C..2 Open Branes

Equipped with the formalism developed in the previous sections, we now wish to consider the extension of the path-integral method to the case of open $p$-branes. As we shall see, the main difference stems from the fact that a gauge invariant action for an open $p$-brane requires the introduction of new gauge fields to compensate for the gauge symmetry ``leakage'' through the boundary [11]. Thus, we replace the system (57-58) with the following one,

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

$$S [ \, A , C , \overline{N} _{\mathrm{e}} \, ] = \int \left[\, -\frac{1}{2} \overline{F} ^{\, (p+2)}\, \overline{F... ..._{(p+1)} - d C _{(p)}\,\right) \overline{N} _{\mathrm{e}} ^{\, (p+1)} \right. +$$

$$\qquad \qquad \qquad\qquad \left. - \kappa \left( d C _{(p)} - A _{\, (p+1)} \right) ^{2} \, \right] \quad .$$ (91)

where $C^ { (p) }$ has canonical dimensions $L ^{2-D/2}$, and $\kappa$ is a constant introduced for dimensional reasons. In this case, the divergence of the $p$-brane current is no longer vanishing, but equals the current $J _{\mathrm{e}} ^{(p)}$ associated with the free boundary of the world-manifold. In other words,

$$\partial \overline{N} _{\mathrm{e}} ^{\, (p+1)} = J _{\mathrm{e}} ^{(p)} \quad .$$ (92)

However, the action (33) is still invariant under the extended gauge transformation:

$$\delta _{\Lambda} A _{\, (p+1)} = d \Lambda _{(p)}$$ (93)

$$\delta _{\Lambda} C _{(p)} = \Lambda _{(p)} \quad .$$ (94)

Indeed, the role of the $C_{(p)}$-field, which is a Stückelberg compensating field, is to restore the gauge invariance broken by the boundary of the $p$-brane. Perhaps it's worth emphasizing that $S[\, A , C , \overline{N} _{\mathrm{e}} \, ]$ depends on $C_{(p)}$ only through its covariant curl $d C _{(p)} \equiv \Theta_{(p+1)}$, which is not gauge invariant, but transforms as follows

$$\delta _{\Lambda} \Theta_{(p+1)} = d \Lambda _{(p)} \quad .$$ (95)

The advantage of the path-integral method for constructing the dual action becomes evident at this point, since we can eliminate the Stückelberg potential $C_{(p)}$ in favor of its curl $\Theta _{\, (p+1)}$ by introducing the dual Stückelberg potential $D^{(D-p-2)}(x)$ as we did in equation (85)

$$\delta \left[\, {}^* d\, \Theta _{\, (p+1)}\, \right] = \int [ {\mathcal{D}}D ] e ^ {\imath\, S [ \, D\ , \Theta \, ] }$$ (96)

$$S [ \, D \ , \Theta \, ] = \int\, D ^{\, (p+2)} \left(\, {}^{\ast}d\,\Theta _{\, (p+1)}\,\right)$$

$$= (-1)^ { Dp } \int \, \Theta ^{\, (p+1)} \left(\, {}^{\ast} K(D)\,\right) _{(p+1)}$$ (97)

$$\left(\, {}^{\ast} K(D)\,\right) _{(p+1)} = \left(\, {}^{\ast} d\, D _{D-p-2}\,\right) \quad .$$ (98)

The resulting vacuum amplitude is

$$Z [ \, \overline{N} _{\mathrm{e}} \, ] = {1 \over Z [ \, ... ...A \ , D \ , \Theta , \overline{N} _{\mathrm{e}} \, ]} \quad ,$$ (99)

where

$$S [ \, A\ , D\ , \Theta\ , \overline{N} _{\mathrm{e}} \, ] = \int\left[\, - { 1\over 2 }\overline{F} _{(p+2)} \,\overline{ F} ... ...- \Theta _{\, (p+1)}\, \right) \overline{N} _{\mathrm{e}} ^{\, (p+1)} \right. +$$

$$\qquad + { \kappa\over 2 }\left. \left( \Theta _{\, (p+1)}- A _{\, (p+1)} ... ...a ^{\, (p+1)} \left(\, {}^{\ast} K( D )\,\right) _{\, (p+1)} \, \right] \quad .$$ (100)

is the ``Stückelberg Dual Action'', in the sense we dualised the $C_{(p)}$ field only.
Translational invariance of the functional integration measure enables us to shift the $I$-field and introduce the gauge invariant field strength $\overline{\Theta}$

$$\overline{\Theta} ^{\, (p+1)} \equiv \Theta^{\, (p+1)} - A ^{\, (p+1)}$$ (101)

as a new integration variable instead of $\Theta$.
Once $\overline{\Theta}$ is integrated out, we find

$$Z[\, \overline{N} _{\mathrm{e}} \, ] = {1 \over Z[ \, 0 \... ...l{D}}D ] e^{-S[\, A\ , D\ , \overline{N} _{\mathrm{e}} \, ]}$$ (102)

with

$$S [ \, A\ , D\ , \overline{N} _{\mathrm{e}} \, ] = \int\left[\, -\frac{1}{2} \overline{F} ^{\, (p+2)}\overline{F} _{... ...} D\, ) ^{\, (p+2)}\left(\, \overline{F} _{(p+2)}+ G _{(p+2)}\,\right)\right. +$$

$$\quad \qquad \left. + \frac{1}{2\kappa}\left(\, e _{p} \, \overli... ...imath\, ( -1 ) ^{Dp}(\, {}^{\ast} K) ^{\, (p+1)}\, \right)^2 \, \right] \quad .$$ (103)

By recognizing that the first line in equation (103) is the same as (81), once ${}^{\ast} D$ is identified with $\overline{N}$, i.e.,

$$e _{p}\, \overline{N} ^{\, (p+2)}_{\mathrm{e}} \rightarrow \imath\, (-1) ^{(D-p)p} \, {}^*(\, D _{\, (D-p-2)}\, ) \quad ,$$ (104)

we can write the Complete Dual Action without repeating all the previous calculations:

$$Z [ \, \overline{N} _{\mathrm{e}} , J _{\mathrm{g}} \, ] = \frac{1}{Z [ \, 0 , 0 \, ]} \int [ {\mathcal{D}}B ] [ {\mathcal{D... ... { D-p-3 } \, \right] e ^{ - S [ \, B , \, D , \overline{N} _{\mathrm{e}} \, ]}$$

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

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

$$- \imath\int \left [\, (-1)^{Dp} N ^{ (D-p-2)}\, D _{(D-p-2)} + g _... ...\, }\, B _{(D-p-3)} \overline{J} _{\mathrm{g}} ^{\, (D-p-3)}\, \right ] \quad .$$ (105)

>From the dual action (105) one gets the field equations

$$\partial\, \left[ \, D ^{(D-p-2)} - (-1)^{Dp} d B ^{(D-p-1)}\,\right]= -\imath (-1)^ { Dp } g_{D-p-4 \, } \overline{J} _{\mathrm{g}} ^{\, (D-p-3)}$$ (106)

$$\partial\, \left[ \,d D ^{\, (D-p-2)} +\imath\, (-1) ^{D- (p+1)^2... ...1)}\, \right] + \kappa D ^{\, (D-p-2)}= -\imath\,\kappa (-1)^{Dp} N ^{ (D-p-2)}$$ (107)

Solving equation (107) one finds

$$D ^{(D-p-2)} - (-1)^{Dp} d B ^{(D-p-1)}= \widehat{D} ^{(D-p-2)} -... ...)^ { Dp } g_{D-p-4 \, } d {1\over\Box } \overline{J}_{\mathrm{g}} ^{\, (D-p-3)}$$ (108)

$$\partial\,\widehat{D} ^{(D-p-2)}=0$$ (109)

After decomposing the magnetic parent current as

$$N ^{ (D-p-2)}= \widehat{N} ^{ (D-p-2)} + g_{D-p-4 \, } d {1... ...} ^{\, (D-p-3)}\ ,\qquad \partial\,\widehat{N} ^{ (D-p-2)}=0$$ (110)

equation (108) becomes

$$\partial\, d\, \widehat{D} ^{\, (D-p-2)} + \kappa \widehat{... ... (D-p-2)}= \imath\, \kappa (-1)^{Dp} \widehat{N} ^{ (D-p-2)}$$ (111)

and gives for $\widehat{D} ^{\, (D-p-2)}$ the following solution

$$\widehat{D} ^{\, (D-p-2)}= -\imath\,\kappa (-1)^{Dp} {1\over \Box +\kappa}\widehat{N} ^{ (D-p-2)}$$ (112)

$$d\, \widehat{D} ^{\, (D-p-2)}=-\imath\,\kappa (-1)^{Dp} d {1\over \Box +\kappa}\widehat{N} ^{ (D-p-2)}$$ (113)

When (112), (113) are inserted back into (105) we find

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

$$+ \int \left[\, { g_{D-p-4 \, }^2\over 2 } \overline{J} _{\mathrm{g... ... \kappa} j _{\mathrm{e}} ^{\,(p)} { 1\over \Box } j _{\mathrm{e}\,(p)}\,\right]$$ (114)

where, short-range bulk and long range boundary interaction are clearly displayed.

 

TABLES

	Spacetime manifold	Embedding functions	Fiducial manifold	Parameters	Source Current	Dimensionality
Type
Electric	${\mathcal{E}}$	$E$	$\Xi$	$\xi ^{i}$	$J$	$p+1$
Magnetic	${\mathcal{M}}$	$M$	$\Sigma$	$\sigma ^{i}$	$\overline{J}$	$D-p-3$
Magnetic Dirac brane	${\mathcal{N}}$	$N$	$\Gamma$	$\gamma ^{i}$	$N$	$D-p-2$
Electric Dirac brane	$\overline{{\mathcal{N}}}$	$\overline{N}$	$\overline{\Gamma}$	$\overline{\gamma} ^{i}$	$\overline{N}$	$p+2$
	${\mathcal{O}}$	$O$	$\Omega$	$\omega ^{i}$	$\Omega$	$D-p-1$
Table 1: Definition of electric/magnetic objects.

	$p$	$p$-brane	$\tilde p=7-p$ dual-brane
	0	point-particle	7-brane
	1	string	6-brane
	2	membrane	5-brane
	3	bag	4-brane
Table 2: Closed $p$-branes duality in $D=11$ spacetime dimensions.

 

Field	Dimension in $\hbar=c=1$ units	Rank	Physical Meaning
$A _{\, \mu _1 \dots \mu _{p+1}}$	$(length) ^{1-D/2}$	$p+1$	``electric'' potential
$F _{\, \mu _{1} \dots \mu _{p+2}}$	$(length) ^{-D/2}$	$p+2$	``electric'' field strength
$G _{\, \mu _{1} \dots \mu _{p+2}}$	$(length) ^{-D/2}$	$p+2$	``magnetic'' field strength
$\overline{F} _{\, \mu _{1} \dots \mu _{p+2}}$	$(length) ^{-D/2}$	$p+2$	``singular'' field strength
$B _{\, \mu _{1} \dots \mu _{D-p-3}}$	$(length) ^{1-D/2}$	$D-p-3$	dual gauge potential
$H _{\, \mu _{1} \dots \mu _{D-p-2}}$	$(length) ^{-D/2}$	$D-p-2$	dual field strength
Table 3: Closed $p$-brane gauge potentials and fields.

 

Current Density	Dimension in $\hbar=c=1$ units	Rank	Physical Meaning
$J _{\, \mu _{1} \dots \mu _{p+1}}$	$(length) ^{p+1-D}$	$p+1$	electric current
$\overline{N} _{\, \mu _{1} \dots \mu _{D-p-2}}$	$(length) ^{-D/2}$	$D-p-2$	parent electric current
$\overline{J} _{\, \mu _{1} \dots \mu _{D-p-3}}$	$(length) ^{-p-3}$	$D-p-3$	magnetic current
$N _{\, \mu _{1} \dots \mu _{D-p-2}}$	$(length) ^{-D/2}$	$D-p-2$	parent magnetic current
$\Omega _{\, \mu _{1} \dots \mu _{D-p-1}}$	$(length) ^{1-D/2}$	$D-p-1$	gauge brane current
Table 4: Closed $p$-brane associated currents.

 

Field	Dimension in $\hbar=c=1$ units	Rank	Physical Meaning
$A _{\, \mu _{1} \dots \mu _{p+1}}$	$(length) ^{1-D/2}$	$p+1$	``electric'' potential
$F _{\, \mu _{1} \dots \mu _{p+2}}$	$(length) ^{-D/2}$	$p+2$	``electric'' field strength
$C _{\, \mu _{1} \dots \mu _{p}}$	$(length) ^{2-D/2}$	$p$	Stückelberg gauge potential
$K _{\, \mu _{1} \dots \mu _{p+1}}$	$(length) ^{1-D/2}$	$p+1$	Stückelberg field strength
$D _{\, \mu_{p+3}\dots \mu_D}$	$(length) ^{-D/2}$	$D-p-2$	dual Stückelberg potential
$I _{\, \mu_1\dots\mu_{p+1}}$	$(length) ^{-1-D/2}$	$p+1$	dual Stückelberg strength
Table 5: Open $p$-brane gauge potentials and fields.

 

Current Density	Dimension in $\hbar=c=1$ units	Rank	Physical Meaning
$\overline{N} _{\mathrm{e}}^{ \, \mu _{1} \dots \mu _{D-p-2}}$	$(length) ^{p+1-D}$	$p+1$	electric parent current
$J _{\mathrm{e}} ^{\, \mu _{1} \dots \mu _{p}}$	$(length)^{p-D}$	$p$	boundary current
$N ^{\, \mu _{1} \dots \mu _{D-p-2}}$	$(length) ^{-D/2}$	$D-p-2$	parent magnetic current
$\overline{J} _{\mathrm{g}}^{ \, \mu _{1} \dots \mu _{D-p-3}}$	$(length) ^{-p-3}$	$D-p-3$	magnetic current
Table 6: Open $p$-brane associated currents

 

Coupling constants	Dimension in $\hbar=c=1$ units
$e$	$(length) ^{D/2-p-2}$
$g$	$(length) ^{p+2-D/2}$
$e \, g$	$1$
$\kappa$	$(length) ^{-2}$
Table 7: Dimension of couplings

 

$p$	$p$-brane	$\tilde{p} = 7-p$ dual-brane
$0$	point-particle	$7$-brane
$1$	string	$6$-brane
$2$	membrane	$5$-brane
$3$	bag	$4$-brane
Table 8: Open $p$-branes duality in $D=10$ spacetime dimensions.