1. Introduction
The charged p-brane is inherently coupled to the abelian (p + 1)-form gauge field A[p+1] through the term
$\begin{eqnarray}{Q}_{p}{\int }_{{V}_{p+1}}{A}_{[p+1]}\end{eqnarray}$
in the worldvolume action. This system remains invariant under the U(1) p-form gauge transformation $\begin{eqnarray}{\tilde{A}}_{[p+1]}={A}_{[p+1]}-{\rm{d}}{{\rm{\Lambda }}}_{[p]}.\end{eqnarray}$
When p = 0, it characterizes the interaction between charged particles and the electromagnetic field. The concept of zero-form gauge symmetry has been extended to non-abelian groups [1] and plays a central role in modern physics. For p > 0, it is equally natural to consider the non-abelian extension of higher form gauge symmetries. For instance, the non-abelian two-form gauge field should appear in the worldvolume theory of M5-brane [2–4]. In the literature, various attempts [5–22] have been made in constructing the non-abelian higher form gauge theory. However, the generic theory of non-abelian p-form gauge symmetry has yet to be established. In this paper, we start with analyzing the p-form gauge symmetry from the standard p-brane point of view, and then construct a toy model for p-branes which would simplify the construction of the target space action.2. p-brane
p-brane action
Let us consider a p-brane moving in a target space ${ \mathcal M }$ with coordinates xM. The tension of the p-brane is ${T}_{p}={l}_{p}^{-(p+1)}$, and its charge is taken to be − Qp. The embedding of the n =p + 1 dimensional worldvolume Vn into ${ \mathcal M }$ is given by xM = XM(σ), where σμ = {τ, σi} are the worldvolume coordinates. The p-brane naturally couples to the target space metric GMN(x) as well as the (p + 1)-form gauge field
$\begin{eqnarray}{A}_{[p+1]}(x)=\frac{1}{(p+1)!}{A}_{{M}_{0}\cdots {M}_{p}}(x){\rm{d}}{x}^{{M}_{0}}\wedge \cdots \wedge {\rm{d}}{x}^{{M}_{p}}.\end{eqnarray}$
The induced quantities on Vn are $\begin{eqnarray}{{\boldsymbol{h}}}_{\mu \nu }={G}_{MN}(X){\partial }_{\mu }{X}^{M}{\partial }_{\nu }{X}^{N},\end{eqnarray}$
$\begin{eqnarray}{{\boldsymbol{A}}}_{{\mu }_{0}\cdots {\mu }_{p}}={A}_{{M}_{0}\cdots {M}_{p}}(X){\partial }_{{\mu }_{0}}{X}^{{M}_{0}}\cdots {\partial }_{{\mu }_{p}}{X}^{{M}_{p}}.\end{eqnarray}$
The dynamics of the p-brane are governed by the action $\begin{eqnarray}{S}_{p}={\int }_{{V}_{n}}{{\rm{d}}}^{n}{\boldsymbol{\sigma }}\left[-{T}_{p}\sqrt{{\boldsymbol{h}}}-\frac{{Q}_{p}}{(p+1)!}{\varepsilon }^{{\mu }_{0}\cdots {\mu }_{p}}{{\boldsymbol{A}}}_{{\mu }_{0}\cdots {\mu }_{p}}\right],\end{eqnarray}$
where ${\varepsilon }^{{\mu }_{0}\cdots {\mu }_{p}}$ is the total antisymmetric with ϵ0⋯p = 1 and $\begin{eqnarray}{\boldsymbol{h}}=-\det ({{\boldsymbol{h}}}_{\mu \nu }).\end{eqnarray}$
The conjugation momentum density of this system is given by
$\begin{eqnarray}{{\rm{\Pi }}}_{M}=\frac{\delta S}{\delta {\dot{X}}^{M}}={T}_{p}{P}_{M}-{Q}_{p}{{ \mathcal A }}_{M},\end{eqnarray}$
where PM is the mechanical momentum density $\begin{eqnarray}{P}_{M}=-{{\boldsymbol{h}}}^{\frac{1}{2}}{{\boldsymbol{h}}}^{0\mu }{G}_{MN}(X){\partial }_{\mu }{X}^{N},\end{eqnarray}$
and the additional term ${{ \mathcal A }}_{M}$ $\begin{eqnarray}\begin{array}{rcl}{{ \mathcal A }}_{M} & = & \frac{1}{p!}{\varepsilon }^{{i}_{1}\cdots {i}_{p}}{A}_{M{M}_{1}\cdots {M}_{p}}(X){\partial }_{{i}_{1}}{X}^{{M}_{1}}\cdots {\partial }_{{i}_{p}}{X}^{{M}_{p}},\\ & & {\varepsilon }^{{i}_{1}\cdots {i}_{p}}\equiv {\varepsilon }^{0{i}_{1}\cdots {i}_{p}},\end{array}\end{eqnarray}$
arises due to coupling with the (p + 1)-form gauge field. The classical dynamics of the p-brane are entirely determined by the constraints of the mechanical momentum $\begin{eqnarray}\begin{array}{rcl}{G}^{MN}(X){P}_{M}{P}_{N} & = & -\det ({{\boldsymbol{h}}}_{ij}),\\ & & {P}_{M}{\partial }_{i}{X}^{M}=0.\end{array}\end{eqnarray}$
p-brane wave functional
At the off-shell level, this system can be formally quantized by the canonical procedure. The Poisson brackets are mapped to the equal-time commutators of the operators 12 ) imply that we can identify the canonical momentum density operator with the functional derivative operator 11 ).
$\begin{eqnarray}\begin{array}{rcl}[{X}^{M}(0,\sigma ),{{\rm{\Pi }}}_{N}(0,\tilde{\sigma })] & = & {\rm{i}}{\delta }_{N}^{M}{\delta }^{p}(\sigma -\tilde{\sigma }),\\ & & [{X}^{M}(0,\sigma ),{X}^{N}(0,\tilde{\sigma })]=0,\\ & & [{{\rm{\Pi }}}_{M}(0,\sigma ),{{\rm{\Pi }}}_{N}(0,\tilde{\sigma })]=0.\end{array}\end{eqnarray}$
These operators act on the p-brane wave functional $\begin{eqnarray}{\rm{\Psi }}[{X}^{M}(\sigma )]=\langle {X}^{M}(\sigma )| {\rm{\Psi }}\rangle ,\end{eqnarray}$
which is defined on the infinite dimensional space {XM(σ)} of p-brane equal time configurations. In the path integral formalism, the p-brane wave functional can be formally expressed as $\begin{eqnarray}{\rm{\Psi }}[X(\sigma )]={\int }_{{\rm{\Psi }}}^{\tilde{X}(0,\sigma )=X(\sigma )}[{ \mathcal D }\tilde{X}]{{\rm{e}}}^{{\rm{i}}{S}_{p}[\tilde{X}]}.\end{eqnarray}$
The canonical commutation relations ( $\begin{eqnarray}{{\rm{\Pi }}}_{M}(\sigma )=-{\rm{i}}\frac{\delta }{\delta {X}^{M}(\sigma )}.\end{eqnarray}$
Thus, the quantum operator for iTpPM is the functional covariant derivative $\begin{eqnarray}{\rm{i}}{T}_{p}{P}_{M}={{\mathscr{D}}}_{M}(\sigma )=\frac{\delta }{\delta {X}^{M}(\sigma )}+{\rm{i}}{Q}_{p}{{ \mathcal A }}_{p}.\end{eqnarray}$
The wave functional $\Psi$[XM(σ)] must satisfies the wave equations which are the operator equations for the Hamiltonian constraints (3. p-form gauge symmetry
3.1. Abelian p-form gauge symmetry
Functional U(1) gauge symmetry
When p = 0, the local gauge symmetry is associated with the phase ambiguity of the corresponding wave functions of point particles. To discuss the higher form gauge symmetry, it is natural to consider the local U(1) phase ambiguity of the p-brane wave functional $\Psi$[XM(σ)]. Generically, it is
$\begin{eqnarray}{\rm{\Psi }}[{X}^{M}(\sigma )]\sim \tilde{{\rm{\Psi }}}[{X}^{M}(\sigma )]={{\rm{e}}}^{{\rm{i}}{Q}_{p}{\rm{\Theta }}[{X}^{M}(\sigma )]}{\rm{\Psi }}[{X}^{M}(\sigma )],\end{eqnarray}$
where the U(1) phase Θ[XM(σ)] itself is a local functional in the p-brane configuration space {XM(σ)}. It is generically non-local in the target space point of view. For simplicity, we set Qp = 1 in the following discussions.As in the p = 0 case, one can introduce the corresponding functional gauge covariant derivative
$\begin{eqnarray}{{\mathscr{D}}}_{M}(\sigma )=\frac{\delta }{\delta {X}^{M}(\sigma )}+{\rm{i}}{{\mathscr{A}}}_{M}(\sigma )[X].\end{eqnarray}$
The functional U(1) gauge field 1-form in the p-brane configuration space {XM(σ)} is $\begin{eqnarray}{\mathscr{A}}[X]={\int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}\sigma \,\delta {X}^{M}(\sigma ){{\mathscr{A}}}_{M}(\sigma )[X],\end{eqnarray}$
where {δXM(σ)} forms a basis of differential forms on {XM(σ)}. Together with the total derivative operator $\begin{eqnarray}\delta ={\int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}\sigma \,\delta {X}^{M}(\sigma )\frac{\delta }{\delta {X}^{M}(\sigma )},\end{eqnarray}$
in {XM(σ)}, we have $\begin{eqnarray}{\mathscr{D}}={\int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}\sigma \,\delta {X}^{M}(\sigma ){{\mathscr{D}}}_{M}(\sigma )=\delta +{\rm{i}}{\mathscr{A}}[X].\end{eqnarray}$
The gauge covariance of ${\mathscr{D}}$ demands that
$\begin{eqnarray}{{\rm{e}}}^{{\rm{i}}{\rm{\Theta }}[X]}{\mathscr{D}}{\rm{\Psi }}[X]=\tilde{{\mathscr{D}}}\tilde{{\rm{\Psi }}}[X].\end{eqnarray}$
It follows that the functional one-form U(1) gauge field must transform as $\begin{eqnarray}\tilde{{\mathscr{A}}}[X]={\mathscr{A}}[X]-\delta {\rm{\Theta }}[X].\end{eqnarray}$
The commutator of the covariant derivatives gives rise to the functional field strength two-form $\begin{eqnarray}\begin{array}{rcl}{\mathscr{F}}[X] & = & -\frac{{\rm{i}}}{2}\left[{\mathscr{D}},{\mathscr{D}}\right]=\frac{1}{2}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}{\sigma }_{1}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}{\sigma }_{2}\\ & & \times \,\delta {X}^{{M}_{1}}({\sigma }_{1})\wedge \delta {X}^{{M}_{2}}({\sigma }_{2}){{\mathscr{F}}}_{{M}_{1},{M}_{2}}({\sigma }_{1},{\sigma }_{2})[X],\end{array}\end{eqnarray}$
where ∧ is the wedge product in the p-brane configuration space {XM(σ)}, and $\begin{eqnarray}\begin{array}{rcl}{{\mathscr{F}}}_{{M}_{1},{M}_{2}}({\sigma }_{1},{\sigma }_{2})[X] & = & \frac{\delta }{\delta {X}^{{M}_{1}}({\sigma }_{1})}{{\mathscr{A}}}_{{M}_{2}}({\sigma }_{2})[X]\\ & & -\frac{\delta }{\delta {X}^{{M}_{2}}({\sigma }_{2})}{{\mathscr{A}}}_{{M}_{1}}({\sigma }_{1})[X].\end{array}\end{eqnarray}$
As in the p = 0 case, ${\mathscr{F}}[X]$ is invariant under the functional U(1) gauge transformation.Abelian p-form gauge symmetry
Under the target space abelian p-form gauge transformation, the (p + 1)-form gauge field transforms as 6 ) 17 ), it is obvious that the p-form gauge symmetry is just the special realization
$\begin{eqnarray}{A}_{[p+1]}(x)\to {\tilde{A}}_{[p+1]}(x)={A}_{[p+1]}(x)-{\rm{d}}{{\rm{\Lambda }}}_{[p]}(x).\end{eqnarray}$
It induces a boundary term in the worldvolume action ( $\begin{eqnarray}{\tilde{S}}_{p}={S}_{p}+{\int }_{{V}_{n}}{\rm{d}}{{\rm{\Lambda }}}_{[p]}={S}_{p}+{\int }_{\partial {V}_{n}}{{\rm{\Lambda }}}_{[p]}.\end{eqnarray}$
Consequently, the p-brane wave functional $\begin{eqnarray}{\rm{\Psi }}[X(\sigma )]={\int }_{{\rm{\Psi }}}^{\tilde{X}(0,\sigma )=X(\sigma )}[{ \mathcal D }\tilde{X}]{{\rm{e}}}^{{\rm{i}}{S}_{p}[\tilde{X}]},\end{eqnarray}$
transforms as $\begin{eqnarray}\tilde{{\rm{\Psi }}}[X(\sigma )]={{\rm{e}}}^{{\rm{i}}{\int }_{{{\rm{\Sigma }}}_{p}=\partial {V}_{n}}{{\rm{\Lambda }}}_{[p]}}{\rm{\Psi }}[X(\sigma )].\end{eqnarray}$
Comparing with ( $\begin{eqnarray}\begin{array}{rcl}{\rm{\Theta }}[X] & = & {\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{\Lambda }}}_{[p]}(X)=\frac{1}{p!}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}\sigma \,{\varepsilon }^{{i}_{1}\cdots {i}_{p}}{{\rm{\Lambda }}}_{{M}_{1}\cdots {M}_{p}}(X(\sigma ))\\ & & \times {\partial }_{{i}_{1}}{X}^{{M}_{1}}(\sigma )\cdots {\partial }_{{i}_{p}}{X}^{{M}_{p}}(\sigma )\end{array}\end{eqnarray}$
of the local U(1) transformation in p-brane configuration space {XM(σ)}.The corresponding functional one-form gauge field in {XM(σ)} is 23 ) correctly reproduces the initial (p + 1)-form gauge field transformation rule (26 ) in this special realization
$\begin{eqnarray}\begin{array}{rcl}{\mathscr{A}}[X] & = & \frac{1}{p!}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}\sigma \,\delta {X}^{M}(\sigma ){\varepsilon }^{{i}_{1}\cdots {i}_{p}}{A}_{M{M}_{1}\cdots {M}_{p}}(X)\\ & & \times {\partial }_{{i}_{1}}{X}^{{M}_{1}}(\sigma )\cdots {\partial }_{{i}_{p}}{X}^{{M}_{p}}(\sigma )\\ & = & {\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{\iota }_{\delta {X}^{M}(\sigma )}{A}_{[p+1]}(X).\end{array}\end{eqnarray}$
By using integration by parts on Σp, one can verify that the generic transformation rule ( $\begin{eqnarray}\begin{array}{rcl}\tilde{{\mathscr{A}}}[X] & = & {\mathscr{A}}[X]-\delta {\rm{\Theta }}[X]\\ & = & {\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{\iota }_{\delta {X}^{M}(\sigma )}\left[{A}_{[p+1]}(X)-{\rm{d}}{{\rm{\Lambda }}}_{[p]}(X)\right].\end{array}\end{eqnarray}$
Furthermore, it is natural to expect that the functional two-form gauge field strength ${\mathscr{F}}[X]$ is related to the (p + 2)-form field strength
$\begin{eqnarray}{F}_{[p+2]}={\rm{d}}{A}_{[p+1]}.\end{eqnarray}$
In fact, given the boundary condition $\delta X(\sigma ){| }_{\partial {{\rm{\Sigma }}}_{p}}=0$, we find that $\begin{eqnarray}\begin{array}{rcl}{\mathscr{F}}[X] & = & \frac{1}{2\,p!}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}\sigma \,\delta {X}^{M}(\sigma )\wedge \delta {X}^{N}(\sigma ){\varepsilon }^{{i}_{1}\cdots {i}_{p}}\\ & & \times {F}_{MN{M}_{1}\cdots {M}_{p}}(X){\partial }_{{i}_{1}}{X}^{{M}_{1}}(\sigma )\cdots {\partial }_{{i}_{p}}{X}^{{M}_{p}}(\sigma )\\ & = & \frac{1}{2}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}\,{\iota }_{\delta {X}^{M}(\sigma )\wedge \delta {X}^{N}(\sigma )}{F}_{[p+2]}(X(\sigma )).\end{array}\end{eqnarray}$
3.2. Non-abelian generalization
Functional non-abelian gauge symmetry
As in the p = 0 case, we can further consider the wave functional multiplet ${{\rm{\Phi }}}^{\hat{A}}[X(\sigma )]$ which forms a linear representation of a non-abelian group ${ \mathcal G }$ with the Lie algebra
$\begin{eqnarray}[{t}_{\hat{a}},{t}_{\hat{b}}]={\rm{i}}{f}_{\hat{a}\hat{b}}^{\hat{c}}{t}_{\hat{c}}.\end{eqnarray}$
At the infinitesimal level, the functional gauge transformation is $\begin{eqnarray}{\mathfrak{U}}[X]=1+{\rm{i}}{{\mathscr{U}}}^{\hat{a}}[X]{({t}_{\hat{a}})}^{\hat{A}}{\,}_{\hat{B}}.\end{eqnarray}$
Correspondingly, we have the functional one-form gauge field $\begin{eqnarray}\begin{array}{rcl}{\mathscr{A}}[X] & = & {{\mathscr{A}}}^{\hat{a}}[X]{({t}_{\hat{a}})}^{\hat{A}}{\,}_{\hat{B}}\\ & = & {\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}\sigma \,\delta {X}^{M}(\sigma ){{\mathscr{A}}}_{M}^{\hat{a}}(\sigma )[X]{({t}_{\hat{a}})}^{\hat{A}}{\,}_{\hat{B}},\end{array}\end{eqnarray}$
as well as the functional gauge covariant derivative similar to the abelian case $\begin{eqnarray}{\mathscr{D}}=\delta +{\rm{i}}{\mathscr{A}}[X].\end{eqnarray}$
The gauge covariance of the functional covariant derivative $\begin{eqnarray}{\mathfrak{U}}[X]{\mathscr{D}}{\rm{\Phi }}[X]=\tilde{{\mathscr{D}}}({\mathfrak{U}}[X]{\rm{\Phi }}[X]),\end{eqnarray}$
implies that $\begin{eqnarray}\tilde{{\mathscr{A}}}[X]={\mathfrak{U}}[X]{\mathscr{A}}[X]{{\mathfrak{U}}}^{-1}[X]-\delta {\mathfrak{U}}[X]{{\mathfrak{U}}}^{-1}[X].\end{eqnarray}$
At the infinitesimal level, it becomes $\begin{eqnarray}{\delta }_{{ \mathcal G }}{\mathscr{A}}[X]=-{\mathscr{D}}{{\mathscr{U}}}^{\hat{a}}[X].\end{eqnarray}$
The functional field strength two-form is
$\begin{eqnarray}\begin{array}{rcl}{\mathscr{F}}[X] & = & -\frac{{\rm{i}}}{2}\left[{\mathscr{D}},{\mathscr{D}}\right]=\frac{1}{2}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}{\sigma }_{1}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}{\sigma }_{2}\\ & & \times \,\delta {X}^{{M}_{1}}({\sigma }_{1})\wedge \delta {X}^{{M}_{2}}({\sigma }_{2})\\ & & \times {{\mathscr{F}}}_{{M}_{1},{M}_{2}}^{\hat{a}}({\sigma }_{1},{\sigma }_{2})[X]{t}_{\hat{a}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{{\mathscr{F}}}_{{M}_{1},{M}_{2}}^{\hat{a}}({\sigma }_{1},{\sigma }_{2})[X] & = & \frac{\delta }{\delta {X}^{{M}_{1}}({\sigma }_{1})}{{\mathscr{A}}}_{{M}_{2}}^{\hat{a}}({\sigma }_{2})[X]\\ & & -\frac{\delta }{\delta {X}^{{M}_{2}}({\sigma }_{2})}{{\mathscr{A}}}_{{M}_{1}}^{\hat{a}}({\sigma }_{1})[X]\\ & & +{\rm{i}}{f}^{\hat{a}}{\,}_{{\hat{b}}_{1}{\hat{b}}_{2}}{{\mathscr{A}}}_{{M}_{1}}^{{\hat{b}}_{1}}({\sigma }_{1})[X]{{\mathscr{A}}}_{{M}_{2}}^{{\hat{b}}_{2}}({\sigma }_{2})[X].\end{array}\end{eqnarray}$
Non-abelian p-form gauge symmetry
Inspired by the abelian case, we can try to realize ${{\mathscr{A}}}^{\hat{a}}[X]$ specifically through a target space non-abelian p-form gauge field ${A}_{M{M}_{1}\cdots {M}_{p}}^{\hat{a}}(X)$ as follows
$\begin{eqnarray}\begin{array}{rcl}{{\mathscr{A}}}^{\hat{a}}[X] & = & \frac{1}{p!}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}\sigma \,\delta {X}^{M}(\sigma ){\varepsilon }^{{i}_{1}\cdots {i}_{p}}{A}_{M{M}_{1}\cdots {M}_{p}}^{\hat{a}}(X)\\ & & \times {\partial }_{{i}_{1}}{X}^{{M}_{1}}(\sigma )\cdots {\partial }_{{i}_{p}}{X}^{{M}_{p}}(\sigma )\\ & = & {\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{\iota }_{\delta {X}^{M}(\sigma )}{A}_{[p+1]}^{\hat{a}}[X].\end{array}\end{eqnarray}$
The corresponding infinitesimal functional gauge transformation is expected to be $\begin{eqnarray}\begin{array}{rcl}{{\mathscr{U}}}^{\hat{a}}[X] & = & {\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{u}_{[p]}^{\hat{a}}[X]=\frac{1}{p!}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}\sigma \,{\varepsilon }^{{i}_{1}\cdots {i}_{p}}{u}_{{M}_{1}\cdots {M}_{p}}^{\hat{a}}(X)\\ & & \times {\partial }_{{i}_{1}}{X}^{{M}_{1}}(\sigma )\cdots {\partial }_{{i}_{p}}{X}^{{M}_{p}}(\sigma ),\end{array}\end{eqnarray}$
which is associated with a target space Lie-algebra valued p-form ${u}_{[p]}^{\hat{a}}(x)$.The infinitesimal functional gauge transformation of ${{\mathscr{A}}}^{\hat{a}}[X]$ then becomes
$\begin{eqnarray}\begin{array}{l}{\delta }_{{ \mathcal G }}{{\mathscr{A}}}^{\hat{a}}[X]=-{\mathscr{D}}{{\mathscr{U}}}^{\hat{a}}[x]=-{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{\iota }_{\delta {X}^{M}(\sigma )}{\rm{d}}{u}_{[p]}^{\hat{a}}\\ -{\rm{i}}{f}^{\hat{a}}{\,}_{\hat{b}\hat{c}}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{\iota }_{\delta {X}^{M}(\sigma )}\,{A}_{[p+1]}^{\hat{b}}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{u}_{[p]}^{\hat{c}}.\end{array}\end{eqnarray}$
It implies that the target space non-abelian p-form gauge transformation of ${A}_{[p+1]}^{\hat{a}}$ is $\begin{eqnarray}{\delta }_{{ \mathcal G }}{A}_{[p+1]}^{\hat{a}}=-{\rm{d}}{u}_{[p]}^{\hat{a}}-{\rm{i}}{f}^{\hat{a}}{\,}_{\hat{b}\hat{c}}{A}_{[p+1]}^{\hat{b}}{\int }_{{{\rm{\Sigma }}}_{p}}{u}_{[p]}^{\hat{c}}.\end{eqnarray}$
Although ${\delta }_{{ \mathcal G }}{{\mathscr{A}}}^{\hat{a}}[X]$ is local in the p-brane configuration space {XM(σ)}, the induced ${\delta }_{{ \mathcal G }}{A}_{[p+1]}^{\hat{a}}(x)$ is non-local from the target space point of view.Under the present special realization, the functional two-form field strength is
$\begin{eqnarray}\begin{array}{rcl}{{\mathscr{F}}}^{\hat{a}}[X] & = & \frac{1}{2}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}{\sigma }_{1}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{{\rm{d}}}^{p}{\sigma }_{2}\,\delta {X}^{{M}_{1}}({\sigma }_{1})\\ & & \times \wedge \delta {X}^{{M}_{2}}({\sigma }_{2}){{\mathscr{F}}}_{{M}_{1},{M}_{2}}^{\hat{a}}({\sigma }_{1},{\sigma }_{2})[X]\\ & = & \frac{1}{2}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}\,{\iota }_{\delta {X}^{{M}_{1}}(\sigma )\wedge \delta {X}^{{M}_{2}}(\sigma )}{\rm{d}}{A}_{[p+1]}^{\hat{a}}(X(\sigma ))\\ & & +\frac{{\rm{i}}}{2}{f}^{\hat{a}}{\,}_{{\hat{b}}_{1}{\hat{b}}_{2}}{\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{\iota }_{\delta {X}^{{M}_{1}}({\sigma }_{1})}{A}_{[p+1]}^{{\hat{b}}_{1}}(X({\sigma }_{1}))\\ & & \times \wedge {\displaystyle \int }_{{{\rm{\Sigma }}}_{p}}{\iota }_{\delta {X}^{{M}_{2}}({\sigma }_{2})}{A}_{[p+1]}^{{\hat{b}}_{2}}(X({\sigma }_{2})).\end{array}\end{eqnarray}$
Unlike the abelian case, we cannot define a local (p + 2)-form gauge field strength since ${{\mathscr{F}}}_{{M}_{1},{M}_{2}}^{\hat{a}}({\sigma }_{1},{\sigma }_{2})[X]$ is typically a bi-local quantity from the perspective of the target space. One can only try to define a semi-local (p + 2)-form field strength F[p+2](x, Σp). The semi-local field strength depends on the spacetime point x as well as the p-brane configuration Σp where x ∈ Σp. For such a field strength, the electric/magnetic dual is expected to be $\begin{eqnarray}{\check{F}}_{[d-p-2]}(x,{\check{{\rm{\Sigma }}}}_{d-p-4})={* }_{{{\rm{\Sigma }}}_{p},{\check{{\rm{\Sigma }}}}_{d-p-4}}{F}_{[p+2]}(x,{{\rm{\Sigma }}}_{p})\end{eqnarray}$
where the dual operator ${* }_{{\rm{\Sigma }},\check{{\rm{\Sigma }}}}$ depends on the choices of the electric p-brane configuration as well as the magnetic (d − p − 4)-brane configuration. In particular, to construct the M5 worldvolume theory, it seems that one needs to impose the Σ1 dependent three-form self-dual condition $\begin{eqnarray}\begin{array}{l}{\check{F}}_{[3]}(x,{\check{{\rm{\Sigma }}}}_{1}={{\rm{\Sigma }}}_{1})\\ \quad =\,{* }_{{{\rm{\Sigma }}}_{1},{\check{{\rm{\Sigma }}}}_{1}={{\rm{\Sigma }}}_{1}}{F}_{[3]}(x,{{\rm{\Sigma }}}_{1})={F}_{[3]}(x,{{\rm{\Sigma }}}_{1}).\end{array}\end{eqnarray}$
Obviously, there are many ambiguities in defining F[p+2](x, Σp) and ${* }_{{\rm{\Sigma }},\check{{\rm{\Sigma }}}}$. We hope to clarify these issues in our future works.
In fact, the presence of spacetime non-locality is very crucial for the existence of non-abelian higher form gauge symmetries. In the discussions of generalized symmetry in QFT [23], a p-form symmetry is defined by a topological operator Q which can be inserted along any codimension-(p + 1) submanifold Σd−p−1. For two topological operators of codimension greater than one, one can use topological deformations to change the ordering. Consequently, the corresponding p-form gauge symmetry must be abelian. Such a description of p-form gauge symmetry is ‘local’in spacetime, in the sense that the group element is simply given by Q(Σd−p−1). However, in our previous discussions, the action of non-abelian p-form gauge transformation intrinsically depends on the p-brane configuration Σp. Thus the non-abelian group element shall be Q(Σd−p−1∣Σp), which also depends on the p-brane configuration Σp. Fixing Σp, one can still expect that a single operator Q(Σd−p−1∣Σp) is topological. However, in this spacetime non-local picture, the additional dependence on Σp becomes an obstruction to performing the topological deformations which change the ordering of two operators.
4. A toy matrix model
In section 3 , it is shown that the target space p-form gauge symmetry can be regarded as a specific construction of the generic zero-form gauge symmetry in the functional space {XM(σ)} of p-brane configurations. For p > 0, the theory is complicated since one needs to deal with infinite dimensional functional space {XM(σ)}. In this section, we develop a much simpler toy model in which the infinite dimensional functional space {XM(σ)} is replaced by a finite dimensional matrix space.
Matrix string model
For simplicity, let us first consider the p = 1 case. The basic idea is to replace the continuous spatial coordinate σ on the string worldsheet by the discrete indices of a 2 × 2 matrix. That is
$\begin{eqnarray}\sigma {\leftrightarrow }_{\,b}^{a},\qquad a,b=1,2.\end{eqnarray}$
For a field φ(σ) on string, the infinite dimensional configuration space {φ(σ)} is replaced by the space ${{ \mathcal M }}^{(2)}$ of 2 × 2 matrices. Under the equal time mode expansion, φ(σ) will be expanded by the ‘eigenfunctions’${({\hat{\phi }}_{\kappa })}^{a}{\,}_{b}$ $\begin{eqnarray}\begin{array}{rcl}{\hat{\phi }}_{0} & = & {\bf{1}},\quad {\hat{\phi }}_{\pm }={\hat{\sigma }}_{\pm }=\frac{1}{2}({\hat{\sigma }}_{1}\pm {\rm{i}}{\hat{\sigma }}_{2}),\\ {\hat{\phi }}_{\odot } & = & {\hat{\sigma }}_{3},\end{array}\end{eqnarray}$
where ${\hat{\sigma }}_{i}$’s are the Pauli matrices. For example, XM(σ) becomes $\begin{eqnarray}\begin{array}{rcl}{X}^{M}(\sigma ) & \to & \,{x}^{M}{\hat{\phi }}_{0}+{l}_{s}\displaystyle \sum _{\kappa =\pm }{\chi }^{M\kappa }{\hat{\phi }}_{\kappa }+{l}_{s}^{2}{w}^{M}{\hat{\phi }}_{\odot }\\ & = & \left(\begin{array}{cc}{x}^{M}+{l}_{s}^{2}{w}^{M} & {l}_{s}{\chi }^{M,+}\\ {l}_{s}{\chi }^{M,-} & {x}^{M}-{l}_{s}^{2}{w}^{M}\end{array}\right),\end{array}\end{eqnarray}$
where ${l}_{s}={l}_{1}={T}_{1}^{-\frac{1}{2}}$ is the string length parameter.The products of functions on string are mapped to
$\begin{eqnarray}\begin{array}{rcl}\phi (\sigma )\psi ({\sigma }^{{\prime} }) & \leftrightarrow & {\phi }^{a}{\,}_{b}{\psi }^{{a}^{{\prime} }}{\,}_{{b}^{{\prime} }},\\ \phi (\sigma )\psi (\sigma )=(\phi \psi )(\sigma ) & \leftrightarrow & {(\phi \psi )}^{a}{\,}_{b}={\phi }^{a}{\,}_{c}{\psi }^{c}{\,}_{b}.\end{array}\end{eqnarray}$
The spatial integration on the string worldsheet becomes the symmetric trace $\begin{eqnarray}\begin{array}{l}\displaystyle \int {\rm{d}}\sigma \,{\phi }_{1}(\sigma )\cdots {\phi }_{m}(\sigma )\leftrightarrow {{\rm{Tr}}}_{{\rm{sym}}}({\phi }_{1}\cdots {\phi }_{m})\\ \quad =\frac{1}{m!}{\rm{Tr}}({\phi }_{1}\cdots {\phi }_{m}+{\rm{permutations}}).\end{array}\end{eqnarray}$
Especially, $\begin{eqnarray}{V}_{1}=\int {\rm{d}}\sigma \,\leftrightarrow {\rm{Tr}}{\boldsymbol{1}}=2.\end{eqnarray}$
Although the matrix product is non-commutative, we do not need to worry about it inside the symmetric trace.The inner product of the functions on string becomes
$\begin{eqnarray}\langle \phi ,\psi \rangle =\frac{1}{{V}_{1}}\int {\rm{d}}\sigma \,{\phi }^{\dagger }(\sigma )\psi (\sigma )\leftrightarrow \frac{1}{2}{\rm{Tr}}({\phi }^{\dagger }\psi ).\end{eqnarray}$
Thus the ‘eigenfunctions’${\hat{\phi }}_{\kappa }$ satisfy the orthonormality condition $\begin{eqnarray}\langle {\hat{\phi }}_{\tilde{\kappa }},{\hat{\phi }}_{\kappa }\rangle =\frac{1}{2}{\rm{Tr}}[{\hat{\phi }}_{\tilde{\kappa }}^{\dagger }(\sigma ){\hat{\phi }}_{\kappa }(\sigma )]={\delta }_{\tilde{\kappa }\kappa }.\end{eqnarray}$
Summing over these ‘eigenfunctions’, we get the δ-function of this matrix string model $\begin{eqnarray}\begin{array}{rcl}\delta (\sigma ;{\sigma }^{{\prime} }) & = & \displaystyle \sum _{\kappa }{\hat{\phi }}_{\kappa }(\sigma ){\hat{\phi }}_{\kappa }^{\dagger }({\sigma }^{{\prime} })\leftrightarrow {\delta }^{a;{a}^{{\prime} }}{\,}_{b;{b}^{{\prime} }}\\ & = & \frac{1}{2}\left[{\delta }^{a}{\,}_{b}{\delta }_{{b}^{{\prime} }}^{{a}^{{\prime} }}+{({\hat{{\boldsymbol{\sigma }}}}^{m})}_{b}^{a}{({\hat{{\boldsymbol{\sigma }}}}_{m})}_{{b}^{{\prime} }}^{{a}^{{\prime} }}\right]={\delta }^{a}{\,}_{{b}^{{\prime} }}{\delta }^{{a}^{{\prime} }}{\,}_{b}.\end{array}\end{eqnarray}$
One can easily confirm that the fundamental property of the δ-function is satisfied $\begin{eqnarray}\begin{array}{l}\displaystyle \int {\rm{d}}{\sigma }^{{\prime} }\,\delta (\sigma ;{\sigma }^{{\prime} })\psi ({\sigma }^{{\prime} })\leftrightarrow {\delta }^{a;{a}^{{\prime} }}{\,}_{b;{c}^{{\prime} }}{\psi }^{{c}^{{\prime} }}\,{\,}_{{a}^{{\prime} }}\\ =\,{\delta }^{a}{\,}_{{c}^{{\prime} }}{\delta }^{{a}^{{\prime} }}{\,}_{b}{\psi }^{{c}^{{\prime} }}{\,}_{{a}^{{\prime} }}={\psi }^{a}{\,}_{b}\leftrightarrow \psi (\sigma ).\end{array}\end{eqnarray}$
It is also obvious that $\begin{eqnarray}\frac{\delta \phi (\sigma )}{\delta \phi ({\sigma }^{{\prime} })}\leftrightarrow \frac{\partial {\phi }_{b}^{a}}{\partial {\phi }_{{b}^{{\prime} }}^{{a}^{{\prime} }}}={\delta }_{{a}^{{\prime} }}^{a}{\delta }_{b}^{{b}^{{\prime} }}\leftrightarrow \delta (\sigma ;{\sigma }^{{\prime} }).\end{eqnarray}$
The spatial derivative operator on the worldsheet is realized by
$\begin{eqnarray}{\partial }_{\sigma }\phi (\sigma )\leftrightarrow {l}_{s}^{-1}[{\hat{\sigma }}_{3},\phi ].\end{eqnarray}$
By construction, it satisfies the Leibniz rule $\begin{eqnarray}\begin{array}{l}{\partial }_{\sigma }(\phi \psi )\leftrightarrow {l}_{s}^{-1}[{\hat{\sigma }}_{3},\phi \psi ]={l}_{s}^{-1}[{\hat{\sigma }}_{3},\phi ]\psi \\ +{l}_{s}^{-1}\phi [{\hat{\sigma }}_{3},\psi ]\leftrightarrow {\partial }_{\sigma }\phi \psi +\phi {\partial }_{\sigma }\psi .\end{array}\end{eqnarray}$
Integrating over the string, we get $\begin{eqnarray}\int {\rm{d}}\sigma \,{\partial }_{\sigma }\phi (\sigma )\leftrightarrow {l}_{s}^{-1}{\rm{Tr}}[{\hat{{\boldsymbol{\sigma }}}}_{3},\phi ]=0,\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}\displaystyle \int {\rm{d}}{\sigma }^{{\prime} }{\partial }_{{\sigma }^{{\prime} }}\delta (\sigma ;{\sigma }^{{\prime} })\phi ({\sigma }^{{\prime} })\\ \quad \leftrightarrow {l}_{s}^{-1}[{({\hat{{\boldsymbol{\sigma }}}}_{3})}_{{c}^{{\prime} }}^{{b}^{{\prime} }}{\delta }_{b;{a}^{{\prime} }}^{a;{c}^{{\prime} }}-{\delta }_{b;{c}^{{\prime} }}^{a;{b}^{{\prime} }}{({\hat{{\boldsymbol{\sigma }}}}_{3})}_{{a}^{{\prime} }}^{{c}^{{\prime} }}]{\phi }_{{b}^{{\prime} }}^{{a}^{{\prime} }}\\ \quad ={l}_{s}^{-1}[{({\hat{{\boldsymbol{\sigma }}}}_{3})}_{b}^{{b}^{{\prime} }}{\phi }_{{b}^{{\prime} }}^{a}-{\phi }_{b}^{{a}^{{\prime} }}{({\hat{{\boldsymbol{\sigma }}}}_{3})}_{{a}^{{\prime} }}^{a}]\\ \quad =-{l}_{s}^{-1}{[{\hat{{\boldsymbol{\sigma }}}}_{3},\phi ]}_{b}^{a}\leftrightarrow -{\partial }_{\sigma }\phi (\sigma ).\end{array}\end{eqnarray}$
Given a target space field Φ(x), its pullback on the matrix worldsheet is defined by the formal Taylor expansion 54 , 55 , 59 , 62 ), all the σ-integrated expressions that appear in the original string model can be transplanted to our matrix models on ${{ \mathcal M }}^{(2)}$. Due to the properties (60 , 61 , 63 , 64 , 65 ), the discussions in section 3 are directly inherited by the present matrix model for p = 1. For example, the target space two-form gauge field BMN(x) is used to construct the one-form gauge field ${{\mathscr{A}}}^{\hat{a}}[X]$ in the matrix string space 3 implies that
$\begin{eqnarray}\begin{array}{l}{\rm{\Phi }}(X)=\displaystyle \sum _{n=0}^{\infty }\frac{1}{n!}{\partial }_{{N}_{n}}\cdots {\partial }_{{N}_{2}}{\partial }_{{N}_{1}}{\rm{\Phi }}(x){({X}^{{N}_{1}})}^{a},\\ \quad {\,}_{{a}_{2}}{({X}^{{N}_{2}})}^{{a}_{2}}{\,}_{{a}_{3}}\cdots {({X}^{{N}_{n}})}^{{a}_{n}}{\,}_{b}.\end{array}\end{eqnarray}$
Now, together with the definitions ( $\begin{eqnarray}\begin{array}{l}{{\mathscr{A}}}^{\hat{a}}[X]={\displaystyle \int }_{{{\rm{\Sigma }}}_{1}}{\rm{d}}\sigma \,\delta {X}^{M}(\sigma ){B}_{MN}^{\hat{a}}(X){\partial }_{\sigma }{X}^{N}(\sigma )\\ \quad \leftrightarrow {{\rm{Tr}}}_{{\rm{sym}}}\left[{\rm{d}}{X}^{M}{B}_{MN}^{\hat{a}}(X){\partial }_{\sigma }{X}^{N}\right].\end{array}\end{eqnarray}$
The target space one-form gauge transformation is realized as the zero-form gauge transformation in the matrix string space $\begin{eqnarray}\begin{array}{rcl}{u}^{\hat{a}}[X] & = & {\displaystyle \int }_{{{\rm{\Sigma }}}_{1}}{\rm{d}}\sigma \,{u}_{N}^{\hat{a}}(X){\partial }_{\sigma }{X}^{N}(\sigma )\\ & \leftrightarrow & {\rm{Tr}}\left({u}_{N}^{\hat{a}}(X){\partial }_{\sigma }{X}^{N}\right).\end{array}\end{eqnarray}$
The same discussion as in section $\begin{eqnarray}\begin{array}{rcl}{\delta }_{{ \mathcal G }}{B}_{MN}^{\hat{a}} & = & -2{\partial }_{[M}{u}_{N]}^{\hat{a}}\\ & & -{\rm{i}}{f}^{\hat{a}}{\,}_{\hat{b}\hat{c}}{B}_{MN}^{\hat{b}}{\rm{Tr}}\left({u}_{{N}_{1}}^{\hat{c}}(X){\partial }_{\sigma }{X}^{{N}_{1}}\right).\end{array}\end{eqnarray}$
Matrix p-brane model
For generic p-branes, one just needs to consider the tensor product of p-copies 2 × 2 matrix
$\begin{eqnarray}{{ \mathcal M }}^{(2,p)}={{ \mathcal M }}^{(2)}\otimes \cdots \otimes {{ \mathcal M }}^{(2)}.\end{eqnarray}$
Thus $\begin{eqnarray}{X}^{M}(\sigma )\leftrightarrow {({X}^{M})}^{{a}_{1}\cdots {a}_{p}}{\,}_{{b}_{1}\cdots {b}_{p}},\,{a}_{i},{b}_{i}=1,2,\end{eqnarray}$
where the i-th worldvolume spatial coordinate σi is identified with the indices ${\,}^{{a}_{i}}{\,}_{{b}_{i}}$.All the definitions (54 , 55 , 59 , 62 , 66 ) above can be straightforwardly generalized to p > 1 cases. In particular, the derivative operators are realized as 60 , 61 , 63 , 64 , 65 ) for p > 1. As a result, the discussions of p-form gauge symmetry in section 3 are fully inherited by the matrix model based on ${{ \mathcal M }}^{(2,p)}$.
$\begin{eqnarray}\begin{array}{lcl}{\partial }_{{\sigma }^{1}}\phi (\sigma ) & \leftrightarrow & [{\hat{\sigma }}_{3}\displaystyle \otimes {\boldsymbol{1}}\displaystyle \otimes {\boldsymbol{1}}\displaystyle \otimes \cdots \displaystyle \otimes {\boldsymbol{1}},\phi ],\\ {\partial }_{{\sigma }^{2}}\phi (\sigma ) & \leftrightarrow & [{\boldsymbol{1}}\displaystyle \otimes {\hat{\sigma }}_{3}\displaystyle \otimes {\boldsymbol{1}}\displaystyle \otimes \cdots \displaystyle \otimes {\boldsymbol{1}},\phi ],\\ & \vdots \end{array}\end{eqnarray}$
such that the property $\begin{eqnarray}{\partial }_{{\sigma }^{i}}{\partial }_{{\sigma }^{j}}\phi (\sigma )={\partial }_{{\sigma }^{j}}{\partial }_{{\sigma }^{i}}\phi (\sigma )\end{eqnarray}$
is satisfied. Besides, one can also easily generalize the properties (5. Summary and discussion
By considering the phase ambiguity of the p-brane wave functional, it is shown that the target space p-form gauge symmetry can be regarded as a special construction of the generic zero-form gauge symmetry in the functional space {XM(σ)} of p-brane configurations. These discussions are valid both for the abelian and non-abelian gauge groups. Furthermore, we develop a toy model for p-form gauge symmetry by replacing the infinite dimensional p-brane configuration space with a finite dimensional matrix space.
To construct a gauge invariant action for p-form gauge theory, it is also natural to start from the p-brane configuration space. We notice that the target space metric induces a natural metric on {XM(σ)}3 . On the other hand, the number of oscillating modes is finite in the matrix model introduced in section 4 . Thus it would be much easier to get a target space p-form gauge theory in the matrix model approach.
$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}_{[X(\sigma )]}^{2} & = & \displaystyle \int {{\rm{d}}}^{p}\sigma \displaystyle \int {{\rm{d}}}^{p}{\sigma }^{{\prime} }\,{{\mathscr{G}}}_{M,N}(\sigma ,{\sigma }^{{\prime} })[X]\\ & & \times \delta {X}^{M}(\sigma )\delta {X}^{N}({\sigma }^{{\prime} })\\ & = & \displaystyle \int {{\rm{d}}}^{p}\sigma \,{G}_{MN}(X)\delta {X}^{M}(\sigma )\delta {X}^{N}(\sigma ).\end{array}\end{eqnarray}$
Then one can naively propose a Yang–Mills type of Lagrangian on {XM(σ)} $\begin{eqnarray}\begin{array}{rcl}{ \mathcal L }[X(\sigma )] & = & \displaystyle \int {{\rm{d}}}^{p}{\sigma }_{1}\displaystyle \int \,\,{{\rm{d}}}^{p}{\sigma }_{2}\displaystyle \int \,\,{{\rm{d}}}^{p}{\sigma }_{1}^{{\prime} }\,\displaystyle \int {{\rm{d}}}^{p}{\sigma }_{2}^{{\prime} }\\ & & \times \,{{\mathscr{G}}}^{{M}_{1},{M}_{1}^{{\prime} }}({\sigma }_{1},{\sigma }_{1}^{{\prime} })[X]{{\mathscr{G}}}^{{M}_{2},{M}_{2}^{{\prime} }}({\sigma }_{2},{\sigma }_{2}^{{\prime} })[X]\\ & & \times {{\rm{Tr}}}_{{ \mathcal G }}\left\{{{\mathscr{F}}}_{{M}_{1},{M}_{2}}({\sigma }_{1},{\sigma }_{2})[X]{{\mathscr{F}}}_{{M}_{1}^{{\prime} },{M}_{2}^{{\prime} }}({\sigma }_{1}^{{\prime} },{\sigma }_{2}^{{\prime} })[X]\right\}.\end{array}\end{eqnarray}$
The action is obtained by integrating over the p-brane configurations $\begin{eqnarray}{S}_{\{X(\sigma )\}}=\int [{ \mathcal D }X(\sigma )]\,{ \mathcal L }[X(\sigma )].\end{eqnarray}$
In terms of the mode expansion $\begin{eqnarray}{X}^{M}(\sigma )={x}^{M}+\displaystyle \sum _{{k}_{i}}{a}_{{k}_{i}}^{M}{{\rm{e}}}^{{\rm{i}}{k}_{i}{\sigma }^{i}},\end{eqnarray}$
the integration measure is converted to $\begin{eqnarray}\int [{ \mathcal D }X(\sigma )]=\int {\rm{d}}x\displaystyle \prod _{{k}_{i}}{\rm{d}}{a}_{{k}_{i}}.\end{eqnarray}$
To derive an effective target space Lagrangian, we need to integrate out the tower of oscillating modes $\begin{eqnarray}{{ \mathcal L }}_{{\rm{eff}}}(x)=\int [\displaystyle \prod _{{k}_{i}}{\rm{d}}{a}_{{k}_{i}}]\,{ \mathcal L }[X(\sigma )].\end{eqnarray}$
Of course, it is rather difficult to perform the infinite dimensional integration in the original p-brane model in section