We construct a hierarchy of integrable systems whose Poisson structure corresponds to
the BMS
Article funded by SCOAP3
1$ have also been discussed along diverse lines in~\cite{Casali-ml-2016atr,Bagchi-ml-2016yyf,Banerjee-ml-2016nio,Lodato-ml-2016alv,Mandal-ml-2010gx,Banerjee-ml-2017gzj,Basu-ml-2017aqn,Fuentealba-ml-2017fck}. Interestingly, nonlinear extensions of the BMS$_{3}$ algebra are known to exist when higher spin bosonic or fermionic generators are included, see~\cite{Afshar-ml-2013vka,Gonzalez-ml-2013oaa,Gary-ml-2014ppa,Matulich-ml-2014hea,Gonzalez-ml-2014tba} and~\cite{Fuentealba-ml-2015jma,Fuentealba-ml-2015wza} respectively. Further generalizations of the BMS$_{3}$ algebra can also be found from suitable expansions of the Virasoro algebra~\cite{Caroca-ml-2017onr}. One of the main purposes of our work is exploring whether the BMS$_{3}$ algebra could be further linked with some sort of integrable systems. There are some hints that can be borrowed from CFTs in two dimensions that suggest to look towards this direction. Indeed, it is known that CFTs in 2D admit an infinite set of conserved charges that commute between themselves, which can be constructed out from suitable nonlinear combinations of the generators of the Virasoro algebra and their derivatives (see e.g.~\cite{DiFrancesco-ml-639405}). Remarkably, these composite operators turn out to be precisely the conserved charges of the KdV equation, which also correspond to the Hamiltonians of the KdV hierarchy. Therefore, since the BMS$_{3}$ algebra can be seen as a limiting case of the conformal algebra in 2D, it is natural to wonder about the possibility of performing a similar construction in that limit. Specifically, one would like to know about the existence of an infinite number of commuting conserved charges that could be suitably recovered from the BMS$_{3}$ generators, as well as its possible relation with some integrable system, or even to an entire hierarchy of them. Here we show that this is certainly the case. Furthermore, and noteworthy, the dynamics of this class of integrable systems can be equivalently understood in terms of Riemannian geometry. Indeed, following similar strategy as the one in~\cite{Perez-ml-2016vqo}, here we show that General Relativity without cosmological constant in 3D can be endowed with a suitable set of boundary conditions, so that in the reduced phase space, the Einstein equations precisely agree with the ones of the hierarchy aforementioned. In other words, the dynamics of our class of integrable system can be fully geometrized, since it can be seen to emerge from the evolution of spacelike surfaces embedded in locally flat spacetimes. As a remarkable consequence, in the geometric picture, the symmetries of the integrable systems correspond to diffeomorphisms that maintain the asymptotic form of the spacetime metric, so that they manifestly become Noetherian. Hence, the infinite set of conserved charges can be readily obtained from the surface integrals associated to the asymptotic symmetries in the canonical approach. In the next section we explicitly construct dynamical (Hamiltonian) systems whose Poisson structure corresponds to the BMS$_{3}$ algebra. In order to analyze their symmetries and conserved charges, in section~\ref{sec:Drinfeld-Sokolov-formulation} we show how the Drinfeld-Sokolov formulation can be adapted to our case, through the use of suitable flat connections for $isl(2,\mathbb{R})$. Section~\ref{sec:Hierarchy-of-integrable} is devoted to a thorough construction and the analysis of an entire bi-Hamiltonian hierarchy of integrable systems with BMS$_{3}$ Poisson structure, labeled by a nonnegative integer $k$. We start with a very simple dynamical system ($k=0$) from which the bi-Hamiltonian structure can be naturally unveiled. The case of $k=1$ is described in section~\ref{subsec:Integrable-bi-Hamiltonian-systemk=00003D00003D1}, where the Abelian infinite-dimensional symmetries and conserved charges are explicitly identified in terms of a suitable generalization of the Gelfand-Dikii polynomials (see also appendices~\ref{sec:Lists} and~\ref{sec:Involution-of-the}). The equivalence between our field equations and the ones of the Hirota-Satsuma coupled KdV system of type ix, is shown to hold by virtue of a suitable field redefinition and time scaling in section~\ref{subsec:Equivalence-with-Hirota-Satsuma}. The hierarchy of integrable systems with BMS$_{3}$ Poisson structure is then explicitly discussed in section~\ref{subsec:The-hierarchyk>1}, where it is shown that the so-called ``perturbed KdV equations'' are included as a particular case. Section~\ref{subsec:Analytic-solutions} is devoted to the construction of a wide interesting class of analytic solutions for a generic value of the label of the hierarchy $k$. In section~\ref{sec:Geometrization-of-the}, we show how the dynamics of the hierarchy of integrable systems can be fully geometrized in terms of locally flat three-dimensional spacetimes. The deep link with General Relativity in 3D is explicitly addressed. We conclude with some remarks about possible extensions of our results in section~\ref{sec:Extensions}. Appendices~\ref{sec:Lists},~\ref{sec:Involution-of-the},~\ref{sec:Verification} and~\ref{sec:Proof} are devoted to some technical remarks. ]]>
1}. In sum, the analysis of this extremely simple dynamical system with BMS$_{3}$ Poisson structure $\mathcal{D}^{(2)}$, being trivially integrable, allows to unveil a naturally related Poisson structure given by $\mathcal{D}^{(1)}$. The presence of both Poisson structures turns out to be the key in order to proceed with the construction of nontrivial integrable systems as well as an entire hierarchy associated to them. This can be seen as follows. One begins verifying that both Poisson structures are ``compatible'' in the sense that any linear combination of $\mathcal{D}^{(2)}$ and $\mathcal{D}^{(1)}$ also defines a Poisson structure. It is then enough proving that the operator $\mathcal{D}^{(3)}=\mathcal{D}^{(2)}+\mathcal{D}^{(1)}$ also defines a Poisson structure (see e.g.~\cite{olver2000applications}). This is so because the Poisson bracket constructed out from $\mathcal{D}^{(3)}$ is clearly antisymmetric, and furthermore, since $\mathcal{D}^{(3)}$ also fulfills the BMS$_{3}$ algebra, but just being shifted by the zero modes of $\mathcal{P}_{0}\rightarrow\mathcal{P}_{0}+\pi$, the Jacobi identity also holds. Besides, by virtue of the fact that our Poisson structure $\mathcal{D}^{(1)}$ is nondegenerate, the hypotheses of a strong theorem for bi-Hamiltonian systems in~\cite{Gelfand-Dorfman}, and further elaborated in~\cite{olver2000applications}, are satisfied, which guarantees the existence of the type of hierarchy of integrable systems that we are searching for. Furthermore, and remarkably, the simple dynamical system described in this section can be seen to be equivalent to the Einstein equations for the reduced phase space that is obtained from a suitable set of boundary conditions for General Relativity in three spacetime dimensions, including its extension with purely geometrical parity-odd terms in the action. This is discussed in section~\ref{sec:Geometrization-of-the}. It is also worth pointing out that our field equations~\eqref{eq:sol-k=00003D00003D0}, in the case of $c_{\mathcal{J}}=a=0$, can be interpreted as the ones of a compressible Euler fluid~\cite{Penna-ml-2017vms}. In the next subsection we carry out the explicit construction and the analysis of a simple, but nontrivial, integrable system with BMS$_{3}$ Poisson structure. \boldmath ]]>
1}, this provides the basis to extend this integrable system to an entire hierarchy. ]]>
1$, not only the field equations in~\eqref{eq:EoMJPk}, but also the consistency condition for the functions $\varepsilon_{\mathcal{J}}$ and $\varepsilon_{\mathcal{P}}$ that parametrize the symmetries in~\eqref{eq:epsilon-punto} become severely more complicated than the simplest cases of $k=0,1$ (see eqs.~\eqref{eq:EoMJP0},~\eqref{eq:epsilonJP0}, and~\eqref{eq:EoMJP1},~\eqref{eq:epsilonJP1}, respectively). However, and remarkably, when $\varepsilon_{\mathcal{J}}$ and $\varepsilon_{\mathcal{P}}$ are assumed to depend only on the dynamical fields and their spatial derivatives, but not explicitly on $t$, $\phi$, by virtue of the theorem in~\cite{Gelfand-Dorfman,olver2000applications}, the general solution of the consistency condition for the parameters in~\eqref{eq:epsilon-punto} for $k\geq0$ turns out to be precisely given by the same expansion in terms of the generalized Gelfand-Dikii polynomials $K^{(j)}$ and $\tilde{K}^{(j)}$, as in~\eqref{eq:sum-polynomials} for $k=1$. This is explicitly verified in appendix~\ref{sec:Verification}. Therefore, as a consequence, the corresponding canonical generators turn out to be given by the two independent sets of conserved charges given by~\eqref{eq:Q-1}, which are in involution, i.e., the commuting charges fulfill~\eqref{eq:involution} for both Poisson structures. Nevertheless, depending on the choice of Poisson structure, $\mathcal{D}^{(2)}$ or $\mathcal{D}^{(1)}$, the energy of the system in~\eqref{eq:Energy}, now corresponds to the Hamiltonian, $H^{(k)}$ or $H^{(k+1)}$, defined through~\eqref{eq:Hk}, respectively. Furthermore, by construction, the field equations for any representative of the hierarchy turn out to be invariant under anisotropic scaling transformations given by \begin{equation} t\rightarrow\sigma^{z}t\,,\quad\quad\phi\rightarrow\sigma\phi\,,\quad\quad\left(\begin{array}{c} \mathcal{J}\\ \mathcal{P} \end{array}\right)\rightarrow\sigma^{-2}\left(\begin{array}{c} \mathcal{J}\\ \mathcal{P} \end{array}\right)\,,\label{eq:Lifshitz-z-scaling} \end{equation} which is of Lifshitz type, and characterized by a dynamical exponent $z=2k+1$. Isotropic scaling then only holds for $k=0$. In terms of the BMS$_{3}$ Poisson structure $\mathcal{D}^{(2)}$, the parameters $\varepsilon_{\mathcal{J}}$, $\varepsilon_{\mathcal{P}}$ that correspond to the infinitesimal anisotropic scaling transformation in~\eqref{eq:Lifshitz-z-scaling}, are then given by \begin{equation} \left(\begin{array}{c} \varepsilon_{\mathcal{J}}\\ \varepsilon_{\mathcal{P}} \end{array}\right)=\lambda zt\left(\tilde{K}^{(k)}+aK^{(k)}\right)+\lambda\phi\tilde{K}^{(0)}\,, \end{equation} and fulfill the consistency conditions in~\eqref{eq:epsilon-punto}. The corresponding conserved charge can then be obtained from $\varepsilon_{\mathcal{J}}=\lambda\frac{\delta D}{\delta\mathcal{J}}$ and $\varepsilon_{\mathcal{P}}=\lambda\frac{\delta D}{\delta\mathcal{P}}$, with \begin{equation} D=ztH^{\left(k\right)}+\int d\phi\,\phi\mathcal{J}\,,\label{eq:D-Lifshitz-k} \end{equation} where $H^{\left(k\right)}$ stands for the Hamiltonian, given by eq.~\eqref{eq:Hk}. As an ending remark of this subsection, it is worth noting that the field equations of the so-called ``perturbed KdV hierarchy'' (see e.g.,~\cite{bib0305-4470-16-16-015,MA199649}) are precisely recovered from~\eqref{eq:EoMJPk} for the particular case of $c_{\mathcal{J}}=a=0$. In this case, for the entire hierarchy, according to~\eqref{eq:Hk}, configurations with ${\cal J}=0$ do not carry energy, which goes by hand with the fact that the Hamiltonian corresponds to the energy of a perturbation described by ${\cal J}$. Indeed, for this class of configurations, the entire set of conserved charges $\tilde{H}^{(j)}$ in~\eqref{eq:Htilde} also vanishes. Note that the remaining ones, given by $H_{\text{KdV}}^{\left(j\right)}$ generically remain nontrivial, which can be interpreted as the conserved charges associated to an arbitrary ``background \mbox{configuration}'' described by ${\cal P}={\cal P}\left(t,\phi\right)$ that solves the field equations of the $k$-th representative of the KdV hierarchy. ]]>
1}. Therefore, in this case, the variation of the surface integrals in~\eqref{eq:deltaqg} integrates precisely as in~\eqref{eq:Q-1}. Note that the energy of a gravitational configuration that fulfills these boundary conditions is then given by the Hamiltonian of the corresponding integrable system as in~\eqref{eq:Energy}, i.e., $E=Q[\partial_{t}]=H^{\left(k\right)}$. In the simplest case of $k=0$, described in section~\ref{subsec:Warming-up-with}, we recover the set of boundary conditions proposed in~\cite{Matulich-ml-2014hea}\footnote{Strictly speaking, we are dealing with the boundary conditions in~\cite{Matulich-ml-2014hea} provided that the higher spin fields and their corresponding chemical potentials are turned-off.} (see also~\cite{Gary-ml-2014ppa}) which contain the boundary conditions in~\cite{Barnich-ml-2006av} for a particular choice of Lagrange multipliers at the boundary. Note that in this case, the BMS$_{3}$ algebra is realized as the asymptotic symmetry algebra. For the remaining cases ($k\geq1$), the new class of boundary conditions is such that the asymptotic symmetry algebra is infinite-dimensional, abelian and devoid of central charges, which is equivalent to the fact that the conserved charges of the hierarchy are in involution (see eq.~\eqref{eq:involution}). It is worth pointing out that, in the metric formalism, our particular choices for the Lagrange multipliers $\mu_{\mathcal{J}}$ and $\mu_{\mathcal{P}}$ correspond to fixing the lapse and shift functions in the ADM decomposition of the metric, as suitable local functionals of the dynamical variables at the boundary. Furthermore, and remarkably, the class of locally flat spacetimes described by this new set of asymptotic conditions inherits the anisotropic Lifshitz scaling of the corresponding integrable systems by virtue of the boundary conditions. This effect then becomes the flat analogue of the one found in~\cite{Perez-ml-2016vqo} for the case of locally AdS$_{3}$ spacetimes, where isotropy is recovered in the particular case of the Brown-Henneaux boundary conditions~\cite{Brown-ml-1986nw}. It is also amusing to verify that trivial solutions of the class of integrable systems described above, like the ones just given by configurations with ${\cal J}$ and ${\cal P}$ constants, actually correspond to the geometries of the flat cosmological spacetimes in~\cite{Cornalba-ml-2002fi}. Hence, from a gravitational point of view, these configurations become certainly non-trivial, not only in a geometric sense, but also in the physical one since they carry Hawking temperature and entropy associated to the cosmological horizon. It would then be worth to explore additional simple configurations for the integrable systems that could naturally become non-trivial in the gravitational framework and vice versa. The results of this section are also related to some interesting issues that will be discussed in a forthcoming work, including the precise details of the asymptotic structure in the metric formulation, as well as the microscopic entropy of flat cosmological spacetimes once they are embedded within the new set of boundary conditions. An intriguing link with the recent results in refs.~\cite{Afshar-ml-2016kjj,Grumiller-ml-2017jft} concerning ``soft hair'' in the sense of~\cite{Hawking-ml-2016msc,Hawking-ml-2016sgy,Strominger-ml-2017aeh} for asymptotically flat spacetimes in 3D, can also be established once the hierarchy of integrable systems with BMS$_{3}$ Poisson structure presented here is suitably extended to the case of $z=0$ in a form akin to the one performed in~\cite{Perez-ml-2016vqo} for the KdV hierarchy. It can also be shown that the geometrization of the class of integrable systems discussed here, but in the generic case with $c_{\mathcal{J}}\neq0$, can be performed through a suitable extension of the analysis in~\cite{Barnich-ml-2014cwa,Fuentealba-ml-2015wza} once suitable parity odd terms in the action are included (see also~\cite{Giacomini-ml-2006dr}). ]]>
2$ was found in~\cite{Gonzalez-ml-2013oaa} (in full agreement with the algebra simultaneously found in~\cite{Afshar-ml-2013vka} for $s=3$). This kind of extensions can be regarded as ``flat $W$-algebras'', since they can be recovered from a suitable In\"on\"u-Wigner contraction of two copies of certain classical $W$-algebras (for a review about $W$-algebras, see e.g.~\cite{Bouwknegt-ml-1992wg}). Noteworthy, preliminary results~\cite{W-Flat-IntegrableSystems} show that new hierarchies of integrable systems whose Poisson structures correspond to flat $W$-algebras indeed exist. In fact, this family of integrable systems turns out to be bi-Hamiltonian, and furthermore, following the lines of section~\ref{sec:Geometrization-of-the}, they can also be geometrized in terms of higher spin gravity without cosmological constant in three spacetime dimensions endowed with a suitable set of boundary conditions. Therefore, the symmetries of this novel class of integrable systems can be seen as combinations of diffeomorphisms and higher spin gauge transformations that preserve the asymptotic form of the three-dimensional configurations. As a consequence, in the three-dimensional geometric setup, the infinite set of conserved charges of the integrable systems emerge as the canonical generators that correspond to the asymptotic symmetries, being described by suitable surface integrals at the boundary, which turn out to be in involution. Further interesting links between certain well-known classes of integrable systems and higher spin gravity on AdS$_{3}$ have been explored in~\cite{Compere-ml-2013gja,Gutperle-ml-2014aja,Beccaria-ml-2015iwa,Perez-ml-2016vqo,Gutperle-ml-2017ewo}. ]]>
n$. Thus, the Poisson bracket associated to the BMS$_{3}$ operator $\mathcal{D}^{(2)}$ of two conserved charges, which reads \begin{equation} \lbrace\bar{H}^{(m)},\bar{H}^{(n)}\rbrace_{(2)}=\int d\phi\left(\begin{array}{cc} \frac{\delta\bar{H}^{(m)}}{\delta\mathcal{J}} & \frac{\delta\bar{H}^{(m)}}{\delta\mathcal{P}}\end{array}\right)\mathcal{D}^{\left(2\right)}\left(\begin{array}{c} \frac{\delta\bar{H}^{(n)}}{\delta\mathcal{J}}\\ \frac{\delta\bar{H}^{(n)}}{\delta\mathcal{P}} \end{array}\right)\,, \end{equation} by virtue of the recursion relation in~\eqref{eq:recrelation}, can be written in terms of the Poisson bracket associated to the ``canonical'' operator $\mathcal{D}^{(1)}$, according to \begin{align} \lbrace\bar{H}^{(m)},\bar{H}^{(n)}\rbrace_{(2)} & = \int d\phi\left(\begin{array}{cc} \frac{\delta\bar{H}^{(m)}}{\delta\mathcal{J}} & \frac{\delta\bar{H}^{(m)}}{\delta\mathcal{P}}\end{array}\right)\mathcal{D}^{\left(1\right)}\left(\begin{array}{c} \frac{\delta\bar{H}^{(n+1)}}{\delta\mathcal{J}}\\ \frac{\delta\bar{H}^{(n+1)}}{\delta\mathcal{P}} \end{array}\right)\nonumber \\ & = \lbrace\bar{H}^{(m)},\bar{H}^{(n+1)}\rbrace_{(1)}\\ & = -\lbrace\bar{H}^{(n+1)},\bar{H}^{(m)}\rbrace_{(1)}\,.\nonumber \end{align} Analogously, making use of the recursion relationship again, one finds that \begin{equation} \lbrace\bar{H}^{(m)},\bar{H}^{(n)}\rbrace_{(2)}=-\lbrace\bar{H}^{(n+1)},\bar{H}^{(m)}\rbrace_{(1)}=\lbrace\bar{H}^{(m-1)},\bar{H}^{(n+1)}\rbrace_{(2)}\,. \end{equation} Therefore, once the procedure is applied $m-n$ times, one obtains \begin{equation} \lbrace\bar{H}^{(m)},\bar{H}^{(n)}\rbrace_{(2)}=\lbrace\bar{H}^{(n)},\bar{H}^{(m)}\rbrace_{(2)}\,, \end{equation} which implies that the conserved charges are involution in both Poisson brackets, i.e., \begin{equation} \lbrace\bar{H}^{(m)},\bar{H}^{(n)}\rbrace_{(2)}=\lbrace\bar{H}^{(m)},\bar{H}^{(n)}\rbrace_{(1)}=0\,. \end{equation} ]]>