserves any prizes it wins from people who can appreciate Mandela's genius and Eastwood and Freeman's skill.
Andrew December 23rd, 2009 | 7:38am
Can't wait to see this movie! Invictus has a really cool Facebook fan page too: check it out: http://www.ropeofsilicon.com/article/reviews-for-eastwoods-invictus-begi...
Whoops- wrong link :) http://www.facebook.com/search/?q=invictus&init=quick#/invictus?v=wall&r...


text:
brother ken / 2017-12-14 09:02
09:02:11 Mike Scott Bring 'em All In Bring 'em All In
09:05:18 Bob Dylan Mixed-Up Confusion Biography
09:07:42 The Rolling Stones You Can Make It If You Try The Rolling Stones
09:10:13 Them Bring 'Em On In Them featuring Van Morrison
09:13:21 Lone Justice After The Flood Lone Justice
09:17:27 John Lennon Look At Me Plastic Ono Band
09:20:18 Paul Simon ME and Julio Down by the Schoolyard Paul Simon
09:23:02 The Guess Who No Time The Best of the Guess Who
09:26:44 The Monkees Daydream Believer Then & Now...
09:29:57 Rilo Kiley Portions for Foxes More Adventurous
09:34:21 Morrissey Certain People I know "Your Arsenal"
09:38:11 The Like We Are Lost Are You Thinking What I'm Thinking?
09:45:15 Lou Reed Sad Song Berlin
09:52:14 Men At Work No Sign Of Yesterday Cargo
09:57:57 The Beatles You've Got To Hide Your Love Away The Beatles 1962-1968
10:00:21 Sly & The Famliy Stone If You Want Me To Stay Dead Presidents
10:03:40 Nikka Costa Everybody Got Their Something Everybody Got Their Something
10:08:22 Stevie Wonder Don't You Worry 'Bout A Thing Innervisions
10:12:49 Prince The Same December Chaos & Disorder
10:16:48 The Police Too Much Information Ghost in the Machine
10:20:19 Echo And The Bunnymen Read It In Books Crocodiles
10:22:51 That Petrol Emotion Can't Stop Manic Pop Thrill
10:25:43 Joe Strummer Passport To Detroit Earthquake Weather
10:31:52 Jonathan Richman and the Modern Lovers I Was Dancing in the Lesbian Bar I, Jonathan
10:36:33 Bongwater Lesbians Of Russia Double Bummer
10:38:02 Sleater-Kinney Dig Me Out Dig Me Out
10:40:58 fIREHOSE Towin' The Line Flyin' The Flannel
10:44:24 The Waterboys Red Army Blues A Pagan Place
10:52:36 Tori Amos Yes, Anastasia Under The Pink
11:02:02 Walter Murphy A Fifth of Beethoven Saturday Night Fever Soundtrack
11:04:59 Queen The Prophet's Song A Night At The Opera
11:13:20 Tommy James And The Shondells Crimson And Clover Crimson And Clover
11:18:48 The Beach Boys In My Room Surfer Girl
11:25:02 Mull Historical Society The Final Arrears Us
11:29:06 Stump Living It Down A Fierce Pancake
11:30:58 Flipper Way Of The World Generic Flipper
11:35:22 Devo Uncontrollable Urge Are We Not Men? We Are Devo
11:40:48 Moby Feeling So Real Everything Is Wrong
11:49:17 Afro Celt Sound System with Peter Gabriel When You're Falling Volume 3: Further in Time.
11:50:33 Lorde A World Alone Pure Heroine
11:55:06 The Soft Boys If You Know Time Live @ WFMU
11:57:27 Fugazi Shut the Door Repeater


text:
/*
 * TelnetEmulator.java
 *
 * Created on 22 Feb 2006, 16:38
 *
 * To change this template, choose Tools | Options and locate the template under
 * the Source Creation and Management node. Right-click the template and choose
 * Open. You can then make changes to the template in the Source Editor.
 */

package guiUI;

import java.awt.BorderLayout;
import java.awt.Color;
import java.awt.Font;
import java.awt.event.KeyAdapter;
import java.awt.event.KeyEvent;
import java.awt.event.WindowEvent;
import java.util.Vector;

import javax.swing.JFrame;
import javax.swing.JPanel;
import javax.swing.JScrollPane;
import javax.swing.JTextArea;
import javax.swing.JTextField;

import core.protocolsuite.tcp_ip.PosixTelnetClient;

/**
 *
 * @author key
 */
public class PosixTelnetClientGUI extends JFrame {
    /**
	 *
	 */
	private static final long serialVersionUID = -471040634350408360L;
	private JPanel panel;
    private JScrollPane scrollpane;
    private JTextArea terminal;
    private JTextField cmdline;
    private String text;
    //private KeyListener kl;
    //private core.NetworkLayerDevice device;
    MainScreen parent;
    private Vector<String> history = new Vector<String>(0);
    private int pos_history=0;
    private final static int max_history = 128;
    private PosixTelnetClient ptc;
    private String Host;
    private int port;
    public boolean busy;

    /** Creates a new instance of TelnetEmulator */
    public PosixTelnetClientGUI(MainScreen parent, core.ApplicationLayerDevice dev, String Host, int port) {
        super("Console: " + dev.NodeProtocolStack.getParentNodeName());
        //device = dev;
        this.Host = Host;
        this.port = port;
        this.parent = parent;

        ptc = (PosixTelnetClient) dev.getApp(10023);

        text = "Connecting to " + Host + ":" + port + "...\n";
        panel = new JPanel();
        scrollpane = new JScrollPane();
        terminal = new JTextArea(text);
        cmdline = new JTextField();
        this.setContentPane(panel);
        panel.setLayout(new java.awt.BorderLayout());
        panel.add(scrollpane);
        panel.add(cmdline, BorderLayout.SOUTH);
        panel.setPreferredSize(new java.awt.Dimension(600,500));
        scrollpane.setViewportView(terminal);
        terminal.setEnabled(true);
        terminal.setEditable(false);
        terminal.setBackground(Color.BLACK);
        terminal.setForeground(Color.WHITE);
        terminal.setFont(new Font("Courier New", Font.PLAIN, 12));
        cmdline.setFocusable(true);

        //cmdline.addKeyListener( kl = new KeyAdapter(){
        cmdline.addKeyListener( new KeyAdapter(){
            @Override
			public void keyPressed(KeyEvent e) {
                char c = e.getKeyChar();
                switch(c) {
                    case 0x0A: {
                        if(cmdline.getText().compareTo("")!=0) {
                            if(history.size()>=PosixTelnetClientGUI.max_history) history.remove(0);
                            history.add(cmdline.getText());
                            pos_history = history.size();
                        }
                        String toSend = cmdline.getText() + "\r\n";
                        try{
                            ptc.SendData(toSend);
                        }catch(Exception exc){

                        }
                        //text += cmdline.getText() + "\r\n";
                        cmdline.setText("");

                        printInfo();
                        break;
                    }
                    case 0x04: {
                        exitWindow();
                        break;
                    }
                    case 0x1B: {
                        cmdline.setText("");
                        break;
                    }
                    default: {
                        if(c>=65535) {
                            int i = e.getKeyCode();
                            switch(i) {
                                case 0x26: {
                                    if(history.size()>0) {
                                        if(pos_history == history.size() && cmdline.getText().compareTo("")!=0) {
                                            if(history.size()>=PosixTelnetClientGUI.max_history) history.remove(0);
                                            history.add(cmdline.getText());
                                            pos_history = history.size()-1;
                                        }
                                        if(pos_history>0) {
                                            cmdline.setText(history.get(--pos_history));
                                        }
                                        else if(pos_history>0) {
                                            cmdline.setText(history.get(pos_history-1));
                                        }
                                    }
                                    break;
                                }
                                case 0x28: {
                                    if(history.size()>0) {
                                        if(pos_history+1<history.size()) {
                                            cmdline.setText(history.get(++pos_history));
                                        }
                                    }
                                    break;
                                }
                                default: //terminal.setText(text+"\n"+Integer.toHexString(i));
                            }
                        }
                        else {
                            pos_history = history.size();
                        }
                        break;
                    }
                }
            }
        });

        this.addWindowListener( new java.awt.event.WindowAdapter() {
            @Override
			public void windowClosing(WindowEvent winEvt) {
                printInfo();
            }
        });

    }

    public void start(){
        try{
            ptc.Telnet(this, Host, port);
        }catch(Exception e){

        }
    }

    public void recvData(String Data){
        busy = true;
        text += Data + "\n";
        terminal.setText(text);
        busy = false;
        //try{
        //    ptc.SendData("");
        //}catch(Exception e){
        //}
    }

    public void exitWindow() {
        printInfo();
        this.dispose();
    }

    public void printInfo(){
        parent.printLayerInfo(false);
        //!!!!!: add more headers here
    }

    private String removeSpaces(String s) {
        while(s.startsWith(" ")) s = s.substring(1);
        while(s.endsWith(" ")) s = s.substring(0, s.length()-1);
        return s;
    }
}

redpajama_set_name: RedPajamaGithub


By default, Shibboleth is configured to log IdP messages at level INFO.
By default, Shibboleth is configured to log SAML messages at level INFO.
By default, Shibboleth is configured to log LDAP messages at level INFO.
Any of these logger's can be configured to log at the following levels.
Based on the log level selected, there should be logging events in the $shibboleth_home/log/ files.


A lifestyle and travel blogger, , loves the platform as it facilitated a lot of sponsored post opportunities for her. This is a great way to increase engagement. For businesses who frequently use Twitter, you can also link your Instagram to your Twitter to reap the benefits on that platform, too. Also, make sure that you are using the same filter across all of your pictures. Really dive deep into your market until your understand their habits better than you understand your own. In their , Ahrefs found that only 22% of pages currently ranking in top 10 are less than an year old. Many will follow you back! This will ensure that people who are already in contact with your brand online can find you on social media.
Snezana Markoski was when she promoted a coffee scrub on her Instagram account with her daughter. But, there are still entrepreneurs out there who would rather work together than try and outcompete every other user. Discover how to create Instagram posts that increase engagement and drive sales. From personal experience, I can tell you that Instagram gets a lot more fun when more people are following you! The apps you use are just tools. My personal favorite strategy for Story giveaways is to get people to post photos using a specific hashtag so you can find them , and then reposting their photos on your Story. Finally, he emailed all the people that showed interest, addressing their objections and offering them an option to buy his course.
How to Fix It A major problem brands discover when they , is the images and videos at the end of their Stories receive less attention. Share customer testimonials, stories and reviews It's one thing for you to tell your followers that your products are awesome and that they should buy them. Of course, it's not enough to simply identify your personality and infuse it within your posts whenever you feel like it. This encourages your fans to engage with your brand, and increases your visibility at the same time. What is an Instagram Nametag? If you understand your audience well and they respond to your endorsements, then you can quote a value-based price instead of a fixed dollar for every post. Join an Instagram community group Instagram is competitive. Pictures and videos see higher rates of engagement, and with Facebook now , images are becoming more and more valuable.
. Post consistently Your existing followers want to see content from you. Use a tool like to combine up to 9 photos or videos. This is where the magic happens! I am quite excited for utilizing Instagram to showcase my Jewelry. From a single dashboard you can schedule and publish your perfectly edited photos directly to Instagram, engage your audience, measure performance, and run all your other social media profiles.
So Instagram presents a great opportunity for smart folks to quickly build an audience and monetize it. Depending on your skill, experience, and pitch, you can close your first client by tomorrow and start minting dollars. People using these hashtags are more likely to enjoy your profile and follow you. When you offer a special discount to your followers. Foundr claims to have reached with their landing pages because of the insane value they offer.
All you need to do is use hashtags to search for people who do the same kind of photography as you. Discover six tools to improve your Instagram marketing. You can find more tips, along with plenty of great brand examples, in our. Follow me on Twitter JaysonDeMers. Here are 5 tips to market your Airbnb: 1.
Search for and follow people who are using popular hashtags like followme and likeforlike. Some of the most popular ones are love, instagood, tbt and photooftheday. One bonus tip is to incentivize people to follow you. You might find that following lots of people is impractical. They work brilliantly for increasing your social following because they directly reward entrants who engage with your business on social media For example, someone who enters your bonus entry contest contest might earn 5 extra entries for following you on Instagram, 3 entries for Tweeting the contest link, or 10 entries for getting a friend to enter the contest, too. It helps people to recognize you across social media if you use the same photo. Better visualize your Stories to see where spikes and dips occur in your content.
Try creating signage with your Instagram Nametag to gain more followers. Do you have any recommendations for making better Instagram Stories? If you offer any special amenities like local guides, maps, cuisine, then share that as well. To give you an idea, here are the. Though you might not necessarily be out to change the marketing game, consider partnering with other brands whose products complement your own to run joint marketing campaigns. And now they sell their magazine issues and products on creating an online store and Instagram marketing among others. Utilizing different content forms allows you to maximize your reach on the platform - it's can often be more interesting for a follower to see a video on their feed amongst hundreds of other photos.
I recommend checking out for inspiration. Use apps:Use photo-editing apps to enhance your photos before you post them Instagram. We trust text to tell us what we see. Remember, your guests are on a vacation unlike you — always being on the road. Having a Business account allows you to get unique metrics on your posts and Stories, insights into your followers, and add information about your company like your business hours, location, and phone number.
You can also use text to add context. If you've just recently posted a new photo or video, use your Story to direct people to it. And after using this technique for a while, your engagement and following will start to grow organically. Make it a limited-time offer to increase urgency, and then link to the product page or your website through your Story. YouTube to creators since 2007.


text:
/****** Object:  StoredProcedure [AddE09_Region_ReChild] ******/
IF EXISTS (SELECT * FROM sys.objects WHERE object_id = OBJECT_ID(N'[AddE09_Region_ReChild]') AND type in (N'P', N'PC'))
    DROP PROCEDURE [AddE09_Region_ReChild]
GO

CREATE PROCEDURE [AddE09_Region_ReChild]
    @Region_ID2 int,
    @Region_Child_Name varchar(50)
AS
    BEGIN

        SET NOCOUNT ON

        /* Insert object into 4_Regions_ReChild */
        INSERT INTO [4_Regions_ReChild]
        (
            [Region_ID2],
            [Region_Child_Name]
        )
        VALUES
        (
            @Region_ID2,
            @Region_Child_Name
        )

    END
GO

/****** Object:  StoredProcedure [UpdateE09_Region_ReChild] ******/
IF EXISTS (SELECT * FROM sys.objects WHERE object_id = OBJECT_ID(N'[UpdateE09_Region_ReChild]') AND type in (N'P', N'PC'))
    DROP PROCEDURE [UpdateE09_Region_ReChild]
GO

CREATE PROCEDURE [UpdateE09_Region_ReChild]
    @Region_ID2 int,
    @Region_Child_Name varchar(50)
AS
    BEGIN

        SET NOCOUNT ON

        /* Check for object existence */
        IF NOT EXISTS
        (
            SELECT [Region_ID2] FROM [4_Regions_ReChild]
            WHERE
                [Region_ID2] = @Region_ID2 AND
                [IsActive] = 'true'
        )
        BEGIN
            RAISERROR ('''E09_Region_ReChild'' object not found. It was probably removed by another user.', 16, 1)
            RETURN
        END

        /* Update object in 4_Regions_ReChild */
        UPDATE [4_Regions_ReChild]
        SET
            [Region_Child_Name] = @Region_Child_Name
        WHERE
            [Region_ID2] = @Region_ID2

    END
GO

/****** Object:  StoredProcedure [DeleteE09_Region_ReChild] ******/
IF EXISTS (SELECT * FROM sys.objects WHERE object_id = OBJECT_ID(N'[DeleteE09_Region_ReChild]') AND type in (N'P', N'PC'))
    DROP PROCEDURE [DeleteE09_Region_ReChild]
GO

CREATE PROCEDURE [DeleteE09_Region_ReChild]
    @Region_ID2 int
AS
    BEGIN

        SET NOCOUNT ON

        /* Check for object existence */
        IF NOT EXISTS
        (
            SELECT [Region_ID2] FROM [4_Regions_ReChild]
            WHERE
                [Region_ID2] = @Region_ID2 AND
                [IsActive] = 'true'
        )
        BEGIN
            RAISERROR ('''E09_Region_ReChild'' object not found. It was probably removed by another user.', 16, 1)
            RETURN
        END

        /* Mark E09_Region_ReChild object as not active */
        UPDATE [4_Regions_ReChild]
        SET    [IsActive] = 'false'
        WHERE
            [4_Regions_ReChild].[Region_ID2] = @Region_ID2

    END
GO


text:
Movies & Shows I want to watch
by christianhernandez | created - 27 Mar 2016 | updated - 4 months ago | Public
Prime Video (3)
IMDb TV (1)
Action (18) Adventure (18) Comedy (16) Drama (16) Sci-Fi (11) Fantasy (10) Thriller (10) Romance (7) Animation (6) Family (6) Mystery (4) Crime (3) Horror (3) Music (3) Biography (1) Musical (1)
Feature Film (32)
Punched In The Face (19)
Slow Motion Scene (16)
No Opening Credits (15)
Fight (14)
Rescue (14)
Surprise Ending (14)
Bare Chested Male (13)
Cell Phone (13)
Falling From Height (13)
Flashback (13)
Near Death Experience (13)
Opening Action Scene (13)
Betrayal (12)
Brawl (12)
Car Crash (12)
Fistfight (12)
Knife (12)
Pistol (12)
Race Against Time (12)
Sequel (12)
Ambush (11)
Double Cross (11)
Held At Gunpoint (11)
Hostage (11)
Husband Wife Relationship (11)
Shot In The Chest (11)
Father Son Relationship (10)
Foot Chase (10)
Hand To Hand Combat (10)
Punched In The Chest (10)
Shot In The Back (10)
Shot To Death (10)
Showdown (10)
Subjective Camera (10)
Disarming Someone (9)
Exploding Car (9)
F Word (9)
Shootout (9)
Tough Guy (9)
Tracking Device (9)
Aerial Shot (8)
Character's Point Of View Camera Shot (8)
Disguise (8)
Final Showdown (8)
Handcuffs (8)
Hit By A Car (8)
Kicked In The Face (8)
Knocked Out (8)
News Report (8)
Shot In The Head (8)
Teenage Girl (8)
Telephone Call (8)
Tough Girl (8)
American Abroad (7)
1. Everest (2015)
PG-13 | 121 min | Action, Adventure, Biography
The story of New Zealand's Robert "Rob" Edwin Hall, who on May 10, 1996, together with Scott Fischer, teamed up on a joint expedition to ascend Mount Everest.
Director: Baltasar Kormákur | Stars: Jason Clarke, Ang Phula Sherpa, Thomas M. Wright, Martin Henderson
Votes: 182,022 | Gross: $43.48M
2. Gravity (2013)
PG-13 | 91 min | Drama, Sci-Fi, Thriller
Two astronauts work together to survive after an accident leaves them stranded in space.
Director: Alfonso Cuarón | Stars: Sandra Bullock, George Clooney, Ed Harris, Orto Ignatiussen
Votes: 713,781 | Gross: $274.09M
3. Hitman: Agent 47 (2015)
R | 96 min | Action, Adventure, Crime
An assassin teams up with a woman to help her find her father and uncover the mysteries of her ancestry.
Director: Aleksander Bach | Stars: Rupert Friend, Hannah Ware, Zachary Quinto, Ciarán Hinds
4. Mad Max: Fury Road (2015)
R | 120 min | Action, Adventure, Sci-Fi
In a post-apocalyptic wasteland, a woman rebels against a tyrannical ruler in search for her homeland with the aid of a group of female prisoners, a psychotic worshiper, and a drifter named Max.
Director: George Miller | Stars: Tom Hardy, Charlize Theron, Nicholas Hoult, Zoë Kravitz
5. Magic Mike XXL (2015)
R | 115 min | Comedy, Drama, Music
Three years after Mike bowed out of the stripper life at the top of his game, he and the remaining Kings of Tampa hit the road to Myrtle Beach to put on one last blow-out performance.
Director: Gregory Jacobs | Stars: Channing Tatum, Joe Manganiello, Matt Bomer, Adam Rodriguez
6. Mission: Impossible - Rogue Nation (2015)
PG-13 | 131 min | Action, Adventure, Thriller
Ethan and team take on their most impossible mission yet, eradicating the Syndicate - an International rogue organization as highly skilled as they are, committed to destroying the IMF.
Director: Christopher McQuarrie | Stars: Tom Cruise, Rebecca Ferguson, Jeremy Renner, Simon Pegg
7. Paper Towns (2015)
PG-13 | 109 min | Comedy, Drama, Mystery
After an all-night adventure, Quentin's lifelong crush, Margo, disappears, leaving behind clues that Quentin and his friends follow on the journey of a lifetime.
Director: Jake Schreier | Stars: Nat Wolff, Cara Delevingne, Austin Abrams, Justice Smith
8. Pete's Dragon (2016)
PG | 102 min | Action, Adventure, Comedy
The adventures of an orphaned boy named Pete and his best friend Elliot, who just so happens to be a dragon.
Director: David Lowery | Stars: Bryce Dallas Howard, Robert Redford, Oakes Fegley, Oona Laurence
9. Silver Linings Playbook (2012)
R | 122 min | Comedy, Drama, Romance
After a stint in a mental institution, former teacher Pat Solitano moves back in with his parents and tries to reconcile with his ex-wife. Things get more challenging when Pat meets Tiffany, a mysterious girl with problems of her own.
Director: David O. Russell | Stars: Bradley Cooper, Jennifer Lawrence, Robert De Niro, Jacki Weaver
10. Spectre (I) (2015)
A cryptic message from 007's past sends him pitted against a mysterious terrorist organization called Spectre, and learns of its involvement in previous events of his most dangerous missions.
Director: Sam Mendes | Stars: Daniel Craig, Christoph Waltz, Léa Seydoux, Ralph Fiennes
11. Spy (2015)
R | 119 min | Action, Comedy, Crime
A desk-bound CIA analyst volunteers to go undercover to infiltrate the world of a deadly arms dealer, and prevent diabolical global disaster.
Director: Paul Feig | Stars: Melissa McCarthy, Rose Byrne, Jude Law, Jason Statham
12. The Kings of Summer (2013)
R | 95 min | Adventure, Comedy, Drama
Three teenage friends, in the ultimate act of independence, decide to spend their summer building a house in the woods and living off the land.
Director: Jordan Vogt-Roberts | Stars: Nick Robinson, Gabriel Basso, Moises Arias, Nick Offerman
Votes: 73,498 | Gross: $1.29M
13. The Longest Ride (2015)
PG-13 | 123 min | Drama, Romance
The lives of a young couple intertwine with a much older man, as he reflects back on a past love.
Director: George Tillman Jr. | Stars: Scott Eastwood, Britt Robertson, Alan Alda, Jack Huston
14. The Perks of Being a Wallflower (2012)
PG-13 | 103 min | Drama
An introvert freshman is taken under the wings of two seniors who welcome him to the real world.
Director: Stephen Chbosky | Stars: Logan Lerman, Emma Watson, Ezra Miller, Paul Rudd
15. The Right Kind of Wrong (2013)
Leo the dishwasher falls in love with a bride on the day of her wedding - to another man.
Director: Jeremiah S. Chechik | Stars: Ryan Kwanten, Kristen Hager, Maria Menounos, Jamani
Votes: 9,007 | Gross: $0.00M
16. This Is the End (2013)
R | 107 min | Comedy, Fantasy
6 Los Angeles celebrities are stuck in James Franco's house after a series of devastating events just destroyed the city. Inside, the group not only have to face the apocalypse, but themselves.
Directors: Evan Goldberg, Seth Rogen | Stars: James Franco, Jonah Hill, Seth Rogen, Jay Baruchel
17. Trainwreck (2015)
Having thought that monogamy was never possible, a commitment-phobic career woman may have to face her fears when she meets a good guy.
Director: Judd Apatow | Stars: Amy Schumer, Bill Hader, Brie Larson, Colin Quinn
18. Mike and Dave Need Wedding Dates (2016)
R | 98 min | Adventure, Comedy, Romance
Two hard-partying brothers place an online ad to find the perfect dates for their sister's Hawaiian wedding. Hoping for a wild getaway, the boys instead find themselves out-hustled by an uncontrollable duo.
Director: Jake Szymanski | Stars: Zac Efron, Adam Devine, Anna Kendrick, Aubrey Plaza
19. Goat (2016)
R | 96 min | Drama
Reeling from a terrifying assault, a 19 year-old enrolls into college with his brother and pledges the same fraternity. What happens there, in the name of "brotherhood," tests him and his loyalty to his brother in brutal ways.
Director: Andrew Neel | Stars: Ben Schnetzer, Nick Jonas, Gus Halper, Danny Flaherty
20. Ash vs Evil Dead (2015–2018)
TV-MA | 30 min | Action, Comedy, Fantasy
Ash has spent the last thirty years avoiding responsibility, maturity, and the terrors of the Evil Dead until a Deadite plague threatens to destroy all of mankind and Ash becomes mankind's only hope.
Stars: Bruce Campbell, Ray Santiago, Dana DeLorenzo, Lucy Lawless
21. Dominion (2014–2015)
TV-MA | 90 min | Action, Drama, Fantasy
Follows the perilous journey of a rebellious young soldier who discovers he's the unlikely savior of humanity.
Stars: Christopher Egan, Tom Wisdom, Roxanne McKee, Luke Allen-Gale
22. Under the Dome (2013–2015)
TV-14 | 43 min | Drama, Mystery, Sci-Fi
An invisible and mysterious force field descends upon a small actual town of Chester's Mill, Maine, USA, trapping residents inside, cut off from the rest of civilization. The trapped townspeople must discover the secrets and purpose of the "dome" or "sphere" and its origins, while coming to learn more than they ever knew about each other and animals too.
Stars: Mike Vogel, Rachelle Lefevre, Alexander Koch, Colin Ford
23. Sherlock (2010– )
24. Popstar: Never Stop Never Stopping (2016)
R | 87 min | Comedy, Music
When it becomes clear that his solo album is a failure, a former boy band member does everything in his power to maintain his celebrity status.
Directors: Akiva Schaffer, Jorma Taccone | Stars: Andy Samberg, Jorma Taccone, Akiva Schaffer, Sarah Silverman
25. Mission: Impossible - Fallout (2018)
Ethan Hunt and his IMF team, along with some familiar allies, race against time after a mission gone wrong.
Director: Christopher McQuarrie | Stars: Tom Cruise, Henry Cavill, Ving Rhames, Simon Pegg
26. Jurassic World: Fallen Kingdom (2018)
PG-13 | 128 min | Action, Adventure, Sci-Fi
When the island's dormant volcano begins roaring to life, Owen and Claire mount a campaign to rescue the remaining dinosaurs from this extinction-level event.
Director: J.A. Bayona | Stars: Chris Pratt, Bryce Dallas Howard, Rafe Spall, Justice Smith
27. Bumblebee (2018)
On the run in the year 1987, Bumblebee finds refuge in a junkyard in a small California beach town. On the cusp of turning 18 and trying to find her place in the world, Charlie Watson discovers Bumblebee, battle-scarred and broken.
Director: Travis Knight | Stars: Hailee Steinfeld, Jorge Lendeborg Jr., John Cena, Jason Drucker
Votes: 99,100 | Gross: $127.20M
28. Ralph Breaks the Internet (2018)
PG | 112 min | Animation, Adventure, Comedy
Six years after the events of "Wreck-It Ralph," Ralph and Vanellope, now friends, discover a wi-fi router in their arcade, leading them into a new adventure.
Directors: Phil Johnston, Rich Moore | Stars: John C. Reilly, Sarah Silverman, Gal Gadot, Taraji P. Henson
29. Incredibles 2 (2018)
PG | 118 min | Animation, Action, Adventure
The Incredibles hero family takes on a new mission, which involves a change in family roles: Bob Parr (Mr Incredible) must manage the house while his wife Helen (Elastigirl) goes out to save the world.
Director: Brad Bird | Stars: Craig T. Nelson, Holly Hunter, Sarah Vowell, Huck Milner
30. Mortal Engines (2018)
PG-13 | 128 min | Action, Adventure, Fantasy
In a post-apocalyptic world where cities ride on wheels and consume each other to survive, two people meet in London and try to stop a conspiracy.
Director: Christian Rivers | Stars: Hera Hilmar, Robert Sheehan, Hugo Weaving, Jihae
31. The Lego Movie 2: The Second Part (2019)
It's been five years since everything was awesome and the citizens are facing a huge new threat: Lego Duplo invaders from outer space, wrecking everything faster than they can rebuild.
Director: Mike Mitchell | Stars: Chris Pratt, Elizabeth Banks, Will Arnett, Tiffany Haddish
32. A Star Is Born (2018)
R | 136 min | Drama, Music, Romance
A musician helps a young singer find fame as age and alcoholism send his own career into a downward spiral.
Director: Bradley Cooper | Stars: Lady Gaga, Bradley Cooper, Sam Elliott, Greg Grunberg
33. Spider-Man: Into the Spider-Verse (2018)
Teen Miles Morales becomes Spider-Man of his reality, crossing his path with five counterparts from other dimensions to stop a threat for all realities.
Directors: Bob Persichetti, Peter Ramsey, Rodney Rothman | Stars: Shameik Moore, Jake Johnson, Hailee Steinfeld, Mahershala Ali
34. How to Train Your Dragon: The Hidden World (2019)
When Hiccup discovers Toothless isn't the only Night Fury, he must seek "The Hidden World", a secret Dragon Utopia before a hired tyrant named Grimmel finds it first.
Director: Dean DeBlois | Stars: Jay Baruchel, America Ferrera, F. Murray Abraham, Cate Blanchett
35. Supernatural: The Animation (2011– )
TV-14 | 23 min | Animation, Drama, Fantasy
After losing their mother to a demon, two brothers grow up fighting supernatural beings.
Stars: Jared Padalecki, Annakin Slayd, Hiroki Touchi, Yûya Uchida
36. Solo: A Star Wars Story (2018)
During an adventure into the criminal underworld, Han Solo meets his future co-pilot Chewbacca and encounters Lando Calrissian years before joining the Rebellion.
Director: Ron Howard | Stars: Alden Ehrenreich, Woody Harrelson, Emilia Clarke, Donald Glover
37. Instant Family (2018)
PG-13 | 118 min | Comedy, Drama
A couple find themselves in over their heads when they foster three children.
Director: Sean Anders | Stars: Mark Wahlberg, Rose Byrne, Isabela Moner, Gustavo Escobar
Other Lists by christianhernandez
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