1 Introduction

The relation between gauge theories and relativistic extended objects is one of the most intriguing open problems currently under investigation in high energy theoretical physics. Gauge symmetry is the inspiring principle underlying unification of fundamental forces at the quantum level, gravity not included. A really unified theory, including a consistent quantum theory of gravitational phenomena as well, forces the introduction of relativistic extended objects as the basic building blocks of matter, space and time. If correct, this picture must be able to account for the low energy role of gauge symmetry. The presence of massless vector excitations, carrying Chan-Paton indices in the massless sector of the open string spectrum, is a first step towards the answer of this problem in a perturbative framework. The recent proposals for a non-perturbative formulation of string theory in terms of matrices and $D$-branes [1] provides further clues in favor of the strings/gauge fields. The problem is equally difficult to deal from the low energy viewpoint, involving non-perturbative aspects of gauge theory. Looking for extended excitations in the spectrum of Abelian gauge theories is a problem dating back to the seminal Dirac's work about strings and monopoles [2]. Recent generalization to higher rank gauge fields has been given in [3], [4], [5]. In the non-Abelian case the problem is even more difficult because of the interplay with confinement [6]. Thus, it can be dealt within some appropriate approximation scheme. Because of the the large value of the gauge coupling constant standard perturbation theory is not available and different computational techniques have to be adopted. One of the most successful is the large-$N$ expansion, where $N$ refers to the number of colors [7]. To match Yang-Mills theory and matrix string theory further approximations are available, i.e. ``quenching'' and ``reduction''. The original $SU(N)$ Yang-Mills field is replaced by the same field at a single point [8], say $x^\mu=0$ (for a recent review see [9]) and represented by a unitary $N \times N$ matrix ${\mbox{\boldmath {$A$}}} _{\mu} {}^{i} _{j}$. 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 [10]. Accordingly, the covariant derivative becomes $\mathrm{i}{\mbox{\boldmath {$D$}}}_\mu = \left[ {\mbox{\boldmath {$p$}}}_\mu + {\mbox{\boldmath {$A$}}}_\mu , \dots \right]$. Thus, the reduced, quenched, Yang-Mills field strength is

$${\mbox{\boldmath {$F$}}}_{\mu\nu}{}^i_j\equiv \left[\mathr... ...$}}}_\mu, \mathrm{i}{\mbox{\boldmath {$D$}}}_\nu \right]^i_j$$

It has been shown in [11] that the dynamics of reduced, quenched, Yang-Mills theory can be formulated in the large-$N$ limit in terms of pure string dynamics. In a recent paper we have generalized this result to include bags and membranes in the spectrum of $4$-dimensional Yang-Mills theory [12]. In this note, we shall look for higher dimensional objects fitting into generalized Yang-Mills theories, in more than $4$-dimensional spacetime.

Yang-Mills theory admits in $D=4k$, $k=1, 2,\dots$ dimensions a generalization preserving both the canonical dimension of the gauge field, i.e. $\left[ {\mbox{\boldmath {$A$}}}_\mu \right]=( \mathrm{length} )^{-1}$, and of the coupling constant $\left[ g_{\mathrm{YM}} \right]=1$. The action we shall use is of the form introduced in [13]

$$S^{\mathrm{GYM}}=-{1\over 2 (2k)! g^2_{\mathrm{YM}}} \int ... ...ot\dots\cdot {\mbox{\boldmath {$F$}}}^{\mu_{2k-1}\mu_{2k}]} ,$$ (1)

where ${\mbox{\boldmath {$F$}}}_{\mu_1\mu_2}\equiv \partial_{[\mu_1} {\mbox{\boldmath ... ...left[ {\mbox{\boldmath {$A$}}}_{\mu_1}, {\mbox{\boldmath {$A$}}}_{\mu_2}\right]$, and the trace operation is over internal indices. The action $S^{\mathrm{GYM}}$ can be supplemented by a topological term extending to $4k$ dimension the usual $\theta$-term:

$$S^{\theta}=-{\theta g^2_{\mathrm{YM}}\over 4\pi^2 4 ^{k}} ... ...ot\dots\cdot {\mbox{\boldmath {$F$}}}_{\mu_{4k-1}\mu_{4k}]} .$$ (2)

The main purpose of this note is to establish a correspondence between the action $S^{\mathrm{GYM}}+ S^{\theta}$ and some appropriate brane action.