text:
\section{Introduction}
\label{sec:intro}
  Seiberg duality \cite{Seiberg} plays an important role
  for understanding phase structures of ${\cal N}=1$ supersymmetric gauge theories in four dimensions.
  Although an ${\cal N}=1$ theory and its Seiberg dual theory are not equivalent,
  the dual theory describes the same infrared physics as that of the original theory.
  For example, in ${\mathcal N}=1$ supersymmetric QCD (SQCD) in conformal window ($3N_c/2 \leq N_f \leq 3 N_c$),
  the Seiberg duality implies the existence of a non-trivial infrared fixed point,
  where an interacting superconformal field theory is realized \cite{Seiberg} (see \cite{ISreview} for a review).
  The original and dual theories flow to the same infrared fixed point.

  On the other hand, some ${\mathcal N}=2$ superconformal gauge theories are known to have an exact S-duality,
  which means that a strong gauge coupling region of a theory is equivalent to a weak
  coupling region of another theory at any energy scale \cite{SW, APShapere},
  like Montonen-Olive duality in the ${\mathcal N}=4$ supersymmetric gauge theory.
  Interestingly, it was proposed in \cite{LS} that the Seiberg duality is associated with the S-duality
  in the ${\cal N}=2$ supersymmetric gauge theory.
  By the mass deformation for the adjoint chiral multiplet,
  the S-dual pair of ${\mathcal N}=2$ superconformal gauge theories flows to the ${\mathcal N}=1$ theories
  which are precisely a Seiberg dual pair.
  (See \cite{APS, HMS, AILS} for related discussions.)

  Recently, Gaiotto proposed a new chain of S-dualities in ${\mathcal N}=2$ superconformal quiver gauge theories \cite{Gaiotto}
  and many related developments have been made \cite{Gaiotto:2009gz, Tachikawa, BBT, AGT, ND, Wyllard}.
  The ${\mathcal N}=2$ superconformal theories associated with the generalized quiver diagrams,
  which we will explain later, are all equivalent, or S-dual to each other,
  if they have the same ``genus" of the quiver diagram and the same global symmetry.
  The theories where the gauge group is $SU(2)^{p}$ are the simplest case.
  In this case, we can explicitly construct the Lagrangians of all the generalized quiver gauge theories and
  their flavor symmetries are generically $SU(2)^n$.\footnote{
    For some $SU(N)$ generalized quiver gauge theories ($N > 2$), the Lagrangian description has not been found.}
  They form a large class of the quiver gauge theories, which is denoted as ${\mathcal T}_{g, n}$, where $3-3g=n-p$,
  because they are proposed to realized on M5 branes wrapped on genus $g$ Riemann surface with $n$ punctures.

  In this paper, from the Gaiotto's S-dualities in ${\mathcal T}_{g, n}$,
  we propose a number of new Seiberg dualities of $\mathcal{N} =1$ quiver gauge theories,
  which implies there are a large number of new non-trivial $\mathcal{N} =1$ superconformal field theories.
  As one gauge group case in \cite{LS},
  ${\mathcal N}=2$ superconformal quiver gauge theories ${\mathcal T}_{g, n}$ are expected to flow to infrared fixed points,
  by the mass deformations for the adjoint chiral multiplets.
  This can be partially verified by turning off the gauge couplings except one.
  In that case, the theory is nothing but ${\mathcal N}=1$ SQCD with four flavors
  which flows to the non-trivial infrared fixed point.

  In general, this deformation produces several quartic terms in the superpotential.
  In the ultraviolet, these are irrelevant operators.
  However, they show non-trivial behavior in the infrared:
  some combinations of these develop to exactly marginal operators,
  whose coupling constants span a manifold of fixed points.
  It is generally difficult to identify the exactly marginal operators.
  However, we can still count the (complex) dimension of the manifold of the fixed points,
  that is the number of the exactly marginal operators by means of the argument in \cite{LS}.
  It reveals that if we concentrate on the operators keeping the flavor symmetry $SU(2)^n$,
  the number of them is $2n$ and universal for the quiver gauge theories
  obtained from ${\mathcal T}_{g, n}$ for fixed $g$ and $n$.

  We focus on the above-mentioned fact that
  the S-duality relates various different looking ${\mathcal N}=2$ superconformal quiver gauge theories.
  We show that this property implies interesting physics:
  many different looking ${\mathcal N}=1$ quiver gauge theories flow to the same infrared fixed point,
  by the mass deformations of the S-dual family.
  As a result, we propose new Seiberg dualities which relate a large number of ${\mathcal N}=1$ quiver gauge theories.
  For instance, two quiver theories obtained from ${\mathcal T}_{1, 2}$ theories are expected to be dual.
  One quiver in this category is a slightly generalized theory of the one considered by Klebanov and Witten \cite{KW},
  i.e., the quiver with a loop and two gauge groups.
  The S-dualities imply that this theory is Seiberg dual to the other theory with a different quiver diagram.
  Notice that the original generalized Klebanov-Witten theory is self-dual under the usual Seiberg duality on one gauge group.
  Therefore, it is very remarkable that the new Seiberg dual theory describes the same infrared physics
  as that of the generalized Klebanov-Witten theory.

  In order to verify our proposal, we give non-trivial consistency checks of these dualities.
  The above counting of the exactly marginal operators could be strong evidence:
  recall that it is universal for fixed $g$ and $n$.
  Also, we show that the 't Hooft anomaly matching holds
  for all the theories obtained from ${\mathcal T}_{g, n}$ for fixed $g$ and $n$,
  although we need a bit care in the case with an enhanced global symmetry.
  Another non-trivial check of these dualities is the matching of several operators.
  We will perform this in the quiver gauge theories obtained from ${\mathcal T}_{1, 2}$ theories mentioned above.
  We demonstrate that chiral operators match between the generalized Klebanov-Witten theory and its dual.
  We also consider the matching of non-linear relations for the chiral
  operators, i.e., the matching of the chiral ring,
  which indicates that the matching of the classical moduli space.

  The organization of this paper is as follows.
  After reviewing the S-dualities in ${\mathcal N}=2$ superconformal quiver gauge theories \cite{Gaiotto} in subsection 2.1,
  we then consider the deformation to ${\mathcal N}=1$ in subsection 2.2.
  We will propose new Seiberg dualities among these theories.
  Also, we check that the global anomalies match in these theories.
  In section 3, we analyze exactly marginal operators in these theories.
  We will see that the number of the exactly marginal operators is universal for the proposed Seiberg dual theories.
  In section 4, we consider a simple model, whose quiver diagram is identical to
  the Klebanov-Witten theory, and its dual.
  We will identify the operators in the dual theory, which correspond to the operators in the generalized Klebanov-Witten theory.
  Section 5 is devoted to conclusion and discussion.

\section{${\mathcal N}=1$ SCFTs from ${\mathcal N}=2$ SCFTs}

\subsection{${\mathcal N}=2$ superconformal quiver gauge theories and S-dualities}
\label{subsec:Gaiotto}
  A large class of ${\mathcal N}=2$ superconformal quiver gauge theories in four dimensions was constructed in \cite{Gaiotto}.
  In this paper, we mainly consider the theories with $SU(2)^p$ gauge group.
  In this case, a class of ${\mathcal N}=2$ superconformal quivers is specified by
  the number of $SU(2)$ flavor symmetries, $n$, and ``genus'' of the quiver diagram,
  $g = 1 + \frac{p - n}{3}$, which is denoted as ${\mathcal T}_{g,n}$.
  Since there exist many Lagrangian descriptions associated with a label $(g,n)$,
  quivers contained in ${\mathcal T}_{g,n}$ are various in shape.
  The different looking quivers with same $g$ and $n$ are related by the S-duality.
  We will review this in this subsection.

  Among these ${\mathcal N}=2$ superconformal gauge theories,
  the $SU(2)$ gauge theory with 4 fundamental hypermultiplets, ${\mathcal T}_{0,4}$, is the simplest and important example.
  Since the fundamental representation of $SU(2)$ is pseudo-real, the flavor symmetry is $SO(8)$.
  We consider  an $SO(4) \times SO(4) \simeq SU(2)^4$ subgroup of the $SO(8)$ flavor symmetry according to \cite{Gaiotto}.
  Motivated by this, we denote this theory as Fig.~\ref{fig:T40}.

  In order to construct the Lagrangian of this theory, we start with four free fundamental hypermultiplets,
  which we will denote as ${\mathcal T}_{0,3}$.
  Let us denote these superfields by $Q^{\alpha a i}$,
  where three indices label fundamental representations
  of the different flavor $SU(2)$'s ($\alpha = 1, 2$, $a = 1, 2$ and $i = 1, 2$).
  By gauging one of three $SU(2)$ flavor symmetries, e.g., $SU(2)$ labeled by $\alpha$,
  the superpotential can be written as
    \begin{eqnarray}
    W
     =     Q^{\alpha a i} (\varepsilon \phi)_{\alpha \beta} \varepsilon_{ab} \varepsilon_{ij} Q^{\beta b j}.
    \end{eqnarray}
  Now we introduce two copies of four free hypermultiplets to construct ${\mathcal T}_{0,4}$ theory.
  The $SU(2)$ gauge group can be obtained
  by gauging a diagonal part of two $SU(2)$ symmetries of different sets of four free hypermultiplets.

    \begin{figure}
    \begin{center}
    \includegraphics[scale=0.6]{quiver_T40.eps}
    \caption{{\small The quiver diagram of $SU(2)$ gauge theory with four fundamental hypermultiplets.}}
    \label{fig:T40}
    \end{center}
    \end{figure}

  In order to consider the S-duality, we introduce the mass parameters associated
  with $SU(2)_{a, b, \ldots}$ flavor symmetries as $m_{a, b, \ldots}$.
  The S-duality of this theory is associated with the triality of $SO(8)$ which
  exchanges $\textbf{8}_v$, $\textbf{8}_s$ and $\textbf{8}_c$ representations of $SO(8)$.
  Under the $SU(2)_a \times SU(2)_b \times SU(2)_c \times SU(2)_d$ subgroup, they decompose as
    \begin{eqnarray}
    \textbf{8}_v
    &=&    (\textbf{2}_a \otimes \textbf{2}_b) \oplus (\textbf{2}_c \otimes \textbf{2}_d),
           \nonumber \\
    \textbf{8}_s
    &=&    (\textbf{2}_a \otimes \textbf{2}_c) \oplus (\textbf{2}_b \otimes \textbf{2}_d),
           \nonumber \\
    \textbf{8}_c
    &=&    (\textbf{2}_a \otimes \textbf{2}_d) \oplus (\textbf{2}_b \otimes \textbf{2}_c).
    \end{eqnarray}
  Therefore, the S-duality permutes four $SU(2)$ symmetries.
  The strongly coupled limits of the original theory are S-dual to
  weakly coupled limits of the theory where $SU(2)$ flavor symmetries are permuted.

  The generalization to the quiver gauge theory is straightforward.
  All possible superconformal quiver gauge theories with $SU(2)$ gauge groups can be constructed
  from the fundamental building block ${\mathcal T}_{0, 3}$ by gauging some of the flavor symmetries.
  The gauging of one $SU(2)$ symmetry of ${\mathcal T}_{0, 3}$
  leads to two fundamental hypermultiplets, as seen above.
  By gauging two $SU(2)$ symmetries of ${\mathcal T}_{0, 3}$,
  we obtain a bifundamental hypermultiplet which has one $SU(2)$ flavor symmetry.
  Let us denote a bifundamental hypermultiplet by $B^{\alpha_1 \alpha_2 i}$,
  where $i$ ($=1, 2$) are the flavor indices and $\alpha_1$ and $\alpha_2$ label two gauged $SU(2)$ respectively.
  The superpotential of this bifundamental can be written as
    \begin{eqnarray}
    W
     =     B^{\alpha_1 \alpha_2 i} \left[ (\varepsilon \phi_1)_{\alpha_1 \beta_1} \varepsilon_{\alpha_2 \beta_2}
         + \varepsilon_{\alpha_1 \beta_1} (\varepsilon \phi_2)_{\alpha_2 \beta_2} \right] B^{\beta_1 \beta_2 j}
           \varepsilon_{ij}.
    \end{eqnarray}
  Also, by gauging three $SU(2)$ flavor symmetries, we obtain a trifundamental multiplet.
  If we denote this multiplet by $T^{\alpha_1 \alpha_2 \alpha_3}$, the superpotential is
    \begin{eqnarray}
    W
     =     T^{\alpha_1 \alpha_2 \alpha_3}
           \left[ (\varepsilon \phi_1)_{\alpha_1 \beta_1} \varepsilon_{\alpha_2 \beta_2} \varepsilon_{\alpha_3 \beta_3}
         + \varepsilon_{\alpha_1 \beta_1} (\varepsilon \phi_2)_{\alpha_2 \beta_2} \varepsilon_{\alpha_3 \beta_3}
         + \varepsilon_{\alpha_1 \beta_1} \varepsilon_{\alpha_2 \beta_2} (\varepsilon \phi_3)_{\alpha_3 \beta_3} \right]
           T^{\beta_1 \beta_2 \beta_3}.
    \end{eqnarray}
  In order to obtain a superconformal gauge theory where $\beta$-functions of the gauge couplings vanish,
  each gauge factor has to couple effectively to four fundamental hypermultiplets.
  This means that each $SU(2)$ gauge group should be obtained
  by gauging of a diagonal subgroup of two $SU(2)$ flavor symmetries of ${\mathcal T}_{0, 3}$'s.
  Collecting these pieces, we can construct various ${\mathcal N}=2$ superconformal quiver gauge theories
  which have $SU(2)^p$ gauge group and $SU(2)^n$ flavor symmetry.
  Corresponding quiver diagrams have genus $g = 1 + \frac{p - n}{3}$.
  Applying the S-duality of ${\mathcal T}_{0, 4}$ to each gauge group,
  we have different quiver gauge theories in ${\mathcal T}_{g, n}$.
  (This procedure corresponds to $s$-$t$ duality
  regarding the $SU(2)$ gauge factor as a propagator and ${\mathcal T}_{0, 3}$ as a vertex.)
  Therefore, all the theories in ${\mathcal T}_{g, n}$ are related by the S-dualities.

  In general, the global symmetry of ${\mathcal T}_{g, n}$ is $SU(2)^n \times SU(2)_R \times U(1)_R$.
  However, in some cases, the flavor symmetry of the gauge theory is further enhanced.
  A trivial case is, of course, ${\mathcal T}_{0, 4}$ where $SU(2)^4$ is enhanced to $SO(8)$.
  The simplest non-trivial case is ${\mathcal T}_{1, 2}$, as depicted in Fig.~\ref{fig:KW}.
  The left quiver can be regarded as an $SO(4)$ gauge theory with four half-hypermultiplets transforming in $\textbf{4}$.
  Since $\textbf{4}$ is real, the flavor symmetry is enhanced to $USp(4)$.
  More concrete observation of the enhancement of the flavor symmetry,
  based on $SU(2) \times SU(2)$ instead of $SO(4)$, will be presented in appendix \ref{sec:USp4}.

  The nontrivial S-dual theory of the above-mentioned one corresponds to the right quiver in Fig.~\ref{fig:KW}.
  The full flavor symmetry can be seen as follows:
  the right trifundamental is charged under the right $SU(2)$ gauge symmetry as
  $\textbf{2} \otimes \textbf{2} = \textbf{3} \oplus \textbf{1}$.
  We can regard this as a $SO(3)$ vector and a singlet.
  Therefore, the trifundamental decomposes into a bifundamental of the $SU(2) \times SO(3)$ gauge symmetry
  and a fundamental of $SU(2)$.
  The latter is mixed with the left fundamentals to form an $SO(5)$ flavor symmetry (Fig.~\ref{fig:KWsymmetry}).
  This matches with the symmetry of the original quiver.
  We will analyze these in the subsequent sections more explicitly.

    \begin{figure}[t]
    \begin{center}
    \includegraphics[scale=0.6]{quiver_T21.eps}
    \caption{{\small Two different quivers ${\mathcal T}_{1,2}$ which are related by the S-duality.}}
    \label{fig:KW}
    \end{center}
    \end{figure}
    \begin{figure}[t]
    \begin{center}
    \includegraphics[scale=0.6]{quiver_T21symmetry.eps}
    \caption{{\small The enhanced flavor symmetry of ${\mathcal T}_{1, 2}$.
             Both quivers have the $USp(4) \cong SO(5)$ flavor symmetry.}}
    \label{fig:KWsymmetry}
    \end{center}
    \end{figure}

  Another example is ${\mathcal T}_{2, 0}$.
  In this case, we can see that the flavor symmetry is enhanced to $SO(2)$ as in Fig.~\ref{fig:T02}.

    \begin{figure}[t]
    \begin{center}
    \includegraphics[scale=0.6]{quiver_T02.eps}
    \caption{{\small The enhanced flavor symmetry of ${\mathcal T}_{2, 0}$.
             Both quivers have the $SO(2)$ flavor symmetry.
             On the left, upper and lower trifundamentals form a $SO(2)$ vector.
             In the middle, each trifundamental decomposes as
             $\textbf{2} \otimes \textbf{2} = \textbf{3} \oplus \textbf{1}$, as the ${\mathcal T}_{1, 2}$ case.
             It produces two fundamental chiral multiplets for the center node (right).
             Thus, the flavor symmetry is $SO(2)$.}}
    \label{fig:T02}
    \end{center}
    \end{figure}

  Finally, we comment on punctured Riemann surfaces which play a role in the study the ${\mathcal N} =2$ quiver gauge theories.
  In \cite{Gaiotto}, it was pointed out that the space of the gauge coupling constants of the theory is identified
  with the complex moduli space of the associated Riemann surface.
  The labels $g$ and $n$ of this family of SCFTs are precisely
  the genus and the number of punctures of the corresponding Riemann surface.
  Let us consider the schematic description of ${\mathcal T}_{0,4}$ for instance,
  which is associated with a sphere with four punctures $C_{0,4}$.
  The four $SU(2)$ flavor symmetries of ${\mathcal T}_{0,4}$ correspond to four punctures on the sphere $C_{0,4}$.
  By the decoupling of the gauge coupling $\tau \rightarrow i \infty$,
  we obtain two copies of ${\mathcal T}_{0, 3}$'s, each of which has an $SU(2)^3$ flavor symmetry.
  This decoupling limit corresponds to the degeneration limit of a sphere into two three punctured-spheres.
  Thus, ${\mathcal T}_{0,3}$ is the building block for the quiver, associated with a sphere with three punctures $C_{0,3}$.
  A decomposition of a punctured surface $C_{g,n}$ into $C_{0,3}$'s corresponds
  to a weakly-coupled gauge theory description of the SCFTs ${\mathcal T}_{g,n}$ as well.

    \begin{figure}[t]
    \begin{center}
    \includegraphics[scale=0.6]{quiver_T40decoupling.eps}
    \caption{{\small Possible degeneration limits of a sphere with four punctures, which correspond to
             the usual weak coupling limit and S-dual weak coupling descriptions.}}
    \label{fig:T40decoupling}
    \end{center}
    \end{figure}

 We can also give a schematic explanation of the S-dualities.
 Let us introduce the mass parameters for ${\mathcal T}_{0,4}$ theory associated with $SU(2)_{a, b, \ldots}$ flavor symmetries.
 In this case, the gauge coupling moduli space is now described by a sphere with four marked punctures.
 Since the punctures are marked, we have three different degeneration limits of a sphere,
 as in Fig.~\ref{fig:T40decoupling}.
 This is precisely the S-duality which permutes four $SU(2)$ symmetries.
 In generic ${\mathcal T}_{g, n}$, all the possible weak coupling S-dual descriptions correspond
 to the possible degeneration limits of the Riemann surface $C_{g, n}$.

\subsection{${\mathcal N}=1$ superconformal quiver gauge theories}
\label{subsec:SCFT}
  In what follows, we will consider the deformation of ${\mathcal N}=2$ superconformal quiver gauge theories ${\mathcal T}_{g, n}$
  to ${\mathcal N}=1$ by the mass terms for the adjoint ${\mathcal N}=1$ chiral multiplets in ${\mathcal N}=2$ vector multiplets.

  Let us consider the ${\mathcal T}_{0, 4}$ case, i.e., the $SU(2)$ gauge theory with four fundamental hypermultiplets.
  (For simplicity, we focus on the massless fundamental case.)
  With the mass deformation, below the energy scale of the mass parameter
  we integrate out the adjoint and the superpotential becomes $W \sim h Q^4$.
  In the ultraviolet, $h$ is an irrelevant coupling.
  However, it is, in the infrared, exactly marginal coupling in this ${\mathcal N}=1$ theory \cite{LS}.
  Indeed, the $\beta$-function for $h$ is proportional to the one for the gauge coupling constant:
    \begin{eqnarray}
    \beta_h \propto \beta_g \propto 1 + 2 \gamma,
    \end{eqnarray}
  where $\gamma$ is the anomalous dimension of $Q$.
  Thus, the solutions to $\beta_g = \beta_h = 0$ form one complex dimensional
  manifold (fixed line) because $\gamma$ is a function of $g$ and $h$.

  The existence of the fixed line can be convinced by considering $h = 0$,
  that is ${\mathcal N}=1$ SQCD with $N_f = 2 N_c$, in which it was shown that a nontrivial infrared fixed point exists \cite{Seiberg}.
  From ${\mathcal N}=2$ point of view, the gauge coupling constant in the ultraviolet region,
  which is the exactly marginal coupling in ${\mathcal T}_{0, 4}$ theory, parametrizes this ${\mathcal N}=1$ fixed line.

  The ${\mathcal T}_{0, 4}$ theory is self-dual in a sense that
  the S-duality does not change the quiver diagram.
  Thus, we obtain a similar theory by the mass deformation \cite{LS}.
  This ``dual'' theory also has a quartic coupling $h_D$, whose value at the infrared fixed point
  is roughly the inverse of that of the original coupling $h$,
  associated with the S-dual transformation of the gauge coupling constant.

  Things become more interesting when we consider higher $g$ and $n$ cases.
  As the ${\mathcal T}_{0, 4}$ case above, ${\mathcal N}=1$ quiver gauge theories obtained from ${\mathcal T}_{g, n}$
  by the mass deformation are expected to flow to the superconformal fixed point.
  Furthermore, it leads to the higher dimensional manifold of fixed points,
  which is associated with the fixed manifold of ${\mathcal N}=2$ superconformal theories ${\mathcal T}_{g, n}$.
  Partial evidence of the superconformal fixed points can be seen, as above,
  by taking $h_{1, 2, \ldots, n} = 0$ and $g_{1, 2, \ldots, n} = 0$ except for $g_i$.
  In this case, the theory reduces to ${\mathcal N}=1$ SQCD and we know the existence of the infrared fixed point.

  Recall that the S-dualities relate many different looking quiver gauge theories in ${\mathcal T}_{g, n}$.
  Hence, we obtain many different looking ${\mathcal N}=1$ quiver gauge theories by the mass deformation.
  These are supposed to describe the same infrared physics.
  Therefore, this implies the existence of dualities among those ${\mathcal N}=1$  quiver gauge theories.
  Of course, it includes the self-dual duality as the ${\mathcal T}_{0, 4}$ case as well.

  The easiest non-trivial check of the existence of the superconformal fixed
  points and this Seiberg duality might be the 't Hooft anomaly matching.
  For generic quivers obtained from ${\mathcal T}_{g, n}$ where the enhancement of the flavor symmetry does not occur,
  the anomaly matching is very simple.
  We perform this at the origin of the moduli space of vacua.
  The global symmetry is $SU(2)_1 \times SU(2)_2 \times \ldots \times SU(2)_n \times U(1)_R$.
  The $SU(2)_i^3$ anomalies ($i = 1, \ldots, n$) vanish,
  because the fundamental representation of $SU(2)$ are pseudo-real.
  Also, the $SU(2)_i \times U(1)_R^2$ anomalies are trivially zero.
  Next note that all the chiral multiplets have the same $U(1)_R$ charge $- 1/2$
  and there exist the same number of such chiral superfields for fixed $n$ and $g$
  if we do not distinguish the $SU(2)$ flavor and gauge indices.
  Note also that the multiplets which have the $SU(2)_i$ flavor symmetry, which are ${\mathcal T}_{0, 3}$,
  have the same number of extra indices of $SU(2)^2$.
  Therefore, $SU(2)_i^2 \times U(1)_R$, $U(1)_R^3$ and $U(1)_R$ anomalies are, respectively,
  the same for fixed $n$ and $g$.

  The non-trivial case is the quivers obtained from ${\mathcal T}_{1, 2}$ where the global symmetries are enhanced
  to $USp(4) (\cong SO(5)) \times U(1)_R$.
  The left quiver of Fig.~\ref{fig:KWsymmetry} has the chiral multiplets in $\textbf{4}$ of $USp(4)$,
  while the right quiver of Fig.~\ref{fig:KWsymmetry} has those in $\textbf{5}$ of $SO(5)$.
  Since these representations are real, the $USp(4)^3$ anomalies are zero for both sides.
  Also, since all the chiral multiplets have the same $U(1)_R$ charge $- 1/2$ as noted above,
  the matching of the $U(1)_R^3$ and $U(1)_R$ anomalies is trivial.
  Finally, note that in terms of $USp(4)$, the quadratic Casimir of $\textbf{5}$ is twice as that of $\textbf{4}$.
  Therefore,
    \begin{eqnarray}
    (USp(4))^2 U(1)_R:
    ~~~~1 \times 4 \times \left( - \frac{1}{2} \right)
     =     2 \times 2 \times \left( - \frac{1}{2} \right).
    \end{eqnarray}
  This confirms that the anomalies of both theories match.

  In the rest of this paper, we will devote to collect other
  non-trivial evidence of the existence of the superconformal fixed
  points and the Seiberg dualities.
  In section \ref{sec:exactlymarginal}, we will consider exactly marginal operators
  in these ${\mathcal N}=1$ quiver gauge theories.
  In section \ref{sec:KW}, we concentrate on a particular example:
  the generalized Klebanov-Witten theory and its dual introduced above.
  We will see the matching of the chiral operators and the nonlinear constraints on them.

\section{Exactly marginal operators}
\label{sec:exactlymarginal}
  In this section, we analyze exactly marginal operators
  in ${\mathcal N}=1$ quiver gauge theories with generic quartic terms in the superpotential.
  First of all, we briefly review the argument of \cite{LS} for the existence of the exactly marginal operators.
  Let us consider a supersymmetric gauge theory with product gauge groups, $\prod_{i=1}^{p} G_i$,
  and chiral multiplets, $\phi_a$, which is transformed as a representation $R_a(G_i)$ of the gauge group $G_i$.
  We consider a superpotential $W = \sum_s h_s W^{(s)}(\phi_a)$,
  where each $W^{(s)}$ is a product of $d_s$ chiral superfields ($s = 1, \ldots, m$).
  As in \cite{LS}, at a superconformal fixed point, the scaling coefficients
    \begin{eqnarray}
    A_{g_i}
    &=&  - \left( 3 C_2(G_i) - \sum_a T(R_a(G_i)) (1 - \gamma_a) \right),
           \\
    A_{h_s}
    &=&    d_s - 3 + \frac{1}{2} \sum_a \gamma_a \frac{\partial \ln W^{(s)}}{\partial \ln \phi_a}
           \label{beta2}
    \end{eqnarray}
  have to vanish \cite{LS}.
  Here $C_2(G_i)$ is the quadratic Casimir, $T(R_a(G_i))$ is the index of the
  representation $R_a(G_i)$ and $\gamma_a$ is the anomalous dimension of the chiral superfield $\phi_a$.
  The derivative of the last term in (\ref{beta2}) counts the number of $\phi_a$ in $W^{(s)}$.
  From these, in general, we obtain $p + m$ equations.
  However, some of the equations would be degenerate.
  Let us denote the number of the linearly independent equations as $q$ ($\leq p + m$).
  These impose the $q$ conditions on $p + m$ coupling constants.\footnote{
    Of course there may be relevant and irrelevant operators which make the equations insolvable.
    In this case, we take these coupling vanish and reconsider the equations
    for the vanishing beta functions forgetting these couplings.}
  Thus, we expect that there is $p + m - q$ dimensional space of the solutions to these equations.
  In other words, there will exist $p + m - q$ exactly marginal operators.
  We will simply assume there indeed exist $p + m - q$ exactly marginal operators in our application.

  Now, we consider ${\mathcal N}=1$ supersymmetric quiver gauge theories
  associated with ${\mathcal N}=2$ superconformal quiver gauge theories ${\mathcal T}_{g, n}$.
  In particular, we consider the theories where the enhancement of the flavor symmetry does not occur.
  (We will analyze the theories with enhanced flavor symmetries, after the general discussion.)
  These theories are obtained as follows.
  First of all, we focus on a particular node of quiver.
  For such a node, the superpotential is
    \begin{eqnarray}
    W
     =     \frac{1}{2} m \mathop{\rm Tr} \phi^2 + \sum_{s = 1,2} h_s \mathop{\rm Tr} \phi X_s,
    \end{eqnarray}
  where $\phi$ is the adjoint chiral superfield of the node we are considering.
  The second term is due to the superpotential of the ${\mathcal N}=2$ supersymmetric gauge theory.
  The couplings $h_s$ are related with the gauge coupling $g$ in the ${\mathcal N}=2$ theory.
  Each $X_s$ is $Q^2$ or $B^2$ or $T^2$, where $Q$, $B$, and $T$ are
  fundamental, bifundamental, and trifundamental superfields, respectively,
  as seen in the subsection \ref{subsec:Gaiotto}.
  The trace is taken over the gauge indices of the node.
  All the other indices of the gauge and flavor symmetries of (bi or tri)fundamentals
  are already contracted as the superpotential is invariant under such symmetries.
  Integrating $\phi$ out, we obtain
    \begin{eqnarray}
    W
    &=&  - \frac{1}{2 m} \left( h_1 \mathop{\rm Tr} X_1^2 + h_2 \mathop{\rm Tr} X_2^2 + 2 h_1 h_2 \mathop{\rm Tr} X_1 X_2 \right).
           \label{W1}
    \end{eqnarray}
  For each node, we add the mass term for the adjoint chiral field
  and we have the superpotential (\ref{W1}) after integrating out it.
  The resulting ${\mathcal N}=1$ supersymmetric gauge theory will be superconformal
  at least if we tune the masses and the gauge coupling constants.
  Actually, all beta functions vanish if $\gamma_a=-1/2$ and there will be exactly marginal operators.

  We then consider how many exactly marginal operators keeping the flavor symmetry $SU(2)^n$ exist for this theory.
  Because $\gamma_a=-1/2$, only quartic superpotential can be marginal.
  As we saw in subsection \ref{subsec:Gaiotto}, each node couples to two different matter multiplets,
  say $P$ and $R$, each of which is two fundamentals or a bifundamental or a trifundamental.
  Let $X_1$ and $X_2$ be products of $P$ and $R$ respectively.
  The generic quartic superpotential keeping the flavor symmetry is a sum of the superpotential
  associated for each node, like (\ref{W1}),
    \begin{eqnarray}
    W
    &=&    H_1 P^4 + H_2 R^4 + H_3 P^2 R^2.
           \label{W}
    \end{eqnarray}
  Note that there is one independent quartic operator which is constructed from one field,
  e.g. $P^4$ or $R^4$ in (\ref{W})\footnote{
    This fact is easily seen by noticing that the chiral field have three global or gauge $SU(2)$ indices,
    let us denote it $Q^{a_1,a_2,a_3}$,
    and there is only one invariant under the three $SU(2)$ constructed from four $Q$'s.}
  and in the sum of the superpotentials for the nodes,
  a $H_1$ or $H_2$-type coupling can appear in the superpotentials for two or three adjacent nodes.
  However, we need a bit care about the operator which is generated by two fields, like $P^2 R^2$ in (\ref{W}).
  Generically, there exists one such independent operator from two fields
  and, therefore, the $H_3$ type coupling appear only once in the superpotential at a node.
  In special cases, however, it is possible to construct two independent operators.
  For a moment, we assume that the number of such independent operator is one
  and treat such cases after general argument.

  In order to count the exactly marginal operators for the ${\mathcal N}=1$ theory,
  it is convenient to consider the relations between scaling coefficients when the mass perturbation is turned off.
  Let $A_{\tilde{g}}$ and $A_{h_s}$ be the scaling coefficients of the gauge coupling constant
  and the couplings $h_s$ before integrating out $\phi$.
  We refer to the anomalous dimensions of $P$, $R$ and $\phi$ as $\gamma_P$, $\gamma_R$ and $\gamma_\phi$, respectively.
  In this case, the scaling coefficients for $h_1$ and $h_2$ are
  $A_{h_1} = \frac{1}{2} (\gamma_\phi + 2 \gamma_P)$ and
  $A_{h_2} = \frac{1}{2} (\gamma_\phi + 2 \gamma_R)$.
  In terms of these, $A_{\tilde{g}}$ can be written as
    \begin{eqnarray}
    A_{\tilde{g}}
     =   - 2 (\gamma_\phi + \gamma_P + \gamma_R)
     =   - 2 (A_{h_1} + A_{h_2}).
           \label{Ag2}
    \end{eqnarray}
  Such relation is satisfied for each node.

  Then, we return to consider the ${\mathcal N}=1$ supersymmetric quiver gauge theory with (\ref{W}).
  The scaling coefficients for $H_1$, $H_2$ and $H_3$ can be evaluated as
    \begin{eqnarray}
    A_{H_1}
    &=&    1 + \left( A_{h_1} - \frac{1}{2} \gamma_\phi \right)
         + \left( A_{h_1} - \frac{1}{2} \gamma_\phi \right)
     =     1 + 2 A_{h_1} - \gamma_\phi,
           \nonumber \\
    A_{H_2}
    &=&    1 + 2 A_{h_2} - \gamma_\phi,
           \nonumber \\
    A_{H_3}
    &=&    1 + A_{h_1} + A_{h_2} - \gamma_\phi
     =     \frac{1}{2} (A_{H_1} + A_{H_2}).
           \label{AH3}
    \end{eqnarray}
  Also, by using (\ref{Ag2}), the scaling coefficient for the gauge coupling $g$ for the node can be calculated as
    \begin{eqnarray}
    A_{g}
     =     A_{\tilde{g}} - 2 (1 - \gamma_\phi)
     =   - 2 (A_{h_1} + A_{h_2} + 1 - \gamma_\phi)
     =   - (A_{H_1} + A_{H_2}).
           \label{Ag1}
    \end{eqnarray}
  Therefore, we see that two equations, (\ref{AH3}) and (\ref{Ag1}), are redundant.

  So far, we only considered a particular node.
  For each node, we can do the same calculation as above.
  As we assumed above,
  the operator associated with the $H_3$-type coupling constant appears exactly one time for each node.
  Therefore, the number of the dependent equations is 2 for each node
  and we conclude that the number of the exactly marginal operator is $2p$,
  where $p$ is the total number of the nodes.

    \begin{figure}[t]
    \begin{center}
    \includegraphics[scale=0.6]{quiver_smallloop.eps}
    \caption{{\small A quiver including ``small loop" which consists of two gauge groups and two tri(or bi)fundamentals.}}
    \label{fig:smallloop}
    \end{center}
    \end{figure}

  At this stage, let us analyze the validity of the assumption.
  We should be careful about the counting of independent $H_3$-type operators
  in the case where the quiver includes ``small loop",
  which consists of two nodes and two bi(or tri)fundamentals as depicted in Fig.~\ref{fig:smallloop}.
  One might think that the $H_3$-type operators
  from the upper node and that from the lower node are the same.
  However, we can construct two independent $H_3$-type operators in this case.
  Therefore, there are two $H_3$-type operators for two nodes of the small loop
  and the conclusion in the previous paragraph is correct.

  There is one more exceptional case which needs care.
  This is the quivers with the leg ending by the small loop, as Fig.~\ref{fig:legsmallloop}.
  Naive consideration leads to that the $H_3$-type operator for the right hand side node is
  the same as the $H_1$- or $H_2$-operator.
  It follows that the number of the exactly marginal operators is reduced by one for such a loop.
  However, this naive conclusion is wrong.
  Indeed, we can see this quiver as the right hand side quiver in Fig.~\ref{fig:legsmallloop}.
  In that, we have three different fields, a bifundamental chiral superfield of $SU(2) \times SO(3)$, $P$,
  a fundamental chiral superfield for $SU(2)$, $q$, and a chiral superfield, $R$.
  Let $\gamma_P$, $\gamma_q$ and $\gamma_R$ be the anomalous dimensions of $P$, $q$ and $R$.
  The scaling coefficients for the gauge couplings are
    \begin{eqnarray}
    A_{g_1}
     =   - \left( 2 + 2 \gamma_R + \frac{3}{2} \gamma_P + \frac{1}{2} \gamma_q \right),
           ~~~
    A_{g_2}
     =   - 2 (1 + 2 \gamma_P),
    \end{eqnarray}
  where $g_1$ and $g_2$ are the gauge couplings of the $SU(2)$ and $SO(3)$ groups.
  Also, the generic quartic superpotential from these fields is schematically
    \begin{eqnarray}
    W
     =     \lambda_1 P^4 + \lambda_2 R^4 + \lambda_3 R^2 P^2 + \lambda_4 q^2 R^2 + \lambda_5 q^2 P^2.
    \end{eqnarray}
  The scaling coefficients are
    \begin{eqnarray}
    A_{\lambda_1}
    &=&    1 + 2 \gamma_P,
           ~~~
    A_{\lambda_2}
     =     1 + 2 \gamma_R,
           ~~~
    A_{\lambda_3}
     =     1 + \gamma_P + \gamma_R,
           \nonumber \\
    A_{\lambda_4}
    &=&    1 + \gamma_q + \gamma_R,
           ~~~
    A_{\lambda_5}
     =     1 + \gamma_q + \gamma_P.
    \end{eqnarray}
  Among them, four constraints are redundant.
  Therefore, there are four exactly marginal operators and it matches with the general rule above.
  We conclude
  that the number of the exactly marginal operators are same for a class of theories which are Seiberg dual each other.

    \begin{figure}[t]
    \begin{center}
    \includegraphics[scale=0.6]{quiver_legsmallloop.eps}
    \caption{{\small The quiver including a loop at the tip of quiver (left).
             As in Fig.~\ref{fig:KWsymmetry}, this quiver can be seen as the right hand side,
             where $1$ means one chiral multiplet.}}
    \label{fig:legsmallloop}
    \end{center}
    \end{figure}

  We note that the Seiberg dual theory contains the gauge singlet, meson, for ${\mathcal N}=1$ SQCD
  and expect that there are theories with mesons for our cases.
  In the discussion of \cite{LS}, the theory containing the singlet meson was introduced and
  was expected to flow to the infrared superconformal fixed point of ${\mathcal N}=1$ SQCD with the quartic superpotential.
  In our case, the quartic superpotential for each node is
    \begin{eqnarray}
    W
     =     \frac{H_1}{2} P^4 + \frac{H_2}{2} R^4 + \frac{H_3}{2} P^2 R^2.
           \label{superpotentialH}
    \end{eqnarray}
  ($H_1$ and $H_2$ type operators will appear in the superpotentials from different nodes.)
  We can also introduce the theory with mesons which has the same chiral operators as follows:
    \begin{eqnarray}
    W
     =     N_1 P^2 + N_2 R^2 + N_3 P R - \frac{1}{2H_1} N_1^2 - \frac{1}{2 H_2} N_2^2 - \frac{1}{2 H_3} N_3^2,
    \end{eqnarray}
  where $N_{1, 2, 3}$ are independent operators
  and, in $P^2$, $R^2$ and $PR$, only the gauge indices of the node we are considering are contracted.
  We expect them to flow to the infrared superconformal fixed point of the theory with no meson (\ref{superpotentialH}),
  although there is no strong evidence to support it.
  At least the chiral ring of the theory with mesons is the same as the one without mesons.

  We have counted the number of the exactly marginal operators keeping the flavor symmetry $SU(2)^n$.
  However, as seen in subsection \ref{subsec:Gaiotto}, the flavor symmetry would be enhanced in some cases.
  The number of the exactly marginal operator keeping this enhanced symmetry could be reduced.
  Let us consider these cases here.
  The first simple example is the $SU(2)$ gauge theory with four flavors (eight fundamental chiral multiplets)
  obtained from ${\mathcal T}_{0, 4}$, where the flavor symmetry is $SO(8)$.
  As noted in subsection \ref{subsec:SCFT},
  the coupling constant of a quartic term in the superpotential is a exactly marginal coupling
  and there is one exactly marginal operator.
  Indeed, the independent quartic operator is unique in this case and
  two conditions $\beta_g = \beta_h = 0$ are linearly dependent.

  A nontrivial case is the quivers obtained from ${\mathcal T}_{1, 2}$ in Fig.~\ref{fig:KWsymmetry}.
  As analyzed in subsection \ref{subsec:Gaiotto}, these two quivers have the $USp(4) \cong SO(5)$ flavor symmetry.
  First of all, let us consider the quiver associated with the left hand side in Fig.~\ref{fig:KWsymmetry},
  which we will call as ``generalized" Klebanov-Witten theory in the following section.
  (The meaning of ``generalized" will be soon clear.)
  As we will analyze in section \ref{sec:KW}, there are two independent quartic operators:
  a bilinear of the mesonic operator and a baryonic one.
  Therefore, the most generic superpotential is
    \begin{eqnarray}
    W
     =     h_1 J_{ik} J_{jl} Q^{\mu i} Q^{\mu j} Q^{\nu k} Q^{\nu l}
         + h_2 \varepsilon_{\mu \nu \rho \sigma} \varepsilon_{ijkl} Q^{\mu i} Q^{\nu j} Q^{\rho k} Q^{\sigma l}.
           \label{superpotentialT12}
    \end{eqnarray}
  In this case, all the scaling coefficients are proportional to $1 + 2\gamma_Q$,
  where $Q$ is the anomalous dimension of $Q$.
  Therefore, there are three exactly marginal operators keeping $USp(4)$ in this theory.\footnote{
    The reader may wonder that this counting is different from that in Klebanov-Witten theory \cite{KW}.
    This is because Klebanov-Witten theory was obtained by a specific mass deformation from ${\mathcal T}_{1,2}$,
    where mass parameters are chosen as $m_1 = - m_2$, as we will see explicitly in section \ref{sec:KW}.
    In that case, the first term in (\ref{superpotentialT12}) vanishes and the flavor symmetry is enhanced to $SU(4)$.
    Hence, the number of the exactly marginal operators is reduced to two.
    This matches with the result in \cite{KW}.
    This enhancement of the flavor symmetry should be seen in the dual theory non-trivially
    although we have not find a mechanism yet.}
  On the other hand, the right hand side quiver in Fig.~\ref{fig:KWsymmetry} has different matter content.
  We will refer to this theory as dual theory in the following.
  This quiver is very similar to the one in Fig.~\ref{fig:legsmallloop}.
  Therefore, the counting of the exactly marginal operators is straightforward
  and we obtain the same answer as the generalized Klebanov-Witten theory.

  We also consider two quivers obtained from the ${\mathcal T}_{2, 0}$ theories (Fig.~\ref{fig:T02}),
  where the flavor symmetry is enhanced to $SO(2)$.
  There are six exactly marginal operators in both two theories.
  (Note that if we keep the $SU(2)^n$ flavor symmetry, that is no flavor symmetry,
  the number of the exactly marginal operators is nine.
  This is only case where the general rule for the number is incorrect.)

  Finally, we comment on the $SU(N)^p$ gauge group case.
  In the case where the Lagrangian descriptions are exist, e.g., $A_p$ and $\hat{A}_{p-1}$ theories
  \cite{Cachazo:2001gh, Cachazo:2001sg, DM}, we can follow above argument.
  There are many exactly marginal operators.\footnote{
    We thank the referee for pointing out the mistakes in the previous version.}
  We expect that the theories flow to non-trivial infrared fixed points.

\section{Generalized Klebanov-Witten theory and its dual}
\label{sec:KW}
  In this section, we consider a simple example which leads to a non-trivial Seiberg duality by the mass deformation.
  Since a theory with one gauge group does not have a non-trivial S-duality which changes matter contents,
  we consider a theory with two gauge groups $SU(2) \times SU(2)$.
  In particular, we consider a theory with genus one,
  which is specified by the generalized quivers illustrated in Fig.~\ref{fig:KW}.
  The claim is that when we consider the mass deformation for each theory, the resulting theories are Seiberg dual.

  In the following, we consider the mass deformation for each theory.
  From the quiver illustrated in the left of Fig.~\ref{fig:KW},
  we obtain the ${\mathcal N}=1$ generalized Klebanov-Witten theory
  as discussed in the previous section.
  The other quiver (the right in Fig.~\ref{fig:KW}) leads to its dual theory.
  We will see that the chiral rings of the resulting theories match non-trivially,
  e.g., a non-linear constraint for the mesons from the F-term equation is dual to a classical trivial constraint.
  Note that we analyze the classical chiral rings and classical moduli spaces.
  Since for $|M| \gg \Lambda$, the gauge group is generically broken to an abelian group for these theories,
  no non-perturbative effect will appear
  unlike the mass deformed ${\mathcal N}=1$ SQCD with $N_f = N_c + 2$ (in the dual description) \cite{Seiberg}.
  Therefore, this matching can be regarded
  as non-trivial evidence of the existence of the superconformal field theory and the Seiberg dual.
  In subsection \ref{subsec:generalizedKW} and \ref{subsec:dual},
  we analyze the generalized Klebanov-Witten theory and its dual respectively.

\subsection{Generalized Klebanov-Witten theory}
\label{subsec:generalizedKW}
  In this subsection, we analyze the classical chiral ring of the generalized Klebanov-Witten theory,
  whose gauge group is $SU(2)_1 \times SU(2)_2$ and the flavor symmetry is $USp(4)$.
  The superpotential of the ${\cal N}=2$ theory before the mass deformation is given by
    \begin{align}
    W
     =     {\rm Tr} (B_1 \phi A_1) + {\rm Tr} (B_2 \phi A_2)
         + {\rm Tr} (A_1 \tilde{\phi} B_1) + {\rm Tr} (A_2 \tilde{\phi} B_2),
    \end{align}
  where $A$ and $B$ are bifundamental chiral superfields and
  $\phi$ ($\tilde{\phi}$) is an adjoint chiral superfields of $SU(2)_1$ ($SU(2)_2$).
  By adding the following mass term
    \begin{align}
    W
     =     \frac{1}{2} m_1 {\rm Tr} \phi^2 + \frac{1}{2} m_2 {\rm Tr} \tilde{\phi}^2,
    \end{align}
  and integrating out the massive adjoint chiral superfields, we obtain
    \begin{align}
    W
     =
    &      \frac{1}{8} \left( \frac{1}{m_1} + \frac{1}{m_2} \right)
           J_{ik} J_{jl} Q^{\mu i} Q^{\mu j} Q^{\nu k} Q^{\nu l} \nonumber \\
    &    - \frac{1}{2\cdot 4!} \left( \frac{1}{m_1} - \frac{1}{m_2} \right)
           \varepsilon_{\mu \nu \rho \sigma} \varepsilon_{ijkl} Q^{\mu i} Q^{\nu j} Q^{\rho k} Q^{\sigma l}.
           \label{W_KW}
    \end{align}
  Here, for later convenience, we renamed the fields as
    \begin{align}
    (A_1, A_2, B_1, B_2) \to (Q^1, Q^2, Q^3, Q^4),
    \end{align}
  and regard the gauge group as $SO(4)$ instead of $SU(2)_1 \times SU(2)_2$.
  We label the indices of the gauge group $SO(4)$ by $\mu,\nu, \cdots$
  and that of the global $USp(4)$ group by $i,j,\cdots$.
  The invariant tensor $J$ is defined as
    \begin{align}
    J_{ij}
     =     \left( \begin{array}{cc}
           {\bf 0} & {\bf 1} \\
           {\bf -1} &  {\bf 0} \\
           \end{array}\right) , \qquad
    J^{ij}
     =     \left( \begin{array}{cc}
           {\bf 0} & {\bf -1} \\
           {\bf 1} &  {\bf 0} \\
           \end{array} \right),
    \end{align}
  where ${\bf -1},{\bf 0},{\bf 1}$ are $2\times 2$ matrices.
  For details about changing the notation from $SU(2)_1 \times SU(2)_2$ to $SO(4)$
  and about integrating out the adjoint superfields, see appendix \ref{sec:massdeformation}.
  Matter contents of this theory are summarized in Table \ref{matter_original}.

    \begin{table}
    \centering
    \begin{tabular}{c|c|c}
      & SO(4) & USp(4) \vspace{-3mm} \\
      &{\scriptsize $\mu, \nu, \cdots$} & {\scriptsize $i,j, \cdots$} \\
      \hline
      $Q^{\mu i}$ & {\bf 4}     &  {\bf 4}   \\
      $(W_{\alpha})^{\mu}{}_{\nu}$ & {\bf 6} & {\bf 1}
    \end{tabular}
    \caption{Matter contents of the generalized Klebanov-Witten theory.
     $W_{\alpha}$ is a field strength.}
    \label{matter_original}
    \end{table}

  It is known that for the $SO(4)$ gauge theory with the superpotential $W=0$,
  the independent operators in the chiral ring are \cite{IS}
    \begin{align}
    &M^{(ij)} = Q^{\mu i} Q^{\mu j}, \\
    &B = \frac{1}{4!}
    \varepsilon_{\mu \nu \rho \sigma} \varepsilon_{ijkl}
    Q^{\mu i} Q^{\nu j} Q^{\rho k} Q^{\sigma l}, \\
    &h^{[ij]}_{\alpha} = \frac{1}{2} \varepsilon_{\mu \nu \rho \sigma}
    \left( Q^{\mu i} Q^{\nu j}
    - \frac{1}{4} (Q^{\mu k} Q^{\nu l} J_{lk}) J^{ij} \right)
    W_{\alpha}{}^{\rho \sigma}, \\
    &h_{\alpha} = \frac{1}{4} \varepsilon_{\mu \nu \rho \sigma}
    J_{ij} Q^{\mu i} Q^{\nu j} W_{\alpha}{}^{\rho \sigma}, \\
    &H = \frac{1}{4} \varepsilon_{\mu \nu \rho \sigma}
    W^{\alpha}{}^{\mu\nu} W_{\alpha}{}^{\rho\sigma}, \\
    &S= {\rm Tr} W^{\alpha} W_{\alpha}.
    \end{align}
  For later convenience, we have decomposed the operator $QQW$ into $h^{[ij]}_{\alpha}$ and $h_{\alpha}$,
  which are in the irreducible representations of the global $USp(4)$ group.

  Since we actually have a non-vanishing superpotential, we have to consider the equations of motion:
    \begin{align}
    (m_1+m_2) J_{ik} J_{jl} Q^{\mu j} Q^{\nu k} Q^{\nu l}
    + \frac{1}{6}(m_1-m_2) \varepsilon_{\mu \nu \rho \sigma} \varepsilon_{ijkl}
    Q^{\nu j} Q^{\rho k} Q^{\sigma l} = 0,
    \label{eom_original}
    \end{align}
  which leads to non-trivial chiral ring relations.
  By multiplying $Q^{\mu m} J^{ni}$ to the equations of motion (\ref{eom_original}), we obtain
    \begin{align}
    (m_1+m_2) M^{mj} J_{jl} M^{ln}
     =     (m_1-m_2) B J^{mn}.
    \end{align}
  This is decomposed into the following irreducible representations of $USp(4)$:
    \begin{align}
    & (m_1+m_2) \left[ M^{ij} J_{jk} M^{kl} - \frac{1}{4} \left( M^{mn} J_{np} M^{pq} J_{qm} \right) J^{il} \right]
     =     0 \label{con5}, \\
    & (m_1+m_2) M^{ij} J_{jk} M^{kl} J_{li} = 4 (m_1-m_2) B.
    \label{con1}
    \end{align}
  The first equations show non-linear constraints on the meson operator $M^{ij}$,
  while the second equation indicates that the baryon operator $B$ is decomposed into the product of the meson operator.
  Only when $m_1-m_2=0$, the baryon operator exists in the chiral ring.

  Other non-linear constraints can be obtained by multiplying
  $\varepsilon_{\mu\kappa\lambda\tau} \varepsilon_{mnpq} Q^{\kappa n}Q^{\lambda p}Q^{\tau q}$
  to the equations of motion.
  Using identities among invariant tensors like
  $\varepsilon^{\mu\nu\rho\sigma} \varepsilon_{\mu\kappa\lambda\tau}
  = \delta^{[\nu}_{\kappa} \delta^{\rho}_{\lambda} \delta^{\sigma]}_{\tau}$,
  we obtain a constraint
    \begin{align}
    (m_1+m_2) B J_{ij} M^{jk} J_{km}
     =     \frac{1}{6} (m_1-m_2) ({\rm cof} M)_{im},
           \label{M3_original}
    \end{align}
  where cofactor ${\rm cof} M$ of the matrix $M$ is defined as
    \begin{align}
    ({\rm cof} M)_{im}
     \equiv
           \varepsilon _{ijkl} \varepsilon_{mnpq} M^{jn} M^{kp} M^{lq}.
           \label{cofM}
    \end{align}
  Since the totally antisymmetric invariant tensor $\varepsilon_{ijkl}$ can be rewritten
  in terms of the invariant tensor $J_{ij}$ as
    \begin{align}
    \varepsilon_{ijkl}
     =   - J_{i[j} J_{kl]},
    \end{align}
  (\ref{cofM}) can also be rewritten as
    \begin{align}
    ({\rm cof} M)_{im}
     =     3 J_{ij} M^{jk} J_{km} (J_{np}M^{pq}J_{qr}M^{rn}) - 6 J_{ij} M^{jk} J_{kl} M^{ln} J_{np} M^{pq} J_{qm}.
    \end{align}
  By using this identity together with (\ref{con1}), the constraint (\ref{M3_original}) becomes
    \begin{align}
    & 4 (m_1 - m_2)^2 J_{ij} M^{jk} J_{kl} M^{ln} J_{np} M^{pq} J_{qm} \nonumber \\
    & -  \left[ 2(m_1-m_2)^2 - (m_1+m_2)^2 \right]
    J_{ij} M^{jk} J_{km} (J_{np}M^{pq}J_{qr}M^{rn})
     =     0.
           \label{JMJMJMJ}
    \end{align}
On the other hand, by multiplying $J_{ri} \times J_{ls} M^{st} J_{tu}$
to (\ref{con5}), we obtain
\begin{align}
(m_1+m_2)
\left[ J_{ri} M^{ij} J_{jk} M^{kl} J_{ls} M^{st} J_{tu}
- \frac{1}{4} \left( M^{mn} J_{np} M^{pq} J_{qm} \right)
J_{rs} M^{st} J_{tu} \right] = 0.
\label{JMJMJMJ2}
\end{align}
Comparing (\ref{JMJMJMJ}) and (\ref{JMJMJMJ2}),
we obtain the following two constraints:
\begin{align}
J_{ij} M^{jk} J_{km} (J_{np}M^{pq}J_{qr}M^{rn}) = 0 \label{M1M2}, \\
J_{qi} M^{ij} J_{jk} M^{kl} J_{lm} M^{mn} J_{np} = 0 \label{M3},
\end{align}
  for generic masses.
  Since we are assuming that $m_1\neq 0$, $m_2\neq 0$ in order that we can integrate out the adjoint field,
  (\ref{JMJMJMJ}) and (\ref{JMJMJMJ2}) cannot be identical constraints.
  Only the special case is $m_1+m_2=0$, where the constraints (\ref{JMJMJMJ2}) vanish and (\ref{JMJMJMJ}) becomes
\begin{align}
J_{ri} M^{ij} J_{jk} M^{kl} J_{ls} M^{st} J_{tu}
- \frac{1}{2} \left( M^{mn} J_{np} M^{pq} J_{qm} \right)
J_{rs} M^{st} J_{tu} = 0.
\end{align}

If we impose the former constraints (\ref{M1M2}),
the latter (\ref{M3}) can be derived from
(\ref{JMJMJMJ2}), which originates in (\ref{con5}).
Thus, the independent constraints for generic masses are (\ref{con5}) and (\ref{M1M2}).

The usual classical constraints
\begin{align}
B^2 = \frac{1}{4!} \det M
\end{align}
can be derived from (\ref{con1}) and (\ref{M3_original}),
and does not lead to a new constraint for the meson operator.

In summary, when the masses are generic,
the independent chiral operators are
$
M^{ij}, h^{[ij]}_{\alpha}, h_{\alpha}, H, S,
$
and the non-linear constraints for the meson operator are
\begin{align}
(MJM)^{ij} - \frac{1}{4} {\rm Tr} (MJMJ) J^{ij} = 0,
\label{sum1} \\
{\rm Tr} (MJMJ) M^{ij} = 0. \label{sum2}
\end{align}
Other non-linear constraints for operators
including $W_{\alpha}$ also exist,
but we do not analyze them here.

\subsection{Dual theory}
\label{subsec:dual}
  In this subsection, we consider the dual of the generalized Klebanov-Witten theory,
  whose generalized quiver is illustrated in Fig.~\ref{fig:KWsymmetry}.

  The superpotential of this dual ${\cal N}=2$ theory before the mass deformation is given by
    \begin{align}
    W
     =     P^a{}_b{}^{\dot{a}} \phi ^b{}_c P^c{}_a{}^{\dot{b}} \varepsilon_{\dot{a}\dot{b}}
         + P^a{}_b{}^{\dot{a}} \varepsilon_{\dot{a}\dot{b}} \tilde{\phi} ^{\dot{b}}{}_{\dot{c}} P^b{}_a{}^{\dot{c}}
         + q^{\dot{a}}{}_I \varepsilon_{\dot{a}\dot{b}} \tilde{\phi}^{\dot{b}}{}_{\dot{c}} q^{\dot{c}}{}_I,
    \end{align}
  where $\phi$ is the adjoint field of $SU(2)_1$ and $\tilde{\phi}$ is the adjoint field of $SU(2)_2$.
  By adding the following mass term
    \begin{align}
    W
     =     \frac{1}{2} m_1 {\rm Tr} \phi^2 + \frac{1}{2} m_2 {\rm Tr} \tilde{\phi}^2,
    \end{align}
  and integrating out the massive adjoint fields, we obtain
    \begin{align}
    W
     =     &-\frac{1}{2} \left( \frac{1}{m_1} + \frac{1}{m_2} \right)
           P^{a}{}_b{}^{\dot{a}} P^{b}{}_a{}^{\dot{b}} \varepsilon_{\dot{b} \dot{c}}
           P^{c}{}_d{}^{\dot{c}} P^{d}{}_c{}^{\dot{d}} \varepsilon_{\dot{d} \dot{a}}
           \nonumber \\
    &    - \frac{1}{m_2}
           q^{\dot{a}}_I \varepsilon_{\dot{a}\dot{b}} P^{a}{}_b{}^{\dot{b}}
           P^{b}{}_a{}^{\dot{c}} \varepsilon_{\dot{c}\dot{d}} q^{\dot{d}}_I - \frac{1}{2m_2}
           q^{\dot{a}}_I \varepsilon_{\dot{a}\dot{b}} q^{\dot{b}}_J
           q^{\dot{c}}_J \varepsilon_{\dot{c}\dot{d}} q^{\dot{d}}_I.
    \end{align}
  Matter contents of this mass deformed dual theory is summarized in Table \ref{matter_dual}.

\begin{table}
\centering
\begin{tabular}{c|cc|c}
 & SU(2) & SU(2) & SO(5) \vspace{-3mm} \\
 & {\scriptsize $a,b,\cdots$} & {\scriptsize $\dot{a},\dot{b},\cdots$}
 & {\scriptsize $I,J,\cdots$} \\
\hline
$P^{a}{}_b{}^{\dot{a}}$   & {\bf 3} & {\bf 2} & {\bf 1} \\
$q^{\dot{a}}{}_{I}$  & {\bf 1} & {\bf 2} & {\bf 5} \\
$W_{\alpha}{}^a{}_b$ & {\bf 3} & {\bf 1} & {\bf 1} \\
$w_{\alpha}{}^{\dot{a}}{}_{\dot{b}}$
& {\bf 1} & {\bf 3} & {\bf 1} \\
\end{tabular}
\caption{Matter contents of the dual theory.}
\label{matter_dual}
\end{table}

As discussed in appendix \ref{sec:chiraloperators}, the independent generators
of the chiral ring of this theory for generic masses are
\begin{align}
&\tilde{M}_{IJ} = q^{\dot{a}}{}_{I} \label{M_IJ}
\varepsilon_{\dot{a} \dot{b}} q^{\dot{b}}{}_{J}, \\
& \tilde{h}^{I}_{\alpha} = q^{\dot{a}}_I \varepsilon_{\dot{a} \dot{b}}
P^{a}{}_b{}^{\dot{b}} (W_{\alpha})^{a}{}_{b}, \\
& \tilde{h}_{\alpha} = P^{a}{}_b{}^{\dot{a}} \varepsilon_{\dot{a}\dot{b}}
P^{b}{}_c{}^{\dot{b}} (W_{\alpha})^c{}_a
\,\, \sim \,\, P^{a}{}_b{}^{\dot{a}} P^{b}{}_a{}^{\dot{b}}
\varepsilon_{\dot{b}\dot{c}} (w_{\alpha})^{\dot{c}}{}_{\dot{a}}, \\
& S_1 = {\rm Tr} \,\, W^{\alpha} W_{\alpha}, \\
& S_2 ={\rm Tr} \,\, w^{\alpha} w_{\alpha}, \label{S_2}
\end{align}
  where we used the equations of motion:
    \begin{align}
    &-2 \left( \frac{1}{m_1} + \frac{1}{m_2} \right)
    P^{b}{}_a{}^{\dot{b}} \varepsilon_{\dot{b} \dot{c}}
    P^{c}{}_d{}^{\dot{c}} P^{d}{}_c{}^{\dot{d}} \varepsilon_{\dot{d} \dot{a}}
    - \frac{2}{m_2}
    q^{\dot{b}}_I \varepsilon_{\dot{b}\dot{a}}
    P^{b}{}_a{}^{\dot{c}} \varepsilon_{\dot{c}\dot{d}} q^{\dot{d}}_I = 0, \label{_eom1_dual}\\
    &- \frac{2}{m_2}
    \varepsilon_{\dot{a}\dot{b}} P^{a}{}_b{}^{\dot{b}}
    P^{b}{}_a{}^{\dot{c}} \varepsilon_{\dot{c}\dot{d}} q^{\dot{d}}_I - \frac{2}{m_2}
    \varepsilon_{\dot{a}\dot{b}} q^{\dot{b}}_J
    q^{\dot{c}}_J \varepsilon_{\dot{c}\dot{d}} q^{\dot{d}}_I = 0. \label{_eom2_dual}
    \end{align}

These operators match to those of the original theory as
\begin{align}
M^{ij} & \sim (\Gamma^{IJ})^i{}_k J^{kj} \tilde{M}_{IJ}, \\
h^{[ij]}_{\alpha} & \sim \tilde{h}^{I}_{\alpha}
(\Gamma^{I})^i{}_k J^{kj}, \\
h_{\alpha} & \sim \tilde{h}_{\alpha}, \\
H, S &\sim S_1, S_2,
\end{align}
where $\Gamma^{I}$ is the gamma matrices for the $Spin(5)$ group,
and chosen such that
$$ (\Gamma^{I})^{(i}{}_k J^{|k|j)} = 0,
\qquad {\rm Tr} \Gamma^I = 0.$$
Certain linear combinations of $S_1$ and $S_2$ correspond to
$H$ and $S$, which cannot be determined from the global charge.
Only when $m_1+m_2=0$, an operator
\begin{align}
P^4 = P^{a}{}_b{}^{\dot{a}} P^{b}{}_a{}^{\dot{b}} \varepsilon_{\dot{b} \dot{c}}
P^{c}{}_d{}^{\dot{c}} P^{d}{}_c{}^{\dot{d}} \varepsilon_{\dot{d} \dot{a}}
\end{align}
appears in the classical chiral ring.
This operator is expected to correspond to the baryon operator $B$ of
the generalized Klebanov-Witten theory.
This matching of the operators in the classical chiral ring
gives a non-trivial consistency check to the duality.

We further investigate the non-linear constraints of the meson
operator.
From the definition of the meson operator $\tilde{M}_{IJ}$,
we obtain non-linear constraints
\begin{align}
\tilde{M}_{I[J} \tilde{M}_{KL]} = 0,
\label{MM}
\end{align}
which follow from the identity
$q^b{}_{[J} q^c{}_K q^d{}_{L]} =0 $,
where the gauge indices of $SU(2)_1$ run $a,b,c=1,2$.
This can also be rewritten as
\begin{align}
\varepsilon^{IJKLM} \tilde{M}_{JK} \tilde{M}_{LM} = 0.
\label{MM0}
\end{align}
Further non-linear constraints can be obtained from
equations of motion (\ref{_eom1_dual}) and (\ref{_eom2_dual}).
By multiplying $P^c{}_d{}^{\dot{c}}P^d{}_c{}^{\dot{d}}$
to the second equations of motion (\ref{_eom2_dual}),
and by using (\ref{P4_qPPq}) and (\ref{qPPq_MM}) in appendix \ref{sec:chiraloperators},
which are also derived from the equations of motion,
we finally obtain
\begin{align}
m_1 (\tilde{M}_{KL} \tilde{M}_{LK}) \tilde{M}_{IJ}
= 2(m_1+m_2) \tilde{M}_{IK} \tilde{M}_{KL} \tilde{M}_{LJ}.
\label{MMM1}
\end{align}
By multiplying $\tilde{M}_{JK}$ to (\ref{MM}),
we find
\begin{align}
(\tilde{M}_{KL} \tilde{M}_{LK}) \tilde{M}_{IJ}
= 2 \tilde{M}_{IK} \tilde{M}_{KL} \tilde{M}_{LJ}.
\label{MMM2}
\end{align}
Comparing (\ref{MMM1}) and (\ref{MMM2}), we obtain
\begin{align}
(\tilde{M}_{KL} \tilde{M}_{LK}) \tilde{M}_{IJ} = 0 \label{MMM3},\\
\tilde{M}_{IK} \tilde{M}_{KL} \tilde{M}_{LJ} = 0 \label{MMM4},
\end{align}
for generic masses.
If we impose the former constraint (\ref{MMM3}),
the latter (\ref{MMM4}) can be derived from
(\ref{MMM2}), which originates in (\ref{MM0}).
Thus, independent constraints are (\ref{MM0}) and (\ref{MMM3}).

  In summary, the non-linear constraints for the meson operator are
    \begin{align}
    \varepsilon^{IJKLM} \tilde{M}_{JK} \tilde{M}_{LM}
     =     0,
           \\
    ({\rm Tr} \tilde{M}^2) \tilde{M}_{IJ}
     =     0.
    \end{align}
  These are equivalent constraints as (\ref{sum1}) and (\ref{sum2}).
  This matching of the non-linear constraints also indicates matching of the classical moduli space.
  This can be strong evidence of the Seiberg duality.

\section{Conclusion and discussion}
  In this paper, we have proposed a large number of new Seiberg dualities of ${\mathcal N}=1$ generalized quiver gauge theories,
  which originate in the S-dualities of ${\mathcal N}=2$ superconformal gauge theories proposed by \cite{Gaiotto}.
  By deforming ${\mathcal N}=2$ S-dual theories with adjoint mass terms, they flow to the Seiberg dual theories,
  where ${\mathcal N}=1$ superconformal field theories are realized.
  We have shown some evidence for the existence of such infrared fixed points and the Seiberg dualities.
  We have found that for generic $SU(2)^n$ quiver gauge theories,
  the numbers of the exactly marginal operators are $2n$, which are universal for the proposed Seiberg dual theories.
  We have checked that the 't Hooft anomaly matching also hold for the Seiberg dual theories.
  As a simple example, we have considered the generalized Klebanov-Witten theory and its dual theory
  and demonstrated that chiral operators match between these two theories.
  We have also shown the matching of non-linear constraints for meson operators.

  In this paper, we have concentrated on the generalized $SU(2)^p$ quiver gauge theories.
  However, it is very interesting problem to generalize the gauge group to generic $SU(N)$.
  It would be possible to deform them to ${\mathcal N}=1$ (see \cite{TW} for related discussion)
  and discuss new Seiberg dualities of $SU(N)$ quiver gauge theories.
  This generalization is quite a non-trivial task
  because the S-dualities of generic $SU(N)$ gauge theory is totally different
  from that of $SU(2)$ as discussed in \cite{Gaiotto}.
  For example, the S-dualities for $SU(3)$ quiver gauge theories are based on the Argyres-Seiberg duality \cite{AS},
  where the $E_6$ superconformal field theory appears, whose $SU(2)$ subgroup is gauged.
  Since the explicit Lagrangian descriptions for the S-dual theories are not known,
  corresponding deformation as our analysis is not straightforward.
  It would also be interesting to generalize to ${\mathcal N}=1$ $SO$-$USp$ quiver gauge theories,
  by using the S-dualities in ${\mathcal N}=2$ superconformal $SO$-$USp$ quiver theories analyzed in \cite{Tachikawa}.

  For the $SU(N)$ gauge group, there are ${\cal N}=1$ SCFTs \cite{ArDo,TeYa} which
  are obtained by adding a superpotential ${\rm Tr} \Phi^N$ to the
  Argyres-Douglas ${\cal N}=2$ SCFT, even if there are no flavors.
  We expect that non-trivial ${\cal N}=1$ SCFTs can also be obtained from
  the ${\cal N}=2$ SCFTs considered in \cite{Gaiotto}
  by adding a superpotential ${\rm Tr} \Phi^N$.
  It will be interesting to study properties of
  this type of SCFT.

\section*{Acknowledgements}
  We would like to thank Yuji Tachikawa for useful comments.
  Discussions during the YITP workshop YITP-W-09-04 on
  ``Development of Quantum Field Theory and String Theory'' were useful to complete this work.
  The research of K.~M.~is supported in part by JSPS Research Fellowships
  for Young Scientists. S.~T.~is partly supported by the Japan Ministry
  of Education, Culture, Sports, Science and Technology.
  M.~T.~is supported by JSPS Grant-in-Aid for Creative Scientific Research, No. 19GS0219.

redpajama_set_name: RedPajamaArXiv


3.The technology can be adjusted according to customer's raw material.
Applicable materials:EPDM, PE, POE, EVA, PVC, materials plate/sheet etc.
Products:car roor, car trunk plate, car mats, car interiors, EVA sheet, insulation sheet for vehicle, interior sound insulation pad, car carpet, front enclosed sheet and the side enclosed sheet;etc.


This post will cover the integration of a Nimble Storage Fibre Channel array in a VMware ESXi environment. The steps are fairly similar to integrating a Nimble iSCSI array but there are some more FC specific settings which need to be set.
First, go ahead and create a new volume on your array. Go to Manage -> Volumes and click on New Volume. Specify the Volume Name, Description and select the proper Performance Policy for proper block alignment. Next, select the initiator group which has the information of your ESXi host. If you don't have an initiator group yet, click on New Initiator Group.
Name your new initiator group and specify the WWNs of your ESXi hosts. This will allow your hosts to connect to the newly created volume.
Also, specify a unique LUN ID. In this case, I have assigned LUN ID 87.
Next, specify the size and reservation settings for the volume.
Specify any protection schedule if required and click on Finish to create the volume.
Now, the volume is created on the array and your initiators are set-up to allow connection from your FC HBA on the host to connect.
After a rescan of the FC HBA, I can see my LUN with the ID 87.
Looking at the path details for LUN 87, you can 8 paths (2HBA's x 4 Target Ports). The PSP should be set to NIMBLE_PSP_DIRECTED.
I have 4 Active(I/O) paths and 4 Standby paths. The Active(I/O) paths are going to the active controller and the Standby paths are for the standby controller.
On the array I can now see all 8 paths under Manage -> Connections.
The volume can now be used as a Raw Device Map or a Datastore. Those were all steps required to get your FC array connected to the an ESXi host, once your zones on your FC switches are configured.
Some of the images have been provided by Rich Fenton, one of Nimble's Sales Engineer from UK.


New! Move in Ready! This lovely 3 bedroom home features a large family room with lots of light. The fully equipped kitchen showcases new, energy efficient kitchen appliances (including refrigerator), beautiful wood cabinets, and stunning granite countertops. The spacious master suite features a large walk in closet and spacious private bath. This home is a must see!
I was searching for a Property and found this listing (MLS® #1354963). Please send me more information regarding 6333 Hibiscus, New Braunfels, TX, 78132-5339. Thank you!
I'd like to request a showing of 6333 Hibiscus, New Braunfels, TX, 78132-5339 (MLS® #1354963). Thank you!


text:
Biologists Ask For Global System To Monitor Biodiversity Loss
An international team of biologists has proposed a global monitoring system that measures and reports on biodiversity change or loss,
AsianScientist (Jan. 21, 2013) – An international team of biologists has proposed a global monitoring system that measures and reports on biodiversity change or loss.
In an article published in Science on Friday, 30 researchers led by Henrique Miguel Pereira, from the Center for Environmental Biology of the University of Lisbon, proposed a global biodiversity monitoring system based on a set of essential variables.
Examples of essential biodiversity variables (EBVs) include the genetic diversity of wild, crop and domestic species, the population abundances of representative groups of species (such as birds, and threatened and problem plants and animals), the cover and three-dimensional structure of habitats, and nutrient use in sensitive ecosystems.
Co-author Associate Professor Melodie McGeoch of Monash University's School of Biological Sciences said that over the past 20 years, biodiversity loss has continued at an alarming rate, but there are critical gaps in scientific knowledge.
"Informed policy decisions are essential to a sustainable future, and a globally harmonized system for monitoring essential components of biodiversity is needed to achieve this," she said.
Lead author Dr. Pereira said it was essential that developing countries are included in the development of a truly global biodiversity monitoring system.
"The biggest gaps in biodiversity monitoring occur in developing countries, in regions receiving some of the largest environmental pressures, and many of these pressures are caused upstream by developed countries," Pereira said.
Source: Monash University; Photo: Tang Yew Chung/Asian Scientist Magazine.
#Australia #Australian Museum #Biodiversity #CSIRO #Japan #Kyoto University #Monash University #Nature
The Secret Sex Lives Of Koalas GPS-enabled tracking collars have given scientists a sneak peak into the little known mating habits of koala bears.
A New Measure Of Biodiversity The CANAPE model unites genetic and spatial information, providing a more comprehensive understanding of biodiversity.
Researchers Discover Genetic Loci Linked To Rheumatoid Arthritis The largest international study to date into the genetic basis of rheumatoid arthritis has shed light on how existing drugs target it.
Study: Human Embryonic Stem Cells Acquire Mutations From Long-Term Culture A large international stem cell collaboration has identified a portion of the genome mutated during long-term culture of human embryonic stem cells.
22nd Annual Australian Museum Eureka Prizes Celebrates That 'Eureka' Moment The world-class achievements of Australia's top scientists were celebrated at the 2011 Australian Museum Eureka Prizes.

redpajama_set_name: RedPajamaCommonCrawl


text:
What are GMOs and are they okay to eat? What does organic farming look like? How are animals like chickens or cattle raised?
snapAG is a series of resources that invite students to explore the hot topics affecting the agriculture industry today. Topics range from organics, biotechnology, GMOs, livestock, and more.
Explore what's trending in agriculture in Canada by browsing the topics below.
animals antibiotics biosecurity chickens cows environment food hormones housing pigs transportation vaccines
Learn about Agriculture Animal Care
What is antibiotic resistance? How does it affect animal health?
What is antibiotic resistance?
Bacteria continuously find ways to grow even when an antibiotic is present. This process is known as antibiotic resistance. Every time a new antibiotic is created, bacteria eventually find a way to defeat it. The existence of antibiotic resistance means that more illnesses are becoming harder to treat.
Not all antibiotics are equal!
Some antibiotics are more powerful than others and many of powerful antibiotics are not approved for use in animals in order to reduce the risk of the most important and powerful antibiotics developing resistance. One category of antibiotics (e.g. ionophores) are not used at all in human medicine, and are often used preventatively in animals. This is important because previous experience has shown that when preventative antibiotics are banned and animals get sick, a higher dose of a more powerful antibiotic is required.
Download pdf to Learn More
Related SnapAG Articles
Grass-Fed and Grain-Finished Beef
Pesticides – What and why?
Today's Farm
Aquaculture in Canada
Websites to Investigate This Topic Further
Alberta Beef Producers
AntibioticAwareness.ca
Chicken Farmers of Canada
Dairy Farmers of Canada
The Food Dialogues
K. Bhullar et al. (2012). "Antibiotic Resistance is Prevalent in an Isolated Cave Microbiome." PLoS ONE 7(4):e347953.
Health Agency of Canada. (July 26, 2007). "Canadian Integrated Program for Antimicrobial Resistance Surveillance (CIPARS)."
Government of Canada. (November 15, 2013). "FoodNet Canada."
Government of Canada. (February 23, 2018). "Canadian Antimicrobial Resistance Surveillance System 2017 Report."
Consultation with Dr. Betty Althouse, Chief Veterinary Officer, Saskatchewan Ministry of Agriculture, November 2017.
H.S Hurd et al. (2004). "Public Health Consequences of Macrolide Use in Food Animals: A Deterministic Risk Assessment." Journal of Food Protection 67: 980-992.
Bhullar, K. et al. (2012). "Antibiotic Resistance is Prevalent in an Isolated Cave Microbiome." PLoS ONE 7(4):e347953.
Hurd, H.S. et al. (2004). "Public Health Consequences of Macrolide Use in Food Animals: A Deterministic Risk Assessment." Journal of Food Protection 67: 980-992.
Farmer with beef cattle, Page 1 – Canada Beef
Piglets, Page 1 – Farm & Food Care Photo Library
Sheep, Page 2 – Farm & Food Care Saskatchewan
All Snap AG Categories
Farming Basics
Farming & The Environment
Growing Crops
animals , antibiotics


text:
Home Cruise Ships
Condo tower to swallow up popular Toronto restaurant and rowhouses
in Cruise Ships
Developers are finding new ways to pack on the density in a housing-starved Toronto, like a new development application calling for a condo tower that would float above an existing student rental building in the Church-Wellesley Village neighbourhood.
The proposal from Podium Developments calls for the redevelopment of 396-398 Church Street, currently home to Kinton Ramen and Kinka Izakaya locations, a trio of rowhouses at 73-77 McGill Street, and a portion of an existing student rental building at 50 Gerrard Street East.
In place of the current buildings, Podium seeks rezoning and an Official Plan Amendment to build a 43-storey condo tower with a height of 141 metres, designed by architects IBI Group.
The development portion of the affected addresses includes the single-storey restaurant building, three-storey houses, and a northern section of the eight-storey student rental, leaving the majority of the building intact.
The latter involves an unusual arrangement where strata rights were purchased from the owner of the rental building in order to allow the new condos to float above its northern end.
A total gross floor area of 28,998 square metres would be almost entirely dedicated to residential space, with 28,782 square metres supporting 463 condominium units proposed in a mix of 47 studios, 229 one-bedrooms, 147 two-bedrooms, and 40 three-bedrooms units.
The remaining 216 square metres of space is proposed as retail fronting Church Street, though it's still far too early to determine which retailer would follow in the footsteps of the popular ramen and Izakaya joints currently operating at this address.
Residents of the tower will be treated to some pretty spectacular city views through a planned 43rd-storey outdoor amenity area at the north side of the building, measuring 153.2 square metres, or 1,649.3 square feet.
For those shy of extreme heights, an even larger outdoor amenity area of 308.6 square metres/ 3,321.9 square feet is located much closer to the ground on the third floor.
window.fbAsyncInit = function () { FB.init({ appId: "100616073343311", xfbml: true, version: "v2.5" });
resolve(); } });
Tags: 396churchstreettoronto
Sad End to Woman Overboard P&O Cruises Ship
Carnival Kids Menus · Prof. Cruise
Royal Caribbean Advises Guests on Parking Issues at Florida Port
Unfinished 9,000-passenger ship Global Dream 'could be completed for Disney Cruise Line'
Cruise Ship Out of Water – How Does It Look?
If the War in Ukraine Goes Nuclear, What Would it Do to the Oceans?


Oh Heck! No I'm not blaspheming, merely telling you about who I'm collaborating on with this post. HECK are probably best known for their fab sausages with flavours you can swear by (and names that will make you smirk) however they've gone green and now have a new range of veggie burgers, bangers and balls all officially approved by the Vegetarian Society. Currently exclusive to Waitrose, as well as being meat free they are also dairy free, gluten free and low in fat. Retailing at £3 for a pack of 9 the Super Greens veggie balls are made with quinoa, spinach, kale and ginger whilst the 9 Thai Bites Veggie balls are made with fragrant thai pesto, sweet potato & sticky rice. Both pack a real punch in terms of flavour.
Following the excitement of Christmas you could be mistaken for thinking that all goes quiet with regards to foodie celebrations however there's pretty much always a day/month/occasion to be celebrated. Amongst the somewhat tenuous Nutella Day, Toast Day, Strawberry Day and Yorkshire Pudding Day, January and February have already brought us Veganuary, Burns Night, Chinese New Year, Valentines Day and Pancake Day. March kicks off with St David's Day and Fairtrade Fortnight with 17th March bringing us to St Patrick's Day. National day of all things green (and black). So when Heck asked me to get creative with their Super Greens there was only one way to go - add more green! Pinterest is full of green food to celebrate the luck of the Irish however much of it owes its vibrancy to copious amounts of artificial food colourings. None of those in this recipe - nature really does have the best colour pallet.
Heck suggest serving up the Super Greens with a crisp beansprout and cashew nut salad #chopchop and stir frying the Thai Bites with bok choi and a sprinkle of chilli #chopsticksoptional I'm also going to give Kate at Veggie Desserts recipe a try for Vegan Noodle Salad with Peanut Sauce and Thai Bites. To add further greenness to my super greens I used avocado, soya beans, thai green curry paste, coriander, cucumber and mint. Oh and a green plate. If you have never grilled (or BBQ'd) avocado before, you are in for an absolute treat. The texture of charred on the outside and smooth and creamy inside is simply divine. As for grilled Halloumi - hot squeaky cheese is my idea of absolute heaven. Without further ado grab your green balls and a pint of the black stuff and give 'em Heck this St Patrick's Day.
If using wooden skewers, soak in water for 30 minutes.
De-stone and cut your Avocado into quarters. Peel and cut each quarter into three.
Cut the Halloumi cheese into bite sized cubes.
Carefully thread the Super Greens, Avocado and Halloumi onto skewers.
To make the Raita grate the cucumber on the largest holes of a grater and place in a bowl along with the yoghurt. Season well with salt and pepper and stir through the cumin seeds and chopped mint.
The skewers can be cooked under the grill, on a griddle pan or if weather permits, on the BBQ.
Cook them on a medium heat for 7-10 minutes turning occasionally until golden brown and warmed through.
Meanwhile stir fry the rice along with the Thai green curry paste and soya beans. When heated through mix in the chopped coriander.
Warm the flatbreads in the oven or toaster and serve filled with the Thai Green Rice, Super Greens, Halloumi and Avocado Skewers and Cucumber & Mint Raita on the side.
Disclosure: This is a commissioned recipe for Heck. As always, all views expressed are my own.
I love anything with halloumi cheese so these are right up my street. They look delicious!
There is something very addictive about Halloumi, I can't get enough of its squeaky goodness!
I've tried the Oh Heck sausages and they are good, so I'm thrilled to see some veggie offerings from them. Love the flavours you've added - it all makes for a very green plate.
Yup I'm a fan of their fab sausages too - especially the chicken ones. Their veggie offerings are equally good.
I love the Heck range and this is my kind of recipe. Push the plate this way! Stumbled, yummed and pinned.
I'm afraid the plate is now well and truly empty Jac, but I have some more in my freezer!
Fantastic colour. So lovely to see some veggie items in their range. Looks delicious.
No need for artificial colouring when you've got nature's paintbox at your disposal.
I impressed myself with the combination of textures and flavours in this one. Green is most definitely good.
Apologies if I shocked you with my colourful language ;-) Do keep an eye out for Heck, they have a fab range of products both meaty and veggie.
Such beautiful green skewers! I love the addition of those Heck veggie balls and the haloumi cheese. Yum!
The addition of Haloumi Cheese makes anything better!
Why is it so much more exciting to eat something off a stick I wonder??
Yum! I love halloumi and of course I love avocado so this sounds delicious to me!
Doesn't everyone love avocado these days or are there some haters out there?!
Wow! These Super Greens look super healthy and yummy! I love that you used halloumi as well! One of my favs!
Super Green. Super Healthy. Super Yummy.
That could indeed be their tag line!


text:
Q: Exception (stacktrace) not available in springboot unit test I am starting to use spring boot framework and stumble on every second step.
Currently I have a situation which i absolutely do not understand.
I am using spring boot 2.3.5.RELEASE building with maven
    <properties>
        <java.version>11</java.version>
        <maven.compiler.source>${java.version}</maven.compiler.source>
        <maven.compiler.target>${java.version}</maven.compiler.target>
    </properties>

    <parent>
        <groupId>org.springframework.boot</groupId>
        <artifactId>spring-boot-starter-parent</artifactId>
        <version>2.3.5.RELEASE</version>
        <relativePath/>
        <!-- lookup parent from repository -->
    </parent>

    <dependencies>
        <dependency>
            <groupId>org.springframework.boot</groupId>
            <artifactId>spring-boot-starter-web</artifactId>
        </dependency>

        <!-- https://mvnrepository.com/artifact/org.hibernate.validator/hibernate-validator -->
        <dependency>
            <groupId>org.hibernate.validator</groupId>
            <artifactId>hibernate-validator</artifactId>
            <version>6.1.6.Final</version>
        </dependency>

        <!-- OpenAPi (swagger)-->
        <dependency>
            <groupId>org.springdoc</groupId>
            <artifactId>springdoc-openapi-ui</artifactId>
            <version>1.2.32</version>
        </dependency>

    </dependencies>

I created a rather simple class RESTCommunication. No @Component, @Controller or similar.
public class RESTCommunication {
    private final Logger m_logger = LoggerFactory.getLogger(this.getClass());
    private String m_contentType;
    private String m_path;
    private String m_body;

    private HashMap<String,String> m_urlParams;
    private HashMap<String,String> m_queryParams;

    public RESTCommunication() {
    m_logger.debug("RESTCommunication constructor");

    m_scheme    = RESTCommunication.SCHEME_DEFAULT;
    m_authority = RESTCommunication.AUTHORITY_DEFAULT;
    m_port      = RESTCommunication.PORT_DEFAULT;
    m_path      = RESTCommunication.PATH_DEFAULT;

    m_urlParams = new HashMap<>();
    // m_queryParams = new HashMap<>();  // I know that this is the reason for the exception
}

@Override
public String toString() {
    StringBuilder sbToString = new StringBuilder();
    sbToString.append(buildURI());
    sbToString.append("|numURLParams:").append(m_urlParams.size());
    sbToString.append("|numQueryParams:").append(m_queryParams.size());  // This has to fail because m_queryParams is not initialized
    return sbToString.toString();
}
    .....

For this class i created a very simple test class RESTCommunicationTest
@SpringBootTest
public class RESTCommunicationTest {
    private final Logger m_logger = LoggerFactory.getLogger(this.getClass());

    @Test
    public void defaultInitTest() {
        RESTCommunication restComm = new RESTCommunication();

        String restCommURL = restComm.toString();  // Expect the nullpointer INSIDE the toString method
        String expectedURL = sbExpected.toString();

        assertEquals(expectedURL, restCommURL, "Expected URL does not match generated");
    }

I run the test with the command: $./mvnw -e -X package -Dtest=RESTCommunicationTest#defaultInitTest
And as expectable i see the Nullpointer in the stdout logging.
BUT in the logging I find this:
webServerFactoryCustomizerBeanPostProcessor
websocketServletWebServerCustomizer
welcomePageHandlerMapping
[ERROR] Tests run: 1, Failures: 0, Errors: 1, Skipped: 0, Time elapsed: 3.532 s <<< FAILURE! - in org.highpots.hippo.core.communication.RESTCommunicationTest
[ERROR] defaultInitTest  Time elapsed: 0.399 s  <<< ERROR!
java.lang.NullPointerException
    at org.ilovespringboot.not.RESTCommunicationTest.defaultInitTest(RESTCommunicationTest.java:36)

2020-12-08 22:18:10.810  INFO 10381 --- [extShutdownHook] o.s.s.concurrent.ThreadPoolTaskExecutor  : Shutting down ExecutorService 'applicationTaskExecutor'
[INFO]
[INFO] Results:
[INFO]
[ERROR] Errors:
[ERROR]   RESTCommunicationTest.defaultInitTest:36 » NullPointer
[INFO]
[ERROR] Tests run: 1, Failures: 0, Errors: 1, Skipped: 0

A nullpointer in RESTCommunicationTest.java:36. Absolutely no hint that the real NullPointer happens in RESTCommunication class.
Can someone explain my that behaviour? Am I doing something wrong?
I know I am some kind of programming dinosaurs. (java projects are about 10+ years in the past). But I would expect to get a hint where the exception is thrown. I would expect a stacktrace with detailed caused by sections.
If can not imagine this is not normal behavior. Because that would mean I have to find all errors by mind-stepping through the complete source... This is not possible
Tanks for your help in advance
Harri E

A: Maven uses surefire plugin to run tests.To see Exception stacktrace you must set trimStackTrace to false in surefire plugin configuration.
Historically surefire plugin use to print complete stacktrace. Though the information in the stack trace was useful for debugging, stack traces took up rather a lot of space in the console output. Particularly if there were a number of failing tests it became easy to get lost in the noise.
Since version 2.13 surefire just prints summary by default.
    <build>
      <plugins>
        <plugin>
            <groupId>org.apache.maven.plugins</groupId>
            <artifactId>maven-surefire-plugin</artifactId>
            <version>${your.maven.surefire.version.here}</version>
            <configuration>
                <trimStackTrace>false</trimStackTrace>
            </configuration>
        </plugin>
      </plugins>
    </build>

redpajama_set_name: RedPajamaStackExchange


"Mom, why do sunflowers grow so tall, "
A chocolate brown horse galloped over the hill towards them.
so we can't eat them when their seeds are ready.
Survival of the fittest, I'd say." responded the horse.
for her son to consider other points of view and to form his own conclusion.
examining them from every angle.
He considered that they, like the tree, were there to provide shade.
so their seeds could be warmed by the sun.
The mother cow praised her young son for thinking such fine thoughts.
She asked him how he was like the sunflower.
He first laughed, and told her was not at all like the sunflower.
He had four legs, and he could move around as he pleased.
so he didn't need to reach high towards the sun.
He provided shade for no one.
and proudly share their gifts with the world.
They are reminders of how wonderful and temporary life is."
and fully breath in the fresh air around you while you are young.
Later you will be wise, yet fat like me.
Her son turned and trotted towards the sunflowers again.
and was eager to try to make a little more sense of the world.


ONE of Brendan Rodgers' first acts as Celtic manager was to have the fences around the first team training pitches at Lennoxtown covered with a protective screen to prevent any unwelcome onlookers from seeing what he was working on with his players.
That has not, however, stopped prying eyes in the sky from having an illicit peek at Rodgers putting his charges through their paces.
The former Watford, Reading, Swansea City and Liverpool manager clearly doesn't share the indignation of many, not least Frank Lampard at Derby County, in the game down south at the actions of his Leeds United counterpart Marcelo Bielsa.
A member of the Argentinian's staff was caught at the Derby training ground last Thursday ahead of a Championship game at Elland Road the following evening – which leaders Leeds won 2-0.
Bielsa, who is now under investigation by the English Football League, revealed on Wednesday that he had spied on every opposing team in the second tier this season.
Yet, Rodgers isn't, unlike Lampard and a lot of others in England, of the opinion that Bielsa has been disrespectful or unethical and revealed that he would fully expect such conduct from overseas coaches.
"It's an interesting one," he said. "He did his best to try and defend it. Marcelo Bielsa is someone I've admired for a long, long time. His methods were renowned long before he came to this country.
"Frank caught someone in the bushes. Things happen in football. But Bielsa's an international manager and it's not illegal. I don't think it's unsporting. I tend not to think about that kind of thing too much.
If Airdrieonians had despatched an undercover operative to covertly watch Celtic training yesterday in an attempt to gain an advantage ahead of their William Hill Scottish Cup fourth round tie at Parkhead tomorrow evening they would have discovered a fair few things.
They would also have seen that Burke and Weah, who have joined the Scottish champions on loan until the end of the 2018/19 campaign from West Brom and Paris Saint-Germain respectively, are ready to make their debuts this weekend.
Likewise, a spy would have realised that Rodgers himself, after reports he is on the radar of Premier League club Leicester City, remains very much in situ. "It (Leicester) is a fantastic club, but at this moment in time my concentration is purely on Celtic and looking forward to the second part of this season," he said.
It would also have been apparent to any passer-by that neither Kieran Tierney, the Scotland left back who has been sidelined with a hip injury since the Europa League game against Salzburg last month, or Odsonne Edouard, who has been struggling with an abductor strain, are still not fit.
Asked if he thought Tierney would be available for the Europa League last 32 double header against Valencia next month, his manager said: "That's too far ahead to say. We'll just have to see over the next few weeks.
"But we've got cover with Emilio. We have options and Emilio is a very good and experienced cover player for us. He will play at the weekend.


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Our General Meeting is held on the 3rd Sunday of the month at 2 PM (except January, February, and March when they begin at 1 PM following Pancake Breakfasts). The April General meeting will be held on the 2nd Sunday at 2 PM due to Easter. The June General Meeting will be held on the 2nd Sunday due to the ESP Pull and Father's Day. The Directors' Meeting is held on the 2nd Saturday of the month except in September and December (at 2 PM for January, February, March, and April; at 7 PM other months) in the club House at 10400 Gillate Rd. You are encouraged to attend either or both meetings. Informed members make for a better club. Bring a dish to pass and stay after the general meeting for a potluck meal.
The purpose of the men and women of the Western New York Gas & Steam Engine Association is to preserve and educate all those it can about antique internal engines, steam traction tractors & engines, and a variety of engines and equipment representing yesteryear.