3.2 Magnetic Stückelberg Mechanism

The procedure of the previous section in which we applied the Stückelberg mechanism to restore gauge invariance in a theory of open extended objects, can be readily generalized to the case in which there is also a magnetic charge present. Then we are at work with the following objects:

1. an electric open $p$-brane, with a electric closed $(p-1)$-boundary;
2. a magnetic open $(D-p-3)$-brane, with a magnetic closed $(D-p-4)$-boundary.

Thus the action for the system is a natural generalization of (33):

$$S = \int \left[ -{1\over 2}\left(\, F _{\, (p+2)} (A)- G _{\, (p+2)} ... ...{p}\, \left(\, A _{(p+1)}- d C _{(p)} \,\right)\, J ^{\, (p+1)} _{e}\,\right] +$$

$$\qquad + { \kappa\over 2 }\, \int\, \left(\, A ^{\, (p+1)} - d C ^{(p)} \, \right) ^{2}$$ (47)

If we proceed as in appendix C.2, the final result for the dual action is

$$Z [ \, \overline{N} _{\mathrm{e}} , \overline{J} _{\mathrm{g}} \, ] = {1 \over Z [ \, 0 , 0 \, ]} \int [ {\mathcal{D}}\overline{D} ] [ {\mathcal{D}}I ] e ^{ - S [ \, \overline{D} ,\overline{J} _{\mathrm{g}} \, ]}$$

$$S [ \, D , B , \overline{N} _{\mathrm{e}} , \overline{J} _{\mathrm{g}} \, ] = - \frac{1}{2 \kappa} \int \left[\, d D ^{\, (D-p-2)} -\imath\, (-1)... ...1)-(p+1)^2} e _{p} \, {}^{\ast} \overline{N} _{e} ^{\, (D-p-1)}\, \right] ^{2}+$$

$$+ {1\over 2}\int \left[\, D ^{\, (D-p-2)} + (-1) ^{Dp} d B ^{\, (D-p-3)}\,\right] ^{2} +$$

$$- \imath\,\int \left [\, (-1)^ {Dp} N ^{ (D-p-2)}\, D _{(D-p-2)} + ... ..., }\, B _{(D-p-3)} \overline{J} _{\mathrm{g}} ^{\, (D-p-3)} \, \right ] \quad .$$ (48)

Let us take a closer look at the dual amplitude (48). The most evident feature of (48) is the role exchange between conpensator and gauge field with respect to the electric phase. Now, the dual Stückelberg strength tensor $D$ has been lifted to the role of gauge potential, while the dual gauge potential $B$ plays the role of conpensator, implementing the new gauge symmetry

$$\delta_\Lambda D _{(D-p-2)}= d \Lambda_{(D-p-3)}$$ (49)

$$\delta_\Lambda B _{(D-p-3)}= (-1) ^{Dp}\Lambda_{(D-p-3)}$$ (50)

Solving the field equations in the dual phase one gets the effective action:

$$S[\,\widehat{ N}\ , \overline{J} _{\mathrm{g}}\ , J_e\,]= - {\kappa\over 2 }\int \widehat{N} ^{ (D-p-2)} { 1\over \Box +\kappa } \widehat{N} _{ (D-p-2)}+$$

$$+ \int \left[\, { 1\over 2 }\, g_{D-p-4 \, }^2 \overline{J} _{\math... ... \kappa} j _{\mathrm{e}} ^{\,(p)} { 1\over \Box } j _{\mathrm{e}\,(p)}\,\right]$$ (51)

where the action of duality transformation becomes even more evident. The bulk interaction is short range while the boundary interaction is still long range. Accordingly, the following pattern of duality relations shows up

$$A_{(p+1)}\longleftrightarrow D ^{\, (D-p-2)}$$ (52)

$$\widetilde{J}^{(p+1)} \longleftrightarrow \widehat{N} _{ (D-p-2)}$$ (53)

$$C_{(p)}\longleftrightarrow B ^{\, (D-p-3)}$$ (54)

$${ e_p\over \sqrt \kappa} j _{\mathrm{e}} ^{\,(p)}\longleftrightarrow g_{D-p-4 }\overline{J} _{\mathrm{g}} ^{\, (D-p-3)}$$ (55)

$A_ { (p+1)}$ which in the original phase is coupled to the electric $p$-brane current, transforms into $D ^{\, (D-p-2)}$ and interacts in a gauge invariant way with a $D-p-3$-brane. The Stuckelberg mechanism provide a mass $\sqrt\kappa$ to $A_ { (p+1)}$ and the same mass to $D ^{\, (D-p-2)}$. Accordingly, the bulk interaction between electric or magnetic currents is short-range. In the original action (47) gauge conpensator is $C_{(p)}$, while in the dual action the Stückelberg field is the dual gauge potential $B^{\, (D-p-3)}$. The dynamics of the model removes the conpensator fields from the bulk spectrum and confines them on the brane boundary, where they mediate a long-range interaction.
Finally, in the dual phase the electric parent brane disappears as a physical object. Only $\widehat{N} _{ (D-p-2)}$ and $\overline{J} _{\mathrm{g}} ^{\, (D-p-3)}$ enter as conserved sources in the field equation. The only physical electric source is the divergence free current $j _{\mathrm{e}} ^{\,(p)}$ providing boundary electric/magnetic duality symmetry. A similar phenomenon was discovered by Nambu to occur in the Dual String Model of Mesons [14]. He showed that the mesonic open string acquires physical reality when propagating in the non-trivial Higgs vacuum because of the interaction between the end points magnetic monopoles and the charge of the scalar field. On our part, we have considered higher dimensional open objects and replaced the Higgs mechanism with the tensor mixing mechanism as the basic engine of mass production at the level of $p$-brane dynamics.