# Actions, equations of motion and boundary conditions for Chern-Simons Branes

Pablo Mora  Centro Universitario Regional Este (CURE), Universidad de la
República, Uruguay, Ruta 9 km. 207, Rocha, Uruguay  [<EMAIL_ADDRESS>
###### Abstract

Actions for extended objects based on Transgression and Chern-Simons forms for
space-time groups and supergroups provide a gauge theoretic framework in which
to embed previously studied String and Brane actions, extending them in
interesting ways that may be useful in a future non perturbative formulation
of String Theory. In this Letter I investigate aspects of the actions of these
theories, including equations of motion and boundary conditions, gauge and
space-time symmetries, and Dirac quantization of tensions. This theoretical
framework is shown to include in certain limit and for a suitable gauge group
the standard Bosonic String Theory.

###### keywords:

###### MSC:

[2010]81T30 , 83D99 , 83E99

††journal: Physics Letters B

url]www.cure.edu.uy

##  1 Introduction

Actions for extended objects (branes) based on Chern-Simons and Transgression
forms for space-time groups were introduced in [1, 2] by H. Nishino and
myself, aiming to build a framework in which previous string and brane models
could be embedded, endowing them with desirable additional features. Those
models are background independent gauge systems for the space-time groups. The
background is not fixed and given a priori, but participates of the dynamics
of the extended objects, which are sources of the gauge fields. The actions of
the branes have the same mathematical form than that of the background, in
what we called ”brane-background democracy”.

The above mentioned work built on, and was inspired by, several previous lines
of research:

  1. 1.

It introduces extended objects in a gauge invariance way in Chern-Simons
(super)gravity theories [3, 4, 5, 6, 7, 8, 9], which had the goal of extending
General Relativity to describe gravity as a gauge theory for some space-time
(super)group.

  2. 2.

It generalizes Dixon et al. (DDS) models [10] for the coupling of extended
objects to non abelian gauge fields (heterotic branes) to space-time groups
and their supersymmetric extensions (treated in the same footing as other
possible gauge groups). It differs from the DDS models in that the background
described is described by Chern-Simons/Transgression gravity or supergravity,
in a way formally analogous to the extended objects themselves, and not by
standard gravity or supergravity; and in the fact that our models allow both
sets of gauge fields involved to be non pure gauge.

  3. 3.

It was motivated by Green’s construction [11] of the superstring action as a
gauged Wess-Zumino-Witten (WZW) model for a supergroup.

  4. 4.

It was also motivated by the Chern-Simons supergravity theory for the
supergroup $OSp(32,1)$ proposed by Troncoso and Zanelli [12, 13] and its
conjectured relationship with eleven dimensional supergravity[14], and by
Horava’s work [15] suggesting a possible relation between a Chern-Simons
supergravity with gauge group $OSp(32,1)\times OSP(32,1)$ and M-theory [16,
17, 18].

In the following years there were several works exploring diverse aspects of
models of Chern-Simons branes [19, 20, 21, 22, 23, 24, 25, 26], including
explicit solutions and conserved charges.

In Ref.[1] we suggested that a model of this class may be useful for of a non
perturbative description of String Theory, or M-theory. I still believe that
possibility is the main reason to study these models. Some of the motivations
to do so, and the features that distinguish them from other extended objects
models are:

  1. 1.

As stated in Ref.[1]111The quote is [1]: ”A heuristic motivation for the kind
of models presented here is that if there is some underlying pregeometric
theory in which the de Rham complex get physical content through some kind of
antighost field variables, we must expect the effective action to be a sum of
pieces of diverse dimensions of the form considered here. We believe that this
simple observation is the key to understand the existence of extended objects
in a fundamental theory of nature.”, considering an action that is a sum of
integrals of differential forms of different orders but similar mathematical
form 222One may think of the action as a sort of generalized integral of a
polynomial, or truncated expansion (as befits to Grassmann variables), on the
differentials $dx^{\mu}$. can be understood as a way to give physical content
to the differential structure of the manifold, as General Relativity had given
physical content to the metric structure, giving another way to justify the
possible existence of fundamental extended objects. The frequent question
”what are strings made of?” is often answered with the question ”what are
particles made of?”, somehow implying the first question is moot, yet from our
point of view the answer may be a key to the foundations of String Theory.

  2. 2.

The only dynamical fields are gauge fields, although some of them are
fermionic in the case of a gauge supergroup, providing the fermions required
for a connection with Superstrings. The p-form fields that would be associated
to those occurring in String Theories are composite, and given by Chern-Simons
forms of our gauge fields, what means that the anomalous gauge transformations
and Bianchi identities of those fields are built-in. The notion that the
fundamental degrees of freedom of String Theory may not be the apparent ones
has been widely accepted for quite some time, see for instance
Ref.[27]333Quoting from this Reference: ”In my opinion, the basic challenge in
string theory is not, as sometimes said, to ’understand non-perturbative
processes in string theory’. In fact, the basic problem does not lie in the
quantum domain at all. The basic problem, now and probably for many years to
come, is to understand the classical theory properly.” And ”So the central
task is to (i) find the right degrees of freedom; (ii) find the right
invariance group; and (iii) formulate the right lagrangian.”. In the field
theory case it has been found that full gauge invariance (instead of quasi-
invariance) leads from Chern-Simons forms to Transgressions, which
automatically dictates the correct boundary terms to regularize Noether
conserved charges and black hole thermodynamics [28, 29, 30]. This can be
taken as evidence for requiring a gauge invariant background independent
theory being the right path to follow.

  3. 3.

These theories are already first quantized, analogous to strings on non-flat
backgrounds. They differ from those string models in that the background is
described by Chern-Simons (super)gravity instead of standard (super)gravity.
It is remarkable that the above mentioned theory with supergroup $OSp(32,1)$
of Troncoso and Zanelli [12, 13] has been shown by Izaurieta and Rodríguez
[31] to reduce in certain limit to a theory very similar to standard eleven
dimensional supergravity [14], with corrections given by a expansion in powers
of the parameter used in the limit. It seems possible that with further
analysis of the correspondences between the fields of both theories, and a
suitable choice of the invariant trace involved, the correspondence may be
more precise, and the additional terms could be coherent with anomalous
M-theoretic corrections from dualities [18].

  4. 4.

In a sort of holographic property, if the gauge fields are pure gauge then
only degrees of freedom at the boundaries of the branes and the space-time
remain.

The plan of this Letter is the following: I review the issue of the equations
of motion and boundary conditions, both in the case of generic and pure gauge
connections, and the symmetries of these theories. I discuss some additional
conditions, leading to desirable properties on these models. These include the
inclusion of the kinetic term, added at the boundary and its coefficient
relative to the bulk Chern-Simons term in the bosonic case, independently of
the argument based in kappa-symmetry given in [1], as well as the exclusion of
additional possible bulk terms. Afterwords I consider the particular case or
sub-sector of the theory that makes contact with the Bosonic String in some
limit. Finally I discuss the existence of Dirac quantization conditions on the
tensions or coupling constants of the branes.

##  2 Transgression Actions for Branes

I consider actions for extended objects (branes) with dynamical variables
given by gauge fields given as differential 1-forms
$A=A_{\mu}^{A}G_{A}dx^{\mu}$ valued on the gauge group $\mathcal{G}$ algebra
with generators $G_{A}$, with $\mu=0,...,D$, with $D$ even, the embedding
coordinates of each brane $X_{(d+1)}^{\mu}(\xi_{(d+1)}^{i})$, where
$\xi_{(d+1)}^{i}$, with $i=0,...,d$, with $d$ even, are the coordinates of the
$d$ brane, and the intrinsic metrics defined at the boundaries of the branes
$\gamma_{(d)rs}$, with $r,s=0,...,d-1$. The space-time in which the branes are
embedded can be consider itself as a brane, and is itself described by an
action of the same class as the actions of the branes, therefore its
coordinates are $\xi_{(D+1)}^{\mu}\equiv x^{\mu}$). While the branes
themselves may or may not have boundaries, the boundaries of the branes have
no boundary. The actions considered, introduced in Ref.[1], are given as a sum
of terms of diverse dimensions corresponding to the various branes and the
space-time in which these branes are embedded

$S=\sum_{d=2,~{}d~{}even}^{D}[S_{d+1}^{(Trans)}+S_{d}^{(Kin)}],$ (1)

where the Transgression action part is given by

$S_{d+1}^{(Trans)}=\alpha_{d+1}\int_{\mathcal{S}^{d+1}}\mathfrak{T}_{d+1},$
(2)

with the Transgression forms given by

$\mathfrak{T}_{d+1}(A,\overline{A})=[\frac{d}{2}+1]\int_{0}^{1}dt<\Delta
AF_{t}^{\frac{d}{2}}>.$ (3)

There $\Delta A=A-\overline{A}$, $A_{t}=tA+(1-t)\overline{A}$ and
$F_{t}=dA_{t}+A_{t}^{2}$. The brackets $<...>$ denote symmetrized
(super)traces on the algebra of the gauge (super)group, corresponding to
symmetric invariant tensors $g_{A_{1}...A_{d}}\equiv<G_{A_{1}}...G_{A_{d}}>$.
These symmetrized traces or invariant tensors, for which the simplest choice
is $<...>=STr[...]$, are not unique. Through all this work we use a notation
where the exterior (wedge) product of differential forms is implied
$\alpha_{p}\wedge\beta_{q}\equiv\alpha_{p}\beta_{q}$.

The Kinetic parts of the action has support on the boundaries of the branes,
if such boundaries exist, and are of the form

$S_{d}^{(Kin)}=\beta_{d}\int_{\mathcal{S}^{d}\equiv\partial\mathcal{S}^{d+1}}d^{d}\xi_{(d)}\sqrt{-\gamma_{(d)}}~{}\gamma_{(d)}^{ij}[<\Delta
A_{i}\Delta A_{j}>-(d-2)]$ (4)

where $\Delta A_{i}=\Delta A_{\mu}^{A}\frac{\partial
X_{(d)}^{\mu}}{\partial\xi_{(d)}^{i}}G_{A}$.

##  3 Equations of motion

###  3.1 Equations of motion associated to the variations of the gauge fields

Under variations of $A$ and $\overline{A}$ the variation of the Transgression
part of the action is

$\displaystyle\delta_{A,\overline{A}}S_{d+1}^{(Trans)}=\sum_{d=2,~{}d~{}even}^{D}\alpha_{d}\int_{\mathcal{S}^{d+1}}(\frac{d}{2}+1)[<F^{\frac{d}{2}}\delta
A>-<\overline{F}^{\frac{d}{2}}\delta\overline{A}>]-$
$\displaystyle-\sum_{d=2,~{}d~{}even}^{N}\alpha_{d}\int_{\mathcal{S}^{d}\equiv\partial\mathcal{S}^{d+1}}\frac{d}{2}(\frac{d}{2}+1)\int_{0}^{1}dt<\Delta
AF_{t}^{\frac{d}{2}-1}\delta_{A,\overline{A}}A_{t}>,$ (5)

where $\delta_{A,\overline{A}}A_{t}=t\delta A+(1-t)\delta\overline{A}$.

The variations of the kinetic terms are

$\delta_{A,\overline{A}}S_{d}^{(Kin)}=2\beta_{d}\int_{\mathcal{S}^{d}\equiv\partial\mathcal{S}^{d+1}}d^{d}\xi_{(d)}\sqrt{-\gamma_{(d)}}~{}\gamma_{(d)}^{ij}<\Delta
A_{i}(\delta A_{j}-\delta\overline{A}_{j})>.$ (6)

We will need that using Dirac deltas we can rewrite

$\displaystyle\int_{\mathcal{S}^{d+1}}d^{d+1}\xi_{(d+1)}~{}\mathcal{L}_{d+1}=$
$\displaystyle=\int_{\mathcal{S}^{D+1}}d^{D+1}x\int_{\mathcal{S}^{d+1}}d^{d+1}\xi_{(d+1)}~{}\delta^{D+1}(x^{\mu}-X_{(d+1)}^{\mu}(\xi_{(d+1)}^{i}))\mathcal{L}_{d+1},$
(7)

with $\mathcal{L}_{d+1}$ of the form

$\mathcal{L}_{d+1}=\epsilon^{i_{1}...i_{d+1}}\Omega_{\mu_{1}...\mu_{d+1}}\frac{\partial
X_{(d+1)}^{\mu_{1}}}{\partial\xi_{(d+1)}^{i_{1}}}...\frac{\partial
X_{(2n+1)}^{\mu_{2n+1}}}{\partial\xi_{(d+1)}^{i_{d+1}}},$ (8)

where $\epsilon^{i_{1}...i_{d+1}}$ is the Levi-Civita symbol, so that the
equations of motion that correspond to variations of the gauge potentials can
be written as a sum of various terms describing the coupled dynamic of the
gauge fields (that for space-time groups include the vielbein and spin
connection) and the brane sources.

####  3.1.1 Case of non zero field strength

If the field strengths (or curvatures) $F$ and $\overline{F}$ are non zero,
the equations of motion corresponding to $\delta A$ are

$J_{A}^{\nu(bulk)}+J_{A}^{\nu(boundary)}+J_{A}^{\nu(kin)}=0,$ (9)

where

$\displaystyle
J_{A}^{\nu(bulk)}=\sum_{d=2,~{}d~{}even}^{D}\alpha_{d+1}[\frac{d}{2}+1]\int_{\mathcal{S}^{d+1}}d^{d+1}\xi_{(d+1)}\frac{\delta^{D+1}(x^{\nu}-X_{(d+1)}^{\nu}(\xi^{i}_{(d+1)}))}{d!}\times$
$\displaystyle\times<G_{A}F^{\frac{d}{2}}>_{\mu_{2}...\mu_{d+1}}\frac{\partial
X_{(d+1)}^{[\nu}}{\partial\xi_{(d+1)}^{i_{1}}}\frac{\partial
X_{(d+1)}^{\mu_{2}}}{\partial\xi_{(d+1)}^{i_{2}}}...\frac{\partial
X_{(d+1)}^{\mu_{d+1}]}}{\partial\xi_{(d+1)}^{i_{d+1}}}\epsilon^{i_{1}...i_{d+1}}$
(10) $\displaystyle
J_{A}^{\nu(boundary)}=\sum_{d=2,~{}d~{}even}^{D}\alpha_{d+1}[-d(\frac{d}{2}+1)]\int_{\mathcal{S}^{d}}d^{d}\xi_{(d)}\frac{\delta^{D+1}(x^{\nu}-X_{(d)}^{\nu}(\xi^{i}_{(d)}))}{(d-1)!}\times$
$\displaystyle\times\int_{0}^{1}dt~{}t<\Delta
AF_{t}^{\frac{d}{2}-1}G_{A}>_{\mu_{2}...\mu_{d}}\frac{\partial
X_{(d)}^{[\nu}}{\partial\xi_{(d)}^{i_{1}}}\frac{\partial
X_{(d)}^{\mu_{2}}}{\partial\xi_{(d)}^{i_{2}}}...\frac{\partial
X_{(d)}^{\mu_{d}]}}{\partial\xi_{(d)}^{i_{d}}}\epsilon^{i_{1}...i_{d}}$ (11)
$\displaystyle
J_{A}^{\nu(kin)}=\sum_{d=2,~{}d~{}even}^{D}2\beta_{d}\int_{\mathcal{S}^{d}}d^{d}\xi_{(d)}\delta^{D+1}(x^{\nu}-X_{(d)}^{\nu}(\xi^{i}_{(d)}))\times$
$\displaystyle\times\sqrt{-\gamma_{(d)}}~{}\gamma_{(d)}^{ij}<\Delta
A_{i}G_{A}>\frac{\partial X_{(d)}^{\nu}}{\partial\xi_{(d)}^{j}}.$ (12)

The equations of motion corresponding to $\delta\overline{A}$ are

$\overline{J}_{A}^{\nu(bulk)}+\overline{J}_{A}^{\nu(boundary)}+\overline{J}_{A}^{\nu(kin)}=0,$
(13)

where

$\displaystyle\overline{J}_{A}^{\nu(bulk)}=-\sum_{d=2,~{}d~{}even}^{D}\alpha_{d+1}[\frac{d}{2}+1]\int_{\mathcal{S}^{d+1}}d^{d+1}\xi_{(d+1)}\frac{\delta^{D+1}(x^{\nu}-X_{(d+1)}^{\nu}(\xi^{i}_{(d+1)}))}{d!}\times$
$\displaystyle\times<G_{A}\overline{F}^{\frac{d}{2}}>_{\mu_{2}...\mu_{d+1}}\frac{\partial
X_{(d+1)}^{[\nu}}{\partial\xi_{(d+1)}^{i_{1}}}\frac{\partial
X_{(d+1)}^{\mu_{2}}}{\partial\xi_{(d+1)}^{i_{2}}}...\frac{\partial
X_{(d+1)}^{\mu_{d+1}]}}{\partial\xi_{(d+1)}^{i_{d+1}}}\epsilon^{i_{1}...i_{d+1}}$
(14)
$\displaystyle\overline{J}_{A}^{\nu(boundary)}=\sum_{d=2,~{}d~{}even}^{D}\alpha_{d+1}[-d(\frac{d}{2}+1)]\int_{\mathcal{S}^{d}}d^{d}\xi_{(d)}\frac{\delta^{D+1}(x^{\nu}-X_{(d)}^{\nu}(\xi^{i}_{(d)}))}{(d-1)!}\times$
$\displaystyle\times\int_{0}^{1}dt~{}(1-t)<\Delta
AF_{t}^{\frac{d}{2}-1}G_{A}>_{\mu_{2}...\mu_{d}}\frac{\partial
X_{(d)}^{[\nu}}{\partial\xi_{(d)}^{i_{1}}}\frac{\partial
X_{(d)}^{\mu_{2}}}{\partial\xi_{(d)}^{i_{2}}}...\frac{\partial
X_{(d)}^{\mu_{d}]}}{\partial\xi_{(d)}^{i_{d}}}\epsilon^{i_{1}...i_{d}}$ (15)
$\displaystyle\overline{J}_{A}^{\nu(kin)}=-\sum_{d=2,~{}d~{}even}^{D}2\beta_{d}\int_{\mathcal{S}^{d}}d^{d}\xi_{(d)}\delta^{D+1}(x^{\nu}-X_{(d)}^{\nu}(\xi^{i}_{(d)}))\times$
$\displaystyle\times\sqrt{-\gamma_{(d)}}~{}\gamma_{(d)}^{ij}<\Delta
A_{i}G_{A}>\frac{\partial X_{(d)}^{\nu}}{\partial\xi_{(d)}^{j}}.$ (16)

####  3.1.2 Case of vanishing field strengths

A somewhat subtle case of the variational problem when the gauge fields are
varied occurs if $F=\overline{F}=0$ in the bulk, implying that both $A$ and
$\overline{A}$ should be pure gauge, a condition that must hold also at the
boundary by continuity. In that case the bulk contribution to the variation of
the action vanishes identically, while the variations of the gauge fields at
the boundary are not the general ones but only those compatible with the pure
gauge conditions, which in turn yields weaker equations of motion (or boundary
conditions) at the boundary. In that case the equations of motion obtained
above for the case in which the gauge fields are not pure gauge must me
replaced by the ones resulting from this new variational problem.

In the pure gauge case $F=\overline{F}=0$ all the bulk equations are
satisfied, and only boundary conditions/equations of motion remain. The
boundary equations associated to variations of both the metric and the
embedding coordinates are also the same, except for the fact that now the
gauge field strengths vanish and the gauge potentials are pure gauge. In the
case considered in this subsection $A=g^{-1}dg$ and
$\overline{A}=\overline{g}^{-1}d\overline{g}$ locally, with some elements $g$
and $\overline{g}$ of the gauge group. In this case $\delta A=D(g^{-1}\delta
g)=d(g^{-1}\delta g)+(g^{-1}dg)(g^{-1}\delta g)-(g^{-1}\delta g)(g^{-1}dg)$
and
$\delta\overline{A}=\overline{D}(\overline{g}^{-1}\delta\overline{g})=d(\overline{g}^{-1}\delta\overline{g})+(\overline{g}^{-1}d\overline{g})(\overline{g}^{-1}\delta\overline{g})-(\overline{g}^{-1}\delta\overline{g})(\overline{g}^{-1}d\overline{g})$.
The required variations of the various terms of the action are, obtained
integrating by parts and discarding total derivatives are

$\displaystyle\delta_{A}S_{d}^{(Kin)}=2\beta_{d}\int_{\mathcal{S}^{d}}d^{d}\xi\sqrt{-\gamma_{d}}\gamma_{d}^{ij}<\mathcal{D}_{j}(\Delta
A_{i})(g^{-1}\delta g)>$ (17)
$\displaystyle\delta_{\overline{A}}S_{d}^{(Kin)}=-2\beta_{d}\int_{\mathcal{S}^{d}}d^{d}\xi\sqrt{-\gamma_{d}}\gamma_{d}^{ij}<\overline{\mathcal{D}}_{j}(\Delta
A_{i})(\overline{g}^{-1}\delta\overline{g})>,$ (18)

where the gauge and world-volume covariant derivatives are
$\mathcal{D}_{i}(\Delta A_{j})=\partial_{i}(\Delta A_{j})+[A_{i},\Delta
A_{j}]-\Gamma^{k}_{ij}\Delta A_{k}$, with $\Gamma^{k}_{ij}$ the Christoffel
symbol corresponding to the world-volume metrics $\gamma_{ij}$, and

$\displaystyle\delta_{A}S_{d}^{(Trans)}=-\alpha_{d}\frac{d}{2}(\frac{d}{2}+1)(d-1)B(1+\frac{d}{2},\frac{d}{2})\int_{\mathcal{S}^{d}}<(\Delta
A)^{d}(g^{-1}\delta g)>$ (19)
$\displaystyle\delta_{\overline{A}}S_{d}^{(Trans)}=+\alpha_{d}\frac{d}{2}(\frac{d}{2}+1)(d-1)B(\frac{d}{2},1+\frac{d}{2})\int_{\mathcal{S}^{d}}<(\Delta\overline{A})^{d}(\overline{g}^{-1}\delta\overline{g})>,$
(20)

where $B(1+\frac{d}{2},\frac{d}{2})=B(\frac{d}{2},1+\frac{d}{2})$ es the Euler
Beta function of the given arguments. With these variations it is
straightforward to proceed as in the previous subsection, using Dirac deltas
and factorizing the $g^{-1}\delta g$ and $\overline{g}^{-1}\delta\overline{g}$
variational parameter to leave group generators $G_{A}$, to obtain the
equations of motion of the branes interacting with the fields for which they
are also sources. These equations have now the generic form
$\mathcal{J}_{A}^{\nu(boundary)}+\mathcal{J}_{A}^{\nu(kin)}=0$, without the
bulk contribution and with the currents implied by the variations given above.

###  3.2 Equations of motion associated to the variation of the world volume
metric

The world-volume metric $\gamma_{(d)kr}$ appears in the action only in the
kinetic term $S_{d}^{(Kin)}$. Its equations of motion are

$\displaystyle<\Delta A_{i}\Delta
A_{j}>-\frac{1}{2}\gamma_{(d)ij}[\gamma_{(d)}^{kl}<\Delta A_{k}\Delta
A_{l}>-(d-2)]=0.$ (21)

The solution of these equations is

$\gamma_{(d)ij}=<\Delta A_{i}\Delta A_{j}>.$ (22)

In $d=2$, just like in the case of the Polyakov String Action, these equations
of motion only imply a local proportionality between both members of the
previous equation, but in that dimension the action is invariant under local
rescaling of the metric, allowing to stablish the equality also in that case.

###  3.3 Equations of motion associated to the variation of the embedding
coordinates of the branes

To determine the equations of motion associated to the $X_{(d+1)}^{\mu}$ we
need that the variation of $S^{(Trans)}_{d+1}$ and $S_{d}^{(Kin)}$ under a
variation of $X_{(d)}^{\mu}$ are

$\displaystyle\delta_{X}S^{(Trans)}_{d+1}=\alpha_{d}\int_{\mathcal{S}^{d+1}}d^{d+1}\xi_{(d+1)}\mathfrak{F}_{(d+2)\nu\mu_{1}...\mu_{d+1}}\frac{\partial
X^{\mu_{1}}_{(d+1)}}{\partial\xi^{i_{1}}_{(d+1)}}...\frac{\partial
X^{\mu_{d+1}}_{(d+1)}}{\partial\xi^{i_{d+1}}_{(d+1)}}\epsilon^{i_{1}...i_{d+1}}\delta
X_{(d+1)}^{\nu}+$
$\displaystyle+\alpha_{d}\int_{\mathcal{S}^{d+1}}d^{d+1}\xi_{(d+1)}\frac{\partial~{}}{\partial\xi_{(d+1)}^{k}}\left[\mathcal{P}_{(d+1)\nu}^{k}\delta
X_{(d+1)}^{\nu}\right]$ (23)
$\displaystyle\delta_{X}S_{d}^{(Kin)}=2\beta_{d}\int_{\mathcal{S}^{d}=\partial\mathcal{S}^{d+1}}d^{d}\xi_{d}\times$
$\displaystyle\times\left[\sqrt{-\gamma_{(d)}}\gamma_{(d)}^{ij}<\Delta
A_{i}\frac{\partial\Delta A_{\mu}}{\partial X_{(d)}^{\nu}}>\frac{\partial
X^{\mu}_{(d)}}{\partial\xi^{j}_{(d)}}-\frac{\partial~{}}{\partial\xi^{j}_{(d)}}\left(\sqrt{-\gamma}\gamma^{ij}_{(d)}<\Delta
A_{i}\Delta A_{\nu}>\right)\right]\delta X_{(d)}^{\nu}+$
$\displaystyle+2\beta_{d}\int_{\mathcal{S}^{d}=\partial\mathcal{S}^{d+1}}d^{d}\xi_{d}\frac{\partial~{}}{\partial\xi^{j}_{(d)}}\left(\sqrt{-\gamma}\gamma^{ij}_{(d)}<\Delta
A_{i}\Delta A_{\nu}>\delta X_{(d)}^{\nu}\right),$ (24)

where $\mathfrak{F}_{(d+2)}$ and the momenta conjugated to $X_{(d+1)}^{\nu}$,
$\mathcal{P}_{(d+1)\nu}^{k}$, are

$\displaystyle\mathfrak{F}_{(d+2)}\equiv
d\mathfrak{T}_{d+1}=<F^{\frac{d}{2}+1}>-<\overline{F}^{\frac{d}{2}+1}>$ (25)
$\displaystyle\mathcal{P}_{(d+1)\nu}^{k}=\frac{1}{d!}[\mathfrak{T}_{d+1}]_{\nu\mu_{2}...\mu_{d+1}}\frac{\partial
X^{\mu_{2}}_{(d+1)}}{\partial\xi^{i_{2}}_{(d+1)}}...\frac{\partial
X^{\mu_{d+1}}_{(d+1)}}{\partial\xi^{i_{d+1}}_{(d+1)}}\epsilon^{ki_{2}...i_{d+1}}.$
(26)

Considering that $X_{(d+1)}$ and $X_{(d)}$ are the same at the same point of
the boundary of the brane, we obtain the equations of motion for the bulk

$\mathfrak{F}_{(d+2)\nu\mu_{1}...\mu_{d+1}}\frac{\partial
X^{\mu_{1}}_{(d+1)}}{\partial\xi^{i_{1}}_{(d+1)}}...\frac{\partial
X^{\mu_{d+1}}_{(d+1)}}{\partial\xi^{i_{d+1}}_{(d+1)}}\epsilon^{i_{1}...i_{d+1}}=0.$
(27)

At the boundary, for those values of $\nu$ with $X^{\nu}$ corresponding to
free boundary conditions we obtain the conditions/equations of motion

$\displaystyle 0=\alpha_{d}~{}\mathcal{P}_{(d+1)\nu}^{k}n_{(d)k}+$
$\displaystyle+2\beta_{d}\left[\sqrt{-\gamma_{(d)}}\gamma_{(d)}^{ij}<\Delta
A_{i}\frac{\partial\Delta A_{\mu}}{\partial X_{(d)}^{\nu}}>\frac{\partial
X^{\mu}_{(d)}}{\partial\xi^{j}_{(d)}}-\frac{\partial~{}}{\partial\xi^{j}_{(d)}}\left(\sqrt{-\gamma}\gamma^{ij}_{(d)}<\Delta
A_{i}\Delta A_{\nu}>\right)\right],$ (28)

where $n_{(d)k}$ is the normal to the boundary from Gauss’s Theorem (see
Ref.[32], section 4.23). For instance, if the boundary coordinates are such
that $\xi^{j}_{(d)}=\xi^{j}_{(d+1)}$ if $j=0,...,d-1$, then
$n_{(d)k}=\delta_{kd}$. No further equations are obtained for those values of
$\nu$ such that Dirichlet boundary conditions $\delta X^{\nu}=0$ are imposed.

##  4 Comments on form of the Action, Symmetries and Equations of Motion

###  4.1 Symmetries

The symmetries of the Chern-Simons action were discussed in the original
article[1]. These are: invariance under general coordinate transformations of
the background space-time and the branes world volumes, and gauge invariance
under gauge transformations that are the same for both $A$ and $\overline{A}$,
given by $A^{h}=h^{-1}[A+d]h$ and $\overline{A}^{h}=h^{-1}[\overline{A}+d]h$
for any element $h$ of the gauge group. Gauge invariance follows from the
properties of the symmetrized trace and the covariance of $F$, $\overline{F}$,
$F_{t}$ and $\Delta A$.

The action is not invariant under gauge transformations for which only one of
the gauge fields transforms, or both of them transform with different gauge
group elements, but these variations yield only boundary terms, therefore the
action is said to be quasi-invariant under these transformations. It is
worthwhile to emphasize that possible bulk terms of the form $<(\Delta
A)^{d+1}>$ are invariant under gauge transformations that transform both gauge
fields in the same way, but they are neither invariant nor quasi-invariant
under gauge transformations of only one of the gauge fields. As $d$ is even,
candidate boundary terms $<(\Delta A)^{d}>$ are in fact identically zero. The
action of Chern-Simons branes satisfies then not only the condition of being
gauge invariant under transformations of both gauge potentials with the same
group element, but also the above mentioned quasi-invariance condition, which
can be taken as a requirement on the form of the action that rules out terms
such as $<(\Delta A)^{d+1}>$. It is also true, and related with the previous
remarks, that if both gauge potentials are pure gauge, then the only local
degrees of freedom that remain are located at the boundaries, and there are no
local degrees of freedom in the bulk, what can be seen as a holographic
property of the action.

The interchange of both gauge potentials transforms the transgression part of
the action by a minus sign, while it leaves the kinetic part unchanged.

###  4.2 Special choices for the Action

The requirement on the form of the action mentioned in the previous
subsection, that it be such that in the pure gauge case only boundary degrees
of freedom remain, and the related condition that under gauge transformations
of only one of the gauge potentials the action must be quasi-invariant,
exclude not only terms of the form $<(\Delta A)^{d+1}>$, but also kinetic
terms with support in the bulk.

Furthermore, if one imposes the condition that the only true degrees of
freedom are those of the gauge fields one must exclude also all the boundary
kinetic terms except for the $d=2$ one, making $\beta_{d}=0$ for $d\neq 2$.
The reason is that in $d=2$, because of Weyl invariance, the metric is an
auxiliary variable with no local degrees of freedom, which can be locally
reduced to the flat metric, while that is not the case for $d\neq 2$, and one
should in principle add also terms involving the curvature of these metrics.
In $d=2$ there are two choices of $\beta_{2}$ that result in special
properties for the action, particularly the decoupling of left and right
movers, and Weyl invariance of the action, in a way analogous to Ref.[33]. It
is interesting that the same relative coefficient between the kinetic and
transgression parts resulted in Ref.[1] from the requirement of kappa
symmetry.

We define

$\displaystyle\mathbb{P}^{ij}_{\pm}\equiv\frac{1}{2}[\gamma^{ij}\pm\frac{\epsilon^{ij}}{\sqrt{-\gamma}}],$
(29)

wich has the properties of a projector:
$\mathbb{P}^{ik}_{\pm}\mathbb{P}^{~{}j}_{k\pm}=\mathbb{P}^{ij}_{\pm}$ (that
is, $\mathbb{P}^{2}_{\pm}=\mathbb{P}_{\pm}$) and
$\mathbb{P}^{ik}_{\pm}\mathbb{P}^{~{}j}_{k\mp}=\mathbb{P}^{ik}_{\mp}\mathbb{P}^{~{}j}_{k\pm}=0$
(that is,
$\mathbb{P}_{\pm}\mathbb{P}_{\mp}=\mathbb{P}_{\mp}\mathbb{P}_{\pm}=0$ ).
Furthermore $\mathbb{P}^{ij}_{\pm}=\mathbb{P}^{ji}_{\mp}$, or
$\mathbb{P}^{T}_{\pm}=\mathbb{P}_{\mp}$. Given vectors $V_{i}$ and $W_{j}$ in
$\mathcal{S}^{2}$ we define the projections
$V^{i}_{\pm}=\mathbb{P}^{ij}_{\pm}V_{j}$,
$W^{i}_{\pm}=\mathbb{P}^{ij}_{\pm}W_{j}$, and it holds that
$\mathbb{P}^{ij}_{\pm}W_{i}V_{j}=W_{\mp i}V^{i}_{\pm}$.

There are two special choices for $\beta_{2}$, which are
$\beta_{2}=\pm\frac{1}{2}\alpha_{2}$. For these choices the variation of the
$d=2$ part of the action under variations of $A$ and $\overline{A}$ are

$\displaystyle\delta_{A}[S_{2}^{(Kin)}+S_{3}^{(Trans)}]=2\alpha_{2}\int_{\mathcal{S}^{3}}<F\delta
A>+2\alpha_{2}\int_{\mathcal{S}^{2}=\partial\mathcal{S}^{3}}d^{2}\xi\sqrt{-\gamma}~{}\mathbb{P}^{ij}_{\pm}<\Delta
A_{i}\delta A_{j}>=$ $\displaystyle=2\alpha_{2}\int_{\mathcal{S}^{3}}<F\delta
A>+2\alpha_{2}\int_{\mathcal{S}^{2}=\partial\mathcal{S}^{3}}d^{2}\xi\sqrt{-\gamma}~{}<\Delta
A_{\mp i}\delta A_{\pm}^{i}>$ (30)
$\displaystyle\delta_{\overline{A}}[S_{2}^{(Kin)}+S_{3}^{(Trans)}]=-2\alpha_{2}\int_{\mathcal{S}^{3}}<\overline{F}\delta\overline{A}>-2\alpha_{2}\int_{\mathcal{S}^{2}=\partial\mathcal{S}^{3}}d^{2}\xi\sqrt{-\gamma}~{}\mathbb{P}^{ij}_{\mp}<\Delta
A_{i}\delta\overline{A}_{j}>=$
$\displaystyle=-2\alpha_{2}\int_{\mathcal{S}^{3}}<\overline{F}\delta\overline{A}>-2\alpha_{2}\int_{\mathcal{S}^{2}=\partial\mathcal{S}^{3}}d^{2}\xi\sqrt{-\gamma}~{}<\Delta
A_{\pm i}\delta\overline{A}_{\mp}^{i}>.$ (31)

In the pure gauge case of vanishing field strengths we have

$\displaystyle\delta_{A}[S_{2}^{(Kin)}+S_{3}^{(Trans)}]=\pm\alpha_{2}\int_{\mathcal{S}^{2}}d^{2}\xi~{}\mathbb{P}^{ij}_{\pm}<\mathcal{D}_{j}(\Delta
A_{i})(g^{-1}\delta g)>=$
$\displaystyle=\pm\alpha_{2}\int_{\mathcal{S}^{2}}d^{2}\xi<\mathcal{D}_{\pm}(\Delta
A_{\mp})(g^{-1}\delta g)>$ (32)
$\displaystyle\delta_{\overline{A}}[S_{2}^{(Kin)}+S_{3}^{(Trans)}]=-\pm\alpha_{2}\int_{\mathcal{S}^{2}}d^{2}\xi~{}\mathbb{P}^{ij}_{\pm}<\overline{\mathcal{D}}_{j}(\Delta
A_{i})(\overline{g}^{-1}\delta\overline{g})>=$
$\displaystyle=-\pm\alpha_{2}\int_{\mathcal{S}^{2}}d^{2}\xi<\overline{\mathcal{D}}_{\pm}(\Delta
A_{\mp})(\overline{g}^{-1}\delta\overline{g})>.$ (33)

For instance, the equation of motion obtained from eq.(32), if there are no
other overlapping branes, is $<\mathcal{D}_{\pm}(\Delta A_{\mp})G_{A}>=0$, but
Weyl invariance allows to choose the world-volume metric to be flat, what
makes the Christoffel symbols zero, and $\mathcal{D}_{\pm}=D_{\pm}$ and
$\mathbb{P}^{ij}_{\pm}\equiv\frac{1}{2}[\eta^{ij}\pm\epsilon^{ij}]$. In that
case $0=<\mathcal{D}_{\pm}(\Delta A_{\mp})G_{A}>=<D_{\pm}(\Delta
A_{\mp})G_{A}>=\partial_{\pm}(\Delta A_{\mp})_{B}<G_{B}G_{A}>$. If we take
$\overline{A}=0$ and $<G_{B}G_{A}>$ is invertible, then we obtain
$\partial_{\pm}(g^{-1}\partial_{\mp}g)_{B}=0$, as in Ref.[33], which implies
the decoupling of right and left movers, as claimed.

##  5 The Bosonic String

The AdS group in $D+1$ dimensions $SO(D-1,2)$, which is also the Conformal
group in $D$ dimensions, corresponds to linear transformations that leave
invariant the metric $\eta_{AB}$, $A,B=0,...,D+1$, with zero off-diagonal
entries and diagonal elements $(-1,+1,+1,...,+1,-1)$. The algebra of the
generators $J_{AB}$ of the AdS group is

$\displaystyle[J_{AB},J_{CD}]=+\eta_{AC}J_{BD}+\eta_{BD}J_{AC}-\eta_{BC}J_{AD}-\eta_{AD}J_{BC}$
(34)

Splitting the indices into $a,b=0,...,d$ and $D+1\equiv*$, and defining the
Lorentz generators $J_{ab}$ and the linear momentum generator
$P_{a}=\frac{J_{a*}}{l}$, where $l$ is a real and positive parameter with the
dimension of length called tha ”AdS radius”, we get

$\displaystyle[J_{ab},J_{cd}]=+\eta_{ac}J_{bd}+\eta_{bd}J_{ac}-\eta_{bc}J_{ad}-\eta_{ad}J_{bc}$
(35) $\displaystyle\left[J_{ab},P_{c}\right]=+\eta_{ac}P_{b}+-\eta_{bc}P_{a}$
(36)
$\displaystyle\left[P_{a},P_{b}\right]=+\frac{1}{l^{2}}\eta_{**}J_{ab}=-\frac{1}{l^{2}}J_{ab}$
(37)

In la limit $l\rightarrow\infty$, known as Inonu-Wigner contraction, this
algebra turns into the Poincaré group algebra in $D+1$ dimensions.

The invariant trace of two generators is

$\displaystyle STr[J_{ab}J_{cd}]=\eta_{ac}\eta_{bd}-\eta_{ad}\eta_{bc}$ (38)
$\displaystyle
Str[J_{ab}P_{c}]=0~{}~{}~{},~{}~{}~{}STr[P_{a}P_{b}]=-\frac{1}{l^{2}}\eta_{ab}$
(39)

It is possible to show that the bosonic string with Polyakov Action is
obtained as part of the space of solutions of Chern-Simons branes in a
background Chern-Simons AdS gravity in the limit of large $l$, by proceeding
as follows.

We consider gauge group elements of the form
$g(x)=\exp[-\mathcal{V}^{a}(x)P_{a}]$, where the gauge parameters
$\mathcal{V}^{a}(x)$ are some function of the space-time point, instead of the
generic $g(x)=\exp[-\mathcal{V}^{a}(x)P_{a}+\mathcal{W}^{ab}(x)J_{ab}]$ .
Notice that $\mathcal{V}^{a}P_{a}=\frac{\mathcal{V}^{a}}{l}J_{a*}$, where the
$J_{a*}$ are dimensionless, and therefore the $\mathcal{V}^{a}$ must have the
dimension of length. Assuming the exponent to be small, which amounts to
require $\mathcal{V}^{a}\ll l$, we can consider only the first terms of the
expansions for the group element
$g=I+\mathcal{V}^{a}P_{a}+\frac{1}{2}\mathcal{V}^{a}\mathcal{V}^{b}P_{a}P_{b}+...$,
its inverse
$g^{-1}=I-\mathcal{V}^{a}P_{a}+\frac{1}{2}\mathcal{V}^{a}\mathcal{V}^{b}P_{a}P_{b}+...$,
its exterior derivative
$dg=d\mathcal{V}^{a}P_{a}+\frac{1}{2}d\mathcal{V}^{a}\mathcal{V}^{b}P_{a}P_{b}+\frac{1}{2}\mathcal{V}^{a}d\mathcal{V}^{b}P_{a}P_{b}+...$
and the Maurer-Cartan form
$g^{-1}dg=d\mathcal{V}^{a}P_{a}+\frac{1}{2l^{2}}d\mathcal{V}^{a}\mathcal{V}^{b}J_{ab}+...$,
with leading order $\mathcal{V}/l$. Considering the form of the gauge
potential in Chern-Simons AdS gravity in terms of the vielbein
$e^{a}=e^{a}_{\mu}dx^{\mu}$ and the spin connection
$\omega^{ab}=\omega^{ab}_{\mu}dx^{\mu}$,
$A=e^{a}P_{a}+\frac{\omega^{ab}}{2}J_{ab}$, then the pure gauge potential
associated to the Maurer-Cartan form $A^{(pure~{}gauge)}=g^{-1}dg$ implies
$e^{a}=d\mathcal{V}^{a}$ and
$\omega^{ab}=-\frac{1}{4l^{2}}\mathcal{V}^{[a}d\mathcal{V}^{b]}$. Notice that
the vielbein part is the one of leading order in the expansion in powers of
$\mathcal{V}^{a}/l$, even considering, as we must, $e^{a}/l$. We make a choice
of coordinates such that $\frac{\partial\mathcal{V}^{a}}{\partial
x^{\mu}}=\delta^{a}_{\mu}+...$, or
$\mathcal{V}^{a}=\delta^{a}_{\mu}x^{\mu}+...$, valid in a neighbourhood of
$x^{\mu}=0$ such that $x^{\mu}\ll l$. For this coordinate choice the vielbein
is $e^{a}_{\mu}=\delta^{a}_{\mu}+...$.

Given the embeddings $X^{\mu}(\xi^{i})$ we obtain the pull-backs
$\partial_{i}\mathcal{V}^{a}=\frac{\partial\mathcal{V}^{a}}{\partial\xi^{i}}=\frac{\partial\mathcal{V}^{a}}{\partial
X^{\mu}}\frac{\partial
X^{\mu}}{\partial\xi^{i}}=\frac{\partial\mathcal{V}^{a}}{\partial
X^{\mu}}\partial_{i}X^{\mu}$. With the approximations and choice of
coordinates considered
$\partial_{i}\mathcal{V}^{a}=\delta^{a}_{i}\partial_{i}X^{\mu}+...=\partial_{i}X^{a}+...$.

We focus now in the case of $F=0$ ($A$ pure gauge), concretely an AdS space-
time, and a 2-brane with Dirichlet boundary conditions and its boundary in the
flat AdS boundary. For the $\overline{A}$ configuration we choose the
”Kounterterms” vacuum of Ref.[30], that has a zero vielbein $\overline{e}=0$,
and a non-zero field strength $\overline{F}\neq 0$, but does nonetheless
satisfy the bulk field equations. In that case, and keeping only the leading
order terms in $1/l$, the bulk equations are satisfied for the bulk of both
the background and any branes, and the only remaining boundary contribution
corresponds to the boundary of the 2-branes. We have
$STr[(g^{-1}\partial_{i}g)(g^{-1}\partial_{j}g)]=-\frac{1}{l^{2}}\eta_{ab}\partial_{i}X^{a}\partial_{j}X^{b}+...$.
This means that $STr[A\overline{A}]=0$ to that order, as it is proportional to
the previous expression contracted with antisymmetric $\epsilon^{ij}$. The WZW
term of the action has contributions of order higher than $1/l^{3}$, and what
remains of the 2-brane action is the Polyakov action

$\displaystyle
S^{(Kin)}_{2}=-\beta_{2}\int_{\mathcal{S}^{2}}d^{2}\xi\sqrt{-\gamma}\gamma^{ij}\frac{1}{l^{2}}\eta_{ab}\partial_{i}X^{a}\partial_{j}X^{b}+...$
(40)

Notice that the coordinate normal to the boundary is fixed for the brane
$\partial_{i}X^{D}=0$ and hence in the previous expression $a,b=0,...,D-1$.

##  6 Quantization of the constants in the Brane Actions

There are at least two ways to see that the constants in front of the brane
actions (tensions or coupling constants) must be quantized.

The first argument, which was originally given in Refs.[34, 35, 33], applies
to branes with boundaries in the pure gauge ($F=0$) case, where the actions
reduce to WZW actions at the boundary, arises from the condition that the
Feynman Path Integral must be invariant under large gauge transformations and
independent of the way in which the boundary is extended into the bulk to
define te WZW term. The existence of non equivalent ways of extending the
branes into de bulk is classified by the homotopy groups
$\pi_{d+1}(\mathcal{G})$ for the d-brane. The space-time group $\mathcal{G}$
is en general non compact, then the relevant result is that its homotopy
groups are those of its maximal compact subgroups. For instance, if
$\mathcal{G}$ is the AdS group $SO(d-1,2)$, then its maximal compact subgroup
is $SO(d-1)\times SO(2)$, and $\pi_{d+1}(SO(d-1,2))=\pi_{d+1}(SO(d-1)\times
SO(2))=\pi_{d+1}(SO(d-1))\oplus\pi_{d+1}(SO(2))=\pi_{d+1}(SO(d-1))$, which can
be looked up in standard tables. In the case of the 2-brane the relevant
homotopy group is $\pi_{3}(SO(d-1))=Z$, implying that its tension constant is
quantized.

A second way to see that the tensions are quantized, valid for closed branes,
is analogous to the Dirac quantization condition, and also to the argument of
Ref.[36]. The idea is that if one swings a d-brane in a loop (a d+1
dimensional manifold), and then shrinks that loop until it vanishes, so that
the brane actually does not move, then the amplitude for this process must be
1, but that amplitude is $\exp{(iS/\hbar)}$, which when the loop shrinks to
zero is $\exp{(\frac{i}{\hbar}\int_{M^{d+2}}<F^{\frac{d}{2}+1}>)}$ (where
$M^{d+2}$ is some manifold with the loop as a boundary, and without boundary
when the loop vanishes) and must be $e^{i2\pi N}$ for some integer N. It
results $\int_{M^{d+2}}<F^{\frac{d}{2}+1}>=2\pi\hbar N$, which is possible if
the symmetrized trace $<...>$ is such that the integral corresponds to some
index theorem, and the tension is quantized. If the integral considered does
not vanish that would signal de presence of a dual ”magnetic” solitonic
(D-d-3)-brane. Notice that these conditions constrain the set of acceptable
symmetrized traces.

##  7 Conclusions and further directions

The class of theories considered in this Letter has several attractive
properties that merit further investigation. Among possible lines for future
research are:

  1. 1.

The detailed study of supersymmetric cases in which the gauge group is a
supergroup, which was started in Ref.[1], with the goal of finding a model
with a suitable symmetrised invariant trace and gauge group to connect with
various known limits of M-theory.

  2. 2.

The analysis of conserved charges and anomalies of these theories.

  3. 3.

Looking for solitonic dual ”magnetic” brane solutions to these theories, in
analogy to [37], and looking for dualities between fundamental and solitonic
branes, reinterpreting the strategy of Refs.[38, 39, 18] in our framework. In
those references it is also pointed out that duality mixes tree level
(classical) effects with one-loop (quantum) effects. It would be interesting
to know if that has a counterpart in our framework, and what is its meaning if
it does.

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