text:
\section{Introduction and Summary}

Recently, there has been significant progress in the counting problem of BPS states in type IIA string theory on a toric Calabi-Yau 3-fold\footnote{See \cite{JM,CJ,OY1,OY2,DG,AOVY,CP}. See also \cite{Szendroi,Young1,Young2,MR,NN,Nagao1,Nagao2} for mathematical discussions}.
In the literature, the Calabi-Yau manifold (which we denote by $X$) is assumed to have no compact 4-cycles, and we consider a BPS configuration of D0/D2-branes wrapping compact holomorphic 0/2-cycles, as well as a single D6-brane filling the entire Calabi-Yau manifold. The question is to count the degeneracy of such BPS bound states of D-branes.

One subtlety in this counting problem is the wall crossing phenomena, stating that the degeneracy of BPS bound states depends on the value of moduli at infinity. Indeed, the closed BPS partition function\footnote{The upper index $c$ stands for `closed'.}
$$
\ensuremath{\mathcal{Z}}^c_{\rm BPS, (\sigma',\theta')},
$$
which is defined in \cite{OY1} as the generation function of the degeneracy of D-brane BPS bound states\footnote{The definition of the partition function $\ensuremath{\mathcal{Z}}_{\rm BPS}$ is the same as the partition function $\ensuremath{\mathcal{Z}}_{\rm BH}$ in \cite{OSV}.},
depends on maps $\sigma',\theta'$ specifying a chamber in the K\"ahler moduli space\footnote{See Appendix \ref{app.Weyl} for details.}.
What is interesting is that in one special chamber $\tilde{C}_{\rm top}$ of the K\"ahler moduli space, the BPS partition function is equivalent the topological string partition function\footnote{Actually, the topological string partition function depends on the choice of the resolution of the singular Calabi-Yau manifold $X$. This is related to the choice of the limit, as will be explained in the main text.} (up to the change of variables, which we do not explicitly show here for simplicity):
\begin{equation}
\ensuremath{\mathcal{Z}}^c_{\rm BPS} \Big|_{\tilde{C}_{\rm top}} =\ensuremath{\mathcal{Z}}^c_{\rm top} \label{ZtoZtop}.
\end{equation}

It is natural to expect that similar story should exist for open BPS invariants as well. Namely, we expect to define
open version of the BPS partition function\footnote{The upper index $o$ stands for `open'.}
$$
\ensuremath{\mathcal{Z}}^o_{\textrm{BPS},(\sigma,\theta)}
$$
depending on maps $\sigma, \theta$ specifying the chamber in the K\"ahler moduli space, such that the partition function reduces to the open topological string partition function in a special chamber $C_{\rm top}$:
\begin{equation}
\ensuremath{\mathcal{Z}}^o_{\textrm{BPS}} \Big| _{C_{\rm top}}=\ensuremath{\mathcal{Z}}^o_{\rm top}.
\end{equation}
The question is how to define open BPS degeneracies such that the generating function follows the conditions above.

As a guiding principle of our following argument, we use the crystal melting model developed recently in \cite{OY1} (see \cite{Szendroi,MR} for mathematical discussions).
This crystal melting model generalizes the result of \cite{ORV} for $\ensuremath{\mathbb{C}}^3$ to an arbitrary toric Calabi-Yau manifold. In the case of $\ensuremath{\mathbb{C}}^3$, the crystal melting partition function with the boundary conditions specified by three Young diagrams $\lambda_1,\lambda_2,\lambda_3$ gives
the topological vertex \cite{AKMV} $C_{\lambda_1,\lambda_2,\lambda_3}$. By using these vertices
as a basic building block, we can compute open topological string partition function with non-compact D-branes wrapping Lagrangian 3-cycles of the topology $\ensuremath{\mathbb{R}}^2\times S^1$ included \cite{AV}.
In this story, generalization from closed to open topological string partition function corresponds to the change of the boundary condition of the crystal melting model for $\ensuremath{\mathbb{C}}^3$.

Now the recent result \cite{OY1} shows that the closed BPS partition function  discussed above can be written as a statistical mechanical partition function of the crystal model. This model applies to any toric Calabi-Yau manifold, and for $\ensuremath{\mathbb{C}}^3$ the BPS partition function coincides with the topological string partition function. Similarly to the case of the topological string story mentioned in the previous paragraph, we hope to define the open version of the BPS invariants by changing the boundary condition of the crystal melting model. The invariants defined in this way will be defined in any chamber in the K\"ahler moduli space, and reduces to the ordinary topological vertex in a special chamber. We call such a generalization of the topological vertex ``the non-commutative topological vertex''\footnote{The word `non-commutative' stems from the mathematical terminologies such as ``non-commutative crepant resolution'' \cite{VdB} and ``non-commutative Donaldson-Thomas invariant'' \cite{Szendroi}. The non-commutativity here refers to that of the path algebra of the quiver. The quiver (together with a superpotential) determines a quiver quantum mechanics, which is the low-energy effective theory on the D-brane worldvolume \cite{OY1}. }, \begin{NB}added\end{NB}%
following ``the orbifold topological vertex'' named in \cite{BCY2}.\begin{NB}I removed the footnote: {See \cite{BCY2} for a related proposal. {\bf Q. Our definition include theirs?}}.\end{NB}%

We will see that this expectation is indeed true. We adopt the
\begin{NB}Removed: combinatorial\end{NB
definition proposed by one of the authors in the mathematical literature \cite{Nagao2,Nagao3}. Our non-commutative topological vertex is defined for a Calabi-Yau manifold $X$ without compact 4-cycles, and a set of representations $\lambda$ assigned to external legs of the toric diagram. As in the case of topological vertex, $\lambda$ encodes the boundary condition of the D4-branes wrapping Lagrangian 3-cycles. We propose our vertex as the building block of open BPS invariants. Here by an open BPS invariant we mean a degeneracy counting the number of BPS bound states of D0/D2/D6-branes wrapping holomorphic 0/2/6-cycles, as well as D2-branes wrapping disks whose boundaries are on D4-branes wrapping non-compact Lagrangian 3-cycles.

We can provide several consistency checks of our proposal (see section \ref{sec.check} for more details).
First, our vertex by definition reduces to the closed BPS invariant when all the representations $\lambda$ are trivial. Second, our vertex shows a wall crossing phenomena as we change the closed string K\"ahler moduli, and the vertex coincides with the topological vertex computation in the chamber where the closed BPS partition function reduces to the closed topological string partition function.
Third, the wall crossing factor is independent of the boundary conditions on D-branes, and is therefore the partition function factorizes into the closed string contribution and the open string contribution, as expected from \cite{OVknot} and the generalization of \cite{AOVY}.

Given a combinatorial definition of the new vertex, the next question is whether we can compute it, writing it in a closed expression. We show that the answer is affirmative, by showing the following statement.
For a Calabi-Yau manifold $X$, the non-commutative topological vertex $\ensuremath{\mathcal{C}}_{\textrm{BPS},(\sigma,\theta;\lambda)}(X)$ is equivalent to the closed BPS partition function $\ensuremath{\mathcal{Z}}^c_{\textrm{BPS},(\sigma',\theta')}(X')$ for an orbifold $X'$ of $X$
, under a suitable identification of variables explained in the main text\footnote{More precisely, we need to specify the resolution of $X$ and $X'$. We also need to impose the condition that two of the representations $\lambda$ are trivial. See the discussions in the main text.}:
\begin{equation}
\ensuremath{\mathcal{C}}_{\textrm{BPS},(\sigma,\theta;\lambda)}(X) =\ensuremath{\mathcal{Z}}^c_{\textrm{BPS},(\sigma',\theta')}(X').
\label{eq.main}
\end{equation}
We will give an explicit algorithm to determine $X'$ and $\sigma', \theta'$, starting from the data on the open side.
Since the infinite-product expression for $\ensuremath{\mathcal{Z}}^c_{\textrm{BPS},(\sigma',\theta')}(X')$ is already known \cite{Nagao1,AOVY}, this gives a closed infinite-product expression for our vertex.

The organization of this paper is as follows. We begin in section 2 with a brief summary of the closed BPS invariants and their wall crossings,
and their relation with the topological string theory.
In section 3 we define our new vertex using the crystal melting model. We also perform several consistency checks of our proposal. Section 4 contains our main result \eqref{eq.main}, which shows the
equivalence of
our new vertex with a closed BPS partition function under suitable parameter identifications. We give an explicit algorithm for
constructing closed BPS partition function starting from our vertex.
In Section 5 we treat several examples in order to illustrate our general results. Section 6 is devoted to discussions. We also include Appendices A-C for mathematical proofs and notations.

\section{Closed BPS Invariants}\label{sec.closed}
Before discussing the open BPS invariants, we summarize in this section the definition and the properties of the closed BPS invariants.

Throughout this paper, we concentrate on the case of
the so-called generalized conifolds.
The reason for this is that wall crossing phenomena is understood
well only in cases without compact 4-cycles, which means $X$ is either a generalized conifold or $\ensuremath{\mathbb{C}}^3/(\ensuremath{\mathbb{Z}}_2\times \ensuremath{\mathbb{Z}}_2)$\footnote{See \cite{AOVY} for the proof of this statement.}. \begin{NB}Removed: The case of $\ensuremath{\mathbb{C}}^3/(\ensuremath{\mathbb{Z}}_2\times \ensuremath{\mathbb{Z}}_2)$ is similar.\end{NB}%

By suitable $SL(2,\ensuremath{\mathbb{Z}})$ transformation, we can assume that the toric diagram of a generalized conifold is a trapezoid with height 1, with \begin{NB}corrected\end{NB}%
length $L_+$ edge at the top and $L_-$ at the bottom (see Figure \ref{gctd})\footnote{The Calabi-Yau manifold is determined by $L_+$ and $L_-$ as $xy=z^{L_+} w^{L_-}$.}. If we denote by $L=L_{+}+L_-$ the sum of the length of the edges on the top and the bottom of the trapezoid, this geometry has $L-1$ independent compact $\ensuremath{\mathbb{P}}^1$'s. We label them by $\alpha_i$, borrowing the language of the root lattice of $\hat{A}_{L-1}$ algebra.

\begin{figure}[htpb]
\centering{\includegraphics[scale=0.4]{gctd.eps}}
\caption{The toric diagram of a generalized conifold, with $L_+=3, L_-=5$.}
\label{gctd}
\end{figure}

The language of the root lattice will be used extensively throughout this paper\footnote{The root lattice of $\hat{A}_{L-1}$ is exploited in \cite{Nagao1,AOVY,Nagao2}. See also Appendix \ref{app.Weyl}.}.
We can also make more $\ensuremath{\mathbb{P}}^1$'s by combining them. For example, combining all the $\ensuremath{\mathbb{P}}^1$'s between $i$-th and $j$-th $\ensuremath{\mathbb{P}}^1$ (assume $i<j$), we have another $\ensuremath{\mathbb{P}}^1$ which we denote by
$$
\alpha_{i,j}:=\alpha_i+\ldots+ \alpha_j. \label{alphaij}
$$
This corresponds to a positive root of $\hat{A}_{L-1}$.

Suppose that we have a Calabi-Yau manifold $X$ without compact 4-cycles.
We also consider a single D6-brane filling the entire $X$ and D0/D2-branes
wrapping compact holomorphic 0/2-cycles specified by $n \in H_0(X;\ensuremath{\mathbb{Z}})$ and $\beta\in H_2(X;\ensuremath{\mathbb{Z}})$, respectively. We can then define the
BPS degeneracy $\Omega(n,\beta)$ counting BPS degeneracy of D-branes\footnote{More precisely, this BPS degeneracy is defined by the second helicity supertrace.}.
The closed BPS partition function is then defined by
\begin{equation}
\ensuremath{\mathcal{Z}}^c_{\textrm{BPS}}(q,Q)=\sum_{n,\beta}\Omega(n,\beta) q^n Q^{\beta}.
\end{equation}

The closed BPS partition function for generalized conifolds is studied in \cite{Nagao1,AOVY}.
To describe the results, let us first specify the resolution (crepant resolution\footnote{Crepant resolution is a resolution $f: Y\to X$ such that $\omega_Y=f^* \omega_X$, where $\omega_X$ and $\omega_Y$ are canonical bundles of $X$ and $Y$.}) of $X$\footnote{This is not essential, since by varying the value of the K\"ahler moduli we can go to the geometry with other choices of resolutions. We just need to specify an arbitrary resolution in order to begin the discussion. See Appendix \ref{app.Weyl} for more about this.}.
Each of the $L-1$ $\ensuremath{\mathbb{P}}^1$'s are either $\ensuremath{\mathcal{O}}(-1,-1)$-curve
or $\ensuremath{\mathcal{O}}(-2,0)$-curve.
In the language of the toric diagram, this is to specify the triangulation of the toric diagram. We specify this choice by a map
\begin{equation}
\sigma: \{ 1/2,3/2,\ldots, L-1/2 \} \to \{\pm 1 \}.
\end{equation}
In the following we sometimes write $\pm $ instead of $\pm 1$.
When $\sigma(i-1/2)=1$ ($\sigma(i-1/2)=-1$), the $i$-th triangle from the left has
one of its edges on the top (bottom) edge of the trapezoid.
This means that the $i$-th $\ensuremath{\mathbb{P}}^1$ is a $\ensuremath{\mathcal{O}}(-1,-1)$-curve ($\ensuremath{\mathcal{O}}(-2,0)$-curve) when $\sigma(i-1/2) = -\sigma(i+1/2)$ ($\sigma(i-1/2) = \sigma(i+1/2)$).
By definition, we have $\big| \sigma^{-1}(\pm 1)\big|=L_{\pm}$.

For example, in the case of Suspended Pinched Point ($L_+=1, L_-=2$) whose toric diagram is shown in Figure \ref{fig.sigmaex}, $L=3$ and there are 3 difference choice of resolutions. This is represented by
\begin{align}
\sigma_1:\{ 1/2,3/2,5/2\}\to \{-,-,+ \}, \\
\sigma_2:\{ 1/2,3/2,5/2\}\to \{-,+,- \}, \\
\sigma_3:\{ 1/2,3/2,5/2\}\to \{+,-,- \}.
\label{SPPsigma}
\end{align}

\begin{figure}[htpb]
\centering{\includegraphics[scale=0.4]{SPPresolution.eps}}
\caption{The choice of resolutions of a generalized conifold ($L_+=1,L_-=2$).}\label{fig.sigmaex}
\end{figure}

Given $\sigma$, the topological string partition function is given by \cite{AKMV,IK}
\begin{equation}
\ensuremath{\mathcal{Z}}_{\textrm{top}, \sigma}(q=e^{-g_s},Q=e^{-t})=\prod_{n=1}^{\infty} (1-q^n Q)^{n N_{\beta}^{g=0}},
\end{equation}
where $N_{\beta}^{g=0}$ is the genus 0 Gopakumar-Vafa (GV) invariant\footnote{Higher genus GV invariants vanish for generalized conifolds.}. For the 2-cycle $\beta=\alpha_i+\ldots +\alpha_j$, the explicit form of $N_{\beta}^0$ depends on $\sigma$ and is given by
\begin{align*}
N_{\beta=\alpha_i+\ldots \alpha_j}^{g=0}&=(-1)^{1+\sharp \{k\mid i\le k\le j, \ \sigma(k-1/2)\ne \sigma(k+1/2) \}} \\
&=(-1)^{1+\sharp \{k\mid i\le k\le j, \ \alpha_k \textrm{ is a } \ensuremath{\mathcal{O}}(-1,-1)-\textrm{curve} \}}.
\end{align*}
By CPT invariance in five dimensions \cite{AOVY}, we have
$
N_{\beta=-(\alpha_i+\ldots \alpha_j)}^{g=0}
=N_{\beta=\alpha_i+\ldots \alpha_j}^{g=0}.
$
We also have $N_{\beta=0}^0=\chi(X)/2$, where the Euler character $\chi(X)$ for a toric Calabi-Yau manifold is the same as twice the area of the toric diagram.

As shown in \cite{AOVY,Nagao1}, the closed BPS partition function is given by
\begin{eqnarray}
\ensuremath{\mathcal{Z}}_{\rm BPS}(q,Q)&=&\ensuremath{\mathcal{Z}}_{\rm top}(q,Q) \ensuremath{\mathcal{Z}}_{\rm top}(q,Q^{-1}) \big|_{\rm chamber} \nonumber \\
&=& \prod_{(\beta,n): Z(\beta,n)>0} (1-q^n Q^{\beta})^{n N_{\beta}^0}, \label{Zchamber}
\end{eqnarray}
where the central charge $Z(\beta,n)$ is given by
$$
Z(\beta,n)=(B(\beta)+n)/R. \label{Z}
$$
Here $1/R$ denotes (up to proportionality constants) the central charge of the D0 brane, and following \cite{AOVY} we choose the complexified K\"ahler moduli to be real. Also, the notation $B(\beta)$ means the B-field flux through the cycle $\beta$\footnote{This was written $\beta B$ in \cite{AOVY}.}.

Now suppose that $1/R$ is positive\footnote{Under this condition we are discussing only half of chambers of the K\"ahler moduli space, which lie between the Donaldson-Thomas chamber and the non-commutative Donaldson-Thomas chamber.
 The other half arises when $1/R$ is negative.}.
From \eqref{Zchamber} and \eqref{Z}, it follows that the wall crossing occurs when the integer part of the value of the B-field through the cycle change. For the cycle $\alpha_{i}+\ldots +\alpha_j$, this is given by
$$
\left[ B(\alpha_i)+\ldots + B(\alpha_j) \right], \label{Balphaij}
$$
Since there are \begin{NB}corrected\end{NB}%
$L-1$ $\ensuremath{\mathbb{P}}^1$'s in $X$, there are \begin{NB}corrected\end{NB}%
$L(L-1)/2$ such parameters.

We can take a special limit $B(\alpha_i)\to \infty$. Let us denote this special chamber by $\tilde{C}_{\rm top}$. As discussed in \cite{AOVY,Nagao1}, in this limit the BPS partition function reduces to the closed topological string partition function:
$$
\ensuremath{\mathcal{Z}}^c_{(\sigma,\theta)}\Big|_{\tilde{C}_{\rm top}} =\ensuremath{\mathcal{Z}}^c_{\rm top},
$$
just as advertised in \eqref{ZtoZtop}.

\medskip
For concreteness, let us discuss an example.
We use the example of the Suspended Pinched Point ($N=3$) using the triangulation $\sigma_1$ in \eqref{SPPsigma}. In this example, the topological string partition function is
$$
\ensuremath{\mathcal{Z}}_{\textrm{top}, \sigma=\sigma_1}(q,Q)=M(q)^{3/2} \prod_{n=1}^{\infty} (1-q^n Q_1)^{-n}
\prod_{n=1}^{\infty} (1-q^n Q_2)^{n} \prod_{n=1}^{\infty} (1-q^n Q_1 Q_2)^{n},
$$
where $M(q)$ is the MacMahon function
$$
M(q)=\prod_{n=1}^{\infty} (1-q^n)^{-n}.
$$
The BPS partition function is given by
\begin{align*}
\ensuremath{\mathcal{Z}}_{\rm BPS}&(q,Q)=M(q)^{3}
\prod_{n=1}^{\infty} (1-q^n Q_1)^{-n}
\prod_{n=1}^{\infty} (1-q^n Q_2)^{n} \prod_{n=1}^{\infty} (1-q^n Q_1 Q_2)^{n} \\%
&\times
\prod_{n>\left[ B(\alpha_1) \right]}^{\infty} (1-q^n Q_1^{-1})^{-n}
\prod_{n>\left[ B(\alpha_2) \right]}^{\infty} (1-q^n Q_2^{-1})^{n} \prod_{n>\left[ B(\alpha_1+\alpha_2) \right]}^{\infty} (1-q^n (Q_1 Q_2)^{-1})^{n}.
\end{align*}

\medskip

The parameters $\left[ B(\alpha_{i}+\ldots +\alpha_j) \right]$ specify the chamber, but as we can see from the definition they are not completely independent parameters. Since we only have \begin{NB}corrected\end{NB}%
$L-1$ real parameters $B_i$, it is likely that this parametrization is redundant. Indeed, as explained in the Appendix \ref{app.Weyl}
we can specify the chamber by a map $\theta$, which is specified by \begin{NB}corrected\end{NB}%
$L$ half-integers, \begin{NB}corrected\end{NB}%
$\theta(1/2),\theta(3/2),\ldots, \theta(L-1/2)$, satisfying one constraint
$$
\sum_{i=1}^L \theta\left(i-\frac{1}{2}\right)=\sum_{i=1}^L \left(i-\frac{1}{2}\right).
$$
This means we can indeed parametrize the chamber by \begin{NB}corrected\end{NB}%
$L-1$ independent (half-)integers, which is what we expected. As discussed in Appendix \ref{app.Weyl}, $\theta$ is an element of the Weyl group of \begin{NB}corrected\end{NB}%
$\hat{A}_{L-1}$.

\section{The Noncommutative Topological Vertex}\label{sec.def}

In this section we give a general definition of the non-commutative topological vertex using the crystal melting model. This definition is equivalent to the one given in \cite{Nagao2} using the dimer model\footnote{See Appendix A of \cite{OY1} for the equivalence between crystal melting model and the dimer model.}. \begin{NB}added\end{NB}%
See \cite{Nagao3} for more conceptual definition in terms of Bridgeland's stability conditions and moduli spaces.

To define our vertex, we need the following set of data:

\begin{itemize}
\item A map
$$
\sigma: \{ 1/2,\ldots, L-1/2\}\to \{\pm \}.
$$
As already explained in section \ref{sec.closed}, this gives a triangulation of the toric diagram, or equivalently the choice of the resolution of the Calabi-Yau manifold.

\item A map $\theta:\ensuremath{\mathbb{Z}}_h \to \ensuremath{\mathbb{Z}}_h$. As explained in Appendix \ref{app.Weyl} in the case of closed BPS invariants, $\theta$ and $\sigma$ specify the chamber structure of the open BPS invariants.

\item A set of Young diagrams $\lambda$, assigned to external legs of the $(p,q)$-web. This specifies the boundary condition of the non-compact D-branes ending on the $(p,q)$-web. We denote by \begin{NB}corrected\end{NB}%
$\lambda_1,\ldots,\lambda_{L}$ the Young diagrams for the top and the bottom edges of the trapezoid, and by $\lambda_+,\lambda_-$ the remaining two. We sometimes write $\lambda=(\mu,\nu)$, where \begin{NB}corrected\end{NB}%
$\mu=(\mu_1,\ldots, \mu_L)=(\lambda_1,\ldots,\lambda_L)$ and $\nu=(\lambda_+,\lambda_-)$.
In the example shown in Figure \ref{lambdaconv}, there are five external legs and we have five representations.

\begin{figure}[htbp]
\centering{\includegraphics[scale=0.4]{lambdaconv.eps}}
\caption{representations assigned to external legs of the $(p,q)$-web. The dotted lines represent the $(p,q)$-web.}
\label{lambdaconv}
\end{figure}

For later purposes, we combine $\mu_1,\ldots, \mu_L$ into a single representation $\mu$ by
\begin{equation}
\mu(i-1/2+kL)=\mu_i(k-1/2). \label{combinedmu}
\end{equation}
In other words, we choose $\mu$ such that L-quotients of $\mu$ give $\mu_1, \ldots, \mu_L$. By abuse of notation, we use the same symbol $\mu$ for a set of representations $\mu_1,\ldots, \mu_L$ as well as a single representation define above.
\end{itemize}

Given $\sigma, \theta$ and $\lambda$, we define the non-commutative topological vertex
$$
\ensuremath{\mathcal{C}}_{\textrm{BPS},(\sigma,\theta;\lambda)}(q,Q).
$$
In the following we drop the subscript BPS for simplicity.

Before going into the general definition, we first illustrate our idea using simple example of the resolved conifold.

\subsection{Example: Resolved Conifold}\label{subsec.conifoldeg}

In this example there is only one $\ensuremath{\mathbb{P}}^1$ and the BPS partition function depends on a single positive integer $N:=\left[ B(\alpha_1) \right]$. In the language of $\theta$,\begin{NB}I change the notation: $\theta\to\theta^{-1}$\end{NB}%
$$
\theta(1/2)=1/2-N, \quad \theta(3/2)=3/2+N. \label{thetaN}
$$
We fix $\sigma$ to be
$$
\sigma(1/2)=+, \quad \sigma(3/2)=-.
$$
Without losing generality we concentrate on $N\ge 0$, since $N<0$ corresponds to a flopped geometry, where $\sigma$ is replaced by $-\sigma$ (see Appendix \ref{app.Weyl}).

The ground state crystal for $N=2$ is shown in Figure \ref{conifoldcrystal}.
This crystal, sometimes called a pyramid, consists of infinite layers of atoms, the color alternating between black and white \begin{NB}added\end{NB}%
(\cite{Szendroi, Young1}).
In the $N$-th chamber there are $N+1$ atoms on the top.

\begin{figure}[htbp]
\centering{\includegraphics[scale=0.25]{conifoldn3crystal.eps}}
\caption{The ground state crystal for the resolved conifold for $N=2$. The crystal consists of an infinite number of layers, and only a finite number is shown here. The ridges of the pyramid are represented by four lines extending to infinity.}
\label{conifoldcrystal}
\end{figure}

The closed BPS partition function is defined by removing a finite set of atoms $\Omega$ from the crystal. When we do this, we follow the melting rule \cite{OY1,MR} such that
whenever an atom is removed from the crystal, we remove all the atom above it.
In other words, since an atom is in one-to-one correspondence with an F-term equivalent class of paths starting from a fixed node of the quiver diagram \cite{OY1},
for an arrow $a$ and and an atom $\alpha$ we can define $a\alpha$. The melting rule then says
\begin{equation}
{\rm If } \quad a\alpha\in \Omega,\quad {\rm then} \quad \alpha\in \Omega.
\label{meltingrule}
\end{equation}

We then define the partition function by summing over such $\Omega$:
\begin{equation}
\ensuremath{\mathcal{Z}}=\sum_{\Omega} (q_0^{(N)})^{w_0(\Omega)} (q_1^{(N)})^{w_1(\Omega)},
\label{Zdefconifold}
\end{equation}
where $w_0(\Omega)$ and $w_1(\Omega)$ are the number of white and black atoms in $\Omega$, respectively.

The weights $q_0^{(N)}$ ($q_1^{(N)}$) assigned to white (black) atoms in the $N$-th chamber are determined as follows. We can slice the crystal by the plane,
and each slice is specified by an integer $i$ (see Figure \ref{crystalslice}). We choose $i$ so that atoms on the top of the crystal is located at $i=0$.

\begin{figure}[htbp]
\centering{\includegraphics[scale=0.25]{conifoldcrystalslice.eps}}
\caption{We can slice of the conifold crystal by an infinite number of parallel planes.}
\label{crystalslice}
\end{figure}

The weight $q_i^{(N)}$ depends on the chamber and is given by\begin{NB}corrected\end{NB}%
\begin{equation}
 q_0^{(N)}=q_0^{-N}q_1^{-N+1},
\quad
 q_1^{(N)}=q_0^{N+1}q_1^{N},
\label{conifoldwt1}
\end{equation}
when $N$ is odd, and
\begin{equation}
 q_0^{(N)}=q_0^{N+1}q_1^{N},
\quad
 q_1^{(N)}=q_0^{-N}q_1^{-N+1},
\label{conifoldwt2}
\end{equation}
when $N$ is even.
For example, $q_0^{(0)}=q_0, q_1^{(0)}=q_1$ when $N=0$, and \begin{NB}corrected\end{NB}%
$q_0^{(1)}=q_0^{-1} , q_1^{(1)}=q_0^2q_1$.
The change of variables arises from the Seiberg duality on the quiver quantum mechanics \cite{CJ}, geometrically mutations in the derived category of coherent sheaves \cite{CJ,NN}, or in more combinatorial language the dimer shuffling \cite{Young1}. The parameters $q_0, q_1$ defined here are related to the D0/D2 chemical potentials introduced in section \ref{sec.closed} by\footnote{
The equation \eqref{conifoldwt2} is the same for $N$ odd and even if we suitably exchange the two nodes of the quiver diagram.
The relation \eqref{chemicalpot} can also be written as
$$
q=q_0^{(N)} q_1^{(N)}, \quad Q=(q_0^{(N)})^{N} (q_1^{(N)})^{N+1},
$$
when $N$ is even, and $q_0^{(N)}$ and $q_1^{(N)}$ exchanged when $N$ odd. This coincides with the expression in \cite{CJ}.
}
\begin{equation}
q=q_0 q_1, \quad Q_1=q_1.
\label{chemicalpot}
\end{equation}

Now let us discuss the open case. When non-trivial representations are assigned to each of the four external legs of the $(p,q)$-web, the only thing we need to do is to change the ground state of the crystal.

The crystal has four ridges, corresponding to four external legs of the $(p,q)$-web. When we assign a representation, we remove the atoms with the shape of the Young diagram. More precisely, we remove the atoms with the shape of the Young diagram in the asymptotic direction of the $(p,q)$-web, as well as all the atoms above them, so that the melting rule is satisfied. See Figure \ref{openconifoldeg} for an example.

\begin{figure}[htbp]
\centering{\includegraphics[scale=0.25]{conifoldn3opencrystal.eps}}
\caption{The pyramid for open BPS invariants. A non-trivial representation
$(1,1)$ is placed on with one of the four external lines. As compared with the previous figure, atoms colored gray, corresponding to the Young diagram, are removed from the crystal. The red atoms have no atoms above them.}
\label{openconifoldeg}
\end{figure}

The partition function is defined in exactly the same way by \eqref{Zdefconifold}, and the result is denoted by $\ensuremath{\mathcal{C}}_{(\sigma, \theta;\lambda)}$.

\bigskip

Several comments are now in order.

\begin{NB}Added\end{NB}%
First, let us explain the origin of the name ``the non-commutative topological {\it vertex}''.
Recall that, in commutative case, topological vertex is defined for $\ensuremath{\mathbb{C}}^3$.
For a general affine toric Calabi-Yau manifold $X$, we divide the polygon into triangles and assign a topological vertex to each trivalent vertex of the dual graph.
We can get the topological string partition function for the smooth toric Calabi-Yau manifold $Y$ by gluing them with propagators.
Similarly, \begin{NB2} added \end{NB2} assume that a polygon is divided into trapezoids.
Then we can assign a non-commutative topological vertex to each vertex of the dual graph and glue them by propagators.
The BPS partition function defined in this way \begin{NB2} changed \end{NB2} is related to the
topological string partition function
via wall-crossing\footnote
Given a devision of a polygon into trapezoids, we get a partial resolution of $X$ and a non-commutative algebra $A$ over the partial resolution, which is derived equivalent to $Y$. The BPS partition function
given by gluing non-commutative topological vertices counts torus invariants $A$-modules.}.
In \cite{BCY2,BCY1} they study the case when a polygon is divided into (not necessary minimal) triangles.

\begin{NB}Removed: First, let us emphasize the difference between ordinary (`commutative') topological vertex and our non-commutative topological vertex.
In commutative case, topological vertex is defined only for $\ensuremath{\mathbb{C}}^3$. For a more general toric Calabi-Yau manifold, we cut the $(p,q)$-web into pieces,
assign a topological vertex to each trivalent vertex and
we glue them by propagators. For example, for the resolved conifold we need to glue two vertices.
However, our NCTV is defined for any toric Calabi-Yau manifold, and thus we do not need to glue them. In this sense, our vertex is not a `vertex'.
\end{NB}

Second, it is possible to give more geometric definition of the vertex \begin{NB}Added\end{NB}%
(see \cite{Nagao3}).
For the closed BPS invariants,
the crystal arises as a torus fixed point of the moduli space of the modules of the path algebra quiver (under suitable $\theta$-stability conditions). The moduli space is the vacuum moduli space of the quiver quantum mechanics arising as the low-energy effective theory of D-branes \cite{OY1}. The similar story exists in our case. Namely, the crystal is in one to one correspondence with the fixed point of the moduli space arising from a quiver diagram. For example, for conifold with $\lambda=${\tiny \yng(1,1)}, the quiver is given in Figure \ref{openquiver}.

\begin{figure}[htbp]
\centering{\includegraphics[scale=0.5]{openquiver.eps}}
\caption{Quiver diagram for the open invariant with
$\lambda=(1,1)$. This is the Klebanov-Witten quiver \cite{KW} with an extra node and extra three arrows starting from it. The three arrows correspond to three red atoms in Figure \ref{openconifoldeg}.}
\label{openquiver}
\end{figure}

Third, in the case of $\ensuremath{\mathbb{C}}^3$, our vertex reproduces the topological vertex of $\ensuremath{\mathbb{C}}^3$
by definition.

\subsection{General Definition from Crystal Melting}

We next give a general definition of the vertex. Readers not interested in the details of the definition of the non-commutative topological vertex
can skip this section on first reading.

Given a boundary condition specified by $\sigma, \theta$ and $\lambda=(\mu,\nu)$, we would like to construct a ground state of the crystal, and
determine the weights assigned to the atoms of the crystal.

The basic idea is the same as in the conifold example. First, the closed string BPS partition function is equivalent to the statistical partition function of crystal melting. The ground state crystal can be sliced by an infinite number of parallel planes parametrized by integers $n\in \ensuremath{\mathbb{Z}}$, just as in Figure \ref{crystalslice}. On each slice, there are infinitely many atoms, labeled by integers $(x,y)\in \ensuremath{\mathbb{Z}}_{\ge 0}^2$. Therefore, the atoms in the crystal are label by $(n,x,y)\in \ensuremath{\mathbb{Z}}\times \ensuremath{\mathbb{Z}}_{\ge 0}^2$.

\bigskip

Let us show this in the example of the Suspended Pinched Point. The crystal in
Figure \ref{SPPcrystal} clearly shows this structure.

\begin{figure}[htbp]
\centering{\includegraphics[scale=0.2]{SPPcrystalslice.eps}}
\caption{The crystal for the Suspended Pinched Point. We can slice the crystal along planes represented by lines, which come with three different colors.}
\label{SPPcrystal}
\end{figure}

Another way of explaining this is to
construct a crystal starting from a bipartite graph on $\ensuremath{\mathbb{R}}^2$, shown in Figure \ref{SPPbipartite}\footnote{This is a universal cover of the bipartite graph on $\ensuremath{\mathbb{T}}^2$, which appears in the study of four-dimensional $\ensuremath{\mathcal{N}}=1$ quiver gauge theories. See \cite{BT1,BT2,BT3} for original references, and \cite{Kennaway,Yamazaki} for reviews.}.
In this example, the bipartite graph consists of hexagons and squares, and
periodically changes its shape along the horizontal directions.

\begin{figure}[htbp]
\centering{\includegraphics[scale=0.25]{SPPbipartitecover.eps}}
\caption{The bipartite graph for Suspended Pinched Point. We here take $\sigma=\sigma_1$ and $\theta=id$. The red undotted (the blue dotted) lines have half-integer (integer) values of the coordinate along the horizontal axis.}
\label{SPPbipartite}
\end{figure}

Now the atoms of the crystal are located at the centers of the faces of the bipartite graph. and it thus follows we can slice the crystal along the horizontal axis. Each slice consist of an infinite number of atoms labeled by two integers $(x,y)\in (\ensuremath{\mathbb{Z}}_{\ge 0})^2$, since there are two directions, the horizontal direction and the perpendicular direction to the paper\footnote{In general the bipartite graph is determined by $\sigma$.
A hexagon (a square) corresponds to $\ensuremath{\mathcal{O}}(-2,0)$-curve ($\ensuremath{\mathcal{O}}(-1,-1)$-curve).
In other words, the $i$ th polygon is a hexagon (square) if
$\sigma(i)=\sigma(i+1) (\sigma(i)\ne \sigma(i+1))$.}.

\bigskip

Now consider the open case. In this case, we construct a new ground state by removing atoms from the closed ground state. By the melting rule, the atoms
removed from the $n$-th plane should be labeled by $(x,y)\in \ensuremath{\mathcal{V}}(n)$, where
$\ensuremath{\mathcal{V}}(n)$ is a Young diagram. Depending on the representations on external legs, $\ensuremath{\mathcal{V}}(n)$ increases or decreases as we change $n$. Thus the ground state crystal for open BPS invariants are determined by such a sequence of Young diagrams $\{\ensuremath{\mathcal{V}}(n)\}$, called transitions below. In the following we make this idea more rigorous.

Let us begin with some notations.
Let $\mu$ and $\mu'$ be two Young diagrams.
We say $\mu\overset{+}{\succ}\mu'$ if the row lengths satisfy
\[
\mu_1\geq\mu'_1\geq\mu_2\geq\mu'_2\geq\cdots,
\]
and $\mu\overset{-}{\succ}\mu'$ if the column lengths satisfy
\[
{}^{\mathrm{t}}\mu_1\geq{}^{\mathrm{t}}\mu'_1\geq{}^{\mathrm{t}}\mu_2\geq{}^{\mathrm{t}}\mu'_2\geq\cdots.
\]
We define a {\it transition} $\mca{V}$ of Young diagrams of type $(\sigma,\theta\hspace{0.5pt};\hspace{0.5pt} \mu,\nu)$ as a map
from the set of integers $\ensuremath{\mathbb{Z}}$ to
the set of Young diagrams such that
\begin{itemize}
\item $\mca{V}(n)=\nu_-$ for $n\ll0$ and $\mca{V}(n)=\nu_+$ for $n\gg0$,
\item $\mca{V}(h-\mu\circ\theta(h)/2)\overset{\sigma\circ\theta(h)}{\succ}\mca{V}(h+\mu\circ\theta(h)/2)$.
\end{itemize}
Then as shown in \cite{Nagao2} there is a transition $\V_{\mathrm{min}}$ of Young diagrams of type $(\sigma,\theta\hspace{0.5pt};\hspace{0.5pt} \mu,\nu)$ such that for any transition $\mca{V}$ of Young diagrams of type $(\sigma,\theta\hspace{0.5pt};\hspace{0.5pt} \mu,\nu)$  and for any $n\in\bb{Z}$ we have $\mca{V}(n)\supseteq \V_{\mathrm{min}}(n)$.
The transition $\V_{\mathrm{min}}(n)$ is the sequence of transitions discussed above.

For a transition $\mca{V}$ of Young diagram of type $(\sigma,\theta\hspace{0.5pt};\hspace{0.5pt} \mu,\nu)$, the ground state crystal can be defined by
\[
A(\mca{V}):=\{a(n,x,y)\mid n\in \ensuremath{\mathbb{Z}}, x,y \in \ensuremath{\mathbb{Z}}_{\ge 0}, \ (x,y)\notin \mca{V}(n)\}.
\]
where $a(n,x,y)$ denotes the atom at position $(n,x,y)$.

Having defined the ground state crystal, the partition functions is defined again as the sum over a subset $\Omega$ of $A(\V_{\mathrm{min}})$ satisfying the following two conditions:\footnote{It is straightforward to show that a subset $\Omega \subset A(\V_{\mathrm{min}})$ satisfies the two conditions if and only if $P(\V_{\mathrm{min}})\backslash \Omega=A(\mca{V})$ for a transition $\mca{V}$ of Young diagram of type $(\sigma,\theta\hspace{0.5pt};\hspace{0.5pt} \mu,\nu)$.}

\begin{itemize}
\item $\Omega$ is finite set, and
\item $\Omega$ satisfies the melting rule \eqref{meltingrule}. In other words, if
$a'\in \Omega$ and $a'=a \alpha$ for an arrow $\alpha$, then $a\in \Omega$.\footnote{
We can also explicitly write down the melting rule using the coordinates $(n,x,y)$.
Let us write $a \sqsupset a'$ when there is a path (a composition of arrows) $\alpha$ such that $a'=a \alpha$.
The partial order $\sqsupset$ is then generated by
$$
\begin{array}{lr}
a(h-1/2,x,y)\sqsupset a(h+1/2,x,y) & \lambda\circ \theta(h)=+,\\
a(h+1/2,x,y)\sqsupset a(h-1/2,x,y) & \lambda\circ \theta(h)=-,\\
a(h+1/2,x,y)\sqsupset a(h-1/2,x+1,y) & \lambda\circ \theta(h)=+, \sigma\circ\theta(h)=-,\\
a(h+1/2,x,y)\sqsupset a(h-1/2,x,y+1) & \lambda\circ \theta(h)=+, \sigma\circ\theta(h)=+,\\
a(h-1/2,x,y)\sqsupset a(h+1/2,x+1,y) & \lambda\circ \theta(h)=-, \sigma\circ\theta(h)=-,\\
a(h-1/2,x,y)\sqsupset a(h+1/2,x,y+1) & \lambda\circ \theta(h)=-, \sigma\circ\theta(h)=+,\\
a(n,x,y)\sqsupset a(n,x+1,y) & \sigma\circ\theta(n-1/2)=\sigma\circ\theta(n+1/2)=+,\\
a(n,x,y)\sqsupset a(n,x,y+1) & \sigma\circ\theta(n-1/2)=\sigma\circ\theta(n+1/2)=-.
\end{array}
$$
}
\end{itemize}

For a crystal $\Omega \in A(\V_{\mathrm{min}})$, we define the weight $w(\Omega)_i$ by the number of atoms with the color $i$ contained in $\Omega$:
\[
w(\Omega)_i:=\sharp\{a(n,x,y)\in \Omega\mid n\equiv i\ (\mr{mod}\,L)\}.
\]
Also, for $\theta$, we put
\[
q^\theta_i:=
\begin{cases}
q_{\theta^{-1}(i-1/2)+1/2}\cdot q_{\theta^{-1}(i-1/2)+3/2}\cdot\cdots\cdot q_{\theta^{-1}(i+1/2)-1/2} &  (\theta^{-1}(i-1/2)<\theta^{-1}(i+1/2)),\\
q_{\theta^{-1}(i-1/2)-1/2}^{-1}\cdot q_{\theta^{-1}(i-1/2)-3/2}^{-1}\cdot\cdots\cdot q_{\theta^{-1}(i+1/2)+1/2}^{-1} & (\theta^{-1}(i-1/2)>\theta^{-1}(i+1/2)),
\end{cases}
\]
where we define $q_i$ for $i\in \ensuremath{\mathbb{Z}}$ periodically,
$$
q_{i+L}=q_i.
$$
We then define the vertex by
\[
\mathcal{C}^{\rm ref}_{(\sigma,\theta\hspace{0.5pt};\hspace{0.5pt} \mu,\nu)}(q_0,\ldots,q_{L-1}):=\sum_{\Omega} (q^{\theta}_0)^{w(\Omega)_0}\cdot\cdots\cdot (q^{\theta}_{L-1})^{w(\Omega)_{L-1}}.
\]
The parameters $q_0, \ldots, q_{L-1}$ defined here are related to the D0/D2 chemical potentials introduced in section \ref{sec.closed} by
\begin{equation}
q=q_0\ldots q_{L-1}, \quad Q_i=q_i\quad (i=1,\ldots, L-1).
\label{chemicalweight}
\end{equation}

\begin{NB2} added the whole section \end{NB2}

\subsection{Refinement}\label{subsec.refined}

We can also generalize the definition to include the open refined BPS invariants\footnote{See \cite{KontsevichS,DG,BBS} for closed refined BPS invariants}.

Let us recall the meaning of the refined BPS counting, first in the closed case.
When the type IIA brane configuration is lifted to M-theory \cite{GV1} and when we use the 4d/5d correspondence \cite{GSY,DVVafa}, the D0/D2-branes are lifted to spinning M2-branes in $\ensuremath{\mathbb{R}}^5$, which has spin under the little group in 5d, namely $SO(4)=SU(2)_L\times SU(2)_R$. The ordinary BPS invariant is defined as an index; it keeps only the $SU(2)_L$ spin, while taking an alternate sum over the $SU(2)_R$ spin.
The refined closed BPS invariants is defined by taking both spins into account.

The situation changes slightly when we consider open refined BPS invariants.
The D4-branes wrapping Lagrangians, when included, are mapped to M5-branes on $\ensuremath{\mathbb{R}}^3$. This means that $SO(4)$ is broken to $SO(2)$, and we have only one spin. However, there is an $SO(2)$ R-symmetry for $\ensuremath{\mathcal{N}}=2$ supersymmetry in three dimensions, and in the definition of the ordinary open BPS invariants we keep only one linear combination of the two, while tracing out the other combination \cite{LMV}. The refined open BPS invariants studied here takes boths of the two charges into account.

In the language of crystal melting used in this paper, the open refined BPS invariants are defined simply by modifying the definition of the weights. Here, we explain how to modify the weights in the case of $\mu=\emptyset$.

For an integer $n$ with
$$n\equiv \pi(n) \quad \textrm{mod}\  L, \quad 0\le \pi(n)\le L-1,$$ define
\[
w(\Omega)_n:=\sharp\{ a(n,x,y)\in \Omega \}.
\]
We also define the weights by
\[
\tilde{q}^\theta_n:=
\begin{cases}
\tilde{q}_{\theta^{-1}(n-1/2)+1/2}\cdot \tilde{q}_{\theta^{-1}(n-1/2)+3/2}\cdot\cdots\cdot \tilde{q}_{\theta^{-1}(n+1/2)-1/2} &  (\theta^{-1}(n-1/2)<\theta^{-1}(n+1/2)),\\
\tilde{q}_{\theta^{-1}(n-1/2)-1/2}^{-1}\cdot \tilde{q}_{\theta^{-1}(n-1/2)-3/2}^{-1}\cdot\cdots\cdot \tilde{q}_{\theta^{-1}(n+1/2)+1/2}^{-1} & (\theta^{-1}(n-1/2)>\theta^{-1}(n+1/2)),
\end{cases}
\]
where
$$
\tilde{q}_n=q_{\pi(n)}
$$
when $n \not\equiv 0 \ \textrm{mod} \ L$,
and
$$
\tilde{q}_n=
\begin{cases}
q_+ & n>0 \\
(q_+ q_-)^{1/2}&  n=0\\
q_- & n<0
\end{cases}
$$
when $n \equiv 0 \ \textrm{mod} \ L$.
We then define the refined vertex by
\[
\mathcal{C}_{(\sigma,\theta\hspace{0.5pt};\hspace{0.5pt} \emptyset,\nu)}(q_+,q_-,q_1\ldots,q_{L-1}):=\sum_{\Omega} \prod_{n\in \ensuremath{\mathbb{Z}}} (\tilde{q}^{\theta}_n)^{w(\Omega)_n}.
\]
By definition, the refined vertex reduces to the unrefined vertex by setting $q_+=q_-=q_0$.
The reader can refer to \cite{Nagao2} for the definition of weights in general cases $\mu\ne \emptyset$.

\subsection{Consistency Checks of Our Proposal}\label{sec.check}

In Appendix \ref{app.proof} we gave a purely combinatorial definition of the non-commutative topological vertex. We now claim that is captures open BPS invariants in the following sense.

Consider a generalized conifold (a toric Calabi-Yau manifold without compact 4-cycles) with representations assigned to each leg of the $(p,q)$-web. Each representation specifies a boundary condition on the non-compact D4-brane wrapping Lagrangian 3-cycle of topology $\ensuremath{\mathbb{R}}^2\times S^1$ \cite{AV}.

In the absence of D4-branes, we are counting particles of D0/D2-branes wrapping 0/2-cycles, which makes a bound state with a single D6-brane filling the entire Calabi-Yau manifold. When the D4-branes are included, D2-branes can wrap disks ending on the worldvolume of D4-branes. The degeneracy of such D-brane configurations is what we mean by the open BPS degeneracies. Note that supersymmetry is broken by half due to the inclusion of D4-branes; our counting of BPS particles makes sense because we are counting BPS states in lower dimensions, where the minimal amount of supersymmetry is lower.

We can provide several consistency checks of our proposal.
First, our vertex by definition reduces to the closed BPS invariant when all the representations $\lambda$ are trivial:
$$
\ensuremath{\mathcal{C}}_{\textrm{BPS}, (\sigma,\theta;\lambda=\emptyset)}=\ensuremath{\mathcal{Z}}^c_{\textrm{BPS},(\sigma,\theta)}.
$$
The second consistency check comes from the wall crossing phenomena. As shown in \cite{Nagao2}, the vertex goes through a series of wall crossings as we move around the closed string moduli space (K\"ahler moduli of the Calabi-Yau manifold), just as in the case of closed invariants.
It was also shown in \cite{Nagao2} that in the chamber $C_{\rm top}$ where the closed BPS partition function reduces to the closed topological string partition function, our vertex gives the same answer as that computed from the topological vertex (in the standard framing):
$$
\ensuremath{\mathcal{C}}_{\textrm{BPS}, (\sigma,\theta;\lambda)} \Big|_{C_{\rm top}}=\ensuremath{\mathcal{C}}_{\textrm{topological vertex}, \lambda}.
$$
The third consistency check comes from the fact that the wall crossing factor is independent of representations. In other words,
\[
\bar{\mathcal{C}}_{(\sigma,\theta\hspace{0.5pt};\hspace{0.5pt} \mu,\nu)}(q_0,\ldots,q_{L-1}):=
\frac{\mathcal{C}_{(\sigma,\theta\hspace{0.5pt};\hspace{0.5pt} \mu,\nu)}(q_0^\theta,\ldots,q_{L-1}^\theta)}{\mathcal{C}_{(\sigma,\theta\hspace{0.5pt};\hspace{0.5pt}\emptyset,\emptyset)}(q_0^\theta,\ldots,q_{L-1}^\theta)}.
\]
does not depend on $\theta$ \cite{Nagao2,Nagao3}\footnote{\label{normalization}To be exact, we have to normalize the generating function by a monomial. See \cite[Corollary 3.21]{Nagao3} for the precise statement.}. This means that the open BPS partition function, which is defined by the sum over representations, takes a factorized form
\begin{equation}
\ensuremath{\mathcal{Z}}^{o}_{\rm BPS} =\frac{\ensuremath{\mathcal{Z}}^{o}_{\rm BPS}}{\ensuremath{\mathcal{Z}}^{o}_{\rm top}} \ensuremath{\mathcal{Z}}^o_{\rm top}
= \frac{\ensuremath{\mathcal{Z}}^{c}_{\rm BPS}}{\ensuremath{\mathcal{Z}}^{c}_{\rm top}} \ensuremath{\mathcal{Z}}^o_{\rm top}.
\label{Zoproduct}
\end{equation}
Since $\ensuremath{\mathcal{Z}}^{c}_{\rm BPS}/\ensuremath{\mathcal{Z}}^{c}_{\rm top}$ takes an infinite product form as explained in section \ref{sec.closed} and $\ensuremath{\mathcal{Z}}^o_{\rm top}$ also takes the infinite product form \cite{OVknot}, $\ensuremath{\mathcal{Z}}^{o+c}_{\rm BPS}$ itself should take an infinite product form,
which is consistent with a suitable generalization of the argument of \cite{AOVY}.

Using \eqref{Zoproduct}, we can compute our vertex using the ordinary topological vertex formalism.
In the next section, we give a yet another way of computing the non-commutative topological vertex. The advantage of our approach is that the final expression manifestly takes a simple infinite product form, and we do not have to worry about the summation of Schur functions.

\section{The Closed Expression for the Vertex}

In this section we give a closed expression for our non-commutative topological vertex. We do this by proving a curious identity stating the equivalence of our vertex for a toric Calabi-Yau manifold $X$ with a closed BPS partition function for an orbifold of $X$\footnote{This is reminiscent of story of \cite{GO1}, where the `bubbling geometry' $X'$ is constructed for given a toric Calabi-Yau manifold $X$ such that the open+closed topological string partition function on $X$ is equivalent to closed topological string partition function on $X'$. However, our story is different in that the vertex computes only a part of the full open BPS partition function; the partition function itself is given by summation of our vertices over representations.}. For another method using vertex operators, see \cite{Nagao3,Sulkowski}.

Start from a non-commutative topological vertex for a generalized conifold $X$, which has $L-1$ compact $\ensuremath{\mathbb{P}}^1$'s. As we discussed above, for the definition of the vertex we need
(1) $\sigma$ for a choice of the crepant resolution of $X$,
(2) a map $\theta$ specifying the chamber together with $\sigma$
(3) a set of representations $\lambda=(\mu,\nu)$,
and the resulting vertex is denoted by $\ensuremath{\mathcal{C}}_{\sigma,\theta;\lambda}(q,Q)$. In the following we consider the special case $\nu=(\lambda_+,\lambda_-)=(\emptyset, \emptyset)$.

Choose an $M$ and consider the $\ensuremath{\mathbb{Z}}_M$ orbifold $X'$ of $X$.
We choose the orbifold action such that when the toric diagram of $X$ is a trapezoid with a top and the bottom edge of length $L_+$ and $L_-$ respectively, $X'$ has length $M L_+$ and $M L_-$.
We also choose map
$$
\sigma': \{1/2,3/2,\ldots, ML-1/2\}\to \{\pm 1\}.
$$
and
$$
\theta': \ensuremath{\mathbb{Z}}_h\to \ensuremath{\mathbb{Z}}_h, \quad \theta'(h+ML)=\theta'(h)+ML
$$
such that
\begin{equation}
\sigma\circ \theta=\sigma' \circ\theta', \quad \mu\circ \theta=\emptyset\circ \theta'.
\label{assumptionmain}
\end{equation}

Then
\begin{equation}
\mathcal{C}_{(\sigma,\theta\hspace{0.5pt};\hspace{0.5pt} \emptyset,\mu)}(q,Q)=
\mathcal{C}_{(\sigma',\theta'\hspace{0.5pt};\hspace{0.5pt} \emptyset, \emptyset)}(q',Q')|_{q^{\theta}_i=q'{}^{\theta'}_i=q'{}^{\theta'}_{i+L}=\cdots q'{}^{\theta'}_{i+(M-1)L}},
\label{main2}
\end{equation}
where $i=0,\ldots, L-1$.
\begin{NB}Modified: See Appendix \ref{app.proof} for the proof of this statement.
The Appendix also discusses \end{NB}%
See Appendix \ref{app.proof} for an explicit method for choosing such $M,\sigma', \theta'$ satisfying \eqref{assumptionmain} as well as generalization of \eqref{main2} to the case of refined BPS invariants.

Since the infinite-product expression for closed BPS partition function for a generalized conifold is already known (section \ref{sec.closed}), we have a closed expression of our vertex when $\nu=\emptyset$.

\section{Examples}
Let us illustrate the above procedure by several examples.

\subsection{$\mathbb{C}^3$}
First, we begin with the non-commutative topological vertex for $\mathbb{C}^3$. Since there is no wall crossing phenomena involved in this case\footnote{There is one wall between $R>0$ and $R<0$, however we do not discuss such a wall since we are specializing to the case $R>0$.}, the vertex should coincide with the ordinary topological vertex, thus providing a useful consistency check of our proposal.
For $\ensuremath{\mathbb{C}}^3$, we have $L=1, \sigma(1/2)=-1, \theta=id$.

Take $\mathbb{C}^3$ with representation $\lambda=(\mu,\nu=\emptyset)$ with $\mu=(N,N-1,\ldots, 1)$ at one leg.
The above-mentioned procedure gives
$M=2$, and thus $X=\ensuremath{\mathbb{C}}^2/\ensuremath{\mathbb{Z}}_2\times \ensuremath{\mathbb{C}}$.
The method in Appendix \ref{app.proof} gives
$$
\theta'(1/2)= 1/2-N,\quad  \theta'(3/2)=3/2+N, \quad  \sigma'=-1,
$$
and thus
$$
\left[ B(\alpha_1) \right]=N.
$$
The weight is given by \eqref{conifoldwt1} or \eqref{conifoldwt2}.
By solving for
$$
q'{}_0^{\theta'}=q'{}_1^{\theta'}=q,
$$
we have
$$
q'{}_0=q^{-2N+1}, \quad q'{}_1=q^{2N+1}
$$
in the case of $N$ odd.
\begin{NB}Modified
(for $N$ odd; the case of $N$ even is similar)
$$
q'{}_0^{\theta'}=q'{}_0^{-k} q'{}_1^{-k+1}, \quad
q'{}_1^{\theta'}=q'{}_0^{k+1} q'{}_1^k
$$
and by solving for
$$
q'{}_0^{\theta'}=q'{}_1^{\theta'}=q
$$
we have
$$
q'{}_0=q^{-2k+1}, \quad q'{}_1=q^{2k+1}
$$
\end{NB}%
Substituting this into the closed BPS partition function
$$
\prod_{n>0}(1-q'{}_0^n q'{}_1^n)^{-2n}
\prod_{n>0}(1-q'{}_0^n q'{}_1^{n+1})^{-n} \prod_{n>N}(1-q'{}_0^n q'{}_1^{n-1})^{-n},
$$
we have
\begin{NB}corrected\end{NB}%
\begin{equation}
M(q)/ \prod_{n=1}^N (1-q^{2n-1})^{(N+1)-n}.
\label{egresult1}
\end{equation}
This coincides with the know expression for the topological vertex \cite{AKMV}\footnote{In the normalization of \cite{AKMV}, \eqref{egresult1} coincides with
$M(q)\, q^{-\| \mu^{{}^{\mathrm{t}}} \| /2 }\, C_{\mu,\emptyset,\emptyset}$}. The case of $N$ even in similar. \begin{NB2}added\end{NB2}

\subsection{Resolved Conifold}
Now let us discuss the next simplest example, the resolved conifold.

\begin{NB}
In this case, $L=2$ and we choose a triangulation given by $\sigma(1/2)=+, \sigma(3/2)=-$. The $N$-th chamber is given by
$\theta_N(1/2)=1/2-N, \theta(3/2)=2/3+N$.
\end{NB}%

Consider the representation $\mu=({\rm \tiny \yng(2,1)},{\rm \tiny \yng(1)})$, with \begin{NB}changed $\theta=\theta_0=id$\end{NB}%
$\theta=id, \sigma(1/2)=+, \sigma(3/2)=-$. In this case, the method in Appendix \ref{app.proof} gives \begin{NB}Modified\end{NB}%
$$
\theta'(1/2)=-7/2,\quad \theta'(3/2)=-1/2,\quad \theta'(5/2)=11/2,\quad
\theta'(7/2)=13/2
$$
with
$$
\sigma'(1/2)=+,\quad \sigma'(3/2)=-,\quad \sigma'(5/2)=-,\quad \sigma'(7/2)=+.
$$
Then we have\begin{NB}Modified\end{NB}%
\begin{equation}\label{eq_Bforconifold}
[B(\alpha_1)]=[B(\alpha_3)]=0,\quad
[B(\alpha_2)]=[B(\alpha_2+\alpha_3)]=1,\quad
[B(\alpha_1+\alpha_2)]=[B(\alpha_1+\alpha_2+\alpha_3)]=2,
\end{equation}
and
\begin{equation}\label{eq_qforconifold}
q'_0=q_0^{-3}q_1^{-3}=q^{-3}, \quad
q'_1=q_0 q_1^{2}=q Q , \quad
q'_2=q_0^{3}q_1^{3}=q^3, \quad
q'_3=q_0=q Q^{-1}.
\end{equation}
The closed BPS partition function corresponding to the B-field \eqref{eq_Bforconifold} is the following:
\begin{NB2} changed: $q<-> Q$\end{NB2}
\begin{align}
\prod_{n>0} (1-q'^n)^{-4n}
\prod_{n>0} (1-q'^n Q'_1)^n \prod_{n>0} (1-q'^n Q'{}_1^{-1})^n
\prod_{n>0} (1-q'^n Q'_2)^{-n} \prod_{n>1} (1-q'^n Q'{}_2^{-1})^{-n}\notag\\
\prod_{n>0} (1-q'^n Q'_3)^n \prod_{n>0} (1-q'^n Q'{}_3^{-1})^n
\prod_{n>0} (1-q'^n Q'_1 Q'_2)^n \prod_{n>2} (1-q'^n Q'{}_1^{-1} Q'{}_2^{-1})^n \notag\\
\prod_{n>0} (1-q'^n Q'_2 Q'_3)^n \prod_{n>1} (1-q'^n Q'{}_2^{-1} Q'{}_3^{-1})^n
\prod_{n>0} (1-q'^n Q'_1 Q'_2 Q'_3)^{-n} \prod_{n>2} (1-q'^n Q'{}_1^{-1} Q'{}_2^{-1} Q'{}_3^{-1})^{-n}.\label{eq_closedBPS}
\end{align}
Substituting \eqref{eq_qforconifold} for \eqref{eq_closedBPS} under the identification \eqref{chemicalweight}, we obtain the open BPS partition function:\begin{NB}I think the notation $Z^c_{\mathrm{NCDT}}$ is not suitable in physics paper, but I have no idea what is the correct notation in physics context. Please modify it.\end{NB}%
\[
\ensuremath{\mathcal{Z}}^c_{\mathrm{NCDT}}(q,Q)\cdot (1-q)^{-3}(1-q^3)^{-1}(1-Q)^2(1-q^2 Q)(1-q^2Q^{-1}),
\]
\begin{NB2}\[
\ensuremath{\mathcal{Z}}^c_{\mathrm{NCDT}}(q,Q)\cdot (1-q)^{-3}(1-q^3)^{-1}(1-Q_1)^2(1-q^2Q_1)(1-q^2Q_1^{-1})
\]\end{NB2}
where
\begin{align*}
\ensuremath{\mathcal{Z}}^c_{\mathrm{NCDT}}(q,Q):&=
\prod_{n>0} (1-q^n)^{-2n}
\prod_{n>0} (1-q^n Q)^n
\prod_{n>0} (1-q^n Q^{-1})^n \\
&=\ensuremath{\mathcal{Z}}^c_{\mathrm{top}}(q,Q) \ensuremath{\mathcal{Z}}^c_{\mathrm{top}}(q,Q^{-1}) .
\end{align*}
This coincides with the expression computed from the result of \cite{IK}
$$
M(q)^2 \prod_{n>0} (1-q^n Q) \frac{1}{(1-q)^3 (1-q^3)} (1-Q)^2 (1-Q q^2)(1-Q q^{-2})
$$
up to the wall crossing factor and the normalization by a monomial as remarked in footntote \ref{normalization}.

\section{Discussions}

Let us conclude this paper by pointing out several interesting problems which require further exploration.

\begin{itemize}

\item In this paper, we only discussed wall crossings with respect to the closed string moduli. However, we expect that there should be wall crossing associated with the open string moduli as well. Therefore, the question arises: At which values of the open string moduli is our vertex defined? How does the vertex change as we change the open string moduli? Some of these issues will be discussed in \cite{AY}.

\item It would be interesting to see if there is a generalization of GV large $N$ duality \cite{GVdual} including the background dependence, and whether there is a Chern-Simons interpretation of the wall crossing phenomena (see \cite{CecottiV} for a related idea).

\item It would be interesting to extend our definition and computation to a Calabi-Yau manifold with compact 4-cycles and with multiple D6-branes. \begin{NB2} adde\end{NB2} The former should be possible by combining our vertices, and will be related to the Nekrasov's partition function \cite{Nekrasov}.

\item
Derivation of open BPS invariants from supergravity viewpoint \cite{DenefM} is another interesting problem. See \cite{Denefhole} for the related discussion in the case of orientifolds.
Another related question is the connection of the crystal melting expansion of $\ensuremath{\mathcal{Z}}^o_{\rm BPS}$ with open version of OSV conjecture \cite{ANV}.

\end{itemize}

\section*{Acknowledgments}
\begin{NB2} changed \end{NB2}
M.~Y. would like to thank M.~Aganagic and D.~Krefl for collaboration on related projects.
He would also like to thank M.~Cheng, J.~Gomis, L~Hollands, D.~Krefl,
T.~Okuda, H.~Ooguri, N.~Saulina, P.~Sulkowski, C.~Vafa and J.~Walcher for stimulating discussions.

K.~N. is supported by JSPS Fellowships for Young Scientists (No.\ 19-2672).
M.~Y. is supported by DOE grant DE-FG03-92-ER40701, by the JSPS fellowships for Young Scientists, by the
World Premier International Research Center Initiative,
and by the
Global COE Program for Physical Sciences Frontier at
the University of Tokyo, both by MEXT of Japan.
M.Y. would also like to thank Centro de Ciencias de Benasque Pero Pascul, Simons Center for Geometry and Physics at Stony Brook and Berkeley Center for Theoretical Physics for hospitality, where part of this work has been performed.

redpajama_set_name: RedPajamaArXiv


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In the last module, we have looked at a number of innovation theories.
we will here in particular also look at the technology delta.
impact and how it relates to the barriers for implementation.
So what do we mean by technology Delta then?
better the new technology is compared to the technology that were presently using.
Even though we can't exactly know how a new technology may influence a system.
the larger the potential to drive change is.
So if we can understand what the technology delta is, and how large it is.
the potential impact and the ability for the new technology to overcome barriers.
Can then technology delta be estimated?
Well, we have of course a number of measurable things, so that's quite easy.
We can measure the performance of the new technology.
the scenario, the future, and the map that we are going to draw.
with what we really know and can measure is a good and rather easy starting point.
But still, of course, this demands a lot of knowledge about the new technology.
So we just start off with what we know about the new technology.
Let's look at the solid state drive versus the hard disk drive.
there are a number of parameters that are quite easy to compare.
durability, form factors, noise, energy consumption etc.
And as you may know, the solid state drive is actually not better in all aspects.
So even here it becomes a bit complex.
But is then the technology delta large enough?
in some instances that's suddenly the case and I use one in my computer.
technology should be applied to answer that question.
a component in the laptop, which then is on a higher level.
components, so those are then on a lower level.
the easier it usually is to estimate the impact.
The impact on an industry or society.
and then gradually move up and enlarge that system.
has a significant impact on that limited system, and then we can move upwards.
there are still barriers to overcome.
the more likely it is that these barriers are managed.
the need for the technology delta may vary.
we demand a really high technology delta for that to happen.
is dependent on how big the technology delta is, but also on the barriers.
But when we do this, we need also to consider that everything is constantly changing.
There is no such thing as a steady state here.
the big ones, usually called the mega trends.
that we study and the system that we analyze.
by the end we'll also look at how the industry is changing over time.
you will be ready to start working on drawing the map.


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Podere Cima a Zano lies on the hilltop of Zano, furthest from the state road but in a perfect spot offering panoramic views and utter tranquility.
This farmhouse has been divided into three apartments: Casa di Alfiero, Tinaia and Fienile. Each has private entrance and outside private space which includes garden furniture and BBQ. Both parking and swimming pool are conveniently nearby.


Rachel is an Associate Solicitor in the Corporate and Energy Teams at Carson McDowell LLP.
She graduated with a degree in Law from Queen's University Belfast in 2009, attended the Institute of Professional Legal Studies later that year and was admitted onto the Roll of Solicitors in 2011.
While working at another leading Belfast firm, Rachel advised in relation to a wide range of corporate and commercial matters and also gained experience in debt recovery, bankruptcy and insolvency. She joined the team at Carson McDowell in March 2014.
Rachel has extensive experience in the corporate legal sector, particularly in relation to company disposals, mergers and acquisitions, shareholder arrangements, joint ventures, corporate reorganisations and management buy-outs. She also regularly advises companies on financing, venture capital investments, corporate governance and general compliance issues.
Rachel also has considerable experience in Energy and Renewables. Having completed a long-term secondment with a leading utility company, Rachel has an in depth understanding of the gas industry and the key issues to be considered when advising a regulated business. She has advised in relation to several wind, AD and biomethane projects, energy supply agreements and other regulatory matters.
Rachel has been Chairperson of the Company and Commercial Lawyers Group in Northern Ireland since 2016 and is also a member of the Generation Innovation Steering Team.


text:
Balfour Beatty completes rail line electrification
Balfour Beatty, as contracted by Network Rail, has successfully completed the electrification of 77km of the Great Western Railway (GWR) line between Bristol and Cardiff.
It comes as part of a wider Greater Wester Electrification Programme from Network Rail to electrify 275km of train line between Cardiff and London.
The electrification will provide greener and more reliable journeys to one of Britain's oldest and busiest railways, improving connections and increasing capacity.
Over 750 Balfour Beatty operatives carried out the complex project throughout the Christmas period to complete the finished product on time.
The series of works included installation of the overhead cables Cardiff Central Station, marking the completion of over 4,200 masts and booms responsible for holding the 643km of electric cabling used to power the 77km of route in this section.
To celebrate the milestone, the first electrified train running out of Cardiff Central Station on Jan 6 2020.
Mick Rayner, Managing Director of Balfour Beatty's Rail business, said
"We are proud to have successfully delivered part of the largest modernization project on the Great Western Railway which we been working on for the past three years. Delivering this important upgrade to the Welsh rail network will provide a more sustainable way of travelling for tens of thousands of commuters".
Mark Langman, Network Rail's Managing Director for Wales and Western said:
"I am delighted that services can now run on electricity between Bristol and Cardiff, linking the two capital cities for the very first time via an electric railway. This is the culmination of the largest modernisation project on the Great Western railway since it was built in Victorian times, which we have been working on over the last 10 years."
Photo: Balfour Beatty/First electric train from Cardiff to London
Director of Business development and communications at Xrail, Monica Barbosa, joins us in the RTM Podcast room to discuss the rail industry from an SME perspective. Monica is a cheerleader for small...
SOURCE: RTM Dec 19/Jan 20 Conor Feighan Secretary General European Rail Freight Association (ERFA) The key challenge for policy makers and industry over the coming five years revolves around...
'the sleepers' blog
Rail Technology Magazine's Matt Roberts explains the significant role data can play within the future development of the rail industry. Standing as a cornerstone of the UK transport network, the rail industry is forever striving to in... more >
Andrew Haines, the Chief Executive of Network Rail, has told the Today programme on Radio 4's BBC's flagship news programme that he would not rule out his organisation issuing future r... more >
08/05/2019Advancing the rail industry with management degree apprenticeships
In answering the pressing questions of how current and future generations of managers can provide solutions to high-profile infrastructure projects across the UK, Pearson Business School, part of... more >

redpajama_set_name: RedPajamaCommonCrawl


text:
Campus media receive journalism awards
By Student Voice
Student journalists for the Student Voice and WRFW at UW-River Falls have been recognized for their work by Wisconsin's two leading professional media organizations.
Both the Wisconsin Newspaper Association (WNA) and the Wisconsin Broadcasters Association (WBA) held their annual conventions at the end of February. Among events at the convention were ceremonies awarding the best in professional and student journalism in the state.
In the Collegiate Better Newspaper Contest sponsored by the WNA Foundation, the Student Voice won three awards for articles published during 2013. The weekly newspaper earned a first place for best editorial. The editorial chastised the Student Senate for not representing students professionally. Ashley (Goettl) Hanley, who graduated last year and now works as a reporter for KEYC-TV in Mankato, Minn., earned second place in general reporting for her story about a UWRF professor acquitted of a sexual assault charge. Jonathan Reid, a professional writing major, earned third place in column writing for his remembrance of Professor Meseret Reta, a journalism adjunct who died of cancer in Nov. 2012.
In the WBA Student Awards for Excellence, a former and a current reporter for WRFW were honored for their radio news reports. Christopher Liermann, a 2013 graduate who majored in journalism, earned first place for a sports story about the UWRF swimming and diving team's effort to find a new coach. Liermann also was awarded first place for sports programming for radio with his hockey pre-game show. Journalism major Maddie Berg's story on the opening of "CHICAGO The Musical" on the UWRF campus received second place recognition.
The Student Voice and WRFW are student-run campus media organizations supported primarily by student fees and advised by faculty in the Department of Journalism. The faculty advisor for the Student Voice is Associate Professor Andris Straumanis, while the news department of WRFW is advised by Professor Sandy Ellis, chair of the department.
For further information, contact Professor Sandy Ellis, chair of the Journalism Department, at (715) 425-3169 or sandra.ellis@uwrf.edu, or Associate Professor Andris Straumanis, andris.straumanis@uwrf.edu.


text:
Valuing volunteers – an opportunity to thank those who give so generously to ECEC
With National Volunteer Week falling next week (18 – 24 May), the South Australian Department of Education has encouraged early childhood education and care (ECEC) services to acknowledge the occasion, and think about ways in which they may thank the caring, creative, diverse and generous volunteers who support their services.
Although the reminder has been issued by the South Australian Department, many of the ideas are transferable across all states and territories, supporting services around the country to acknowledge the efforts of their volunteers.
The department has a volunteer workforce of more than 25,000 people, who give their time and share their skills to help children of all ages learn and grow. Comprising a diverse range of parents, family, caregivers and people from the community, the volunteers bring their life skills, expertise, knowledge and diversity to schools, preschools and children's centres across South Australia.
With the recent summer of fires, followed by the unprecedented conditions brought about by the COVID-19 pandemic, volunteers are perhaps more valuable now than ever. Although many face to face events may have to be repositioned to online, it is still important to ensure volunteers are recognised.
"Get creative, and consider ways you can still acknowledge the volunteers who make a difference to your communities during this celebratory week" was the advice from the Department.
Some of the ways in which volunteers will be recognised by the Department this year include:
SA Minister for Education, John Gardner, will deliver a video message to recognise the department's volunteer workforce, and acknowledge the contribution they make in the education and wellbeing of children and young people.
Resources on mental health and wellbeing will be shared with volunteers
Accepting nominations for the Minister's Volunteering Award for Outstanding Service
The Minister's Award aims to recognise individuals or groups who make a significant contribution to others by volunteering in schools and preschools. Recipients get a certificate personally signed by the Minister for Education, thanking them for their service.
Suggestions for recognising volunteers at service level include:
Giving a gift
Volunteers can be acknowledged with a small gift of appreciation from the department, with coffee keep-cups, pens, calico bags and lanyards available free to schools, preschools and centres to pass on to their volunteers. Contact education.volunteers@sa.gov.au for more details.
"From thank-you morning teas online to children drawing/writing thank you letters or social media posts, we hope volunteers will be made to feel special" a spokesperson said.
Services are encouraged to share the creative ways in which they will be saying thank you by emailing Heidi Unferdorben at education.volunteers@sa.gov.au
Member for Bendigo calls on Federal Government to do more to keep ECEC open
ECEC volunteer Val rewarded for her thirty year commitment to children and families
ELAA recognises volunteer value in McArthur award ceremony
Volunteer Day is a great time to say thank you, Playgroup Australia says
Goodstart goes in to bat for Australia's volunteer firefighters, offering free care


COLIN TIZZARD is assured Finian's Oscar will be prepared for the Cheltenham Festival.
Tizzard's Grade 1-profitable hurdler has had a wind operation since being pulled up in the Cleeve Hurdle.
He has a variety of choices over fences and hurdles at Prestbury Park together with the Stayers' Hurdle.
Tizzard stated: "He has had his palate cauterised and is again cantering.
"With a minor operation like that, you'll be able to have them again operating in a month. He ought to be high quality for Cheltenham."
Nevertheless, Tizzard was unable to offer a constructive update on Pingshou.
Tizzard added: " He isn't fairly proper. He missed two years earlier than we had him and he isn't straightforward to maintain sound."


Are You Part of the "IRS Tax Lien Explosion"?
Reading this article can leave the typical taxpayer feeling a little nervous. If you've already had trouble with the IRS, or are currently trying to resolve tax problems, that article, which charts the dramatic increase in IRS tax liens, might make you very nervous indeed!
Earlier this month, National Taxpayer Advocate Nina Olsen released her annual report to Congress. The Advocate expressed continuing concern that the IRS's increasing use of hard-core enforcement actions, particularly tax liens, is inflicting unnecessary harm on financially struggling taxpayers.
Over the past seven years, the IRS has filed more than five million tax liens. 1.1 million liens were filed in 2009! The impact of these liens is tremendous. The individual tax payer will have their credit negatively impacted, making it harder for them to borrow money to buy a house, car, or fund a child's education. A tax lien can affect a taxpayer's employability. Not every company is willing to hire someone who's had trouble with the IRS.
If you have received a notice from the IRS about a potential or impending federal tax lien, don't panic! There is hope for your situation. Begin by realizing you need to get help right away! Resolving tax problems requires expert, experienced assistance. Don't try to go it alone!
There are many routes to settling old tax debt and eliminating federal tax liens. Don't trust anyone who claims to know the solutions to your tax problem before they understand your unique circumstances! Every tax problem is different. You want a qualified tax professional to work with you to discover the best way to resolve your tax issues.


text:
"Joy" is a song by English electronic musician Mark Ryder, released as a single in 2001. It contains samples from Soft Cell's version of "Tainted Love", and features vocals from Special MC and Heidi Rydquist. The song reached the UK top 40, peaking at No. 34 on the UK Singles Chart, and also reached No. 1 on the UK Dance Singles Chart in early 2001.

In November 2016, UK duo Gorgon City compiled a list of their top UK garage songs for Billboard, with "Joy" at #28.

Track listing
UK 12"
A1. "Joy" (Original Version)
A2. "Joy" (Rapapella)
B1. "Joy" (Ruff Ryder Garage Remix)
B2. "Joy" (Waveform Breakbeat Remix)

Charts

References

2000 songs
2001 singles
UK garage songs
Relentless Records singles
Songs written by Ed Cobb

redpajama_set_name: RedPajamaWikipedia


text:
Q: Check if constructor does exist at compile time I am mapping a hibernate query to an object using a constructor like this :
Constructor<Product> contructor = Product.class.getConstructor(String.class, String.class, Integer.class, String.class, String.class, String.class,
                        String.class,String.class, String.class,Boolean.class);
criteria.setResultTransformer(new AliasToBeanConstructorResultTransformer(contructor));

If someone modify the constructor by adding a parameter, this code will fail but at runtime. It is possible to insure at compile time that the Contructor that we need here realy exists ?

A: This is basically what tests are for.
For example:
@Test
public void constructorAvailableTest() {

    try {
        Constructor<Product> contructor = Product.class.getConstructor(String.class, String.class, Integer.class, String.class, String.class, String.class,
            String.class,String.class, String.class,Boolean.class);
    } catch( NoSuchMethodException nsme ) {
       fail( "The specified " + Product.class.getName() + " constructor does not exist!");
    }

}

Better yet, you could put the code that reflexively constructs the Product in a static method and just call that in your test!

redpajama_set_name: RedPajamaStackExchange


text:
Build Progress
Coming to Life
W194 History
Rallye Car
History of the Marque
w194.com
Authentic Horns
It's not easy finding all the real components for a car that is more than 60 years old. Mark Passarelli has been collecting parts for the W194 for nearly three decades, and his search has never stopped. In June of 2012 he finally found the correct horns for this remarkable automobile, allowing for one more step closer to the authenticity of the reconstruction of car Number One.
Mark Passarelli's Commitment to Perfection
Authenticity. That is what makes our w194 Project so special. Mark Passarelli, artist, engineer and visionary, is creating a true classic. This is not a new car, this will be a 1952 Mercedes Benz automobile, identical in every way to the hand-crafted original built by Mercedes Benz racing engineers in Germany more than a half-century ago.
Each of the many thousands of parts that will go into the w194 are exact. Mark has spent decades sourcing and collecting the components, including some from original W194 cars, that will make this reconstruction identical to the original number 1 vehicle. The photo above shows Mark and one of his team of experts creating the painstakingly measured and calibrated body form that will be used to reconstruct the aluminum body that was integral to the racing success of the w194.
Doing It Old School
Creating hand-formed components, including the precisely formed body, takes incredible skill and patience. Above, Mark uses an English Wheel to re-create the W194 in the same manner as the engineer/artists at Mercedes Benz did in 1951. The process is almost unbelievably time-consuming. Fortunately, other automotive artisans, some trained by Mark, will work on various components, under his supervision and to his exacting standards.
A Rare Motor Roars to Life
The powerplant of the original w194 number one car was created from a specially modified coupe/cabriolet motor from the w188 1952 300S . The Passarelli Restorations w194 has one of these exceptionally rare modified M188 motors. This is the actual motor as modified by Mercedes Benz racing engineers in 1952. It has been completely rebuilt by Mark Passarelli personally and is stamped 194/1A. This commitment to authenticity means that the fortunate owner of this automobile will be driving a true classic, exactly as it was created in 1952. In the background is an extremely rare Gullwing rally car, nearing the completion of a 24 month restoration.


text:
Audio of my Interview with Gun Expert John Lott
Getty Images by: John Moore
Today I interviewed Dr. John R. Lott, Jr.
Dr. Lott is an economist and a world recognized expert on guns and crime. Dr. Lott has held research or teaching positions at various academic institutions including the University of Chicago, Yale University, the Wharton School of the University of Pennsylvania, Stanford University, and Rice University, and was the chief economist at the United States Sentencing Commission during 1988-1989. He is currently a Fox News columnist. He holds a Ph.D. in economics from UCLA.
I reached out to Dr. Lott after the mass shooting in Las Vegas and one of my listeners calling into my show yesterday and making the statement that America has "astronomically" higher gun violence than the rest of the world. I asked him for his source in which he is basing that statement on and he could not answer me.
He may be right or he may be wrong, I just want facts to guide all of our opinions and thoughts on such controversial issues. So I reached out to Dr. Lott to find out those facts.
We spoke about the gun violence statistics he has researched and have been published, to find out what he said please click on the following segments:
Interview Dr John Lott gun violence statistic expert 10-3-17 1 of 2