1 Introduction

Electric-magnetic duality for closed $p$-branes embedded in a $D$-dimensional target space was established many years ago[1], leading to the general correspondence that the dual of a $p$-brane is a $\widetilde p$-brane with $\widetilde p=D-p-4$. Table 2 illustrates this correspondence. For instance, the brane solution of $(D=11)$ supergravity is the ``magnetic dual'' of an electric membrane, and there is a consensus that this type of solutions may represent the basic geometric elements of a unified theory of all fundamental interactions. Among the open questions that such a ``final'' theory must address, a preeminent one concerns the mechanism of mass generation in the universe. The celebrated Higgs mechanism was invented and successfully applied within the framework of local quantum field theory, i.e., a ``low energy'' framework dealing with the interactions of point-particles in the Standard Model. Since point-particles are currently thought of as low energy manifestations of the underlying dynamics of strings and higher dimensional objects, it seems pertinent to ask what is the ``engine'' of mass production at the level of $p$-brane dynamics, say at the string scale of energy and beyond. Recent developments in non-perturbative string theory [2] suggest a geometrical picture of the Higgs mechanism in terms of open strings stretched between pairs of $D$-branes.
With such a picture in mind, we shall extend the notion of electric-magnetic duality to the case of open $p$-branes. As a spin-off effect, this extension will enable us to illustrate a new mechanism of mass generation which stems directly from a non-trivial interaction between different rank bulk and boundary gauge fields .
On the mathematical side, there are at least two novel aspects in our approach which stem from the application of path-integral techniques to open $p$-branes. It is only recently that the path integral for open $p$-branes has been considered in the literature, mainly in connection with $D$-branes physics.
The first element of novelty, in our approach, is a modified field strength formulation of higher rank gauge theories, including both ``electric'' and ``magnetic'' objects. The presence of extended magnetic monopoles requires a careful treatment of the Bianchi Identities, which are instrumental in the dualisation process.
A second original feature of our approach is that dualisation proceeds in two different steps. To protect the gauge invariance of the model against `` leakage '' across the $p$-brane boundary one introduces a Stückelberg compensating field. Then duality transformation is first applied to the Stückelberg sector of the model, and then to the remaining gauge part of the partially dualised action. The final output is a massive, gauge invariant theory for higher rank tensor and currents, written in terms of dual gauge potential and dual Stückelberg field.
The interaction between $(p+1)$-rank gauge potential $A_ { (p+1)}$ and the $(p)$-rank conpensator $C_{(p)}$ leads to interesting physical effects: the most evident one is a gauge invariant mass term for $A_ { (p+1)}$, but it is not the only one. The main result of our work is a pair of (semi)classical effective action describing, in a gauge invariant way, the interaction among electric and magnetic branes both in the original, ``electric'', phase and in the dual, ``magnetic'', phase. In the two actions bulk and boundary dynamics are clearly splitted. While the bulk interaction is screened by the mass of the gauge tensor field, the boundary interaction is still long range, as a memory of the Stückelberg massless gauge conpensator. This is an unexpected result suggesting some sort of holographic principle is at work: even when the Stückelberg field is ``eaten'' in a Goldstone-type way by the gauge potential and disappear as a physical excitation in the bulk of the brane, a footprint of the associated long range interaction is recorded on the boundary of the brane itself.
We planned the paper by confining technical computational details to the appendices.
In Sect.II we give a brief introduction to the duality symmetry among electric and magnetic $p$-branes. We recover the quantization condition of electric and magnetic charges as a consequence of the arbitrariness in the choice of a the Dirac-type `` parent brane '' matching with the physical object world manifold.
In Sect.III we first introduce the Stückelberg mechanism for open, electric, $p$-branes. Then, we perform a duality rotation and study the Stückelberg mechanism in the ``magnetic phase'' of the model.
Sect.IV is devoted to a summary and discussion of the results we obtained.
In App.A we enlist the basic definitions we use throughout the paper.
In App.B we introduce the ``magnetic field strength'' and discuss its symmetry properties.
In App.C we provide the details of the computations leading to the dual action for a system of interacting electric and magnetic $p$-branes.
the dual equations and references [3] with the duality 3. 5 mass In order to avoid an unpleasant proliferation of indices we shall adopt the following notation:

$$A^ {( p+1) } \equiv A^ {\mu_1\dots \mu_ { p+1} }$$

$$A_ {( p+1 )} \equiv A_ {\mu_1\dots \mu_ { p+1} }$$

$$A_ {( p+1 )} J^ { ( p+1 )}\equiv {1\over (p+1)! } A_ {\mu_1\dots \mu_ { p+1} } J ^{\mu_1\dots \mu_ { p+1} }$$

$$\partial J^{ (p+1) }\equiv\partial_ { \mu_1 }J ^{\mu_1\dots \mu_ { p+1} }$$

$$d J^{ (p+1) }\equiv\partial^ {[\, \nu_1 }J ^{\mu_1\dots \mu_ { p+1}\,] }$$

$$d A_{ (p+1) }\equiv\partial_ {[\, \mu_1 }A _{\mu_1\dots \mu_ { p+1}\,] }$$

$${ 1\over \Box} J^{ (p+1) }(x)\equiv \int dy\, G(\, x-y\, ) J^{ (p+1) }(y)$$

$${ }^* K^ { (p+1) } \equiv {1\over (D-p-1 ) !} \epsilon ^{\mu_1\dots\mu_{p+1} \mu_ {p+2}\dots\mu_d}K_{\mu_ {p+2}\dots\mu_d }$$

$$D_ { (D-p-2) } {} ^* d I_{ (p+1) }= (-1)^ { Dp } I^ { (p+1) }{} ^* dD^ { (D-p-2) }$$

$$N ^ { (p+1) }= (-1)^ { D(p+1)- (p+1)^2 } {} ^*[\, {} ^*( N ^ { (p+1) }\,)\,]$$ (1)

which means that the upper and lower indices are reminder of the actual number of indices carried by the corresponding tensor. The full index notation is restored only in the appendices where we provide, for the interested reader, detailed calculations leading to the results presented in the various sections. To our knowledge, these computational details cannot be found elsewhere in the literature, especially in the case of open branes. We think the approach we shall introduce deserves some attention in view of the of the increasing importance of extended open objects, like $D$-branes, in the formulation of non-perturbative string theory. Moreover, the derivation of (7) is an original extension of the method introduced in [5] for pointlike charges and magnetic monopoles.
As a further notational simplification, we shall understand $d^Dx$ in all spacetime integration.
Finally, we shall assume $D> p+1$ in order to deal with dynamical extended objects. The limiting case $D=p+1$ requires a separate discussion which will be given elsewhere [6].