So this past weekend I went to SxSW as part of my new role as Director of Enterprise Services for EllisLab (Makers of ExpressionEngine). This was the second time I have traveled to Austin for SxSW. Last year was a blast so I had a motto for this year: "pace yourself". There is plenty of opportunity to go crazy during the festival and I was determined to not crash and burn too quickly. I think I can attribute this motto to having an extremely productive week.
Les Camacho and I arrived on Thursday night and made our way to the hotel. Tired but hungry we decided to eat at the Mexican joint next door. This is where I learned that Les loves good tequila. So if you are wanting a feature or bug fixed in EE and want to bribe him perhaps a bottle Riazul Repasado or Don Julio 1942 or similar will do the trick.
On Friday night we went out to meet the EECMS folks at Red 7. Another chance to get together with users of ExpressionEngine and have some fun. We met the guys from We Are Paper Tiger as well as share some drinks with Simon Collison, Robert Earhardt, Andrew Milne of BV02, Kelly Packer, Jeremy Gimbal and others. I had never heard of the guys from We are Paper Tiger (Marc, Jason and Aaron) before so I think it was really cool that they got the group a place to meet. Apparently they used to run a record label out of New York City and 2 years ago started building websites. Their work is really good.
I also met with Andrew from BV02 out of Ottowa. I have to say that I have seen some of their work and it is amazing. They created the website for the Ottowan Transit system (Transpo). I had lunch with him on Monday and we talked about how he is going to be using EE as a pathway for GPS information coming in from the buses and trains. Pretty amazing stuff. He is plugged in up there and I would definitely like to see him getting in on the Enterprise stuff that we are putting together at EllisLab.
On Sunday I had lunch with Todd Garland from Buy Sell Ads. Todd is a former Hubspot employee that got an idea while working there and has boot-strapped his current endeavor to awesome results. He is a really really nice guy.
Mid day was spent at the Hilton working on a special project with Greg Aker and Pascal Kriete for the ExpressionEngine community. More info soon.
Later that night (Sunday) Les, Greg, Leslie and I had dinner together. Afterwards Leslie and I walked over to my favorite coffee shop, Halcyon. We started drinking with the We Are Paper Tiger folks. Unfortunately I was drinking Insomniacs with Aaron. I seem to have lost my desire to ever have a cup of coffee ever. ever. again. Ugh. Also that was 3 days ago and I think I am still dehydrated from the 8 or so shots of espresso I had that night in the mixed drinks. They don't call it the Insomniac for nothin folks.
I spent Monday walking to Mellow Johnnys. I love this place. Adam Wiggall showed it to me last year and I promise that every year that I attend I will make the trek to Mellow. Such an awesome bike shop. Full of history and drool worthy gear.
Monday night was the Happy Cog Hosting party. It was a pretty cool crowd. I found myself talking with Jason Santa Maria out on the back deck. He was super friendly, and more importantly he is a fan of ExpressionEngine. I also talked to Matt Harris a developer from twitter and his wife Cindy Li who does some awesome work at Yahoo for IM and Yahoo Mail. It wasn't till I got back to a computer to look her up that I realized that I have admired her work before. Got a chance to talk with Zeldman for a couple of minutes as well. I definitely had a blast!
After the Happy Cog Hosting Party we went to dinner with Aaron and Kelly Gustafson (SUPER NICE COUPLE!!), Jenn Lukas, Les, Leslie and Greg. Aaron has a resume to impress. He is a member of the Web Standards Project, Technical Editor on A List Apart and most importantly a huge fan of ExpressionEngine. And Jenn Lukas is a fabulously knowledgeable and utterly genuine front end developer from Happy Cog. I enjoyed getting to know all three of them.
All in all the mood at SxSW was a positive one. Pretty much everyone I talked with asked me what EllisLab was planning for Enterprise Services. I see a blog post going out possibly this week that should at least hint as to what we are thinking. But notice that once again I did not really mention sessions? Yeah. SxSW is all about the people.


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text:
Uploaded: Wed, Nov 10, 2010, 5:58 am
Veteran settles for $250K in Palo Alto VA claim
by Sue Dremann	/ Palo Alto Online
A veteran who claimed he was blinded because of negligent medical care at the VA Palo Alto Health Care System has settled with the U.S. government for $250,000, according to court records.
David E. Woodward Sr., 87, was receiving eye care at the hospital in 2003 and was diagnosed with suspected glaucoma a year later.
But four hospital optometrists did not treat him for the disease during a nearly 5-year period, despite finding progression of the disease. An immediate need for treatment to avoid blindness was not provided, according to the complaint filed in U.S. District Court in San Jose in 2009.
The optometrists also violated hospital protocol when they did not send him to an ophthalmologist, who specializes in eye diseases, according to the complaint.
Woodward began receiving treatment in July 23, 2009, after his case was taken over by the ophthalmology department.
But just eight days later, on July 31, he and six other eye patients were informed by hospital officials that an internal breakdown within the Palo Alto VA had resulted in flawed evaluation and treatment that could have led to their blindness, according to letters from the VA.
The discovery was made after an intensive review of treatment of more than 300 glaucoma patients and of those at risk for the disease.
In 2005, Woodward had lost 30 percent of vision in his right eye and 40-percent loss in his left eye. By 2009, he had lost 90 percent of his vision in the right eye and 85 percent in the left, according to court papers.
Woodward has become legally blind, has suffered physical and emotional pain and experienced anxiety and depression, according to the complaint.
"As we discussed, we have recently reviewed your eye care and have determined that some of the vision loss you suffered may have been preventable had you received a different course of therapy.
"I deeply regret that you did not receive the very best possible care, and are making changes to ensure that every veteran receive care of the highest possible standard," Dr. Stephen Ezeji-Okoye, deputy chief of staff, wrote to Woodward in August 2009, apologizing for the improper care.
Ezeji-Okoye invited Woodward to file an administrative claim for damages, but the government rejected his claim in September 2009.
Kim Staskus, Woodward's attorney, said Woodward battles a number of service-related disabilities resulting from his service as a submariner in the Navy during World War II.
"His submarine sank 14 enemy ships in the Sea of Japan and the South China seas, and he served his country with distinction.
"I'm sure you can appreciate that he did not want to spend the precious time he has left embroiled in litigation.
"While he has lost most of his peripheral vision due to glaucoma, he can still see with reduced vision in a straight-ahead direction. The VA paid a fair settlement in the situation and Mr. Woodward is pleased with the result."
Kerri Childress, spokeswoman for the Palo Alto VA, said she could not speak about the settlement, but did provide a written statement regarding changes to patient eye care:
"VA is dedicated to the quality and safety of the health care provided to America's veterans. It only took one veteran to trigger our investigations and medical evaluations that have now ensured the proper care for every other veteran receiving eye care at the VA Palo Alto Health Care System.
"Following the discovery of this information, the chief of the optometry section was put on administrative leave. The optometry section was moved directly under the ophthalmology section for closer supervision and all glaucoma patients are now being seen by the ophthalmology section.
"We want to ensure all our veterans that we care deeply and monitor our programs thoroughly.
"However, VAPAHCS encourages any veteran who is concerned about their eye care to speak with their physician or provider to ensure they have been appropriately evaluated and treated, or call the patient advocate at 650-493-5000, extension 63543.
"We appreciate the sacrifice you made for our country and will do all we can to ensure your health care is second to none."
Posted by Doug Grayson
on Nov 10, 2010 at 6:26 pm
They call themselves doctors, but never went to medical school. OD's are not MD's !
Posted by Joe
The issue of using ophthalmologist over optometrists in treating glaucoma in our state is a political hot potato. The article doesn't say this, but California is one of the few states where optometrists are restricted in treating glaucoma. The reason is simple: money.
Ophthalmologists tend to charge higher fees for care and office visits. Optometrists are more plentiful, readily available and charge lower fees. In many other states, both can dispense identical medications and give the same level of treatment. No doubt, a board certified ophthalmologist is trained to do many more wonderful and amazing things in managing a complex glaucoma case. But not every case is complex. Visiting a qualified health professional for glaucoma treatment is much better than no treatment, or delayed treatment. Restrictions on insurance coverage or the delay in getting an appointment at a busy ophthalmologist office shouldn't be reasons to put off treatment in California. But, it happens everyday in our state.
While Mr. Woodward's case is written in a compelling and heart tugging manner, it doesn't fill in the details of where things went wrong at the Palo Alto VA. On the surface, the article makes it appear that both the VA and optometrists, in general, are unqualified in providing proper care. That's nonsense, and especially not true in this case. Mr. Woodward repeatedly refused additional visual field tests which would have helped uncover his problem and Mr. Woodward never followed up on referrals to VA ophthalmologists from the VA optometrists. This is typical of patients in advanced age. Elderly patients believe that need better eye glasses, not a medical doctor. There are no tell-tale symptom of glaucoma in its early stages, so elderly don't want to visit an MD. In end, the VA optometrist should have refused to even allow a visit by Mr Woodward and forced him to see an ophthalmologist. But that wasn't possible the way the departments were organized at the VA during this time, so Mr. Woodward gets a $250,000 payday.
In the end, it's not about OD or MD, it's about $D.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
\label{Intoduction}

Superstrings on AdS spaces have been widely studied by virtue of their
applications to the holographic duality, i.e. the AdS/CFT correspondence
\cite{Ma}; and it has become clear that the structure of supergroup
$\sigma $-models is of great importance to investigate superstrings
in these spaces. For instance, the supergroup
PSU(2,2$|$4) turns out to be important to construct superstring theory
on $AdS_5 \times S^5$ \cite{MT}. Besides, superstring theory on $AdS_3 \times
S^3$ can be described in terms of the PSL(1,1$|$2) WZNW model \cite{BVW}.
However, in spite of its importance, quantizing supergroup $\sigma $-models
is a quite difficult problem, and hence solving superstring theory
on AdS spaces exactly still remains as an unsolved question.

Fortunately, there is a simpler type of duality for which string
world-sheet theory is still described by a supergroup WZNW model. It
has been established in \cite{TT,DKKMMS} that two-dimensional
superstring (type 0 string) can be holographically described by a
simple Hermitian matrix model. At present, this is the only
dynamical model of string theory which is non-perturbatively
well-defined and is exactly solvable even at finite temperature. The
two-dimensional type 0 string theory is originally defined by string
world-sheet theory with the $\hat{c}=1$ matter coupled to ${\cal N}
= 1$ super Liouville theory. Boosting the linear dilaton with
Liouville potential kept the same, this theory can be extended to
$\hat{c}\le 1$ type 0 string theory as it has been done for bosonic
string in \cite{Ta,Takayanagi}. Note that dual matrix model can be
constructed even for $\hat{c} < 1$ case, as shown in \cite{Ta}.

In this paper, we argue that these $\hat{c}\leq 1$  superstring
theories can be described by utilizing the supergroup
OSP(1$|$2).\footnote{Current superalgebra of OSP type
also appears in an attempt
\cite{OT} to generalize heterotic string so as to be dual to
Type I string theory with a OSP gauge symmetry.}
 Precisely speaking, we propose that the
$\hat c \leq 1$ superstring is equivalent to topological string
on ${\cal N}=2$ superconformal coset OSP(1$|$2)/U(1).%
\footnote{Topological strings on cosets based on sugerpgroups
have been studied for the analysis of Maldacena conjecture
via world-sheet theory in \cite{Berkovits1,Berkovits2,Bonelli1,Berkovits3,Bonelli2}.}
This
relation can be thought of as a supersymmetric version of the known
relation between $c \leq 1$ bosonic string theory and topological
string on SL(2)/U(1) \cite{MV,Takayanagi}. This extension might
be guessed from the quantum Hamiltonian reduction since
OSP(1$|$2) WZNW model is reduced to ${\cal N} = 1$ super Liouville
theory \cite{BO}, just like SL(2) WZNW model is reduced to bosonic Liouville theory \cite{BO2}.

This paper is organized as follows.
In the next section, we explicitly
construct the ${\cal N}=2$ superconformal coset OSP(1$|$2)/U(1)
as a natural extension of Kazama-Suzuki model for bosonic cosets
\cite{KS,KS2}.%
\footnote{Generic construction of Kazama-Suzuki model for
cosets of supergoups was given in \cite{CRS,Creutzig} very recently.}
We analyze it in
the free field theory and show that the $\hat{c}\leq 1$ string
world-sheet appears after the topological twisting.
In particular, we show that free fields in the coset model
become the matter contents of $\hat{c} \leq 1$ superstring,
and the chiral primaries of the coset model are identified with
the physical operators of $\hat{c} \leq 1$ superstring.
In section 3,
we review and extend the map between the correlation
functions in OSP(1$|$2) WZNW model and those in the ${\cal N}=1$
Liouville theory.
This relation was originally obtained in \cite{HS2} as an
generalization of $H_3^+$-Liouville relation \cite{RT,HS3}.
In section 4, after briefly reviewing
$\hat{c}\le 1$ superstrings, we apply this map to study the
scattering S-matrices. We explicitly show that the
correlation functions of physical operators in the topological
model are mapped to those of physical operators in the
$\hat{c} \leq 1$ superstring.
In section 5, we summarize the conclusion.
In the appendix, we discuss correlation
functions of  OSP(1$|2$) WZNW model in the free field
representation.

\section{${\cal N}=2$ Coset OSP(1$|$2)/U(1) and 2D Superstring}
\label{coset}

In this section we construct and analyze ${\cal N}=2$
supersymmetric coset
(Kazama-Suzuki model \cite{KS,KS2}) based on OSP(1$|$2)/U(1). After its
topological twisting, we show explicitly from the free field
theory analysis that the world-sheet theory of $\hat{c}\leq 1$
superstring indeed appears. We also discuss chiral primary
states which are the physical states in the topologically twisted
theory.

\subsection{OSP(1$|$2) Current Algebra}

The current algebra of OSP(1$|$2) includes SL(2) bosonic subalgebra,
which is generated by $J^{3} (z)$ and $J^{\pm} (0)$ with their OPEs%
\footnote{For a while we concentrate on the holomorphic part.}
\begin{align}
 J^+(z) J^-(0) \sim  \frac{k}{z^2} - \frac{2J^3(0)}{z} , \qquad
 J^3(z) J^{\pm}(0) \sim \pm \frac{J^\pm (0)}{z} ,\qquad
 J^3(z)J^3(0)\sim -  \frac{k}{2z^2} .
\end{align}
In addition to these bosonic generators, there are fermionic ones with
\begin{align}
 &J^3(z) j^{\pm}(0)\sim \pm\frac{j^\pm (0)}{2z} , \qquad
 J^{\pm} (z) j^\mp (0) \sim  \mp \frac{j^\pm(0)}{z} , \\  \nonumber
 &j^+(z)j^-(0) \sim  \frac{2k}{z^2} - \frac{2J^3(0)}{z} , \qquad
 j^{\pm}(z) j^{\pm}(0) \sim - \frac{2J^{\pm}(0)}{z} .
\end{align}
The energy momentum tensor is given by Sugawara construction and the
central charge is $c = 2k/(2k-3)$. These are the definition of
 OSP(1$|$2) current algebra with level $k$.

In a free field representation \cite{Wakimoto,BO,ERS,ERS2} the above
currents may be expressed as
\begin{align}
\label{freefield}
 &J^- = \beta , \qquad J^+ = \beta \gamma ^2 - \frac{1}{b}
 \gamma \partial \phi + \gamma \theta p + k \partial \gamma
  - (k -1 ) \theta \partial \theta  , \\
 &J^3 = \beta \gamma - \frac{1}{2b} \partial \phi + \frac12 \theta p  , \qquad
 j^- = p - \beta \theta , \qquad j^+ =
 \gamma p - \beta \gamma \theta
+ \frac{1}{b} \theta \partial \phi - (2k -1) \partial \theta ,  \nonumber
\end{align}
where the OPEs of these free fields are
\begin{equation}
 \phi (z) \phi (0) \sim - \ln z , \qquad
 \beta (z) \gamma (0) \sim \frac{1}{z} , \qquad
 p(z) \theta (0) \sim \frac{1}{z} ~.  \label{Propagators}
\end{equation}
The field $\phi$ has the background charge $Q_\phi =b$
and the central charge is $c = 1 + 3 Q_{\phi }^2$.
Here the parameter $b$ is related to the level $k$ as $1/b^2 = 2k - 3$.
The bosonic fields $(\beta,\gamma)$ have conformal weights
$(1,0)$ and the central charge of this system is $c=2$.
On the other hand, $(p,\theta)$ are fermions with conformal weights
$(1,0)$ and central charge $c = -2$.
In the following analysis, it is useful to bosonize the fermionic fields $(p,\theta)$ as
\begin{equation}
\theta = e^{i Y} , \qquad p = e^{ - i Y} . \label{Bosonization}
\end{equation}
For instance, the $J^3$ current takes the form
\begin{equation}
J^3  = \beta \partial \gamma -\frac{1}{2b} \partial \phi +\frac{i}{2}
\partial Y .\label{J3Y}
\end{equation}
The energy momentum tensor is given by
\begin{align}
 T = \beta \partial \gamma - \frac12 \partial \phi \partial \phi
 + \frac{b}{2} \partial^2 \phi - p \partial \theta
  = \beta \partial \gamma - \frac12 \partial \phi \partial \phi
 + \frac{b}{2} \partial^2 \phi -  \frac12 \partial Y \partial Y
 + \frac{i}{2} \partial^2 Y  \label{TheT}
\end{align}
in terms of these free fields.

One of the merits to utilize the free field representation is
that vertex operators can be expressed in a simple form.
Namely, the vertex operators of OSP(1$|$2) model can be
written in terms of free fields as
\begin{align}
 \Phi^s_{j,m}  \sim e^{i s Y} \gamma^{ j - s/2 +m }
 e^{ - 2 b j \phi} ,
 \label{freev}
\end{align}
whose conformal weight with respect to (\ref{TheT}) is
\begin{equation}
\Delta= - 2 b^2 j(j+\tfrac12) + \tfrac12 s (s-1) .  \label{ConformalDimension}
\end{equation}
Here we set $s=0,1/2,1$. For $s=0,1$ we can express the vertex operator
even in terms of $\theta$, but not for $s=1/2$. The operator with $s=1/2$
corresponds to a twist operator in R-sector, whose role was argued
for GL(1$|$1) WZNW model in \cite{LeClair}, see also \cite{PT}.
In order to compute correlation functions, the overall normalization
of vertex operators should
be fixed. More precise definition will be given in section \ref{ribault}.
Correlation functions of vertices (\ref{freev}) are discussed in the appendix, where the Coulomb gas prescription is given.

Notice that the expression (\ref{ConformalDimension}) is invariant under the Weyl
transformation $j\to -j-1/2$ and under $s\to 1-s$. This
allows us to consider a second contribution to (\ref{freev}) which
goes like $\sim e^{2b(j+1/2)\phi }$ and it would dominate the large $\phi $
regime for $j>-1/4$.
In addition, there exist conjugate representations which are similar to
those that exist in the free field realization of SU(2) model \cite%
{Dotsenko, Dotsenko2, ERS, ERS2}. For instance, one finds the operator
\begin{align}
\hat{\Phi }_{j,j+\frac{1}{2}}^{0}\sim \beta ^{k-2-2j}e^{b(2k-3-2j)\phi }
\end{align}
which represents a Kac-Moody primary of conformal dimension (\ref%
{ConformalDimension}), with $m=j+1/2$ and $s=0$. It can be also thought of
as a conjugate representation for the state with $m=j$, $s=1$.

In order to define the  OSP(1$|$2)/U(1) coset theory,
we utilize the
representation introduced in \cite{BK} and \cite{DVV} to realize the
$SL(2)/U(1)$ coset theory.
This amount to introduce
a boson $X^3(z)$ with $X^3 (z) X^3 (0) \sim - \ln z$,
as well as a $(b,c)$ ghost system, which are
used to mode out the $U(1)$ factor.
Then the vertex operators of the coset theory are given by
\begin{align}
\Phi^s_{j,m} =
 \Psi^s_{j,m}    e^{  - i \sqrt{\frac{2}{k}} m X^3 }  .
 \label{parafermion}
\end{align}
As we will discuss below, the OSP(1$|$2) current algebra admits
the symmetry under the spectral flow action as in the case of
SL(2) WZNW model \cite{MO}.
For  OSP(1$|$2) model, spectrally flowed states are defined
in this form as
\begin{align}
\Phi^{s,w}_{j,m} =
 \Psi^s_{j,m} e^{ - i \sqrt{\frac{2}{k}} (m + \frac{k}{2}w) X^3 }
 \label{spectralflow}
\end{align}
with the index of spectral flow $w$.

\subsection{${\cal N} = 2$ Supersymmetric Coset Model}

In this subsection we would like to construct
 ${\cal N} = 2$ supersymmetric model based on
the coset OSP(1$|$2)/U(1). For the purpose we first generalize
the OSP(1$|$2) current algebra into ${\cal N} = 1$ supersymmetric
version, therefore we need superpartners of currents
$(J^3,J^{\pm},j^{\pm})$. While we introduce fermions
$(\psi^3,\psi^{\pm})$ with spin $1/2$ for the bosonic currents
$(J^3,J^{\pm})$, we include bosons $\varphi^{\pm}$ with spin $1/2$
for the fermionic currents $j^{\pm}$. We assume the OPEs of these
fields as
\begin{align}
 \psi^+ (z) \psi^- (0) \sim \frac{1}{z} , \qquad
 \psi^3 (z) \psi^3 (0) \sim \frac{1}{z} , \qquad
 \varphi^+ (z) \varphi^- (0) \sim - \frac{1}{z} .
\end{align}
Notice that the extra bosons  $\varphi^{\pm}$ satisfy
wrong spin-statistic relation.
With these new fields we can define ${\cal N}=1$ supercurrents as
\begin{align}
 \hat{J}^\pm  = J^\pm + \sqrt{2} \psi^\pm  \psi^3 +
  \frac{1}{2}\varphi^\pm \varphi^\pm , \qquad
 \hat{j}^\pm = j^{\pm} \pm i \sqrt{2} \psi^\pm \varphi^\mp
  \pm i \psi^3\varphi^\pm ,
 \label{ospsuper}
\end{align}
along with
\begin{align}
 \hat{J}^3 =J^3+\psi^+\psi^-  + \frac{1}{2}\varphi^+ \varphi ^- .
\end{align}
We construct the coset model by using the last current as well as removing
one of the fermions $\psi^3$ to preserve ${\cal N}=1$ world-sheet supersymmetry.

We can show that the coset model actually has enhanced ${\cal N}=2$ supersymmetry as Kazama-Suzuki models for bosonic cosets
\cite{KS,KS2}.
We find that the generators of ${\cal N}=2$ superconformal
symmetry are
\begin{align}
\label{supercon}
&J_R= - \frac{1}{2k - 3}\left(2J^3+(2k-1)\psi^+\psi^- + (2k-2)\varphi^+\varphi^-\right) ,
\\
&G^{\pm} = \frac{1}{\sqrt{2k-3}} \left( 2 J^\pm \psi^\mp
 \pm \sqrt{2} j^\pm \varphi^\mp
 + (\varphi^\mp)^2 \psi^\pm \right) , \nonumber
\\
&T=  \frac{1}{2k-3}\Bigl[J^+J^-+J^-J^++\frac{1}{2}(j^-j^+-j^+j^-)
 + 4J^3\psi^+\psi^- + 2J^3\varphi^+\varphi^-
+ 2\psi^+\psi^-\varphi^+\varphi^- \nonumber \\
& \qquad - \frac{2k+1}{2}(\psi^+\partial\psi^-+\psi^-\partial\psi^+)
 - (k-1)(\varphi^+\partial\varphi^- - \varphi^-\partial
\varphi^+) + \frac{1}{2}(\varphi^+)^2(\varphi^-)^2\Bigr] . \nonumber
\end{align}
In fact, we can compute the OPEs of generators as
\begin{align}
\label{algentwo}
&
T(z)T(0)\sim \frac{c/2}{z^4}+\frac{2T(0)}{z^2}+\frac{\partial T(0)}{z} ~ , \qquad
T(z)G^\pm(0)\sim \frac{\frac{3}{2}G^\pm(0)}{z^2}
 +\frac{\partial G^\pm (0)}{z} , \\
&T(z)J_R(0)\sim \frac{J_R(0)}{z^2}+\frac{\partial J_R (0)}{z} , \qquad
 J_R(z)G^\pm (0)\sim \pm \frac{G^\pm (0)}{z} , \qquad
 J_R(z)J_R(0)\sim \frac{c/3}{z^2} , \nonumber \\
 & G^\pm (z)G^\mp (0) \sim \frac{\frac{2}{3}c}{z^3} \pm \frac{2J_R  (0)}{z^2}
+\frac{2T (0)}{z}
\pm \frac{\partial J_R (0)}{z}  ,
\qquad G^\pm (z)G^\pm (0) \sim 0 .
\nonumber
\end{align}
In this way we have explicitly shown that
these generators satisfy the ${\cal N}=2$ superconformal
algebra with central charge $\hat{c}= c/3 = 1/(2k-3)$.

\subsection{Topological Twisting}

To realize the U(1) quotient more explicitly, we combine a free
scalar field $X$ with the ${\cal N}=1$ OSP current algebra and
finally take a quotient by the complexified U(1) (i.e. the complex
plane ${\mathbb C}$). This quotient can be done by taking BRST
invariant state about the BRST operator
\begin{align}
\label{brst}
Q_{B}=\int dz C(z)J_{g}(z) ,
\end{align}
where we
introduced fermionic ghosts $(B,C)$ with
the conformal weights $(1,0)$.
Here the BRST current $J_{g}$ is defined by
\begin{align}
\label{defghi}
J_g =\hat{J}^3  - \frac{i}{2b}\partial X ,
\end{align}
and it is easy to see that
$J_g(z)J_g(0)\sim 0$ which guarantees $Q_{B}^2=0$.

Notice that in this formalism we can always set $J_g(z)$ to zero
since it is gauged.
Using this fact we can use the following expression of the
R-current of ${\cal N}=2$ superconformal algebra as
\begin{align}
\label{rtcurrent}
J'_R =J_R -\frac{4k - 8}{2k - 3}J_g
=-2J^3-3\psi^+\psi^- - 2\varphi^+\varphi^-
+ i \frac{2k- 4}{\sqrt{2k-3}}\partial X.
\end{align}
In the anti-holomorphic part, we use the same expression with bars.%
\footnote{
This choice means that we gauge the vector $U(1)$ current instead of the
axial $U(1)$ current. The former may produce a trumpet like
geometry and the latter a black hole like geometry \cite{DVV}.
We choose the vector gauge just for the simplicity of expression.}
We will find that this form of R-current is useful to construct
the topological model as in the bosonic case \cite{MV,Takayanagi}.

Now, we perform topological twists \cite{EY,WTST} by
using the expression \eqref{rtcurrent} of R-current.
Namely, we redefine the energy momentum tensor by
$T^{top}=T+\frac{1}{2}\partial J'_R$ and $\bar T^{top} = \bar T +
\frac{1}{2} \bar \partial \bar J '_R$.
Employing the free field representation \eqref{freefield},
we then find the following maps of fields.
First of all, the background charge of the field $\phi$ is
shifted from $Q_\phi = b$ to $Q_\phi = b+1/b$. After the
twist, the field $\phi$ corresponds to the Liouville field.
Recall that the central charge is
written as $c = 1 + 3 Q^2$ in terms of background charge $Q$.
Next, the field $X$ would have background charge
$Q_X = i (1/b - b)$ after the twist, and this field
becomes the bosonic part of the $\hat c \leq 1$ matter.
The conformal weights of fermions $(\theta, p)$ are shifted
from $(0,1)$ to $(1/2,1/2)$ and they become superpartners
of the above bosonic fields. The other fields $\psi^\pm$ and
$\varphi^\pm$ are mapped to the  superghosts
$(b,c)$ and $(\beta ' , \gamma ')$ of type 0 superstring theory.%
\footnote{Here we use the notation $(\beta ' , \gamma ')$
to represent superghosts of superstring in order to
distinguish them from the ones in \eqref{freefield}.}
In table \ref{ba} the changes of conformal weights are summarized.
In the end we expect that $(\beta,\gamma)$ would be canceled out with
$(B,C)$ as in the bosonic string case \cite{MV}. In this way
we obtain the same field contents as the world-sheet theory of
the type 0 $\hat{c} \leq 1$ string including ghosts.
More detailed explanation of the type 0 string
will be given in subsection \ref{secc}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
 ~ & \multicolumn{2}{c|}{Before twisting}
 & \multicolumn{2}{c|}{After twisting} \\
 \cline{2-5}
 ~ & Central charge& Conformal weights &
  Central charge & Conformal weights \\
\hline
$(\theta , p)$ & $-2$ & $(0,1)$ &
$1$ & $(1/2,1/2)$ \\
\hline $(\psi^+,\psi^-)$ & $1$ & $(1/2,1/2) $ &
$-26$ & $(2,-1)$ \\
\hline
$(\varphi^+,\varphi^-)$ &
$-1$ & $(1/2,1/2)$ &
$11$ & $(3/2,-1/2)$ \\
\hline
$(\beta,\gamma)$ &
$2$ & $(1,0)$ &
$2$ & $(0,1)$ \\
\hline
$(B,C)$ &
$-2$ & $(1,0)$ &
$-2$ & $(1,0)$ \\
\hline
\end{tabular}
\end{center}
\caption{Changes of central charges and conformal weights
after topological twisting.}
\label{ba}
\end{table}

\subsection{Chiral Primaries}

In the previous subsection we have shown that free fields
in the ${\cal N} = 2 $ coset are mapped to the matter contents
of the $\hat c \leq 1$ superstring theory after the topological twist.
In fact we can identify physical operators
of the topological model with those of the $\hat c \leq 1$ superstring,
which is the subject of this subsection.
In
order to define the coset model we have introduced two spin $1/2$ fermions
$\psi^{\pm}$ and two spin $1/2$ bosons $\varphi^\pm$. With the bosonization
formula, they can be written as
\begin{align}
 \psi^+ = e^{iH} , \qquad \psi^- = e^{-iH}  , \qquad
 \varphi^+ =  e^{-  \chi}  \partial \xi  ,  \qquad
 \varphi^- =  e^{ \chi} \eta  ,
 \label{bosonized}
\end{align}
where $H , \chi$ are free bosons without background charges
and free fermions $\xi,\eta$ are with
$\Delta_\xi = 0, \Delta_\eta = 1$. The non-trivial OPEs are
given as
\begin{align}
 H(z) H(0) \sim - \ln z , \qquad
 \chi (z) \chi (0) \sim - \ln z , \qquad
  \eta (z)  \xi (0) \sim \frac{1}{z} .
\end{align}
These bosonized expressions of fermions are useful to
define vertex operators.
Since the operators of the coset model must be invariant
under the BRST charge \eqref{brst}, they should take the
form
\begin{align}
 e^{i r H + u \chi} \Phi_{j,m}^s e^{  2 i b(m+r - \frac{u}{2}) X} ,
 \label{brstvo}
\end{align}
whose conformal weight is
\begin{align}
\Delta = - 2b^2j(j+\tfrac12) + \frac{s(s-1)}{2} + \frac{r^2}{2}
 - \frac{u^2}{2} + 2 b^2 \left(m+r-\frac{u}{2}\right)^2 .
\end{align}
Here we have used $\Phi_{j,m}^s$ as the vertex operator of
OSP(1$|$2) WZNW model as defined in \eqref{freev}.

Physical operators of the topological model can be constructed from
chiral primaries of the ${\cal N}=2$ coset model.
Here we review how to perform the topological twist to the
chiral primaries by following \cite{MV,Takayanagi}.
First we find chiral primary states of the coset in NS-sector,%
\footnote{Notice
that there are three different spin structures that appear in this
paper. One is for the OSP(1$|$2) current algebra, which is
defined such that an
an integer $s$ in \eqref{freev} means the NS-sector,
while a half integer $s$ implies R-sector.
The second spin structure is the
ordinary one for the $N=2$ superconformal field theory. The third
one is for the $\hat{c}\le 1$ superstring.
In this section the notion of NS,R is with respect to the second
spin structure.}
which satisfy
\begin{align}
 G^+_{r - 1/2} | { \rm NS } \rangle =  G^-_{r + 1/2} | { \rm NS } \rangle = 0
\end{align}
for $r = 0,1,\cdots$.
Among the vertex operators of the form \eqref{brstvo},
there are chiral primary operators ${\cal O}^{NS,s}_j$
corresponding to the above chiral primary states.
These chiral primaries can be mapped to
R-ground states by spectral flow operation.
Redefining the U(1)$_R$ current \eqref{supercon} as
\begin{align}
 J_R
 =  - 2 b^2 \hat J^3  - \psi^+ \psi^-
 -  \varphi^+ \varphi^- = - i b \partial X_R,
\end{align}
with $X_R(z) X_R(0) \sim \ln z$, the R-ground states are obtained by
${\cal O}^{R,s}_j =  e^{\frac{i}{2} b X_R} {\cal O}^{NS,s}_j$.
Finally, the elements of cohomology for the topological theory are
obtained by the topological twist as ${\cal
O}^{s=1}_{j} = e^{- \frac{i}{2} \sqrt{k} b X'_R}   {\cal
O}^{R,s}_{j}$.
Here we define $X'_R$ as
\begin{align}
 J_{R}' = - 2 J^3 - 3 \psi^+ \psi^- - 2 \varphi^+ \varphi^-
  + 2 ib (k-2) \partial X =: - i \sqrt{k} b \partial X_R '
\end{align}
from the expression of R-current \eqref{rtcurrent}.

Among the chiral primaries of the ${\cal N}=2$ coset model, we focus
on the following two types of operators in NS-sector;
\begin{align}
 {\cal O}^{NS,\frac{1}{2}}_j =
 e^{- \frac{1}{2} \chi} \Phi^{\frac{1}{2}}_{j,j-\frac{1}{4}} e^{2ibjX} , \qquad
 {\cal O}^{NS,1}_j = \Phi^{1  }_{j,j} e^{2ibjX} ,
 \label{cp}
\end{align}
which satisfy  $\Delta = q_R/2 = - j b^2 -1/4$
and $\Delta = q_R/2 = - j b^2$, respectively.
The R-sector ground states are constructed as
\begin{align}
 {\cal O}^{R,\frac{1}{2}}_j =  e^{\frac{i}{2} H}
 \Phi^{\frac{1}{2}}_{j,j-\frac{1}{4}} e^{2ib(j+\frac{1}{4})X} , \qquad
 {\cal O}^{R,1}_j = e^{\frac{i}{2} H + \frac{1}{2} \chi}
  \Phi^{1}_{j,j} e^{2ib(j+\frac{1}{4})X} .
 \label{rg}
\end{align}
After the topological twist we finally obtain
\begin{align}
 {\cal O}^{\frac{1}{2}}_j =  e^{ - i  H - \chi}
 \Phi^{\frac{1}{2}, w = 1}_{j,j-\frac{1}{4}} e^{2ib(j+\frac{1}{4 b^2})X} , \qquad
 {\cal O}^{1}_j = e^{ - i  H - \frac{1}{2} \chi}
  \Phi^{1,w=1}_{j,j} e^{2ib(j+\frac{1}{4 b^2})X} .
  \label{cpt}
\end{align}
Notice that vertex operators are spectrally
flowed in the sense of OSP(1$|$2) WZNW model as in \eqref{spectralflow}
during the procedure of topological twist.
Under the spectral flow action we may identify
$
\Phi_{j, j + \frac{s -1}{2} }^{s,w =  1}
 = \Phi_{-j-\frac{k}{2}+\frac{1}{4} ,
 j + \frac{s-1}{2} + \frac{k}{2}}^{s - \frac{1}{2}}
$.
Combining with the free field representation of vertex operators
\eqref{freev}, we find
\begin{align}
{\cal O}^{\frac{1}{2}}_j
  \sim c e^{ - \chi}
 e^{ 2ib(j+\frac{1}{4 b^2})X + 2 b (j + \frac{k}{2} - \frac14) \phi} ,
 \qquad
  {\cal O}^{1}_j = c e^{ - \frac{1}{2} \chi}
   e^{ \frac{i}{2} Y + 2ib(j+\frac{1}{4 b^2})X
   + 2 b ( j + \frac{k}{2} - \frac14) \phi} .  \label{phyo}
\end{align}
In the above, we renamed $c=\exp(-iH)$ as suggested by the previous
discussion. Moreover the $\beta$-ghost in the superstring
should be written as $\beta' = \partial \xi \exp(-\chi)$.
Therefore, we can say that these operators have
one $c$-ghost and picture $-1$.%
\footnote{The operators corresponding to those in the
other picture may be obtained by the action of operator similar
to the picture changing operator.}

Notice that the above two operators (\ref{phyo}) indeed coincide with the
tachyon and RR field vertex operators in the two dimensional type 0
superstring, respectively (see subsection \ref{secc}).
Actually they complete the list of physical operators since
there are no massive stringy modes in two dimensional superstring.
This fact may be seen by taking the light-cone gauge.
In this way, we have learned that the
physical states (chiral primary states) in the topological string on
OSP(1$|$2)/U(1) are mapped into the physical states in the two
dimensional type 0 string. We will study the relation between these
two theories in more detail below.

\section{OSP(1$|$2)/U(1) Coset from ${\cal N}=1$ Super Liouville}
\label{ribault}

In references \cite{RT,HS3} it was shown that arbitrary correlation
functions of primary fields in SL(2) WZNW model can be written in
terms of correlation functions of Liouville field theory.
This property may be useful to show the
equivalence between the scattering amplitudes in $c\leq 1$ bosonic
string and the topological string on $SL(2)/U(1)$.
The agreement for three-point functions between them has been shown in
\cite{Takayanagi}, and this is generalized by \cite{NN}
to arbitrary tree level amplitudes by utilizing the generalized
relation of \cite{Ribault}.
Recently it was shown in \cite{HS2} that correlation
functions of OSP(1$|$2) WZNW model can be written in terms of those
of ${\cal N}=1$ super Liouville field theory.
Later we would like to show the equivalence between
${\cal N} = 2$ coset model
of OSP(1$|$2)/U(1) and the $\hat c \leq 1$ superstring in the
level of amplitudes.  For the purpose we
generalize the relation such as to include
RR-sectors of fermions and spectrally flowed sectors of OSP(1$|$2)
model. In this section we derive the generalized relation in
the path integral formulation following \cite{HS3,HS2}.

\subsection{OSP(1$|$2) WZNW Model}

Let us start from the action of OSP(1$|$2) WZNW model.
In terms of free fields the action may be written as\footnote{
A derivation of this action can be found in \cite{HS2}.
This action hare is a bit different from the one in \cite{HS2},
but it is easy to see the equivalence between the two expressions.}
\begin{align}   \label{OSPaction}
 S^\text{WZNW} (g) = \frac{1}{2\pi} \int d^2 z
  \Bigl[ \frac12 \partial \phi  \bar \partial \phi
  + \frac{b}{8} \sqrt{g} {\cal R} \phi
  + \beta \bar \partial \gamma
  + \bar \beta \partial \bar \gamma
  + p \bar \partial \theta + \bar p \partial \bar \theta
  \\ \nonumber
  - \frac{1}{k}\beta \bar \beta e^{2b\phi }
   - \frac{1}{2k} (p + \beta \theta )
   ( \bar p + \bar \beta \bar \theta ) e^{b \phi } \Bigr] ,
\end{align}
where $\phi,\gamma,\bar \gamma , \theta , \bar \theta$
are related to the parameters of elements
$g \in \,$ OSP(1$|$2) and $\beta , \bar \beta , \theta
, \bar \theta$ are conjugate variables.
The generators of current algebra symmetry
are written as in \eqref{freefield}
in these variables.
Here we use the form of vertex operator as
\begin{align}
  V_j^{s,\bar s} (\mu | z ) =
  \mu^{j + \frac12 + \frac{s}{2}} \bar \mu^{j + \frac12 + \frac{\bar s}{2}}
 e^{i s Y  + i \bar s \bar Y }
 e^{\mu \gamma - \bar \mu \bar \gamma} e^{2 b (j + \frac{1}{2} )\phi } ~. \label{vmu}
\end{align}
For the NSNS-sector with $s , \bar s = 0 ,1$ these vertex operators
are the same as in \cite{HS2}.
The conformal weights are given as $\Delta =  - 2 b^2 j (j + 1/2)$.
The vertex operators in the RR-sector are given by spin fields
with $s = \bar s = 1/2$, and
the conformal weights are $\Delta =  - 2 b^2 j (j + 1/2) + 1/8$.
The above expression in so-called $\mu$-basis is useful for our purpose,
and it can be mapped to the $m$-basis expression given in \eqref{freev}
by%
\footnote{In order to compare with the previous notation, we may need
to perform a flip $j  \to -j - 1/2$. Moreover, it might be natural to
multiply the factor $
 N^{s.\bar s}_{j,m,\bar m} =
 \frac{\Gamma(-j + 1/2 -s/2 + m)}{\Gamma (j+ 1/2 + \bar s/2 - \bar m)}$
 as, e.g., in \cite{RT}; see also \cite{Giribet2006}. Here we remove it since it may diverge
 in our case.
}
\begin{align}
 \Phi^{s,\bar s}_{j,m,\bar m} =
  \int \frac{d^2 \mu}{|\mu|^2} \mu ^{ - m} \bar \mu^{ - \bar m}
V^{s,\bar s}_j (\mu | z) .
\end{align}
In some sense, the $\mu$-basis expression can be thought of
generating function of the $m$-basis expression.

Since the operators of topological model in \eqref{cpt} are
written in terms of OSP(1$|$2) vertex operators with spectral flow
index $w=1$, it is important to understand
the symmetry under the spectral flow.
The spectral flow action $\rho^w$ can be defined as
\begin{align}
 \rho^w ( J_n^3) = J^3_n - \frac{k}{2} w \delta_{n,0} , \qquad
 \rho^w (J_n^{\pm}) = J^{\pm}_{n \pm w}  , \qquad
 \rho^w (j_r^{\pm} ) = j^{\pm}_{r \pm \frac{w}{2}} ,
\end{align}
where the mode expansions are
$J^A (z) = \sum_n J^A_n z^{-n-1}$ with $A=\pm,3$ and
$j^\pm (z) = \sum_r j^\pm_r z^{-r-1}$.
We can easily see that the new currents satisfy the
same (anti-)commutation relations as before,
which implies that the spectral flow is the symmetry of
the current algebra OSP(1$|$2).
The vacuum state is defined such as
\begin{align}
\rho^w (J^A_n) | w \rangle = 0 , \qquad
\rho^w (j^{\pm}_r ) | w \rangle = 0
\end{align}
for $n,r \geq 0$. In terms of free fields, the vacuum state
  $|w \rangle = |w \rangle_{(\beta,\gamma)}
  \otimes  | w \rangle_{\phi  }
  \otimes  | w \rangle_{Y  } $ is characterized  as
\begin{align}
 \beta_{n - w} |w \rangle_{(\beta,\gamma)} = 0 , \qquad
 \gamma_{n+w} |w \rangle_{(\beta,\gamma)} = 0
 \label{cond1}
\end{align}
for $n \geq 0$, and moreover
\begin{align}
 | w \rangle_{\phi  } =
 e^{\frac{w}{2b} \phi }  | 0 \rangle_{\phi  } , \qquad
 | w \rangle_{Y  } = e^{ - \frac{i w}{2} Y }  | 0 \rangle_{Y  } .
 \label{cond2}
\end{align}
In the following we assume $w \geq 0$ and
denote $v^w (0)$ as the operator
corresponding to the state $|w \rangle$.

As discussed in \cite{Ribault,HS}, generic $N$-point functions
of operators with spectral flow can be reduced to
$N$-point functions
of \eqref{vmu} with the inversion of $v^w(\xi)$.
We choose the position of insertion $v^w (\xi)$ as $\xi = 0$
since it does not affect the following discussion.
In the path integral formulation they are given as
\begin{align}
 \left \langle \prod_{\nu = 1}^N
 V^{s_\nu , \bar s_\nu}_{j_\nu} (\mu_\nu | z_\nu ) v^w (0)
 \right \rangle
 = \int_{(w)} {\cal D} \phi {\cal D} ^2 \beta
{\cal D} ^2 \gamma {\cal D} ^2 \theta {\cal D} ^2 p
 e^{- S^\text{WZNW} (g) }
 \times \\ \times
  \prod_{\nu = 1  }^N
V^{s_\nu , \bar s_\nu } (\mu_\nu | z_\nu )
e^{w ( \phi (0)/2b - i Y (0) /2 ) } .
 \nonumber
\end{align}
The effects of insertion $v^w (0)$ appears in the right hand
side in two ways. One is the extra insertion of vertex operator
$e^{w ( \phi (0)/2b - i Y (0) /2 ) }$, and the other is the
restriction to the integration domain of $\beta , \bar \beta$
such that $\beta , \bar \beta$ have a zero of order $w$ at
$\xi = 0$. For more detail see \cite{HS}.

\subsection{OSP(1$|$2)--Super Liouville Correspondence}

Now that we have the OSP(1$|$2) WZNW model, we can derive
the relation between the correlation functions of OSP(1$|$2) WZNW
model and those of ${\cal N}=1$ super Liouville field theory
by following the analysis of \cite{HS2}.
For this purpose we first integrate $\beta,\gamma$ as in \cite{HS2}.
Integrations over $\gamma$ and $\bar \gamma$ lead to delta
functionals for $\beta$ and $\bar \beta$, which replace the fields
$\beta, \bar \beta$ by
\begin{align}
 \beta (x) = \sum_{\nu = 1}^N \frac{\mu_\nu}{x - z_\nu}
   = u \frac{x^w \prod_{i=1}^{N-2-w}(x-y_i)}
            {\prod_{\nu = 1}^N (x-z_\nu)}
   = : u {\cal B} (x) .
    \label{separation}
\end{align}
The insertion of $v^w(0)$ forces $\beta(x)$ to have a zero of order
$w$ at $x=0$ and this requirement gives constraints
\begin{align}
 \sum_{\nu = 1}^N \mu_\nu z_\nu^{- n} = 0
\end{align}
for $n = 0, 1, \cdots , w$. Since a 1-form with $N$ poles
on a sphere has $N-2$ zero's, $\beta$ can be represented as in the right hand
side by the positions of $N-2-w$ more zero's $y_i$.
In other words, the parameters $y_i$ are defined by the
equation \eqref{separation}, and the new parameters are
essential to relate the model to super Liouville theory
as seen below.
Moreover, we can see that the number of spectral flow $w$
is restricted as $ w \leq N - 2$.

After the integration over $\beta,\gamma$,
the action becomes something similar to super Liouville theory,
but the coefficients  include functions ${\cal B} (z), \bar {\cal B}(\bar z)$.
Following \cite{HS2,HS} these can be removed by the
redefinition of fields as
\begin{align}
 \phi ' : = \phi  + \frac{1}{2 b} \ln | u {\cal B } | ^2 ,
 \qquad Y ' : =  Y  - \frac{i}{2} \ln | u {\cal B } | ^2 .
\end{align}
Moreover, after some manipulations we find the relation
\begin{align}
 & \left \langle \prod_{\nu = 1}^N
 V^{s_\nu,\bar s_\nu}_{j_\nu} (\mu_\nu | z_\nu)
  v^{w}(0)\right\rangle  \nonumber \\
  & \qquad = \prod_{n=0}^w \delta ^2 (\sum_\nu \mu_\nu z^{-n} )
  |u|^{2-\frac{w}{2b^2}+\frac{w}{2}} | \Theta^w_N |^2
 \left\langle \prod_{\nu = 1}^N
  V^{s_\nu-\frac{1}{2},\bar s_\nu - \frac{1}{2}}_{\alpha_\nu} ( z_\nu)
   \prod_{j=1}^{N-2-w} V^{\frac{1}{2},\frac{1}{2}}_{-\frac{1}{2b}} (y_j) \right\rangle
   \label{Ribault}
\end{align}
with $\alpha_\nu = 2b (j_\nu + 1/2 ) +1/2b $.
The right hand side is computed by the sum of ${\cal N}=1$
super Liouville theory $(\phi ',\psi,\bar \psi)$  and massless fermions $(\psi_X,\bar \psi_X)$
\begin{align}
   S [\phi ',\psi,\psi_X]
  & = \  \frac{1}{4\pi} \int d^2 z \, \Bigl[\,  \partial \phi ' \bar \partial \phi '
   + \frac{Q_{\phi ' }}{4} \sqrt g R \phi ' + \frac{2}{k} e^{ 2 b \phi '}\, +
   \nonumber \\[2mm]
 & \hspace*{2cm} +
   \psi \bar \partial \psi + \bar \psi \partial \bar \psi
 + \psi_X \bar \partial \psi_X + \bar \psi_X \partial \bar \psi_X
 - \frac{2}{k} \psi \bar \psi e^{ b \phi '}\,  \Bigr]
 \label{sLaction}
\end{align}
with $Q_{\phi ' } = b + 1/b$.
The fermions are defined by
\begin{align}
 \psi \pm i \psi_X = \sqrt{2} e^{ \pm i Y '} , \qquad
 \bar  \psi \pm i \bar \psi_X = \sqrt{2} e^{ \pm i \bar Y '} ,
\end{align}
and the vertex operators are
\begin{align}
V^{s,\bar s}_{\alpha} ( z )
 = e^{i s Y + i s \bar Y}
   e^{\alpha \phi '}
\end{align}
with conformal weights $\Delta = \alpha ( Q_{\phi ' } - \alpha )/2
 + s^2 /2 $.
The twist factor is
\begin{align}
 \Theta^{w}_N = \prod_{\mu < \nu}^N (z_{\mu} - z_{\nu})
  ^{\frac{1}{4b^2} -\frac{1}{4}}
   \prod_{i<j}^{N-2-w} (y_{i} - y_{j})^{\frac{1}{4b^2} - \frac{1}{4}}
    \prod_{\nu=1}^N \prod_{i=1}^{N-2-w} (z_\nu - y_j)^{-\frac{1}{4b^2} + \frac{1}{4}} .
\end{align}
Here we should notice that the operators in the NSNS(RR)-sector are mapped
to those in the RR(NSNS)-sector. Moreover, if the winding number
is violated maximally as $w = N -2$, then there is no extra insertion of
operator at $z = y_i$.

\subsection{Amplitudes of OSP(1$|$2)/U(1) Coset Model}

Utilizing the fundamental relation \eqref{Ribault}, we can
rewrite correlation functions of OSP(1$|$2)/U(1)
coset in terms of ${\cal N}-1$ super Liouville theory with a supersymmetric pair of free boson and fermion
in a manner similar to the bosonic case \cite{HS}.
Here the vertex operators of coset model are defined as
in \eqref{parafermion}%
\footnote{Only in this subsection we construct the coset model by gauging
the axial U(1) symmetry in order to use the trick of \cite{HS}.
}
\begin{align}
 \Psi^{s,\bar s}_{j,m,\bar m} (z,\bar z)  =
 V^{X^3}_{m , \bar m} (z , \bar z)
 \Phi^{s,\bar s}_{j,m,\bar m}  (z , \bar z)
\end{align}
with
\begin{align}
 V^{X^3}_{m,\bar m} (z,\bar z) =
 e^{i \sqrt{\frac{2}{k}}( - m X^3 + \bar m \bar X^3)} .
\end{align}
Moreover, the vertex operators of OSP(1$|$2) model
with spectral flow index $w$
are related to vertex operators of OSP(1$|$2)/U(1) model as
(see \eqref{spectralflow})
\begin{align}
 \Psi^{s,\bar s}_{j,m,\bar m} (z,\bar z)  =
 V^{X^3}_{m + \frac{kw}{2}, \bar m + \frac{kw}{2}} (z , \bar z)
 \Phi^{s,\bar s , w}_{j,m,\bar m}  (z , \bar z) .
 \label{sf2}
\end{align}
Therefore we can also obtain a formula for correlation functions
of OSP(1$|$2) model with non-trivial spectral flow actions.
In particular, we will
be interested in a specific $N$-point function in OSP(1$|$2) WZNW model
as
\begin{align}
 {\cal M} = \left \langle \Phi^{s_1,\bar s_1}_{j_1,m_1,\bar m_1}  (z_1)
 \Phi^{s_2,\bar s_2}_{j_2,m_2,\bar m_2}  (z_2)
\prod_{\nu = 3}^N
 \Phi^{s_\nu ,\bar s_\nu , w_\nu = 1}_{j_\nu,m_\nu,\bar m_\nu}  (z_\nu )
 \right  \rangle .
 \label{MWV0}
\end{align}
Since the amplitude has $N-2$ number of winding violation, it should be
written in terms of $N$-point function of ${\cal N}=1$ super Liouville theory.

Let us first study $N$-point function of OSP(1$|$2)/U(1) coset model.
As before we introduce a new field $\hat X^3$ by
\begin{align}
 \hat X^3_L : = X^3_L - i \sqrt{\frac{k}{2}}
\ln  ( u {\cal B} ) ,
 \end{align}
and the right mover defined by its complex conjugate.
By closely following \cite{HS3} and utilizing the
formula \eqref{Ribault}, we finally obtain
\begin{align} \label{cigar}
&  \left \langle \prod_{\nu = 1}^N
 \Psi^{s_\nu , \bar s_\nu}_{j_\nu, m_\nu , \bar m_\nu}
 \right  \rangle =
  \int \frac{\prod_{i=1}^{N - 2 - w} d ^2 y_i}{(N-2-w)!}
 \times \\
& \qquad \qquad \times
 \left  \langle \prod_{\nu=1}^N
 V^{\hat X^3}_{m_\nu + \frac{k}{2}, \bar m_\nu + \frac{k}{2}} (z_\nu)
  V^{s_\nu - \frac{1}{2}, \bar s_\nu- \frac{1}{2}}_{\alpha_\nu} (z_\nu  )
  \prod_{i =1}^{N-2 - w} V^{\hat X^3}_{- \frac{k}{2},- \frac{k}{2}}(y_i)
 V^{\frac{1}{2},\frac{1}{2}}_{- \frac{1}{2b}} (y_i  )
 \right  \rangle .
  \nonumber
\end{align}
The label $w$ is related to the winding number violation as
$\sum_\nu m_\nu = \sum_\nu \bar m_\nu = - \frac{kw}{2}$.
The right hand side should be computed by the theory
with the action $S[\phi ',\psi,\psi_X]$ for ${\cal N} = 1$ super
Liouville theory and a free fermion $(\psi_X, \bar \psi_X)$
and a free boson $\hat X^3$ with background charge
$Q = - i \sqrt{k}$ for its dual field.

With the formula \eqref{cigar} and \eqref{sf2}
we can write down
generic correlation functions in the OSP(1$|$2) model
with spectral flow action considered
in terms of super Liouville theory.
Here we only compute the amplitude \eqref{MWV0} since it
is the case  used in the later analysis.
With the formula \eqref{sf2} we can relate the amplitude
\eqref{MWV0} to a $N$-point function of the coset as
\begin{align}
 \left \langle \prod_{\nu = 1}^N
 \Psi^{s_\nu , \bar s_\nu}_{j_\nu, m_\nu , \bar m_\nu}
 \right  \rangle & = {\cal M} \times \left \langle
 V^{X^3}_{m_1 , \bar m_1 } (z_1 )
 V^{X^3}_{m_2 , \bar m_2 } (z_2 )
\prod_{\nu = 3}^N
 V^{X^3}_{m_\nu + \frac{k}{2}, \bar m _\nu + \frac{k}{2}} (z_\nu )
\right  \rangle .
\end{align}
Then, by combining with the formula \eqref{cigar}, we find
\begin{align}
{\cal M} = |\Theta_s (z_\nu )|^2
 \left  \langle \prod_{\nu=1}^N
  V^{s_\nu - \frac{1}{2}, \bar s_\nu- \frac{1}{2}}
  _{\alpha_\nu} (z_\nu  )
 \right  \rangle ,
 \label{MWV}
\end{align}
where the right hand side is computed by the ${\cal N}=1$
super Liouville theory and a free fermion with the action
\eqref{sLaction}. The coefficient is given by
\begin{align}
\Theta_s (z_\nu ) = (z_1 - z_2)^{\frac{k}{2} + m_1 + m_2 }
 \prod_{\nu = 3}^N [ (z_1 - z_\nu ) (z_2 - z_\nu ) ]
 ^{\frac{k}{2} + m_\nu} ~,
\end{align}
and bared expression for $\bar \Theta_s (\bar z_\nu )$.
This formula will be important to relate amplitudes of
the topological model and the $\hat c \leq 1$ superstring theory.

\section{Correspondence to $\hat c \leq 1$ Superstring Theory}

In section \ref{coset}, we have studied the topological model
based on the ${\cal N}=2$ supersymmetric coset OSP(1$|$2)/U(1).
In particular, we have observed that free fields and chiral primaries
of the coset model are mapped to matter contents and physical
operators in the $\hat c \leq 1$ superstring theory.
In this section, we establish the relation in more detail.
After briefly reviewing the $\hat c \leq 1$
superstring theory and the method to compute
amplitudes in topological models, we compare
the amplitudes of both theories.

\subsection{$\hat c \leq 1$ Superstring Theory}
\label{secc}

In section \ref{coset} we have already encountered the
$\hat c \leq 1$ superstring theory during constructing the
topological model of the coset OSP(1$|$2)/U(1).
In this subsection we define the $\hat c \leq 1$ superstring
theory in a more precise way.%
\footnote{Notice that we are setting $\alpha'=2$ in this
paper.} The matter part consists of a linear dilaton $X$ with
background charge $Q_X = i(1/b - b)$ and a free fermion $\psi_X$.
The action of these fields is given by
\begin{align}
 S_X = \frac{1}{4 \pi} \int d^2 z \left[ \partial X \bar \partial X
  + \frac{Q_X}{4}\sqrt{g} {\cal R} X + \psi_X \bar \partial \psi_X
  + \bar \psi_X \partial \bar \psi_X \right] .
\end{align}
The theory also includes the ${\cal N } = 1$ super Liouville theory,
whose action is
\begin{align}
 S = \frac{1}{4 \pi} \int d^2 z \left[ \partial \phi  \bar \partial \phi
  + \frac{Q_{\phi } }{4}\sqrt{g} {\cal R} \phi + \psi \bar \partial \psi
  + \bar \psi \partial \bar \psi + \mu_L \psi \bar \psi e^{b \phi}
 \right]
\end{align}
with $Q_\phi = b + 1/b$.
The total central charge is now
$c = 3/2 + 3 Q_X^2 + 3/2 + 3 Q_\phi^2 = 15$,
and hence we can construct a critical superstring theory by
coupling the world-sheet superghosts
$(b,c)$ and $(\beta ',\gamma ')$.

Primary operators of this theory may take the form
$\exp ( \alpha X + \beta \phi)$, which
has the conformal weight $\Delta = \alpha (
Q_X - \alpha ) /2 +  \beta ( Q_\phi - \beta ) /2 $.
Following the standard method to construct BRST invariant
operators, we can find out physical operators in
the $\hat c \leq 1$ superstring theory.
The tachyon vertex operator is given as
\begin{align}
c \bar c {\cal T}^{(-1)}_p = c \bar c e^{- ( \chi + \bar \chi ) }
e^{i k_X (X + \bar X) + k_\phi^\pm  \phi  } ,
\label{m1picture}
\end{align}
where the momenta run over
$ i k_X = Q_X /2 + i p$ and $k_\phi^\pm = Q_\phi / 2 \pm p$
with $p \in \mathbb{R}$.
We bosonize the superghost $\beta ' , \gamma '$ like in
\eqref{bosonized}, which yields the new fields $\chi$.
In other words, the above expression is in
$(-1,-1)$ picture; and
in $(0,0)$ picture it is written as
\begin{align}
c \bar c {\cal T}^{(0)}_p = c \bar c ( i k_X \psi_X  + k_\phi^\pm \psi  )
  ( i k_X \bar \psi_X  + k_\phi^\pm \bar \psi  )
 e^{ i k_X (X + \bar X ) + k_\phi^\pm \phi } .
 \label{0picture}
\end{align}
There are other physical operators in the RR-sector.
The Ramond vertex operator in $(-1/2 , -1/2)$ picture
is written as
\begin{align}
 c \bar c {\cal R}^{(-1/2)}_p = c \bar c
 e^{- \frac12 ( \chi + \bar \chi ) }
 e^{  \pm \frac{i}{2} (Y + \bar Y ) + i k_X (X_L + X_R ) + k^\pm_\phi \phi } .
 \label{rrv}
\end{align}
Indeed, these vertex operators (\ref{m1picture}) and (\ref{rrv}) are
the same as those obtained from the topological string on
OSP(1$|$2)/U(1) as observed in (\ref{phyo}).

If $X$ direction is compactified with radius $R$, then the momentum
takes discrete values $p = n /R$ with $n \in {\mathbb Z}$.
For winding modes we should replace
$X_R \to - X_R$, $\bar H \to - \bar H$ and $p = w R/2$ with $w \in {\mathbb Z}$.
After the proper GSO projection, type 0B theory has the tachyon modes and the
momentum modes in the RR-sector, On the other hand, type 0A theory has the tachyon
modes and the winding modes in the RR-sector.
See, e.g. \cite{TT,DKKMMS} for more detail.

\subsection{Amplitudes of Topological Model}
\label{atm}

Before dealing with the specific case of OSP(1$|$2)/U(1),
we give generic arguments on
amplitudes in topological models.
Let us consider a topological field theory obtained by the
topological twist $T^{top} (z) = T (z) + 1/2 \partial J_R (z) $ of a
${\cal N} = 2$ super conformal field theory \cite{EY,WTST}. We
consider the B model, namely twist the same way for the
anti-holomorphic part as $\bar T^{top} (\bar z) = \bar T (\bar z) +
1/2 \bar \partial \bar J_R (\bar z) $. Then the physical spectrum
can be computed from the cohomology of BRST operator $Q = \oint G^+
(z) dz$. Let us write a basis of physical operators (in NS sector)
as $\phi_i$, then we can obtain other types of physical operators as
\begin{align}
 \oint dz G^-_{- \frac12} \cdot \phi_i , \qquad
 \oint  d \bar z \bar G^-_{- \frac12} \cdot \phi_i , \qquad
 \int d^2 z G^-_{- \frac12} \bar  G^-_{- \frac12} \cdot \phi_i .
\end{align}
Following the arguments on \cite{DVV2,WTST,BCOV},
we compute amplitudes of the form
\begin{align}
 {\cal F}
 =  \left \langle \phi_{i_1} (z_1 )
 \phi_{i_2} (z_2 ) \phi_{i_3} (z_3 )
 \prod_{\nu =4}^N \left [ \int d ^2 z_\nu \tilde \phi_{i_\nu}(z_\nu) \right ]   \right \rangle ,
 \label{topamp}
\end{align}
which would give us
non-trivial information of the topological model.
Here we have defined
$\tilde \phi_i = G^-_{-\frac{1}{2}} \bar G^-_{-\frac{1}{2}} \phi_i$.

An important fact is that
the above amplitudes of topological model can be computed in
the original untwisted model \cite{WTST}. After the topological
twist the $U(1)_R$ current becomes anomalous, and hence we should
insert $U(1)_R$ fields into correlators of original model to reproduce
the topological amplitudes. Here we insert the operator
$\mu (z ,
\bar z) = e^{ \frac{i}{2} \sqrt{\frac{c}{3}}( X _R (z)) + \bar X _R
( \bar z))}$ at two points $z = z_1,z_2$, where
the free boson $X_R$ is related to the R-current as
$J_R (z) = - i
\sqrt{\frac{c}{3}} \partial X _R (z)$. This operator maps the
physical operators $\phi_i$ of the topological model to operators
$\phi_i^R$ in the R-ground states of the original model.
In this way, we can write the topological amplitude \eqref{topamp} as
\begin{align} \label{orgamp}
 {\cal F}
 &= |z_1 - z_2 |^{q_1 + q_2}
 ( |z_1 - z_3 | |z_2 - z_3 | )^{q_3}
  \times \\ &\times \nonumber
   \left \langle \phi^R_{i_1} (z_1 )
 \phi^R_{i_2} (z_2 ) \phi_{i_3} (z_3 )
 \prod_{\nu =4}^N \left [ \int d ^2 z_\nu
 ( | z_1 - z_\nu | | z_2 - z_\nu | )^{q_\nu - 1}
 \tilde \phi_{i_\nu}(z_\nu) \right ]   \right \rangle  ,
\end{align}
where the right hand side is computed in the original model
before the topological twisting.
Here we have denoted
$q_\nu$ as the $U(1)_R$ charge of $\phi_{i_\nu}$.

\subsection{Comparison of Correlation Functions}
\label{correlation}

After the preparation we can now compare correlation functions
of topological model on OSP(1$|$2)/U(1) and of the $\hat c \leq 1$
superstring. We start from the amplitude of the topological
model, and then show the equivalence by using the formula
\eqref{MWV} obtained above.
 Here we
only consider the amplitudes of operator of the first type in
\eqref{cpt}, which corresponds to the tachyon operator
in the $\hat{c} \leq 1$ superstring. The case with the second type
in \eqref{cpt} can be analyzed in a similar way.

Since the conservation of U(1) current $J
=\partial \chi$ is violated by the amount of $-2$, non vanishing
amplitudes may be given as
\begin{align}
 {\cal F}
 =  \left \langle {\cal O}^{\frac12}_{j_1} (z_1 )
 {\cal O}^{\frac12}_{j_2} (z_2 ) {\cal O}^{-\frac12}_{j_3}(z_3 )
 \prod_{\nu =4}^N \left [ \int d ^2 z_\nu \tilde
  {\cal O}^{-\frac12}_{j_\nu} (z_\nu) \right ]   \right \rangle .
  \label{topamp2}
\end{align}
This violation corresponds to the fact that the sum of picture
must be $-2$ in superstring theory.
The operator ${\cal O}^{\frac{1}{2}}_j$ is defined in
\eqref{cpt} as
\begin{align}
 {\cal O}^{\frac{1}{2}}_j  =  e^{ - i  (H + \bar H ) - ( \chi + \bar \chi ) }
 \Phi^{\frac{1}{2},\frac{1}{2}, w = 1}
    _{j,j-\frac{1}{4},j-\frac{1}{4}}
 e^{2ib(j+\frac{1}{4 b^2}) (X + \bar X )} .
\end{align}
Following the argument in the previous subsection, this
operator would be mapped to the R-ground state operator of
the original model
\begin{align}
 {\cal O}^{R,\frac{1}{2}}_j =  e^{\frac{i}{2} (H + \bar H )}
 \Phi^{\frac{1}{2},\frac{1}{2}}
 _{j,j-\frac{1}{4},j-\frac{1}{4}} e^{2ib(j+\frac{1}{4})(X+ \bar X)} .
\end{align}
Another operator $ {\cal O}^{- \frac{1}{2}}_j$
is given by a linear combination of
\begin{align}
  e^{ - i  (H + \bar H ) }
 \Phi^{s, \bar s, w = 1}
   _{j,j + \frac{1}{4},j+\frac14} e^{2ib(j+\frac{1}{4 b^2})(X+ \bar X)}
\end{align}
with $s , \bar s = -1/2 ,3/2 $.
This operator should correspond to picture $(0,0)$ tachyon, and
can be constructed
by following the analysis in section \ref{coset}.
The other operator $ \tilde {\cal O}^{- \frac{1}{2}}_j$
is then generated by the action of $G^-_{-1/2} \bar G^{-}_{-1/2}$
as mentioned before
and written as a linear combination of
\begin{align}
 \Phi^{s, \bar s, w = 1}
   _{j,j - \frac{3}{4},j-\frac34} e^{2ib(j+\frac{1}{4 b^2})(X+ \bar X)}
\end{align}
with $s , \bar s = -1/2 ,3/2 $.

As argued in the previous subsection,
we first map the amplitude of topological model
\eqref{topamp2} to that of original model before twisting as in
\eqref{orgamp}.
Then we can use the formula \eqref{MWV} to relate it to the
amplitude of ${\cal N}=1$ super Liouville theory.
Following the analysis of \cite{NN,Takayanagi} we can then show that
\begin{align}
 {\cal F}
 =
 \left \langle c \bar c {\cal T}^{(-1)}_{j_1} (z_1 )
 c \bar c {\cal T}^{(-1)}_{j_2} (z_2 )
 c \bar c {\cal T}^{(0)}_{j_3}(z_3 )
 \prod_{\nu =4}^N \left [ \int d ^2 z_\nu
  {\cal T}^{(0)}_{j_\nu} (z_\nu) \right ]   \right \rangle
\end{align}
up to some coefficients.
Here, the operators ${\cal T}^{(p)}$ are tachyon operators
in the $p$-th picture and they are given by
\begin{align}
&{\cal T}^{(-1)}_p = e^{- ( \chi + \bar \chi ) }
e^{i k_X (X + \bar X) + k_\phi^+ \phi  } ~, \\
&{\cal T}^{(0)}_p =  ( i k_X \psi_X  + k_\phi^+ \psi  )
  ( i k_X \bar \psi_X  + k_\phi^+ \bar \psi  )
 e^{ i k_X (X + \bar X ) + k_\phi^+ \phi }
\end{align}
with $k_X = (1/b - b)/2 + 2b ( j + 1/4)$ and
$k_\phi^+ = (1/b + b )/2 + 2b ( j + 1/4 )$.
In this way we have shown that the amplitude of topological
string on OSP(1$|$2)/U(1) can be identified with that of
the $\hat c \leq 1$ superstring.

\section{Conclusion and Discussions}
\label{conclusion}

In this paper we have proposed an equivalence between the
topological string theory based on the coset OSP(1$|$2)/U(1)
and the $\widehat{c}\leq 1$
superstring theory. The latter is constructed by coupling a
$\hat{c}\leq 1$ matter to the $\mathcal{N}=1$
super Liouville theory. This can be regarded as a
supersymmetric version of the equivalence between the topological
string on $SL(2)/U(1)$ and the $c\leq 1$ bosonic string, which was
originally discovered by Mukhi and Vafa \cite{MV} for the case $c=1$ and was
later generalized to the $c<1$ case in \cite{Takayanagi}.
First we showed in the free field description that the field contents and the physical operators of the world-sheet theories of both string theories
match. Moreover, we
investigated the proposed equivalence at the level of scattering amplitudes
by applying the map \cite{HS2} between correlation functions in the
OSP(1$|$2) WZNW model and in super Liouville field theory.

This map
is a supersymmetric version of the one found by Ribault and Teschner
to relate correlation functions in the SL(2) WZNW model and those in
Liouville theory \cite{RT,HS3}. In the last years, the result has been
used with great success to investigate different dualities between
non-rational conformal field theories. In particular, it has led to the
proof of Fateev-Zamolodchikov-Zamolodchikov conjecture in \cite{HS}.
The duality between different non-rational two-dimensional conformal field
theories has a long story, and now a considerable list of examples is
available: quantum Hamiltonian reduction \cite{BO2},
Mukhi-Vafa duality \cite{MV,Takayanagi}%
, and Fateev-Zamolodchikov-Zamolodchikov duality
\cite{FZZunpublished} (and its supersymmetric version \cite{HoKa})
are probably the most renowned examples. These examples were, in
fact, very useful to study string theory. For instance, it was the
Fateev-Zamolodchikov-Zamolodchikov duality what really permitted to
construct a dual matrix model for strings in the the 2D black hole
background \cite{KKK}. It is our hope that the new equivalence
between conformal theories we studied in this paper will be relevant
to understand new aspects of superstring dualities as well.

There are a number of issues which should be understood in the future.
First we would like to understand better the spin structure and the
picture changing operation of the topological string theory on the
supercoset. It is also important to prove the complete equivalence
of physical states between these two theories. An exhaustive analysis of the cohomology of the theory is needed to this end. Finally, it would be
nice if we could understand a geometrical interpretation of the
supercoset OSP(1$|$2)/U(1) in terms of a certain
(maybe super) Calabi-Yau manifold, as SL(2)/U(1) coset
model is related to the conifold.

\subsection*{Acknowledgement}

We are very grateful to K. Hori for a useful discussion. GG thanks
the members of the IPMU for their hospitality during his stay.
YH would like to thank T.~Creutzig, H.~Irie and P.~B.~R{\rm ord}nne for
useful discussions. The
work of GG has been partially supported by ANPCyT grant
PICT-2007-00849, by UBACyT grants X861 and X432, and by JSPS-CONICET
cooperation programme.
The work of YH is supported by JSPS Research Fellowship.
TT is supported by World Premier
International Research Center Initiative (WPI Initiative), MEXT,
Japan.
 The work of TT is supported in part by JSPS Grant-in-Aid for
Scientific Research No.20740132 and by JSPS Grant-in-Aid for
Creative Scientific Research No. 19GS0219.

redpajama_set_name: RedPajamaArXiv


The fight for Transgender Equality, "America's Next Civil Rights Frontier," is alive and well in the state of Florida. Last year, Equality Florida launched our Transgender Inclusion Initiative with three broad objectives: To educate Florida's major employers on effective transgender inclusion protocol, to advocate for the trans community in a broader and more structured way, and to be an integral part of all public policy that involved transgender rights. To achieve these, and even broader goals across the Sunshine State, we formed TransAction Florida, which consists of transgender community leaders from all parts of our state.
In the coming year, TransAction Florida will be taking transgender education and inclusion to the next level by engaging and educating more of Florida's major employers. We will be releasing a statewide transgender resource guide, and working to pass a fully inclusive Human Rights Ordinance in Jacksonville. We will be mounting an education and inclusion program for trans veterans, and we will be working to ensure that our school boards are practicing approved federal protocols in supporting our transgender youth. And, we will be working to ensure that no more transgender discrimination "bathroom" bills are an issue in Tallahassee.
But most of all, we ask that our Florida Transgender Community stay engaged, stay visible and stay vocal in demanding Transgender Equality in all aspects of our society.
In the words of Jazz Jennings, "All we ask is that the world see us as we see ourselves!"


How the family justice system works changed in April 2014, but separating parents are still unsure how to go about resolving their disputes over the care of their children.
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text:
Briana Williams murder: Pretty Little Mamas films sixth anniversary of friend's tragic death
Fri Aug 31, 2018 at 5:22am ET
Pretty Little Mamas stars mourning their friend Briana Williams — Pic Credit: MTV
Pretty Little Mamas just debuted on MTV and already there are plenty of questions surrounding the show. The group of five friends is navigating life together and they are all connected to one special person who was mentioned on the show but isn't appearing on it.
Who is Briana Williams?
Briana Williams was called Bree by her close friends and family members. She was the fifth friend that completed a group of young women who have been friends for years. Just a year after high school, Williams was met with an untimely demise.
What happened to Bree?
On Pretty Little Mamas, the girls talked about their friend Bree. She was one of the best friends they had and fit into their social circle with no issues. Unfortunately, Briana Williams was murdered along with her mother.
During one of the confessionals on Pretty Little Mamas, it was revealed that Briana Williams was murdered. Upon further digging, the story of her death was unearthed and was proven to be devastating to everyone who knew her well.
A post shared by Alyssa Myers (@lyssbrenbren)
Back in July 2011, Briana Williams was murdered by her brother alongside her mother. Brian Rockwell Williams stabbed both his mother and sister, and they died from the injuries sustained.
It was something that rocked the family to their core. With Bree's mother being gone, her siblings were left to pick up the pieces of the double murder that occurred.
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Personal power. Ego. Often, the path the car takes in the dream is representative of the path the dreamer's life is on. If the car is speeding out of control, the dreamer may feels as if his/her life is also speeding out of control. Some other associations to keep in mind: To many, driving is a solitary time. With the windows rolled up and the music playing, one is left alone with one's thoughts. For someone who's been in a car accident, driving can represent a dangerous risk. As cars seem to represent one's life, when your life is out of control, don't be surprised if you have a dream where you're driving an out-of-control car, or a car whose breaks are malfunctioning. Another common variation is controlling the car from a great height above the street, making it difficult to steer without collisions, or even trying to drive the car from the backseat. Whatever the variation, when you're having trouble driving, you probably feel like you're out of control in your waking life. Get your life under control, and you'll have better luck in your dream driving adventures. Please see above definition of "car." Often, the path the car takes in the dream is representative of the path the dreamer's life is on. If the car is speeding out of control, the dreamer may feels as if his/her life is also speeding out of control. Some other associations to keep in mind: To many, driving is a solitary time. With the windows rolled up and the music playing, one is left alone with one's thoughts. For someone who's been in a car accident, driving can represent a dangerous risk. As cars seem to represent one's life, when your life is out of control, don't be surprised if you have a dream where you're driving an out-of-control car, or a car whose breaks are malfunctioning. Another common variation is controlling the car from a great height above the street, making it difficult to steer without collisions, or even trying to drive the car from the backseat. Whatever the variation, when you're having trouble driving, you probably feel like you're out of control in your waking life. Get your life under control, and you'll have better luck in your dream driving adventures. Please see above definition of "car."
> Find more dreams containing 'car'
> Find the meaning in Gematria for 'car'
Preservation. Restraint. Feelings need to be resolve.
> Find more dreams containing 'frozen'
> Find the meaning in Gematria for 'frozen'
Violence. Aggression. Threat. Danger ahead. This is a dream of distress. Hearing the sound of a gun, denotes loss of employment, and bad management to proprietors of establishments. If you shoot a person with a gun, you will fall into dishonor. If you are shot, you will be annoyed by evil persons, and perhaps suffer an acute illness. For a woman to dream of shooting, forecasts for her a quarreling and disagreeable reputation connected with sensations. For a married woman, unhappiness through other women.
> Find more dreams containing 'gun'
> Find the meaning in Gematria for 'gun'
Fellowship. Work on community. Ready to join others in a project. To see your neighbors in your dreams, denotes many profitable hours will be lost in useless strife and gossip. If they appear sad, or angry, it foretells dissensions and quarrels.
> Find more dreams containing 'neighbor'
> Find the meaning in Gematria for 'neighbor'
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working on some additional fixes to improve this.
Fixed and released in last weeks version.
Understand the need. Unfortunately, we don't have that data set for now.
Yes will be doing more updates as I'll be more active with this portfolio. Will keep you posted on the blog and email.


Game description: Burst all the bubbles within a limited time by simply clicking on them. Load your finger and become a Bubble Popper Champ!
this game is great and very relaxing to play after work.
Seems to be a problem with your site - can not play a single bloody game!
Fun and a good time killer!


WILSHIRE LLC owns, operates and continually renovates, downtown Dubois apartment buildings. WILSHIRE LLC is committed to providing outstanding service and value to renters by providing brand new, energy efficient and affording dwellings for professionals. We are recognized as one of the preeminent rental apartment companies in DuBois, PA. We are known for constructing quality dwellings for our tenants and are committed to doing our part to improve downtown DuBois. Stop by today to take a look at the Wilshre II building located at 132 South Brady Street in DuBois!
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Tenants responsible for: Electric, Water, Sewage, Cable, and Phone. We cover garbage.


This morning, October 13, The Rockettes kicked off rehearsals for their 2016 production of "The Christmas Spectacular Starring the Radio City Rockettes" and gave a sneak peek of the number "New York at Christmas." Scroll down for photos!
"The Christmas Spectacular Starring the Radio City Rockettes" will bring joy to audiences of all ages and help you create countless memories with your family, friends and loved ones this holiday season. The Christmas Spectacular Starring the Radio City Rockettes, presented by Chase, will run from November 11, 2016 - January 2, 2017. Tickets for the 2016 Christmas Spectacular are on sale now at www.rockettes.com/christmas and the Radio City box office.
Embraced by generations of families, the Christmas Spectacular Starring the Radio City Rockettes is a one-of-a-kind tradition where the unmistakable spirit of Christmas in New York comes alive. The 2016 production of the Christmas Spectacular features stunning Rockettes dance numbers, spectacular costumes and state-of-the-art technology. The Radio City Rockettes are a legendary Dance Company and have been a part of the fabric of New York City for decades. Since their debut in the Christmas Spectacular in 1933, The Rockettes have danced into the hearts of millions and spread Christmas cheer to people of all ages. Their precision technique continues to inspire fans of all ages and their talent and athleticism is unrivaled.
The Radio City Rockettes kick off rehearsals for the 2016 Radio City Music Hall Christmas Spectacular.


text:
'use strict';

var chai = require('chai'),
  pp = require('../lib/preprocess');

chai.should();

describe('@ifndef directive shall be preprocessed', function () {
  var input;

  describe('in html', function () {
    it('success case, should be included', function () {
      input = "a<!-- @ifndef NONEXISTANT -->b<!-- @endif -->c";
      pp.preprocess(input, {}).should.equal("abc");
    });

    it('fail case, should not be included', function () {
      input = "a<!-- @ifndef NODE_ENV -->b<!-- @endif -->c";
      pp.preprocess(input, {NODE_ENV: 'dev'}).should.equal("ac");
    });

    it('success case, should be included (hidden by default syntax, end tag)', function () {
      input = "a<!-- @ifndef NONEXISTANT !>b<!-- @endif -->c";
      pp.preprocess(input, {}).should.equal("abc");
    });

    it('fail case, should not be included (hidden by default syntax, end tag)', function () {
      input = "a<!-- @ifndef NODE_ENV !>b<!-- @endif -->c";
      pp.preprocess(input, {NODE_ENV: 'dev'}).should.equal("ac");
    });

    it('success case, should be included (hidden by default syntax, end tag and start tag)', function () {
      input = "a<!-- @ifndef NONEXISTANT !>b<! @endif -->c";
      pp.preprocess(input, {}).should.equal("abc");
    });

    it('fail case, should not be included (hidden by default syntax, end tag and start tag)', function () {
      input = "a<!-- @ifndef NODE_ENV !>b<! @endif -->c";
      pp.preprocess(input, {NODE_ENV: 'dev'}).should.equal("ac");
    });

    it('should support nesting', function () {
      input = "a<!-- @ifndef NDEF -->b<!-- @ifndef NODE_ENV -->bad<!-- @endif -->c<!-- @endif -->d";
      pp.preprocess(input, {NODE_ENV: 'dev'}).should.equal("abcd");
    });
  });

  describe('in javascript', function () {
    it('success case, should be included (line)', function () {
      input = "a\n" +
        "// @ifndef NONEXISTANT\n" +
        "b\n" +
        "// @endif\n" +
        "c";
      pp.preprocess(input, {}, 'js').should.equal("a\nb\nc");
    });

    it('fail case, should not be included (line)', function () {
      input = "a\n" +
        "// @ifndef NODE_ENV\n" +
        "b\n" +
        "// @endif\n" +
        "c";
      pp.preprocess(input, {NODE_ENV: 'dev'}, 'js').should.equal("a\nc");
    });

    it('success case, should be included (block)', function () {
      input = "a/* @ifndef NONEXISTANT */b/* @endif */c";
      pp.preprocess(input, {}, 'js').should.equal("abc");
    });

    it('fail case, should not be included (block)', function () {
      input = "a/* @ifndef NODE_ENV */b/* @endif */c";
      pp.preprocess(input, {NODE_ENV: 'dev'}, 'js').should.equal("ac");
    });
  });

  describe('in coffeescript', function () {
    it('success case, should be included', function () {
      input = "a\n" +
        "# @ifndef FLAG\n" +
        "b\n" +
        "# @endif\n" +
        "c";
      pp.preprocess(input, {}, 'coffee').should.equal("a\nb\nc");
    });

    it('success case, should be included (multiple hashes)', function () {
      input = "a\n" +
        "## @ifndef FLAG\n" +
        "b\n" +
        "## @endif\n" +
        "c";
      pp.preprocess(input, {}, 'coffee').should.equal("a\nb\nc");
    });

    it('fail case, should not be included', function () {
      input = "a\n" +
        "# @ifndef FLAG\n" +
        "b\n" +
        "# @endif\n" +
        "c";
      pp.preprocess(input, {FLAG: 1}, 'coffee').should.equal("a\nc");
    });
  });
});

redpajama_set_name: RedPajamaGithub


I can't say I blame her for doing so. How many years has it been now? Seems like he lives his life in the past. Not healthy!!
Yea but Lisa, isnt that part of the court system that it takes so long? I say go Jay all the way. She has put her spin on all the lies. Let the truth come out. Next she will have one on slimey as he does know the truth.
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Racing enthusiasts are, no doubt, aware that radical improvements are being made to the grand stands and members' boxes, as well as giving the jockeys and stewards a spacious room each with all modern conveniences. Even the ponies have not been forgotten. They will have a stable that other steeds would envy.
Fronting the long row of buildings, the members' and public stands, with their tiers of concrete steps showed up very prominently. The whole view is most picturesque, and notwithstanding the fact that drab workmen are still busily moving to and fro, and the grime and mire that are inevitably present in a building under construction, the entire race course has been already beautified.
Beginning from the North end, which is opposite to the Colonial Cemetery, a model stable has sprung up on the same piece of ground where the booths used to be.
Before the great fire these booths were mere matsheds, and after the disaster, a cumbersome concrete building was erected. This latter building was torn down on March 21, and in its place, a home for the ponies has come into being.
This home is built on most sanitary line and affords excellent accommodation for the ponies.
A two-storey building of reinforced concrete, there are 60 stalls on each floor.
The stalls are divided in two rows—30 on each side—with an alley between them. The entire block of building is divided in three sections; being separated by fire-proof steel doors.
The entrance to the stable is from the paddock. Two broad ramps give access to the first floor, while access to the ground floor is directly from the paddock. The interior is well ventilated and the stalls are only to be alloted to owners of race ponies participating in a meeting.
This stable although up-to-date in all respects, is not intended to provide the ponies with a permanent home. They will be housed there only temporarily during a race meeting. Their permanent domicile will still be at the Jockey Club's stables at Causeway Bay.
Crossing over from the cinder paddock is the members' stand which is about 150 feet in length. The stand has 18 tiars of concrete steps with a huge concrete canopy overhead. This is 20 feet in width and affords ample protection to the spectators from sun and rain.
On the first and second floors are to be found the private boxes for members. These boxes can either be had single or double.
They are designed with the view of giving the members the utmost comfort. Each box has its own verandah which commands a full view of the course. Immediately at the rear is a private kitchen and lavatory for each box.
Where the main entrance cuts off the ground floor of the north block from the south block, a clock tower will be built. It will have a height of 120 feet from the ground. On a level with the first floor, where the members will have their lounge an alley on both sides of the tower will be found, so as to join the two blocks of buildings together.
In the middle section of the first floor will be found the members' lounge and luncheon room. The members' kitchen will be found on the second floor.
The luncheon room is very spacious and is a great improvement over the old one while the lounge will be tastefully furnished so as to provide a pleasant resting place to the members after an exciting race meeting.
At the center of the second floor above the members' stand, there will be a spacious box for members of the European Press. The position is most central, and the broad verandah commands a full view of the course. A telephone will be installed in this room for the use of the Press men.
Entrance to the public stand will be from the south end—opposite the Parsee Cemetery. This block of building is also two-storey high, and as explained, it is joined to the block on the north end. On the first and second floors are boxes for members. Each is fitted out in the same manner and each has its own lavatory and kitchen.
The stand for the public is about 270 feet long with a tier of 18 steps of concrete. A wide canopy hangs overhead so as to give protection from sun and rain to the spectators.
On the ground floor there will be the cash sweep counters and pari-muteul for the public. They are so arranged that no delay and inconvenience will be caused.
In front of the public stand there will be the usual boards to announce the entries and the winner and also the cash sweep results and the same thing is to be found on the members' side.
During the past few days a numbers of coolies have been engaged in removing the mound of earth from the side of the course in the straight. This will afford about five feet more room from the 'WongNeiChong 'Village Bend (Sanatorium Bend) to the winning post. This is no improvement which in much needed ,for it will make a lot of difference when twenty or more ponies are coming into the straight practically together.
On the whole the Hong Kong Jockey Club is to be congratulated. The new stands with their appertaining buildings will no doubt be much appreciated by members and the public. It is obvious that the Club has spared no expense in combining beauty with service ability. And in this, it is fair to mention that both the Hon. Mr. B. D. F. Beith and Mr. T. E. Pearce two hard-working Stewards of the Club have spent much time, in connection with the work. But to Mr. C. G. S. Gordon Mackie the scheme owes its birth.
Messrs. Palmer, and Turner who are the architects, are responsible for the beautiful design of the building, and, under their supervision, work has been going on rapidly. Mr. K. K. Staple and Mr. A. J. Linge are supervising the construction.
1929-03-21, he booths on the site of the new stable were pulled down and the new building was completed and handed over to the Jockey Club in August.
1929-05-22, two old stands with their appertaining portions were demolished and that in new foundation had to be dug and built in, it can safely be said that the work constitutes a record for speed in Hong Kong.
1929-08-16, had it not been for the typhoon, the Racecourse rebuilt would have been completed earlier. Considerable havoc to the scaffolding work was done which had necessarily delayed the work for about a fortnight.
The sanitary fittings were carried out by Messrs. Dodwell & Co. The wiring system is in the hands of Messrs. Jardine Engineering Corp. Ltd., while the General; Electric Co. are responsible for the fittings. All the windows are of metal suppled by Messrs, Crittal Manufacturing & Co. London whose local agents are Messrs. Arnhold & Co.
1866 Jacob Arnhold, a German Jewish businessman and Peter Karberg, a Danish merchant, as a German-registered trading company founded Arnhold, Karberg & Co., a company in Canton.
1868 founded by architect William Salway, P & T Architects & Engineers Ltd, (simplified Chinese: 公和洋行; traditional Chinese: 巴馬丹拿集團 abbreviated to P & T ), formerly known as Palmer and Turner Hong Kong is a firm of architects who have designed many landmark buildings in Hong Kong, Shanghai and elsewhere in southeast Asia.
1898 as Manor Works, the works team of the Crittall Window Company formed the Braintree Town Football Club, nickname IRON . in the North Essex near London.


text:
A blog about making culture. Since 1999.
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