1 Introduction

Diverse action functionals have been proposed to describe dynamics of a relativistic, bosonic, $p$-brane [1]. The first brane action, proposed in 1975, was a generalization of the Nambu-Goto action for strings, i.e. the measure of the brane world history [2]

$$S_{DNG}[ Y ] = -m_{p+1}\int_\Sigma d^{ p+1} \sigma\sq... ... det\left( \partial_m Y^\mu \partial_n Y_\mu \right) \quad ,$$ (1)

where $m_{p+1}$ is the ``$p$-tension''. We use ``$p$'' for the spatial dimensionality of the brane; thus, the coordinates $\sigma^m$, $m=0 , 1 ,\dots p$, span the $(p+1)$-dimensional world manifold $\Sigma$. The $D$ functions $Y^\mu(\sigma)$, $\mu=0 , 1 ,\dots D$, are the brane coordinates in the $D$-dimensional target spacetime. The special case $p=2$ and $D=4$ was already introduced in 1962 by Dirac in an attempt to resolve the electron-muon puzzle in terms of a relativistic membrane [3].
An alternative description, preserving world manifold reparametrization invariance, can be achieved by introducing an auxiliary world manifold metric $g_{ m n}(\sigma)$ and a ``cosmological term'' [4], [5],

$$S_{HTP}[ Y , g ]= -{m_{p+1}\over 2}\int_\Sigma d^{ p+1... ...artial_m Y^\mu \partial_n Y_\mu -(p-1) \right] \quad ,$$ (2)

where $g\equiv \det( g_{ m n} )$. In both functionals (1) and (2) the brane tension $m_{p+1}$ is a pre-assigned parameter.
The two actions (1) and (2) are classically equivalent as the ``field equations'' ${\delta S/\delta g^{ m n}(\sigma)}=0$ require the auxiliary world metric to match the induced metric, i.e. $g_{ m n}=\gamma_{ m n}=\partial_m Y^{ \mu} \partial_n Y_{ \mu}$. Moreover they are also complementary: $S_{DNG}$ provides an ``extrinsic'' geometrical description in terms of the embedding functions $Y^\mu(\sigma)$ and the induced metric $\gamma_{mn}$, while $S_{HTP}$ assigns an ``intrinsic'' geometry to the world manifold $\Sigma$ in terms of the metric $g_{ m n}$ and the ``cosmological constant'' $m_{p+1}$; the $Y^\mu(\sigma)$ functions enter as a ``multiplet of scalar fields'' propagating on a curved $(p+1)$-dimensional manifold.
More recently new action functionals have been proposed where the brane tension, or world manifold cosmological constant, is not an a priori assigned parameter, but follows from the dynamics of the object itself and can attain both positive and vanishing values. Either Kaluza-Klein type mechanism [6] and modified integration measure [7] have been proposed as candidate dynamical processes to produce tension at the classical level.