2 Electric and Magnetic branes

In this section we briefly review the way in which the interaction of a $p$-brane with an antisymmetric tensor field can be described. This is nothing but a higher dimensional generalization of what is usually done in the electromagnetic theory: there we have a $0$-dimensional object, namely a point particle, sweeping a $1$-dimensional world-line. The most natural (and ``geometric'') way to couple the particle with a field is through its tangent element; this is a vector field in the tangent bundle and can be coupled easily with a $1$-form. If instead of the $p=0$ case we generalize this method to arbitrary values of $p$ we have $p$-dimensional objects with $(p+1)$-dimensional tangent elements, interacting with $(p+1)$-forms. As in the point-particle case we can associate a current with the point-charged-particle, even in higher dimensions giving a charge to the (now extended) object can be formalized with a current (but now with $p+1$ indices): we then have a charged-extended object.

One could think, at this stage, that all properties of electromagnetism should be translated readily to the extended case without any problem; and indeed it is so, provided all the property of the extended object on which our higher dimensional generalization relies are alla and the same mathematical properties of the point-particle case. But one of these properties, and an important topological one, is that a particle has no boundary, i.e. it is a closed object. So all the point-particle results can be straightforwardly extended only to closed extended objects; or at least we expect a different behavior for closed and open ones.

Let us call $J ^{\, (p+1)}$ the current associated with the $p$-brane ( see (59) in appendix A for precise definitions of the various quantities involved) and $A _{\, (p+1)}$ the $p+1$-dimensional gauge potential. If the $p$-brane is closed, that is, if its world-manifold, let us call it ${\mathcal{E}}$, has no boundary, $\partial {\mathcal{E}} = \emptyset$, then the current $J$ is divergenceless,

$$\partial J ^{\, (p+1)} = 0$$ (2)

and the action for the system (as written in equation (58) of appendix A) is invariant under the tensor gauge transformation:

$$\delta _{\Lambda} A _{(p+1)} = d \Lambda _{(p)} \quad .$$ (3)

Note how charge conservation is now associated to a property of the extended object, namely that it has no boundary, thanks to the geometric way in which we defined the interaction. Moreover this is the way which requires less additional assumptions, apart from covariance. At the same time the cohomological framework plays a deep role at the mathematical level, because we will have at hand forms of different order, in passing from currents and gauge potentials, to fields.

In what follows we will be interested in objects carrying electric as well as magnetic charge. They are described by the following action, whose origin, along the guidelines already traced by Dirac for the magnetic monopole are summarized for completeness in appendix B:

$$S [ \, A , G ; J \, ] = \int \left [ \, - \frac{1}{2} \... ...2} + e _{p}\, A _{(p+1)} J ^{\, (p+1)} \, \right] \quad ,$$ (4)

In the expression (4) we have explicitly separated the standard electromagnetic contribution to the field strength, $F$, from the magnetic field strength $G$, which is the singular part of the electromagnetic field due to the presence of magnetic charge ($x= n(\gamma)$ represents the parent $(D-p-3)$-Dirac-brane). The electric field strength $F = d A$ originates as the exterior differential of the electromagnetic gauge potential $A$, which in turn is coupled to the $(p+1)$-dimensional history of the extended object (an electric $p$-brane) through the source current $J$. As emphasized before since this brane is a closed object, namely without boundary, its source current $J$ is conserved and the full action is gauge invariant under (3). Of course, thanks to the properties of the exterior differential, $d F = 0$, i.e. Bianchi identities are satisfied so that $F$ (as given by $A$) cannot describe magnetic sources. If we insist in having magnetic charges of the same type as the electric ones, it is exactly these Bianchi identities that we have to break: this task is performed by the $G$ field, since the Bianchi identities stemming from (4) (they are explicitly written in appendix B, equation (62)) are not satisfied on ${\mathcal{M}}$, i.e. the magnetic brane. What is important from the point of view we are proposing here is to summarize what kind of extended objects are part of the game. We have a closed electric $p$-brane source, $J$, coupled to the tensor potential $A$ from which a standard $F$ fields stems, and a closed magnetic $(D-p-4)$-Dirac-brane, with source current $\overline{J}$, responsible for the violation of the Bianchi identities for $F$. We also spend a few words to give a simple intuitive explanation of the dimensionality: if we have a $p$-brane, its world manifold is $(p+1)$-dimensional and its source current is a $(p+1)$-vector, which couples to a $(p+1)$-form. Then the field is a $(p+2)$-form whose dual (which is relevant for Bianchi identities) is a $(D-p-2)$-form. The divergence of the dual field equals the magnetic current, which is thus a $(D-p-3)$-vector associated with the world-history of a $(D-p-4)$-brane. Moreover note that the magnetic brane, in Dirac's description is the boundary of a parent brane ${\mathcal{N}}$ and thus necessarily closed simply by topological arguments.

The full action (4) exhibits two distinct gauge symmetries: we already commented a few lines above about the original gauge symmetry (3) associated with the fact that the electric brane is closed and under which $G$ is inert; in addition, there is a new magnetic gauge invariance under the combined transformations

$$G ^{\, (p+2)} ( x ; {\mathcal{U}} ) \longrightarrow G ^{\, (p+2)} ( x ; {\mathcal{V}} ) + g _{D-p-4 \, }\, d (\, {}^{\ast} \,\Omega\,) ^{\, (p+1)} ( x ; {\mathcal{O}} )$$ (5)

$$A ^{\, (p+1)} \longrightarrow A ^{\, (p+1)} + g _{D-p-4 \, } \, (\, {}^{\ast} \Omega\, ) ^{\, (p+1)} ( x ; {\mathcal{O}} ) \quad ,$$ (6)

which is nothing but the freedom in the choice of the parent Dirac brane. Since in the extended objects description there is the described correspondence between branes and the corresponding source currents, then invariance properties of geometrical character associated with the choice of a brane can be translated in mathematical properties of the corresponding current. In this case, since the magnetic brane is a boundary, then its current is a boundary current, i.e the divergence of a higher rank form; clearly we have some freedom in the choice of the bulk current, of which the magnetic current is the divergence and this can be interpreted in terms of the gauge transformation given above provided that the electric and magnetic charges satisfy the Dirac quantization condition

$$e _{p} \, g _{D-p-4 \, } = 2 \pi n \quad , \qquad (\hbox{in units $\hbar = c = 1$})$$ (7)

where, $n=1,2,3,\dots$. In the absence of the electric current, the action (4) is gauge invariant under (5) and (6). However, in the presence of the $(A \, J)$-interaction term, the action $S [ \, A , G ; J \, ]$ is shifted by a quantity

$$\delta _{\Omega} S = (-1) ^{p(D-p)} \, e _{p} \, g _{... ...1)} (x ; {\mathcal{O}}) J ^{\, (p+1)} (\, x\ ; E \,) \quad .$$ (8)

The target space integral of ${}^{\ast} \Omega J$ is an integer equal to the number of times that the two branes intersect. Thus,

$$\delta _{\Omega} S = (-1) ^{p(D-p)} \, e _{p} \, g _{D-p-4 \, } \, \times \hbox{integer number} \quad .$$ (9)

If the Dirac quantization condition holds, then

$$\delta _{\Omega} S = \pm 2 \pi \times \hbox{integer number} \quad .$$ (10)

Accordingly, the (Minkowskian) path-integral is unaffected by a phase shift $S \rightarrow S \pm 2 \pi n$, and the gauge transformation (5), (6), has no physical consequences on the interacting system (4).

Our specific purpose, now is to extend the ``dualization'' procedure for a generalization of the system described by (4), i.e. for a system of open $p$-branes coupled to higher rank gauge potentials. By this extension we are led to a relationship between open and closed objects that is ultimately responsible for the mass generation mechanism that we are proposing here.

As a preliminary step we will comment about the same result for closed objects: it is derived in subsection C.1 of appendix C with a method different than usual, namely using path-integrals. In this description it is particularly transparent the role played by boundary and bulk currents, and by the field equations/Bianchi identities: these can be implemented as constraints in the path-integral formalism ( see appendix (C.1) for the detailed computation) and it is about this point that we will briefly comment now. In particular, if we consider the path integral for the partition function of a system described by (4) we see that the gauge potential $A$ appears only through its field strength $F(A)$ for the interaction term can be written as $N\cdot F(A)$ after an integration by parts, where $N\equiv \partial J$ is an ``electric parent current'':

$$S _{\mathrm{INT}} = - e _{p} \int \overline{N} ^{\, (p+2)} \, F_{(p+2)}( A) \quad .$$ (11)

Accordingly, one would expect to be able to write the path integral in term of the gauge invariant variable $F$ in place of the gauge potential $A$. However, in switching to such a field strength formulation one has to be careful if extended magnetic objects are present. In this case, the field strength is the sum of the curl $dA$ and a ``singular magnetic field strength '' $G$ :

$$F(A)\longrightarrow \overline{F}=dA -G$$ (12)

$G$ is chosen such that the monopole current $\overline{J}$ enters as a source in the ``Bianchi Identities'' for $\overline{F}$:

$${ }^* d \overline{F}= g \overline{J}$$ (13)

The ``new'' Bianchi Identities (13) are encoded into the path integral through the following general relation:

$$\int [\,{\cal{D}} A\,]W[\, dA\ , J\ , \overline{J}\,]= \int ... ...verline{J} \,\right] W[\, \overline{F}\ , N\ , \overline{J}\,]$$ (14)

where, $n(\xi)$ and $\overline{n}(\sigma)$ are the embedding functions of the electric and magnetic parent branes respectively. We suppressed all the non essential labels in (14). Indices, coupling constants, numerical factors has been reinserted in (83).
Performing the computations described in appendix C.1 we arrive at the result for the dual action of (4), which is

$$S [ \, H , G , B , \overline{N} , \overline{J} \, ] = - \int \, \left[ \, d B ^{\, (D-p-3)} - \imath\, (-1) ^{D(p+1)} e _{p} \, (\, {}^{\ast} \overline{N}\, ) ^{\, (D-p-2)} \, \right] ^{2} +$$

$$\qquad - \imath\, \int\left[\, g _{D-p-4 \, }\, B _{(D-p-3)}\overline{J} ^... ...} -e _{p}(\, {}^{\ast} N\, ) ^{\, (p+2)} \overline{N} _{(p+2)}\,\right] \quad .$$ (15)

The first term in (15) is the kinetic term for the dual field strength. The regular part is the curl of the dual gauge potential, while the singular part is the dual of the electric brane current carrying the electric monopole on its boundary. It is the dual of the original kinetic term in (4).
The second term in (15) represents the coupling between the dual potential and the magnetic brane current. The strength of the coupling is represented by the magnetic charge $g _{D-p-4 \, }$. It is worth observing that the strength of this coupling is the inverse of the original electric coupling thanks to the Dirac quantization condition (10). Thus, an effect of the dualization procedure is to reverse the value of the coupling constant. Hence, a system of strongly coupled electric branes can be mapped into a system of weakly coupled magnetic branes and viceversa. In current parlance, this is an example of ``S-duality'' connecting the strong-weak coupling phases of a physical system [10].

Finally, the third term in (15) describes a contact interaction between ``parent'' branes. This term raises a potential problem: as we have seen the Dirac brane is a ``gauge artifact'' in the sense that its motion can be compensated by an appropriate gauge transformation. However, when the electric or magnetic brane coordinates are varied, extra contributions may enter the equation of motion leading to physical effects. In order to avoid this inconsistency we have to invoke the ``Dirac veto'', namely, the condition that the Dirac brane world surface should not intersect the world surface of any other charged object. With this condition, no extra contribution comes to the equation of motion. In this connection, we must emphasize that the physical object is the dual brane, coupled to the $B$-potential, and not the Dirac brane which still maintains its pure gauge status.
Finally, note that the dual current $\overline{J} ^{\, (D-p-3)}$ is divergenceless. Hence, it has support over the world-manifold of a closed $(D-p-4)$-brane and acts as a conserved source in the r.h.s. of the $B$-field Maxwell equation

$$\partial\left[\, d\, B^{\, (D-p-3)} - \imath\, (-1) ^{p+1} ... ... \imath\, g _{D-p-4 \, }\, \overline{J} ^{\, (D-p-3)} \quad .$$ (16)

In order to check the consistency of the above results, consider the two systems defined by equations (4) and (15) in a vacuum, i.e., when the sources $J$ and $\overline{J}$ are switched off. In this case, we have two sets of field equations and Bianchi identities. The first set is given by

$$\partial\, F ^{\, (p+2)} ( A ) = 0 \quad , \qquad \hbox{Maxwell equations}$$ (17)

$$d\, F ^{\, (p+2)} ( A ) = 0 \quad , \qquad \hbox{Bianchi identities}$$ (18)

while the second set involves the $B$-field

$$\partial\, d\, B^{\, (D-p-3)} = 0 \quad , \qquad \hbox{Maxwell equations}$$ (19)

$$d\, d\, B^{\, (D-p-3)} = 0 \quad , \qquad \hbox{Bianchi identities} \quad .$$ (20)

Thus, we recover the familiar result that the (classical) ``duality rotation''

$$F ^{\, (p+2)} = {} ^{\ast} d\, B ^{\, (D-p-3)}$$ (21)

exchanges the role of Maxwell equations and Bianchi identities:

$$\hbox{$A$-Maxwell equations} \longleftrightarrow \hbox{$B$-Bianchi identities}$$ (22)

$$\hbox{$B$-Maxwell equations} \longleftrightarrow \hbox{$A$-Bianchi identities}$$ (23)

Of course, the same relations hold true if we switch on both the electric and magnetic sources. In conclusion, by manipulating the path-integral representation of $Z [ \, J \, ]$ we have obtained the duality relation between relativistic extended objects of different dimensionality [1]

$$\widetilde{p} = D-p-4 \quad ,$$ (24)

and the code of correspondence between dual quantities can be summarized as follows

$$A ^{\, (p+1)} \longleftrightarrow B ^{\, (D-p-3)}$$ (25)

$$\hbox{Field equations (in vacuum)} \longleftrightarrow \hbox{Bianchi identities}$$ (26)

$$J ^{\, (p+1)} \longleftrightarrow (\overline{J}) ^{\, (D-p-3)}$$ (27)

$$\hbox{closed\ } \hbox{$p$-brane} \longleftrightarrow \hbox{closed\ } \hbox{$(D-p-4)$-brane}$$ (28)

$$Z [ \, J \, ] \longleftrightarrow Z [ \, I \, ] \quad .$$ (29)

For the convenience of the reader, a ``dictionary'' of the various fields and currents in our model is listed in table 3 and table 4. Finally, it seems worth observing that the dual action (15) is gauge invariant under ``magnetic gauge transformations''

$$\delta _{\tilde{\Lambda}} B ^{\, (D-p-3)} = d \widetilde{\Lambda} ^{(D-p-4)} \quad .$$ (30)

Accordingly, $B$ is a massless field which is a solution of the field equation (19).