1 Introduction

The pivotal role played by gauge field theories in a unified description of fundamental interactions proposed one of the most challenging questions of modern high energy theoretical physics: if Nature likes so much gauge symmetry why gravity cannot fit into such an elegant and, would be, universal blueprint?
Before the advent of string theory this was a question without an answer. After that, it became clear that all field theories, including Yang-Mills type models, must be seen as low energy, effective approximations of some more fundamental theory where the dynamical degrees of freedom are carried by relativistic extended objects. Furthermore, even the low-energy effective gauge theories require a ``stringy'' approach in the strong coupling regime, where standard perturbation series breaks down. Color confinement in QCD is a remarkable example of a phenomenon where the string tenet meets gauge symmetry. The stringy aspects of confinement have long been studied, but are not yet fully understood⁴. Several different models have been proposed as phenomenological descriptions of the quark-gluon bound states, including color flux tubes, three-string of various shapes, bag-models [4]. To promote some of them to a deeper status one would like to derive extended objects as non-perturbative excitations of an underlying gauge theory [5]. The first remarkable achievement of this program was to obtain stringy objects from $SU(N)$ Yang-Mills theory in the large-$N$ limit [6].

These results have been extended to the case of a self-dual membrane in the $SU(\infty)$ Toda model [7]. In a nutshell, the problem is to establish a formal correspondence between a Yang-Mills connection, $A^i{}_j{}_\mu(x)$, and the string coordinates, $X^\mu(\sigma^0\ ,\sigma^1)$, i.e. one has to ``get rid of'' the internal, non-Abelian, indices $i$, $j$ and replace the spacetime coordinates $x^\mu$ with two continuous coordinates $\left(\, \sigma^0\ ,\sigma^1\,\right)$. With hindsight, the ``recipe'' to turn a non-Abelian gauge field into a set of Abelian functions describing the embedding of the string world-sheet into target spacetime can be summarized as follows:
a) transform the original field theory into a sort of ``matrix quantum mechanics'';
b) use the Wigner-Weyl-Moyal map to build up the symbol associated to the above matrix model;
c) take the ``classical limit'' of the theory obtained in b).

Stage a) requires two sub-steps:
a1) take the large-$N$ limit, i.e. let the row and column labels $i$, $j$ to range over arbitrarily large values; a2) dispose of the spacetime coordinate dependence through the so called ``quenching approximation'', i.e. a technical manipulation which is formally equivalent to collapse the whole spacetime to a single point.
After stage a) the original gauge theory is transformed into a quantum mechanical model where the physical degrees of freedom are carried by large coordinate independent matrices. Then, stage b) associates to each of such big matrices its corresponding symbol, i.e. a function defined over an appropriate non-commutative phase space. The resulting theory is a deformation of an ordinary field theory, where the ordinary product between functions is replaced by a non-commutative $\ast$-product. The deformation parameter, measuring the amount of non-commutativity, results to be $1/N$, and the classical limit corresponds exactly to the large-$N$ limit. The final result, obtained at the stage c), is a string action of the Schild type, which is invariant under area-preserving reparametrization of the world-sheet.
More recently, we have also shown that bag-like objects fit the large-$N$ spectrum of Yang-Mills type theories as well, both in four [8] and higher dimensions. We started from the Yang-Mills action for an $SU(\,N\,)$ gauge theory supplemented by a topological term

$$S\equiv -\int d^4x\, \left(\, {N\over 4g^2_{YM}}\,\mathrm{Tr... ...}F}}_{\mu\nu} \, \mbox{\textit{\bf {}F}}_{\rho\sigma}\,\right)$$ (1)

and went trough the steps from a) to c). As a final result we obtained the following action

$$W= -{\mu^4_0\over 16}\,\int_\Sigma d^4\sigma\, \left\{\, X^\mu\ , X^\nu\,\right\}_{PB} \left\{\, X_\mu\ , X_\nu\,\right\}_{PB} +$$

$$\qquad -\kappa\,\epsilon_{\mu\nu\nu\rho\sigma} \,\int_{\partial\Sigma} d^3s X^\mu\, \left\{\, X^\nu\ , X^\rho\ ,X^\sigma\right\}_{NPB} \quad ,$$ (2)

where

$$\left\{\, X^\mu\ , X^\nu\,\right\}_{PB}\equiv \epsilon^{mn}\,\partial_m \, X^\mu\,\partial_n\, X^\mu$$

is the Poisson Bracket and

$$\left\{\, X^\nu\ , X^\rho\ ,X^\sigma\right\}_{NPB}\equiv \ep... ...jk} \partial_i X^\nu \, \partial_j X^\rho\,\partial_k X^\sigma$$

is the Nambu-Poisson Bracket. The first term in (2) describes a bulk three-brane, or bag, which in four dimensions is a pure volume term characterized by a pressure $\mu^4_0$. All the dynamical degrees of freedom are carried by the second term in (2), where $\kappa$ is the membrane tension; this term encodes the dynamics of the boundary, Chern-Simons membrane, enclosing the bag. Tracing back the bulk and boundary terms in the original action (1) it is possible to establish the following formal correspondence

$$\hbox{\lq\lq glue''} \longleftrightarrow \hbox{bulk $3$-brane}$$

$$\hbox{\lq\lq instantons''} \longleftrightarrow \hbox{Chern--Simons boundary $2$-brane} .$$

This scheme, which has been generalized to Yang-Mills theories in higher dimensional spacetime [9], points out that not only strings but bag-like objects fit the large-$N$ spectrum of $SU(N)$ gauge theories. However, a non-trivial dynamics for these spacetime filling objects comes only from boundary effects, described by Chern-Simons terms in the original gauge action.
Against this background, we would like to investigate the existence of non-topological membrane-like excitations in the large large-$N$ spectrum of a $SU(N)$ Yang-Mills theory in four dimensions. Clues suggesting the existence of these objects come from earlier Abelian models [10] and the recent conjectures about $M$-Theory. $M$(atrix) theory is the, alleged, ultimate non-perturbative formulation of string theory. More in detail, two models have been constructed as possible non-perturbative realization of Type IIA [11] and Type IIB [12] string theory. The matrix formulation of Type IIB strings is provided by a large-$N$, $10$-dimensional super Yang-Mills theory reduced to a single point

$$S_{IKKT}= -{\alpha\over 4}\, \mathrm{Tr} \left[\,\mbox{\text... ...f {}A}}_\mu\ , \mbox{\textit{\bf {}A}}_\nu\, \right]^2+\dots .$$ (3)

The dots refers to the fermionic part of the action which is not relevant to our discussion. The model (3) has a rich spectrum of extended objects. Our investigations in [8] and [9] has been initially triggered by the formal analogy between (3) in $10D$ and quenched Yang-Mills theory in $4D$.
Matrix description of Type IIA strings is given in terms of $0$-branes quantum mechanics

$$S_{BFSS}= {1\over 2g_s}\, \mathrm{Tr}\, \left(\,{ d\mbox{\te... ...^i\ , \mbox{\textit{\bf {}X}}^j\, \right]^2 +\dots \,\right) ,$$ (4)

where $i=1 , \dots , 9$. Again, the $0$-branes matrix coordinates can be seen as Yang-Mills fields reduced to a line.
In this paper we would like to ``reverse the path'' leading Type IIA strings to the matrix model (4) and show how a $3D$ version of (4) can be obtained from the canonical formulation of an $SU(\,N\,)$ Yang-Mills theory through a modified quenching prescription. Then, we are going to extract a non-relativistic, dynamical $2$-brane from the large-$N$ spectrum of the model by following the procedure introduced in [8] and [9].

The paper is organized as follows: in section 2 we start from $SU(N)$ Yang-Mills theory in four dimensions and obtain a corresponding matrix theory through the quenching approximation; two different type of quenching are discussed in section 2.1 and section 2.2; in section 3 we study the large-$N$ limit of the matrix model introduced in section 2.2 and use the Weyl-Wigner-Moyal map to get the action for a membrane; we conclude the paper by computing the mass spectrum of this membrane in the WKB approximation.