C..1 Closed Branes

The precise meaning of the term ``duality'' is implicitly assigned by the procedure employed in this subsection. It can be summarized thus:

1. to exchange the gauge potential $A$ in favor of the field strength $F$, by introducing the Bianchi identities as a constraint in the path-integral measure;
2. to introduce a ``dual'' gauge potential $B$ as the ``Fourier conjugate'' field to the Bianchi identities;
3. to integrate out $F$, and identify the dual current as the object linearly coupled to $B$.

Step(1). In order to write the current-potential interaction in terms of $F$, we note that, since $J$ has vanishing divergence, it is a boundary current and can be written as the divergence of a ``parent'' electric current. In other words, there exists a $(p+2)$-rank current $\overline{N} ^{\, \mu _{1} \dots \mu _{p+2}} \left( x\ ; \overline{n}\right)$ such that

$$\overline{N} ^{\, \mu _{1} \dots \mu _{p+2}} ( x \ ; \overline{n}... ...\delta ^{D)}\left[\, x - \overline{n} \left( \overline{\gamma} \right)\,\right]$$

$$\partial _{\, \mu _{1}} \overline{N} ^{\, \mu _{1} \mu _{2} \dots \mu _{p+2}} = J ^{\, \mu _{2} \dots \mu _{p+2}} \quad .$$

To understand the role of these currents, it is useful to recall once again Dirac's construction in which a particle anti-particle pair can be interpreted as the boundary of an ``electric string'' connecting them [7]. In our case, which involves higher dimensions, the analogue of a particle anti-particle pair is a closed $p$-brane which we interpret as the boundary of an open, $(p+1)$-dimensional, parent brane [8]. From this vantage point, the interaction term in the action can be written as follows

$$S _{\mathrm{INT}} = - {e _{p} \over (p+2) \, !} \int \overline{N} ^{\, (p+2)} \, F _{(p+2)}(A) \quad .$$ (80)

Incidentally, we note here that the same procedure could have been employed directly in the expression for the ``non-magnetic'' action (58), to write it only in terms of the field strength $F$ as

$$S [\, A\ , G\ , \overline{N} \, ] = \int \left[ \, - \frac{1}{2} ... ...(p+2)}\, \left(\, \overline{F} _{(p+2)} + G_{(p+2)}\,\right) \, \right] \quad .$$ (81)

$$\overline{F} _{(p+2)} ( A )\equiv F _{(p+2)} ( A )- G_{(p+2)}$$ (82)

After this remark, that we will use later on, we can proceed with the dualization procedure: since we have a quantity (the action (4)) with the interaction term written as (11) we would also like to use $\overline{F}$ as integration variable in the functional integral. We can do that, i.e. treat $\overline{F}$ as an independent variable, imposing the Bianchi identities as a constraint. The relationship between $\overline{N}$ and $J$ must also be encoded in the path-integral . This can be achieved by performing an integration over the parent brane coordinates $\overline{N} = \overline{N} ^{\mu} \left( \overline{\gamma} ^{i} \right)$, which are constrained to satisfy equation (79). These steps are implemented by inserting the following (functional) equivalence relation into the path-integral

$$\int [ {\mathcal{D}}A ] \,W [ \, dA\ , G\ , J\ , \overline{J}\, ] = \int [ {\mathcal{D}}\... ...rline{F} _{\, (p+2)} - g _{D-p-4 \, } \overline{J} ^{\, (D-p-3)} \right] \times$$

$$\times \delta\left[\, \partial N ^{\, (D-p-2)} - g _{D-p-4 \, }\overline... ...J ^{\, (p+2)} \, \right]\, W[\, \overline{F}\ , N \ , \overline{N} \, ] \quad .$$ (83)

The first Dirac delta-distribution takes into account the presence of the magnetic brane as the singular surface where the Bianchi identities are violated. The second and third delta functions encode the relationship between the boundary currents $J$, $\overline{J}$ and their respective bulk counterparts $\overline{N}$ and $N$. It may be worth emphasizing that the electric and magnetic parent branes enter the path-integral as ``dummy variables'' to be summed over. In other words, the physical sources are the boundary currents $J$ and $\overline{J}$ alone. After performing the above operations, the generating functional, written in terms of the total electric-magnetic field strength $\overline{F}$, takes the form

$$Z [ \, J , \overline{J} \, ] = {1 \over Z [ \, 0 \, ]} \int [ {\mathcal{D}}\overline{F} ] [ {\ma... ...^{\ast}d \overline{F} ^{\, (p+2)} - \overline{J} ^{\, (D-p-3)} \,\right] \times$$

$$\qquad \times \delta \left[\, \partial \overline{N} ^{\, (p+2)}- ... ...^{\, (D-p-2)} \, \right] \, e ^{ - S [ \, \overline{F} , G , \overline{N} \, ]}$$ (84)

Step(2). The ``Bianchi identities Dirac delta-distribution'' can be Fourier transformed by means of the functional representation

$$\delta \left(\, {}^* d \overline{F} ^{\, (p+2)} - g _{D... ...nt \, L ( B , \overline{F} , \overline J) \right) \quad ,$$ (85)

where

$$L ( \, B , \overline{F} , \overline{J} \, ) = B _{\, (D-p-... ...2)}- \, g _{D-p-4 \, } \overline{J} ^{(p+3)}\,\right] \quad .$$ (86)

The net result is a ``field strength formulation'' of the model (57, 58) in terms of a dynamical field $F$, and a Lagrange multiplier $B$:

$$Z [ \, J , \overline{J} \, ] = {1 \over Z [ \, 0 \, ]} \int [ {\mathcal{D}}N ] [ {\mathcal{D}}\o... ...ial N ^{\, (D-p-2)} - g _{D-p-4 \, } \overline{J} ^{\, (D-p-3)} \right ] \times$$

$$\qquad \qquad \times \delta \left [ \partial \overline{N} ^{\, (p... ...)} \right ] \, e ^ { - S [ \, \overline{F} , G , B , \overline{N} \, ]} \quad ,$$ (87)

where

$$S [ \, \overline{F}\ , G\ , B\ , \overline{N}\ , \overline{J} \, ] = - \int\left[\, { 1\over 2}\overline{F} ^{\, (p+2)} \overline{F} _{(... ...t(\overline{F} _{(p+2)}+ G _{(p+2)} \right) \overline{N} ^{\, (p+2)} + \right .$$

$$- \imath\, (-1)^ { D(p+1) }\left . {}^* H ^{\, (p+2)}(B)\,\overline... ...} -\imath\, g _{D-p-4 \, } \, B _{(D-p-3)}\, \overline{J} ^{(D-p-3)} \, \right]$$ (88)

and ${}^{\ast} H$ represents the dual field strength

$$H _{\, (D-p-2)}(B)\equiv d B _{\, (D-p-3)} \quad .$$ (89)

Step(3). Finally, we are ready to switch to the dual description of the model by integrating away the field strength $\overline{F}$. Since the path-integral is Gaussian in $\overline{F}$, the integration can be carried out in a closed form:

$$Z [ \, J , \overline{J} \, ] = {1 \over Z[ \, 0 \, ]} \int [ {\ma... ...ne{N} - J \right] \, e ^{ - S [ \, G\ , B\ , \overline{N}\ , \overline{J} \, ]}$$

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

$$- \int\, \left[\, e _{p}\, G _{(p+2)} \overline{N} ^{\, (p+2)} + i\, g _{D-p-4 \, }\, B _{(D-p-3)}\, \overline{J} ^{(D-p-3)} \,\right]$$

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

$$+ \imath\, \int \left[\, g _{D-p-4 \, } B _{(p+3)}\, \overline{J} ^... ...)} + e _{p}(\, {}^{\ast} N) ^{\, (p+2)} \overline{N} _{(p+2)}\, \right] \quad .$$ (90)