A. Definitions

The vacuum amplitude in the presence of an external $p$-brane current can be written as a path-integral over the gauge field configurations

$$Z [ \, J \, ] = {1 \over Z[ \, 0 \, ]} \int [ {\mathcal{D}}A ] \, e^{- S [ \, A , J \, ]}$$ (57)

$$S [\, A , J \, ] = \int \left[ \, - \frac{1}{2} F _{(p+2)} ( A ) F ^{\, (p+2)} ( A ) + e _{p}\, A _{(p+1)} J ^{\, (p+1)} \, \right] \quad ,$$ (58)

where $F _{(p+2)}\equiv d A _{(p+1)}$, and the $p$-brane current is given by

$$J ^{\, (p+1)} \left(\, x\, ;\, E(\xi)\, \right) = \int _{... ...} \, \delta ^{D)} \left[ \, x - E ( \xi ) \, \right] \quad ,$$ (59)

where $E$ is the image manifold in the target spacetime, through the embedding $x=E(\xi)$ , of a ``fiducial'' $(p+1)$-dimensional domain $\Xi$, parametrized by $p+1$ world coordinates $\xi ^{i}$.
The tensor gauge potential $A_{\, \mu_1\dots \mu_{p+1}}(x)$ is assumed to be a single-valued and integrable function,

$$\partial _{\, [\, \lambda _{1}} \partial _{\lambda _{2} \, ]} A _{\, \mu _{1} \dots \mu _{p+1}} = 0$$ (60)

$$\partial _{\, [\, \lambda _{1}} \partial _{\lambda _{2} \, ]} \Lambda _{\, \mu _{2} \dots \mu_{p+1}}=0$$ (61)

where $\Lambda_{(p+1)}$ is the gauge function.

Following Dirac[7], we can introduce extended magnetic objects as those singular manifolds where the Bianchi are broken:

$$(\, {}^{\ast} d \overline{F}) ^{\, (D-p-3)} = g _{D-p-4 \, }\, \overline{J} ^{\, (D-p-3)} \quad .$$ (62)

Here

$$\overline{J} ^{\, \mu _{1} \dots \mu _{D-p-3}} \left(\, x \... ..., \delta ^{D)} \left[ \, x - M ( \sigma ) \, \right] \quad ,$$ (63)

and

For the benefit of the reader, the symbols associated with the definition of the world-manifold, fiducial space, parametrization, embedding functions and tangent elements of the various objects are listed in Table 1. Thus, in a way completely analogous to the electric case, ${\mathcal{M}}$ represents the world-manifold of the magnetic brane, with fiducial manifold $\Sigma$, parameters $\sigma ^{i}$, embedding functions $M ^{\mu}$, and tangent element $\dot{M} ^{\, \mu _{1} \dots \mu _{D-p-3}}$.

The violation of the Bianchi identities implies that one cannot associate a single valued gauge potential with a magnetic brane. Then, the simplest way to incorporate an extended magnetic object in the action (58) is to follow Dirac's prescription and introduce an open $(D-p-2)$-brane ${\mathcal{N}}$ having as its only boundary the magnetic brane trajectory ${\mathcal{M}}$:

$$N ^{\, \mu _{1} \dots \mu _{D-p-2}} \left( x \ ; n(\gamma) \right) = g _{D-p-4 \, } \int _{\Gamma} d ^{D-p-2} \gamma\, n^{\mu _1}\wedg... ... _{D-p-2}}\, \delta ^{D)} \left[ \, x - n \left( \gamma ^{i} \right) \, \right]$$ (64)

$$\partial_{\, \mu _1 } N ^{\, \mu _{1} \mu _{2}\dots \mu _{D-p-2}} = g _{D-p-4 \, } \overline{J} ^{\mu _1 \mu _{2} \dots \mu _{D-p-2} } \quad .$$ (65)