---
abstract: 'We develop the dynamics of the chiral superconducting membranes (with null current) in an alternative geometrical approach either making a Lagrangian description and a Hamiltonian point of view.  Besides of this, we show the equivalence of the resulting descriptions to the one known Dirac-Nambu-Goto (DNG) case.  Integrability for chiral string model is obtained using a proposed light-cone gauge.  In a similar way, domain walls are integrated by means of a simple ansatz.  We compare the results with recently works appeared in the literature.' 
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ø c ł[L]{} 0 mm 0 mm -10 mm 215 mm 155 mm

SU-4252-741\

3em [**CHIRAL SUPERCONDUCTING** ]{} 3em [**MEMBRANES** ]{} 3em [Rubén Cordero${}^{(1)}$ and Efraín Rojas${}^{(2, 3)}$\
]{} *[ ${}^{(1)}$Departamento de Física,\
Escuela Superior de Física y Matemáticas del IPN\
Edificio 9, 07738, México D.F., MEXICO]{}\
${}^{(2)}$ Physics Department, Syracuse University\
Syracuse, NY 13244-1130, USA\
${}^{(3)}$ Departamento de Física\
Centro de Investigación y de Estudios Avanzados del I.P.N.\
Apdo Postal 14-740, 07000 México, D. F., MEXICO\
*

3em

PACS: 98.80.Cq, 98.80Hw, 11.27+d

Introduction
============

It is believed that cosmic strings are fundamental bridges in the understanding of the Universe formation due to that several cosmological phenomena can be described by means of the cosmic strings properties.  Besides of these there are other kinds of cosmic objects possesing different properties of those inherited to ordinary cosmic strings, for example: domain walls, hybrid structures like domain walls bounded by strings and so forth.  For an extense review in the context of cosmology, see the Ref. [ @Vilenkin1].  They can arise in several Grand Unified Theories whenever there exists an appropriate symmetry breaking scheme.  However, there is other class of cosmological objects that can emerge with the ability to carry some sort of charge.  For instance, as was suggested by Witten [@Witten] in the middle of the eighties, cosmic strings could behave like superconductors. 

Since that time, the vast research on superconducting strings has thrown a new variety of cosmic objects.  The cosmological result of supersymmetric theories (SUSY) was also considered, yielding to another class of cosmic strings, namely chiral cosmic strings.  These objects are the result of a symmetry breaking in SUSY where a $U(1)$ symmetry is broken with a Fayet-Iliopoulos $D$ term, turning out a sole fermion zero mode traveling in only one direction in the string core [@Davis2].  In other words, when the current along the superconducting string shows a light-like causal structure then we have a chiral string.  Carter and Peter [@Carter] have made an exhaustive study of this kind of cosmic strings and some time later on, it has been continued by other authors [@Vilenkin; @Davis1; @Steer; @Olum].  They have found solutions for chiral cosmic strings taking advantadge of different gauge choices showing new cosmological properties.  The fermionic zero mode is traveling at the speed of light, whereby chiral strings have a different evolution from DNG strings.  The dynamics of the chiral string model has been recognized to be an intermediate stage between DNG model and that of the generic elastic model [@Carter; @Davis1; @Carter2], which has interesting cosmological implications.  The microphysics of this kind of topological defects has been investigated, opening up the possibility to have chiral vortons more stable than vortons of other kinds [@Carter]. 

The mathematical generalization of the chiral string model for an arbitrary number of dimensions is irresistible and possible which in turn gives a new understanding of these string features.  Recently, the idea of extra dimensions has attracted a lot of interest because we can think that our universe is like a membrane embedded in a higher dimensional spacetime [@Randall].  The study of this ideas is currently under way but, actually, it is an idea arose from general relativity (GR) a long time ago which has been pursued by some authors [@Regge], whose aim was to reformulate GR by means of an embedding of the spacetime in a higher dimensional space. 

The purpose of this paper is extend in an alternative geometrical way the dynamical results for chiral strings reported in [@Carter; @Vilenkin] using a Kaluza-Klein (KK) reduction mechanism [@Nielsen] and following closely the variational techniques developed in [@Defo; @Edges; @Defoedges; @Supconmem].  Bearing in mind the KK idea, and assuming our original background spacetime to be 4-dimensional, the generalization to higher dimensional objects (membranes) tracing out worldsheets is possible.  From this assumption we found that the dynamics of the chiral membranes resemble that of a five-dimensional DNG case.  However, our analysis can applied to any dimensional fixed background spacetime and for the membrane with fitting dimension.  We describe now our membrane with an extended embedding.  This description has the advantadge of treat the new membrane on the same footing as an ordinary DNG membrane [@Defo; @Defoedges].  It is often the case that while an existing theory admits a number of equivalent descriptions, one of them suggets generalizations and simplicities more readily than others.  This is our goal. 

Our membrane described by a higher embedding, exhibits a dynamics on the brane caused only by dynamical deformations on the membrane itself.  The equations of motion are a generalization of those of the DNG type, i.e., the motion of the chiral membranes looks like minimal surfaces in a KK space but subject to a particular condition.  In addition, we have the current conservation on the membrane which emerges as the remain equation of motion for the extra variable.  Furthermore, to complete the geometric analysis we shall describe briefly chiral membranes from a Hamiltonian point of view taking into account the results in [@Supconmem]. 

The paper is organized as follows.  In section 2 we develop the essential mathematical features to describe the superconducting chiral membranes.  This is done by exploting the theory of deformations achieved in [@Defo; @Defoedges; @Supconmem].  In section 3, we present a simplified version for the dynamics of chiral extended objects, in contrast with others approaches.  In section 4 we specialize in the chiral string model reproducing the content of [@Carter; @Vilenkin; @Davis1] showing consistency of our description.  Moreover, we found a new method of integrability for chiral string model using a light-cone gauge.  In section 5 we found integrability for a simple chiral domain wall model.  A Hamiltonian approach for chiral membranes is developed in section 6.  Finally, we give conclusions and perspectives of the work. 

Geometry for Chiral Membranes
=============================

In this section we describe both the intrinsic and extrinsic geometry for chiral membranes, i.e., possesing null currents on the worldsheet ($\omega =
\gamma^{ab}\, \phi_{,a} \phi_{,b} = 0$), based in the Kaluza-Klein approach achieved in [@Nielsen].  The present development is close to the conceptual framework made in [@Defo; @Edges; @Defoedges; @Supconmem].  To begin with, we consider a relativistic membrane of dimension $d$, whose worldsheet $\{ m, \Gamma_{ab} \}$ is an oriented timelike $d+1$-dimensional manifold, embedded in a 5-dimensional extended arbitrary fixed background spacetime $\{ {\cal M}, g_{{\bar{\mu} \bar{\nu}}} \}$, $\bar{\mu}= 0,1,...,4$. We shall describe the worldsheet by the extended embedding $$X^{\bar{\mu}}= 
\left(
\begin{array}{l}
X^\mu (\xi^a)\\
\,\,\,\phi (\xi^a)
\end{array}
\right)\,\,,
\label{eq:extembb}$$ where $\phi$ is a field living on the worldsheet $m$; $a,b= 0,1,2$, and $\xi^a$ are coordinates on the worldsheet.  With the former embedding, we can make contact with the KK description for the background space-time metric $$g_{\bar{\mu} \bar{\nu}}= 
\left(
\begin{array}{ll}
g_{\mu \nu}&0 \\
0&g_{44} 
\end{array}
\right)\,\,,
\label{eq:newmetric}$$ where $g_{\mu \nu}$ is the metric of the original background spacetime, and $g_{44}$ is a constant.  The tangent basis for the worldsheet is defined by $$e_a := X^{\bar{\mu}}{}_{,a} \partial_{\bar{\mu}} 
= e^{\bar{\mu}}{}_a \partial_{\bar{\mu}}$$ where the prime denotes a partial derivative with respect to the coordinates $\xi^a$. The tangent vectors $e^{\bar{\mu}}{}_a$, associated with the embedding (\[eq:extembb\]), can be written as $$e^{\bar{\mu}}{}_a= 
\left(
\begin{array}{l}
e^\mu {}_a\\
\phi_{,a}
\end{array}
\right)\,\,. 
\label{eq:extvect}$$ The metric induced on $m$ is given by $$\Gamma_{ab}= g_{\bar{\mu} \bar{\nu}}
e^{\bar{\mu}}{}_a e^{\bar{\nu}}{}_b 
= \gamma_{ab} + g_{44} \phi_{,a}\phi_{,b}\,\,,
\label{eq:newmetric}$$ where $\gamma_{ab}= g_{\mu \nu}e^\mu {}_a e^\nu {}_b$ is the standard metric for the worldsheet without the field $\phi$. The normal basis for the worldsheet is denoted by $n^{\bar{\mu}\,I}$ which is intrinsically defined by $$g_{\bar{\mu} \bar{\nu}}n^{\bar{\mu}\,I} 
n^{\bar{\nu}\,J}
= \delta^{IJ}\,, \hspace{1cm} g_{\bar{\mu} 
\bar{\nu}}n^{\bar{\mu}\,I} e^{\bar{\nu}}{}_a 
= 0\,,$$ where $I,J = 1,2,...,N - d$. We can write explicitly the complete orthonormal basis, which we label as $n^{\bar{\mu}\,I} =
\{ n^{\bar{\mu}\,i} , n^{\bar{\mu}\,(4)}\}$ as follows, $$n^{\bar{\mu}\,i}=
\left(
\begin{array}{l}
n^{\mu \,i}\\
0
\end{array}
\right)\,\,,
\hspace{2cm}
n^{\bar{\mu}\,(4)}=
\sqrt{g_{44}} 
\left(
\begin{array}{l}
e^\mu {}_a \phi^{,a} \\
- g^{44}
\end{array}
\right)\,\,,
\label{eq:extvect}$$ where we have assumed that $n^{\mu \,i}$ satisfy $g_{\mu \nu} n^{\mu \,i}n^{\nu \,j}= \delta^{ij}$, and $i$ take the values $i= 1,...,4-d$. We write down the corresponding Gauss-Weingarten equations for the embedding (\[eq:extembb\]), $$\begin{aligned}
D_a e^{\bar{\mu}}{}_b &=& \gamma_{ab} ^c 
e^{\bar{\mu}}{}_c - K_{ab} ^I n^{\bar{\mu}}{}_I \,\,,
\\
D_a n^{\bar{\mu}\,I} &=& K_{ab} ^I e^{\bar{\mu}\,b}
+ \omega_{a}{}^{IJ} n^{\bar{\mu}}{}_J\,\,,\end{aligned}$$ where $D_a := e^{\bar{\mu}}{}_a D_{\bar{\mu}} $ is the gradient along the tangential basis, and $ D_{\bar{\mu}} $ is the covariant derivative compatible with $g_{\bar{\mu}\bar{\nu}}$, and $\gamma_{ab} ^c$ denote the connection coefficients compatibles with $\Gamma_{ab}$. The extrinsic curvature $K_{ab} ^I$ along the normal basis is defined as $$K_{ab} ^I = - n_{\bar{\mu}} {}^I D_a e^{\bar{\mu}}{}_b \,.$$ The last expression can be split as follows: [*i*]{}) For $I=i$ we have, $K_{ab} ^i= - n_\mu {}^i 
D_a e^{\mu}{}_b$ which is the well known expression for the extrinsic curvature for the worldsheet of the membrane [@Defo], and [*ii*]{}) for $I=4$, $K_{ab} ^{(4)}
= \sqrt{g_{44}} \,\nabla_a \nabla_b \phi$, where $\nabla_a$ is the covariant derivative compatible with $\gamma_{ab}$. The index $(4)$ denotes the direction along the normal $n^{\bar{\mu}\,(4)}$. The extended extrinsic twist, $$\omega_a {}^{IJ} = g_{\bar{\mu} \bar{\nu}}
n^{\bar{\mu}J}D_a n^{\bar{\nu}I}\,,$$ is the connection associated with covariance under normal rotations.  With respect to the last adapted basis, it is simple to check out that $\omega_a {}^{ij} = \tilde{\omega}_a {}^{ij}$ (it reduces to an extrinsic twist potential in four dimensions) and $\omega_a {}^{i\,(4)} = \sqrt{g_{44}}\,\phi^{,\,b}
K_{ab} ^i$, and $\omega_a {}^{(4)(4)} =0$. Note that the mixed twist is constructed from the projection of original worldsheet extrinsic curvature along the conserved current. 

Chiral Membrane Dynamics
========================

In this section we will show the equivalence between the chiral membrane dynamics and the DNG dynamics in an extended background spacetime plus a chirality condition.  The starting point to discuss the dynamics of chiral membranes is the DNG-like action which is invariant under reparametrizations of the worldsheet $m$, $$S= - \mu_0 \,\int_m d^{d+1} \xi \,\sqrt{- \Gamma} \,\,,
\label{eq:chiaction}$$ where $\Gamma$ is the determinant of the induced metric (\[eq:newmetric\]) from the space-time by the embedding (\[eq:extembb\]), and $\mu_0$ is a constant.  The determinant is straightforwardly computed and given by $\Gamma = \gamma \,(1 + g_{44}\,\omega)$, where $\gamma$ is the determinant of the old induced metric on the worldsheet, $\gamma_{ab}$. The action (\[eq:chiaction\]) turns into $$S= - \mu_0 \,\int_m d^{d+1} \xi \sqrt{- \gamma}\,
(1 + g_{44}\,\omega)^{1/2} \,. 
\label{eq:cuasi-act}$$ Observe that the resulting action from the DNG like action (\[eq:chiaction\]), is the one for superconducting strings involving the Nielsen model, where ${\cal L}(\omega)= 
\sqrt{1 + g_{44}\,\omega}$, [@Nielsen].  In other words, the superconducting string theory with the Nielsen model is equivalent to DNG like action (\[eq:chiaction\]). 

An important issue that deserves attention is that of the equations of motion which are already known in [@Carter; @Vilenkin].  We want rebound here the geometrical framework introduced in Sec.  2 in the attainment of chiral membrane dynamics.  Using similar variational techniques to that developed in [@Defo; @Defoedges] we can get the equations of motion from the action (\[eq:chiaction\]).  It is worthy to mention that this method is very graceful because rebound the geometrical nature of the worldsheet.  The variation of the action (\[eq:chiaction\]) gives $$\begin{aligned}
\delta S &=& -\mu_0 \int_m d^{d+1} \xi \,\frac{1}{2}\,
\sqrt{ -\Gamma}\,\Gamma^{ab} \,\delta \Gamma_{ab} 
\nonumber \\
&=& -\mu_0 \int_m d^{d+1} \xi \,\sqrt{ -\Gamma}\, 
\Gamma^{ab} K_{ab} ^I \Phi_I = 0\,\,,\end{aligned}$$ where we have considered only normal deformations to the worldsheet[^1], $\Phi^I$ are the deformation normal vector fields and $\Gamma^{ab}$ is the inverse metric of $\Gamma_{ab}$ given by $\Gamma^{ab}
= \gamma^{ab} - g_{44} \nabla^a \phi \nabla^b \phi$. We can immediately read the equations of motion $$\mu_0 \Gamma^{ab} K_{ab} ^I = 0 \,\,. 
\label{eq:eqmotion}$$ It is worth noticing the similarity of these equations with those ones arising for minimal surfaces, namely, $\gamma^{ab} K_{ab} ^i = K^i = 0$, [@Defo].  In fact, in our description $\Gamma^{ab}$ play the role of a metric.  Let us now decode the important cases involved in the Eq. ( \[eq:eqmotion\]).  a) $I=i$. The equations of motion take the form, $$\mu_0 \gamma^{ab} K_{ab} ^i - \mu_0 g_{44}\nabla^a \phi 
\nabla^b \phi K_{ab} ^i = 0\,. 
\label{eq:eqmotion1}$$ On other hand, in the generic superconducting membranes picture, the strees-energy-momentum tensor adquires the form $T_{ab} = {\cal L}(\omega) \gamma_{ab} 
- 2 \,(d {\cal L}/d \omega)\, \nabla_a \phi 
\nabla_b \phi$, where ${\cal L}(\omega)$ is a function of $\omega$, depending on the particular models [@Carter2; @Cordero].  When the chiral current limit is taken into account, the quantities ${\cal L}(\omega)$ and ${d {\cal L}}/{d \omega}$, adquire constant values.  If we define $g_{44}:= 2(d {\cal L}/d \omega)|_{\omega =0}$ and $\mu_0 = {\cal L}(\omega)|_{\omega =0}$, we can identify the Eq. ( \[eq:eqmotion1\]) with the standard equations of motion, namely: ${ T}^{ab} K_{ab} ^i = 0$, [@Carter2; @Cordero].  b) $I=(4)$. In this case we have now directly, $$\Gamma^{ab} K_{ab} ^{(4)} =0= \nabla_a \nabla^a
\phi \,,$$ which is a wave equation for $\phi$, corresponding to a conserved current carrying onto the worldsheet for chiral currents. 

Chiral String Model
===================

With the purpose of make contact with previous works [@Carter; @Vilenkin], we specialize now to the case of chiral strings.  In the next section we will study the case of chiral domain walls.  We illustrate the chiral string model from a Lagrangian point of view. 

Gauge Choices
-------------

The presence of the gauge symmetry in a field theory means that not all of the field components, $X^{\bar{\mu}}
(\xi^a)$, are dynamical.  In our case, the reparameterization invariance allow us to choose a gauge in which the dynamical equations are tractable.  Due to we have considered a DNG like action, (\[eq:chiaction\]), we have the freedom of choice of an acceptable gauge condition. 

[**[a)]{}**]{} Recently, Carter and Peter presented a solution for the chiral string model.  This is an ansatz useful in the solution for the transonic string model [@Carter].  We reproduce the same result via a particular choice for the embedding of the worldsheet in the extended background.  Assuming the embedding $$X^{\bar{\mu}}= 
\left(
\begin{array}{l}
X^\mu (q,\eta)\\
\,\,\,\,\,\, \alpha \eta
\end{array}
\right)\,\,,
\label{eq:nextembb}$$ where $q$ and $\eta$ are coordinates on the worldsheet and $\alpha$ is a constant of proportionality.  With this choice, the internal scalar field is promoted as $\phi = \alpha \eta$. In such a case, and according to our notation, we demand the condition $$\sqrt{-\Gamma}\, \Gamma^{ab}= 
\left(
\begin{array}{ll}
0&  1 \\
1 &0 
\end{array}
\right)\,\,. 
\label{eq:ngaugeinvgamma}$$ The induced metric $\gamma_{ab}$, will take the form $$\gamma_{ab}= 
\left(
\begin{array}{ll}
\,\,\,0&  \sqrt{- \Gamma} \\
\sqrt{- \Gamma} &- g_{44}
\end{array}
\right)\,\,,$$ with the inverse matrix given by $$\gamma \,\gamma^{ab}= 
\left(
\begin{array}{ll}
\,\,- g_{44}& - \sqrt{- \Gamma} \\
-\sqrt{- \Gamma} &\hspace{.4cm}\,\,0
\end{array}
\right)\,\,.$$ Using the last information we observe easily the condition for chirality to be $$\gamma^{\eta \eta} = 0\,. 
\label{eq:newcondit}$$ The corresponding equations of motion for (\[eq:nextembb\]) taking into account the gauge (\[eq:ngaugeinvgamma\]), result in $$\partial_q \partial_\eta \,X^{\bar{\mu}} = 0\,,$$ which have been recently reported in [@Carter]. 

[**[b)]{}**]{} In a short time later, Blanco-Pillado et al., [ @Vilenkin], gave an alternative description and a solution for the same model by means of a different gauge choice of that proposed in [@Carter].  In our description, a convenient gauge is the well known conformal gauge, $$\sqrt{- \Gamma}\, \Gamma^{ab}= \eta^{ab}\,\,.$$ Following the standard string theory notation[^2], we attempt to solve the chiral string model from (\[eq:chiaction\]), which is accomplished taking into account the next form for the metric $$\gamma_{ab}= 
\left(
\begin{array}{ll}
-\sqrt{-\Gamma} - g_{44}\,{\phi '}^2 &  
\,\,\,\,\,\,\,\,-g_{44}\,\dot{\phi}\,\phi ' \\
\,\,\,\,\,\,\,\,\,- g_{44}\,\dot{\phi}\,\phi '  &
\sqrt{-\Gamma} - g_{44}\,\dot{\phi}^2 
\end{array}
\right)\,\,,
\label{eq:gaugegamma}$$ [i.e.,]{} this corresponds to our gauge choice.  In fact, this choice is equivalent to that studied in [@Vilenkin].  To complement the calculation we write the inverse metric $\gamma^{ab}$, easily obtained, $$\gamma \gamma^{ab}= 
\left(
\begin{array}{ll}
\sqrt{-\Gamma} - g_{44}\,\dot{\phi}^2 &  
\,\,\,\,\,\,\,\,\,\,\,\,g_{44}\,\dot{\phi}\,\phi ' \\
\,\,\,\,\,\,\,\,\,\,g_{44}\,\dot{\phi}\,\phi '  &
-\sqrt{-\Gamma} - g_{44}\,{\phi '}^2 
\end{array}
\right)\,\,. 
\label{eq:gaugeinvgamma}$$ Taking into account (\[eq:gaugeinvgamma\]), the condition for chirality $(\omega=0)$ will take the form $$\dot{\phi}^2 - {\phi'} ^2 = 0\,,
\label{eq:wavephi}$$ which is a condition for the field $\phi$ in this gauge.  The equations of motion for the system, using the refered gauge, are obtained in a straightforward way from those of DNG case, $\partial_a ( \sqrt{- \Gamma}
\Gamma^{ab}\,X^{\bar{\mu}}{}_b ) = 0$, namely, $\ddot{X}^{\bar{\mu}} - {X} ^{''\bar{\mu}} = 0$, whose solutions are $X^{\bar{\mu}} = \frac{1}{2}a^{\bar{\mu}}
(\tau + \sigma) + \frac{1}{2}b^{\bar{\mu}}(\tau - \sigma)$. The conformal gauge imposes the following constraints over $a^{\bar{\mu}}$ and $b^{\bar{\mu}}$, namely, $a^{\bar{\mu}}`a`_{\bar{\mu}}= 1$ and $b^{\bar{\mu}}`b`_{\bar{\mu}}= 1$, where ‘ denotes derivation with respect to their arguments.  These are the same conditions like in the DNG case but now in the extended space.  Explicitly they take the form $|\vec{a} `| ^2 + g_{44} \phi_1 ^{`\,\,2} = 1$ and $|\vec{b} `| ^2 + g_{44} \phi_2 ^{`\,\,2} = 1 $, where $\phi_1 $ and $\phi_2$ are the last components of the vectors, $a^{\bar{\mu}}= ( \vec{a} \,,\,\phi_1)$ and $b^{\bar{\mu}}= ( \vec{b} \,,\,\phi_2)$. However, the condition (\[eq:wavephi\]) tell us that either $\phi_1
(\tau + \sigma) =0$ or $\phi_2(\tau - \sigma) =0$, and we are able to reproduce the results of [@Vilenkin].  The vorton states are obtained when $\dot{X} ^\mu = X`^{\mu} $ i.e., $\vec{b}` =0$ and $\phi_1 =0$.

[**[c)]{}**]{} Now we turn to consider the light-cone gauge over the spacetime coordinates adapted to our description.  When we study non-superconducting strings, it is well known that the orthonormal gauge do not fully fix the gauge because there is residual reparametrization invariance.  A favorite gauge choice that fix the gauge and allow us to solve the constraints is the light-cone gauge [@GSW].  In string theory, this gauge is convenient for its quantization because it allow us eliminate all unphysical degreees of freedom and unitarity is guaranteed.  In other context, this gauge was used by Hoppe in the search for explicit solutions for the classical equations of motion of relativistic membranes [@Hoppe].  We shall use this orthonormal light-cone gauge in the search of integrability for the chiral string model. 

To proceed further, we assume the original background metric to be flat, $g_{\mu \nu} = \eta_{\mu \nu}$, with signature $(-,+,+,+)$ and the embedding (\[eq:extembb\]).  For the KK spacetime we define light-cone coordinates, $X^+$ and $X^-$, as $$\begin{aligned}
X^+ &=& \frac{1}{\sqrt{2}}\,(X^0 + 
       \sqrt{g_{44}}\,\phi) \,, 
\label{eq:X+}       \\
X^- &=& \frac{1}{\sqrt{2}}\,(X^0 - 
       \sqrt{g_{44}}\,\phi)\,,
\label{eq:X-}      \\      
\vec{X} &=& (X^1,X^2,X^3)\,.  \end{aligned}$$ The light-cone gauge points $\tau$ along $X^+$, $$X^+ = X^+ _0 + P^+ \,\tau \,,$$ where $X^+ _0$ and $P^+$ are constants.  The idea is solve for $X^-$ leaving the $X^i$ variables, where $i=1,2,3$. Laying hold of the orthonormal light-cone gauge [@Hoppe], $$\Gamma_{ab}= 
\left(
\begin{array}{ll}
\Gamma_{\tau \tau} & \,\, 0\\
\,0  & \Gamma_{AB}
\end{array}
\right)\,\,,
\label{eq:algo}$$ where $\Gamma_{ab}$ is given by (\[eq:newmetric\]) and $A,B=1,...,d$, besides of $\sqrt{-\Gamma}\,\Gamma^{\tau \tau} = -1$, we can simplify the equations of motion, $\partial_a(\sqrt{-\Gamma}\,
\Gamma^{ab}\,X^{\bar{\mu}}{} _{,b})= 0$ in the set of equations $$\begin{aligned}
{\cal D} \,X^{\bar{\mu}} &=& 0 \,,
\label{eq:lcemotion} 
\\
2 P^+ \dot{X} ^- &=& \dot{\vec{X}}\cdot \dot{\vec{X}}
                    + \bar{\Gamma} \,,
\label{eq:constraint1}                    
\\
P^+ X^- {}_{,A}&=& \dot{\vec{X}}\cdot \vec{X}_{,A}
\label{eq:constraint2}\end{aligned}$$ where we have defined $\bar{\Gamma}:= {\mbox{det}}
(\Gamma_{AB}) = - \Gamma_{\tau \tau}$ and we have defined the differential operator $${\cal D} := - \partial_\tau ^2 + \partial_A 
(\bar{\Gamma}\,\Gamma^{AB}\,\partial_B )\,.$$ Equation (\[eq:lcemotion\]) represents the equations of motion in this gauge, (\[eq:constraint1\]) and (\[eq:constraint2\]) are the constraints relations for the system.  Deriving with respect to $\tau$ the Eq. ( \[eq:constraint1\]), we can rewrite it as $P^+ {\cal D} X^- = \dot{\vec{X}}\cdot {\cal D}
\vec{X}$; so if $${\cal D} \vec{X} =0 \,,
\label{eq:lcemotion1}$$ we get the condition ${\cal D} X^- = 0$, which we can observe from (\[eq:lcemotion\]).  Thus, we have reduced the problem to solve the set (\[eq:constraint1\]), (\[eq:constraint2\]) and (\[eq:lcemotion1\]).  So far the results are general for minimal surfaces of arbitrary dimension.  Now we specialize to the case of chiral strings.  In order to get integrability for the chiral string model, besides Eqs. ( \[eq:constraint1\]), (\[eq:constraint2\]) and (\[eq:lcemotion1\]), the condition $\omega=0$ must be considered.  Using the stringy notation, the appropiate expressions are $$\Gamma_{ab}= 
\left(
\begin{array}{ll}
- \vec{X}' \cdot \vec{X}' & \,\,\,\,\,\,\,\,0\\
\,\,\,\,\,\,\,\,0  & \vec{X}' \cdot \vec{X}'
\end{array}
\right)\,\,,
\label{eq:algomas}$$ with $\Gamma_{\tau \tau}= - \Gamma_{\sigma \sigma}$. The equations of motion (\[eq:lcemotion1\]) transform in the wave equation: ${\cal D} \vec{X}
= (- \partial_\tau ^2 + \partial_\sigma ^2 )\vec{X}
= -\ddot{\vec{X}} +\vec{X}'' =0$, whose general solution is given by $$\vec{X} = \vec{a} (\tau + \sigma) + \vec{b}(\tau -
\sigma) \,.$$ The constraints (\[eq:constraint1\]) and (\[eq:constraint2\]) adquire the form $$\begin{aligned}
P^+ \dot{X}^- &=& |\vec{a}`|^2 + |\vec{b}`|^2 \,,
\label{eq:constraint3} \\
P^+  X^{- '}  &=& |\vec{a}`|^2 - |\vec{b}`|^2 \,,
\label{eq:constraint4}\end{aligned}$$ where we have used the notation $\vec{a}`$ and $\vec{b}`$ to denote derivatives with respect to their arguments.  From the Eqs. ( \[eq:X+\]) and (\[eq:X-\]) as well as Eq. ( \[eq:newmetric\]), we can separate the metric $\gamma_{ab}$, $$\gamma_{ab}= 
\left(
\begin{array}{ll}
- \vec{X}' \cdot \vec{X}' - \frac{1}{2}
\,(P^+ - \dot{X} ^-)^2 & \,\,\,\frac{1}{2}
(P^+ - \dot{X} ^-) X^{- '}\\
\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{2}\,(P^+ - 
\dot{X} ^-) X^{- '}  & 
\vec{X}' \cdot \vec{X}' - \frac{1}{2}\,(X^{-'})^2
\end{array}
\right)\,\,,
\label{eq:algomas1}$$ and its inverse $$\gamma \gamma^{ab}= 
\left(
\begin{array}{ll}
\vec{X}' \cdot \vec{X}' - \frac{1}{2}\,(X^{-'})^2 & 
\,\,\,\,\,\,\,\,-\frac{1}{2}\,(P^+ - 
\dot{X} ^-) X^{- '}\\
-\frac{1}{2}\,(P^+ - \dot{X} ^-) X^{- '}  & 
- \vec{X}' \cdot \vec{X}' - \frac{1}{2}
\,(P^+ - \dot{X} ^-)^2 
\end{array}
\right)\,\,. 
\label{eq:algomas2}$$ Now, the chirality condition becomes $$\begin{aligned}
\omega &=& \gamma^{ab}\,\nabla_a \phi
\nabla_b \phi \nonumber \\
&=& \frac{1}{2\gamma g_{44}}\,[
(P^+ - \dot{\vec{X}}^2 )^2 - (X^{-'})^2]
(\vec{X}' \cdot \vec{X}') =0 \,,\end{aligned}$$ from which is deduced that $$X^{-'}= \pm (P^+ - \dot{\vec{X}}^2)\,. 
\label{eq:solution}$$ Plugging the Eq. ( \[eq:solution\]) in the constraint (\[eq:constraint2\]) we get the conditions that should satisfy $\vec{a}$ and $\vec{b}$ in the chiral string solution, namely, $$\pm (P^+)^2 = (\vec{a}` + \vec{b}`)\cdot [ \vec{a}` 
- \vec{b}` \pm 
P^+ (\vec{a}` + \vec{b}`)]\,. 
\label{eq:condition}$$ Thus, this equation suggest to consider some cases. 

For instance, if we assume $P^+=1$, we get $$1= 2(\vec{a}` + \vec{b}`)\cdot \vec{a}`  \,,$$ which is different for the values for $\vec{a}$ and $\vec{b}$ reported in [@Vilenkin]. 

It is worthy to mention that the integrability for the last cases was reached in absence of electromagnetic field coupled to superconducting strings.  From the equations defining $X^+$, $X^-$ and the contraints (\[eq:constraint3\]) and (\[eq:constraint4\]) we can find the value for $\phi$. Again the vortons states are obtained when $\vec{b} `=0$ and in such case $\vec{a}$ satisfy the relation $\pm P^{+2} = \vec{a}`\cdot [\vec{a} ` ( 1 \pm P^+)] $. 

Chiral Domain Wall Model
========================

In this section applying a similar mechanism to that for conformal gauge chiral string model we get integrability for a chiral domain wall model by means of a special ansatz.  For such an intention, with all the former ingredients, we study a worldsheet described by the embedding [@Vilenkin1], $$X^{\bar{\mu}}(\tau,\xi^1,\xi^2)= 
\left(
\begin{array}{l}
\,\,\tau\\
\vec{\tilde{X}}
\end{array}
\right)\,\,,
\label{eq:dwembedd1}$$ where $$\vec{\tilde{X}}= 
\left(
\begin{array}{l}
\vec{{X}} \\
\phi
\end{array}
\right)\,. 
\label{eq:dwembedd2}$$ Now we can choose a gauge similar to the conformal gauge for strings , as follows $$\Gamma_{ab}= 
\left(
\begin{array}{ll}
\Gamma_{\tau \tau} & \,\, 0\\
\,0  & \Gamma_{AB}
\end{array}
\right)\,\,,
\label{eq:metric1}$$ where $\Gamma_{AB} = g_{\bar{\mu} \bar{\nu}} 
X^{\bar{\mu}}{} _{,A}X^{\bar{\nu}}{}_{,B}$ and $\Gamma_{\tau A} =0$. We assume the special form for the embedding (\[eq:dwembedd1\]), as $$\vec{\tilde{X}}= \hat{\tilde{n}}\,\xi^2 + 
\vec{\tilde{X}}_{\perp} (\tau,\xi^1) \,,
\label{eq:dwgauge}$$ with the conditions on $\hat{\tilde{n}}$ to be a unit vector and perpendicular to $\vec{\tilde{X}}_{\perp}$. So, in this conformal gauge we have the constraints \_ ’\_ &=& 0 , \[eq:const1\]\
\_ \_ + ’\_ ’\_ &=& 1 , \[eq:const2\] and the condition $\sqrt{- \Gamma}\,\Gamma^{\tau \tau} 
= - 1$. It is straightforward to demonstrate that $\bar{\Gamma}:= {\mbox{det}}(\Gamma_{AB}) = - 
\Gamma_{\tau \tau}$. In this part of the work, $^.$ and $'$ denote derivatives with respect to $\tau$ and $\xi^1$, respectively. 

Assuming $g_{\mu \nu}$ to be Minkowski’s metric and resting in the constraints (\[eq:const1\]) and (\[eq:const2\]), the induced metric $\Gamma_{ab}$ takes the form, $$\Gamma_{ab}= 
\left(
\begin{array}{lll}
-|\vec{\tilde{X}} _\perp '|^2 & \hspace{4mm} 0 & 0\\
\hspace{6mm} 0  & |\vec{\tilde{X}} _\perp '|^2 & 0 \\
\hspace{6mm} 0 & \hspace{4mm} 0 & 1
\end{array}
\right)\,. 
\label{eq:dwmetric}$$ According to the standard DNG equations of motion, in our present case the corresponding ones are promoted as $\ddot{\vec{\tilde{X}}} _\perp  - 
\vec{\tilde{X}} '' _\perp =0$, whose solutions have the form $$\vec{\tilde{X}} _\perp = \frac{1}{2}\vec{\tilde{a}} 
( t + \xi^1) + \frac{1}{2}\vec{\tilde{b}} 
( t - \xi^1) \,. 
\label{eq:dwsolution}$$ Imposition of the chirality for superconducting domain walls, lead us to the relation |\_’|\^2 (n\^4)\^2 = \_\^2 - ”\_\^2 , \[eq:dwcondition\] where $n^4$ is the four component of the vector $\hat{\tilde{n}}$. Furthermore, the constraints (\[eq:const1\]) and (\[eq:const2\]) read as $ |\vec{\tilde{a}}{}{}'|^2 
=1$ and $ |\vec{\tilde{b}}{}{}'|^2 =1$, or explicitly they are given by $$\begin{aligned}
|\vec{a}{}'|^2  + g_{44} \phi ' {}^2 &=& 1\,, \\
|\vec{{b}}{}'|^2  + g_{44} \bar{\phi} ' {}^2 &=&1 \,,\end{aligned}$$ where we have considered the notation $\vec{\tilde{a}} = 
(\vec{a} ,\,\, \phi)$ and $\vec{\tilde{b}} = 
(\vec{b} ,\,\, \bar{\phi})$, [*i.e.,*]{} $$\vec{\tilde{X}}_\perp= 
\left(
\begin{array}{l}
\vec{{X}}_\perp = \frac{1}{2}\vec{{a}} ( t + \xi^1) + 
\frac{1}{2}\vec{b} ( t - \xi^1)
\\
\phi_\perp = \frac{1}{2}\phi( t + \xi^1) + 
\frac{1}{2}\bar{\phi}( t - \xi^1)
\end{array}
\right)\,. 
\label{eq:dwvect1}$$ Plugging (\[eq:dwvect1\]) in the condition (\[eq:dwcondition\]), the chirality condition is expressed now as ‘ | ‘ \[1 + g\_[44]{} (n\^4)\^2\] = (n\^4) \^2 (1- ‘ ‘ ) .  In a similar way as in the chiral string model case, the last equation suggests some cases.  For example, the solution considering $n^4 =0$ and $\phi =0$ (or $\bar{\phi} =0$), correspond to a straight superconducting domain wall with a carrying current arbitrary cross-section, but not including current along the $\xi^2$ direction. 

Hamiltonian Dynamics for Chiral\
Membranes
================================

In this part of the work we want to make a Hamiltonian approach to the dynamics for chiral membranes.  Nowadays, canonical formulation is successfully used as a starting point for quantization but, also is used as a tool for tackling dynamical problems.  Bearing in mind the last use as well as its possible aplication to brane universe scenario, we first review briefly the Hamiltonian framework already developed in [@Supconmem].  Before going to the canonical analysis we shall mention something about the useful mathematical issues necessary for this purpose.  Taking into account the established notation of Sect.  2, we will consider a superconducting relativistic membrane of dimension $d$ whose worldsheet $\{ m',\gamma_{ab} \}$ is $d+1$-dimensional which is embedded in a given fixed background spacetime $\{ {\cal{M}}',g_{\mu ,\nu} \}$. According to the ADM procedure for the canonical general relativity, we assume that $m'$ has an adequate topology such that we can foliate to the worldsheet into $d$-dimensional spacelike hypersurfaces $\Sigma_t$, parametetrized by constant values of a function $t$. Each slice of the foliation represents the system at an instant of time and each one is diffeomorphic to each other.  In order to describe the evolution of the leaves of the foliation is convenient to introduce the worldsheet time vector field \[eq:timevect\] t\^a = \^a:= N \^a + N\^A \^a \_A , where $N$ and $N^A$ denote the well-known lapse function and shift vector, respectively, $\eta^a $ denotes the unit future-oriented timelike normal vector field to the slice $\Sigma_t$ and $\epsilon^a {}_A$ the corresponding tangent vector. 

The decomposition of the several geometric quantities involved in the theory in the normal and tangential parts to the slice, taking advantadge of the deformation vector field (\[eq:timevect\]), is primordial in the Hamiltonian treatment.  The various geometrical quantities that characterize the intrinsic and extrinsic geometry of $\Sigma_t$, can be decomposed based in the formalism of deformations.  For instance, the worldsheet metric $\gamma_{ab}$ in the ADM decomposition looks like \_[ab]{}=(

[ll]{} - N\^2 + N\^A N\_A & N\_A\
N\_A & h\_[AB]{}

) , \[eq:ggamma\] and for the inverse $$\gamma^{ab} =
{1 \over N^2} \left(
\begin{array}{ll}
 - 1  & \hspace{1.5cm}N^A  \\
 N^A   & (h^{AB} N^2 - N^A N^B )
\end{array}
\right)\,.$$ Note that from (\[eq:ggamma\]) it follows that: $\gamma = - N^2 \,h,$ where $h$ is the determinant of the metric $h_{AB}$ induced on the hypersurface $\Sigma_t$, via the embedding $x^\mu=
X^\mu(u^A)$, and we have assumed that $u^A$ are local coordinates on $\Sigma_t$. For a general treatment concerning this formalism, see the Refs. [ @Supconmem; @ADM].  As the starting point for our Hamiltonian study we have the generic action $$S = \int_{m'} d^{d+1}\xi \,\sqrt{- \gamma}\,{\cal L}(\omega) \,,
\label{eq:genericact}$$ where ${\cal L}(\omega)$ is a function depending of the internal and external fields acting on the worldsheet.  The split action (\[eq:genericact\]) in time and space is performed according to the geometric canonical procedure.  The Lagrangian density in the ADM fashion is given by $$L= N\sqrt{h}\,{\cal L}(\omega) \,,$$ where we should understand that $ {\cal L}
(\omega)$ has been split in space and time where $\omega$ has the expression: $\omega = - (1/N^2) \,[ \dot{\phi} - N^A
\bar{{\cal D}}_A \phi + (t\cdot A) ]^2 +
\tilde{\omega}$, where $\bar{\omega} = h^{AB} \phi_{,A}\phi_{,B}$.

The phase space ${\mathcal{A}}= \{ X,P;\phi,\pi \}$ is closely related to the geometry of $\Sigma_t$, which is seen from the equations defining the momenta $\pi =- 2 \sqrt{h} 
(\bar{\omega} - \omega )^{1/2}
\,d {\cal L}/d \omega $ and $P_\mu = 
[ - \sqrt{h} {\cal L}(\omega) + (\bar{\omega} -
\omega )^{1/2}\,\pi ]\,\eta_\mu - \pi \,
\tilde{{\cal D}}_A \phi \,\,
\epsilon_\mu {}^A  + \,\,\,\pi  A_\mu $, conjugate to $\phi$ and $X^\mu$, respectively. 

The canonical Hamiltonian vanishes identically which stems from the requirement of reparametrization invariance of the worldsheet.  For this field theory, the constraints are given by $$\begin{aligned}
{\cal C}_0 &=& g^{\mu \, \nu}\Upsilon_\mu \Upsilon_\nu + 
h\,\left( {\cal L}(\omega ) + \frac{\pi ^2}{2\,h\,
(d{\cal L}/d\omega )}
\right)^2  -  \pi^2 \,\left( \omega + \frac{\pi^2}
{4\,h\,(d{\cal L}/d\omega)^2}
\right) 
\label{C0} \\
{\cal C}_A&=& \Upsilon_\mu \,\epsilon^\mu {}_A + \pi\,
\tilde{{\cal D}}_A \phi \,,
\label{C1} \end{aligned}$$ where the kinetic momentum is $\Upsilon_\mu := 
P_\mu - \pi A_\mu $, with $A_\mu$ being a background electromagnetic potential. 

We turn now to consider the canonical description for chiral membranes except the background electromagnetic field, from two different alternatives, in like manner to that for canonical DNG case.  First, we follow the standard way close to that described previously.  Next, we benefit from the canonical approach developed at the beggining of this section using the constraints (\[C0\]-\[C1\]) directly.  For the Hamiltonian approach for the action (\[eq:chiaction\]), it would be first decomposed in the ADM fashion as S= \_R \_[\_t]{} [N]{} , \[eq:chiralADM\] where we have defined a new lapse function ${\cal N}:= N \,(1 + g_{44}\,\omega)^{1/2}/(1 + g_{44}\,
\bar{\omega})^{1/2}$ and ${\cal H}$ is the determinant of the induced metric on the hypersurface, ${\cal H}_{AB}$, from the embedding (\[eq:extembb\]).  Explicitly, it is given by ${\cal H} = h\,(1 + g_{44}\,\bar{\omega})$. The next step is the computation of the momenta conjugate to the configuration variables.  They are given by $$\begin{aligned}
\pi &:=& -  \frac{\sqrt{\cal{H}}\,g_{44}\,(\bar{\omega} 
- \omega)^{1/2}}{(1 + g_{44}\,\bar{\omega})^{1/2}
(1 + g_{44}\,\omega)^{1/2}} 
\label{eq:chiralpi}
\\
P_\mu &:=& \left[ - \sqrt{\cal{H}}\,\frac{(1 + g_{44}\,\omega)^{1/2}}
{(1 + g_{44}\,\bar{\omega})^{1/2}} + \pi \,(\bar{\omega} -
\omega)^{1/2} \right] \,\eta_\mu - \pi {\cal{D}}_A \phi \, 
\epsilon_\mu {}^A\,. 
\label{eq:chiralP}\end{aligned}$$ Taking into account the equations defining the momenta (\[eq:chiralpi\]-\[eq:chiralP\]) and the chiral limit ($\omega =0$), we get the constraints $$\begin{aligned}
C_0 &=& g^{\bar{\mu} \, \bar{\nu}} P_{\bar{\mu}} 
P_{\bar{\nu}} + {\cal H}\,\,,
\label{eq:newC0}
\\
C_A  &:=& P_{\bar{\mu}} \epsilon^{\bar{\mu}}{}_A \,\,,
\label{eq:newCA}\end{aligned}$$ where we have defined the extended momenta, $P_{\bar{\mu}} := (P_\mu , \pi)$.

The second way to get the constraints for the chiral case is straightforward from the constraints (\[C0\]-\[C1\]) governing the superconducting membrane theory because those encode the geometrical information for such.  For chiral membranes, it is enough to make $\omega =0$ in the expressions for ${\cal L}(\omega)$ and $d {\cal L} /\omega $ which become constants $c_1$ and $c_2$, respectively, whose inclusion in the constraints provide the corresponding constraints for chiral membranes, $$\begin{aligned}
C_0 = {\cal C}_0 \mid _{\omega =0} &=&
g^{\mu \, \nu}P_\mu P_\nu + h\,c_1 ^2 + 
\frac{c_1}{c_2}\,\pi^2 
:= g^{\bar{\mu} \, \bar{\nu}} P_{\bar{\mu}} 
P_{\bar{\nu}} + {\cal H}\, c_1 ^2 \,\,,
\label{eq:chiC0}
\\
C_A = {\cal C}_A \mid _{\omega =0} &:=& P_
{\bar{\mu}} \epsilon^{\bar{\mu}}{}_A \,\,,
\label{eq:chiC1}\end{aligned}$$ where the expression defining the momentum $\pi$, namely, $\pi= -2
\sqrt{h} \,({\bar{\omega}})^{1/2} \, 
(d{\cal L}/d\omega)$ has been used, and we have defined $g^{44}$ as the quotient $c_1 /2c_2$. Note the similarity of (\[eq:chiC0\]) and (\[eq:chiC1\]) with the DNG constraints in the canonical language where $c_1$ plays the role of tension of the membrane [@ADM].  The preceding description show us that the dynamical behaviour for chiral membranes is similar to DNG case, but their internal structures are different.  In fact, for the chiral string model it is an intermediate stage between the transonic string model [@Carter1] (possesing two internal degrees of freedom) and that of DNG case (without internal degrees of freedom). 

Conclusions
===========

In this paper we have developed the dynamics of chiral membranes using geometrical techniques.  We are able to reproduce the results of [@Carter; @Vilenkin; @Davis1] showing consistency of our description.  In fact, our scheme is resemble to DNG theory in five dimensions using a Kaluza-Klein approach but different internal structure.  However, our study is valid in any number of dimensions for the background spacetime and for the embedded membrane.  Integrability for both chiral string model and chiral domain wall model was obtained using a simple ansatz.  The full physical description is not over yet because a deep understanding of integration of equations of motion has not been accompplished.  Finally, we remark that our study might be useful in the study of brane universe scenarios, at least at canonical level, where we could think our world embedded in a higher-dimensional space, where the Hamiltonian analysis is demanded for quantization.  Besides this, the search of new solutions for the case of chiral superconducting membranes is part of a forthcoming paper. 

Acknowledgements
================

The authors are indebted to R. Capovilla, X. Martin and J.J. Blanco-Pillado for many valuable discussions and suggestions.  E. Rojas expresses grateful thanks to Professors A. P. Balachandran and Alberto García for the encouragement to the paper likewise to the Department of Physics of Syracuse University for hospitality.  R.C. thanks to J. Mendoza and R. Rodriguez for stimulating discussions.  The support in part from SNI (México) and CONACYT is grateful. 

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[^1]: The tangential deformations can be identified with the actions of worldsheet diffeomorphisms so we can ignore them since we are interested in quantities invariant under reparameterizations of the worldsheet.  These tangential deformations are important in the study of composite objects [@Defoedges; @Cordero4]. 

[^2]: As is well known, in such a case the worldsheet is parametrized by the coordinates $\xi^0 = \tau$ and $\xi^1 = \sigma$. The symbols $^.$ and $'$ denote partial derivatives with respect to $\xi^0$ and $\xi^1$, respectively.