4 Conclusion

There are several properties of the dual action (48) which summarize the results of our discussion. To begin with, in Table 5 we have summarized the relevant properties of fields, currents and coupling constants for the open case. In this connection, we observe that the current $K$ has support on the world manifold of a $(D-p-2)$-brane. Hence, in contrast to the case of closed $p$-branes, we conclude that the spatial dimensionality of dual open branes is given by

$$\tilde{p} = D - p - 3 .$$ (56)

Dual open objects in $D=10$ spacetime are listed in Table 8. That list is consistent with the results reported in Ref.[16] through a different approach. By comparing table 8 with table 2, one sees at a glance that the pattern of dual objects in $D=10$ looks like the one for closed objects in $D=11$ supergravity. Far from being an accident, this similarity follows from the relative dimensionality of the boundary of an open brane and the dimensionality of the brane itself. The point is that a closed $p$-dimensional surface can always be considered as the boundary of an open $(p+1)$-dimensional volume. With hindsight, it is not surprising that the relation (56) can be obtained from (24) by replacing $p$ with $p+1$. However, the formal relationship (56) reflects a deeper physical phenomenon, namely, the occurrence of a mass term in the dual action (48). The origin of this term can be traced back to the introduction of the Stuckelberg field which is necessary to compensate for the leakage of symmetry through the boundary of the $p$-brane [17]. Geometrically, the introduction of the compensating field is tantamount to ``closing the surface'', thereby restoring gauge invariance, albeit in an extended form. The net result of the whole procedure is that the gauge field acquires a mass through the ``mixing'' of different gauge potentials. In this connection, it is worth emphasizing the different role of the dual gauge potential $B$ in the closed and open case. If the $p$-brane is closed, $B$ is the physical field interacting with the dual brane, as it is manifest from the action (15). On the other hand, if the brane is open, $B$ represents only the gauge part of $\overline{D}$, according to equation (110). Moreover, $B$ is massless, and remains such in the closed case. However, in the open case it provides the ``longitudinal degree of freedom'' which endows the physical field $\overline{D}$ with a mass $\kappa$. This sort of Higgs mechanism, involving simultaneously gauge invariance and topology, is a peculiarity of relativistic extended objects. As a matter of fact, a precursor of this mechanism was discussed, in the prehistory of string theory, by Kalb and Ramond for an open string with a quark-antiquark pair at the end points [17], and later extended to the case of an open membrane having a closed string as its boundary [9]. It seems interesting, and suggestive of some deeper topological effect, that this peculiar mechanism for generating mass finds a natural setting in the geometric properties of open and closed $p$-branes.