%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                %
%                   研究会「場の理論 2004」                      %
%                template for the proceeding                     %
%                                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% この template は short talk および poster 用です。
% １ページ以内に収まるようにお願いします。

\documentclass[11pt]{article}

%% 使いたい人はどうぞ
%\usepackage{amsmath,amssymb}
%\usepackage{graphicx}

\oddsidemargin 0cm \evensidemargin 0cm
\textwidth 16cm

%\renewcommand{\title}[1]{%
%	\begin{center} \Large \bf #1 \end{center}%
%	}
%\renewcommand{\author}[3]{%
%	\begin{center} #1　#2 \\ %
%	  {\small E-mail: \texttt{#3}}%
%	\end{center}%
%	\addvspace{\baselineskip}%
%	}

%%%%%%%%%%%%%%%%%%%%%%%%%%　ここから　%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\baselineskip 5mm
%%% title %%%
\begin{center}
 {\Large \bf Fuzzy CP$^{2}$ or S$^{2}$ --- which is the true vacuum ?}
\end{center}

%%% author %%%
\begin{center}
 Takehiro Azuma (KEK)\\
 {\small \tt azumat@post.kek.jp}
\end{center}
%
 This poster session introduces our work [1], in
 which we have discussed the bosonic matrix model as a
 toy model that has the curved-space classical solution:
\begin{eqnarray}
 S = N \rm{tr} \left( - \frac{1}{4} \sum_{\mu,\nu=1}^{8} [ A_{\mu}, A_{\nu}
  ]^{2}
  + \frac{2i \alpha}{3} \sum_{\mu,\nu,\rho=1}^{8} f_{\mu \nu \rho} A_{\mu}
 A_{\nu} A_{\rho} \right) \nonumber
\end{eqnarray}
 This matrix model accommodates the following two classical solutions. One is
 the fuzzy S$^{2}$ manifold spanned in the 123 direction. The other is the
 more interesting four-dimensional fuzzy CP$^{2}$ manifold.
 To scrutinize the properties of these two fuzzy manifolds,
 we have studied this matrix model numerically via the heat bath algorithm
 of the Monte Carlo simulation. The results are recapitulated as follows. 

 We have found that this matrix model undergoes the first-order
 phase transition at a certain critical point $\alpha_{\rm cr}$, as
 we change the coefficient $\alpha$ of the cubic term in the action.
 This phase transition is similar to that of the three-dimensional
 bosonic model studied in our previous work [2].
 At $\alpha < \alpha_{\rm cr}$, the effect of the cubic term is
 negated and the model behaves like the pure Yang-Mills model ($\alpha = 0$).
 On the other hand, at $\alpha > \alpha_{\rm cr}$, the above-mentioned
 classical solutions retain the (meta)stability due to the meager quantum
 effect. In addition, we have elucidated that the observables calculated
 numerically are close to the one-loop perturbative calculation.
 The critical point is also close to the value predicted from the one-loop
 effective action.
 In this sense, we proclaim that this matrix model has the ``one-loop
 dominance''. 

 The one-loop dominance plays a pivotal role in determining the true vacuum 
 of this matrix model. By the comparison of the one-loop effective action,
 we have concluded that the two-dimensional fuzzy S$^{2}$ is the true
 vacuum of this model, and that the four-dimensional
 fuzzy CP$^{2}$ is a metastable state. 

 This poster session is based on the collaboration with
 Subrata Bal, Keiichi Nagao and Jun Nishimura. 

\paragraph{References} \hspace{0mm} 

[1] T.~Azuma, S.~Bal, K.~Nagao and J.~Nishimura,
``Dynamical aspects of the fuzzy CP$^{2}$ in the large N reduced model with a
cubic term,''
{\tt hep-th/0405277}.
%%CITATION = HEP-TH 0405277;%%

 
[2] T.~Azuma, S.~Bal, K.~Nagao and J.~Nishimura,
 ``Nonperturbative studies of fuzzy spheres in a matrix model with the
Chern-Simons term,''
JHEP {\bf 0405}, 005 (2004)
[{\tt hep-th/0401038}]. 

 

\end{document}




