text:
\section{Motivation}

The QCD phase diagram is conjectured to have a rich phase structure, but only little is known from lattice QCD due to the sign problem.
The available Monte Carlo methods are all limited to $\mu/T \lesssim 1$. Lattice QCD in the strong coupling limit,
$\beta=\frac{2{N_{c}}}{g^2} \rightarrow 0$, and in a dual representation is a model where the sign problem is mild
enough to study the full $\mu$-$T$ phase diagram.
This virtue crucially depends on the order of integration. The following three orders are common:
(1) Integrating out fermions first. This results in the fermion determinant $\det M[U]$. The Monte Carlo simulation is over gauge fields, $\beta$ can be varied continuously.
However, there is the severe sign problem at finite $\mu$, and it is expensive to approach the chiral limit.
(2) Integrating out spatial gauge links first then the fermions. The remaining temporal gauge links are mapped on Polyakov loops to obtain a 3-dim. heavy quark effective theory \cite{DePietri2007,Fromm2013}.
This is applicable to Wilson fermions, where backtracking of fermion world lines is prohibited.
The fermion determinant is factorized into a kinetic and a static part. Corrections to the static limit are treated analytically (expansion in hopping parameter and gauge action up to some order $\mathcal{O}(\kappa^n u(\beta)^m))$.
(3) Integrating out all gauge links first, then the fermions. For staggered fermions, this leads to the Monomer-Dimer-System \cite{Rossi1984}, which has a mild sign problem, and the chiral limit is cheap.
There is no fermion determinant and it can be studied e.g.~via Worm algorithms \cite{Forcrand2010}.
For Wilson fermions, results only consist for the Schwinger model so far \cite{Salmhofer1991,Scharnhorst1996}.
Moreover, incorporating the gauge action requires additional gauge integrals and introduces plaquette occupation numbers \cite{Unger2014}.

I will focus here on strategy (3) and explain its combinatorial interpretation. Lattice QCD at strong coupling (SC-LQCD) shares important features with continuum QCD:
it is ``confining'' in the sense that only color singlet d.o.f.~survive gauge integration, the mesons and baryons. These are point-like objects in the strong coupling limit, but become extended objects
away from the strong coupling limit and mix with gluons. SC-LQCD also has a (nuclear) liquid gas transition from
the vacuum to a baryonic crystal, where all lattice sites are occupied by baryons. Since the lattice spacing at strong coupling is maximally coarse, the degrees of freedom are on a hypercubic crystal and
saturation is due to the Pauli principle.
For staggered fermions, there is also spontaneous chiral symmetry breaking and its restoration at some critical temperature $aT_c$. In contrast to Wilson fermions, there is a remnant
chiral symmetry $U_{55}(1)\subset SU_L({N_{f}})\times SU_R(N_f)$ that is not broken by the finite lattice spacing.
The ultimate goal is to study the QCD phase diagram and the nuclear transition away from the strong coupling limit.
A first step into that direction, the $O(\beta)$ corrections to the strong coupling phase diagram, has already been undertaken \cite{Unger2014}.

A dimer/flux representation is possible for both lattice actions, but they differ qualitatively.
For staggered fermions: a partition function in the monomer-dimer representation is valid for any quark mass;
there is an exact chiral symmetry, hence it is adequate to study chiral properties (also, simulations in the chiral limit are cheap); however, staggered fermions are spinless in the strong coupling limit.
Contrast this with Wilson fermions: the flux representation involves spin, but since backtracking of fermions is not allowed, $(\mathbbm{1}-\gamma_\mu)(\mathbbm{1}+\gamma_\mu)=0$,
it poses a complicated combinatorial problem and expansion in spatial hadronic hoppings is required.
Both discretizations have very different lattice artifacts. The main motivation for this analysis is the question whether they share a
``physical'' content at strong coupling or at $\Ord{\beta}$ which could be isolated from lattice artifacts. The combinatorial perspective may help to shed light on this question.

\section{Gauge Integrals, Invariants and Combinatorics}

Combinatorics can give additional insight into lattice QCD, when formulated in a
dual, color singlet representation based on integer variables. The combinatorial paradigm I want to utilize is the question of how many ways there are to put $n$  balls into $k$ boxes.
Many combinatorial problems reduce to this question, and the answer will depend on the permutation symmetries in the problem, i.e.~whether balls or boxes are distinguishable or not, and which restrictions on the
placements are made (e.g.~the 12 canonical answers known as ``twelvefold way'').
Combinatorial formulae amount to integer sequences, which are listed in the On-Line Encyclopedia of Integer Sequences \cite{OEIS}. In the following I quote the A numbers from OEIS for further explanations and proofs.

First consider the SU$({N_{c}})$ one-link integral \cite{Creutz1978,Eriksson1981}
which can be evaluated both for staggered and Wilson fermions:
\begin{align}
z(x,\mu)&=\int dU_\mu(x) e^{{\rm tr}_c \left[U_\mu(x)M^\dagger + U_\mu(x)^\dagger M\right]},\\
(M^{\rm stagg.})_{ij}&=\chi^{f}_i(x){\bar{\chi}}^{f}_j(x+{\hat{\mu}}),\qquad
(M^{\rm Wilson})_{ij}&=\psi^{\beta,f}_{i}(x)(\mathbbm{1}-\gamma_\mu)_{\alpha\beta}{\bar{\psi}}^{\alpha,f}_{j}(x+{\hat{\mu}}),
\end{align}
with $i,j\in\{1,\ldots {N_{c}}\}$, $f\in\{1,\ldots {N_{f}}\}$ and $\alpha$, $\beta$ Dirac indices.
In both cases, the link integrals are gauge invariants, which can be expressed by linear combinations of traces and determinants.
\begin{align}
z(x,\mu)=\sum_{k_1,\ldots k_{{N_{c}}+1}}\alpha_{k_1\ldots k_{{N_{c}}+1}}\det_c\left[M\right]^{k_1}\det_c\left[M^\dagger\right]^{k_2}{\rm tr}_c\left[MM^\dagger\right]^{k_3}\ldots {\rm tr}_c\left[(MM^\dagger)^{{N_{c}}-1}\right]^{k_{{N_{c}}+1}}
\end{align}
The prefactors $\alpha_{k_1\ldots k_{{N_{c}}+1}}$ can be determined via Grassmann identities, e.g.~for ${N_{f}}=1$ staggered fermions
(with $y=x+{\hat{\mu}}$ and $B(x)=\frac{1}{{N_{c}}!}\epsilon_{i_1\ldots i_{{N_{c}}}}\chi_{i_1}(x)\ldots \chi_{i_{{N_{c}}}}(x)$):
\begin{align}
e^{{\bar{\chi}}_y\chi_y}&=\int d\chi_xd{\bar{\chi}}_x\int dU e^{{\bar{\chi}}_x \chi_x+{\bar{\chi}}_xU \chi_y-{\bar{\chi}}_yU^{\dagger} \chi_x}=\sum_{l=0}^{{N_{c}}} \alpha_k \frac{{N_{c}}!}{({N_{c}}-k)!}({\bar{\chi}}_x \chi_x{\bar{\chi}}_y \chi_y)^k\\
\Rightarrow \qquad z(x,y)&=\sum_{k=0}^{{N_{c}}}\frac{({N_{c}}-k)!}{{N_{c}}!k!}(M_x M_{y})^k +\bar{B}(x)B(y)+ (-1)^{{N_{c}}} \bar{B}(y)B(x).
\end{align}

\noindent \begin{minipage}{0.65\textwidth}
The prefactors can also be determined via combinatorics (labeled balls into labeled boxes, see Fig.~\ref{ballsboxes}). This strategy can be generalized to also apply to ${N_{f}}>1$
and for Wilson fermions, where meson hoppings $(M_x M_{x+{\hat{\mu}}})$ and baryon hoppings $\bar{B}(x)B(x+{\hat{\mu}})$ carry flavor and spin.
The corresponding integrals have been determined for ${N_{c}}\leq 3$ in \cite{Eriksson1981} and are in agreement with the combinatorial
determination.
\end{minipage}
 \qquad
\begin{minipage}{0.3\textwidth}
\vspace{-6mm}
\hspace{-6mm}
\includegraphics[width=1.1\textwidth]{ballsboxesdimer2.pdf}
\captionof{figure}{
\label{ballsboxes}
$k=2$ balls ($\bar{\chi}\chi$) into ${N_{c}}=3$ boxes (mesons).\\[-2mm]}

\end{minipage}\\

\begin{table}
\begin{minipage}{0.6\textwidth}
\begin{tabular}{cccccc}
 \hline
           &  $1$ {\tiny{(mesonic)}} & $L^{3}$ & $L^{6}$ & $L^{9}$ & $L^{12}$ \\
 \hline
\hspace{4mm}$1$ {\tiny{(baryonic)}}\hspace{-4mm} &   1 & 1 & 5 & 42 & 462 \\
$(LL^*)$   &   1 & 3 & 21 & 210 & 2574 \\
$(LL^*)^2$ &   2 & 11 & 98 & 1122 & 15015 \\
$(LL^*)^3$ &   6 & 74 & 498 & 6336 & 91091 \\
$(LL^*)^4$ &  23 & 225 & 2709 & 37466 & 571428 \\
 \hline
\end{tabular}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\includegraphics[width=\textwidth]{young2.pdf}
\end{minipage}
\captionof{table}{
\label{Table}
List of SU(3) gauge integrals $\langle(LL^*)^n(L^3)^m\rangle$,
which enumerate the number of restricted permutations patterns, which is the number representations of bounded height, see Eq.~(2.8).
}
\end{table}

Another type of gauge integrals are those over the trace of closed loops of gauge links: let
$P=\prod\limits_{(x,\mu)\in\mathcal{C}} U_\mu(x)={\rm diag}\left(e^{i\phi_1},\ldots e^{i\phi_{{N_{c}}-1}},e^{i\phi_{{N_{c}}}}\right)$ with
$\phi_{{N_{c}}}=-\sum\limits_{k=1}^{{N_{c}}-1}\phi_k$
be any closed loop of gauge links along contour $\mathcal{C}$ (e.g. Polyakov loop, Wilson loop), then, for SU(${N_{c}}$) there are ${N_{c}}-1$ gauge invariants, such as $L={\rm tr}_c [P]$ and $L^*$.
Only the mesonic $\mathcal{M}=LL^*$, baryonic $\mathcal{B}=L^{{N_{c}}}$, and mixed operators $\mathcal{O}=\mathcal{M}^n\mathcal{B}^m$ for $n,m\in \mathbb{N}$) have non-vanishing expectation values:
\begin{align}
\expval{\mathcal{O}(L,L^*\ldots)}=\frac{1}{(2\pi)^{{N_{c}}-1}}\int d\phi_1 \ldots d\phi_{{N_{c}}-1}\, V(L,L*,\ldots) \mathcal{O}(L,L^*,\ldots),
\end{align}
where  $V(L,L^*\ldots)$ is obtained from the invariant Haar measure:
$d\mu(\boldsymbol{\phi})=\prod\limits_{i>j}|e^{i\phi_i}-e^{i\phi_j}|^2\prod\limits_i d\phi_i$.
The result for SU(2), where $L^*=L$, gives rise to the so-called Catalan numbers (A000108), which play a prominent role in combinatorics, e.g.
as the number of 123-avoiding permutation patterns:
\begin{align}
\expval{L^{2n}}=\frac{1}{2\pi}\int_0^{2\pi} d\phi 2 \sin^2 \phi (2\cos \phi)^{2n}=\frac{1}{2\pi}\int_{-2}^2 dL \sqrt{4-L^2}L^{2n}=C_n\equiv\frac{1}{n+1}{2n \choose n}
\end{align}
For SU(3), where ${\rm tr}[P^2]=L^2-2{L^*}^2$, the various results listed in Tab.~\ref{Table} can be mapped on representations of the permutation group.
The invariants of higher moments in $L$, $L^*$, which are needed to express the static limit of Wilson fermions (see below), can be characterized as restricted permutation patterns,
which correspond to dimensions of standard young tableaux of bounded height.
\begin{align}
\label{bound}
m_{{N_{c}}}(n)=\sum\limits_{h(\lambda_n)\leq {N_{c}}} d_{\lambda_n}^2,\qquad   b_{{N_{c}}}(n)= d_{n\times{N_{c}}},\qquad mix_{{N_{c}}}(n_m,n_b)=
\sum\limits_{h(\lambda_{n_m})\leq {N_{c}}}
d_{n_b\times {N_{c}},\lambda_{n_m}} d_{\lambda_{n_m}}.
\end{align}
To compute these invariants in the general case of SU(${N_{c}}$) or U(${N_{c}}$) and ${N_{f}}>1$, one needs to evaluate ${\rm tr}[P^n]$, where $n=1,\ldots, {N_{c}}{N_{f}}$. These can be obtained via generalized Lucas polynomials.
The Lucas $n$-step numbers are $F_k^{(n)}=\sum\limits_{i=1}^{n}F_{k-i}^{(n)}$ (which is Fibonacci-like for $n=2$).
Related to SU(3) are the 3-step Lucas numbers
$F_{k}^{(3)}=F_{k-1}^{(3)}+F_{k-2}^{(3)}+F_{k-3}^{(3)}$
with seeds $F_{0}^{(3)}=3$, $F_{1}^{(3)}=1$, $F_{2}^{(3)}=3$,
from which the following 3-step polynomials $F$ in the variables $x$, $y$, $z$ are obtained :
\begin{align}
F^{(3)}_{n}(x,y,z)=
  {\rm tr} \left[\left(
  \begin{array}{ccc}
   x & y & z\\
   1 & 0 & 0\\
   0 & 1 & 0\\
  \end{array}
  \right)^n
 \right],\quad
 \tilde{F}^{(3)}_{n}(x,y,z)=
  {\rm tr} \left[\left(
  \begin{array}{ccc}
   x & y & z\\
   -1 & 0 & 0\\
   0 & -1 & 0\\
  \end{array}
  \right)^n
 \right].
\end{align}
It turns out that the signed verison $\tilde{F}_n$ is directly related to ${\rm tr}[P^n]=\tilde{F}_n(P)$  by identifiying $x\equiv L={\rm tr}[P]$, $y\equiv L^*={\rm tr}[P^\dagger]$, $z\equiv D=\det[P]$ (=1 for SU(${N_{c}}$)). The first orders are
 \begin{align*}
 {\rm tr}[P^0]&=3, &
 {\rm tr}[P^1]&=L, \\
 {\rm tr}[P^2]&=L^2-2L^*, &
 {\rm tr}[P^3]&=L^3-3L L^* +3D,\\
 {\rm tr}[P^4]&=L^4-4L^2L^*+2L^{*2}+4LD, &
 {\rm tr}[P^5]&=L^5-5L^3L^*+5L L^{*2}+5L^2D-5L^{*}D.
 \end{align*}
The corresponding versions for arbitrary ${N_{c}}$ is obtained by considering the signed ${N_{c}}\times{N_{c}}$ matrix $\tilde{F}(x_1,\ldots, x_{N_{c}})$.
I have used the generalized Lucas polynomials to determine the flavor dependence of the static limit.

\section{Strong Coupling Partition Functions}

The final partition functions at strong coupling are obtained after
Grassmann integration,
which introduces site weights $w_x$. For staggered fermions:
\begin{align}
 w_x=\int\prod\limits_{c}[d\chi_{c,x}d{\bar{\chi}}_{c,x}]e^{2am_q{\bar{\chi}}_{c,x}\chi_{c,x}}({\bar{\chi}}_{c,x}\chi_{c,x})^{k_x}=\frac{{N_{c}}!}{n_x!}(2am_q)^{n_x},
\end{align}
with monomers $n_x={N_{c}}-k_x$, determined by the Grassmann constraint $k_x=\sum\limits_{\pm{\hat{\mu}}}k_{\pm{\hat{\mu}}}(x)$, hence \linebreak $n_x\in \{0,\ldots {N_{c}}\}$, and no monomers at baryonic sites.
The well-known staggered partition function (${N_{f}}=1$) valid to all orders in the hopping parameter $\kappa=\frac{1}{2am_q}$ is
\begin{align}
Z_{SC}^{\rm stagg.}(m_q,\mu,\gamma)= \sum_{\{k_b,n_x,\ell\}}
\underbrace{\prod_{b=(x,\mu)}\frac{({N_{c}}-k_b)!}{{N_{c}}!k_b!}\gamma^{2 k_b\delta_{\mu 0}}}_{\text{meson hoppings}\, M_xM_y}
\underbrace{\prod_{x}\frac{{N_{c}}!}{n_x!}(2am_q)^{n_x}}_{
\text{chiral condensate}\, M_x}
\underbrace{\prod_\ell w(\ell,\mu)}_{\text{baryon hoppings}\, \bar{B}_xB_y},
\end{align}
with $k_b\in \{0,\ldots {N_{c}}\}$, $n_x \in \{0,\ldots {N_{c}}\}$, $\ell_b \in \{0,\pm 1\}$.
The weight $w(\ell,\mu)$ and sign $\sigma(\ell)=\pm 1$ for an oriented loop $\ell$ depend on loop geometry.
The anisotropy $\gamma=a/a_t$ is needed to vary the temperature continuously at $\beta=0$ \cite{Unger2012}.
\\
For Wilson fermions, Grassmann integration amounts to
spin and flavor conservation.
The site weights (almost) cancel link weights.
Only when spatial hoppings of color neutral states occur, the site weights are non-trivial. The partition function can generally be mapped on a vertex model.
This has been done for the Schwinger model, which maps on a 7-vertex model for ${N_{f}}=1$ \cite{Salmhofer1991} and on a modified 3-state 20-vertex model for ${N_{f}}=2$ \cite{Scharnhorst1996}.
Grassmann integration for ${N_{c}}>1$ is too complicated to do by hand but can be automatized using computer algebra.
The Wilson fermion partition function has the general structure
\begin{align}
 Z_{SC}^{\rm Wilson}(\kappa,\mu)= \sum_{\{k_b,n_x,\ell_j\}} N(\{k_b,\ell_j\}) v_i^{C_i}
\prod_x \frac{1}{(2\kappa)^{n_x}}\prod_{\ell_j} w(\ell_j,\mu).
\end{align}
$C_i$ counts how often vertices of type $i$ occur and $N(\{k_b,\ell_j\})$ counts multiplicities of loops. There are various baryonic loops $\ell_j$ (depending on spin and flavor).
The Grassmann constraint allows mesonic and baryonic world lines to intersect even for ${N_{f}}=1$. The vertex weights $v_i$ still need to be determined in general via Grassmann integration.

In the static limit, i.e. in the absence of spatial fermion hoppings, the strong coupling partition function is $Z_{SC}^{\rm static}=\prod\limits_{\vec{x}}Z_1(\vec{x})$, where $Z_1$ is the sum over all possible hadronic quantum states
$|\psi\rangle$.
This describes SC-LQCD in the high temperature and/or high density regime, see Fig.~\ref{Configs}.
For staggered fermions,
the chiral restoration takes place when the number of spatial dimers reaches a critical value. The nuclear and chiral transition coincide, because
$\expval{\bar{\chi}\chi}$ vanishes as a baryonic crystal forms. The number of hadronic states $|\psi\rangle=|P_u,P_d,\ldots P_{N_{f}},Q_{\pi^+},Q_{K^+},\ldots\rangle$ is
\begin{align}
\qquad Z_1(\mu,T)=\quad &{2{N_{f}} \choose {N_{c}}{N_{f}}}_{{N_{c}}}+ \sum\limits_{n=1}^{N_t{N_{c}}/2} t_n (2am_q)^{2n}+2
{{{N_{c}}+{N_{f}}-1} \choose {{N_{f}}-1}}
\cosh(\mu_B/T),
\end{align}
where the terms $\mathcal{O}(2am_q)$ are suppressed at high $T$ (for ${N_{c}}=3$ in the chiral limit, the prefactors $t_n$ are related to Tribonacci numbers (A000073), a generalization of Fibonacci).
The degeneracies of the mesonic states are given by central polynomial coefficients (see A077042) which number the possibilities to put $n$ unlabeled balls into $k$ labeled boxes,
allowing at most ${N_{c}}$ balls in each box.

For Wilson fermions,
with $K_t=(2\kappa)^{{N_{\tau}}}$ and hopping parameter $\kappa=\frac{1}{2d+2am_q}$, for ${N_{f}}=1$:
\begin{align}
\label{WilsonStat}
Z_1(\mu,T)= \sum\limits_{k=0}^{2{N_{c}}} T(k) K_t^{2k} +
\sum\limits_{k=0}^{{N_{c}}}P(k)K_t^{(2k+{N_{c}})} 2\cosh(\mu_B/T)+
K_t^{2{N_{c}}} 2\cosh(2\mu_B/T).
\end{align}
The combinatorics of the mesonic sector is given by the so-called tetrahedal numbers (A133826) \linebreak $T(k)=\sum\limits_{q=0}^{k}d_{D^{0q}}={3+\min(k,2{N_{c}}-k) \choose 3 }$, with $D^{0q}$ the mesonic irreducible representations of SU(${N_{c}}$),
and product numbers $P(k)=(1+k)(1+{N_{c}}-k)$. This can be generalized for ${N_{f}}>1$, e.g.\linebreak $T(k)={(2{N_{f}})^2-1+\min(k,2{N_{c}}-k) \choose (2{N_{f}})^2-1 }$, and Eq.~(\ref{WilsonStat}) contains $2{N_{f}}+1$ sums $\sim K_t^{2k+n{N_{c}}}$.
To conclude, the quantum number degeneracies of all static states can be listed via combinatorial formulae.

\begin{figure}
\vspace{-5mm}
\begin{minipage}{0.12\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{config_Nt8_HT.pdf}\\[-4mm]
{\scriptsize $\mu=0,T\gg T_c$ }\\[4mm]
\includegraphics[width=\textwidth]{config_Nt8.pdf}\\[-4mm]
{\scriptsize$\mu=0,T<T_c$}\\
\end{center}
\end{minipage}
\hspace{-3mm}
\begin{minipage}{0.35\textwidth}
\centerline{{\footnotesize Staggered Fermions}}
\vspace{2mm}
\includegraphics[width=\textwidth]{phasediag_gamma4.pdf}
\end{minipage}
\hspace{-3mm}
\begin{minipage}{0.12\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{config_Nt8_HT2.pdf}\\[-4mm]
{\scriptsize$\mu > \mu_c,\,T\gg T_c$}\\[4mm]
\includegraphics[width=\textwidth]{config_Nt8bar.pdf}\\[-4mm]
{\scriptsize$\mu > \mu_c,\,T=0$}\\
\end{center}
\end{minipage}\;
\begin{minipage}{0.01\textwidth}
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
|\\[-3mm]
\end{minipage}
\begin{minipage}{0.4\textwidth}
\vspace{4mm}
\centerline{{\footnotesize Static Wilson Fermions}}
\vspace{3mm}
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{config_Nt8_HTW2.pdf}\\[-3mm]
\centerline{{\footnotesize various $\mathcal{O}(K_t)$}}\\[4mm]
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{config_Nt8_HTW.pdf}\\[-3mm]
\centerline{{\footnotesize various spins}}\\[4mm]
\end{minipage}
\end{minipage}\hspace{-3mm}
\vspace{-2mm}
\caption{
\label{Configs}
At $\beta=0$, both staggered and Wilson fermions become static at large $T$ and/or $\mu$. Left: The staggered SC phase diagram, it has been measured in [4,7].
Right: Static Wilson fermions described by monomers/dimers/fluxes, multiplicities are due to spin and $K_t$.
}
\end{figure}

There are two kinds of corrections to the static limit, which can be both addressed systematically via an expansion:
(1) The (spatial) hopping parameter expansion in $\kappa_s$ allows to approach
the chiral limit, with the number of spatial mesonic and baryonic hoppings being controlled by the quark mass.
In a finite volume, this expansion always terminates due to the Grassmann constraint!
(2) The expansion in $\beta$ (the inverse gauge coupling) allows to approach the continuum limit. The staggered strong coupling partition function is in fact valid for all quark masses
(with the chiral limit being cheapest when addressed with a worm algorithm), whereas the Wilson partition function is restricted to rather large quark masses.
In both lattice discretizations, the gauge action can be incorporated order by order, which gives rise to higher order link integrals.

The strategy to study both lattice discretizations on a par is to expand around the static limit by making use of the Hamiltonian formulation that can be derived in the continuous time limit,
${N_{\tau}}\rightarrow \infty$ \cite{Unger2012}.
In this limit, the partition function simplifies further as only single meson hoppings need to be considered. The static lines are the in and out states of the transfer matrix:
\begin{align}
Z&={\rm Tr}[e^{\beta \mathcal{H}}],& \mathcal{H}&=
\frac{1}{2}\sum_{\langle x,y \rangle}
\sum_{Q_i}J^+_{Q_i(x)} J^-_{Q_i(y)},&
J^{-}_{Q_i}&=(J^{+}_{Q_i})^\dagger,
\end{align}
where the generalized quantum numbers $Q_i$ (spin, parity, flavor) are globally conserved, and spatial dimers represented by
$J^+_{Q_i(x)} J^-_{Q_i(y)}$ raise quantum number $Q_i$ at site $x$ and lower them at a neighboring site $y$ (see \cite{Unger2012} for the case of
${N_{f}}=1,2$ for staggered fermions). For both staggered and Wilson fermions, the matrices $J^{\pm}_{Q_i}$ contain vertex weights. They are the crucial input to sample
the corresponding partition function with a quantum Monte Carlo algorithm to all orders in $\kappa_s$, e.g.~via a continuous time Worm algorithm.
However, whether also for Wilson fermions in four dimensions all vertex weights are positive to evade the sign problem is still an open question, although due to the continuous time limit, only a small
set of vertices need to be considered. For the simulation of the Schwinger model with Wilson fermions at strong coupling with the Worm algorithm see \cite{Wenger2008}.

\begin{figure}[t!]
\vspace{-5mm}
\hspace{-7mm}
\includegraphics[width=0.58\textwidth]{ladder2.pdf}\qquad
\vspace{-28mm}\\
\vphantom{.}\hspace{0.53\textwidth}
\includegraphics[width=0.5\textwidth]{weights5.pdf}
\vspace{-5mm}
\caption{\emph{Left:} Systematic expansion in the spatial hoppings in the Hamiltonian formulation, where hadrons are emitted/absorbed at events in continuous time, depending on vertex weights $v_i$.
\emph{Right:} Example of a gauge correction to the SC-limit, hadrons become extended objects.}
\end{figure}
Also the gauge corrections could be included in this Hamiltonian formulation.
So far, the gauge corrections have been studied for finite ${N_{\tau}}=4$ and for ${N_{f}}=1$ staggered fermions.
In a collaboration with J.~Langelage, P. de Forcrand and  O.~Philipsen, we have determined the phase diagram of SC-LQCD at $\mathcal{O}(\beta)$ \cite{Unger2014},
where the gauge action is linearized
and a new set of one-link integrals (those along an excited plaquette) have to be evaluated and new invariants with a combinatorial interpretation arise.
We find that the second order phase boundary in the $\mu$-$T$ plane is shifted to lower temperatures with increasing $\beta$, but that the tricritical point and the first order transition is invariant at $\mathcal{O}(\beta)$.
In contrast, the critical endpoint of the first order nuclear transition, which coincides with the chiral transition at $\beta=0$, moves down along the chiral first order line.

\emph{Acknowledgement} - I would like to thank Philippe de Forcrand and Owe Philipsen for numerous discussions.
This works was supported by the Helmholtz International Center for FAIR
within the LOEWE program launched by the State
of Hesse.


text:
\section{Introduction}
\label{sec:intro}

Extractive summarization remains a simple
and fast approach to produce summaries which
are grammatical and accurately represent the source text.
In the news domain, these systems are able to
use a dominant signal: the position of a sentence
in the source document.
Due to journalistic conventions which place important information
early in the articles, the lead sentences often contain key information. In this paper, we explore how systems can look beyond this simple trend.

Naturally, automatic
systems have all along exploited position cues in news
as key indicators of important content  \citep{Schiffman:2002:EMS,hong2014improving,ext_bert}. The `lead' baseline is rather strong in single-document news summarization \citep{brandow1995automatic,nenkova2005automatic},
with automatic systems only modestly improving the
results.
Nevertheless, more than 20-30\% of
summary-worthy sentences come from the second half of news documents \citep{data2_nallapati2016abstractive,kedzie2018content},
and the lead baseline, as shown in Table \ref{tab:lead_ex}, does not always produce convincing summaries.
So, systems must balance the position bias with representations of the semantic content
throughout the document. Alas, preliminary studies
\citep{kedzie2018content} suggest that even the most recent neural
methods predominantly pick sentences from the lead, and
that their content selection performance drops greatly
when the position cues are withheld.

\begin{table}[t]
    \centering
    \small
    \begin{tabular}{|p{0.94\linewidth}|}
        \hline
        \textbf{Lead-3:} Bangladesh beat fellow World Cup quarter-finalists Pakistan by 79 runs in the first one-day international in Dhaka. Tamim Iqbal and Mushfiqur Rahim scored centuries as Bangladesh made 329 for six and Pakistan could only muster 250 in reply. Pakistan will have the chance to level the three-match series on Sunday when the second odi takes place in Mirpur. \\ \hline
        \textbf{Reference:} Bangladesh beat fellow World Cup quarter-finalists Pakistan by 79 runs. Tamim Iqbal and Mushfiqur Rahim scored centuries for Bangladesh. Bangladesh made 329 for six and Pakistan could only muster 250 in reply. Pakistan will have the chance to level the three-match series on Sunday. \\ \hline \hline
        \textbf{Lead-3}: Standing up for what you believe. What does it cost you? What do you gain? \\ \hline
        \textbf{Reference:} Indiana town's Memories Pizza is shut down after online threat. Its owners say they'd refuse to cater a same-sex couple's wedding. \\
        \hline
    \end{tabular}
    \caption{`Lead' (first 3 sentences of source) can produce extremely faithful (top) to disastrously inaccurate (bottom) summaries. Gold standard summaries are also shown.}
    \label{tab:lead_ex}
\end{table}

In this paper, we verify that
sentence position and lead bias dominate the learning signal for  state-of-the-art neural extractive summarizers in
the news domain.
We then present techniques to improve content selection in
the face of this bias.
The first technique makes use of `unbiased data'
created by permuting the order of sentences in the training
articles. We use this shuffled dataset for pre-training, followed by
training on the original (unshuffled) articles. The second method
introduces an auxiliary loss which encourages the model's scores
for sentences to mimic an estimated score distribution over the
sentences, the latter computed using ROUGE overlap with the gold standard.
We implement these techniques for two recent reinforcement learning based systems, RNES \citep{DBLP:conf/aaai/WuH18} and BanditSum \citep{dong2018banditsum}, and evaluate them on the CNN/Daily Mail dataset \citep{hermann2015teaching}.

 We find that our
auxiliary loss achieves significantly better ROUGE scores
compared to the base systems, and that the improvement is even
more pronounced when the true
best sentences appear later in the
article. On the other hand, the pretraining approach
produces mixed results.
We also confirm that when summary-worthy sentences appear late, there is a large performance discrepancy between the oracle summary and state-of-the-art summarizers,
indicating that learning to balance lead bias with other features of news text is a noteworthy issue to tackle.

\section{Related Work}\label{sec:related_work}

Modern summarization methods for news are typically
based on neural network-based sequence-to-sequence learning
\cite{cnn1_kalchbrenner2014convolutional,cnn2_kim2014convolutional,rnn2_chung2014gru,ext2_2015Yin,ext3_cao2015learning,ext4_cheng2016neural,ext5_summarunner,narayan2018don,neusum}.
In MLE-based training, extractive summarizers are trained with gradient ascent to maximize the likelihood of heuristically-generated ground-truth binary labels \citep{ext5_summarunner}. Many MLE-based models do not perform as well as their reinforcement learning-based (RL) competitors that directly optimize ROUGE \cite{abs5_paulus2017deep,DBLP:Narayan/2018,dong2018banditsum,DBLP:conf/aaai/WuH18}.
As RL-based models represent the state of the art for extractive summarization, we analyze them in this paper.

The closest work to ours is a recent study by \citet{kedzie2018content} which showed that
MLE-based models
learn a significant bias for selecting early sentences
when trained on news articles as opposed
to other domains. As much as 58\% of selected
summary sentences come directly from the lead.
Moreover, when these models
are trained on articles whose sentences
are randomly shuffled, the performance drops considerably for news domain only.
While this drop
could be due to the destruction of position cues,
it may also arise because the article's coherence
and context were lost.

In this paper, we employ finer control on the distortion
of sentence position, coherence, and context, and confirm that performance
drops are mainly due to the lack of position cues.
We also propose the first techniques to
counter the effects of lead bias in neural extractive systems.

\section{Base Models for Extractive Summarization}

In supervised systems, given a document $D = \{ s_1, . . . , s_n \}$ with $n$ sentences,
a summary can be seen as
set of binary labels $y_1,\dots, y_n \in \{0, 1\}$, where $y_i = 1$
indicates that the $i$-th sentence is included in the summary.

We choose to experiment with two state-of-the-art RL-based extractive models:
{\bf RNES} \cite{DBLP:conf/aaai/WuH18} and {\bf BanditSum} \citep{dong2018banditsum}. Both employ an encoder-decoder structure, where the encoder extracts sentence features into fixed-dimensional vector representations $h_1, \dots,h_n$, and a decoder produces the labels $y_1,\dots, y_n$ based on these sentence representations. RNES uses a CNN+bi-GRU encoder, and
BanditSum a hierarchical bi-LSTM. RNES's decoder is
\textit{auto-regressive}, meaning it predicts the current sentence's label based on decisions made on previous sentences; i.e., $y_t = f(D,h_t, y_{1:t-1})$. In BanditSum, there is no such dependence: it produces affinity
scores for each sentence and the top scoring sentences are then selected.

\section{Lead Bias of News Systems}\label{sec:lead_bias}

\begin{table*}[h]
\centering
\small
\begin{tabular}{c|ccccc|cc}
\toprule
\diagbox{train setting}{test setting} &original & random & reverse & insert-lead & insert-lead3 & Mean & Std. Dev.  \\ \hline
Lead-3 baseline &32.68 & 22.81&17.94&27.67&27.68 &25.76 &5.00\\ \hline
original & \textbf{33.85} & 26.18 &20.71 &31.71 & 31.11 &28.71 & 4.72\\
random & 30.88 & \textbf{29.70} & 29.79 & 29.97 & 30.09 &\textbf{30.09}& \textbf{0.42}\\
reverse & 21.35 & 26.32 & \textbf{33.59} & 21.63 & 21.65 &24.91 & 4.72 \\
insert-lead & 33.21 & 26.07 & 20.70 & \textbf{33.41} & 31.59 &29.00 &4.93 \\
insert-lead3 & 32.29 & 25.57 & 20.22 & 32.92 & \textbf{32.15} &28.63&4.98 \\ \bottomrule
\end{tabular}

\caption{BanditSum's performance---calculated as the average between ROUGE-1,-2, and -L F1---on the validation set of the CNN/Daily Mail corpus. The sentence position information is perturbed at different levels, as explained in Section \ref{sec:lead_bias}.}
\label{tab:data_manipulation}
\end{table*}

First, we investigate the impact of sentence
position on our models. We manipulate the \textbf{original}
CNN/Daily Mail dataset to
preserve sentence position information
at different levels. In the \textbf{random} setting, sentences are shuffled randomly; in \textbf{reverse}, they are in reverse order; in \textbf{insert-lead} and \textbf{insert-lead3}, we insert an out-of-document
sentence (chosen randomly from the corpus) as the first sentence or randomly as one of the first three sentences, respectively.

In Table \ref{tab:data_manipulation}, we show BanditSum's performance,\footnote{We notice the same trends on  RNES.
} when trained and tested on the various datasets.
All models (except random)  perform worse when tested on a mismatched data perturbation.
Even when the distortion is at a single lead
position in \textbf{insert-lead} and \textbf{insert-lead3},
the performance on the original data is significantly
lower than when trained without the distortion.
These results corroborate \citet{kedzie2018content}'s findings for RL-based
systems. Interestingly, the \textbf{random} model has the
best mean performance and the lowest variation indicating that completely removing
the position bias may allow a model to focus on learning robust sentence semantics.

\section{Learning to Counter Position Bias}

We present two methods which encourage models
to locate key phrases at diverse
parts of the article.

\subsection{Multi-Stage Training}
This technique is inspired by the robust results from the
\textbf{random} model in section  \ref{sec:lead_bias}.
We implement a
multi-stage training method for both BanditSum and RNES where
in the first few epochs, we
train on an `unbiased' dataset where the sentences in every training document
are randomly shuffled. We then fine-tune
the models by training on the original training articles.
The goal is to prime the model to learn
sentence semantics independently of position, and then
introduce the task of balancing semantics and positional cues.

\subsection{ROUGE-based Auxiliary Loss}
We observed that BanditSum tends to converge to a low-entropy policy, in the sense that the model's affinity scores are either 1 or 0 at the end of training.
Furthermore, over 68\% of its selections are from the three leading sentences of the source.
Regularizing low-entropy policies can increase a model's propensity to explore potentially good states
or stay close to a known good policy \citep{nachum2017improving,galashov2019information}.
We extend this idea to summarization by introducing a ROUGE-based loss
which regularizes the model policy using an estimate of the value of individual sentences.

These sentence-level estimates are computed as a \textit{distribution} $P_R$:

\begin{equation}
    P_R( x = i ) = \frac{r(s_i, \mathcal{G})}{\sum_{j=1}^{n}{r(s_j, \mathcal{G})}},
\end{equation}

\noindent where $r$ is the average of ROUGE-1, -2 and \mbox{-L} F\textsubscript{1} scores between sentence $s_i$ in the article
and the reference summary $\mathcal{G}$. We would like the model's predictive distribution $P_\mathcal{M}$ to approximately match
$P_R$. To compute $P_\mathcal{M}$,
we normalize the predicted scores from a non-auto-regressive model. In an auto-regressive model such as RNES,  the decision of including a sentence
depends on those selected so far. So a straightforward KL objective is hard to implement, and
we use this technique for BanditSum only.

Our auxiliary loss is defined as the KL divergence: $\mathcal{L}_{\KL} = D_{\KL} \infdivx{P_R}{P_\mathcal{M}}$. The update rule then becomes:
\begin{equation}
\label{eq:kl_loss}
    \theta^{(t+1)} = \theta^{(t)} + \alpha \left( \nabla \mathcal{L}_{\mathcal{M}}(\theta^{(t)}) + \beta \nabla \mathcal{L}_{\KL}(\theta^{(t)}) \right)
\end{equation}

where $\theta^{(t)}$ represents the model's parameters at time step $t$, $\mathcal{L}_{\mathcal{M}}$ is the original model's loss function, and $\beta$ is a hyperparameter.

\section{Experimental Setup}
We use the CNN/Daily Mail dataset \citep{hermann2015teaching} with the standard train/dev/test splits of 287,227/13,368/11,490.
To avoid inconsistencies, we built on top of the author-provided implementations for
BanditSum and our faithful reimplementation of RNES.

To reduce training time, we pre-compute and store the average of ROUGE-1, -2, and -L for every sentence triplet of each article, using a HDF5 table and PyTables \cite{pytables, hdf5}. This allows for a considerable increase in training speed. We limit the maximum number of sentences considered in an article to the first 100.

All the models were trained for 4 epochs. For the multi-stage training,
we pretrain for 2 epochs, then train on the original articles for 2 epochs.
 We set the auxiliary loss hyperparameters $\alpha=1e-4$ and $\beta=0.0095$ in
eq. \ref{eq:kl_loss} based on a grid search using the Tune library \cite{tune}.

We also train a baseline {\bf entropy} model by replacing $\mathcal{L}_{\KL}$ with the negated entropy of $P_\mathcal{M}$ in eq. \ref{eq:kl_loss}.
This loss penalizes low entropy, helping the model explore, but it is `undirected' compared to our proposed method. We present the results of Lead-3 baseline (first 3 sentences), and two other competitive
models---Refresh\footnote{We are unable to evaluate this model on the lead overlap measure due to lack of access to the model outputs.} \citep{narayan2018don} and NeuSum \citep{neusum}.

Lastly, we include results from an \textit{oracle} summarizer, computed as the triplet of source sentences with the highest average of ROUGE-1, -2 and -L scores against the abstractive gold standard.

\section{Results and Discussion} \label{sec:result}

Table \ref{tab:results} reports the F1 scores for ROUGE-1,-2 and -L \cite{eva1_lin:2004:ACLsummarization}.
We use the \emph{pyrouge}\footnote{\url{www.github.com/bheinzerling/pyrouge}} wrapper library to evaluate the final models, while training
with a faster Python-only
implementation\footnote{\url{www.github.com/Diego999/py-rouge}}.

\begin{table}[t]
    \centering
    \small
    \begin{tabular}{l|ccc|c}
        \toprule
        Model	&	\multicolumn{3}{c}{ROUGE}	& Overlp	\\
        	&	1	&	2	&	L	&	\%	\\ \hline
        Lead-3	&	40.06	&	17.53	&	36.18	&   100.0	\\
        Oracle	&	56.53	&	32.65	&	53.12	&	27.24	\\
        Refresh	&	40.0	&	18.2	&	36.6	&	--	\\
        NeuSum	&	40.15	&	17.80	&	36.63	&	58.24	\\\hline
        RNES	&	41.15	&	18.81	&	37.75	&	68.44	\\
        RNES+pretrain & 41.29    &  18.85         &    37.79       &     68.22   \\ \hline
        BanditSum	    &	41.68	&	18.78	&	38.00	&	69.87	\\
        B.Sum+pretrain  &   41.68   &   18.79   &   37.99   &   70.77   \\
        B.Sum+entropy	&	41.71	&	18.87	&	38.04	&	64.83	\\
        BanditSum+KL	&	\textbf{41.81*}	&	\textbf{18.96*}	&	\textbf{38.16*}	&	65.13	\\
        \bottomrule
    \end{tabular}
    \caption{ROUGE scores for systems. `Overlp' denotes the model's overlap in extraction choices with the lead-3 baseline. Scores significantly higher than
    BanditSum  with $p<0.001$ (bootstrap resampling test) are marked with *.}
    \label{tab:results}
\end{table}

We test for significance between the baseline models and our proposed techniques using the bootstrap method. This method was first recommended for testing significance in ROUGE scores by \citet{eva1_lin:2004:ACLsummarization}, and has subsequently been advocated as an appropriate measure in works such as \citet{dror-hitchhikers} and \citet{berg-kirkpatrick}.

The simple entropy regularizer has a small but not significant improvement, and pretraining
has a similar improvement only for RNES. But the auxiliary ROUGE loss significantly ($p<0.001$)
improves over BanditSum, obtaining an extra 0.15 ROUGE points on average.
The last column
reports the percentage of summary sentences which overlap with the lead. The auxiliary
loss leads to a 4.7\% absolute decrease in such selections compared
to the base system, while also reaching a better ROUGE score.
Figure \ref{fig:train_curves} shows that the reward (average ROUGE-1,-2,-L) for the
auxiliary loss model is consistently above the base.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/final_dev_fig.png}
    \caption{Training curves for BanditSum based models. Average ROUGE is the average of ROUGE-1, -2 and -L F1.}
    \label{fig:train_curves}
    \vspace{2mm}
\end{figure}

We also examined the auxiliary loss model on documents where the summary is mostly comprised of
lead sentences $D_{\mathrm{early}}${}, mostly sentences much later in the article $D_{\mathrm{late}}${}, and a dataset
at the midway point, $D_{\mathrm{med}}${}.
To create these sets, we rank test articles using the average index of its
summary sentences in the source document. The
100 test articles with lowest average index are $D_{\mathrm{early}}${}, the 100 with highest value
are $D_{\mathrm{late}}${} and the 100 closest to the median are $D_{\mathrm{med}}${}.
In Table~\ref{d_early}, we can see that the auxiliary loss model's improvements are even
more amplified on $D_{\mathrm{med}}${} and $D_{\mathrm{late}}${}.

On the other hand, our pretraining results are mixed. We hope to employ more controlled multi-tasking methods \citep{kiperwasser2018scheduled}
in the future to deal with the issue.

The second line in Table \ref{d_early} reports the oracle ROUGE scores of the best possible extractive summary.
While all systems are quite close to the oracle on $D_{\mathrm{early}}${}
they only reach half the performance on $D_{\mathrm{late}}${}. This gap indicates that our improvements only scratch the surface, but also that this problem is worthy and challenging to explore.

It is worth noting that we have attempted to build a single model which can summarize both lead-biased articles and those
whose information is spread throughout. Our aim was to
encourage the model to explore useful regions as a way of
learning better document semantics. But we hypothesize that
our models can be further improved by  learning to automatically predict when the lead paragraph
suffices as a summary, and when
the model should look further in the document.

\begin{table}
\centering
\small
\begin{tabular}{l|l|l|l}
\toprule
Model	&	$D_{\mathrm{early}}$	& $D_{\mathrm{med}}$ &	$D_{\mathrm{late}}$	\\ \hline
Lead-3	&	46.17	&	30.90	&	20.18	\\
Oracle	&	50.52	&	47.92	&	42.21	\\
RNES	&	41.76 &32.11 &20.62			\\
RNES+pretrain	& 41.66 & 32.38 & 20.64	\\ \hline
BanditSum	&	43.10	&	32.65	&	21.63	\\
BanditSum+entropy	&	41.96	&	32.59	&	22.12	\\
BanditSum+KL	&	42.63	&	33.05	&	21.96	\\
\bottomrule
\end{tabular}
\caption{Average ROUGE-1, -2 and -L F1 scores on $D_{\mathrm{early}}$, and $D_{\mathrm{med}}$, $D_{\mathrm{late}}$. Each set contains 100 documents.}
\label{d_early}

\end{table}
\vspace{2mm}

\section{Conclusion}

In this paper, we have presented the first approaches for learning a summarization system by countering the strong effect of summary-worthy lead sentences.
We demonstrate that recent summarization systems over-exploit the inherent lead bias present in news articles, to the detriment of their summarization capabilities. We explore two techniques aimed at learning to better balance positional cues with semantic ones.
While our auxiliary loss method achieves
significant improvement, we note that there is a large gap which better methods can
hope to bridge in the future.

One approach, building on ours, is to examine other ways to combine loss signals  \cite{maml}, and to encourage exploration \cite{sac}.
We will also carry out deeper study of the properties
of $D_{\mathrm{early}}${} and $D_{\mathrm{late}}${} type
documents and use them to inform new solutions.
On cursory analysis, the most frequent terms in $D_{\mathrm{early}}${} tend to be about UK politics, while  in $D_{\mathrm{late}}${} they are often related to British soccer.

\section*{Acknowledgments}
This work is supported by the Natural Sciences and Engineering Research Council of Canada, the Institute for Data Valorisation (IVADO), Compute Canada, and the CIFAR Canada AI Chair program.

redpajama_set_name: RedPajamaArXiv


text:
\section{\textbf{Introduction}}

The Poincar\'{e} algebra and Poincar\'{e} group describe the symmetries of
empty Minkowski spacetime. It is known since $1970$ \cite{bacry}, that the
presence of a constant electromagnetic field in Minkowski spacetime leads to
the modification of Poincar\'{e} symmetries.

The presence of a constant classical electromagnetic field in Minkowski
spacetime modifies the Poincar\'{e} algebra into the so-called Maxwell
algebra \cite{schrader},\cite{beckers},\cite{soroka},\cite{bonanos}, \cit
{bonanos1}, \cite{bonanos2}, \cite{gomis}. This algebra can also be obtained
from the anti-de Sitter (AdS) algebra and a particular semigroup $S$ by
means of the $S$-expansion procedure introduced in Refs. \cite{hat}, \cit
{salg2}, \cite{salg3}, \cite{azcarr}. Using this method it is possible to
obtain more general modifications to the Poincar\'{e} algebra (see, for
example, \cite{irs}, \cite{salg4}). An interesting modification to the
Poincar\'{e} symmetries, obtained by the aforementioned expansion procedure,
is given by the so-called Lie $\mathfrak{B}$ algebra also known as
generalized Poincar\'{e} algebra, whose generators satisfy the commutation
relation shown in Eq. (7) of Ref. \cite{gomez}.

The Einstein-Chern-Simons (EChS) gravity \cite{irs} is a gauge theory whose
Lagrangian density is given by a $5$-dimensional Chern-Simons form for the
\mathfrak{B}$ algebra. The field content induced by the $\mathfrak{B}$
algebra includes the vielbein $e^{a}$, the spin connection $\omega ^{ab}$,
and two extra bosonic fields $h^{a}$ and $k^{ab}.$ The EChS gravity has the
interesting property that the $5$-dimensional Chern-Simons Lagrangian for
the $\mathcal{B}$ algebra, given by \cite{irs
\begin{eqnarray}
L_{ChS}^{(5)}[e,\omega ,h,k] &=&\alpha _{1}l^{2}\varepsilon
_{abcde}R^{ab}R^{cd}e^{e}  \label{1t} \\
&&+\alpha _{3}\varepsilon _{abcde}\left( \frac{2}{3
R^{ab}e^{c}e^{d}e^{e}+2l^{2}k^{ab}R^{cd}T^{\text{ }e}+l^{2}R^{ab}R^{cd}h^{e
\right) ,  \notag
\end{eqnarray
where $R^{ab}=\mathrm{d}\omega ^{ab}+\omega _{\text{ }c}^{a}\omega ^{cb}$
and $T^{a}=\mathrm{d}e^{a}+\omega _{\text{ }c}^{a}e^{c}$, leads to the
standard general relativity without cosmological constant in the limit where
the coupling constant $l$ tends to zero while keeping the Newton's constant
fixed. It should be noted that there is an absence of kinetic terms for the
fields $h^{a}$ and $k^{ab}$ in the Lagrangian $L_{ChS}^{(5)}$ (for details
see Ref. \cite{salg5}).

Recently was shown in Ref. \cite{bra1} that the $5$-dimensional EChS gravity
can be consistent with the idea of a $4$-dimensional spacetime. In this
Reference was replaced a Randall-Sundrum type metric \cite{randall} \cit
{randall1} in the EChS gravity Lagrangian (\ref{1t}) to get (see Appendix)

\begin{eqnarray}
\tilde{S}[\tilde{e},\tilde{h}] &=&\int_{\Sigma _{4}}\tilde{\varepsilon
_{mnpq}\left( -\frac{1}{2}\tilde{R}^{mn}\tilde{e}^{p}\tilde{e}^{q}\right. +
\notag \\
&&\left. +C\tilde{R}^{mn}\tilde{e}^{p}\tilde{h}^{q}-\frac{C}{4r_{c}^{2}
\tilde{e}^{m}\tilde{e}^{n}\tilde{e}^{p}\tilde{h}^{q}\right) ,  \label{s3}
\end{eqnarray
which is an gravity action with a cosmological constant for a $4
-dimensional brane embedded in the $5$-dimensional spacetime of the EChS
theory of gravity.\textbf{\ }$\tilde{\varepsilon}_{mnpq}$\textbf{, }$\tilde{
}^{m}$, $\tilde{R}^{mn}$\textbf{\ }and\textbf{\ }$\tilde{h}^{m}$\textbf{\
represent, respectively, the\textbf{\ }$4$\textbf{-}dimensional versions of
the Levi-Civita symbol,\textbf{\ }the vielbein, the curvature form and
\textbf{\ }matter field. It is of interest to note that the field $h^{a}$, a
bosonic gauge field from the Chern-Simons gravity action, which gives rise
to a form of positive cosmological constant, appears as a consequence of
modification of the Poincar\'{e} symmetries, carried out through the
expansion procedure.

On the other hand\textbf{, }$C$\textbf{\ }and\textbf{\ }$r_{c}$\textbf{\ (
the "compactification radius") are constants\textbf{. }The corresponding
version in tensor language (see Appendix) is given by

\begin{equation}
\tilde{S}[\tilde{g},\tilde{h}]=\int d^{4}\tilde{x}\sqrt{-\tilde{g}}\left[
\tilde{R}+2C\left( \tilde{R}\tilde{h}-2\tilde{R}_{\text{ }\nu }^{\mu }\tilde
h}_{\text{ }\mu }^{\nu }\right) -\frac{3C}{2r_{c}^{2}}\tilde{h}\right] ,
\label{31t'}
\end{equation
where we can see that when $l\rightarrow 0$ then $C\rightarrow 0$ and hence
\ref{a}) becomes the $4$-dimensional Einstein-Hilbert action. \

In this paper we introduce the geometric framework obtained by gauging of
the so called\textbf{\ }$\mathfrak{B}$ algebra. Besides the vierbein $e_{\mu
}^{a}$ and the spin connection $\omega _{\mu }^{ab}$, our scheme includes
the fields\textbf{\ }$k_{\mu }^{ab}$ and $h_{\mu }^{a}$ whose dynamic is
described by the field equation obtained from the corresponding actions. The
application of the cosmological principle shows that the field $h^{a}$\ has
a similar behavior to that of a cosmological constant, which leads to the
conjecture that the equations of motion and their accelerated solutions are
compatible with the era of dark energy.

It might be of interest to note that, according to standard GR (Einstein
framework in a FLRW background), a simple way to describe dark energy (also
dark matter) is through an equation of state that relates density ($\rho $)
of a fluid and its pressure ($p$) through the equation $p=\omega \rho $,
where $\omega $ is the parameter of the equation of state. Dark energy is
characterized by $-1\leq \omega <-1/3$\textbf{, \ }$\omega =-1$ represents
the cosmological constant and $\omega <-1$ corresponds to the so-called
phantom dark energy. This means that in the context of general relativity
the parameter $\omega $ is "set by hand" and then contrasted with
observational information.

In the present work, the cosmological constant is not "set by hand" but
rather arises from the framework that we present. An example is shown where
a quintessence-type evolution as well as a phantom evolution are equally
possible. This means that a possible geometric origin of dark energy can be
conjectured in the context of the so-called Einstein-Chern-Simons gravity.

The article is organized as follows: in Section\textbf{\ }$II$\textbf{, }we
rewrite the action\textbf{\ }(\ref{31t'})\textbf{\ }by introducing a scalar
field associated to the field\textbf{\ }$\tilde{h}_{\mu \nu }$, we find the
corresponding equations of motion, and then we discuss the cosmological
consequences of this scheme. In Section\textbf{\ }$III$\textbf{, }a kinetic
term is added in the action and its effects on cosmology are studied.\
Finally, Concluding Remarks are presented in Section\textbf{\ }$IV$\textbf{.
}An Appendix is also included where we review the derivation of the actio
\textbf{\ }(\ref{31t'}).

\section{\textbf{Cosmological consequences}}

In this Section we will study the cosmological consecuences associated with
the action\ (\ref{31t'})\textbf{. }If we consider a maximally symmetric
spacetime (for instance, the de Sitter's space), the equation 13.4.6 of Ref.
\cite{weinberg} allows us to write the field\textbf{\ }$\tilde{h}_{\mu \nu }
\textbf{\ }as\textbf{\ }

\begin{equation}
\tilde{h}_{\mu \nu }=\frac{\tilde{F}(\tilde{\varphi})}{4}\tilde{g}_{\mu \nu
},  \label{66t}
\end{equation
where $\tilde{F}$ is an arbitrary function of an $4$-scalar field $\tilde
\varphi}=$ $\tilde{\varphi}(\tilde{x})$.\ This means

\begin{equation}
\tilde{R}_{\text{ }\nu }^{\mu }\tilde{h}_{\text{ }\mu }^{\nu }=\frac{\tilde{
}(\tilde{\varphi})}{4}\tilde{R}\text{ \ \ },\text{\ \ }\tilde{h}=\tilde{h
_{\mu \nu }\tilde{g}_{\text{ }}^{\mu \nu }=\tilde{F}(\tilde{\varphi}),
\end{equation
so that the action (\ref{31t'}) takes the form (see Appendix)
\begin{equation}
\tilde{S}[\tilde{g},\tilde{\varphi}]=\int d^{4}\tilde{x}\sqrt{-\tilde{g}
\left[ \tilde{R}+C\tilde{R}\tilde{F}(\tilde{\varphi})-\frac{3C}{2r_{c}^{2}
\tilde{F}(\tilde{\varphi})\right] ,  \label{32t}
\end{equation
which corresponds to an action for the $4$-dimensional gravity coupled
non-minimally to a scalar field. Note that this action has the for
\begin{equation}
\tilde{S}_{B}=\tilde{S}_{g}+\tilde{S}_{g\varphi }+\tilde{S}_{\varphi },
\end{equation
where, $\tilde{S}_{g}$ is a pure gravitational action term, $\tilde{S
_{g\varphi }$ is a non-minimal interaction term between gravity and a scalar
field, and $\tilde{S}_{\varphi }$ represents a kind of scalar field
potential. In order to write down the action in the usual way, we define the
constant $\varepsilon $ and the potential $V(\varphi )$ as (removing the
symbols $\sim $ in (\ref{32t})). In fact
\begin{equation}
\varepsilon =\frac{4\kappa r_{c}^{2}}{3}\text{ \ \ },\text{\ \ }V(\varphi )
\frac{3C}{4\kappa r_{c}^{2}}F(\varphi ),  \label{100t}
\end{equation
where $\kappa $ is the gravitational constant. This permits to rewrite the
action for a $4$-dimensional brane non-minimally coupled to a scalar field,
immersed in a $5$-dimensional space-time a
\begin{equation}
S[g,\varphi ]=\int d^{4}x\sqrt{-g}\left[ R+\varepsilon RV(\varphi )-2\kappa
V(\varphi )\right] .  \label{33t}
\end{equation
The corresponding field equations describing the behavior of the $4
-dimensional brane in the presence of the scalar field $\varphi $ are given
by

\begin{equation}
G_{\mu \nu }\left( 1+\varepsilon V\right) +\varepsilon H_{\mu \nu }=-\kappa
g_{\mu \nu }V,  \label{z1}
\end{equation
\begin{equation}
\frac{\partial V}{\partial \varphi }\left( 1-\frac{\varepsilon R}{2\kappa
\right) =0,  \label{z2}
\end{equation
wher
\begin{equation}
H_{\mu \nu }=g_{\mu \nu }\nabla ^{\lambda }\nabla _{\lambda }V-\nabla _{\mu
}\nabla _{\nu }V.
\end{equation}

In order to construct a model of universe based on Eqs. (\ref{z1}-\ref{z2}),
we consider the Friedmann-Lema\^{\i}tre-Robertson-Walker metri
\begin{equation}
ds^{2}=-dt^{2}+a^{2}(t)\left( \frac{dr^{2}}{1-kr^{2}}+r^{2}\left( d\theta
^{2}+\sin ^{2}\theta d\psi ^{2}\right) \right) ,
\end{equation
where $a(t)$ is the so called "cosmic scale factor" and $k=0,+1,-1$
describes flat, spherical and hyperbolic spatial geometries, respectively.
Following the usual procedure, we find the following field equations

\begin{eqnarray}
3\left( H^{2}+\frac{k}{a^{2}}\right) \left( 1+\varepsilon V\right)
+3\varepsilon H\dot{\varphi}\frac{\partial V}{\partial \varphi } &=&V,
\label{iii1} \\
\left( 2\dot{H}+3H^{2}+\frac{k}{a^{2}}\right) \left( 1+\varepsilon V\right)
+\varepsilon \left( \dot{\varphi}^{2}\frac{\partial ^{2}V}{\partial \varphi
^{2}}+\left( \ddot{\varphi}+2H\dot{\varphi}\right) \frac{\partial V}
\partial \varphi }\right) &=&V,  \label{iii2} \\
\frac{\partial V}{\partial \varphi }\left[ 1-3\varepsilon \left( \dot{H
+2H^{2}+\frac{k}{a^{2}}\right) \right] &=&0,  \label{iii3}
\end{eqnarray
where $H=\dot{a}/a$ is the Hubble parameter and we have used natural unities
$\kappa =8\pi G=c=1$. Dot means derivative with respect to time.

From (\ref{iii1}, \ref{iii2}, \ref{iii3}) we see that when $\varepsilon =0$
and $V=$ $const.$, we have a de Sitter behavior governed by $H=\sqrt{V/3}.$
On the another hand, from equation (\ref{100t}) we can see that $\varepsilon
$ is a positive quantity. This fact allows us to define an effective
cosmological constant a
\begin{equation}
\Lambda _{eff}=\frac{1}{2\varepsilon },
\end{equation
which will play an important role in the cosmological consequences that we
will show below.

In the flat case, the Eqs. (\ref{iii1}, \ref{iii2}, \ref{iii3}) are given by

\begin{eqnarray}
3H^{2} &=&-\frac{1}{V+2\Lambda _{eff}}\left( 3H\frac{dV}{dt}-2\Lambda
_{eff}V\right) ,  \label{iii4} \\
2\dot{H}+3H^{2} &=&-\frac{1}{V+2\Lambda _{eff}}\left( \frac{d^{2}V}{dt^{2}
+2H\frac{dV}{dt}-2\Lambda _{eff}V\right) ,  \label{iii5}
\end{eqnarray
where the equation\textbf{\ (}\ref{iii3}) was not considered because it is
not an independent equation. In fact, subtracting\textbf{\ (}\ref{iii5}
\textbf{\ }from\textbf{\ }(\ref{iii4})\textbf{\ }we obtai
\begin{equation}
2\dot{H}=-\frac{1}{V+2\Lambda _{eff}}\left( \frac{d^{2}V}{dt^{2}}-H\frac{dV}
dt}\right) .  \label{iii5'}
\end{equation
Deriving\textbf{\ (}\ref{iii4}\textbf{) }with respect to time and using
\textbf{(}\ref{iii5'})\textbf{\ }we find\textbf{\ (}\ref{iii3}), when\textbf
\ }$k=0$\textbf{. }Bear in mind that, at the end of this Section, we will
study an interesting consequence derived from this equation.

We write now the Eqs. (\ref{iii4}, \ref{iii5}) in the "standard" for
\begin{eqnarray}
3H^{2} &=&\rho \text{ \ \ },\text{ \ }\rho =-\frac{1}{V+2\Lambda _{eff}
\left( 3H\frac{dV}{dt}-2\Lambda _{eff}V\right) ,  \label{iii6} \\
\dot{H}+H^{2} &=&-qH^{2}=-\frac{1}{6}\left( \rho +3p\right) ,  \notag \\
\frac{1}{6}\left( \rho +3p\right) &=&\frac{1}{4\Lambda _{eff}}\frac{1}
\left( 1+V/2\Lambda _{eff}\right) }\left( \frac{d^{2}V}{dt^{2}}+H\frac{dV}{d
}-\frac{4\Lambda _{eff}}{3}V\right) ,  \label{iii7}
\end{eqnarray
being $q$ the deceleration parameter defined by $q=-1-\dot{H}/H^{2}$ and $p$
the pressure associated to $\rho $ given by

\begin{equation}
p=\frac{1}{V+2\Lambda _{eff}}\left( \frac{d^{2}V}{dt^{2}}+2H\frac{dV}{dt
-2\Lambda _{eff}V\right) ,  \label{iii8}
\end{equation
which allows to write the barotropic equation $p=\omega \rho $, where

\begin{equation}
\omega =-\left( \frac{2\Lambda _{eff}V-2HdV/dt-d^{2}V/dt^{2}}{2\Lambda
_{eff}V-3HdV/dt}\right) ,  \label{iii9}
\end{equation
and we note that $V=const.$ leads to $\omega =-1$, i.e., a de Sitter
evolution.

Considering again Eqs. (\ref{iii4}, \ref{iii5}) and defining\ $x=V/2\Lambda
_{eff}$,\ we write the field equations in the form

\begin{eqnarray}
3\left( H^{2}+H\frac{d}{dt}\ln \left( 1+x\right) \right) &=&2\Lambda
_{eff}\left( \frac{x}{1+x}\right) ,  \label{iii11} \\
3\left( qH^{2}-H\frac{d}{dt}\ln \left( 1+x\right) \right) &=&-2\Lambda
_{eff}\left( \frac{x}{1+x}\right) +\frac{3}{2}\left( \frac{1}{1+x}\right)
\frac{d^{2}x}{dt^{2}},  \label{iii12}
\end{eqnarray
and we discuss some examples:

\textbf{(a) }$x=x_{0}$\ $=$\textbf{\ }const.\textbf{\ }If $x$\ behaves as a
constant\textbf{,} the solution for the Hubble parameter is given by
\begin{equation}
H=\sqrt{\frac{2\Lambda _{eff}}{3}\frac{x_{0}}{1+x_{0}}},  \label{iii14}
\end{equation
i.e., a de Sitter evolution for all time.

\textbf{(b)} $x=t/t_{0}$. In this case, the solution is\textit{\ }
\begin{equation}
H\left( t\right) =\frac{1}{2t_{0}\left( 1+t/t_{0}\right) }\left( \sqrt{1
\frac{8\Lambda _{eff}t_{0}}{3}\left( 1+t/t_{0}\right) t}-1\right) ,
\label{iii15}
\end{equation
where we can see that
\begin{equation}
H\left( t\rightarrow \infty \right) \rightarrow \sqrt{\frac{2\Lambda _{eff}}
3}}\ \text{\ \ },\text{\ \ }\rho \left( t\rightarrow \infty \right)
\rightarrow 2\Lambda _{eff}\text{ \ \ \ }and\text{\ \ \ }q\left(
t\rightarrow \infty \right) =-1,  \label{iii16}
\end{equation
which means that we have a late de Sitter evolution.

\textbf{(c) }$x=$ $\exp \left( t/t_{0}\right) $. Here, the Hubble parameter
turns out to be
\begin{equation}
H\left( t\right) =\frac{1}{2t_{0}\left[ 1+\exp \left( t/t_{0}\right) \right]
}\left( \sqrt{1+\frac{8\Lambda _{eff}t_{0}^{2}}{3}\exp \left( t/t_{0}\right)
\left[ 1+\exp \left( t/t_{0}\right) \right] }-1\right) ,
\end{equation
and
\begin{equation}
\text{ \ \ \ }H\left( t\rightarrow \infty \right) \rightarrow \sqrt{\frac
2\Lambda _{eff}}{3}}\text{ \ \ },\text{ \ }\rho \left( t\rightarrow \infty
\right) \rightarrow 2\Lambda _{eff}\text{ \ \ }and\text{ \ \ }q\left(
t\rightarrow \infty \right) \rightarrow -1,
\end{equation
and, as in the previous case, we have a late de Sitter evolution.

From (\ref{iii9}) the parameter of the equation of state takes the form

\begin{equation}
\omega \left( t\right) =-\left( \frac{2\Lambda _{eff}x-2Hdx/dt-d^{2}x/dt^{2
}{2\Lambda _{eff}x-3Hdx/dt}\right) ,  \label{iii19}
\end{equation
and if $x=t/t_{0}$ one find
\begin{equation}
\omega \left( t\right) =-\left( \frac{t-H\left( t\right) /\Lambda _{eff}}
t-3H\left( t\right) /2\Lambda _{eff}}\right) ,
\end{equation
so that if we identify $t_{0}$\ as the current time, then

\begin{equation}
\text{\ }\omega \left( t_{0}\right) =-\left( \frac{t_{0}-H\left(
t_{0}\right) /\Lambda _{eff}}{t_{0}-3H\left( t_{0}\right) /2\Lambda _{eff}
\right) <-1\text{ \ \ \ }and\text{ \ \ \ }\omega \left( t\rightarrow \infty
\right) =-1,\text{ \ \ \ }
\end{equation
and we have a transient phantom evolution (not ruled out by the current
observational data). Theoretical frameworks where this type of evolution is
discussed can be seen in \cite{lepe1}, \cite{zar}, and \cite{lepe2}.

The shown examples above have a common characteristic, namely they show a
late de Sitter evolution like, for instance, $\Lambda CDM$ at late times,
but we do not know if this characteristic comes from the formalism that we
are inspecting or from the choice (Ansatz) that we make for $V\left(
t\right) $. Since we do not have something to guide us towards a form for
V\left( t\right) $ from first principles, we are tied to playing with
different Ansatze for that potential. At least those shown here, give us
interesting results, in particular, that obtained from the Ansatz given in (
b$), a transient phantom evolution.

Previously, we have seen that equation\textbf{\ }(\ref{iii3})\textbf{\ }is
not independent, and therefore it was not analyzed in the first instance.
However, it reveals an interesting fact of our scheme, the presence of a
cosmological bounce. In case that\textbf{\ }$k=0$\textbf{,\ }$\Lambda
_{eff}= $ $1/2\varepsilon $ and $\partial V/\partial \varphi \neq 0$,\textbf
\ }the equation\textbf{\ }(\ref{iii3})\textbf{\ }takes the form

\begin{equation}
\frac{2}{3}\Lambda _{eff}-\left( \dot{H}+2H^{2}\right) =0,  \label{v1}
\end{equation
which leads to the following solution for the Hubble parameter

\begin{equation}
H\left( t\right) =\sqrt{\Lambda _{eff}/3}\left( \frac{\exp \left[ 4\sqrt
\Lambda _{eff}/3}\left( t-t_{0}\right) \right] -1/\Delta \left( t_{0}\right)
}{\exp \left[ 4\sqrt{\Lambda _{eff}/3}\left( t-t_{0}\right) \right]
+1/\Delta \left( t_{0}\right) }\right) ,  \label{v2}
\end{equation
wher
\begin{equation}
\Delta \left( t_{0}\right) =\frac{\sqrt{\Lambda _{eff}/3}+H_{0}}{\sqrt
\Lambda _{eff}/3}-H_{0}}.
\end{equation
Note that the equation (\ref{v2})\textbf{\ }can be written in terms of an
hyperbolic tangent a
\begin{equation}
H\left( t\right) =\sqrt{\frac{\Lambda _{eff}}{3}}\tanh \left( 2\sqrt{\frac
\Lambda _{eff}}{3}}\left( t-t_{0}\right) +\frac{1}{2}\ln \left[ \Delta
\left( t_{0}\right) \right] \right) ,  \label{v3}
\end{equation
which reveals a cosmological bounce i
\begin{equation}
t_{b}=t_{0}-\frac{1}{4}\sqrt{\frac{3}{\Lambda _{eff}}}\ln \Delta \left(
t_{0}\right) \Longrightarrow H\left( t_{b}\right) =0.  \label{v4}
\end{equation
Moreover, the expression (\ref{v3}) is also showing that $H<0$ for $t<t_{b}
, and $H>0$ for $t>t_{b}$; i.e., there is a contraction for $t<t_{b}$ and an
expansion for $t>t_{b}$. In fact, from equation (\ref{v3}) the cosmic scale
factor is obtaine
\begin{equation}
a(t)=a_{b}\cosh ^{1/2}\left( 2\sqrt{\frac{\Lambda _{eff}}{3}}\left(
t-t_{b}\right) \right) ,  \label{v5}
\end{equation
where $a_{b}$ is the minimum of the scale factor, which occurs at $t_{b}$,
and it is given b
\begin{equation}
a_{b}=a(t_{b})=a\left( 0\right) \cosh ^{-1/2}\left( 2\sqrt{\frac{\Lambda
_{eff}}{3}}\left( t_{0}-t_{b}\right) \right) .  \label{v6}
\end{equation
Behaviors of this kind have been frequently studied in the literature in the
context of cosmological bouncing; see e.g., \cite{Biswas1}, \cite{Biswas},
\cite{Bamba}. Finally, we note that
\begin{equation}
H(t\rightarrow \infty )\rightarrow \sqrt{\frac{\Lambda _{eff}}{3}},
\label{v7}
\end{equation
i.e., a late de Sitter evolution.

\section{\textbf{Introduction of the kinetic term }$\left( 1/2\right) \dot
\protect\varphi}^{2}$}

In Ref. \cite{salg4} it was found that the surface term $B_{\text{EChS
}^{(4)}$ in the Lagrangian (\ref{1t}) is given b
\begin{align}
B_{EChS}^{(4)}& =\alpha _{1}l^{2}\epsilon _{abcde}e^{a}\omega ^{bc}\left(
\frac{2}{3}d\omega ^{de}+\frac{1}{2}\omega _{\text{ \ }f}^{d}\omega
^{fe}\right)  \notag \\
& \quad +\alpha _{3}\epsilon _{abcde}\left[ l^{2}\left( h^{a}\omega
^{bc}+k^{ab}e^{c}\right) \left( \frac{2}{3}d\omega ^{de}+\frac{1}{2}\omega _
\text{ \ }f}^{d}\omega ^{fe}\right) \right.  \notag \\
& \qquad \qquad \qquad \left. +l^{2}k^{ab}\omega ^{cd}\left( \frac{2}{3
de^{e}+\frac{1}{2}\omega _{\text{ \ }f}^{d}e^{e}\right) +\frac{1}{6
e^{a}e^{b}e^{c}\omega ^{de}\right] .  \label{8'}
\end{align}

From (\ref{1t}) and (\ref{8'}) we can see that kinetic terms corresponding
to the fields $h^{a}$ and $k^{ab}$, absent in the Lagrangian, are present in
the surface term. This situation is common to all Chern-Simons theories.
This has the consequence that the action (\ref{33t}) does not have the
kinetic term for the scalar field $\varphi $.

It could be interesting to add a kinetic term to the $4$-dimensional brane
action. In this case, the action (\ref{33t}) takes the for
\begin{equation}
S[g,\varphi ]=\int d^{4}x\sqrt{-g}\left[ R+\varepsilon RV(\varphi )-2\kappa
\left[ \frac{1}{2}\left( \nabla _{\mu }\varphi \right) \left( \nabla ^{\mu
}\varphi \right) +V(\varphi )\right] \right] .  \label{34t}
\end{equation
The corresponding field equations are given by

\begin{equation}
G_{\mu \nu }\left( 1+\varepsilon V\right) +\varepsilon H_{\mu \nu }=\kappa
T_{\mu \nu }^{\varphi },  \label{40t}
\end{equation
\begin{equation}
\nabla _{\mu }\nabla ^{\mu }\varphi -\frac{\partial V}{\partial \varphi
\left( 1-\frac{\varepsilon R}{2\kappa }\right) =0,  \label{41t}
\end{equation
where $T_{\mu \nu }^{\varphi }$ is the energy-momentum tensor of the scalar
fiel
\begin{equation}
T_{\mu \nu }^{\varphi }=\nabla _{\mu }\varphi \nabla _{\nu }\varphi -g_{\mu
\nu }\left( \frac{1}{2}\nabla ^{\lambda }\varphi \nabla _{\lambda }\varphi
+V\right) ,  \label{37t}
\end{equation
and the rank-2 tensor $H_{\mu \nu }$ is defined a
\begin{equation}
H_{\mu \nu }=g_{\mu \nu }\nabla ^{\lambda }\nabla _{\lambda }V-\nabla _{\mu
}\nabla _{\nu }V.  \label{38t}
\end{equation}

Following the usual procedure, we find that the FLRW type equations are
given b
\begin{equation}
3\left( \frac{\dot{a}^{2}+k}{a^{2}}\right) \left( 1+\varepsilon V\right)
+3\varepsilon \frac{\dot{a}}{a}\dot{\varphi}\frac{\partial V}{\partial
\varphi }=\kappa \left( \frac{1}{2}\dot{\varphi}^{2}+V\right) ,  \label{48x}
\end{equation
\begin{equation}
\left( 2\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}+k}{a^{2}}\right) \left(
1+\varepsilon V\right) +\varepsilon \left[ \dot{\varphi}^{2}\frac{\partial
^{2}V}{\partial \varphi ^{2}}+\left( \ddot{\varphi}+2\frac{\dot{a}}{a}\dot
\varphi}\right) \frac{\partial V}{\partial \varphi }\right] =-\kappa \left(
\frac{1}{2}\dot{\varphi}^{2}-V\right) ,  \label{49x}
\end{equation
\begin{equation}
\ddot{\varphi}+3\frac{\dot{a}}{a}\dot{\varphi}+\frac{\partial V}{\partial
\varphi }\left[ 1-\frac{3\varepsilon }{\kappa }\left( \frac{\ddot{a}}{a}
\frac{\dot{a}^{2}+k}{a^{2}}\right) \right] =0.  \label{50x}
\end{equation
In the case $k=0$, \textbf{and using }$\kappa =1$, Eqs. (\ref{48x},\ref{49x}
\ref{50x}) takes the form

\begin{eqnarray}
3H^{2}\left( 1+\epsilon V\right) +3\epsilon H\frac{dV}{dt} &=&\frac{1}{2
\dot{\varphi}^{2}+V,  \label{iv1} \\
\left( 2\dot{H}+3H^{2}\right) \left( 1+\epsilon V\right) +\epsilon \left(
\frac{d^{2}V}{dt^{2}}+2H\frac{dV}{dt}\right) &=&-\left( \frac{1}{2}\dot
\varphi}^{2}-V\right) ,  \label{iv2} \\
\left( \ddot{\varphi}+3H\dot{\varphi}\right) \dot{\varphi}+\frac{dV}{dt
\left[ 1-3\epsilon \left( \dot{H}+2H^{2}\right) \right] &=&0,  \label{iv3}
\end{eqnarray
and here, the equation\textbf{\ }(\ref{iv3})\textbf{\ }it is not an
independent equation. In fact, subtracting the equation\textbf{\ }(\ref{iv2}
\textbf{\ }from the\textbf{\ }equation\textbf{\ }(\ref{iv1})\textbf{\ }we
obtai
\begin{equation}
2\dot{H}\left( 1+\varepsilon \right) =\varepsilon H\frac{dV}{dt}-\varepsilon
\frac{d^{2}V}{dt^{2}}+\dot{\varphi}^{2}.  \label{iv3'}
\end{equation
Deriving the equation\textbf{\ }(\ref{iv1})\textbf{\ }with respect to time
and using the equation\textbf{\ }(\ref{iv3'})\textbf{\ }we find the equatio
\textbf{\ }(\ref{iv3})\textbf{.}

The combination $\left( 1/2\right) \dot{\varphi}^{2}+V\left( \varphi \right)
$, together with the combination $\left( 1/2\right) \dot{\varphi
^{2}-V\left( \varphi \right) $, reminds us that in a standard scalar field
theory

\begin{equation}
\rho =\frac{1}{2}\dot{\varphi}^{2}+V\text{ \ },\text{ \ }p=\frac{1}{2}\dot
\varphi}^{2}-V,  \label{iv4}
\end{equation
which is recovered at the limit $\varepsilon \rightarrow 0$. In fact, when
\varepsilon \rightarrow 0$ the equations (\ref{iv1}, \ref{iv2}, \ref{iv3}
\textbf{\ }takes the for
\begin{eqnarray}
3H^{2} &=&\rho ,  \label{iv5} \\
\text{ \ }2\dot{H}+3H^{2} &=&-p, \\
\dot{\rho}+3H\left( p+\rho \right) &=&0,  \label{iv7}
\end{eqnarray
where we have used\textbf{\ }(\ref{iv4})\textbf{\ }to obtain\textbf{\ }(\re
{iv7})\textbf{\ }from\textbf{\ }(\ref{iv3})\textbf{\ }when\textbf{\ }
\varepsilon \rightarrow 0$\textbf{.}

The equation (\ref{iv4}) allows us to write (\ref{iv1},\ref{iv2}) in the for
\begin{eqnarray}
3H^{2} &=&\frac{1}{1+\varepsilon V}\left( \rho -3\varepsilon H\dot{\varphi
\frac{\partial V}{\partial \varphi }\right) ,  \label{iv17} \\
2\dot{H}+3H^{2} &=&-\frac{1}{1+\varepsilon V}\left( p-\varepsilon \left[
\left( \rho +p\right) \frac{\partial ^{2}V}{\partial \varphi ^{2}}+\left(
\ddot{\varphi}+2H\dot{\varphi}\right) \frac{\partial V}{\partial \varphi
\right] \right) ,  \label{iv18}
\end{eqnarray
and if we choose $p=-\rho $, by thinking in a de Sitter evolution, we obtai
\begin{equation}
3H^{2}=\frac{\rho }{1+\varepsilon V}-3H\frac{d}{dt}\ln \left( 1+\varepsilon
V\right) ,  \label{(a)}
\end{equation
\begin{equation}
\dot{H}=\frac{1}{2}\left( \frac{\ddot{\varphi}}{\dot{\varphi}}+5H\right)
\frac{d}{dt}\ln \left( 1+\varepsilon V\right) .  \label{(b)}
\end{equation
According to (\ref{(b)}), $\dot{H}=0$ say us

\begin{equation}
\frac{\ddot{\varphi}}{\dot{\varphi}}+5H=0\text{ \ \ }or\text{ \ \ }V\left(
t\right) =const.,
\end{equation
and, according to (\ref{(a)}), $V\left( t\right) =const.$ implies $\rho
=const.$ i.e., $H=const$., i.e., an usual de Sitter evolution. But, if
\ddot{\varphi}/\dot{\varphi}+5H=0$ and $V\left( t\right) \neq const.$, after
to see (\ref{(a)}) we have, with $H=H_{0}=const$.,

\begin{equation}
\rho \left( t\right) =3H_{0}\left[ H_{0}+\frac{d}{dt}\ln \left(
1+\varepsilon V\left( t\right) \right) \right] \left( 1+\varepsilon V\left(
t\right) \right) ,
\end{equation
and we have a de Sitter evolution although $\rho \neq const$. One more
detail, the equation $\ddot{\varphi}/\dot{\varphi}+5H=0$ has the solution
\dot{\varphi}\sim a^{-5}$ and so, the kinetic term $\left( 1/2\right) \dot
\varphi}^{2}$ dissolves very quickly with evolution leading us to $\rho \sim
V\left( t\right) $ and $p\sim -V\left( t\right) $ at late times, i.e., a de
Sitter evolution.

Writing (\ref{iv17},\ref{iv18}) in the form

\begin{eqnarray}
3H^{2} &=&\frac{1}{1+\varepsilon V}\left( \rho -3\varepsilon H\frac{dV}{dt
\right) ,  \label{iv25} \\
\dot{H}+H^{2} &=&-\frac{1}{6\left( 1+\varepsilon V\right) }\left[ \left(
\rho +3p\right) +3\varepsilon \left( \frac{d^{2}V}{dt^{2}}+H\frac{dV}{dt
\right) \right] ,  \label{iv26}
\end{eqnarray
we see that when $\varepsilon =0$ we recover the results of General
Relativity, i.e., $3H^{2}=\rho $ and $\dot{H}+H^{2}=-\left( 1/6\right)
\left( 1+3\omega \right) \rho $. By following this reminder, we write (\re
{iv25}, \ref{iv26}) in the standard form

\begin{eqnarray}
3H^{2} &=&\rho _{tot}, \\
\text{ \ \ }\dot{H}+H^{2} &=&-\frac{1}{6}\left( \rho _{tot}+3p_{tot}\right) ,
\\
\rho _{tot} &=&\frac{1}{1+\varepsilon V}\left( \rho -3H\frac{d\left(
\varepsilon V\right) }{dt}\right) , \\
\text{\ \ }p_{tot} &=&\frac{1}{1+\varepsilon V}\left( p+2H\frac{d\left(
\varepsilon V\right) }{dt}+\frac{d^{2}\left( \varepsilon V\right) }{dt^{2}
\right) ,  \label{iv28}
\end{eqnarray
we can see that we can build a barotropic equation $p_{tot}=\omega
_{tot}\rho _{tot}$, where

\begin{equation}
\omega _{tot}=\frac{p+2Hd\left( \varepsilon V\right) /dt+d^{2}\left(
\varepsilon V\right) /dt^{2}}{\rho -3Hd\left( \varepsilon V\right) /dt}\text{
\ \ }and\text{ \ \ }q=\frac{1}{2}\left( 1+3\omega _{tot}\right) .
\label{iv29}
\end{equation
By doing $p=\omega \rho $, we can write (\ref{iv29}) as

\begin{equation}
\omega _{tot}=\omega +\frac{\left( 2+3\omega \right) Hd\left( \varepsilon
V\right) /dt+d^{2}\left( \varepsilon V\right) /dt^{2}}{\rho -3Hd\left(
\varepsilon V\right) /dt}.  \label{iv30}
\end{equation
Here we can see that if $d^{2}\left( \varepsilon V\right) /dt^{2}=0$ and
d\left( \varepsilon V\right) /dt\neq 0$, then $\omega =-2/3$, $\omega
_{tot}=-2/3$ and $q=-1/2.$ This means that $\omega _{tot}$ belongs to the
quintessence zone. So, with $\left( \alpha ,\beta \right) $ constants,
\varepsilon V\left( t\right) =\alpha \left( t/t_{0}\right) +\beta $ is an
obvious choice for $\varepsilon V\left( t\right) $.

On the other hand, it is direct to show tha
\begin{equation}
\text{ \ }\dot{\rho}_{tot}+3H\left( 1+\omega _{tot}\right) \rho _{tot}=0,
\label{iv32}
\end{equation
so tha
\begin{eqnarray}
\rho _{tot}\left( a\right) &=&\rho _{tot}\left( a_{0}\right) \left( \frac
a_{0}}{a}\right) ^{3}\exp \left( -3\int_{t_{0}}^{t}\omega _{tot}\left(
t\right) d\ln a\right) ,  \label{iv33} \\
and\text{ \ \ \ }\omega _{tot} &=&-\frac{2}{3}\rightarrow \rho _{tot}\left(
a\right) =\rho _{tot}\left( a_{0}\right) \left( \frac{a_{0}}{a}\right) ,
\notag
\end{eqnarray
and the same is true for $\rho \left( a\right) $, that is, $\rho \left(
a\right) =\rho \left( 0\right) \left( a_{0}/a\right) $.

Note that, if $V\left( t\right) =V_{0}=const.
\begin{equation}
\rho _{tot}=\frac{1}{1+\epsilon V_{0}}\rho \text{ \ \ },\text{ \ \ }p_{tot}
\frac{1}{1+\epsilon V_{0}}p\text{ \ \ \ }and\text{ \ \ }\omega _{tot}=\omega
,
\end{equation
and if $\omega =0$ then $\omega _{tot}=0$ and then $p_{tot}=0$. This means
that $\omega _{tot}=0$ plays the role of the usual dark matter ($\omega =0
), although $\rho _{tot}\neq \rho $.

Finally, we have been using the quantity\textbf{\ }$\Lambda
_{eff}=1/2\varepsilon $\textbf{, }where\textbf{\ }$\varepsilon =4\kappa
r_{c}^{2}/3=const.$\textbf{\ }is a parameter derived from the mechanism of
dimensional reduction under consideration, which depends on the
gravitational constant\textbf{\ }$\kappa $\textbf{\ }and the
compactification radius\textbf{\ }$r_{c}$\textbf{. }This parameter plays the
role of an effective cosmological constant (its inverse) recalling that in
the action\textbf{\ }$S[g,\varphi ]=\int d^{4}x\sqrt{-g}\left[ R+\left(
\varepsilon R-2\kappa \right) V(\varphi )\right] $\textbf{\ }there is no a
"bare" cosmological constant. This fact could lead us to conjecture that th
\textbf{\ }$h$\textbf{-}field\textbf{\ (}or\textbf{\ }$\tilde{h}$\textbf{-
field\textbf{), }in some way, manifests itself as dark energy. If so, the
next step will be to submit the present outline to the verdict of
observation.

\section{\textbf{Concluding remarks}}

We have considered a modification of the Poincar\'{e} symmetries known as
given $\mathfrak{B}$ Lie algebra also known as generalized Poincar\'{e}
algebra, whose generators satisfy the commutation relation shown in Eq. (7)
of Ref. \cite{gomez}. Besides the vierbein $e_{\mu }^{a}$ and the spin
connection $\omega _{\mu }^{ab}$, our scheme includes the fields\textbf{\ }
k_{\mu }^{ab}$ and $h_{\mu }^{a}$ whose dynamic is described by the field
equation obtained from the corresponding actions.

We have used the field equations for a $4$-dimensional brane embedded in the
$5$-dimensional spacetime of \cite{bra1} to study their cosmological
consequences. The corresponding FLRW equations are found by means of the
usual procedure and cosmological solutions are shown and discussed. We
highlight two solutions, by choosing $\partial V/\partial t=const.$, a
transient phantom evolution (not ruled out by the current observational
data) is obtained and if $\partial V/\partial t\neq const.$ we obtain a
bouncing solution.

Since the kinetic terms corresponding to the fields $h^{a}$ and $k^{ab}$ are
present in the surface term (see (\ref{1t}) and (\ref{8'})) it was necessary
to introduce a kinetic term to the $4$-dimensional action. As a consequence
of this, in the corresponding cosmological framework we highlight a de
Sitter evolution even when the energy density involved is not constant.

Whatever it is, and since we do not have something to guide us towards a
form for $V\left( t\right) $ from first principles, we are tied to playing
with different Ansatze for that potential. At least in the cases that were
considered give us interesting results. But, we must insist, we are
completely dependent on the Ansatze for $V\left( t\right) $. If we are
thinking on cosmology, the results shown here suffer from this "slavery".
The hope, a common feeling, is that what is shown can be a contribution that
guides us towards a better understanding of the present formalism and its
chance of being a possible alternative to General Relativity. It is evident
that the observational information will be key when it comes to
discriminating between both models\textbf{. }To extract information that
leads us to\textbf{\ }$V\left( t\right) $\textbf{\ }in order to visualize if
the scalar field\textbf{\ }philosophy has a viable chance of being real when
it comes to doing cosmology is the challenge to face.

\section{\textbf{Appendix. Derivation of the} \textbf{action for a }$4
\textbf{-dimensional brane embedded in the }$5$\textbf{-dimensional spacetim
}}

In this Appendix we briefly review the derivation of the action\textbf{\
\ref{31t'}). }In order to find it, we will first consider the following $5
-dimensional Randall-Sundrum \cite{randall} \cite{randall1} type metri
\begin{eqnarray}
ds^{2} &=&e^{2f(\phi )}\tilde{g}_{\mu \nu }(\tilde{x})d\tilde{x}^{\mu }
\tilde{x}^{\nu }+r_{c}^{2}d\phi ^{2},  \notag \\
&=&\eta _{ab}e^{a}e^{b},  \notag \\
&=&e^{2f(\phi )}\tilde{\eta}_{mn}\tilde{e}^{m}\tilde{e}^{n}+r_{c}^{2}d\phi
^{2},  \label{5t2}
\end{eqnarray
where $e^{2f(\phi )}$ is the so-called "warp factor", and $r_{c}$ is the
so-called "compactification radius" of the extra dimension, which is
associated with the coordinate $0\leqslant \phi <2\pi $. The symbol $\sim $
denotes $4$-dimensional quantities$.$ We will use the usual notatio
\begin{eqnarray}
x^{\alpha } &=&\left( \tilde{x}^{\mu },\phi \right) ;\text{ \ \ \ \ }\alpha
,\beta =0,...,4;\text{ \ \ \ \ }a,b=0,...,4;  \notag \\
\mu ,\nu &=&0,...,3;\text{ \ \ \ \ }m,n=0,...,3;  \notag \\
\eta _{ab} &=&diag(-1,1,1,1,1);\text{ \ \ \ \ }\tilde{\eta
_{mn}=diag(-1,1,1,1),  \label{6t2}
\end{eqnarray
which allows us to write the vielbein
\begin{equation}
e^{m}(\phi ,\tilde{x})=e^{f(\phi )}\tilde{e}^{m}(\tilde{x})=e^{f(\phi )
\tilde{e}_{\text{ }\mu }^{m}(\tilde{x})d\tilde{x}^{\mu };\text{ \ \
e^{4}(\phi )=r_{c}d\phi ,\text{\ }  \label{s22}
\end{equation
where\textbf{\ }$\tilde{e}^{m}$\textbf{\ }is the vierbein.

From the vanishing torsion conditio
\begin{equation}
T^{a}=de^{a}+\omega _{\text{ }b}^{a}e^{b}=0,  \label{2t}
\end{equation
we obtain the connections
\begin{equation}
\omega _{\text{ }b\alpha }^{a}=-e_{\text{ }b}^{\beta }\left( \partial
_{\alpha }e_{\text{ }\beta }^{a}-\Gamma _{\text{ }\alpha \beta }^{\gamma }e_
\text{ }\gamma }^{a}\right) ,  \label{3t}
\end{equation
where $\Gamma _{\text{ }\alpha \beta }^{\gamma }$ is the Christoffel symbol.

From Eqs. (\ref{s22}) and (\ref{2t}) we fin
\begin{equation}
\omega _{\text{ }4}^{m}=\frac{e^{f}f^{\prime }}{r_{c}}\tilde{e}^{m},
\label{102t}
\end{equation
and the $4$-dimensional vanishing torsion condition
\begin{equation}
\tilde{T}^{m}=\tilde{d}\tilde{e}^{m}+\tilde{\omega}_{\text{ }n}^{m}\tilde{e
^{n}=0,  \label{1030t}
\end{equation
where\textbf{\ \ }$f^{\prime }=\frac{\partial f}{\partial \phi }$\textbf{, }
\tilde{\omega}_{\text{ }n}^{m}=\omega _{\text{ }n}^{m}$\textbf{\ }and\textbf
\ }$\tilde{d}=d\tilde{x}^{\mu }\frac{\partial }{\partial \tilde{x}^{\mu }}.$

From (\ref{102t}), (\ref{1030t}) and the Cartan's second structural
equation, $R^{ab}=d\omega ^{ab}+\omega _{\text{ }c}^{a}\omega ^{cb}$, we
obtain the components of the $2$-form curvatur
\begin{equation}
R^{m4}=\frac{e^{f}}{r_{c}}\left( f^{\prime 2}+f^{\prime \prime }\right)
d\phi \tilde{e}^{m},\text{ \ }R^{mn}=\tilde{R}^{mn}-\left( \frac
e^{f}f^{\prime }}{r_{c}}\right) ^{2}\tilde{e}^{m}\tilde{e}^{n},\text{\ }
\label{105t}
\end{equation
where the $4$-dimensional $2$-form curvature is given b
\begin{equation}
\tilde{R}^{mn}=\tilde{d}\tilde{\omega}^{mn}+\tilde{\omega}_{\text{ }p}^{m
\tilde{\omega}^{pn}.
\end{equation}

The torsion-free condition implies that the third term in the EChS action,
given in equation (\ref{1t}), vanishes. This means that the corresponding
Lagrangian is no longer dependent on the field $k^{ab}$. So, the Lagrangian
\ref{1t}) has now two independent fields, $e^{a}$ and $h^{a}$, and it is
given by
\begin{equation}
L_{ChS}^{(5)}[e,h]=\alpha _{1}l^{2}\varepsilon
_{abcde}R^{ab}R^{cd}e^{e}+\alpha _{3}\varepsilon _{abcde}\left( \frac{2}{3
R^{ab}e^{c}e^{d}e^{e}+l^{2}R^{ab}R^{cd}h^{e}\right) .  \label{4t}
\end{equation}

From Eq. (\ref{4t}) we can see that the Lagrangian contains the Gauss-Bonnet
term $L_{GB}$, the Einstein-Hilbert term $L_{EH}$ and a term $L_{H}$ which
couples geometry and matter. In fact, replacing (\ref{s22}) and (\ref{105t})
in (\ref{4t}) and using $\tilde{\varepsilon}_{mnpq}=\varepsilon _{mnpq4}$,
we obtain

\begin{eqnarray}
\tilde{S}[\tilde{e},\tilde{h}] &=&\int_{\Sigma _{4}}\tilde{\varepsilon
_{mnpq}\left( A\tilde{R}^{mn}\tilde{e}^{p}\tilde{e}^{q}+B\ \tilde{e}^{m
\tilde{e}^{n}\tilde{e}^{p}\tilde{e}^{q}+\right.  \notag \\
&&\left. +C\tilde{R}^{mn}\tilde{e}^{p}\tilde{h}^{q}+E\tilde{e}^{m}\tilde{e
^{n}\tilde{e}^{p}\tilde{h}^{q}\right) ,  \label{999}
\end{eqnarray
wher
\begin{equation}
h^{m}(\phi ,\tilde{x})=e^{g(\phi )}\tilde{h}^{m}(\tilde{x}),\text{ \
h^{4}=0,  \label{o}
\end{equation
and\textbf{\
\begin{equation}
A=2r_{c}\int_{0}^{2\pi }d\phi e^{2f}\left[ \alpha _{3}-\frac{\alpha _{1}l^{2
}{r_{c}^{2}}\left( 3f^{\prime 2}+2f^{\prime \prime }\right) \right] ,
\label{12t}
\end{equation
\begin{equation}
B=-\frac{1}{r_{c}}\int_{0}^{2\pi }d\phi e^{4f}\left[ \frac{2\alpha _{3}}{3
\left( 5f^{\prime 2}+2f^{\prime \prime }\right) -\frac{\alpha _{1}l^{2}}
r_{c}^{2}}f^{\prime 2}\left( 5f^{\prime 2}+4f^{\prime \prime }\right) \right]
,  \label{13t}
\end{equation
\begin{equation}
C=-\frac{4\alpha _{3}l^{2}}{r_{c}}\int_{0}^{2\pi }d\phi e^{f}e^{g}\left(
f^{\prime 2}+f^{\prime \prime }\right) ,  \label{14t}
\end{equation
\begin{equation}
E=\frac{4\alpha _{3}l^{2}}{r_{c}^{3}}\int_{0}^{2\pi }d\phi
e^{3f}e^{g}f^{\prime 2}\left( f^{\prime 2}+f^{\prime \prime }\right) ,
\label{15t}
\end{equation
with\textbf{\ }$f(\phi )$\textbf{\ }and\textbf{\ }$g(\phi )$\textbf{\
representing functions that can be chosen (non-unique choice) as\textbf{\ }
f(\phi )=g(\phi )=ln(K+sin\phi )$\textbf{\ }with\textbf{\ }$K=constant>1
\textbf{; }and therefore we hav
\begin{equation}
A=\frac{2\pi }{r_{c}}\left[ \alpha _{3}r_{c}^{2}\left( 2K^{2}+1\right)
+\alpha _{1}l^{2}\right] ,  \label{201t}
\end{equation
\begin{equation}
B=\frac{\pi }{2r_{c}}\left[ \alpha _{3}\left( 4K^{2}+1\right) -\frac{\alpha
_{1}l^{2}}{2r_{c}^{2}}\right] ,  \label{21t}
\end{equation
\begin{equation}
C=-4r_{c}^{2}E=\frac{4\pi \alpha _{3}l^{2}}{r_{c}}.  \label{75t}
\end{equation
$\ \ \ \ \ \ \ $Taking into account that $L_{ChS}^{(5)}[e,h]$ flows into
L_{EH}^{(5)}$ when $l\longrightarrow 0$ \cite{irs}, we have that action (\re
{999}) should lead to the action of Einstein-Hilbert when $l\longrightarrow
0 $. From (\ref{999}) it is direct to see that this occurs when $A=-1/2$ and
$B=0$. In this case, from Eqs. (\ref{201t}), (\ref{21t}) and (\ref{75t}), we
can see tha
\begin{equation}
\alpha _{1}=-\frac{r_{c}\left( 4K^{2}+1\right) }{2\pi l^{2}\left(
10K^{2}+3\right) },  \label{24t}
\end{equation
\begin{equation}
\alpha _{3}=-\frac{1}{4\pi r_{c}\left( 10K^{2}+3\right) },  \label{25t}
\end{equation}

\begin{equation}
C=-4r_{c}^{2}E=-\frac{l^{2}}{r_{c}^{2}\left( 10K^{2}+3\right) },
\end{equation
and therefore the action (\ref{999}) takes the form
\begin{eqnarray}
\tilde{S}[\tilde{e},\tilde{h}] &=&\int_{\Sigma _{4}}\tilde{\varepsilon
_{mnpq}\left( -\frac{1}{2}\tilde{R}^{mn}\tilde{e}^{p}\tilde{e}^{q}\right. +
\notag \\
&&\left. +C\tilde{R}^{mn}\tilde{e}^{p}\tilde{h}^{q}-\frac{C}{4r_{c}^{2}
\tilde{e}^{m}\tilde{e}^{n}\tilde{e}^{p}\tilde{h}^{q}\right) ,  \label{a}
\end{eqnarray
corresponding to a $4$-dimensional brane embedded in the $5$-dimensional
spacetime of the EChS gravity. We can see that when $l\rightarrow 0$ then
C\rightarrow 0$ and hence (\ref{a}) becomes the $4$-dimensional
Einstein-Hilbert action.

Finally, it is convenient to express the action\textbf{\ }(\ref{a})\textbf{\
}in tensorial language. To achieve this, we write\textbf{\ }$\tilde{e}^{m}
\tilde{x})=\tilde{e}_{\text{ }\mu }^{m}(\tilde{x})d\tilde{x}^{\mu }$\textbf
\ }and\textbf{\ }$\tilde{h}^{m}=\tilde{h}_{\text{ \ }\mu }^{m}d\tilde{x
^{\mu }$\textbf{, }and then we compute the individual terms in\textbf{\ }
\ref{a})\textbf{\ }a
\begin{eqnarray}
\tilde{\varepsilon}_{mnpq}\tilde{R}^{mn}\tilde{e}^{p}\tilde{e}^{q} &=&-
\sqrt{-\tilde{g}}\tilde{R}d^{4}\tilde{x}, \\
\tilde{\varepsilon}_{mnpq}\tilde{R}^{mn}\tilde{e}^{p}\tilde{h}^{q} &=&2\sqrt
-\tilde{g}}\left( \tilde{R}\tilde{h}-2\tilde{R}_{\text{ }\nu }^{\mu }\tilde{
}_{\text{ }\mu }^{\nu }\right) d^{4}\tilde{x}, \\
\tilde{\varepsilon}_{mnpq}\tilde{e}^{m}\tilde{e}^{n}\tilde{e}^{p}\tilde{h
^{q} &=&6\sqrt{-\tilde{g}}\tilde{h}d^{4}\tilde{x},
\end{eqnarray
where it has been defined\textbf{\ }$\tilde{h}\equiv \tilde{h}_{\text{ }\mu
}^{\mu }$\textbf{. }So, the $4$-dimensional action for the brane immersed in
the $5$-dimensional space-time of the EChS gravitational theory is given by

\begin{equation}
\tilde{S}[\tilde{g},\tilde{h}]=\int d^{4}\tilde{x}\sqrt{-\tilde{g}}\left[
\tilde{R}+2C\left( \tilde{R}\tilde{h}-2\tilde{R}_{\text{ }\nu }^{\mu }\tilde
h}_{\text{ }\mu }^{\nu }\right) -\frac{3C}{2r_{c}^{2}}\tilde{h}\right] .
\end{equation}

\textbf{Acknowledgements}

This work was supported in part by\textit{\ }FONDECYT Grant\textit{\ }No
\textit{\ }1180681 from the Government of Chile. One of the authors (FG) was
supported by Grant \# R12/18 from Direcci\'{o}n de Investigaci\'{o}n,
Universidad de Los Lagos.