B. Magnetic Field Strength

We shall exploit the arbitrariness in the choice of ${\mathcal{N}}$ by introducing a new gauge symmetry of the system in the following way. Suppose ${\mathcal{U}}$ and ${\mathcal{V}}$ represent two distinct $(D-p-2)$-branes ``attached'' to the magnetic brane. Let ${\mathcal{O}}$ be a $(D-p-1)$-open brane having ${\mathcal{U}}$, ${\mathcal{V}}$ and the magnetic brane history as its boundary. If

$$\Omega ^{\, \mu _{1} \dots \mu _{D-p-1}} ( x ; {\mathcal{O}} ) = \, \int _{\Omega} d ^{D-p-1}\, \omega\, O^{\, \mu _{1}}\wedge \do... ...^{\mu _{D-p-1}} \, \delta ^{D)} \left[\, x - O \left( \omega \right) \, \right]$$

is the current with support over the $(D-p-1)$-dimensional history $\Omega$, then, by definition

$$g _{D-p-4}\, \partial \, \Omega ^{\, (D-p-1)} \left( x \, ... ... N ^{\, (D-p-2)} \left( x \, ; {\mathcal{V}} \right) \quad .$$ (66)

Moreover, the condition

$$\partial _{\, \mu _{1}} \partial _{\, \mu _{2}} \Omega ^{\, \mu _{1} \mu _{2} \dots \mu _{D-p-1}} = 0$$ (67)

is guaranteed by the following equalities

$$\partial N ^{\, (D-p-2)} \left( x \, ; {\mathcal{U}} \right... ...{J} ^{\, (D-p-3)} \left( x \, ; {\mathcal{M}} \right) \quad .$$ (68)

Thus, we are led to the following relation between the currents associated with two different Dirac branes sharing the same boundary:

$$N ^{\, (D-p-2)} \left( x ; {\mathcal{U}} \right) = N ^{\, (D-p-2)} \left( x ; {\mathcal{V}} \right) + g _{D-p-4 \, }\, \partial \Omega ^{\, (D-p-1)} \left( x ; {\mathcal{O}} \right) \quad ,$$ (69)

$$\partial {\mathcal{O}} = {\mathcal{U}} - {\mathcal{V}} \quad .$$

Equation (69) takes the more conventional form of a ``gauge transformation'' if one switches to the dual currents:

$$(^{\ast} N) ^{\, (p+2)} \left( x \, ; {\mathcal{U}} \right)... ...) ^{\, (p+1)}\,) \left(x \, ; {\mathcal{O}} \right) \quad .$$ (70)

Since it is convenient to work with the dual of the $N$ field, we introduce a magnetic brane field strength, $G ^{\, (p+2)}$, as the dual of the magnetic current $N ^{\, (D-p-2)}$:

$$G _{(p+2)} \left(\, x \ ; W(\gamma)\, \right) \equiv (- 1)^ { D(p+1)} \left(\, {} ^{\ast} N \,\right)_{(p+2)}$$

$$\equiv { g _{D-p-4 }\over (D-p-2)! } \, \epsilon _{\mu _{1} \dots \mu _{p+2} \nu_{1} \dots \nu _{D-p-2}} \times$$

$$\qquad \qquad \times \int _{\Gamma} d ^{D-p-2} \gamma\, n^{\nu_1}... ...-2}} \delta ^{D)} \left[ \, x - n \left( \gamma ^{i} \right) \, \right] \quad .$$ (71)

The main property of the magnetic field strength is the way it transforms under (70), i.e.,

$$G _{\, (p+2)} ( x ; {\mathcal{U}} ) = G _{\, (p+2)} ( x ;... ...^{\ast} ( d \Omega _{\, (p+1)}) ( x ; {\mathcal{O}} ) \quad .$$ (72)

The transformation law (72) allows us to introduce a new gauge symmetry which we shall discuss in a short while. Presently, we can write the action for a system of electric and magnetic branes as follows

$$S [ \, A , G ; J \, ] = \int \left [ \, - \frac{1}{2} ... ...+ e _{p}\, A _{\, (p+1)} J ^{\, (p+1)} \, \right] \quad ,$$ (73)

where, in the magnetic case, we recall that $\overline{F} = F - G$, and that the tensor potential accounts only for the non-singular part of $\overline{F}$, i.e. $F = d A$. The action (4) leads to classical field equations for $A$ which are dual to the Bianchi identities for the whole field strength $\overline{F} = F - G$:

$$\partial \left[ \, F ^{\, (p+2)} ( A ) - G ^{\, (p+2)} ( x ; {\mathcal{N}} ) \, \right] = e _{p}\, J ^{\, (p+1)}$$ (74)

$$\partial \left[ \, (^{\ast} F) ^{\, (D-p-2)} ( A ) - (^{\ast} G) ^{\, (D-p-2)} ( x ; {\mathcal{N}} ) \, \right] = g _{D-p-4 \, } \overline{J} ^{\, (D-p-3)} \quad .$$ (75)

The symmetry of the two sets of equations (74) and (75) under the transformations

$$F ^{\, (p+2)} ( A ) - G ^{\, (p+2)} ( x ; {\mathcal{N}} ) \longleftrightarrow (^{\ast} F) ^{\, (D-p-2)} ( A ) - (^{\ast} G) ^{\, (D-p-1)} ( x ; {\mathcal{N}} )$$ (76)

$$e _{p}\, J ^{\, (p+1)} \longleftrightarrow g _{D-p-4 \, } \overline{J} ^{\, (D-p-3)}$$ (77)

is now manifest.