3 Conclusion

In summary, we have shown that reduced, quenched, $SU(\, N\,)$ gauge theory can fit in four dimensions, in the large-$N$ limit, not only strings but three-branes with a dynamical boundary, as well. The WWM correspondence maps the original matrix action (2) into the phase space action (16) where the gauge field strength is replaced by a Moyal bracket for the Weyl symbol of matrix gauge field. The large-$N$, or classical, limit of (16) reproduces the Dolan-Tchrakian action $S_{\mathrm{DT}}$ for a three brane, in the conformally flat background geometry (21), and a Chern-Simons action for its boundary. In even dimensional target spacetime the $S_{\mathrm{DT}}$ functional matches the Schild action (23). In analogy with the string case, the action (23) is not invariant under reparametrization but it is only invariant under residual world volume preserving diffeomorphisms. With hindsight, this is not a surprise: the reduced symmetry seems to be the memory of the constant inverse volume factor $\left(\, {2\pi\over a}\,\right)^4$ in front of the original reduced quenched action (2). A similar conclusion was obtained in [22] where the light-cone gauge choice leads to a residual $p$-volume preserving diffeomorphisms, while we have $p+1$ world volume preserving diffeomorphisms. In the limiting case $p=3$, $D=4$ the $S_{DT}$ action degenerates into a pure volume term with no proper dynamics. All the physical degrees of freedom are carried by the Chern-Simons membrane enclosing the bag. By tracing back the bulk and boundary terms to the original Yang-Mills action, the following correspondence will show up:

$$\hbox{\lq\lq glue'' :}\qquad S^{\mathrm{qYM}}_{\mathrm{red}} \longleftrightarrow S_{DT}\propto \hbox{ Bulk Volume}$$

$$\hbox{ instantons :}\qquad S^{\mathrm{q\, \theta}}_{\mathrm{red}} \longleftrightarrow S_{CS}\propto \hbox{ Boundary Membrane}$$ (33)

Finally, an order of magnitude estimate of the induced bag constant results to be in agreement with the phenomenological value, and suggests a model for hadrons as $QCD$ vacuum bubbles bounded by Chern-Simons membranes. This new formulation of the hadronic bag model, and its generalization to the case of higher dimensional branes is currently under investigation [23].