text:
\section*{Abstract}
	{\bf

		We propose nonabelian higher-rank gauge theories in 2+1D and 3+1D.  The gauge group is constructed from the volume-preserving diffeomorphisms of space. We show that the intriguing physics of the lowest Landau level (LLL) limit can be interpreted as the consequences of the symmetry. We derive the renowned Girvin-MacDonald-Platzman (GMP) algebra  as well as the topological Wen-Zee term within our formalism. Using the gauge symmetry in 2+1D, we derive the LLL effective action  of vortex crystal in rotating Bose gas as well as Wigner crystal of electron in an applied magnetic field. We show that the nonlinear sigma models of ferromagnets in 2+1D and 3+1D  exhibit the higher-rank gauge symmetries that we introduce in this paper. We interpret the fractonic behavior of the excitations on the lowest Landau level and of skyrmions in ferromagnets as the consequence of the higher-rank gauge symmetry.

	}

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\section{Introduction}

Recently, considerable interest has been drawn to ``higher-rank gauge
theories,'' i.e., theories where the gauge potential is not a one-form,
but a tensor of higher rank~\cite{Pretko:2016kxt,higherrank1,higherrank2,higherrank3,Zhai:2020kpj,higherrank5,higherrank6,higherrank7,higher8,Seiberg1,Seiberg2,Fractonreview1,Gromov:Multipole,Gromov:ChiralEalstic}.  The physical motivation of the higher-rank gauge theories is the discovery of a new class of topological matter known as \textit{fractons}~\cite{Chamon2005,Bravyi2011,Haah2011,Vijay:2015mka,Pretko:2020cko},
one feature of which is the existence of excitations with restricted mobility. The excitations either cannot move at all or can only move in lower-dimensional sub-spaces, while composites of elementary excitations can move freely. The restricted mobility of the fractonic excitations can be interpreted as the consequence of the higher-rank theories' conservation laws. In particular, the tensor Gauss's
law leads to the conservation of not only the electric charge but
also the electric dipole moment and, in some cases, higher moments of the
charge distribution \cite{Pretko:2016kxt,higherrank1,Pretko:2020cko}.  This has the physical effect of rendering the
electric charge immobile but leaving the dipoles mobile or partially
mobile.  One representative example is the so-called ``traceless
scalar charge theory,'' in which the electric charges are immobile and
the electric dipole can move only in the direction perpendicular to
the dipole moment \cite{higherrank1}.   The
higher-rank gauge theories have also been applied to describe defects
in solids~\cite{PretkoRadzihovsky:2018,Cvetkovic:2006,Beekman:2017,kleinert1982duality,kleinert1983dual},
supersolids~\cite{Pretko:2018tit}, superfluid
vortices~\cite{Doshi:2020jso,Nguyen:2020yve},
smectics~\cite{Nguyen:2020yve,Zhai:2020kpj,Radzihovsky:smectic}, and other systems.

All the higher-rank gauge symmetries in the physical models mentioned above are abelian.
There were attempts to
generalize these symmetries to nonabelian symmetries \cite{ higherrank5, Wang:nonabelian, Juven:non-abelian1, Juven:non-abelian2}; however, physical systems that realize those symmetries have not been explicitly proposed.  In this paper,
we propose nonabelian higher-rank symmetry theories for condensed matter systems of physical relevance. We reformulate the tensor gauge transformation in the traceless scalar charge theory in 2+1D \cite{Pretko:2016kxt,higherrank1}, showing that the gauge transformation is nothing but the volume-preserving diffeomorphism (VPD) in the linearized form.  We then construct nonlinear higher-rank gauge theories with VPDs as the symmetry group.  The nonabelian
nature of the gauge symmetry has a nontrivial consequence: the
 charge density operators at different positions do not commute.
Instead, they form the long-wavelength limit of the Girvin-MacDonald-Platzman (GMP)
algebra~\cite{Girvin:1986zz,Cappelli:1992yv,Iso:1992aa}, which
 reveals a connection to the lowest
Landau level (LLL).

It is easy to notice  that many features of the
physics on the LLL bear a close
resemblance to that of the field-theory models with higher rank
symmetries \cite{Cappelli:Higherspin}: for example,
electric charges are pinned to one place by the
large magnetic field, and neutral excitations (e.g., the
composite fermion in the half-filled Landau level~\cite{PH-dipole})
carry an electric dipole moment and can move in the direction
perpendicular to the direction of motion.
In this paper, we show that this resemblance is
not accidental; in fact, the nonlinear higher-rank symmetry is
realized as a symmetry of the problem of charged particles on the
LLL.  The relation between the dipole moment and momentum of an excitation in models with higher-rank symmetry was noticed in Refs.~\cite{Gromov:Multipole,Doshi:2020jso}.

We also will present several physical systems that enjoy the nonabelian higher-rank symmetry. In 2+1D systems, the symmetry originates from the lowest-Landau-level limit, where the volume-preserving nature of the diffeomorphisms comes from the restriction that the transformations should preserve the background magnetic field. In addition to the derivation of the GMP algebra, we draw a connection between the topological
Wen-Zee term \cite{Wen:1992ej} with the Chern-Simons term in a higher-rank gauge theory.

Furthermore, we will use the gauge symmetry to derive the effective theories of the Wigner crystal of electrons in a strong magnetic field and the vortex crystal in a rotating Bose gas.

Finally, we find that the nonlinear sigma models describing ferromagnetism in 2+1D and 3+1D also exhibit the global higher-rank symmetry.  The higher rank gauge symmetry provides a new interpretation of the conservation of multipole moments in ferromagnets~\cite{Papanicolaou:1990sg}; it also explains the close resemblance between the behaviors of skyrmions in ferromagnets and charged particles in a magnetic field~\cite{Stone:1996}.

\section{Review of the traceless scalar charge theory}
For the paper to be self-contained, in this Section we will review the symmetric tensor gauge theory proposed by Pretko~\cite{Pretko:2016kxt,higherrank1}.
We consider a higher-rank gauge theory called ``traceless scalar charge
theory,'' where the gauge potential is a symmetric rank-2 tensor $A_{ij}$: $A_{ij}=A_{ji}$.  Its conjugate
momentum is the electric field $E_{ij}$.  They satisfy the canonical
commutation relation
\begin{equation}\label{Pretko-comm}
  [E_{ij}(\x), \, A_{kl}(\y)] =  i(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}) \delta(\x-\y).
\end{equation}
One imposes the Gauss law and the traceless constraint:
\begin{align}
\label{eq:constr1}
  \d_i \d_j E_{ij} &= \rho,\\
  \label{eq:constr2}
  E\equiv E_{ii} &= 0,
\end{align}
which lead to the conservation of the following charges:
\begin{equation}\label{conserved-quant}
  \int\!d\x\, \rho(\x), \quad \int\!d\x\, \x\rho(\x), \quad
  \int\!d\x\, x^2\rho(\x).
\end{equation}
The conservation of the quantities listed in in Eq. \eqref{conserved-quant} imply that a charge cannot move, and a dipole can move only
along the direction perpendicular to the dipole moment~\cite{Pretko:2016kxt}.

The constraints \eqref{eq:constr1} and \eqref{eq:constr2} generate the gauge transformations on the gauge potential
\begin{align}
  A_{ij} &\to A_{ij} + \d_i \d_j \lambda , \\
  A_{ij} &\to A_{ij} + \delta_{ij}\mu . \label{gauge-trace}
\end{align}

For our purpose, it is convenient to use the second gauge
transformation~(\ref{gauge-trace}) to explicitly fix the gauge $A\equiv
A_{ii}=0$ and eliminate the trace of $A_{ij}$ from the set of dynamical
degrees of freedom.  Since the two constraints $E=0$ and $A=0$ do not
commute according to the commutation relation~(\ref{Pretko-comm}), following
Dirac one should modify the commutators, replacing them by
the Dirac brackets~\cite{Dirac:1950}, which in our case is
\begin{equation}
  [O_1, \, O_2] \to [O_1,\, O_2]_{\text{D}} = [O_1,\, O_2]
  + [O_1,\, E] [E,\, A]^{-1} [A,\, O_2]
  - [O_1,\, A] [E,\, A]^{-1} [E,\, O_2]
\end{equation}
The new commutator is then
\begin{equation}
  [E_{ij}(\x),\, A_{kl}(\y)] = i\left(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}
  - \frac2d \delta_{ij}\delta_{kl}\right) \delta(\x-\y).
\end{equation}
The Gauss constraint $\d_i\d_j E_{ij}=\rho$ generates now the gauge
transformation
\begin{equation}\label{Aij-gauge}
  A_{ij} \to A_{ij} + \d_i\d_j \lambda - \frac1d \delta_{ij}\d^2\lambda,
  \qquad \d^2\equiv \d^k\d_k .
\end{equation}

To construct a gauge-invariant Lagrangian, we introduce the field
strengths.  The electric field
\begin{equation}
  E_{ij} =  \d_i\d_j A_0 - \frac1d\delta_{ij}\d^2 A_0 - \d_t A_{ij} ,
\end{equation}
is obviously gauge invariant.  One notices that
\begin{equation}
  \omega_i= - \d_j A_{ij}
\end{equation}
transforms like a $U(1)$ vector potential,
\begin{equation}
\label{eq:omegatran}
  \omega_i \to \omega_i - \d_i\tilde\lambda, \qquad
  \tilde{\lambda}= \left( 1-\frac 1d \right)\partial^2 \lambda ,
\end{equation}
using which one can define the magnetic field,
\begin{equation}
  H_{ij} = \d_i \omega_j - \d_j \omega_i
  = - \d_i \d_k A_{jk} + \d_j \d_k A_{ik} ,
\end{equation}
that is manifestly gauge invariant.
The simplest Lagrangian for the gauge field is then the ``Maxwell theory,''
\begin{equation}
  L = c_1 E_{ij}^2 - c_2 H_{ij} H_{ij} .
\end{equation}
However, as noticed in
Ref.~\cite{Pretko:2017xar}, in (2+1)D, another possible term in the
Lagrangian is the Chern-Simons term which, up to an overall coefficient,
reads
\begin{equation}\label{LCS}
  L_{\rm CS} = \vareps^{ij} (A_0 H_{ij} - A_{ik} \dot A_{jk}).
\end{equation}
As for a Chern-Simons term, the Lagrangian density is gauge-invariant
only up to a total derivative.  This term is more relevant than the
Maxwell term. The higher-rank Chern-Simons theory in 3 spatial dimensions was also discussed previously in Ref.~\cite{You:CShigher}.

Note that one does not have to require the theory to contain dynamical
gauge fields in order to have the conserved quantities~(\ref{conserved-quant}). A theory coupled to background
gauge fields $(A_0, A_{ij})$ and is invariant under the gauge symmetry
under which the gauge fields transform as
\begin{equation}\label{phi-gauge}
  A_0\to A_0+\dot\lambda, \qquad
  A_{ij} + \d_i\d_j \lambda - \frac1d \delta_{ij}\d^2\lambda,
\end{equation}
will have the conservation law
\begin{equation}
\label{eq:cont}
  \d_t \rho - \d_i \d_j J_{ij} = 0,
\end{equation}
where $\rho$ and $J_{ij}$ are the operators that couple to $A_0$ and
$A_{ij}$, respectively, and $J_{ii}=0$.  This is sufficient to derive
the conservation of the quantities~(\ref{conserved-quant}).  In fact,
in most of this paper, we will consider theories coupled to
nondynamical background gauge fields.

\section{Nonlinear higher-ranked symmetry}
\label{sec:nonlinear}
\subsection{Traceless scalar charge theory in (2+1)D as a theory of linearized gravity}

We now show that the theory presented in the previous section can be
interpreted as a theory of linearized gravity, and the higher-rank
symmetry is the linearized version of VPD.
Instead of $A_{ij}$ we introduce an equivalent field $h_{ij}$ defined as
\begin{equation}
  h_{ij} = - \ell^2 (\vareps_{ik} A_{jk} + \vareps_{jk} A_{ik}),
\end{equation}
where $\ell$ is some constant of the dimension of length\footnote{We
assume $A_0$ has dimension 1 and $A_{ij}$ is of dimension 2, so
$h_{ij}$ is dimensionless.}.  Note that $h_{ij}$ is also symmetric and
traceless.   The
gauge transformation for $h_{ij}$ is inherited from \eqref{phi-gauge}
\begin{equation}
  h_{ij} \to h_{ij} - \ell^2 (\vareps_{ik}\d_j \d_k + \vareps_{jk}\d_i\d_k)\lambda .
\end{equation}
If we define
\begin{equation}
\label{xilambda}
  \xi^i = \ell^2 \vareps^{ik}\d_k\lambda ,
\end{equation}
then the transformation law of $h_{ij}$ can be reformulated as
\begin{equation}
\label{eq:metrictrans}
  h_{ij} \to h_{ij} - \d_i \xi_j - \d_j \xi_i .
\end{equation}
The transformation \eqref{eq:metrictrans} is nothing but the transformation of the metric
under the volume-preserving (or in 2D, area-preserving) diffeomorphism $x^i\to x^i + \xi^i$,  since $\d_i\xi^i=0$ due to  the definition \eqref{xilambda}. The connection between a higher-rank gauge theory and a linearized gravity theory was proposed previously in Ref.~\cite{xu2006novel}.

\subsection{Nonlinear higher-rank symmetry}
\label{sec:nonlinearizing}

The fact that the gauge symmetry resembles the transformation law of a
metric under VPDs allows one to devise a
nonlinear version of the gauge symmetry.  Namely, in our nonlinear
theory, instead of a gauge field $h_{ij}$ (or $A_{ij}$ for that
matter), the degree of freedom is the metric $g_{ij}$.  The
tracelessness of $h_{ij}$ translates into the statement that the
metric is unimodular: $\det g=1$.  The linear theory is restored when one
expands the metric around the flat metric: $g_{ij} = \delta_{ij} +
h_{ij} + O(h^2)$.

Under an infinitesimal VPD $x^i\to
x^i+\xi^i=x^i+\ell^2\vareps^{ij}\d_j\lambda$, the metric transforms as
\begin{equation}\label{g-vpd}
  \delta_\lambda g_{ij} = -\xi^k \d_k g_{ij} - g_{kj}\d_i\xi^k - g_{ik}\d_j\xi^k
  = -\ell^2 \vareps^{kl} (\d_k g_{ij}  + g_{kj}\d_i
    + g_{ik} \d_j) \d_l\lambda .
\end{equation}
Now we need to write down the nonlinear version of the transformation
laws for $A_0$.  One notices that the VPDs do not commute: from
Eq.~(\ref{g-vpd}) one reads
\begin{equation}\label{delta-comm}
  [\delta_\alpha, \delta_\beta] = \delta_{[\alpha,\beta]} ,
\end{equation}
with $[\alpha,\beta]=\ell^2\vareps^{ij}\d_i\alpha\d_j\beta$.
This means that our gauge symmetry is nonabelian;
in this paper, we will use ``nonlinear'' and ``nonabelian'' interchangeably.
The transformation of $A_0$ must satisfy the commutation relation~(\ref{delta-comm}).  One
can check that this can be accomplished by the following simple
modification

\begin{equation}\label{phi-vpd}
  \delta_\lambda A_0 = \d_t\lambda - \xi^k \d_k A_0
    = \d_t \lambda - \ell^2\vareps^{kl} \d_k A_0\, \d_l \lambda .
\end{equation}

The transformation of  $A_0$ in Eq. \eqref{phi-vpd} was motivated by the symmetries of the lowest Landau level that will be discussed subsequently. Nonetheless, it is the unique nonlinear generalization of \eqref{phi-gauge}  given that the transformation is at most linear in $A_0$ and respects rotational invariance. Furthermore, the second term of \eqref{phi-vpd} means that $A_0$ transforms as a scalar field under time-independent spatial diffeomorphism, which is expected.
We leave the detailed discussion on the uniqueness of the nonlinear transformation \eqref{phi-vpd} to Appendix \ref{sec:A0}.
Equations~(\ref{phi-vpd}) and (\ref{g-vpd}) give the transformation
laws of a nonlinear higher-rank symmetry, which we collect here for
convenience:
\begin{align}\label{A0gij-vpd}
  \delta_\lambda A_0
   &= \d_t \lambda - \ell^2\vareps^{kl} \d_k A_0\, \d_l \lambda , \\
   \label{A0gij1-vpd}
  \delta_\lambda g_{ij} &=
  -\ell^2 \vareps^{kl} (\d_k g_{ij}  + g_{kj}\d_i
    + g_{ik} \d_j) \d_l\lambda .
\end{align}
A nonlinear transformation similar to Eq.~\eqref{A0gij1-vpd} was considered in Ref.~\cite{Gromov:ChiralEalstic} within a dynamical gauge model of the traceless scalar charge theory.
One can derive the Ward identity from Eqs.~(\ref{A0gij-vpd}) and \eqref{A0gij1-vpd}.  Let us
define the charge density $\rho$ and the stress tensor $T^{ij}$ by
varying the logarithm of the partition function
\begin{equation}
  \delta \ln Z = \int\!d^3x
  \left( \rho\delta A_0 + \frac12 T^{ij}\delta h_{ij} \right).
\end{equation}
The Ward identity is then
\begin{equation}
  \dot\rho - \ell^2 \vareps^{kl}
  \d_l \left[\rho \d_k A_0 + \tfrac12 T^{ij}\d_k g_{ij}
    + \d_i (T^{ij} g_{jk}) \right] = 0
\end{equation}
In the presence of the background field, only the total charge is
conserved, but not the higher multipoles in~(\ref{conserved-quant}).

Since the total charge is conserved, it is possible to introduce a
vector potential $A_i$ so that the theory is invariant under the usual
$U(1)$ gauge symmetry $A_\mu\to A_\mu+\d_\mu\alpha$.  In this case
$A_0$ plays a double role: it is the
temporal component of a U(1) gauge field $(A_0, A_i)$, and also as the
scalar component of the gauge potential of a higher-spin symmetry,
$(A_0, g_{ij})$.  The two sets of gauge potentials share one scalar
component, see Fig.~\ref{fig:venn-diagram1}.  We will see an example
when we consider ferromagnets (Sec.~\ref{sec:ferro}).
\begin{figure}[ht]
 \centering
  \includegraphics[width=0.3 \textwidth]{venn-diagram1.pdf}
 \caption{$A_0$ is shared by two sets of gauge potentials.}
  \label{fig:venn-diagram1}
\end{figure}

The nonabelian nature of the gauge symmetry, in some cases, allows us
to derive the algebra satisfied by the charge density of the matter
coupled to the gauge field.  Imagine that the action
$S_{\text{m}}(\psi, A_0, g_{ij})$ describing the coupling of the
matter fields $\psi$ with the gauge fields $(A_0, g_{ij})$ does not
contain the time derivatives of any fields, $A_0$, $g_{ij}$, or $\psi$.  In this case, if one
promotes the gauge fields to dynamical fields by adding to the action
a pure gauge action $S_{\text{g}}$,
\begin{equation}
  S = S_{\text{g}}[A_0, g_{ij}] + S_{\text{m}}[\psi,A_0,g_{ij}],
\end{equation}
then upon quantization, the canonical commutation relations in the
gauge sector are set by $S_{\text{g}}$ and in the matter sector by
$S_{\text{m}}$. The left-hand side of the Gauss constraint,
\begin{equation}
  \frac{\delta S_{\text{g}}}{\delta A_0} + \frac{\delta
  S_{\text{m}}}{\delta A_0} = 0,
\end{equation}
is the generator that generates gauge transformations.
In particular, in the matter sector,
the commutator of the charge
density $\rho(\x)$ with any matter field $O(\x)$ will give the change
of $O$ under infinitesimal gauge transformation:
\begin{equation}
  \left[ \int\!d\y\, \lambda(\y)\rho(\y),\,
   O(\x) \right] = i \delta_\lambda O(\x).
\end{equation}
But diffeomorphisms do not commute, so we conclude that the charge density at different points does not commute in our theory.  We find
\begin{equation}
  [\rho(\x), \, \rho(\y)] = i\ell^2\vareps^{ij} \d_i \rho(\x)
  \d_j \delta(\x-\y).
\end{equation}
Here we recover the long-wavelength version of the
Girvin--MacDonald--Platzman (GMP) algebra~\cite{Girvin:1986zz} (or the
$w_\infty$ algebra), which suggests that the symmetry described here
is related to the physics of the LLL.

\section{Connection to quantum Hall effect}

To establish the connection with the physics of the LLL,
we recall the symmetry of the problem.  A system of particles
interacting with an electromagnetic field can also be put in curved space:
\begin{equation}
  S = \int\!dt\, d\x\, \sqrt g \left( \frac i2 \psi^\+ \dlr_t \psi
  + A_0\psi^\+ \psi - \frac{g_{ij}}{2m} D_i\psi^\+ D_j \psi +
  \cdots \right),
\end{equation}
where $\cdots$ includes interaction terms.
(Strictly speaking, the discussion here corresponds to the $g=2$, $s=1$ version
of the LLL symmetry~\cite{Geracie:2014nka}.)
One can check that the
classical action is invariant with respect to time-dependent spatial
diffeomorphisms (i.e., all diffeomorphism transformations which
preserve the time slices)~\cite{Hoyos:2011ez,Geracie:2014nka}:
\begin{align}\label{NR-diff}
  \delta A_0 &= -\xi^k \d_k A_0 - A_k \dot \xi^k, \\
  \delta A_i &= -\xi^k\d_k A_i - A_k \d_i \xi^k - m g_{ik}\dot\xi^k,\\
  \delta g_{ij} &= -\xi^k \d_k g_{ij} - g_{ik} \d_j\xi^k - g_{ik}\d_j \xi^k .
\end{align}
The LLL limit corresponds to taking $m\to0$.  The term
proportional to $m$ disappears from the transformation law for $A_i$;
now $A_\mu$ simply transforms like a one-form under spatial diffs:
\begin{equation}
  \delta A_\mu = -\xi^k\d_k A_\mu - A_\lambda \d_\mu \xi^\lambda,
  \qquad \xi^\lambda = (0, \xi^i) .
\end{equation}
The metric $g_{ij}$ also transforms like a covariant tensor.  The
nontrivial feature of the states on the LLL that sets it apart from
other states in a magnetic field is that, although a time-varying
diffeomorphism generates the $g_{0i}$ components of the metric tensor
from $g_{ij}$: $\delta g_{0i} = \cdots + g_{ij}\dot\xi^j$, the
partition function of the theory does not depend at all on $g_{0i}$.

The fractional quantum Hall effect exists in a finite magnetic field
$B=\d_1 A_2-\d_2 A_1$. Suppose one is not interested in computing the electric current by varying the partition function with respect to $A_i$. In that case, one can assume that $A_i$ has some fixed value, for example, $A_i=-\frac12B\vareps_{ij}x^j$, and only consider $A_0$ and $g_{ij}$ as external backgrounds. Then it is natural to ask if the partition function of the theory is symmetric under any gauge transformation that touches only $A_0$ and $g_{ij}$, and explore the Ward-Takahashi identities that follow.
To keep $B$ unchanged, we need to restrict ourselves to
VPDs. These correspond to $\xi^k =\ell^2\varepsilon^{kl}\d_l\lambda$, where we chose $\ell$ to be the magnetic length $\ell=1/\sqrt B$.  For VPDs, the change of the
spatial components of gauge potential $A_i$,
\begin{multline}
  \delta A_i = -\ell^2\varepsilon^{kl}\d_l\lambda \d_k A_i
  -\ell^2 A_k\varepsilon^{kl}\d_i \d_l \lambda
  = - \ell^2\varepsilon^{kl}\d_l \lambda (\d_k A_i - \d_i A_k)
  - \ell^2\d_i (\varepsilon^{kl} A_k \d_l\lambda)\\
  = -\d_i (\lambda + \ell^2 \varepsilon^{kl} A_k \d_l\lambda),
\end{multline}
can be compensated by a gauge transformation $A_\mu\to A_\mu +
\d_\mu\alpha$ with $\alpha= \lambda +\ell^2\varepsilon^{kl}A_k\d_l\lambda$.
Under this combination of coordinates and gauge transformations,
$A_0$ transforms as
\begin{equation}\label{dA0}
  \delta A_0 = - \ell^2\varepsilon^{kl}\d_l\lambda \d_k A_0
  - \ell^2 A_k \d_t(\varepsilon^{kl}\d_l \lambda)
  + \dot\lambda + \ell^2 \d_t(\varepsilon^{kl}A_k \d_l\lambda)
  = \dot\lambda - \ell^2 \varepsilon^{kl} \d_k A_0 \d_l\lambda .
\end{equation}
We see that the transformation law for $A_0$ has exactly the form that
we have postulated in Eq.~(\ref{phi-vpd}).  The metric, of course,
transforms as in Eq.~(\ref{g-vpd}).

\subsection{Higher-rank symmetry in the lowest Landau level limit}

Within the context of the LLL, it is possible to give an intuitive
interpretation of the higher-rank conservation law.  We write down the
current conservation
\begin{equation}
\label{eq:conrho}
  \frac{\partial\rho}{\d t} + \bm{\nabla}\cdot \mathbf{j} = 0,
\end{equation}
and the law of conservation of momentum,
\begin{equation}
\label{eq:conT}
  \frac{\d \pi_i}{\d t} + \d_j T_{ij} = E_i \rho + \vareps_{ik} j_k B,
\end{equation}
where $\pi_i$ is the momentum density. In a Galilean-invariant theory
with particles of mass $m$, the momentum density is proportional to
the particle number flux $\pi_i=m j_i$, and vanishes in the LLL limit
$m\to0$.  Now the conservation of momentum becomes simply the
equation of balance of force, which in the absence of the electric
field simply reads
\begin{equation}
  \d_j T_{ij} = \vareps_{ik} j_k B,
\end{equation}
and can be solved to yield for the current
\begin{equation}
  j_i = - \frac1B \vareps_{ij} \d_k T_{jk} .
\end{equation}
The equation for the conservation of charge is now
\begin{equation}
\label{eq:conser}
  \frac{\d\rho}{\d t} - \frac1{2B} \d_i\d_j
  (\vareps_{ik} T_{kj} + \vareps_{jk} T_{ik}) = 0 .
\end{equation}
One notices that the conservation law \eqref{eq:conser} is in the same form as \eqref{eq:cont} in the earlier version of the symmetric tensor gauge theory of fracton. Thus, the conservation of charge has a ``higher-rank'' form due to the fact that, on the LLL, the current density is no longer independent but can be expressed through the derivative of the stress tensor. The same conservation law was derived previously in Ref.~\cite{nguyen2014lowest} using a LLL field theory formalism.  The connection between volume-preserving diffeomorphism and quantum Hall physics was also noticed in Refs.~\cite{Gromov:ChiralEalstic,Doshi:2020jso,Cappelli:1992yv,Iso:1992aa,Cappelli1,Cappelli2,Cappelli3,Kogan}.

Some comments are in order. We began with two independent Ward's identities, \eqref{eq:conrho} and \eqref{eq:conT}. The conservation law  \eqref{eq:conser} can be considered as the linear combination of the charge conservation \eqref{eq:conrho} and the massless limit of \eqref{eq:conT}. One then recognizes that we end up with two independent conservation laws, one being \eqref{eq:conser} and the other being the original charge conservation. It is the same conclusion that we arrived at in Section \ref{sec:nonlinear}.

\subsection{The Wen-Zee term}

One possible term in the effective action for the fractional quantum
Hall is the Wen-Zee term~\cite{Wen:1992ej}.  To introduce this
term, we need to define the Newton-Cartan geometry and the
spin connection that comes with it.  We only give the relevant formulas here; for details, see, e.g., Refs.~\cite{Son:2013rqa,Geracie:2014nka}.  The
Newton-Cartan geometry structure is given by a one-form $n_\mu$ (in
the simplest version of the geometry $dn=0$), a vector $v^\mu$, and a
symmetric contravariant metric tensor $g^{\mu\nu}$ satisfying $n_\mu
v^\mu=1$, $g^{\mu\nu}n_\nu=0$.  In our case,
\begin{equation}
  n_\mu = (1, \mathbf{0}), \qquad
  v^\mu = \begin{pmatrix} 1\\ v^i \end{pmatrix}, \qquad
  g^{\mu\nu} = \begin{pmatrix} 0 & 0 \\ 0 & g^{ij} \end{pmatrix} .
\end{equation}
where
\begin{equation}
  v^i=\ell^2\vareps^{ij}\d_j A_0,
\end{equation}
and $g^{ij}$ is the inverse matrix
of $g_{ij}$.  One then defines the covariant metric tensor
$g_{\mu\nu}$ so that $g_{\mu\nu}v^\nu=0$ and
$g_{\mu\nu}g^{\nu\lambda}=\delta_\mu^\lambda-n_\mu v^\lambda$,
\begin{equation}
  g_{\mu\nu} = \begin{pmatrix} g_{ij} v^i v^j & - v_j\\
  - v_i & g_{ij} \end{pmatrix} ,
\end{equation}
where $v_i\equiv g_{ij}v^j$,
together with the Christoffel symbol which can be used to define covariant derivatives
\begin{equation}\label{Christoffel}
  \Gamma^\mu_{\nu\lambda} = v^\mu\d_\nu n_\lambda + \frac12 g^{\mu\rho}
  ( \d_\nu g_{\rho\lambda} + \d_\lambda g_{\rho\nu} - \d_\rho g_{\nu\lambda}).
\end{equation}
One further defines the vielbein $e^a_\mu$ so that
\begin{equation}
  g^{\mu\nu} = e^{a\mu} e^{a\nu} ,
\end{equation}
and the spin connection is defined as
\begin{equation}
  \omega_\mu = \frac12 \vareps^{ab} e^{a\nu}\nabla_\mu e^b_\nu ,
\end{equation}
which, in components, reads
\begin{align}
  \omega_0 &= \frac12 (\vareps^{ab} e^{aj}\d_0 e^b_j
  +\vareps^{ij} \d_i v_j),\\
  \omega_i &= \frac12 (\vareps^{ab} e^{aj}\d_i e^b_j-\vareps^{jk}\d_jg_{ik}).
\end{align}
The spin connection transforms like the gauge potential under the
local O(2) rotation of the vielbein: $e^a(x)\to
e^a(x)+\alpha(x)\vareps^{ab}e^b(x)$.
The Wen-Zee term \cite{Wen:1992ej} is given by
\begin{equation}
  \frac\kappa{4\pi}\vareps^{\mu\nu\lambda}\omega_\mu \d_\nu A_\lambda
  = \frac\kappa{4\pi} \left(
  \omega_0 B + A_0 \vareps^{ij}\d_i\omega_j - \vareps^{ij}\omega_i \d_0 A_j\right)
  = \frac\kappa{4\pi}\left(\frac{\omega_0}{\ell^2} + \frac12 A_0 R\right) ,
\end{equation}
where we have put $\vareps^{ij}\d_i A_j=\ell^{-2}$ and $\d_0 A_i=0$. The coefficient $\kappa$ is related to the filling fraction $\nu$ and the Wen-Zee shift $\mathcal{S}$ of a fraction quantum Hall (FQH) system by the relation
\begin{equation}
	\kappa=\nu \mathcal{S}.
\end{equation}
Up to quadratic order and ignoring the total derivative terms, we can rewrite the Wen-Zee term as
\begin{equation}
\label{eq:WZ}
  \frac\kappa{8\pi} \left( A_0 R - \frac1{4\ell^2} \vareps^{ij} h_{ik} \dot h_{jk}\right) ,
\end{equation}
which is exactly the Chern-Simons term~(\ref{LCS}).

A remark can be made here. We demonstrated that the Wen-Zee term in the FQH literature is nothing but the Chern-Simon term in the higher-rank gauge theory. One can think of this in the reversed order. The higher-rank gauge symmetry dictates the relationship between the Wen-Zee shift $\mathcal{S}$ and the Hall viscosity $\eta_H$, which enter two separated components of the Chern-Simons Lagrangian in the higher-rank gauge theory \eqref{eq:WZ}.

\section{Generalization to (3+1) dimensions}

\subsection{Construction of the (3+1)D nonlinear higher-rank symmetry}

This section will generalize the nonabelian higher-rank symmetry to (3+1) dimensions.
To do that, we imagine a three-dimensional version of the LLL.  Instead
of a background magnetic field, we imagine a background Kalb-Ramond field.
Concretely, we imagine a nonrelativistic theory living in
background metric $g_{ij}$ and a Kalb-Ramond field
$B_{\mu\nu}=-B_{\nu\mu}$.  The field strength of the latter is
\begin{equation}
  H_{\mu\nu\lambda} = \d_\mu B_{\nu\lambda} + \d_\nu B_{\lambda\mu}
    + \d_\lambda B_{\mu\nu} ,
\end{equation}
and we assume that there is gauge symmetry with one-form gauge
parameter $\alpha_\mu$ under which
\begin{equation}\label{KR-gauge}
  \delta B_{\mu\nu} = \d_\mu\alpha_\nu - \d_\nu \alpha_\mu .
\end{equation}
and $H_{\mu\nu\lambda}$ is invariant.
Most crucially, we assume that
our theory is invariant symmetry under time-dependent spatial
diffeomorphisms,
\begin{align}\label{3D-diff1}
  \delta g_{ij} &= -\xi^k \d_k g_{ij} -g_{kj}\d_i \xi^k -g_{ik} \d_j \xi^k,\\
  \label{3D-diff2}
  \delta B_{ij} &= -\xi^k \d_k B_{ij} -B_{kj}\d_i \xi^k -B_{ik} \d_j \xi^k,\\
  \label{3D-diff3}
  \delta B_{i0} &= -\xi^k \d_k B_{i0} -B_{k0}\d_i\xi^k - B_{ik}\dot\xi^k.
\end{align}
which is a 3D version of the $m\to0$ (i.e., LLL) limit of the
nonrelativistic diffeomorphism~(\ref{NR-diff}).  We do not have a concrete
example of a well-defined theory with the symmetry~\eqref{3D-diff1}, \eqref{3D-diff2} and \eqref{3D-diff3}.
As the LLL can be thought of as the massless limit for particles in a
magnetic field, one can imagine a theory of massless strings coupled
to a Kalb-Ramond field.  The details (or even the existence) of such a theory are not important for our further discussion.

Following our discussion of the LLL in (2+1)D, we
assume that our system lives in a finite Kalb-Ramond field $H_{ijk}=
\ell^{-3}\vareps_{ijk}$, and restrict ourselves to VPDs
with $\d_k\xi^k=0$ or
\begin{equation}
  \xi^k = \ell^3\vareps^{klm}\d_l \lambda_m .
\end{equation}
The change of $B_{ij}$ under this VPD,
\begin{equation}
  \delta B_{ij} = -\ell^3\vareps^{klm}( \d_k B_{ij}
  + B_{kj} \d_i
  + B_{ik} \d_j) \d_l \lambda_m,
\end{equation}
again can be compensated by a gauge
transformation~(\ref{KR-gauge}) with the gauge parameter
\begin{equation}
  \alpha_i = \lambda_i - \ell^3\vareps^{klm}\ B_{ik}\d_l\lambda_m .
\end{equation}
The combined VPD and gauge transformation changes only the $B_{0i}$ components
of the Kalb-Ramond potential and the metric.  This is a higher-rank
symmetry that transforms our set of gauge fields $(B_{0i}, g_{ij})$ like
\begin{subequations}\label{3D-higher-rank-1}
\begin{align}
  \delta B_{0i} & =  \dot\lambda_i - \ell^3\vareps^{klm}
  (\d_k B_{0i} + B_{0k} \d_i) \d_l \lambda_m , \\
  \delta g_{ij} &= -\ell^3 \vareps^{klm}
  (\d_k g_{ij}  + g_{kj}\d_i + g_{ik}\d_j) \d_l \lambda_m .
\end{align}
\end{subequations}
In addition, we also inherit from the gauge symmetry~(\ref{KR-gauge})
those transformations which leave $B_{ij}$ invariant.  These correspond
to gauge parameters with vanishing spatial components: $\alpha_\mu=(\alpha_0,
\mathbf{0})$.  Under these gauge transformations,
\begin{subequations}\label{3D-higher-rank-2}
\begin{align}
  \delta B_{0i} & = -\d_i\alpha_0 ,\\
  \delta g_{ij} & = 0 .
\end{align}
\end{subequations}
Equations~(\ref{3D-higher-rank-1}) and (\ref{3D-higher-rank-2})
represent the full group of higher-rank symmetries.  Though we have
used an analogy with the LLL physics in 2D as a motivation, the
resulting transformation laws do not require any LLL-type microscopic
physics.  In fact, we will see that the higher-rank symmetry appears
in the context of 3D ferromagnets.

As in 2D, it is possible to ``complete'' $B_{0i}$ by adding the
spatial components $B_{ij}$ so that $B_{\mu\nu}$ form a set of
Kalb-Ramond gauge potentials. This would make $B_{0i}$ the shared
components of the two sets of gauge potentials: the Kalb-Ramond gauge
potentials and the gauge potentials of the higher-rank symmetry of
VPDs (see Fig.~\ref{fig:venn-diagram2}).
\begin{figure}[ht]
 \centering
  \includegraphics[width=0.3 \textwidth]{venn-diagram2.pdf}
 \caption{$B_{0i}$ is shared by two sets of gauge potentials.}
  \label{fig:venn-diagram2}
\end{figure}

\subsection{Linearized higher-ranked symmetries and conservation laws}

At the  linearized level, the higher-rank gauge invariance is
\begin{align}\label{3d-linearized1}
  \delta B_{0i} &= \dot\lambda_i -\d_i \alpha_0,\\
  \label{3d-linearized2}
  \delta h_{ij} &= -\ell^3 (\vareps_{jkl}\d_i
                   + \vareps_{ikl}\d_j) \d_k\lambda_l .
\end{align}
As far as we know, this linear higher-rank symmetry has not been considered
previously in the literature.

Let us now assume that the currents coupled to $B_{0i}$ and $h_{ij}$
are $J^i$ and $T^{ij}$,
\begin{equation}
  \delta \ln Z = \int\!d^4x
  \left( J^i\delta B_{0i} + \frac12 T^{ij}\delta h_{ij} \right).
\end{equation}

Two conservation laws follow from Eqs.~\eqref{3d-linearized1} and \eqref{3d-linearized2}.  First,
the gauge invariance with gauge parameter $\alpha_0$ implies that the
``current'' $J^i$ is divergence-free:
\begin{equation}
  \bm{\nabla}\cdot\mathbf{J} = 0.
\end{equation}
On the other hand the VPD invariance, generated by $\lambda_i$, leads to
\begin{equation}
\label{eq:continuity3D}
  \frac{\d J_i}{\d t} -  \ell^3\vareps_{ijk} \d_j \d_l T_{kl} = 0.
\end{equation}
This means the following quantities are conserved
\begin{align}
  \mathbf{I}_1 &= \int\!d\x\, \mathbf{J},\\
  I_2 &= \int\!d\x\, (\x\cdot \mathbf{J}), \quad
  \mathbf{I}_3 = \int\!d\x\, (\x \times \mathbf{J}), \quad
  I_4^{ij} = \int\!d\x\, x^{\{i} J^{j\}}, \\
  \mathbf{I}_5 &= \int\!d\x\, \x (\x\cdot\mathbf{J}),\qquad
  \mathbf{I}_6 = \int\!d\x\, \frac{x^2}2\mathbf{J},\\
  I_7^{ijk} &= \int\!d\x\, x^{\{i} x^j J^{k\}} .
\end{align}
where the $\lbrace ij\cdots \rbrace$ denotes symmetrization over the indices $i,j,\cdots$ \footnote{
Explicitly
\begin{equation}
	A^{\{i} B^{j\}}=\frac{1}{2}\left(A^i B^j+A^j B^i\right),
\end{equation}
and
\begin{equation}
	A^{\{i} B^j C^{k\}}=\frac{1}{3}\left(A^iB^{\{j} C^{k\}}+B^iC^{\{j} A^{k\}}+C^iA^{\{j} B^{k\}}\right).
\end{equation}}.
On the other hand,
\begin{equation}
  I_8^{ij} = \int\!d\x\, x^{\{i} (\x\times\mathbf{J})^{j\}}
\end{equation}
is not conserved; its time derivative is proportional to $\int\!d\x\,
T^{ij}$.

Assume that the ``current'' $\mathbf{J}$ is nonzero only in a finite region of space, then because it is divergence-free, one can express it as the curl of a vector field: $\mathbf{J}=\bm{\nabla}\times\bm{\mu}$, where
$\bm{\mu}$ vanishes at infinity.  Then among the conserved quantities,
only the following are nonzero: $\mathbf{I}_3$, $\mathbf{I}_5$, and
$\mathbf{I}_6$, and the last two quantities are not independent:
\begin{align}
  \mathbf{I}_3 &= \int\!d\x\, \bm{\mu} ,\\
  \mathbf{I}_5 &= - \mathbf{I}_6 = \int\!d\x\, (\x\times\bm{\mu}) .
\end{align}
One can think of $\bm{\mu}$ as the magnetic moment density and
$\mathbf{J}$ as the magnetization current.  Let us assume that there
is a quasiparticle that carries a magnetic moment.  Then the
conservation of $\mathbf{I}_3$ means that the total magnetic moment does not change its value.  The conservation of $\mathbf{I}_5$ implies that a particle can move only along the direction of its magnetic moment, but not along the two perpendicular directions.

Note that the (3+1)D higher-rank symmetry presented above, even in the
linearized version, differs from that of the vector charge theory
proposed in Ref.~\cite{higherrank1}.  Our motivation was to generalize
the area-preserving diffeomorphism of the LLL in (2+1)D to
volume-preserving diffeomorphism of (3+1)D.  We have generalized the
charge density $\rho$, the one-form gauge potential $A_\mu$ and the
constant magnetic field $B$ to the vector charge density $J^i$, the
two-form gauge potential $B_{\mu\nu}$ and the constant Kalb-Ramond
field strength $H_{ijk}$. We then arrive at a different gauge
transformation rather than the one in Ref.~\cite{higherrank1}. One can
define the electric field and magnetic field that are invariant under
the gauge transformations and write down a generalized Maxwell
action.  We will not do that here, as the aim of this paper is to investigate the
conservation laws and their physical consequences\footnote{Since
the gauge transformations
\eqref{3d-linearized1} and \eqref{3d-linearized2} differ from the ones in Ref.~\cite{higherrank1}, the definitions of the gauge-invariant
electric field and magnetic
field should be modified accordingly, namely
$E^{ij}=\left(\varepsilon_{jkl}\partial_i+\varepsilon_{ikl}\partial_j \right)\partial_k B_{0l}+\partial_t h_{ij}$
and
$H=\partial_i\partial_j h_{ij}$.
We will not discuss them further.}.

The conservation law \eqref{eq:continuity3D}, though similar to the one of vector charge theory in Ref \cite{higherrank1}, is not the same because of the different gauge transformations.  Recall that the gauge transformation of the symmetry tensor gauge in the vector charge version is \cite{higherrank1,Pretko:2017xar}
  \begin{equation}
  	\delta A_{ij}= \partial_i \lambda_j + \partial_j \lambda_i , \quad \delta \phi_i=\partial_t \lambda_i
  \end{equation}
which leads to a conservation law with only one spatial derivative acting on the current density $J^{ij}$
  \begin{equation}
  	\partial_t \rho^i+\partial_j J^{ij}=0
  \end{equation}

instead of an equation with two spatial derivative.
Furthermore, inherited from a system of vector matter field coupled with the Kalb-Ramond field, we have an \textit{extra} ``conservation law'' $\vec{\nabla}\cdot \vec{J}=0$, which does not have a counterpart in the vector charge theory proposed in Ref.~\cite{higherrank1}.

We will show in Section~\ref{sec:ferro} that the symmetry that has
been proposed is realized in the nonlinear sigma model describing 3D
ferromagnets.

\section{Examples of theories with volume-preserving diffeomorphism invariance}

For the abelian higher-rank symmetry, one of the simplest ways to couple a matter
field to the gauge field $(A_0, h_{ij})$ is to introduce a Goldstone
boson $\varphi$ which transforms under the gauge transformation as
$\varphi \to \varphi + \lambda$ and write
\begin{equation}
  \mathcal L = \frac{c_1}2 (\d_0\varphi - A_0)^2
  - \frac {c_2}2 (\d_i\d_j\varphi - h_{ij})^2  .
\end{equation}
However, we were not able to find
a nonlinear version of this
transformation.  For example, if one postulates
\begin{equation}
  \delta_\lambda \varphi = \lambda - \ell^2\vareps^{kl}\d_k\varphi \d_l \lambda,
\end{equation}
then one can check by direct calculation that $[\delta_\alpha,
\delta_\beta]\varphi \neq \delta_{[\alpha,\beta]}\varphi$, so such
transformation law would be inconsistent.  We have to devise other
ways to couple the matter field to the gauge field.

\subsection{Composite fermions}

In the fractional quantum Hall effect at filling fractions $\nu=1/2$,
$\nu=1/4$ etc., the quasiparticle is electrically neutral.  One can
consistently couple such a particle to the higher-rank gauge field.\
For example, a Lagrangian for a nonrelativistic particle with
dispersion relation $\omega=k^2/2m$ would be
\begin{equation}
  L = \frac i2 v^\mu \psi^\+ \dlr_\mu \psi - \frac{1}{2m} g^{\mu\nu} \d_\mu \psi^\+ \d_\nu\psi
  = \frac i2 \psi^\+ \dlr_0 \psi
  + \frac i2 \ell^2\vareps^{ij} \d_j A_0\, \psi^\+ \dlr_i \psi
  - \frac{g^{ij}}{2m} \d_i\psi^\+ \d_j \psi .
\end{equation}

One can interpret the coupling of the electric field $\d_i A_0$ with
the particle momentum as a dipole moment, perpendicular to the
direction of its motion. This is consistent with the constraints that
follow from conservation laws.  In fact, that is how a composite
fermion in the fractional quantum Hall state at $\nu=1/2$ or $\nu=1/4$
should couple to the
external potential: the composite fermion is neutral but possesses an
electric dipole moment.

\subsection{Crystal on the lowest Landau level}

Another way to realize the higher-rank symmetry is through an
effective theory of a solid.  Such a solid may be realized as a
Wigner crystal, which is expected to be the ground state of electrons
on the LLL at small filling fractions.  A solid is
parametrized by a map from the physical coordinates $x^i$ to the
coordinate system $X^a$ frozen into the solid:
$X^a=X^a(x^i)$~\cite{SoperBook}. In the ground state
$X^a=x^i\delta_{ai}$, the displacement $u^a$ are defined as $X^a =
x^a-u^a$.  On the LLL the coordinates of a lattice site
do not commute: $[x,\, y]=-i\ell^2$.
The Lagrangian should thus contain the following term
\begin{equation}
  S_{\text{Berry}}= \frac{n_0}{2 \ell^2} \vareps^{ab} X^a \d_t X^b ,
\end{equation}
where $n_0$ is the equilibrium particle number density.
Under a VPD, $X^a$ transforms as
\begin{equation}
  \delta_\lambda X^a = - \ell^2\vareps^{ij} \d_i X^a \d_j \lambda ,
\end{equation}
and so
\begin{equation}
  \delta_\lambda S_{\text{Berry}}
  = - \frac{n_0}2 \vareps^{ij}\vareps^{ab} X^a \d_i X^b \d_j\dot\lambda
  = - \frac{n_0}2 \dot\lambda \vareps^{ij}\vareps^{ab}\d_i X^a \d_j X^b .
\end{equation}
This change of the action can be compensated by including a term
proportional to $A_0$ into the Lagrangian.  The full Lagrangian is then
\begin{equation}
  \mathcal L = \frac{n_0}{2\ell^2} \vareps^{ab} X^a \d_t X^b
  + \frac{n_0}2 A_0 \vareps^{ab}\vareps^{ij} \d_i X^a \d_j X^b
  - \vareps(O^{ab}),
\end{equation}
with $O^{ab}=g^{ij}\d_i X^a \d_j X^b$ and $\vareps(O^{ab})$ is the energy associated with elastic deformations.
The spectrum of this theory can be obtained by expanding the action to
quadratic order over the displacement $u^a$.  The presence of a term
with a first time derivative implies that the dispersion relation of the
lattice sound wave has the quadratic form $\omega\sim q^2$ rather than
the linear form.

Thus, we have been able to construct the effective theory of a Wigner
crystal on the LLL starting from the higher-rank symmetry.

\subsection{Vortex crystal}

Another type of matter is the ``vortex crystal,'' which is realized, for
example, in a rotating Bose gas~\cite{Moroz:2019qdw}.  The crystal is
formed by the zeros of the condensate wave function.  In this case,
the lattice fields $X^a$ do not couple to $A_0$ directly, but through
a dynamical gauge field $a_\mu$, which is the dual of the superfluid
phonon.  Under VPD $a_\mu$ transforms as a one-form,
\begin{equation}
  \delta a_\mu = - \ell^2\vareps^{kl} (\d_k a_\mu - a_k \d_\mu) \d_l\lambda .
\end{equation}
The field $a_\mu$ couples to the background $A_0$
through the following term
\begin{equation}\label{aA-coupling}
   \frac1{2\pi} \int\!d^3x \left(\frac{a_0}{\ell^2} + A_0 b\right),
\end{equation}
where $b=\varepsilon^{ij}\partial_i a_j$ is the emergent magnetic field. One can check directly that this term is invariant under VPDs.  It is
also obviously invariant under gauge transformations $a_\mu\to
a_\mu+ \d_\mu\alpha$.  In fact, (\ref{aA-coupling}) can be obtained
from the Chern-Simons term $\frac1{2\pi} \vareps^{\mu\nu\lambda}
a_\mu \d_\nu A_\lambda$ by setting $A_i$ to be the static background
with $\vareps^{ij}\d_i A_j=\ell^{-2}$.

The Lagrangian of the vortex crystal
is then determined by the symmetry and
reads
\begin{equation}
  \mathcal L = - \vareps^{\mu\nu\lambda} \vareps^{ab} a_\mu \d_\nu X^a \d_\lambda X^b
  - \vareps (O^{ab}) - \vareps_b(b)
  +  \frac1{2\pi} \left( \frac{a_0}{\ell^2} + A_0 b \right) ,
\end{equation}
where $\vareps(O^{ab})$ and $\vareps_b(b)$ represent
the energies of the lattice and the condensate, respectively.
Assuming $a_\mu$ transforms like a one-form under VPD,
it can be checked that the Lagrangian above is
invariant with respect to this symmetry.  The eigenmode of this theory
is again a Tkachenko mode with a quadratic dispersion
relation~\cite{Moroz:2019qdw}.

Note that the above Lagrangian contains only leading-derivative terms
and does not include next-to-leading terms considered in
Ref.~\cite{Moroz:2019qdw}.

\begin{comment}
\subsection{Bimetric theory}
 \DN{I don't know what do you want to achieve or to demonstrate in this sub-section }

The bimetric theory~\cite{Gromov:2017qeb} is a model describing the
spin-2 magnetoroton in the fractional quantum Hall effect.  The
dynamical degree of freedom in this theory is a dynamical metric $\hat
g_{ij}$.  One also introduces the vielbein $\hat e^a_i$ so that
\begin{equation}
  \hat g_{ij} = \hat e^a_i \hat e^b_j \delta_{ab}
\end{equation}
and the inverse vielbein $\hat E^i_a$.  The  \textit{ambient spin connection}, defined as
\begin{equation}
  \hat \omega_\mu = \frac12 \vareps^{ab} \hat E^j_a \nabla_\mu \hat e^b_j ,
\end{equation}
transforms like an abelian gauge field under the O(2) rotation of the
vielbein and as a one-form under VPD.
One needs a connection to define the covariant derivative
$\nabla_\mu$; two (out of many) possibilities are to use the
Newton-Cartan connections~(\ref{Christoffel}) using the frame vector
$v^\mu=(1,\vareps^{ij}\d_j A_0)$ with the spatial metric
$g^{ij}$ or $\hat g^{ij}$.  The leading terms in the bimetric action is
\begin{equation}
  L = \hat \omega_0 + A_0 \hat R - V(\hat g_{ij}, g_{ij}),
\end{equation}
where $\hat R = \frac12 \vareps^{ij} \d_i\hat \omega_j$, and $V$ is a
potential energy term which is minimized when $\hat g_{ij}=g_{ij}$.

In Ref.~\cite{Gromov:2017qeb} the $\hat\omega_\mu$ is defined using
the Christoffel symbol~(\ref{Christoffel}) constructed from the
dynamical metric $\hat g_{ij}$.  In this case, up to a total
derivative, $\hat\omega_0=\frac12\vareps^{ab}\hat E^i_a\d_0\hat
e^b_i$, and the effective Lagrangian contains no time derivatives of the
external gauge field $(A_0, g_{ij})$.  As explained in
Sec.~\ref{sec:nonlinearizing}, this means that the charge density
satisfies the commutation relation of the long-distance version of the
GMP algebra (the $w_\infty$ algebra), which is exactly what was found
in Ref.~\cite{Gromov:2017qeb}.  On the other hand, if one defines the
Christoffel symbols through external metric $g_{ij}$, then $\omega_0$
contains the time derivative of $g_{ij}$, so the matter Lagrangian
violates the assumptions assumed in Sec.~\ref{sec:nonlinearizing} and
so the charge density computed from the effective field theory may not
be expected to form a $w_\infty$ algebra, which is the signature of the LLL physics.
\end{comment}

\subsection{Ferromagnets}
\label{sec:ferro}
\subsubsection{Ferromagnets in 2+1 D}

We now show that the ferromagnet in 2+1D secretly possesses a higher-rank
symmetry similar to the models with particles on the LLL \footnote{In fact, one can show that the dynamical equation of a single skyrmion in a 2-dimensional ferromagnet is the same as the equation of motion of a charged particle in a constant magnetic field \cite{HanBook}. }.
At the long-wavelength limit, a ferromagnet is described by a nonlinear sigma (NLS) model \cite{FradkinBook},
written in terms of an $O(3)$ unit vector $n^a$, $n^a n^a=1$,
with the action
\begin{equation}
  S= S_{\text{Berry}} + S_{\text{NLS}}
  = S_0\! \int\limits_0^1\!d\sigma\! \int\!dt\,d\x\,
  \vareps^{abc} n^a \d_t n^b \d_\sigma n^c
  - \frac J2\! \int\!dt\, d\x\, \delta^{ij} \d_i n^a \d_j n^a .
\end{equation}
The first term is a Wess-Zumino topological term in spin's action, which is induced by the Berry phase \cite{FradkinBook}.
The second term is the energy term of the nonlinear sigma
model and can be made invariant under VPD by
replacing $\delta^{ij}$ with $g^{ij}$.  The first term is, however not
invariant:
\begin{equation}
  \delta_\lambda S_{\text{Berry}} = - S_0\ell^2\! \int\limits_0^1\!d\sigma\! \int\!dt\,d\x\,
  \vareps^{abc} \vareps^{ij} n^a \d_i n^b \d_\sigma n^c \d_j \dot\lambda .
\end{equation}
Integrating by parts, taking into account that $\epsilon^{abc}\d_j n^a
\d_i n^b \d_\sigma n^c=0$ (this is because all the three O(3) vectors
$\d_i n^a$, $\d_j n^a$, $\d_\sigma n^a$
are perpendicular to $n^a$ and hence are linearly dependent) we find
\begin{multline}
  \delta_\lambda S_{\text{Berry}} = S_0\ell^2\! \int\limits_0^1\!d\sigma\! \int\!dt\,d\x\,
  \vareps^{abc} \vareps^{ij} n^a \d_i n^b \d_j \d_\sigma n^c \dot\lambda \\
  = \frac{S_0}2\ell^2\! \int\limits_0^1\!d\sigma\! \int\!dt\,d\x\,
  \vareps^{abc} \vareps^{ij} \d_\sigma (n^a \d_i n^b \d_j  n^c) \dot\lambda
  = \frac{S_0}2\ell^2\! \int\!dt\,d\x\,
  \vareps^{abc} \vareps^{ij} n^a \d_i n^b \d_j  n^c \dot\lambda .
\end{multline}
We now choose
\begin{equation}\label{ell-S0}
  \ell^2 = \frac 1{4\pi S_0}\,,
\end{equation}
and add the following term to the action
\begin{equation}\label{SA0}
  S_{\!A_0} = - \frac{1}{8\pi}\!  \int\!dt\,d\x\, A_0
  \vareps^{abc}\vareps^{ij} n^a \d_i n^b \d_j n^c .
\end{equation}
Then $\delta_\lambda (S_{\text{Berry}} + S_{\!A_0})=0$.
Thus if we couple the ferromagnetic order parameter with the gauge
fields $A_0$, $g_{ij}$ in the following way
\begin{equation}
  S = S_{\text{Berry}} + S_{\!A_0} -
 \frac{J}{2} \! \int\!dt\,d\x\, g^{ij} \d_i n^a \d_j n^a,
\end{equation}
then the action is invariant under VPD
with $\ell^2$ defined in Eq.~(\ref{ell-S0}).

One can further introduce into the theory the vector gauge
potential $A_i$, promoting Eq.~(\ref{SA0}) to
\begin{equation}
  S_{\!A_\mu} = - \frac{1}{8\pi}\!  \int\!dt\,d\x\, A_\mu
  \vareps^{abc}\vareps^{\mu\nu\lambda} n^a \d_\nu n^b \d_\lambda n^c .
\end{equation}
with $A_i$ transforming as a one-form under VPDs.
In this case, the potential $A_0$ is simultaneously the scalar component of the
U(1) gauge field $(A_0, A_i)$ and the
scalar component of the gauge potential of a higher-spin symmetry,
$(A_0, g_{ij})$ (Fig.~\ref{fig:venn-diagram1}).

Now the scalar potential $A_0$ is coupled to the topological charge
density $\rho(\x)$.
This means that a skyrmion behaves like a particle in an effective magnetic field with the magnitude
\begin{equation}
  B_{\text{eff}} = -4\pi S_0 q,
\end{equation}
where $q=\int\!d\x\, \rho(\x)$ is the topological charge of the
skyrmion.  This fact can be derived by calculating the Berry phase
associated with the motion of a skyrmion~\cite{Stone:1996}.  One also
finds that the following quantities
\begin{equation}
  \int\! d\x\, \rho, \quad \int\! d\x\, x^i \rho, \qquad \int\! d\x\, x^2\rho,
\end{equation}
are conserved.  This fact is again well
known~\cite{Papanicolaou:1990sg}.\footnote{In the presence of the
Dyaloshinskii-Morya interaction, which breaks the higher-rank symmetry,
only the first two quantities are conserved~\cite{Komineas:2015vba}.}

It is instructive to rewrite the ferromagnet in the $\mathbb{CP}^1$
parametrization, where
\begin{equation}
  n^a = z^\+ \sigma^a z, \qquad
  z = \begin{pmatrix} z_1 \\ z_2 \end{pmatrix},
  \qquad z^\+ z = 1.
\end{equation}
The action of the ferromagnet is then~\cite{HanBook}
\begin{equation}
S = S_{Berry} + S_{NLS} = 2i S_0 \! \int\!dt\,d\x\, z^\+ \d_t z  -2 J\! \int\!dt\, d\x\,  D_i z^\+  D_i z ,
\end{equation}
where $D_i z \equiv (\d_i-ia_i) z$, and $a_i=-i z^\dagger \partial_i z$ is promoted to a dynamical field.

Now we couple the $\mathbb{CP}^1$ model to the external probes $(g_{ij}, A_0)$.  We assume that under VPD, $z$
transforms as
\begin{equation}
  \delta_\lambda z =  - \ell^2\vareps^{kl} \d_k z \d_l \lambda,
\end{equation}
therefore,
\begin{equation}
  \delta_\lambda S_{\text{Berry}} = -2i
  S_0 \ell^2 \! \int\!dt\,d\x\, \vareps^{ij} z^\+ \d_i z \d_j\dot\lambda
  = -2i S_0 \ell^2 \! \int\!dt\,d\x\, \dot\lambda \vareps^{ij} \d_i
  z^\+ \d_j z .
\end{equation}
We now add the following term to the action:
\begin{equation}
  S_{A_0} =  2i S_0\ell^2\! \int\!dt\,d\x\, A_0 \vareps^{ij} \d_i z^\+ \d_j z .
\end{equation}
Then $ \delta_\lambda (S_{Berry} + S_{A_0}) = 0 $, therefore the
coupling of the ferromagnetic model with the gauge field $g_{ij}$,
$A_0$ is
\begin{equation}
    S = S_{\text{Berry}} + S_{A_0}
    -2 J\! \int\!dt\, d\x\, g^{ij} D_i z^\+  D_j z ,
\end{equation}
and respects the higher-rank symmetry.

\subsubsection{Ferromagnets in 3+1 D}
For a ferromagnet in (3+1)D, the term $S_{\!A_0}$ that couples $A_0$
to the topological charge density is replaced by the coupling of
$B_{0i}$ to the density of a one-form current,
\begin{equation}\label{SB0i}
  S_{B_{0i}} =  - \frac1{8\pi} \int\!dt\,d\x\,
  \vareps^{abc}\vareps^{ijk} B_{0i} n^a \d_j n^b \d_k n^c ,
\end{equation}
and this has the (3+1)D version of VPD invariance with
$\ell^3= (4\pi S_0)^{-1}$.
Again one can promote~(\ref{SB0i}) to
\begin{equation}\label{SBmunu}
  S_{B_{\mu\nu}} =  - \frac1{8\pi} \int\!dt\,d\x\,
  \vareps^{abc}\vareps^{\mu\nu\lambda\rho} B_{\mu\nu} n^a \d_\lambda n^b \d_\rho n^c .
\end{equation}
In this case, $B_{0i}$ are the shared components of a
Kalb-Ramond gauge field and
the set of gauge potentials of a higher-rank gauge symmetry
$(B_{0i}, g_{ij})$ (see Fig.~\ref{fig:venn-diagram2}).

It was found in Ref.~\cite{Papanicolaou:1990sg} that
$\mathbf{I}_3$ and $\mathbf{I}_5=-\mathbf{I}_6$ are conserved.  We have
given this fact a new interpretation in terms of a hidden higher-rank
symmetry.

\section{Conclusion}

We have presented a nonlinear version of a higher-rank gauge symmetry.
The symmetry is basically that of volume-preserving diffeomorphism.
We show several examples of coupling of matter with the higher-rank
gauge potential that respects the symmetry.  Many examples are taken
from the physics of the LLL, which we show to naturally have
volume-preserving diffeomorphism invariance.  We also show that the nonlinear sigma models of ferromagnetism also exhibit this symmetry if one couples a gauge potential of the higher-rank symmetry with the topological charge density.

\begin{comment}
The symmetry of the fractional quantum Hall effect is a global
symmetry acting on the background scalar potential and the metric,
which are non-dynamical.  One can in principle promote this symmetry
into a gauge symmetry.  This can be done also partially---one can
promote only the metric part $g_{ij}$ to a dynamical field, leaving
the scalar potential $A_0$ a background field.  This is essentially
what has been down in the bimetric model of fractional quantum Hall
states, which tries to describe the lowest ``magnetoroton'' mode with
a field theory.  The effective action still contains the background
metric, and a potential energy term (proportional to the square of the
difference between the dynamical metric and the background metric)
gives the dynamical graviton a mass.

In Ref.~\cite{Gromov:2017qeb} it has been observed that the charge
density, computed from the model by the standard way of
differentiating the action with respect to the scalar potential,
satisfies the long-wavelength version of the GMP algebra.  The direct
calculation of the commutator in Ref.~\cite{Gromov:2017qeb} seems
quite nontrivial, but as we have seen, it is an automatic consequence
of the invariance of the theory with respect to volume-preserving
diffeomorphism.
\end{comment}

We have shown that, under certain conditions, the charge densities
satisfy the commutation relation of the VPD, i.e., the $w_\infty$ algebra.
In this way, one can easily understand why this algebra is realized in the bimetric model of the FQH effect~\cite{Gromov:2017qeb}, without explicit calculations.
One interesting question is whether the $w_\infty$ symmetry can be
``upgraded'' to the quantum $W_\infty$ symmetry~\cite{Cappelli:1992yv,Iso:1992aa}.  This question, relevant
for the fractional quantum Hall effect, is deferred to future work.
\section*{Acknowledgements}
The authors thank Adrey Gromov and Sergej Moroz for discussions and comments on the earlier draft of this paper. This work is
supported, in part, by the U.S.\ DOE grant No.\ DE-FG02-13ER41958, a
Simons Investigator grant and by the Simons Collaboration on
Ultra-Quantum Matter, which is a grant from the Simons Foundation
(651440, DTS). DXN is supported by the Brown Theoretical Physics Center.