2.3 Induced Bulk and Boundary Actions

In this subsection we wish to discuss those features of the brane classical dynamics which are instrumental for the subsequent evaluation of the quantum path-integral.
In agreement with the restricted reparametrization invariance of the action (10), as discussed in the previous subsection, we first set up a ``canonical formulation'' which preserves that same symmetry through all computational steps. This means that all the world indices $m\ , n\ , \dots$ are raised, lowered and contracted by means of the center of mass metric $\overline{g}_{\, mn}$. From the brane action (10) we extract the brane relative momentum $P^m{}_\mu$ and the corresponding Hamiltonian $H$:

$$P^m{}_\mu \equiv {\partial L\over \partial \partial_m\, Y^\mu(\, \sigma\, )}= - \overline{g}^{\, mn}\, \partial_n\, Y_\mu(\, \sigma\, )\,$$

$$H \equiv P^{ m}{}_\mu\, \partial_m\, Y^\mu -L = -{1\over 2}\, \left[ \, \overline{g}_{ \, m n}\, P^{ m}{}_\mu\, P^{n}{}^\mu + m_{p+1} \, p\, \right]\ .$$ (11)

Thus, we can write the action in the following canonical form

$$S = \int_{\Sigma_{p+1}} d^{ p+1}\sigma\, \sqrt{\, \overline g\,} \, \left(\, P^{ m}{}_\mu \, \partial_m \, Y^{\mu} - H\, \right)$$

$$= \int_{\Sigma_{p+1}} d^{ p+1}\sigma\, \sqrt{\, \overline g\, }\, \... ...e{g}_{\, ab}\, P^{ a}{}_\mu \, P^{ b}{}^\mu + p\, m_{p+1}\, \right) \, \right].$$ (12)

The first term in Eq.(12) can be rewritten as follows

$$\int_{\Sigma_{p+1}} d^{ p+1}\sigma\, \sqrt{\, \overline g\, ... ...,\sqrt{\,\overline g\,} \, P^{ m}{}_\mu\, \right)\, \right]\ .$$ (13)

According to Eq. (13) we can write the total action as the sum of a boundary term plus a bulk term

$$S = S_B[\, \partial {\Sigma_{p+1}} \, ] + S_J[\, {\Sigma_{p+1}} \, ]$$

$$= \int_{\Sigma_{p}} d^{p} s \, \sqrt{h}\, N _{n}\, p^n{}_\mu\, y^\m... ... m}{}_\mu - \int_{\Sigma_{p+1}} d^{p+1}\sigma\, \sqrt{\, \overline g\, }\, H\ ,$$ (14)

where $p^n{}_\mu$ and $y^\mu$ are the momentum and coordinate of the boundary, $d^{p} s \sqrt{h} N _{n}$ represents the oriented surface element of the boundary, and $\nabla_m$ stands for the covariant derivative with respect to the metric $\overline{g}_{\, mn}$. The distinctive feature of this rearrangement is that the bulk coordinates $Y ^\mu(\sigma)$ enter the action as Lagrange multipliers enforcing the classical equation of motion:

$${\delta S\over\delta \, Y^\mu(\, \sigma\, ) }=0 \quad \Longr... ...\left(\, \sqrt{\, \overline g\, }\, P^{m}{}_\mu\, \right)=0\ .$$ (15)

The general solution of Eq. (15) may be expressed as follows

$$P^{ m}{}_\mu = \overline{g}^{ \, mn}\,\partial_n\, \phi_\mu + {1\over p!}\, \ove... ..., Y^{ \mu_{p+1}} + \partial_{[\, m_2}\, A_{\mu m_3 \dots m_{p+1}\, ] }\,\right)$$

$$= \overline{g}^{ \, mn}\, \partial_n \, \phi_\mu + {1\over p!}\, \o... ...\, P^{0)}{}_{\mu m_2\dots m_{p+1}}+ F(\, A\, )_{\mu m_2\dots m_{p+1}}\, \right)$$ (16)

with

$$\Box_{\overline g} \, \phi^\mu(\, \sigma\, ) = 0\,$$ (17)

$$\partial_m \, P^{0)}{}_{\mu\mu_2\dots \mu_{p+1}} = 0\ .$$ (18)

Here, the components $\phi_\mu$ represent local harmonic modes on the bulk, and $\overline{\epsilon}^{\, mm_2\dots m_{p+1}}\equiv (\overline g)^{-1/2}\, \delta^{[\, m m_2\dots m_{p+1}\, ]}$ stands for the totally antisymmetric tensor. Moreover, the constant antisymmetric tensor $P^{0)}{}_{\mu\mu_2\dots \mu_{p+1}}$ represents the volume momentum zero-mode, or collective-mode, that describes the global volume variation of the brane.

In order to be able to treat $\phi$, $P^{0)}$ and $A$ as independent oscillation modes, we demand that the following orthogonality relations are satisfied:

$${\mathcal{C}} _{\mathrm{I}} \equiv \int_{\Sigma_{p+1}} d^{p+1}\sigma\, \sqrt{\, \overline g\, }\, \o... ...hi^\mu\, \partial_{m_2}\, Y^{\mu_2}\dots \partial_{m_{p+1}}\, Y^{\mu_{p+1}} =0,$$ (19)

$${\mathcal{C}} _{\mathrm{II}} \equiv \int_{\Sigma_{p+1}} d^{p+1}\sigma \,\sqrt{\,\overline g\,} \overl... ..._m\, \phi^\mu\, \partial_{[\, m_2}\, P ^{0)} _{\mu m_2 \dots m_{p+1}\, ]} =0\ ,$$ (20)

$${\mathcal{C}} _{\mathrm{III}} \equiv \int_{\Sigma_{p+1}} d^{p+1}\sigma\, \sqrt{\,\overline g\,}\, \ove... ...\, \partial_m\, \phi^\mu\, \partial_{[\, m_2}\, A_{m_3 \dots m_{p+1}\, ]} =0\ .$$ (21)

The three orthogonality constraints on the bulk determine the field behavior on the boundary through Stokes' theorem. In particular ${\mathcal{C}} _{\mathrm{I}}$ gives

$${\mathcal{C}} _{\mathrm{I}}=\int_{\Sigma _p}\, \phi^\mu \, d... ...\mu_{p+1}}=0\quad\Longrightarrow\quad \phi^\mu(\,\vec s\, )=0,$$ (22)

which is a Dirichlet boundary condition, whereas in ${\mathcal{C}} _{\mathrm{III}}$ two integrations remain:

$${\mathcal{C}} _{\mathrm{III}} = \int_{\Sigma _{p}} d^{p}s \, F_\mu{}^{m_2\dots m_{p+1}}\, y^\mu\partial_{m_2}\, y^{\mu_2}\dots \partial_{m_{p+1}}\, y^{\mu_{p+1}}$$

$$-\int_{\Sigma_{p+1}} d^{p+1}\sigma\, \sqrt{\, \overline g\, } \le... ...\partial_{ m_3}\, Y^{\mu_3}\dots \partial_{ m_{p+1}}\, Y^{\mu_{p+1}} \right)\ .$$ (23)

Since the two integrations are carried over the boundary and the bulk respectively, the orthogonality condition can be satisfied only if each integral is identically vanishing,

$${\mathcal{C}} _{\mathrm{III}} =0 \quad \Longrightarrow A^\mu{}_{m_3\dots m_{p+1}}(\, \vec s\, )=\partial_{[\, m_3 }\, \Lambda^\mu\, {}_{m_4\dots m_{p+1}\, ] }$$

$$\mathrm{and} \quad \partial_{m_2}\, F_\mu{}^{m_2 m_3\dots m_{p+1}}=0 \ .$$ (24)

Thus, $A$ must solve free Maxwell-type equations on the bulk and reduce to a pure gauge configuration on the boundary. Note that under these conditions for $\phi$ and $A$, ${\mathcal{C}} _{\mathrm{II}}$ is satisfied as well.
In summary, the classical solution for the brane momentum reduces the original field content of the model to:

• a multiplet $\phi^\mu$ of world, harmonic, scalar fields (target spacetime vector) which satisfy Dirichlet boundary conditions;
• a multiplet $A^\mu{}_{m_3\dots m_{p+1} }$ of world, Kalb-Ramond fields (target spacetime vector) which reduce to a pure gauge configuration on the boundary;
• a world-manifold (cosmological) constant $P^{0)}{}_{\mu\mu_2\dots \mu_{p+1}}$ (target spacetime constant tensor) corresponding to a constant energy background along the brane world-manifold.