2 Our proposal

In order to address the above question, let us consider the gauge invariant action for a gauge field $A$, of rank $p=d-1$, interacting with an external current $j$ in a $d$-dimensional Minkowski spacetime,

$$S\equiv -{1\over 2d! }\int d^d x \, F_{\,\mu_1\dots\mu_d \,}... ...1\dots \mu_{d-1}\,}\left(\, x\, ;\partial V \, \right) \quad .$$ (1)

The external current

$$j^{\,\mu_1\dots\mu_{d-1}\,}\left(\, x\, ; \partial V \,\righ... ...igma)\, \right] \, dy^{\mu_1}\wedge \dots\wedge dy^{\mu_{d-1}}$$ (2)

may be thought of as originating from the timelike history of a $(d-2)$-brane, and can be viewed as the boundary current of a $d$-dimensional bulk volume, or ``bag''. Accordingly, the boundary $\partial V$ is parametrized by $d-1$ coordinates $\{\,\sigma^a\, : a = 1 , \dots , d-1 \, \}$. Gauge invariance of the action requires the current to satisfy the condition

$$\partial_{\mu_1}j^{\,\mu_1\dots\mu_{d-1}\,}=0 \quad .$$ (3)

The conservation of the boundary current, in turn, implies that $j^{\,\mu_1\dots\mu_{d-1}\,}$ can be written as the divergence of the bulk current $J$ :

$$j^{\,\mu_1\dots\mu_{d-1}\,}\left(\, x\, ; \partial V \,\righ... ...al_{\mu_d}J^{\mu_1\,\mu_2\dots\mu_d}\left(\, x\, ; V \,\right)$$ (4)

where

$$J^{\mu_1\,\mu_2\dots\mu_d}\left(\, x\, ; V \,\right)\equiv \... ...(\xi)\,\right] dz^{\mu_1}\wedge\dots \wedge dz^{\mu_d} \quad .$$ (5)

In the case of the extremal theory encoded in the action functional (1), the field strength $F$ is an antisymmetric tensor of maximum rank $d$ for which no Bianchi Identities can be formulated. We propose to circumvent this obstacle by using the first order formalism. Technically, this means that the dualization procedure should be implemented using the constraint $\delta\left[\, F - dA\, \right]$ instead of $\delta\left[\, dF \,\right]$ within the path integral approach. In other words, as in the case of electrodynamics in two spacetime dimensions, we impose that the ``Maxwell tensor'' is, in fact, the covariant curl of the gauge potential. This leads us to the partition functional corresponding to the original action (1),

$$Z[j]= Z[0]^{-1}\int [{\mathcal{D}}F][{\mathcal{D}}A]\,\delta\left... ...{\mu_1\dots\mu_d }-\partial_{[\,\mu_1} A_{\mu_2\dots \mu_d\, ]}\, \right]\times$$

$$\times\exp\left\{{i\over 2 \, d\, ! }\int d^4 x \, F^{\mu_1\dots\... ...)\, j^{\,\mu_1\dots\mu_{d-1}\,}\left(\, x\, ;\partial V \right)\right\} \quad .$$ (6)

Next, we write the Dirac-delta function in the exponential form with the help of a Lagrange multiplier $B^{\,\mu_1\dots\mu_d\,}\,(x)$,

$$\delta\left[\, F_{\mu_1\dots\mu_d}- \partial_{[\,\mu_1}\, A_... ...\,\mu_1 }\, A_{\mu_2 \dots\mu_d\,]}\,\right)\, \right\}\quad .$$ (7)

From here, the dual partition functional is obtained by integrating out independently both $F$ and $A$.

First, the path integral (6), being Gaussian in $F$, is easily evaluated, and one finds

$$\int [\, {\mathcal{D}}F\, ] \exp\left\{{i\over 2 d\, !}\int d^4 x... ... 2F^{ \, \mu_1\dots\mu_d\, }\, \, B_{\,\mu_1\dots\mu_d\,}\, \right]\, \right\}=$$

$$\qquad \qquad \qquad = \exp\left\{\, {i\over 2d!}\int d^4x\, B_{\,\mu_1\dots\mu_d\,}\,B^{\,\mu_1\dots\mu_d\,}\, \right\} \quad .$$ (8)

As for the $A$-integration, we note that the key feature of the first order formalism is to introduce the original gauge potential $A$ linearly into the path integral. In this way, the potential $A$ appears as an additional Lagrange multiplier which, after integration, yields the following condition

$$\int [{\mathcal{D}}A]\exp\left\{\,{i\over (d-1)!} \int d^4x \,\, ... ..., B^{\mu_1\mu_2\dots \mu_d } - \,e\, j^{\mu_2\dots \mu_d }\, \right)\,\right\}=$$

$$\qquad \qquad \qquad=\delta\left[\,\partial_{\mu_1} \, B^{\mu_1\mu_2\dots \mu_d } -\,e\, j^{\mu_2\dots \mu_d } \,\right] \quad .$$ (9)

The effect of the above delta-function is to restrict the ``trajectories''in the path integral to the family of classical field equation for the dual field $B$. In order to extract the physical meaning of the dual theory, it may be helpful to rewrite the above delta function in terms of the bulk current $J$, as follows

$$\delta\left[\,\partial_{\mu_1 } B^{ \mu_1\mu_2\dots \mu_d} -... ...ts \mu_d }_0 -\,e\, J^{\mu_1\mu_2\dots \mu_d}\,\right] \quad .$$ (10)

It should be noted that in trading the boundary current for the bulk current in the delta function (10), we have introduced an arbitrary constant field $B^{\mu_1\mu_2\dots \mu_d }_0$. As mentioned in the introduction, this arbitrary constant represents the solution of the homogeneous equation for the $B$-field, and corresponds to a cosmological term in the action.

Performing the integration over $B$ with the help of the above delta-function leads to the following expression of the dual partition functional,

$$\exp \left\{\,i\, W\left[\,J\,\right]\, \right\}= \exp\left\... ...u_d}_0 -e\, J^{\mu_1\dots\mu_d }\, \right)^2\,\right\} \quad .$$ (11)

Without loss of generality, our discussion can be simplified by resetting the constant $B_0$ to zero, and by choosing the normalization factor $Z[0]$ in such a way to cancel the determinant factor appearing in (10). With such redefinitions, equation (11) represents a direct current-current interaction within the bulk which is dual to the original theory (1) whose interaction takes place between elements of the boundary through the mediating agency of a $(d-1)$-index potential.

From the above result, one might be deceived into thinking that there are physical quanta being exchanged between $p$-brane elements in this limiting case. However, the truth of the matter is that a $(d-1)$-index potential in $d$-dimensions does not represent a genuine ``radiation'' field, in the sense that there are no propagating degrees of freedom. As we have emphasized earlier, the field strength, in this case, merely represents a constant background disguised as a gauge field. However, if there are no physical degrees of freedom in the original theory, the same must be true for the dual theory, and this begs the question: what is the nature of the interaction that we have uncovered here? In this connection, it is instructive to play the game in reverse, and write the dual partition functional $W\left[\, J\,\right]$ in terms of the boundary current $j$. Using equation (5), one finds

$$\exp \left\{\, i\, W\left[\, j\,\right]\,\right\}=\exp\left\... ...}}\, \frac{1}{\Box}\, j_{\mu_1\dots\mu_{d-1}}\,\right\}\quad .$$ (12)

The boundary current $j$, on the other hand, can be expressed through its Hodge dual

$$j^{\mu_1\dots\mu_{d-1}}=\epsilon^{\mu_1\dots\mu_{d-1}\lambda}\, j_\lambda$$ (13)

so that equation (12) can be rewritten as follows

$$\exp \left\{\, i\, W\left[\, j\,\right]\, \right\}=\exp\left... ...nt d^dx\, j^{\mu}\, \frac{1}{\Box}\, j_{\mu}\,\right\} \quad .$$ (14)

The above expression closely resembles the analogous formula for the interaction between point charges in four dimensions. However, that analogy is formal. More to the point, the physical content of equation (14) is exactly analogous to that of ``electrodynamics'' in two dimensions. Indeed, the four vector $j_\mu$, in the limiting case of $(d-1)$-branes, has no transverse, spatial, components. This is due to the conservation property (3) of the original boundary current. In terms of the Hodge dual $j_\mu$, that conservation property implies the following relations

$$j_\lambda=\partial_\lambda \,\phi$$ (15)

$$j_0 = \partial_0\, {\partial^i\, j_{Li} \over \nabla^2}$$ (16)

where $j_i^L$ represents the longitudinal component of the spatial part of $j^\mu$. In terms of that longitudinal component one can rewrite equation (14) as follows

$$\exp \left\{\, i\, W\left[\, J\,\right]\,\right\}= \exp\left... ... 2} \, \int d^dx\, j^i_L\,{1\over\nabla^2 }\, j_{Li}\,\right\}$$ (17)

and this equation, in turn, can be rewritten in terms of the original brane current

$$\exp \left\{\,i\, W\left[\, J\, \right]\,\right\}=\exp\left\... ...\over\nabla^2 }\, j_{0\mu_2\dots\mu_{d-1}} \,\right\} \quad .$$ (18)

While the result (18) emphasizes the role of the ``zero-component'' of the brane current, thereby violating manifest covariance, it has the advantage of describing a static, long range, interaction between surface elements of the boundary. This is clearly reminiscent of the fact that the original gauge potential $A$ has no propagating degrees of freedom, much as electromagnetism in two dimensions, and actually represents its generalization for extended objects in higher dimensions. With hindsight, ``electrodynamics in two dimensions'' may be reinterpreted as a theory of two dimensional bags [7].

Summing up our discussion so far, we have shown how to extend the $p$-dualization procedure, within a path integral approach, using a first order formalism instead of Bianchi Identities in order to include the limiting case of rank $p=d-1$ fields. Equation (11) shows that the absence of a dual potential results in a local (contact) interaction of the bulk current $J$ in the dual theory, while the original potential $A$ induces a non-local, though static, long range interaction on the boundary. The success of the procedure relies on the fact that it dualizes the field strength $F(A)$ to a field $B$, which, by itself, is not necessarily the covariant curl of a dual potential. In this sense, even if the dual field of the gauge potential $A$ does not exist, one can still construct a dual theory for its field strength. This would be impossible using the second order formalism since, in that formalism, it is the gauge potential $A$ that is dualized. Furthermore, it can be seen from equation (18) that a physically meaningful dual theory exist only in case of an interacting theory, as anticipated in the introduction. With the above results in hand, we turn now to the following question that arises naturally from our preceding discussion: can one include the usual formulation of $p$-duality within the extended dualization procedure outlined above for the limiting theory? Clearly, this would be desirable in order to have a unified approach to $p$-duality for all values of $p$.

Our purpose in the remaining part of this letter, then, is to show that it is indeed possible to reformulate the whole $p$-duality approach without using the Bianchi identities. The road to a unified formulation starts from the ``parent Lagrangian'' for a $p$-form $A$ and an external current $K$ (in the case of an interacting theory) coupled to a field $B$, which we later identify as the dual field. Our strategy, then, is to construct the parent Lagrangian in such a way that the dualization procedure is applied to the field strength rather than the gauge potential $A$. In this approach, the procedure will turn out to be equivalent to the first order formalism. What remains to be seen is that in a non-limiting case the new procedure gives the same result as the standard approach based on Bianchi identities.
We take the parent Lagrangian to be of the form⁷

$$L_P=-{1\over 2}\left(B - e\, J^* \right)^2 + B\, F^{\,*}(A) +g\, K\, B \quad ,$$ (19)

where we have introduced both an ``electric brane'' current $J$, a ``magnetic brane'' current $K$, and the corresponding ``electric'' and ``magnetic'' charges $e$ and $g$. The introduction of two distinct currents is designed to reproduce within our formalism, among other dualitites, also the well known Dirac electric-magnetic duality. Furthermore, $F(A)$ is assumed, at the outset, to be the curl of the gauge potential $p$-form $A$, while the dual field $B$ is a $(d-p-1)$-form to be determined in the course of dualization. Our procedure, which is encapsulated in the diagram of Fig.1 involves two distinct steps which we discuss separately.

Figure 1: Summary of the dualization procedure

1. Variation with respect to the $B$ field in the parent Lagrangian yields the equation of motion
   

   $$\delta_B\, L_P=0 \quad \longrightarrow \quad B= F^{\,*} (A) +g\, K +e\, J^* \quad ,$$ (20)

   
   
   which, when inserted back into equation (19), gives the Lagrangian of the interacting theory for the field $A$ as follows
   

   $$L_A= -{1\over 2}\left(\, F(A) +g(-1)^ { (p+1)(d-p-1) } K^* \, \right)^2 +e\, A\, j_e + e\,g\, K \, J^* \quad .$$ (21)

   
   
   The ``electric boundary'' current $j_e$, coupled to the gauge potential $A$, can be expressed, as before, in terms of the ``electric'' bulk current $J$
   

   $$j_e= \partial J \quad .$$ (22)

   
   
   Inspection of equation (21) shows that there is a current-current contact term $K^2$, as well as a mixed term $K \, J^*$. Such contact terms are unavoidable consequences of the dualization procedure, and were noticed for the first time in Ref.[11]. The mixed contact term $K \, J^*$ seems to be especially relevant since it leads to the Dirac charge quantization condition (Ref.[12]). Their overall importance for our present discussion will become clear at the end of this letter. Next, we redefine the field strength, in order to absorb the magnetic bulk current: $\widetilde F= F +(-1)^ { (p+1)(d-p-1) } K^{\, *}$ and this leads to the simple expression for the Lagrangian of the $A$ field
   

   $$L_A= -{1\over 2} \widetilde F\, {}^2 +e\, A\, j_e + e\,g\, K J^*$$ (23)

   
   
   and to the corresponding field equation
   

   $$\partial \widetilde F= e\, j_e \quad .$$ (24)

   
   

2. Variation of the parent Lagrangian with respect to $A$,
   

   $$\delta_A\, L=0 \quad \longrightarrow \quad F^{\,*}(B)= 0 \quad .$$ (25)

   
   
   In all cases for which $p\le d-2$, the above ``equation of motion'' , it turns out, has a solution which defines the dual field $B$ as the field strength of a dual potential $C$, in agreement with the result of the standard approach. This proves the equivalence of the two procedures for non limiting cases. When equation (25) is inserted back in the expression (19), one obtains the dual theory for the field $C$ coupled to the external magnetic current $K$, as described by the Lagrangian
   

   $$L_B=-{1\over 2}\left(\, B(C) -e\, J^*\,\right)^2 +g\, K\, B(C) \quad .$$ (26)

   
   
   The above Lagrangian leads to the field equations
   

   $$\partial\widetilde B = g\,j_m \quad ,$$ (27)

   
   
   where, once again, we have used the redefinition $\widetilde B\equiv B -e\, J^*$. Similarly, the magnetic boundary current can be expressed as $j_m=\partial K$.

As a check on the consistency of our procedure, it seems worth observing that equations (24) and (27) in the case $p=1$, $d=4$, reproduce the Dirac electric-magnetic duality. Furthermore, one can easily check that our procedure reproduces the well known scalar-tensor duality between an interacting scalar field in four dimensions, and an interacting two-index antisymmetric gauge field. Indeed, with the choice $p=0$ ( $\Rightarrow F_\mu(\phi)=\partial_\mu\phi$, $B(C) \longrightarrow B^{\mu\nu\rho}(C)$) and $J=0$, one finds a Lagrangian for a scalar field

$$L(\phi)= -{1\over 2}\left(\,\partial_\mu\phi + g\, K^*_\mu\,\right)^2$$ (28)

which, in turn, leads to the equation of motion

$$\Box\,\phi=g\,\partial_\mu K^{*\,\mu}\equiv g\, F^*(K) \quad ,$$ (29)

while the dual field Lagrangian which follows from (26) is

$$L_C= -{1\over 2\cdot 3!}\, B^{\mu\nu\rho}(C)\, B_{\mu\nu\rho}(C) +{g\over 3!} \, B_{\mu\nu\rho}(C)\, K^{\mu\nu\rho}$$ (30)

and gives the equation of motion

$$\partial_\mu B^{\mu\nu\rho}(C)=-g\, j^{\nu\rho}\equiv -g\, \partial_\mu K^{\mu\nu\rho} \quad .$$ (31)

Along the same lines one can prove that the above procedure gives all known dual interacting theories whose current-free version are described, for example, in Ref.[6]. Finally, in order to prove the equivalence to the first order formalism described previously using the path integral approach, we take the limiting value $p=d-1$ and $K=0$ in equation (21). This choice immediately leads to

$$L_A=-{1\over 2\cdot d!}\, F^2_{\mu_1\dots\mu_d}(A) +{e\over... ...)!}\, A_{\mu_1\dots\mu_{d-1} } \, j^{\mu_1\dots\mu_{d-1} } \ .$$ (32)

This is the same expression derived from the action functional (1). The dual theory follows instead from equation (25). As already mentioned, normally that equation defines $B$ as the field strength of some potential $C$, except in the limiting case where it gives the condition

$$\epsilon^{\lambda\mu\nu\rho}\, \partial_\rho\, B=0\quad\longrightarrow\quad B = \mathrm{const.}\equiv B_0 .$$ (33)

This is the same constant field encountered in (10). The dual theory is obtained via equation (26),

$$L_B=-{1\over 2}\, (B_0- e J^*)^2$$ (34)

which is the same result as in equation (11) once we identify the Hodge duals $B^{\mu_1\dots\mu_d}_0=\epsilon^{\mu_1\dots\mu_d}B_0$ and $J^{\mu_1\dots\mu_d}=\epsilon^{\mu_1\dots\mu_d}J^*$. This shows the asserted equivalence of the path integral and algebraic procedure in the limiting case, thus providing us with the non trivial result that the modified $p$-duality procedure is applicable to any interacting theory. As a matter of fact, we can proceed one step further with our extension of the $p$-duality procedure, and show that the algebraic procedure includes massive Abelian topological theories as well.

In order to substantiate the above statement, we include in the parent Lagrangian a massive topological term, $m\, A\, F^{\, *}(A)$, for the $A$ field. The index structure of such topological terms imposes the following restriction on the dimensionality of spacetime: $d=2p+1$, showing, as is well known, that topological terms can exist a priori only in odd dimensions. The addition of the topological term in the parent Lagrangian does not affect equation (20), which is obtained by variation of the Lagrangian with respect to the $B$ field. Thus, inserting equation (20) back into the parent Lagrangian yields the following Lagrangian for the massive topological theory for the $A$ field

$$L_A= -{1\over 2}\left(\, F(A) + g\, K^* \, \right)^2 + m\, A\, F^{\,*}(A)\quad .$$ (35)

Here we have kept only the ``magnetic'' current $K$ for an easier comparison with the results in Ref.[11]. In this connection, we also note that variation of the topological mass term alone with respect to $A$ gives a term of the form, $m\,[\, 1+(-1)^{p+1}]\, F^{\,*}(A)$. This term contributes to the equation of motion only if $p=2k-1$, while for $p=2k$ the topological term is a total derivative. Taking into account the previous restriction to odd $p$, leads to the number of dimensions $d=4k-1$ quoted in Ref.[11]. In a spacetime with such dimensions, the equation of motion for $A$ becomes

$$\partial \widetilde F= -2m\, F^{\,*}(A) \quad .$$ (36)

On the other hand, varying the full parent Lagrangian with respect to $A$, yields the equation of ``motion'' for the dual field $B$

$$F^{\,*}(B)=-2m\, F^{\,*}(A)$$ (37)

which represents an extension of the result (25) for the topological term. It is also worth mentioning that equation (25) previously was used to define the dual field as the field strength of the dual potential $C$, while equation (37) implies that $B$ has to be of the same rank as $A$. Inserting equation (37) into the parent Lagrangian leads to the dual Lagrangian for the massive ``topological'' theory

$$L_B=-{1\over 2}\, B^2 +g\, K\, B + {1\over 4m}\, B\, F^*(B)$$ (38)

from which we derive the following equation of motion

$$B= -{1\over 2m}\, F^{\,*} (B) + g\, K \quad .$$ (39)

In this way we have shown that the dual version of an interacting, topologically massive, Abelian gauge theory discussed, for instance, in Ref.[10], is an integral part of the modified $p$-duality approach, thus generalizing the results reported in Ref.[11] to arbitrary dimensions $d=4k-1$.