We present new non-Ricci-flat K\"ahler metrics with $U(N)$ and $O(N)$ isometries as target manifolds of superconformally invariant sigma models with an anomalous dimension. They are so-called Ricci solitons, special solutions to a Ricci-flow equation. These metrics explicitly contain the anomalous dimension and reduce to Ricci-flat K\"ahler metrics on the canonical line bundles over certain coset spaces in the limit of vanishing anomalous dimension.
hep-th0411149.tex
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{\normalsize
\begin{flushright}
{\tt hep-th/0411149} \\
TIT/HEP-532\\
\end{flushright}
}

\vs{2}

\begin{center}
{\huge\bf 
Conformal Sigma Models with Anomalous Dimensions
and Ricci Solitons}

\vs{30}
{\renewcommand{\thefootnote}{\fnsymbol{footnote}}
{\Large\bf 
Muneto Nitta\footnote{
     E-mail: {\tt nitta@th.phys.titech.ac.jp}}
}}


\vs{2}

{\it Department of Physics, Tokyo Institute of 
Technology \\
Tokyo 152-8551, JAPAN}
 

\end{center}

\setcounter{footnote}{0}


%%%%%%%%% abstract %%%%%%%%

\vs{7}

\begin{abstract}

We present new non-Ricci-flat K\"ahler metrics 
with $U(N)$ and $O(N)$ isometries as target manifolds of 
superconformally invariant sigma models with 
an anomalous dimension.  
They are so-called Ricci solitons, 
special solutions to a Ricci-flow equation. 
These metrics explicitly contain the anomalous dimension 
and reduce to Ricci-flat K\"ahler metrics 
on the canonical line bundles over certain coset spaces 
in the limit of vanishing anomalous dimension. 





\end{abstract}

\end{titlepage}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
String compactification with consistent 
non-trivial background 
is one of important subjects for long time.
Sigma model approach gives 
a set of equations of motion for such backgrounds 
by vanishing condition on 
the beta function~\cite{Polchinski:1998rq}.
Calabi-Yau manifolds, Ricci-flat K\"ahler manifolds, 
are required as compactified manifolds  
with constant dilaton backgrounds.
However no explicit metrics for 
compact Calabi-Yau manifold are known. 
On the other hand, if we allow non-compact manifolds, 
some explicit metrics for non-compact 
Calabi-Yau manifolds can be 
constructed~\cite{Ca}--\cite{Nitta:2003yk}.
Explicit solutions for non-Ricci-flat K\"ahler manifold
with a non-trivial dilaton 
background were obtained by 
Kiritsis, Kounnas and Lust \cite{Kiritsis:1993pb}
as a generalization of two dimensional 
Euclidean black hole~\cite{Wi,Hori:2001ax}. 

We can consider nonlinear sigma models whose 
scalar fields have an anomalous dimension 
instead of a constant dilaton background,  
because they can be transformed to each other 
by a field redefinition for the scalar fields 
to absorb the dilaton field~\cite{Hori:2001ax}. 
Using the Wilsonian renormalization group equation, 
the beta function of the ${\cal N}=(2,2)$ 
supersymmetric nonlinear sigma models consisting of 
the complex scalar fields $(\varphi^i,\varphi^{*i})$ 
with an anomalous dimension $\gamma$ was obtained in 
\cite{Higashijima:2002mh,Higashijima:2003ki,Higashijima:2003rp} 
(see \cite{Itou:2004px} for a review) 
as\footnote{
To derive this equation, 
it was useful to expand the Lagrangian in terms of 
K\"ahler normal coordinates~\cite{KNC} which 
are natural extension of Riemann normal coordinates 
to K\"ahler manifolds.
}
\beq
- {\del g_{ij^*} \over \del t} 
 = \beta(g_{i j^*})
 = \frac{1}{2\pi} R_{i j^*} 
  + \gamma \Big(\varphi^k g_{lj^*}\Gamma^l{}_{ik} 
  + \varphi^{* k} g_{i l^*}\Gamma^{l^*}{}_{j^*k^*} 
  + 2g_{i j^*} \Big) \label{CFT}
\eeq
with $g_{ij^*}$, $R_{i j^*}$ and $\Gamma^i{}_{ij}$ being  
the K\"ahler metric, the Ricci-form and 
the connection, respectively. 
Conformally invariant models are defined by 
the condition of vanishing beta function 
\beq
 R_{i j^*} + 2 \pi \gamma \Big(\varphi^k g_{lj^*}\Gamma^l{}_{ik} 
  + \varphi^{* k} g_{i l^*}\Gamma^{l^*}{}_{j^*k^*} 
  + 2g_{i j^*} \Big) 
  =0 .
   \label{RFeq}
\eeq 
A $U(N)$-invariant solution for this equation 
was obtained in \cite{Higashijima:2003ki}. 
It was shown to be equivalent to 
the $U(N)$-invariant solution with a dilaton background 
constructed in \cite{Kiritsis:1993pb}. 
This is due to the equivalence of this model to 
the dilaton model.
%as a higher dimensional extension of 
%two dimensional Euclidean black hole~\cite{Wi,Hori:2001ax}.  
In this note we derive another $U(N)$-invariant solution 
and its extension to $O(N)$.

Eq.~(\ref{CFT}) is a so-called Ricci-flow equation 
which has attracted much attention recently in mathematics 
(see \cite{Bakas:2004hn} for a review by a physicist). 
In the Riemann manifold with a metric $g_{\mu\nu}$ 
and some vector field $\xi^{\mu}$, 
the general Ricci-flow equation is written as
\beq
 - {\del g_{\mu\nu} \over \del t}  
 = R_{\mu\nu} + \nabla_{\mu} \xi_{\nu} 
    + \nabla_{\nu} \xi_{\mu} . \label{Ricci-flow}
\eeq
If we take $\xi^{\mu} = 2\pi \gamma (\ph^i, \ph^{*i})$ 
as a special case in the K\"ahler manifold, 
the Ricci-flow equation (\ref{Ricci-flow}) 
reduces to Eq.~(\ref{CFT}).
Solutions to ${\del g_{\mu\nu} \over \del t} = 
R_{\mu\nu} + \nabla_{\mu} \xi_{\nu} 
    + \nabla_{\nu} \xi_{\mu}=0$ are called Ricci solitons 
and play a central role in classification of manifolds.
Our new solutions presented in this note 
are K\"ahler Ricci solitons and we hope that 
these solutions 
are useful for classification of K\"ahler manifolds.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$U(N)$ Invariant Model}
We prepare an $N$-vector 
$\vec{\phi} = (\phi^1, \cdots, \phi^N)$ belonging 
to the fundamental representation of $U(N)$. 
Let us assume that the K\"ahler potential $K$ 
is written as a function of the $U(N)$-invariant as
\beq
  K = K(X) \,, \hs{5} 
  X \equiv \vec{\phi}\dagg \vec{\phi} .
\eeq
Geometric quantities can be calculated~\cite{Higashijima:2003ki}, 
to yield
\beq
 g_{ij^*} &=& K' \delta_{ij} + K'' \phi^{*i} \phi^j \\
 R_{i j^*} 
 &=& -\partial_i \partial_{j^*} \tr \log g_{k l^*} \nonumber\\
&=& -\left[(N-1)\frac{K''}{K'} +\frac{2K''+ K''' X}{K'+K'' X}  \right] 
         \delta_{ij} \nonumber\\
&& - \left[(N-1) \bigg(\frac{K'''}{K''} - \frac{(K'')^2}{(K')^2} \bigg) 
   + \frac{3K''' + K'''' X} {K' + K'' X} 
   - \frac{(2K'' + K''' X)^2}{(K' + K''X)^2} \right]
        \phi^{*i} \phi^{j}, \\
 g_{lj^*} \Gamma^l{}_{ik} &=& g_{ij^*},_{k} 
 = K'' (\phi^{*k} \delta_{ij} + \phi^{*i} \delta_{kj})
   + K''' \phi^{*k} \phi^{*i} \phi^j 
\eeq
with the prime denoting a differentiation with respect to $X$. 
Substituting these into Eq.~(\ref{RFeq}), 
we obtain an ordinary differential equation, 
from a term proportional to $\delta_{ij}$,
\beq
 (N-1)\frac{K''}{K'} +\frac{2K''+ K''' X}{K'+K'' X}
  + a (K' + K''X) = 0  \label{diffeq1}
\eeq
with $a$ a constant defined by the anomalous dimension 
as $a \equiv - 4 \pi \gamma$,  
and the derivative of this equation with respect to $X$ 
from a term proportional to $\phi^{*i} \phi^j$. 
The differential equation (\ref{diffeq1}) can be integrated to give
\beq
 (K')^{N-1} (K' + K''X) = c \, e^{- a K'X}  
  \label{eq-U}
\eeq
with $c$ an integration constant. 
Defining the function
\beq
 F \equiv K' X ,
\eeq
Eq.~(\ref{eq-U}) can be rewritten as
\beq
 F^{N-1} F' X^{1-N} = c \, e^{- a F} .
\eeq
Again this can be integrated to give the algebraic equation
\beq
  e^{a F}  
  \sum_{r=0}^{N-1} (-1)^r 
  {(N-1)!\, F^{(N-1)-r} \over (N-1-r)!\, a^{r+1} } 
   = {c \over N} X^N + b  \;  \label{eq-F}
\eeq
with $b$ an integration constant.
This reproduces the solution 
found in \cite{Kiritsis:1993pb,Higashijima:2003ki} with 
a boundary condition $F(0) = 0$ 
which implies $K(0)=$ const.   
It reduces to $F = \1{a} \log (1 + a X)$ for $N=1$ 
which defines the two dimensional 
Euclidean black hole~\cite{Wi,Hori:2001ax} 
for $a>0$. 


We now construct a new $U(N)$-invariant solution.
To this end it is useful to define new coordinates by 
\beq
 \vec{\phi}^{\,T} = \sigma (1, \vec{z}^{\,T}) .
\eeq
We label $z$ by the same indices $i,j,\cdots$ with $\phi$
in the following. 
Then the invariant can be rewritten as
\beq
 X = \vec{\phi}\dagg \vec{\phi} 
  = |\sigma|^2 (1 + |\vec{z}|^2) 
   \equiv |\sig|^2 Z \, .
\eeq
It is useful to write down the metric in these coordinates as
\beq 
&& g = \left(
   \begin{array}{cc}
    g_{\sigma\sigma} & g_{\sigma j^*} \\
    g_{i \sigma^*}   & g_{ij^*}
     \end{array} 
    \right) , \non
&& g_{\sigma\sigma} = K'Z + K'' |\sigma|^2 Z^2 , \non
&& g_{\sigma j^*} = K' \sigma^* \del_{j^*} Z 
    + K'' \sigma^* |\sigma|^2 Z \del_{j^*} Z , \non
&& g_{ij^*} = K' |\sigma|^2 \del_i \del_{j^*} Z 
          + K'' |\sigma|^4 \del_i Z \del_{j^*} Z .
          \label{gen-metric}
\eeq
Using a solution $F$ of the same equation (\ref{eq-F}) 
with a boundary condition $F(0)= {\rm const.} \neq 0$ 
{\it different} from the previous one, 
the metric can be calculated 
in these coordinates as
\beq
&& ds^2 = 
    c \, e^{- a F} F^{1-N} Z^N |\sig|^{2N-2} |d \sig|^2
  + [c e^{- a F} F^{1-N} Z^{N-1} 
    |\sig|^{2N-2} \sig^* \del_{j^*} Z d \sig dz^{*j} 
      + {\rm c.c.}] \non
&& \hs{7} 
 + [ F (Z^{-1}\del_i \del_{j^*} Z - Z^{-2} \del_i Z \del_{j^*}Z)
    + c e^{- a F} F^{1-N} Z^{N-2} 
      |\sig|^{2N} \del_i Z \del_{j^*} Z] dz^i dz^{*j}. 
\eeq
Due to that boundary condition, 
this has a coordinate singularity in the limit $\sig \to 0$.
We perform the coordinate transformation 
\beq 
 \rho = \sigma^N / N   \label{coord}
\eeq
to remove this coordinate singularity.
We thus obtain the final form 
\beq
&& ds^2 = 
    c \, e^{- a F} F^{1-N} Z^N |d \rho|^2
  + [c N e^{- a F} F^{1-N} Z^{N-1} 
    \del_{j^*} Z \rho^* d \rho dz^{*j} + {\rm c.c.}] \non
&& \hs{7} 
 + [ F (Z^{-1}\del_i \del_{j^*} Z - Z^{-2} \del_i Z \del_{j^*}Z)
    + c N^2 e^{- a F} F^{1-N} Z^{N-2} 
      |\rho|^2 \del_i Z \del_{j^*} Z] dz^i dz^{*j}  
      \label{U-sol}
\eeq
with $Z = 1 + |\vec{z}|^2$. 


The metric at the $\rho=0$ surface 
\beq
 ds^2|_{\rho =0} = F(0) 
  (Z^{-1} \del_i \del_{j^*} Z - Z^{-2} \del_i \del_{j^*}Z) dz^i dz^{*j}  
  =  F(0) \del_i \del_{j^*} \log (1 + |z|^2) dz^i dz^{*j}  
  \label{CPN}
\eeq
is the Fubini-Study metric on 
${\bf C}P^{N-1} \simeq SU(N)/[SU(N-1) \times U(1)]$. 
Therefore the metric (\ref{U-sol}) is 
the (canonical) line bundle over ${\bf C}P^{N-1}$, 
${\bf C} \ltimes {\bf C}P^{N-1}$. 
In fact, if we take $a=0$ we recover the Ricci-flat 
K\"ahler metric of Calabi~\cite{Ca} (see also \cite{HKN4}).


In the limit of 
the boundary condition constant $F(0)$ tending to zero, 
${\bf C}P^{N-1}$  with the metric (\ref{CPN}) shrinks and 
the whole metric (\ref{U-sol}) contains a singularity. 
It is an orbifold singularity in the orbifold 
${\bf C}^N/{\bf Z}_N$ defined by 
the identification (\ref{coord}).
Therefore $F(0)$ is a blow-up parameter 
for the orbifold singularity.

Since the asymptotic form of $F$ is 
$F \simeq {N\over a} \log aX$ for large $X$,
the metric becomes asymptotically 
\beq
&& ds^2 \simeq {c \over a X} (NX \log aX)^{1-N} Z^N |d \rho|^2
 + \left[ {c N \over a X} (NX \log aX )^{1-N} Z^{N-1} 
  \del_{j^*} Z \rho^* d\rho dz^{*j} + {\rm c.c.} \right] \non
&&\hs{7} 
 + \left[{N\over a} \log aX 
    (Z^{-1}\del_i \del_{j^*} Z - Z^{-2} \del_i Z \del_{j^*}Z) 
    + c N^2 \left({\log aX \over aX}\right)^{1-N} Z^{N-2} |\rho|^2 
     \del_i Z \del_{j^*} Z \right] dz^i dz^{*j} . \non
\eeq 

The difference between the previous solution 
\cite{Kiritsis:1993pb,Higashijima:2003ki} 
and the present solution is locally 
just the boundary condition. 
For the previous case, they required regularity on 
$K$ at $X=0$, $K(0)=$ const. (or $F(0)=0$), and therefore 
$K = k_0 + k_1 X + k_2 X^2 + \cdots$.
It is, however, not necessary for regularity on  
the metric as seen above. 
For our case the condition is $F(0) =$ const., and therefore 
$K = k_{-1} \log X + k_0 + k_1 X +   k_2 X^2 + \cdots$ 
is not regular at $X \to 0$. 
As a result the topology is drastically changed. 
The previous solution has topology ${\bf C}^N$, 
but the present solution has topology 
${\bf C} \ltimes {\bf C}P^{N-1}$ blowing up the orbifold singularity 
in ${\bf C}^N/{\bf Z}_N$. 

%%%%%%%%%%%%%%%%%%%%
\section{$O(N)$ Invariant Model} 
Let us generalize the solution obtained in the last section 
to a $O(N)$-invariant solution.
We prepare an $N$-vector 
$\vec{\phi} = (\phi^A, \cdots, \phi^N)$ again  
and put a constraint 
\beq
 \vec{\phi}^{\,2} = \sum_{A=1}^N (\phi^A )^2 = 0 
  \label{constraint}
\eeq
to define a conifold~\cite{conifold}.
It is convenient to rewrite this constraint as
\begin{align}
 \vec{\phi}^{\, T} J \vec{\phi} = 0 , \hs{5}
J \ & \equiv \ \left(
\begin{array}{ccc}
0 & {\bf 0} & 1 \\
{\bf 0} & {\bf 1}_{N-2} & {\bf 0} \\
1 & {\bf 0} & 0 
\end{array} \right) 
\end{align}
with $J$ the rank-2 $O(N)$ invariant tensor.
The constraint (\ref{constraint}) 
can be solved as \cite{HN,Higashijima:2000rp}
\beq
 \vec{\phi}^{\, T} 
  = \sigma \left(1, z^i, -\1{2} \vec{z}^{\,2} \right)\; . 
   \label{QN-sol}
\eeq
Ricci-flat metrics on conifolds 
with the singularity deformed by $\vec{\phi}^{\,2} = r$
were constructed in \cite{conifold} for $N=3$ and 
\cite{Higashijima:2001yn} for general $N$.
Ricci-flat metrics on conifolds without deformation 
was constructed in \cite{HKN2,HKN4} 
which is still regular by an integration constant.
Our solution here is a non-Ricci-flat deformation 
of the latter one.

The $O(N)$-invariant can be written as 
\beq
 X \equiv \vec{\phi}\dagg\vec{\phi} 
  = |\sig|^2 \left( 1 + |\vec{z}|^2 + \1{4} |\vec{z}^{\,2}|^2 \right)
  \equiv |\sig|^2 Z . \label{Z-O}
\eeq
The expression of the metric is the same as the metric 
(\ref{gen-metric}) but with $X$ and $Z$ in Eq.~(\ref{Z-O}).
Components of the Ricci-form are calculated as
\beq
 && R_{\sig\sig^*} 
  = - L^{-1} L' Z - L^{-2} [L L'' - (L')^2] |\sig|^2 Z^2 ,\non
 && R_{\sig j^*} = 
  - L^{-1} L' \sig^* \del_{j^*} Z 
  - L^{-2}  [L L'' - (L')^2] \sig^* |\sig|^2 Z \del_{j^*} Z , \non
 && R_{ij^*} 
  = - L^{-1} L' |\sig|^2 \del_i \del_{j^*} Z 
   - L^{-2}  [L L'' - (L')^2] |\sig|^4 \del_i Z \del_{j^*} Z
\eeq
with $L \equiv (K')^{N-2} (K'' X^2 + K' X)$. 
Components of the connection are 
\beq
 && g_{\sig\sig^*,\sig^*} = K'' 3 \sig Z^2 + K''' \sig |\sig|^2 Z^3 ,\non
 && g_{\sig\sig^*,i^*} = K' \del_{i^*} Z 
  + K'' 3 |\sig|^2 Z \del_{i^*} Z + K'''|\sig|^4 Z^2 \del_{i^*}Z ,\non
 && g_{ij^*,\sig^*} = K' \sig \del_i \del_{j^*} Z 
    + K'' (2 \sig |\sig|^2 Z \del_i \del_{j^* }Z 
        + \sig |\sig|^2 \del_iZ \del_{j^*}Z) 
    + K''' \sig |\sig|^4 Z \del_i Z \del_{j^*} Z ,\non
&& g_{ij^*,k^*} = K' |\sig|^2 \del_i \del_{j^*} \del_{k^*} Z   
    + K''|\sig|^4 (\del_{k^* }Z \del_i \del_{j^* }Z 
        + \del_i Z \del_{j^*} \del_{k^* }Z 
        +  \del_{j^* }Z \del_i \del_{k^* }Z ) \non 
  && \hs{15} 
   + K''' |\sig|^6 \del_i Z \del_{j^*} Z  \del_{k^*} Z .
\eeq
Substituting all these quantities to Eq.~(\ref{RFeq}), 
we get 
\beq
 (\log L)' = - a (K' X)' .
\eeq
By integrating this we find that 
$F = K' X$ satisfies the equation similar to 
the $U(N)$ case,
\beq
 F^{N-2} F' X^{3-N} = c \, e^{- a F} 
\eeq
with an integration constant $c$.
Again this can be integrated to yield
\beq
  e^{a F} \sum_{r=0}^{N-2} (-1)^r 
  {(N-2)!\, F^{N-2-r} \over (N-2-r)!\, a^{r+1} } 
   = {c \over N-2} X^{N-2} + b  \;  \label{eq-F-O}
\eeq
with an integration constant $b$. 
The coordinate transformation 
\beq
 \rho = {\sigma^{N-2} \over N-2}
\eeq
is needed to remove the coordinate singularity. 
We thus obtain the final form of the metric 
\beq
&& ds^2 = 
    c \, e^{- a F} F^{2-N} Z^{N-2} |d \rho|^2
  + [c (N-2) e^{- a F} F^{2-N} Z^{N-3} 
     \del_{j^*} Z \rho^* d \rho dz^{*j} + {\rm c.c.}] \non
&& \hs{7} 
 + [ F (Z^{-1}\del_i \del_{j^*} Z - Z^{-2} \del_i Z \del_{j^*}Z)
    + c \, (N-2)^2  e^{- a F} F^{2-N} Z^{N-4} 
      |\rho|^2 \del_i Z \del_{j^*} Z] dz^i dz^{*j}  
      \label{O-sol}
\eeq
with $Z = 1 + |\vec{z}|^2 + \1{4} |\vec{z}^{\,2}|^2$. 

The metric at the $\rho=0$ surface is 
\beq
 ds^2|_{\rho =0} = F(0) 
  (Z^{-1} \del_i \del_{j^*} Z - Z^{-2} \del_i \del_{j^*}Z) dz^i dz^{*j}  
  =  F(0) \del_i \del_{j^*} 
    \log \left(1 + |z|^2 + \1{4} |\vec{z}^{\,2}|^2 \right) 
      dz^i dz^{*j}  .
  \label{QN}
\eeq
This is the metric on the quadric surface 
$Q^{N-2} \simeq SO(N)/[SO(N-2) \times U(1)]$~\cite{HN,Higashijima:2000rp}. 
The metric (\ref{O-sol}) is thus 
the (canonical) line bundle over $Q^{N-2}$. 
The Ricci-flat metric on a conifold \cite{HKN2,HKN4} 
is obtained by taking $a=0$ in the metric (\ref{O-sol}).
Our metric is non-Ricci-flat deformation of that conifold.

The asymptotic form of 
the metric (\ref{O-sol}) is ($F\simeq {N-2 \over a} \log aX$)  
\beq
&& ds^2 \simeq c [(N-2)X \log aX]^{2-N} Z^{N-2} |d \rho|^2 \non
&&\hs{7} 
   + \left[ c (N-2) \{ (N-2)X \log aX \}^{2-N} Z^{N-3} 
   \del_{j^*} Z \rho^* d\rho dz^{*j} + {\rm c.c.} \right] \non
&&\hs{7} 
 + \left[{N-2 \over a} \log aX 
    (Z^{-1}\del_i \del_{j^*} Z - Z^{-2} \del_i Z \del_{j^*}Z) \right. \non
&& \left. \hs{20}   
 + c (N-2)^2 \left\{ (N-2) X {\log aX}\right\}^{2-N} Z^{N-4} 
     |\rho|^2 \del_i Z \del_{j^*} Z \right] dz^i dz^{*j} . 
\eeq 

%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
We have given two new metrics (\ref{U-sol}) and 
(\ref{O-sol}) with $U(N)$ and $O(N)$ symmetries, respectively  
as solutions of Eq.~(\ref{RFeq}) 
for target spaces of 
conformally invariant sigma models with 
an anomalous dimension $\gamma$. 
They are canonical line bundles over 
the projective space ${\bf C}P^{N-1}$ and 
the quadric surface $Q^{N-2}$, respectively.
These metrics contain the anomalous dimension 
explicitly through the parameter $a = - 4 \pi \gamma$. 
In the limit of vanishing anomalous dimension $a \to 0$, 
they reduce to those for Calabi-Yau manifolds~\cite{HKN2,HKN4}.
Generalization to other base coset spaces \cite{HKN4,HKN5} 
is straightforward. 
These new solutions give examples of K\"ahler Ricci solitons, 
singular solutions to the Ricci-flow equation (\ref{Ricci-flow}). 

A paper \cite{NS} recently posted to ArXiv has 
overlap with the present work. 
They have obtained explicit solutions 
with a dilaton background which also reduce to 
the same Calabi-Yau manifolds \cite{HKN2}--\cite{HKN6} 
with the present work.
Therefore solutions in the present paper and 
those in \cite{NS}
would be equivalent to each other 
by some field redefinition. 


%%%%%%%%%
\section*{Acknowledgements}
The author would like to thank 
Kiyoshi Higashijima and Etsuko Itou 
for a collaboration in the early stages of this work 
and for explanation of their works 
\cite{Higashijima:2002mh,Higashijima:2003ki}. 
He is also grateful to Takashi Maeda for explanation of 
the Ricci-flow equation. 
His work is supported by Japan Society for the Promotion 
of Science under the Post-doctoral Research Program. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% reference %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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%\href{http://www.slac.stanford.edu/spires/find/hep/www?irn=4634799}{SPIRES entry}
%\cite{Polchinski:1998rr}
%\bibitem{Polchinski:1998rr}
%J.~Polchinski,
``String theory. Vol. 2: Superstring theory and beyond''.
%\href{http://www.slac.stanford.edu/spires/find/hep/www?irn=4634802}{SPIRES entry}

\bibitem{Ca}
E.~Calabi, Ann. Scient. Ec. Norm. Sup. {\bf 12}, 269 (1979).

\bibitem{conifold}
P.~Candelas and X.~C.~de la Ossa,
Nucl. Phys. B{\bf 342}, 246 (1990);  
R.~Minasian and D.~Tsimpis,
Nucl. Phys. B{\bf 572}, 499 (2000), [arXiv:hep-th/9911042]; 
K.~Ohta and T.~Yokono, JHEP {\bf 0002} (2000) 023,
[arXiv:hep-th/9912266].

%\cite{Higashijima:2001yn}
\bibitem{Higashijima:2001yn} 
M.~B.~Stenzel, Manuscripta Math. {\bf 80}, 151 (1993);
%\cite{Cvetic:2000db}
%\bibitem{Cvetic:2000db}
M.~Cvetic, G.~W.~Gibbons, H.~Lu and C.~N.~Pope,
%``Ricci-flat metrics, harmonic forms and brane resolutions,''
Commun.\ Math.\ Phys.\  {\bf 232}, 457 (2003)
[arXiv:hep-th/0012011];
%%CITATION = HEP-TH 0012011;%% 
K.~Higashijima, T.~Kimura and M.~Nitta,
%``Supersymmetric nonlinear sigma models on Ricci-flat Kaehler manifolds  with
%O(N) symmetry,''
Phys.\ Lett.\ B {\bf 515}, 421 (2001)
[arXiv:hep-th/0104184].
%%CITATION = HEP-TH 0104184;%% 

\bibitem{PP}
D.~N.~Page and C.~N.~Pope, Class. Quantum Grav. {\bf 4} (1987) 213.

%\cite{Higashijima:2001de}
\bibitem{HKN2}
K.~Higashijima, T.~Kimura and M.~Nitta,
%``A note on conifolds,''
Phys.\ Lett.\ B {\bf 518}, 301 (2001)
[arXiv:hep-th/0107100].
%%CITATION = HEP-TH 0107100;%%

%\cite{Higashijima:2001vk}
\bibitem{HKN3}
K.~Higashijima, T.~Kimura and M.~Nitta,
%``Ricci-flat Kaehler manifolds from supersymmetric gauge theories,''
Nucl.\ Phys.\ B {\bf 623}, 133 (2002)
[arXiv:hep-th/0108084].
%%CITATION = HEP-TH 0108084;%%

%\cite{Higashijima:2001fp}
\bibitem{HKN4}
K.~Higashijima, T.~Kimura and M.~Nitta,
%``Gauge theoretical construction of non-compact Calabi-Yau manifolds,''
Annals Phys.\  {\bf 296}, 347 (2002)
[arXiv:hep-th/0110216].
%%CITATION = HEP-TH 0110216;%%

%\cite{Higashijima:2002px}
\bibitem{HKN5}
K.~Higashijima, T.~Kimura and M.~Nitta,
%``Calabi-Yau manifolds of cohomogeneity one as complex line bundles,''
Nucl.\ Phys.\ B {\bf 645}, 438 (2002)
[arXiv:hep-th/0202064].
%%CITATION = HEP-TH 0202064;%%

%\cite{Higashijima:2002jt}
\bibitem{HKN6}
K.~Higashijima, T.~Kimura and M.~Nitta,
%``Construction of supersymmetric nonlinear sigma models on noncompact
%Calabi-Yau manifolds with isometry,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 117}, 867 (2003)
[arXiv:hep-th/0210034].
%%CITATION = HEP-TH 0210034;%%

%\cite{Nitta:2003yk}
\bibitem{Nitta:2003yk}
M.~Nitta,
%``Non-compact Calabi-Yau metrics from nonlinear realizations,''
arXiv:hep-th/0309004.
%%CITATION = HEP-TH 0309004;%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\cite{Kiritsis:1993pb}
\bibitem{Kiritsis:1993pb}
E.~Kiritsis, C.~Kounnas and D.~Lust,
%``A Large class of new gravitational and axionic backgrounds for
%four-dimensional superstrings,''
Int.\ J.\ Mod.\ Phys.\ A {\bf 9}, 1361 (1994)
[arXiv:hep-th/9308124].
%%CITATION = HEP-TH 9308124;%%

\bibitem{Wi}
E. Witten, Phys. Rev. D {\bf 44}, 314 (1991).

%\cite{Hori:2001ax}
\bibitem{Hori:2001ax}
K.~Hori and A.~Kapustin,
%``Duality of the fermionic 2d black hole and N = 2 Liouville theory as  mirror
%symmetry,''
JHEP {\bf 0108}, 045 (2001)
[arXiv:hep-th/0104202]; 
%%CITATION = HEP-TH 0104202;%%
%\cite{Hori:2002cd}
%\bibitem{Hori:2002cd}
%K.~Hori and A.~Kapustin,
%``Worldsheet descriptions of wrapped NS five-branes,''
JHEP {\bf 0211}, 038 (2002)
[arXiv:hep-th/0203147].
%%CITATION = HEP-TH 0203147;%%


%\cite{Higashijima:2002mh}
\bibitem{Higashijima:2002mh}
K.~Higashijima and E.~Itou,
%``Wilsonian renormalization group approach to N = 2 supersymmetric sigma
%models,''
Prog.\ Theor.\ Phys.\  {\bf 108}, 737 (2002)
[arXiv:hep-th/0205036].
%%CITATION = HEP-TH 0205036;%%

%\cite{Higashijima:2003ki}
\bibitem{Higashijima:2003ki}
K.~Higashijima and E.~Itou,
%``A new class of conformal field theories with anomalous dimensions,''
Prog.\ Theor.\ Phys.\  {\bf 109}, 751 (2003)
[arXiv:hep-th/0302090].
%%CITATION = HEP-TH 0302090;%%

%\cite{Higashijima:2003rp}
\bibitem{Higashijima:2003rp}
K.~Higashijima and E.~Itou,
%``Three dimensional nonlinear sigma models in the Wilsonian  renormalization
%method,''
Prog.\ Theor.\ Phys.\  {\bf 110}, 563 (2003)
[arXiv:hep-th/0304194].
%%CITATION = HEP-TH 0304194;%%

%\cite{Itou:2004px}
\bibitem{Itou:2004px}
E.~Itou,
to appear in the proceedings of 12th International Conference on Supersymmetry and Unification of Fundamental Interactions (SUSY 04), Tsukuba, Japan, 17-23 Jun 2004,   
%``New conformal field theories with anomalous dimensions,''
[arXiv:hep-th/0409220].
%%CITATION = HEP-TH 0409220;%%


\bibitem{KNC}
%\cite{Higashijima:2000wz}
%\bibitem{Higashijima:2000wz}
K.~Higashijima and M.~Nitta,
%``Kaehler normal coordinate expansion in supersymmetric theories,''
Prog.\ Theor.\ Phys.\  {\bf 105}, 243 (2001)
[arXiv:hep-th/0006027]; \\
%%CITATION = HEP-TH 0006027;%%
%K.~Higashijima E.~Itou and M.~Nitta, 
%\cite{Higashijima:2002fq}
%\bibitem{Higashijima:2002fq}
K.~Higashijima, E.~Itou and M.~Nitta,
%``Normal coordinates in Kaehler manifolds and the background field  method,''
Prog.\ Theor.\ Phys.\  {\bf 108}, 185 (2002)
[arXiv:hep-th/0203081].
%%CITATION = HEP-TH 0203081;%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\cite{Bakas:2004hn}
\bibitem{Bakas:2004hn}
I.~Bakas,
%``Ricci flows and their integrability in two dimensions,''
contributed to Annual International Conference on Strings, Theory and Applications (Strings 2004), Paris, France, 28 Jun - Jul 2, 2004, 
[arXiv:hep-th/0410093].
%%CITATION = HEP-TH 0410093;%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{HN}
%\cite{Higashijima:1999ki}
%\bibitem{Higashijima:1999ki}
K.~Higashijima and M.~Nitta,
%``Supersymmetric nonlinear sigma models as gauge theories,''
Prog.\ Theor.\ Phys.\  {\bf 103}, 635 (2000)
[arXiv:hep-th/9911139];
%%CITATION = HEP-TH 9911139;%%
%\cite{Higashijima:1999js}
%\bibitem{Higashijima:1999js}
%K.~Higashijima and M.~Nitta,
%``Quantum equivalence of auxiliary field methods in supersymmetric
%theories,''
Prog.\ Theor.\ Phys.\  {\bf 103}, 833 (2000)
[arXiv:hep-th/9911225];
%%CITATION = HEP-TH 9911225;%% 
in Proceedings of Confinement 2000 (World Scientific, 2001),
Eds. H.Suganuma et al., pp. 279 - 286, [arXiv:hep-th/0006025];
in Proceedings of ICHEP2000 (World Scientific 2001),
Eds. C.S.Lim et al., pp. 1368 - 1370, [arXiv:hep-th/0008240].
%K.~Higashijima, T.~Kimura, M.~Nitta and M.~Tsuzuki,
%Prog.~Theor.~Phys. {\bf 105} (2001) 261, {\tt hep-th/0010272}.

%\cite{Higashijima:2000rp}
\bibitem{Higashijima:2000rp}
K.~Higashijima, T.~Kimura, M.~Nitta and M.~Tsuzuki,
%``Large-N limit of N = 2 supersymmetric Q**N model in two dimensions,''
Prog.\ Theor.\ Phys.\  {\bf 105}, 261 (2001)
[arXiv:hep-th/0010272].
%%CITATION = HEP-TH 0010272;%%

\bibitem{NS}
N.~Nakayama and K.~Sugiyama
arXiv:hep-th/0411143. 

\end{thebibliography}

\end{document}





hep-th0411149.extracted_bib
\begin{thebibliography}{99}
%\cite{Polchinski:1998rq}
\bibitem{Polchinski:1998rq}
J.~Polchinski, 
``String theory. Vol. 1: An introduction to the bosonic string''; 
%\href{http://www.slac.stanford.edu/spires/find/hep/www?irn=4634799}{SPIRES entry}
%\cite{Polchinski:1998rr}
%\bibitem{Polchinski:1998rr}
%J.~Polchinski,
``String theory. Vol. 2: Superstring theory and beyond''.
%\href{http://www.slac.stanford.edu/spires/find/hep/www?irn=4634802}{SPIRES entry}

\bibitem{Ca}
E.~Calabi, Ann. Scient. Ec. Norm. Sup. {\bf 12}, 269 (1979).

\bibitem{conifold}
P.~Candelas and X.~C.~de la Ossa,
Nucl. Phys. B{\bf 342}, 246 (1990);  
R.~Minasian and D.~Tsimpis,
Nucl. Phys. B{\bf 572}, 499 (2000), [arXiv:hep-th/9911042]; 
K.~Ohta and T.~Yokono, JHEP {\bf 0002} (2000) 023,
[arXiv:hep-th/9912266].

%\cite{Higashijima:2001yn}
\bibitem{Higashijima:2001yn} 
M.~B.~Stenzel, Manuscripta Math. {\bf 80}, 151 (1993);
%\cite{Cvetic:2000db}
%\bibitem{Cvetic:2000db}
M.~Cvetic, G.~W.~Gibbons, H.~Lu and C.~N.~Pope,
%``Ricci-flat metrics, harmonic forms and brane resolutions,''
Commun.\ Math.\ Phys.\  {\bf 232}, 457 (2003)
[arXiv:hep-th/0012011];
%%CITATION = HEP-TH 0012011;%% 
K.~Higashijima, T.~Kimura and M.~Nitta,
%``Supersymmetric nonlinear sigma models on Ricci-flat Kaehler manifolds  with
%O(N) symmetry,''
Phys.\ Lett.\ B {\bf 515}, 421 (2001)
[arXiv:hep-th/0104184].
%%CITATION = HEP-TH 0104184;%% 

\bibitem{PP}
D.~N.~Page and C.~N.~Pope, Class. Quantum Grav. {\bf 4} (1987) 213.

%\cite{Higashijima:2001de}
\bibitem{HKN2}
K.~Higashijima, T.~Kimura and M.~Nitta,
%``A note on conifolds,''
Phys.\ Lett.\ B {\bf 518}, 301 (2001)
[arXiv:hep-th/0107100].
%%CITATION = HEP-TH 0107100;%%

%\cite{Higashijima:2001vk}
\bibitem{HKN3}
K.~Higashijima, T.~Kimura and M.~Nitta,
%``Ricci-flat Kaehler manifolds from supersymmetric gauge theories,''
Nucl.\ Phys.\ B {\bf 623}, 133 (2002)
[arXiv:hep-th/0108084].
%%CITATION = HEP-TH 0108084;%%

%\cite{Higashijima:2001fp}
\bibitem{HKN4}
K.~Higashijima, T.~Kimura and M.~Nitta,
%``Gauge theoretical construction of non-compact Calabi-Yau manifolds,''
Annals Phys.\  {\bf 296}, 347 (2002)
[arXiv:hep-th/0110216].
%%CITATION = HEP-TH 0110216;%%

%\cite{Higashijima:2002px}
\bibitem{HKN5}
K.~Higashijima, T.~Kimura and M.~Nitta,
%``Calabi-Yau manifolds of cohomogeneity one as complex line bundles,''
Nucl.\ Phys.\ B {\bf 645}, 438 (2002)
[arXiv:hep-th/0202064].
%%CITATION = HEP-TH 0202064;%%

%\cite{Higashijima:2002jt}
\bibitem{HKN6}
K.~Higashijima, T.~Kimura and M.~Nitta,
%``Construction of supersymmetric nonlinear sigma models on noncompact
%Calabi-Yau manifolds with isometry,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 117}, 867 (2003)
[arXiv:hep-th/0210034].
%%CITATION = HEP-TH 0210034;%%

%\cite{Nitta:2003yk}
\bibitem{Nitta:2003yk}
M.~Nitta,
%``Non-compact Calabi-Yau metrics from nonlinear realizations,''
arXiv:hep-th/0309004.
%%CITATION = HEP-TH 0309004;%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\cite{Kiritsis:1993pb}
\bibitem{Kiritsis:1993pb}
E.~Kiritsis, C.~Kounnas and D.~Lust,
%``A Large class of new gravitational and axionic backgrounds for
%four-dimensional superstrings,''
Int.\ J.\ Mod.\ Phys.\ A {\bf 9}, 1361 (1994)
[arXiv:hep-th/9308124].
%%CITATION = HEP-TH 9308124;%%

\bibitem{Wi}
E. Witten, Phys. Rev. D {\bf 44}, 314 (1991).

%\cite{Hori:2001ax}
\bibitem{Hori:2001ax}
K.~Hori and A.~Kapustin,
%``Duality of the fermionic 2d black hole and N = 2 Liouville theory as  mirror
%symmetry,''
JHEP {\bf 0108}, 045 (2001)
[arXiv:hep-th/0104202]; 
%%CITATION = HEP-TH 0104202;%%
%\cite{Hori:2002cd}
%\bibitem{Hori:2002cd}
%K.~Hori and A.~Kapustin,
%``Worldsheet descriptions of wrapped NS five-branes,''
JHEP {\bf 0211}, 038 (2002)
[arXiv:hep-th/0203147].
%%CITATION = HEP-TH 0203147;%%


%\cite{Higashijima:2002mh}
\bibitem{Higashijima:2002mh}
K.~Higashijima and E.~Itou,
%``Wilsonian renormalization group approach to N = 2 supersymmetric sigma
%models,''
Prog.\ Theor.\ Phys.\  {\bf 108}, 737 (2002)
[arXiv:hep-th/0205036].
%%CITATION = HEP-TH 0205036;%%

%\cite{Higashijima:2003ki}
\bibitem{Higashijima:2003ki}
K.~Higashijima and E.~Itou,
%``A new class of conformal field theories with anomalous dimensions,''
Prog.\ Theor.\ Phys.\  {\bf 109}, 751 (2003)
[arXiv:hep-th/0302090].
%%CITATION = HEP-TH 0302090;%%

%\cite{Higashijima:2003rp}
\bibitem{Higashijima:2003rp}
K.~Higashijima and E.~Itou,
%``Three dimensional nonlinear sigma models in the Wilsonian  renormalization
%method,''
Prog.\ Theor.\ Phys.\  {\bf 110}, 563 (2003)
[arXiv:hep-th/0304194].
%%CITATION = HEP-TH 0304194;%%

%\cite{Itou:2004px}
\bibitem{Itou:2004px}
E.~Itou,
to appear in the proceedings of 12th International Conference on Supersymmetry and Unification of Fundamental Interactions (SUSY 04), Tsukuba, Japan, 17-23 Jun 2004,   
%``New conformal field theories with anomalous dimensions,''
[arXiv:hep-th/0409220].
%%CITATION = HEP-TH 0409220;%%


\bibitem{KNC}
%\cite{Higashijima:2000wz}
%\bibitem{Higashijima:2000wz}
K.~Higashijima and M.~Nitta,
%``Kaehler normal coordinate expansion in supersymmetric theories,''
Prog.\ Theor.\ Phys.\  {\bf 105}, 243 (2001)
[arXiv:hep-th/0006027]; \\
%%CITATION = HEP-TH 0006027;%%
%K.~Higashijima E.~Itou and M.~Nitta, 
%\cite{Higashijima:2002fq}
%\bibitem{Higashijima:2002fq}
K.~Higashijima, E.~Itou and M.~Nitta,
%``Normal coordinates in Kaehler manifolds and the background field  method,''
Prog.\ Theor.\ Phys.\  {\bf 108}, 185 (2002)
[arXiv:hep-th/0203081].
%%CITATION = HEP-TH 0203081;%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\cite{Bakas:2004hn}
\bibitem{Bakas:2004hn}
I.~Bakas,
%``Ricci flows and their integrability in two dimensions,''
contributed to Annual International Conference on Strings, Theory and Applications (Strings 2004), Paris, France, 28 Jun - Jul 2, 2004, 
[arXiv:hep-th/0410093].
%%CITATION = HEP-TH 0410093;%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{HN}
%\cite{Higashijima:1999ki}
%\bibitem{Higashijima:1999ki}
K.~Higashijima and M.~Nitta,
%``Supersymmetric nonlinear sigma models as gauge theories,''
Prog.\ Theor.\ Phys.\  {\bf 103}, 635 (2000)
[arXiv:hep-th/9911139];
%%CITATION = HEP-TH 9911139;%%
%\cite{Higashijima:1999js}
%\bibitem{Higashijima:1999js}
%K.~Higashijima and M.~Nitta,
%``Quantum equivalence of auxiliary field methods in supersymmetric
%theories,''
Prog.\ Theor.\ Phys.\  {\bf 103}, 833 (2000)
[arXiv:hep-th/9911225];
%%CITATION = HEP-TH 9911225;%% 
in Proceedings of Confinement 2000 (World Scientific, 2001),
Eds. H.Suganuma et al., pp. 279 - 286, [arXiv:hep-th/0006025];
in Proceedings of ICHEP2000 (World Scientific 2001),
Eds. C.S.Lim et al., pp. 1368 - 1370, [arXiv:hep-th/0008240].
%K.~Higashijima, T.~Kimura, M.~Nitta and M.~Tsuzuki,
%Prog.~Theor.~Phys. {\bf 105} (2001) 261, {\tt hep-th/0010272}.

%\cite{Higashijima:2000rp}
\bibitem{Higashijima:2000rp}
K.~Higashijima, T.~Kimura, M.~Nitta and M.~Tsuzuki,
%``Large-N limit of N = 2 supersymmetric Q**N model in two dimensions,''
Prog.\ Theor.\ Phys.\  {\bf 105}, 261 (2001)
[arXiv:hep-th/0010272].
%%CITATION = HEP-TH 0010272;%%

\bibitem{NS}
N.~Nakayama and K.~Sugiyama
arXiv:hep-th/0411143. 

\end{thebibliography}