2.4 Effective Bulk and Boundary Actions

By inserting Eq. (18) into the action (12), and taking into account the conditions (19), (20), (21), (22), (23), we can write an effective classical action for the three types of oscillation modes,

$$S^{eff} \equiv S_B + S_J$$ (25)

$$S_B = {1\over (p + 1) !}\, P^{0)}{}_{\mu\mu_2\dots\mu_{p+1}}\, \int_{\p... ...+ \int_{\Sigma_{p}} d^p s \, y^\mu \, N^m(\, \vec s\, )\, \partial_m\, \phi_\mu$$ (26)

$$= {1\over (p + 1) !}\, P^{0)}{}_{\mu\mu_2\dots\mu_{p+1}}\, \sigma^{... ...+ \int_{\Sigma_{p}} d^p s \, y^\mu \, N^m(\, \vec s\, )\, \partial_m\, \phi_\mu$$ (27)

$$S_J = -{1\over 2 m_{p+1}}\, \int_{\Sigma_{p+1}} d^{p+1}\sigma\, \sqrt{\... ...1\over p!}\, F^\mu{}_{m_2\dots m_{p+1}}\, F_\mu{}^{m_2\dots m_{p+1}} \, \right)$$

$$\qquad +{1\over 2 m_{p+1}}\, \left[\, {1\over (p+1)!} P^{0)}{}_{\... ...}\, P^{0)}{}^{\mu m_2\dots m_{p+1}} - m_{p+1}^2\, p\, \right]\, \Omega_{p+1}\ ,$$ (28)

where $N^m(\, \vec s\, )$ represents the normal to the boundary and

$$\sigma^{\mu\mu_2\dots\mu_{p+1}} =\int_{\partial\Sigma}\, y... ...gma^{\mu\mu_2\dots\mu_{p+1}}(\,\vec s\, )\ , \qquad p\ge 1$$ (29)

stands for the volume tensor of the brane in target spacetime, while $d\sigma^{\mu\mu_2\dots\mu_{p+1}}(\, \vec s\, )$ represents the oriented volume element attached to the original $p$-brane at the contact point $x^\mu=y^\mu(\,\vec s\,)$. Finally, by definition, we set

$$\Omega_{p+1}\,\equiv \int_{\Sigma_{p+1}} d^{ p+1}\sigma \, \... ...} d^p\sigma\, \sqrt h \equiv V_p\, \int_0^T d\tau\, e(\tau)\ .$$ (30)

Expression (29) allows us to establish a relation between functional derivatives in $p$-loop space,

$${\delta\over \delta y^\mu( \,\vec s \, )}= y^{\mu_2\dots \mu... ..._{m_2}\, y^{\mu_2}\dots \partial_{m_{p+1}}\, y^{\mu_{p+1}}\ .$$ (31)

The above relationship can be used to describe the shape deformations, or local distortions of the $p$-brane in terms of the Jacobi equation in $p$-loop space [21]. The boundary effective action $S_B$ leads us to define the boundary momentum density as the dynamical variable canonically conjugated to the boundary coordinate $y^\mu(\vec s)$:

$${\delta S_B\over \delta y^\mu(\, \vec s\, )}\equiv P_\mu(\, ... ...ial_{m_{p+1}}\, y^{\mu_{p+1}} +N^m \,\partial _m\, \phi_\mu\ .$$ (32)

Here, $P_\mu(\, \vec s\, )$ describes the overall response of the $p$-brane boundary to local volume deformations encoded into $d\sigma^{\mu_1\dots \mu_{p+1} }(\, \vec s\, )$, as well as to induced harmonic deformations, orthogonal to the boundary, described by the normal derivative of $\phi_\mu$. In a similar way, we can define the energy density of the system as the dynamical variable canonically conjugated to the $p$-brane history volume variation

$${\partial S_J\over\partial \Omega_{p+1}} = {1\over 2 m_{p+1}... ...} P^{0)}{}^{ \mu\mu_2\dots\mu_{p+1}} -\frac{m_{p+1}}{2}\, p\ .$$ (33)

Finally, from the anti-symmetry of $P^{0)}{}_{\mu\mu_2\dots \mu_{p+1}}$ under index permutations we deduce the following identity

$$P_\mu\, P^\mu \equiv { h\over (p+1) ! }\, P^{0)}{}_{\mu\mu_2\dots\mu_{p+1}}\, P^{0)}{}^{ \mu\mu_2\dots\mu_{p+1}}\ .$$ (34)

Thus, we arrive at the main result of the classical formulation in the form of a reparametrization invariant, relativistic, effective Jacobi equation

$${1\over 2 m_{p+1}V_p}\, \int_{\Sigma_{p}}{ d^{ p}s\over \s... ...+1}}{2}\, p = {\partial S^{eff}\over\partial \Omega_{p+1}}\ .$$ (35)

This Jacobi equation encodes the boundary dynamics of the $p$-brane with respect to an evolution parameter represented by the world-volume of the $p$-brane history. This is a generalization of the areal string dynamics originally introduced by Eguchi [22] via reparametrization of the Schild action [20], [23].