1 Introduction

Recent proposals for a non-perturbative formulation of String Theory [1] has renewed the interest for matrix models of non-Abelian gauge theories. Large $N$ Yang-Mills theories on a $d$-dimensional spacetime [2] have been shown to be equivalent to reduced matrix models, where the original $N\times N$ matrix gauge field ${\mbox{\boldmath {$A$}}}_\mu {}^i_j\left(\, x\,\right)$ is replaced by the same field at a single point, say $x^\mu=0$ [3] (for a recent review see [4]). Partial derivative operators are replaced by commutators with a fixed diagonal matrix ${\mbox{\boldmath {$p$}}}_\mu {}^i_j$, playing the role of translation generator and called the quenched momentum [5]. Accordingly, the covariant derivative becomes $i {\mbox{\boldmath {$D$}}}_\mu = \left[\, {\mbox{\boldmath {$p$}}}_\mu\ ,\dots\, \right] + {\mbox{\boldmath {$A$}}}_\mu$. Thus, the reduced, quenched, Yang-Mills field strength is

$${\mbox{\boldmath {$F$}}}_{\mu\nu}{}^i_j\equiv \left[\,i {\mb... ...th {$D$}}}_\mu\ , i {\mbox{\boldmath {$D$}}}_\nu\, \right]^i_j$$ (1)

leading to the matrix Yang-Mills action in four dimensions

$$S^{\mathrm{qYM}}_{\mathrm{red}}=-{1\over 4}\left(\, {2\pi\o... ...oldmath {$F$}}}_{\mu\nu}\, {\mbox{\boldmath {$F$}}}^{\mu\nu},$$ (2)

where, $g_{\mathrm{YM}}$ is the strong coupling constant, and $a\equiv 2\pi/ \Lambda$ is an inverse momentum cut-off, or lattice spacing. $S^{\mathrm{qYM}}_{\mathrm{red}}$ is the starting point to study large-$N$ Yang-Mills theory. For our purposes, it is important to supplement $S^{\mathrm{qYM}}_{\mathrm{red}}$ with the contribution from topologically non-trivial field configurations, i.e. instantons, which is accounted for by the topological term $\epsilon^{\lambda\mu\nu\rho}\, F_{\lambda\mu}\, F_{\nu\rho}$. The reason for this choice will be evident later on:

$$S^{\mathrm{qYM}}_{\mathrm{red}}\longrightarrow S^q_{red}\equiv S^{\mathrm{qYM}}_{\mathrm{red}} +S^{\mathrm{q\, \theta}}_{\mathrm{red}}$$ (3)

$$S^{\mathrm{q\, \theta}}_{\mathrm{red}}= -{\theta\,N\, g^2_{\mathr... ...r\,{\mbox{\boldmath {$F$}}}_{\mu\nu}\, {\mbox{\boldmath {$F$}}}_{\rho\sigma},\$$ (4)

where, $\theta$ is the vacuum angle.
The matrix model (2) in the limit $N\to\infty$ has been shown to describe strings [6]. Here we would like to extend this result and show that the model encoded by $S^q_{red}$ includes, in the large-$N$ limit, not only strings, but also an open $3$-brane with a Chern-Simons $2$-brane as its dynamical boundary. These higher dimensional objects would provide appropriate models for hadronic bags embedded in a four dimensional target spacetime. In this way, we would hopefully fill the gap between $QCD$, as the fundamental quantum field theory of strong interactions, and more phenomenologically oriented models for strongly interacting, confined, objects. To make evident the relationship between matrix gauge fields and extended objects we shall implement the effective, and simple, correspondence among unitary operators in Hilbert space and ordinary functions in a non-commutative phase space [7], which has been originally established by the Wigner-Weyl-Moyal (WWM) formulation of quantum mechanics (see [8] for recent applications of this quantization method).
The eigenvalues of the unitary operator ${\mbox{\boldmath {$D$}}}_\mu {}^i_j$ span a $D$-dimensional eigen-lattice [9] with $0 \le D \le 4$. Thus, ${\mbox{\boldmath {$D$}}}_\mu$ can be expressed in terms of $2D$ independent matrices ${\bf p}_i$, ${\bf q}_j$, $i\ , j = 1\ ,\dots D$, through the WWM relation

$${\mbox{\boldmath {$A$}}}_\mu \equiv {1\over (2\pi)^D}\int d^... ...h {$p$}}}_i + i\, p^j \, {\mbox{\boldmath {$q$}}}_j\,\right) ,$$ (5)

where the operators ${\mbox{\boldmath {$p$}}}_i$, ${\mbox{\boldmath {$q$}}}_j$ satisfy the Heisenberg algebra

$$\left[\, {\mbox{\boldmath {$p$}}}_i\ , {\mbox{\boldmath {$q$}}}_j\,\right]= -i\, \hbar\, \delta_{ij}\ ,$$ (6)

and $\left(\, q^i\ , p^j\,\right)$ play the role of coordinates in a Fourier dual space. $\hbar$ is the deformation parameter, which for historical reason is often represented by the same symbol as the Planck constant.
The basic idea under this approach is to identify the Fourier space as the dual of a $(D+D)$-dimensional world manifold of a $p=2D-1$ brane. The case $D=1$, corresponding to strings, has been widely investigated because of the relation between the $\mathit{su}(\,\infty\,)$ Lie algebra and the area preserving diffeomorphism algebra over a two-dimensional manifold $\Sigma$, $\mathit{sdiff}\left(\,\Sigma \,\right)$. Much less attention has been given, in this framework, to higher dimensional objects (a remarkable exception is [10]).
In this letter, we shall consider the case $D=2$ and show that in the limit $N\to\infty$ the action $S^{\mathrm{Q}}_{\mathrm{red}}$ becomes a bag action endowed with a dynamical boundary.