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%   Three Dimensional Nonlinear Sigma Models
%   in the Wilsonian Renormalization Method
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%   by K. Higashijima and E. Itou
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%   1st draft 2003/04/21 by E. Itou
%   2nd draft 2003/04/22 by K. Higashijima
%   3rd draft 2003/04/23 by K. Higashijima
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OU-HET 439\\
{\tt hep-th/0304194}\\
April 2003
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{\LARGE\bf
Three Dimensional Nonlinear Sigma Models
in the Wilsonian Renormalization Method
}
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{\renewcommand{\thefootnote}{\fnsymbol{footnote}}
{\large\bf Kiyoshi Higashijima\footnote{
     E-mail: {\tt higashij@phys.sci.osaka-u.ac.jp}} and
 Etsuko Itou\footnote{
     E-mail: {\tt itou@het.phys.sci.osaka-u.ac.jp}}
}}

\vspace{4mm}

{\sl
Department of Physics,
Graduate School of Science, Osaka University,\\ 
Toyonaka, Osaka 560-0043, Japan \\
}

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\begin{abstract}
The three dimensional nonlinear sigma model is unrenormalizable in perturbative method. By using the $\beta$ function in the nonperturbative Wilsonian renormalization group method, we argue that ${\cal N}=2$ supersymmetric nonlinear $\sigma$ models are renormalizable in three dimensions.
When the target space is an Einstein-K\"{a}hler manifold with positive scalar curvature, such as C$P^N$ or $Q^N$, there are nontrivial ultraviolet (UV) fixed point, which can be used to define the nontrivial continuum theory. 
If the target space has a negative scalar curvature, however, the theory has only the infrared Gaussian fixed point, and the sensible continuum theory cannot be defined. We also construct a model which interpolates between the C$P^N$ and $Q^N$ models with two coupling constants. This model has two non-trivial UV fixed points which can be used to define the continuum theory. 
Finally, we construct a class of conformal field theories with ${\bf SU}(N)$ symmetry, defined at the fixed point of the nonperturbative $\beta$ function. These conformal field theories have a free parameter corresponding to the anomalous dimension of the scalar fields. If we choose a specific value of the parameter, we recover the conformal field theory defined at the UV fixed point of C$P^N$ model and the symmetry is enhanced to ${\bf SU}(N+1)$.

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\section{Introduction}
Nonlinear sigma models (NL$\sigma$Ms) are renormalizable in two dimensions. They are unrenormalizable, however, in three dimensions in the perturbation theory. Therefore, we have to use nonperturbative methods to study the renormalizability of NL$\sigma$Ms. One of such nonperturbative methods is the large-N expansion which has been applied to some examples of NL$\sigma$Ms\cite{Arefeva}. For example, the $U(N)$ invariant NL$\sigma$M, in which field variables take values on the complex projective space $CP^{N-1}$, is renormalizable in the leading and next-to-leading orders of the large-N expansion\cite{Inami}. In fact, the next-to-leading contribution to the $\beta$ function for the C$P^N$ model with ${\cal N}=2$ supersymmetry vanishes in the $1/N$ expansion\cite{Inami}.


The Wilsonian renormalization group (WRG) offers another powerful tool suitable for the nonperturbative study. The $2$- or $3$-dimensional NL$\sigma$M describes the theory of superstring or supermembrane, and the target spaces of these theories correspond to the real space-time. Since the consistency of these models requires conformal invariance, it is important to investigate the fixed point of NL$\sigma$M with ${\cal N}=2$ supersymmetry. In a previous paper, we used the WRG method to show that the fixed points of the $2$-dimensional NL$\sigma$Ms provide novel conformal field theories whose target spaces are non-Ricci flat \cite{SU2dim}. These target spaces are interpreted as consistent backgrounds of superstrings in the presence of the dilaton \cite{KKL}.

In the WRG approach, the renormalizability of NL$\sigma$Ms is equivalent to the existence of the nontrivial continuum limit $\Lambda \rightarrow \infty$. When the UV cutoff $\Lambda$ tends to infinity, we have to fine tune the coupling constant to the critical value at the ultraviolet (UV) fixed point, so as to keep the observable quantities to be finite. Therefore, it is important to show the existence of the UV fixed point without using the perturbation theory. 

The WRG equation describes the variation of the general effective action when the cutoff scale is changed. The most general effective action depends on infinitely many coupling constants, and in practice, we have to introduce some kind of symmetry and truncation to solve the WRG equation. The simplest truncation is the local potential approximation, in which only the potential term without derivative is retained. In ${\cal N}=2$ supersymmetric NL$\sigma$Ms we are going to discuss, the field variables take values in complex curved spaces called K\"{a}hler manifolds,  whose metrics are specified completely by the K\"{a}hler potentials. Since the higher derivative terms are irrelevant in infrared (IR) region, we assume that we can safely drop such terms. Furthermore, we assume that ${\cal N}=2$ supersymmetry is maintained in the renormalization procedure.  Then, our action is completely fixed by the K\"{a}hler potential. With these assumptions, we have already derived the WRG equation for $3$-dimensional NL$\sigma$M \cite{HI}, and the $\beta$ function for the target metric is 
\beq
\beta (g_{i \bar{j}})=\frac{1}{2 \pi^2} R_{i \bar{j}}+\gamma [\varphi^k g_{i \bar{j},k} +\varphi^{* \bar{k}}g_{i \bar{j}, \bar{k}}+2 g_{i \bar{j}}  ]+\frac{1}{2}[\varphi^k g_{i \bar{j},k} +\varphi^{* \bar{k}}g_{i \bar{j}, \bar{k}}].
\eeq
In this paper, we investigate $3$-dimensional ${\cal N}=2$ supersymmetric NL$\sigma$M using this equation.

First, we study renormalization group flows for some simple models.
One of them is the theory whose target space is the Einstein-K\"{a}hler manifold, including the C$P^N$ and $Q^N$ models.
We will show that if the target manifold has a positive scalar curvature, the theory has a nontrivial UV fixed point together with an IR fixed point at the origin. It is possible to take the continuum limit by using the UV fixed point, so that these NL$\sigma$M are renormalizable in three dimensions, at least in our truncation method. We also study a model whose target manifold is not Einstein-K\"{a}hler manifold. This model has two coupling constant, and connects C$P^N$ and $Q^N$ models through the renormalization group flow. The theory spaces are divided into four phases.


Next, we construct the conformal field theories at the fixed point of the nonperturbative $\beta$ function in more general theory spaces. To simplify, we assume ${\bf SU}(N)$ symmetry for the theory, and obtain a class of ${\bf SU}(N)$ invariant conformal field theories with one free parameter. If we choose a specific value of the free parameter, we obtain a conformal field theory defined at the UV fixed point of C$P^N$ model and the symmetry is enhanced to ${\bf SU}(N+1)$ in this case.
The conformal field theory reduces to a free field theory if the parameter is set equal to zero, therefore the free parameter describes a marginal deformation from the IR to UV fixed points of the C$P^N$ model in the theory spaces.





This paper is organized as follows:
In \S \ref{review}, we have short a review of the derivation of the WRG equation for NL$\sigma$M.
In \S \ref{E-K} and \S \ref{RGflow}, we study renormalization group flow for some models.
We construct a class of conformal field theories which has ${\bf SU}(N)$ symmetry in \S \ref{solution}.
Such class has one free parameter corresponding to the anomalous dimension of the scalar field.
In \S \ref{corres-CPN}, we show that the conformal theory is equal to UV fixed point of C$P^N$ model, if we take a specific value of the parameter.

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\section{Wilsonian Renormalization Group (WRG) equation}\label{review}

In general, WRG eq. shows the variation of the effective action $S$ when the cutoff energy scale $\Lambda$ is changed to $\Lambda (\delta t)=\Lambda e^{-\delta t}$ in $D$ dimensional field theory \cite{Wilson Kogut, Wegner and Houghton, Morris, Aoki}:
\beq  
\frac{d}{dt}S[\Omega; t]&=&\frac{1}{2\delta t} \int_{p'} \tr \ln \left(\frac{\delta^2 S}{\delta \Omega^i \delta \Omega^j}\right)\nonumber\\
&&-\frac{1}{2 \delta t}\int_{p'} \int_{q'} \frac{\delta S}{\delta \Omega^i (p')} \left(\frac{\delta^2 S}{\delta \Omega^i (p')\delta \Omega^j (q')} \right)^{-1} \frac{\delta S}{\delta \Omega^j (q')} \nonumber\\
&&+ \left[D-\sum_{\Omega_i} \int_p \hat{\Omega}_i (p) \left(d_{\Omega_i}+\gamma_{\Omega_i}+\hat{p}^{\mu} \frac{\partial}{\partial \hat{p}^{\mu}} \right) \frac{\delta}{\delta \hat{\Omega}_i (p)} \right] \hat{S},\nonumber\\ \label{WRG-1} 
\eeq
where $d_{\Omega}$ and $\gamma_{\Omega}$ denote the canonical and anomalous dimensions of the field $\Omega$, respectively.
The carat indicates dimensionless quantities.
The first and second terms in eq.(\ref{WRG-1}) correspond to the one-loop and dumbbell diagrams, respectively.
The remaining terms come from the rescaling of fields.
We always normalize the coefficient of kinetic term to unity.


This WRG equation consists of an infinite set of differential equations for various coupling constants in the most general action $S$.
In practice, we usually expand the action in powers of derivatives and retain the first few terms.
We often introduce symmetry to further reduce the number of independent coupling constants.


We consider the simplest ${\cal N}=2$ supersymmetric theory which is given by the K\"{a}hler potential term.
Then the action is written as
\beq
S&=&\int d^2 \theta d^2 \bar{\theta} d^3 x K[\Phi, \Phi^\dag]\nonumber\\
&=&\int d^3 x \Bigg[g_{n \bar{m}}\left(\partial^{\mu} \varphi^n \partial_{\mu} \varphi^{* \bar{m}} +\frac{i}{2} \bar{\psi}^{\bar{m}} \sigma^{\mu}(D_{\mu} \psi)^n +\frac{i}{2} \psi^{n} \bar{\sigma}^{\mu}(D_{\mu} \bar{\psi})^{\bar{m}} +\bar{F}^{\bar{m}} F^{n}\right) \nonumber\\
&&-\frac{1}{2} K_{,nm \bar{l}} \bar{F}^{\bar{l}} \psi^n \psi^m -\frac{1}{2} K_{,n \bar{m} \bar{l}} F^{n} \bar{\psi}^{\bar{m}} \bar{\psi}^{\bar{l}}+\frac{1}{4} K_{,nm \bar{k} \bar{l}} (\bar{\psi}^{\bar{k}} \bar{\psi}^{\bar{l}})(\psi^n \psi^m)\Bigg],\label{action}
\eeq
where $\Phi^n$ denotes chiral superfields, whose components fields are complex scalars $\varphi^n (x)$, Dirac fermions $\psi^n (x)$ and complex auxiliary fields $F^n (x)$.
In the usual perturbative method, this model is unrenormalizable.
Then we examine it using WRG equation, which is one of nonperturbative methods.


When we substitute this action for eq.(\ref{WRG-1}), the one-loop correction term cannot be written in covariant form \cite{HI}.
We use the K\"{a}hler normal coordinates (KNC) to obtain a covariant expression for the loop correction term
{\renewcommand{\thefootnote}{\fnsymbol{footnote}}{\footnote[1]{
The coordinate transformation from the original coordinate to KNC is holomorphic\cite{HN}.
The Jacobian for path integral measure cancels between bosons and fermions\cite{HN2}.}.



Finally, we obtain the WRG equations for the scalar part as follows:
\beq
&&\frac{d}{dt}\int d^3 x g_{i \bar{j}} (\partial_\mu \varphi)^i (\partial^\mu \varphi^*)^{\bar{j}}\nonumber\\
&&=\int d^3 x \Big[-\frac{1}{2 \pi^2} R_{i \bar{j}} \nonumber\\
&&-\gamma \Big(\varphi^k g_{i \bar{j},k}+ \varphi^{*\bar{k}}g_{i \bar{j},\bar{k}} +2g_{i \bar{j}} \Big) -\frac{1}{2} \Big( \varphi^k g_{i \bar{j},k}+ \varphi^{*\bar{k}}g_{i \bar{j},\bar{k}} \Big) \Big] (\partial_\mu \varphi)^i  (\partial^\mu \varphi^*)^{\bar{j}}, \nonumber\\
\eeq
where the scalar fields $\varphi^n (x)$ are assumed to be independent of $t$ through a suitable rescaling, which introduces the anomalous dimension $\gamma$.
From this WRG equation, the $\beta$ function of K\"{a}hler metric is
\beq
\frac{d}{dt}g_{i \bar{j}}&=&-\frac{1}{2 \pi^2}R_{i \bar{j}} -\gamma \Big[\varphi^k g_{i \bar{j},k} +\varphi^{* \bar{k}}g_{i \bar{j},\bar{k}}+2g_{i \bar{j}} \Big] -\frac{1}{2} \Big[\varphi^k g_{i \bar{j},k} +\varphi^{* \bar{k}}g_{i \bar{j},\bar{k}} \Big]  \nonumber\\
&\equiv&-\beta (g_{i \bar{j}}).\label{beta}
\eeq

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\section{Einstein-K\"{a}hler manifolds}\label{E-K}
Let us consider the theories whose target spaces are Einstein-K\"{a}hler manifolds.
The Einstein-K\"{a}hler manifolds satisfy the condition
\beq
R_{i \bar{j}}=\frac{h}{a^2}g_{i \bar{j}},\label{EKcond}
\eeq
where $a$ is the radius of the manifold, which is related to the coupling constant $\lambda$ by
\beq
\lambda =\frac{1}{a}.
\eeq
A special class of Einstein-K\"{a}hler manifolds is called the Hermitian symmetric space, if it is a symmetric coset space ($G/H$), namely, if the coset space is invariant under a parity operation. If the manifold is the Hermitian symmetric space, the positive constant $h$ in eq.(\ref{EKcond}) is the eigenvalue of the quadratic Casimir operator in  the adjoint representation of global symmetry $G$, as shown in Table $1$.

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	\begin{table}[h]
	
	\begin{center}
	\begin{tabular}{|c|c|c|}
	\hline
	G/H                              & Dimensions (complex)  &$h$\\
	\hline \hline
	$SU(N)/[SU(N-1) \otimes U(1)]=$C$P^{N-1}$   & $N-1$        & $N$\\
	$SU(N)/[SU(N-M) \otimes U(M)]$   & $M(N-M)$                & $N$\\
	$SO(N)/[SO(N-2) \otimes U(1)]=Q^{N-2}$   & $N-2$           & $N-2$\\
	$Sp(N)/U(N)$                     & $\frac{1}{2}N(N+1)$   & $N+1$\\
	$SO(2N)/U(N)$                    & $\frac{1}{2}N(N+1)$   & $N-1$\\
	$E_{6}/[SO(10) \otimes U(1)]$    &$16$                     &$12$\\
	$E_{7}/[E_6 \otimes U(1)]$        &$27$                     &$18$\\
	\hline
	\end{tabular}
	\caption{The values of $h$ for Hermitian symmetric spaces}
	\end{center}
	\end{table}
	
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When the manifolds have the radius $a=\frac{1}{\lambda}$, the scalar part of the SNL$\sigma$M Lagrangian can be represented in the following form:
\beq
{\cal L}_{scalar}&=&g_{i \bar{j}}(\varphi,\varphi^*) \partial_{\mu} \varphi^i \partial^{\mu} \varphi^{* \bar{j}} \nonumber\\
&\stackrel{\varphi,\varphi^* \approx 0}{\longrightarrow}& \frac{1}{\lambda^2} \delta_{i \bar{j}} \partial_{\mu} \varphi^i \partial^{\mu} \varphi^{* \bar{j}}.\label{lag-1}
\eeq
To normalize the kinetic term, we rescale the scalar fields as follows:
\beq
\varphi \rightarrow \tilde{\varphi}=\frac{1}{\lambda} \varphi.\label{rescale-1}
\eeq
Then, the Lagrangian (\ref{lag-1}) has the normalized kinetic term
\beq
{\cal L}_{scalar}=\tilde{g}_{i \bar{j}}(\lambda \tilde{\varphi},\lambda \tilde{\varphi}^*) \partial_{\mu} \tilde{\varphi}^i \partial^{\mu} \tilde{\varphi}^{* \bar{j}}
\eeq
with
\beq
\tilde{g}_{i \bar{j}}|_{\tilde{\varphi},\tilde{\varphi}^*=0}=\delta_{i \bar{j}}.
\eeq
Rescaling the WRG Eq.(\ref{beta}) and comparing the coefficient of $\partial_{\mu} \tilde{\varphi}^i \partial^{\mu} \tilde{\varphi}^{* \bar{j}}$, we have
\beq
\frac{\partial}{\partial t} \tilde{g}_{i \bar{j}} (\lambda \tilde{\varphi},\lambda \tilde{\varphi}^*)&=&-\frac{1}{2\pi^2} \tilde{R}_{i \bar{j}} \nonumber\\
&&-\gamma [\tilde{\varphi}^k \tilde{g}_{i \bar{j},k }+\tilde{\varphi}^{* \bar{k}} \tilde{g}_{i \bar{j},\bar{k}}+2 \tilde{g}_{i \bar{j}}]-\frac{1}{2}[\tilde{\varphi}^k \tilde{g}_{i \bar{j},k }+\tilde{\varphi}^{* \bar{k}} \tilde{g}_{i \bar{j},\bar{k}}],\nonumber
\eeq
where $\tilde{R}_{i \bar{j}}$ is the rescaled Ricci tensor and can be written
\beq
\tilde{R}_{i \bar{j}}=h\lambda^2 \tilde{g}_{i \bar{j}}
\eeq
using the Einstein K\"{a}hler condition (\ref{EKcond}).


Because only $\lambda$ depends on $t$, this differential equation can be rewritten as
\beq
\frac{\dot{\lambda}}{\lambda} \tilde{\varphi}^k \tilde{g}_{i \bar{j},k } +\frac{\dot{\lambda}}{\lambda} \tilde{\varphi}^{* \bar{k}} \tilde{g}_{i \bar{j},\bar{k}}&=&-\left(\frac{h \lambda^2}{2\pi^2}+2\gamma  \right) \tilde{g}_{i \bar{j}}-(\gamma+\frac{1}{2})[\tilde{\varphi}^k \tilde{g}_{i \bar{j},k }+\tilde{\varphi}^{* \bar{k}} \tilde{g}_{i \bar{j},\bar{k}}].\nonumber\\. 
\eeq
Because the left-hand side vanishes for $\varphi,\varphi^* \approx 0$, the coefficient of $\tilde{g}_{i \bar{j}}$ must vanish on the right-hand side. Thus, we obtain the anomalous dimension of scalar fields (or chiral superfields) as
\beq
\gamma=- \frac{h \lambda^2}{4\pi^2}.\label{gamma}
\eeq
Comparing the coefficient of $\tilde{\varphi}^k \tilde{g}_{i \bar{j},k }$ (or $\tilde{\varphi}^{* \bar{k}} \tilde{g}_{i \bar{j},\bar{k}}$), we also obtain the $\beta$ function of $\lambda$:
\beq
\beta(\lambda)&\equiv&-\frac{d \lambda}{dt}=-\frac{h}{4\pi^2}\lambda^3+\frac{1}{2} \lambda .\label{beta}
\eeq
We have a IR fixed point at 
\beq
\lambda=0,
\eeq
and we also have a UV fixed point at 
\beq
\lambda^2=\frac{2 \pi^2}{h}\equiv \lambda_c^2,
\eeq
for positive $h$. Therefore, if the constant $h$ is positive, it is possible to take the continuum limit by choosing the cutoff dependence of the bare coupling constant as
\beq
\lambda(\Lambda) \stackrel {\Lambda \rightarrow \infty}{\longrightarrow} \lambda_c-\frac{M}{\Lambda},\label{continuum}
\eeq
where $M$ is a finite mass scale\footnote{Since our WRG equation is derived by using KNC, the renormalizable Lagrangian has to be written by using the KNC.}.

When the constant $h$ is positive, the target manifold is compact Einstein-K\"{a}hler manifold \cite{Page and Pope}.
In this case, the anomalous dimension at the fixed points are given by
\beq
\gamma_{IR}&=&0 :\mbox{IR fixed point (Gaussian fixed point)}\\
\gamma_{UV}&=&-\frac{1}{2} :\mbox{UV fixed point}
\eeq
At UV fixed point, the scaling dimension of the scalar fields ($x_{\varphi}$) is canonical plus anomalous dimension:
\beq
x_{\varphi}&\equiv& d_{\varphi} + \gamma_{\varphi}=0.
\eeq
Thus the scalar fields and the chiral superfields are dimensionless in the UV conformal theory as in the case of two dimensional field theories.
Above the fixed point, the scalar fields have mass, and the symmetry restores \cite{HKNT,HIT}.

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\section{Renormalization group flows}\label{RGflow}
In this section, we study the renormalization group flows for three examples, C$P^N$, $Q^N$ and a new model.

\subsection{C$P^{N}$ and $Q^N$ models}
\begin{enumerate}
	\item C$P^N$ model: $SU(N+1)/[SU(N) \otimes U(1)]$\\
	Consider the following $SU(N+1)$ symmetric K\"{a}hler potential using $(N+1)$-dimensional homogeneous coordinates:
	\beq
	K[\Phi, \Phi^\dag]&=& \frac{1}{\lambda^2} \ln ( |\Phi^1|^2 + \cdots +|\Phi^N|^2 +|\Phi^{N+1}|^2).
	\eeq
	The complex projective space, C$P^N$, is defined by identifying two point related by
	\beq
	\Phi^i &\sim& a \Phi^i, \hspace{1cm} (i=1, \cdots, N+1),
	\eeq
	where $a$ is a complex chiral superfield, so that this is a complexified gauge symmetry. 	We obtain the K\"{a}hler potential for C$P^N$ model by choosing a gauge $\Phi^{N+1}=1$ as
	\beq
	K[\Phi, \Phi^\dag]=\frac{1}{\lambda^2} \ln (1+ \vec{\Phi}\vec{\Phi}^{\dag}),
	\eeq
	where $\vec{\Phi}$ denotes a set of chiral superfields $\vec{\Phi}=(\Phi^1, \cdots ,\Phi^{N})$.
	Hereafter, we rescale scalar fields 
	\beq
	\varphi \rightarrow \tilde{\varphi} =\frac{1}{\lambda} \varphi\label{rescale},
	\eeq
	to normalize the kinetic term,
	and simply write the rescaled scalar fields $\tilde{\varphi}$ as $\varphi$.  
	
	From this K\"{a}hler potential, we can obtain the K\"{a}hler metric and Ricci tensor:
	\beq
	g_{i \bar{j}}&\equiv& \partial_i \partial_{\bar{j}}K= \Big( \frac{\delta_{i \bar{j}}}{1+\lambda^2 \vec{\varphi} \vec{\varphi}^*}- \frac{\lambda^2 \varphi^i \varphi^{* \bar{j}}}{(1+\lambda^2 \vec{\varphi} \vec{\varphi}^*)^2} \Big),\\
	R_{i \bar{j}}&\equiv&-\partial_{\bar{j}} \partial_i (\ln \det g_{k \bar{l}})=(N+1) \lambda^2 g_{i \bar{j}}.\label{CPN-Ricci}
	\eeq
	Equation (\ref{CPN-Ricci}) shows that this target manifold is Einstein-K\"{a}hler manifold with $h=N+1$.
	Hence eqs. (\ref{gamma}) and (\ref{beta}) give us immediately
	\beq
	\gamma&=&-\frac{(N+1)\lambda^2}{4 \pi^2},\\
	\beta (\lambda)&=& -\frac{(N+1)\lambda^3}{4 \pi^2}+\frac{1}{2}\lambda.
	\eeq
	This $\beta$ function is consistent with the large $N$ analysis \cite{Inami}.
	
	\item $Q^N$ model: $SO(N+2)/[SO(N) \otimes SO(2)]$\\
	Another example of the Einstein-K\"{a}hler manifold is the coset manifold $SO(N+2)/[SO(N) \otimes SO(2)]$ called $Q^N$.	We consider the K\"{a}hler potential with homogeneous $(N+2)$-dimensional coordinates:
	\beq
	K[\Phi,\Phi^\dag]=\frac{1}{\lambda^2} \ln ( |\Phi^1|^2 + \cdots +|\Phi^N|^2 +|\Phi^{N+1}|^2+|\Phi^{N+2}|^2).
	\eeq
	Now, we impose two conditions, identification and $O(N)$ symmetric conditions:
	\beq
	\Phi^i &\sim& a \Phi^i, \hspace{1cm} (i=1, \cdots, N+2),\\
	(\Phi^1)^2 + &\cdots& +(\Phi^N)^2 +(\Phi^{N+1})^2 +(\Phi^{N+2})^2=0,
	\eeq
	on the K\"{a}hler potential.
	By these condition, the dimensions of target space becomes $N$, and the K\"{a}hler potential for $Q^N$ can be rewritten as \footnote{We choose the same gauge as the one in the next subsection.}
	\beq
	K[\Phi,\Phi^\dag]=\frac{1}{\lambda^2} \ln \Big( 1+\vec{\Phi} \vec{\Phi}^\dag +\frac{1}{4} \vec{\Phi}^2 \vec{\Phi}^{\dag 2} \Big),
	\eeq
	where $\vec{\Phi}=(\Phi^1, \cdots ,\Phi^N)$.
	
	Hereafter we use rescaled fields (\ref{rescale}).
	From this K\"{a}hler potential, the K\"{a}hler metric and Ricci tensor are given by
	\beq
	g_{i \bar{j}}&=&\frac{\delta_{i \bar{j}}}{1+\lambda^2 \vec{\varphi} \vec{\varphi}^* +\frac{1}{4} \lambda^4 \vec{\varphi}^2 \vec{\varphi}^{* 2}}\nonumber\\
	&&+\frac{\lambda^2 \varphi^i \varphi^{* \bar{j}} \left(1+\lambda^2 \vec{\varphi} \vec{\varphi}^* \right)-\lambda^2 \left(\varphi_i^* \varphi_{\bar{j}} + \frac{1}{2}\lambda^2 \vec{\varphi}^2 \varphi_i^* \varphi^{* \bar{j}} + \frac{1}{2}\lambda^2 \vec{\varphi}^{* 2} \varphi^i \varphi_{\bar{j}} \right) }{ \left(1+\lambda^2 \vec{\varphi} \vec{\varphi}^* +\frac{1}{4}\lambda^4 \vec{\varphi}^2 \vec{\varphi}^{* 2}  \right)^2 }  , \label{Q-metric}\nonumber\\
	\\
	R_{i \bar{j}}&=&N \lambda^2 g_{i \bar{j}} \label{Q-Ricci}.
	\eeq
	Equation (\ref{Q-Ricci}) shows that this manifold is also an Einstein-K\"{a}hler manifold with $h=N$.
	Employing the same argument as in the case of the C$P^N$ model, we obtain the anomalous dimension and $\beta$ function for the coupling constant:
	\beq
	\gamma&=&-\frac{N \lambda^2}{4 \pi^2},\\
	\beta (\lambda)&=& -\frac{N \lambda^3}{4 \pi^2}+\frac{1}{2}\lambda.
	\eeq
	
	Next example shows that there are renormalization group flows which connect C$P^N$ and $Q^N$ models.
	
	\end{enumerate}
	
	\subsection{A new model}
	
	\begin{enumerate}
	\item Construction
	
	Again, we consider the K\"{a}hler potential with homogeneous $(N+2)$-dimensional coordinates:
	\beq
	K[\Phi,\Phi^\dag]=\frac{1}{\lambda^2} \ln ( |\Phi^1|^2 + \cdots +|\Phi^N|^2 +|\Phi^{N+1}|^2+|\Phi^{N+2}|^2).\label{homo-kahler}
	\eeq
	As in C$P^N$ and $Q^N$ models, we identify two points related by
	\beq
	\Phi^i &\sim& a \Phi^i. \hspace{1cm} (i=1, \cdots, N+2) \label{identify}
	\eeq
	Now we deform the $O(N)$ symmetric condition to
	\beq
	b[(\Phi^1)^2 + &\cdots& +(\Phi^N)^2] +(\Phi^{N+1})^2 +(\Phi^{N+2})^2=0,\label{deform-condition}
    \eeq
	where $b$ is an arbitrary complex parameter.
	
	
	\begin{enumerate}
		\item $b=0$ case: \\
		The deformed condition (\ref{deform-condition}) is rewritten as
		\beq
		(\Phi^{N+1})^2 +(\Phi^{N+2})^2 =0.
		\eeq
		We fixed $\Phi^{N+1}$ and $\Phi^{N+2}$ by using the two conditions (\ref{identify}) and (\ref{deform-condition}) as follows:
		\beq
		\Phi^{N+1}&=&\frac{1}{{\sqrt{2}}},\\
		\Phi^{N+2}&=&\pm \frac{i}{\sqrt{2}}.
		\eeq
		Substituting these values for the K\"{a}hler potential (\ref{homo-kahler}), we obtain the K\"{a}hler potential of C$P^N$:
		\beq
		K[\Phi, \Phi^\dag]=\frac{1}{\lambda^2} \ln (1 + |\Phi^1|^2 + \cdots +|\Phi^N|^2).
		\eeq
		
		Thus, the target space is double cover of C$P^N$ located at $\Phi^{N+2}= \pm \frac{i}{\sqrt{2}}$.
		This target manifold has isometry $SU(N+1)$.
		
		
		\item $b \neq 0$ case: \\
		Using the two conditions, we can choose a gauge
		\beq
		\Phi^{N+1}+i \Phi^{N+2}=\sqrt{2},
		\eeq
		and
		\beq
		\Phi^{N+1} -i \Phi^{N+2}=\frac{-b}{\sqrt{2}} \Big( (\Phi^1)^2 +\cdots +(\Phi^{N})^2 \Big).
		\eeq
		Then, the K\"{a}hler potential is rewritten
		\beq
		K[\Phi, \Phi^\dag]=\frac{1}{\lambda^2} \ln \Big( 1+|\Phi^1|^2 +\cdots +|\Phi^{N}|^2 +\frac{|b|^2}{4} | \sum^N_{i=1} (\Phi^i)^2 |^2  \Big).\label{new-kahler} \nonumber\\
		\eeq
		If we take $|b|=1$, this K\"{a}hler potential is equal to that of $Q^N$ model.
		Thus for this special value of $b$, the target manifold has isometry $SO(N+2)$.
		
		
		\item $b=\infty$ case: \\
		The $O(N)$ symmetric condition reduces to
		\beq
		(\Phi^1)^2 + \cdots +(\Phi^N)^2=0.
		\eeq
		The remaining fields $\Phi^{N+1}$ and $\Phi^{N+2}$ can take arbitrary values.
		Using the identification condition, we can fix 
		\beq
		\Phi^{N-1} + i \Phi^{N}=\sqrt{2}.
		\eeq
		Then we obtain the K\"{a}hler potential as follow:
		\beq
		&&K[\Phi,\Phi^\dag]\nonumber\\
		&&\hspace{-1cm}=\frac{1}{\lambda^2} \ln \Big( 1+|\Phi^1|^2 +\cdots +|\Phi^{N-2}|^2 +|\Phi^{N+1}|^2 +|\Phi^{N+2}|^2 + \frac{1}{4} |\sum^{N-2}_{i=1} (\Phi^i)^2 |^2 \Big).\label{b=infty}\nonumber\\
		\eeq
		
		
		
		\end{enumerate}
	
	\item Strong-weak duality\\
	We have very interesting duality for $N=2$.
	For $b=0$, the target space is the double cover of C$P^2$.
	For $|b|=1$, the target space is $Q^2$, which is isomorphic to C$P^1 \times$ C$P^1$.
	
	
	For $b=\infty$, if we choose 
	\beq
	\Phi^1 = \frac{1}{\sqrt{2}},\hspace{1cm}\Phi^2=\pm \frac{i}{\sqrt{2}},
	\eeq
	the K\"{a}hler potential (\ref{b=infty}) reads
	\beq
	K[\Phi,\Phi^\dag]=\frac{1}{\lambda^2} \ln (1+|\Phi^3|^2+|\Phi^4|^2),
	\eeq
	which is the K\"{a}hler potential of C$P^2$.
	Therefore, the target space is again the double cover of C$P^2$, and coincides with that of $b=0$ case exactly.
	
	Let us replace the coordinates $\Phi^1, \Phi^2$ with $\Phi^3, \Phi^4$ in the constraint (\ref{deform-condition}). With this operation, the deformation parameter $b$ is replaced by $1/b$
	\beq
	b &\leftrightarrow& \frac{1}{b}.
	\eeq
Although the K\"{a}hler potential (\ref{new-kahler}) for $N=2$ has completely different form for the deformation parameter $b$ and $1/b$, these two theories are equivalent.	
	Thus, this model for $N=2$ has strong-weak duality.
	The strong coupling region of the new model with (\ref{new-kahler}) corresponds to the weak coupling region of the dual model.
	At the self-dual point $|b|=1$, we have a model on $Q^2 \simeq $ C$P^1 \times $ C$P^1$. At $b=0, \infty$, the target space of this theory is the double cover of C$P^2$.
	 
	\item Renormalization group flow
	
	Now, we will see the renormalization group flow for general value of $b$.
	We use the K\"{a}hler potential 
	\beq
	K[\Phi,\Phi^\dag]=\frac{1}{\lambda^2} \ln \Big( 1+\vec{\Phi} \vec{\Phi}^\dag + g \vec{\Phi}^2 \vec{\Phi}^{\dag 2} \Big),
	\eeq
	where $g=\frac{|b|^2}{4}$ in eq.(\ref{new-kahler}).
	This K\"{a}hler potential gives the following K\"{a}hler metric and Ricci tensor:
\beq
g_{i \bar{j}}&=&\frac{\delta_{i \bar{j}}}{1+\lambda^2 \vec{\varphi} \vec{\varphi}^* +g(t)\lambda^4 \vec{\varphi}^2 \vec{\varphi}^{*2}}\nonumber\\
&&+\frac{4 g(t)\lambda^2 \varphi^i \varphi^{*\bar{j}}(1+\lambda^2 \vec{\varphi} \vec{\varphi}^*)-\lambda^2(\varphi_i^* \varphi_{\bar{j}} +2g(t)\lambda^2 \vec{\varphi}^2 \varphi_i^* \varphi^{*\bar{j}}+2g(t)\lambda^2 \vec{\varphi}^{*2}\varphi^i \varphi_{\bar{j}})}{(1+\lambda^2 \vec{\varphi} \vec{\varphi}^* +g(t) \lambda^4 \vec{\varphi}^2 \vec{\varphi}^{*2})^2},\nonumber\\
R_{i \bar{j}}&=&(N+1)\lambda^2  g_{i \bar{j}}-\Bigg[\frac{4g(t)\lambda^2 \delta_{i \bar{j}}}{1+4g(t)\lambda^2 \vec{\varphi} \vec{\varphi}^* +g(t)\lambda^4 \vec{\varphi}^2 \vec{\varphi}^{*2}} \nonumber\\
&&+\frac{16g^2 (t)\lambda^4 \varphi^i \varphi^{* \bar{j}} \vec{\varphi} \vec{\varphi}^{*} -16 g^2 (t) \lambda^2 \varphi_i^* \varphi_{\bar{j}} -8g^2 (t)\lambda^4 \left(\vec{\varphi}^2 \varphi_i^* \varphi^{*\bar{j}}+\vec{\varphi}^{*2}\varphi^i \varphi_{\bar{j}} \right)}{(1+4g(t)\lambda^2 \vec{\varphi} \vec{\varphi}^* +g(t)\lambda^4 \vec{\varphi}^2 \vec{\varphi}^{*2})^2} \Bigg].\label{new-Ricci}\nonumber\\
\eeq
Here we use the rescaled fields as before.
Note that Eq.(\ref{new-Ricci}) shows that this manifold is not an Einstein K\"{a}hler manifold unless $g$ takes specific values: $g=0,\frac{1}{4}$.

	Substituting these metric and Ricci tensor for eq.(\ref{beta}), we obtain
	\beq
	\gamma&=&-\frac{\lambda^2}{4 \pi^2} [(N+1)-4g],\\
	\beta(\lambda)&=&-\frac{\lambda^3}{4 \pi^2}[(N+1) +8g(2g-1)]+\frac{\lambda}{2},\\
	\beta(g)&=&\frac{4 \lambda^2}{\pi^2} g^2 (4g-1).
	\eeq
	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\psfrag{g}{\Large$g$}
\psfrag{lambda}{\Large$\lambda$}
\includegraphics[width=13cm]{flow4.eps}
\caption{Renormalization group flows (The arrows point toward the infrared region.)}
\label{fig:flow}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	Figure \ref{fig:flow} shows renormalization group flow.  If we use the perturbation theory, the flow diagram is reliable only in the vicinity of the origin. Because we do not use the perturbation theory to derive the $\beta$ function, our flow diagram is reliable in the entire region. The nontrivial UV fixed points of the flow are indicated by points A and B. Any points on the $g$-axis ($\lambda=0$) are IR fixed points.  The curve BAE shows the critical line, along which the direction of the flow is tangential to the line. The lines FBG($g=0$) and CAD($g=1/4$) both represent the renormalized trajectories. The critical line intersects with the renormalized trajectories at UV fixed points. We can define the continuum theories by using these UV fixed points. In this sense, NL$\sigma$Ms are renormalizable in three dimensions, at least in our truncated WRG equation.
	
	The theory has different symmetry at the right- and left-side of the critical line.
	First, we consider continuous symmetry.
	The global symmetry on the renormalized trajectories, FBG($g=0$) and CAD($g=1/4$), is enhanced to $G=${\bf SU}$(N+1)$ and $G=${\bf SO}$(N+2)$, respectively.  In other region of the flow diagram, the global symmetry is {\bf SO}$(N)\otimes ${\bf U}$(1)$. The global symmetry is realized manifestly in the right of the critical line BAE.  At the left of the critical line, however, the enhanced global symmetries are spontaneously broken and there are Numbu-Goldstone bosons, although the {\bf SO}$(N)\otimes ${\bf U}$(1)$ symmetry remains manifest. 
	Next, we consider a discrete transformation 
	\beq
	\psi(t,x_1,x_2) &\rightarrow& \psi' (t,x_1,x_2) =\gamma^2 \psi(t, x_1,-x_2),\nonumber\\
	x_2 &\rightarrow& x_2'=-x_2. \label{parity}
	\eeq
	The Lagrangian (\ref{action}) is invariant under this transformation.
	This transformation forbids fermion mass terms, then there are spontaneous symmetry breakdown at the massive phase.
	At the right of the critical line, the fermion is massive and the discrete symmetry is spontaneously broken. At the left of the critical line, the discrete symmetry protect the fermion to be massless, and the supersymmetry keeps the bosons also massless \cite{HIT}.
	
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The SU(N) symmetric solution of WRG equation}\label{solution}
In this section, we investigate the conformal field theories defined as the fixed point of the $\beta$ function
\beq
\beta&=&\frac{1}{2 \pi^2}R_{i \bar{j}} +\gamma \Big[\varphi^k g_{i \bar{j},k} +\varphi^{* \bar{k}}g_{i \bar{j},\bar{k}}+2g_{i \bar{j}} \Big] +\frac{1}{2} \Big[\varphi^k g_{i \bar{j},k} +\varphi^{* \bar{k}}g_{i \bar{j},\bar{k}} \Big]\nonumber\\
&=&0. 
\eeq
To simplify, we assume ${\bf S}{\bf U}(N)$ symmetric K\"{a}hler potential
\beq
K[\Phi,\Phi^\dag]&=&\sum_{n=1}^{\infty} g_n (\vec{\Phi} \cdot \vec{\Phi}^\dag)^n \equiv f(x),\label{potential}
\eeq
where the chiral superfields $\vec{\Phi}$ have $N$ components, $g_n$ plays the role the coupling constant and $x \equiv \vec{\Phi} \cdot \vec{\Phi}^\dag$.
Using the function $f(x)$, we derive the K\"{a}hler metric and Ricci tensor as follow:
\beq
g_{i \bar{j}}&\equiv&\partial_i \partial_{\bar{j}} K[\Phi,\Phi^\dag]=f' \delta_{i \bar{j}}+f'' \varphi_i^* \varphi_{\bar{j}},\label{metric}\\
R_{i \bar{j}}&\equiv&-\partial_i \partial_{\bar{j}} \tr \ln g_{i \bar{j}}\nonumber\\
&=&-\Big[(N-1)\frac{f''}{f'} +\frac{2f''+f''' x}{f'+f'' x}  \Big]\delta_{i \bar{j}} \nonumber\\
&&-\Big[(N-1)\bigg(\frac{f^{(3)}}{f''}-\frac{(f'')^2}{(f')^2} \bigg)+\frac{3f^{(3)}+f^{(4)} x}{f'+f'' x} -\frac{(2f''+f'''x)^2}{(f'+f''x)^2} \Big]\varphi^*_{i}\varphi_{\bar{j}},\nonumber\\
\eeq
where 
\beq
f'=\frac{df}{dx}.
\eeq
To normalize the kinetic term, we set 
\beq
f'|_{x \approx 0}=1
\Rightarrow g_1=1. \label{normalization}
\eeq
We substitute these metric and Ricci tensor for the $\beta$ function (\ref{beta}) and compare the coefficients of $\delta_{i \bar{j}}$ and $\varphi^i \varphi^{* \bar{j}}$ respectively.
\beq
\frac{\partial}{\partial t}f'&=&\frac{1}{2\pi^2}\Big[(N-1)\frac{f''}{f'} +\frac{2f''+f''' x}{f'+f'' x}  \Big]- 2\gamma(f'+f''x)\label{f'}-f''x,\label{beta-f}\\
\frac{\partial}{\partial t}f''&=&\frac{1}{2\pi^2}\Big[(N-1)\bigg(\frac{f^{(3)}}{f''}-\frac{(f'')^2}{(f')^2} \bigg)+\frac{3f^{(3)}+f^{(4)} x}{f'+f'' x} -\frac{(2f''+f'''x)^2}{(f'+f''x)^2} \Big]\nonumber\\
&&-2\gamma(2f''+f'''x)-(f'''x+f'').\label{f''}
\eeq
The second equation (\ref{f''}) is equivalent to the derivative of the first equation with respect to $x$, so that we use only the first equation.


To obtain a conformal field theory, we must solve the differential equation
\beq
\frac{\partial}{\partial t}f'=\frac{1}{2\pi^2}\Big[(N-1)\frac{f''}{f'} +\frac{2f''+f''' x}{f'+f'' x}  \Big]- 2\gamma(f'+f''x)\label{f'}-f''x =0\label{beta=0}.
\eeq
The function $f(x)$ is a polynomial of infinite degree, and it is hard to solve it analytically. So we truncate the function $f(x)$ at order $O(x^4)$.
From the normalization (\ref{normalization}), the function $f(x)$ is 
\beq
f(x)=x+g_2 x^2 +g_3 x^3 +g_4 x^4.\label{order4}
\eeq
We substitute it for WRG eq.(\ref{beta-f}) and expand it around $x \approx 0$, then the equation (\ref{beta=0}) can be written 
\beq
\frac{\partial}{\partial t}f'&=&\frac{1}{2 \pi^2} \Big[2(N+1)g_2 + \Big( 6(N+2)g_3 -4(N+3)g_2^2 \Big)x \nonumber\\
&&-\Big(18(N+7)g_2 g_3 -8(N+7)g_2^3 -12 (N+2)g_4 \Big)x^2  \Big]\nonumber\\
&&-2\gamma(1+4g_2 x +9 g_3 x^2 )-(2g_2 x +6 g_3 x^2 ) +O(x^3)\nonumber\\
&=&0.
\eeq
We choose the coupling constants and the anomalous dimension, which satisfy this equation.
\beq
\gamma&=&\frac{N+1}{2 \pi^2} g_2,\\
g_3&=&\frac{2(3N+5)}{3(N+2)}g_2^2 +\frac{2\pi^2}{3(N+2)} g_2,\label{g_3}\\
g_4&=&3g_2 g_3 -\frac{2(N+7)}{3(N+3)}g_2^3 +\frac{\pi^2}{N+3}g_3\nonumber\\
&=&\frac{1}{3(N+2)(N+3)}\Big( (16N^2 +66N+62)g_2^3 + 2\pi^2 (6N+14)g_2^2+2\pi^4 g_2 \Big).\nonumber\\
\label{g_4}
\eeq
Note that all coupling constant is written in terms of $g_2$ only.
Similarly, we can fix all coupling constant $g_n$ using $g_2$ order by order.
The function $f(x)$ with such coupling constants describe the conformal field theory and has one free parameter $g_2$.
In other words, if we fix the value of $g_2$, we obtain a conformal field theory.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The explicit example of the novel conformal field theories}\label{corres-CPN}
For general $g_2$, the power series (\ref{order4}) is a complicated function.
In this subsection, we take a specific value to $g_2$, for which $f(x)$ takes a especially simple form.


We take 
\beq
g_2&=&-\frac{1}{2} \cdot \frac{2\pi^2}{N+1} \equiv -\frac{1}{2}a,
\eeq
where
\beq
a\equiv \frac{2\pi^2}{N+1}.
\eeq
We can express all other coupling constants from eqs.(\ref{g_3}),(\ref{g_4})
\beq
g_3&=&\frac{1}{3} a^2,\nonumber\\
g_4&=&-\frac{1}{4}  a^3,\nonumber\\
\vdots\nonumber
\eeq
These coupling constants show the function $f(x)$ is 
\beq
f(x) =\frac{1}{a } \ln (1+ a x ),\label{CP-UV}
\eeq
and this is the K\"{a}hler potential of C$P^N$ model.
In fact, the function (\ref{CP-UV}) satisfies the condition (\ref{beta=0}) exactly.

From this discussion, one of the novel ${\bf SU}(N)$ symmetric conformal field theory is equal to the UV fixed point theory of C$P^N$ model for the specific value of $g_2$.
In this case, the symmetry of this theory enhances to ${\bf SU}(N+1)$ because the C$P^N$ model has the isometry ${\bf SU}(N+1)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and Discussions}
The nonlinear $\sigma$ model in three dimensions is unrenormalizable in perturbation theory.
We discussed ${\cal N}=2$ supersymmetric NL$\sigma$Ms by using WRG equation, which is one of nonperturbative methods. 

First, we examined the sigma models whose target spaces are the Einstein-K\"{a}hler manifolds.
We have shown that the theories whose target spaces are the compact Einstein-K\"{a}hler manifolds with positive scalar curvature have two fixed points.
One of them is the Gaussian IR fixed point, while the other is the nontrivial UV fixed point. We can define the continuum limit at this UV fixed point by the fine-tuning of the bare coupling constant. In this sense, NL$\sigma$Ms on the Einstein-K\"{a}hler manifolds with positive scalar curvature are renormalizable in three dimensions. At this point, the scaling dimension of all superfields is zero, as in the two dimensional theories. On the other hand, the theories whose target spaces are the Einstein-K\"{a}hler manifolds with negative scalar curvature (for example $D^2$ with Poincar\'{e} metric) have only an Gaussian IR fixed point, and cannot have the continuum limit.

Secondly, we studied a new model with two parameters, whose target space is not the Einstein-K\"{a}hler manifolds. The theory has two nontrivial fixed points corresponding to the UV fixed points of C$P^N$ and $Q^N$ models.
In the theory spaces of this model, there are a critical surface and two renormalized trajectories, and the theory has four phases.
We have also shown that the model has strong-weak duality for $N=2$. In order to study the phase structure, we have to introduce the auxiliary fields\cite{HN3}, which is left for future work.

Finally, we constructed a class of the ${\bf SU}(N)$ symmetric conformal field theory by using the WRG equation. This has one free parameter $g_2$ corresponding to the anomalous dimension of the scalar fields. If we choose a specific value of the parameter, we recover the conformal field theory defined at the UV fixed point of C$P^N$ model and the symmetry is enhanced to ${\bf SU}(N+1)$.

We argued that the ${\cal N}=2$ supersymmetric NL$\sigma$Ms are renormalizable in three dimensions. Similar argument will be also valid for ${\cal N}=1$ supersymmetric NL$\sigma$Ms and NL$\sigma$Ms without supersymmetry. Especially, NL$\sigma$Ms for the Einstein manifolds with positive scalar curvature will be renormalizable as long as we use the Riemann normal coordinates, although it is not easy to write the explicit Lagrangian in the Riemann normal coordinates.

\section*{Acknowledgements}
This work was supported in part by Grants-in-Aid for Scientific
Research (\#13640283 and \#13135215) and Sasakawa Scientific Research Grant from The Japan Science Society. We would like to thank Muneto Nitta, Tetsuji Kimura and Makoto Tsuzuki for enlightening discussions.


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\end{document}
