---
title: 3D Maxwell Chern-Simons Gravity
url: https://www.emergentmind.com/topics/three-dimensional-maxwell-chern-simons-gravity
type: topic
---

# 3D Maxwell Chern-Simons Gravity

Searching arXiv for recent and foundational papers on three-dimensional Maxwell Chern-Simons gravity and closely related asymptotic/boundary/higher-spin extensions.
Three-dimensional Maxwell Chern-Simons gravity is a \(2+1\)-dimensional Chern-Simons gauge theory whose gauge algebra is the Maxwell algebra rather than \(\mathfrak{iso}(2,1)\). In this framework, the gauge sector enlarges the Lorentz generators \(J_a\) and translations \(P_a\) by an Abelian triplet \(Z_a\), so that translations cease to commute and instead satisfy \([P_a,P_b]\sim Z_c\). The resulting theory retains the topological character of three-dimensional Chern-Simons gravity, but its field content, asymptotic symmetry algebra, boundary dynamics, and conserved charges differ from those of ordinary asymptotically flat Einstein gravity [1805.08834]. In the standard Maxwell formulation, all locally flat three-dimensional Einstein geometries remain present as particular solutions, while an additional gravitational Maxwell gauge field modifies the vacuum structure, boundary charges, and null-infinity symmetry algebra [1805.08834].

## 1. Algebraic foundation and geometric interpretation

The three-dimensional Maxwell algebra is generated by
\[
\{J_a,P_a,Z_a\},\qquad a=0,1,2,
\]
with Lie brackets
\[
[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,\qquad [J_a,Z_b]=\epsilon_{abc}Z^c,
\]
\[
[P_a,P_b]=\epsilon_{abc}Z^c,\qquad [Z_a,Z_b]=0,\qquad [Z_a,P_b]=0.
\]
Its characteristic deviation from the Poincaré algebra is precisely the nonvanishing commutator of translations, \([P_a,P_b]\sim Z_c\), while the \(Z_a\) span an Abelian ideal [1805.08834]. In tensorial notation this same extension may be written as \([P_a,P_b]=Z_{ab}\), with \(Z_{ab}=\epsilon_{abc}Z^c\) in three dimensions [2506.07651; 1411.1780].

This algebra is physically associated with deformations of Poincaré symmetry by constant electromagnetic-background-type structures, and in the gravitational setting it introduces an extra gauge field \(\sigma^a\) or, in another notation, \(\sigma^{ab}\), that modifies both the vacuum and the asymptotic charges [1805.08834; 1411.1780]. The Maxwell algebra can also be obtained from other parent structures. One route is as a contraction of the AdS-Lorentz algebra [1805.08834; 2107.05716]. Another is through \(S\)-expansion of AdS or \(\mathfrak{sl}(3,\mathbb R)\), which is especially important for higher-spin extensions because it simultaneously produces the invariant tensor needed for the Chern-Simons action [1411.1780; 1712.09975].

A central structural point is that three-dimensional Maxwell Chern-Simons gravity is a genuine gauge-theoretic extension of three-dimensional gravity, not merely a Chern-Simons-like reformulation with auxiliary one-forms. This distinguishes it from broader classes of CS-like massive gravities, where extra Lorentz-vector-valued one-forms encode higher-derivative dynamics rather than gauge fields of an enlarged spacetime algebra [1402.1688; 1405.6213]. By contrast, in Maxwell gravity the new one-form belongs directly to the gauge connection of the enlarged algebra [1805.08834].

## 2. Chern-Simons formulation and field equations

In the standard spin-2 Maxwell theory, the gauge connection is
\[
A=e^a P_a+\omega^a J_a+\sigma^a Z_a,
\]
where \(e^a\) is the dreibein, \(\omega^a\) the dualized spin connection, and \(\sigma^a\) the gravitational Maxwell gauge field [1805.08834]. An equivalent \(\mathfrak{sl}(2)\)-valued notation is
\[
\mathcal A=(\Omega,E,\Sigma),
\]
with \(\Omega=\Omega^a t_a\), \(E=E^a t_a\), and \(\Sigma=\Sigma^a t_a\) [2506.07651].

The nonvanishing invariant bilinear form is
\[
\langle J_a J_b\rangle=\alpha_0\,\eta_{ab},\qquad \langle J_a P_b\rangle=\alpha_1\,\eta_{ab},\qquad \langle P_a P_b\rangle=\alpha_2\,\eta_{ab},
\]
\[
\langle J_a Z_b\rangle=\alpha_2\,\eta_{ab},\qquad \langle Z_a Z_b\rangle=0,\qquad \langle Z_a P_b\rangle=0,
\]
or, in the notation of the boundary-dynamics analysis,
\[
\langle J_a,J_b\rangle=\kappa_1\eta_{ab},\qquad \langle J_a,P_b\rangle=\kappa_2\eta_{ab},\qquad \langle J_a,Z_b\rangle=\kappa_3\eta_{ab},\qquad \langle P_a,P_b\rangle=\kappa_3\eta_{ab}
\]
with all other independent pairings vanishing [1805.08834; 2506.07651]. The three coefficients parameterize the Lorentz or exotic sector, the Einstein-Hilbert sector, and the genuinely Maxwell coupling.

The Chern-Simons action is
\[
I[A]=\frac{k}{4\pi}\int_{\mathcal M}\left\langle A\,dA+\frac23 A^3\right\rangle,\qquad k=\frac1{4G},
\]
and explicitly becomes
\[
\begin{aligned}
I[A]=\frac{k}{4\pi}\int_{\mathcal M}\Big[ &\alpha_0\left(\omega^a d\omega_a+\frac13 \epsilon^{abc}\omega_a\omega_b\omega_c\right) +2\alpha_1 R_a e^a \\
&+\alpha_2\left(T^a e_a+2R^a\sigma_a\right) -d\!\left(\alpha_1\omega^a e_a+\alpha_2\omega^a\sigma_a\right) \Big].
\end{aligned}
\]
Here
\[
R^a=d\omega^a+\frac12\epsilon^{abc}\omega_b\omega_c,\qquad T^a=De^a=de^a+\epsilon^{abc}\omega_b e_c
\]
[1805.08834]. In equivalent notation,
\[
I^{\rm Maxwell_3}_{\rm CS} = -\int_M\bigg[ \kappa_1\,CS(\Omega) +2\kappa_2\,E^a\wedge R_a +\kappa_3\Big(2\Sigma^a\wedge R_a+E^a\wedge T_a\Big) \bigg]
\]
[2506.07651].

Gauge transformations generated by
\[
\Lambda=\varepsilon^a P_a+\chi^a J_a+\gamma^a Z_a
\]
act as
\[
\delta_\Lambda e^a=D\varepsilon^a-\epsilon^{abc}\chi_b e_c,\qquad
\delta_\Lambda \omega^a=D\chi^a,
\]
\[
\delta_\Lambda \sigma^a=D\gamma^a+\epsilon^{abc}\left(e_b\varepsilon_c-\chi_b\sigma_c\right),
\]
and on shell diffeomorphisms are gauge transformations with \(\Lambda=\iota_\xi A\) [1805.08834].

For \(\alpha_2\neq 0\), the equations of motion reduce to
\[
T^a=0,\qquad R^a=0,\qquad D\sigma^a+\frac12\epsilon^{abc}e_b e_c=0.
\]
Thus the metric sector is torsionless and locally flat, exactly as in flat three-dimensional Einstein gravity, while the Maxwell field is generically nonzero on shell [1805.08834]. This is why all Einstein geometries remain embedded as particular solutions, even though the full field content is larger [1805.08834]. The same conclusion is emphasized in the boundary-dynamics construction: the geometry determined by \(E^a\) and \(\Omega^a\) is the same as in the Poincaré theory, while the novel ingredient is the extra flatness equation for \(\Sigma\), sourced by the \(\frac12[E,E]\) term in \(F\) [2506.07651].

A parallel formulation appears in the transgression-based derivation of a Maxwell gauged Wess-Zumino-Witten model. There the connection is written as
\[
\mathcal{A}=e^{a}\, \mathsf{P}_{a}+\frac12\, \omega ^{ab}\, \mathsf{J}_{ab}+\frac12\, \sigma ^{ab}\, \mathsf{Z}_{ab},
\]
and the field equations are
\[
R_{ab}=0,\qquad T_c=0,\qquad D_{\omega}\sigma_{ab} +2e_{a} \wedge e_{b}=0
\]
[1411.1780]. This is the same geometric content in tensorial notation.

## 3. Asymptotic structure and Maxwell-\(\mathfrak{bms}_3\)

The asymptotic analysis of the spin-2 theory is performed at null infinity in BMS gauge. The metric is written as
\[
ds^2=\mathcal M\,du^2-2\,du\,dr+\mathcal N\,du\,d\phi+r^2 d\phi^2,
\]
with
\[
\mathcal M=\mathcal M(\phi),\qquad \mathcal N=\mathcal J(\phi)+u\,\mathcal M'(\phi)
\]
[1805.08834]. A compatible asymptotic Maxwell branch is
\[
\sigma^0=\frac12(\mathcal F-r^2)d\phi+\frac12\mathcal N\,du,\qquad \sigma^1=0,\qquad \sigma^2=r\,du,
\]
where
\[
\mathcal F=\mathcal Z+u\,\mathcal J'+\frac{u^2}{2}\mathcal M''.
\]
The asymptotic data are therefore \(\mathcal M(\phi)\), \(\mathcal J(\phi)\), and \(\mathcal Z(\phi)\) [1805.08834].

In radial gauge,
\[
A=h^{-1}dh+h^{-1}ah,\qquad h=e^{-rP_0},
\]
the asymptotic connection becomes
\[
a= \frac12(\mathcal M\,du+\mathcal N\,d\phi)P_0 +du\,P_1 +\frac12\mathcal M\,d\phi\,J_0 +d\phi\,J_1 +\frac12(\mathcal N\,du+\mathcal F\,d\phi)Z_0.
\]
Residual transformations are controlled by three arbitrary boundary functions \(Y(\phi)\), \(T(\phi)\), and \(R(\phi)\), inducing
\[
\delta\mathcal M=\mathcal M'Y+2\mathcal M Y'-2Y''',
\]
\[
\delta\mathcal J=\mathcal M' T+2\mathcal M T'-2T'''+\mathcal J'Y+2\mathcal J Y',
\]
\[
\delta\mathcal Z=\mathcal M'R+2\mathcal M R'-2R'''+\mathcal J' T+2\mathcal J T'+\mathcal Z'Y+2\mathcal Z Y'
\]
[1805.08834].

The canonical charge integrates to
\[
Q[Y,T,R] = \frac{k}{4\pi}\int d\phi\, \Big[ Y(\alpha_2\mathcal Z+\alpha_0\mathcal M+\mathcal J) +T(\alpha_2\mathcal J+\mathcal M) +\alpha_2 R\,\mathcal M \Big].
\]
The independent charges are
\[
j[Y]=\frac{k}{4\pi}\int d\phi\,Y(\alpha_2\mathcal Z+\mathcal J+\alpha_0\mathcal M),
\]
\[
p[T]=\frac{k}{4\pi}\int d\phi\,T(\alpha_2\mathcal J+\mathcal M),
\]
\[
z[R]=\frac{k}{4\pi}\int d\phi\,\alpha_2 R\,\mathcal M
\]
[1805.08834].

In Fourier modes \(\mathcal J_m\), \(\mathcal P_m\), \(\mathcal Z_m\), the Poisson algebra is
\[
i\{\mathcal J_m,\mathcal J_n\} =(m-n)\mathcal J_{m+n}+\frac{c_1}{12}m^3\delta_{m+n,0},
\]
\[
i\{\mathcal J_m,\mathcal P_n\} =(m-n)\mathcal P_{m+n}+\frac{c_2}{12}m^3\delta_{m+n,0},
\]
\[
i\{\mathcal P_m,\mathcal P_n\} =(m-n)\mathcal Z_{m+n}+\frac{c_3}{12}m^3\delta_{m+n,0},
\]
\[
i\{\mathcal J_m,\mathcal Z_n\} =(m-n)\mathcal Z_{m+n}+\frac{c_3}{12}m^3\delta_{m+n,0},
\]
\[
i\{\mathcal P_m,\mathcal Z_n\}=0,\qquad i\{\mathcal Z_m,\mathcal Z_n\}=0,
\]
with
\[
c_1=12k\alpha_0,\qquad c_2=12k,\qquad c_3=12k\alpha_2
\]
[1805.08834]. This is the defining Maxwell deformation of \(\mathfrak{bms}_3\): supertranslations no longer commute but close on the new Abelian generators \(\mathcal Z_m\), so the algebra is an enlargement and deformation rather than a direct sum with decoupled currents. The 2025 boundary-dynamics derivation reproduces the same structure in the notation \((j_m,p_m,z_m)\), with \(c_i=48\pi\kappa_i\) [2506.07651].

The same algebraic pattern persists in generalized asymptotic conditions with chemical potentials. In the torsionless theory, the temporal component may be chosen as
\[
a_t=\lambda_{(0)}[\mu,\xi,\nu],
\]
and the canonical densities are
\[
\mathcal J=\frac{k}{4\pi}\left(\alpha_2\mathcal F+\alpha_1\mathcal N+\alpha_0\mathcal M\right), \qquad \mathcal P=\frac{k}{4\pi}\left(\alpha_2\mathcal N+\alpha_1\mathcal M\right), \qquad \mathcal Z=\frac{k}{4\pi}\alpha_2\mathcal M
\]
[2510.25559]. This suggests that the Maxwell charge sector can be treated on the same footing as mass and angular momentum once chemical potentials are included.

## 4. Stationary solutions, vacuum structure, and thermodynamic sector

The stationary sector of the torsionless theory can be written in ADM form as
\[
ds^2=-N^2 dt^2+\frac{dr^2}{N^2}+r^2(d\varphi+N_\varphi dt)^2,
\]
with
\[
N^2=-M+\frac{J^2}{4r^2}, \qquad N_\varphi=-\frac{J}{2r^2}.
\]
A corresponding dreibein is
\[
\bar e^0=N\,dt,\qquad \bar e^1=N^{-1}dr,\qquad \bar e^2=r(d\varphi+N_\varphi dt),
\]
with torsionless spin connection
\[
\bar\omega^0=N\,d\varphi,\qquad \bar\omega^1=\frac{J}{2r^2N}\,dr,\qquad \bar\omega^2=-\frac{J}{2r}\,d\varphi
\]
[1805.08834]. The Maxwell field is nontrivial and depends on constants \(b,c\) and gauge functions \(S,W,X\), with a convenient gauge choice
\[
S=-\frac{1}{N}\left(\frac{r^3}{2J}+br\right), \qquad W=\frac{S}{r}, \qquad X=\frac{r}{J N^2}
\]
[1805.08834].

The diffeomorphism or Noether charge is
\[
Q[\xi]=\frac{k}{4\pi}\int_{\partial\Sigma}\langle A\,\iota_\xi A\rangle.
\]
For stationary configurations this yields
\[
m\equiv Q[\partial_t] = \frac{k}{2}\left[M+\alpha_2\left(2Mc-\frac{J}{2}\right)\right],
\]
\[
j\equiv Q[\partial_\varphi] = \frac{k}{2}\left(-\frac{J}{2}+\alpha_0 M-2\alpha_2 bJ\right).
\]
Thus the Maxwell field modifies both energy and angular momentum, and even a static geometry can carry nontrivial total charges because the Maxwell sector contributes [1805.08834].

For Minkowski geometry,
\[
(M,J,b,c)=(-1,0,b,c),
\]
the vacuum charges are
\[
m_0=-\frac{k}{2}(1+2c\alpha_2), \qquad j_0=-\frac{k\alpha_0}{2}.
\]
Hence the vacuum energy is shifted by the Maxwell parameter \(c\), while the vacuum angular momentum is controlled by the gravitational Chern-Simons coupling \(\alpha_0\) [1805.08834].

Recent thermodynamic analyses extend this sector by allowing the most general temporal components of the gauge fields compatible with the asymptotic Maxwell algebras and chemical potentials. In the torsionless theory, the admissible stationary configurations are locally flat cosmological spacetimes, while in the torsional deformation they become BTZ-like black hole geometries [2510.25559]. The same work states that both theories admit solutions carrying mass, angular momentum, and an additional global spin-2 charge, and that the entropy can be expressed in terms of the horizon area and its spin-2 analogues as a reparametrization-invariant integral of the induced spin-2 fields on the spacelike section of the horizon [2510.25559]. A plausible implication is that the Maxwell field plays a thermodynamic role analogous to an additional higher-spin-like charge sector, although the paper formulates this specifically as an extra global spin-2 charge rather than as a propagating higher-spin field [2510.25559].

## 5. Boundary dynamics and induced two-dimensional theories

Three-dimensional Maxwell Chern-Simons gravity admits a boundary reduction to a chiral Wess-Zumino-Witten theory and, after imposing asymptotically flat conditions, to a Maxwellian extension of flat Liouville theory [2506.07651]. In radial gauge,
\[
\mathcal A=h^{-1}\alpha\,h+h^{-1}dh,
\]
with reduced boundary connection
\[
\alpha= \left(\frac{\mathcal M}{2}\,d\phi\right)J_0+d\phi\,J_1 +\frac12\left(\mathcal M\,du+\mathcal N\,d\phi\right)P_0+du\,P_1 +\frac12\left(\mathcal N\,du+\mathcal R\,d\phi\right)Z_0,
\]
and
\[
\mathcal M=\mathcal M(\phi),\qquad \mathcal N=\mathcal T(\phi)+u\,\mathcal M'(\phi),\qquad \mathcal R=\mathcal Y(\phi)+u\,\mathcal T'(\phi)+\frac{u^2}{2}\mathcal M''.
\]
Residual gauge transformations act on \((\mathcal M,\mathcal T,\mathcal Y)\) by the Maxwellian extension of the usual \(\mathfrak{bms}_3\) laws [2506.07651].

After solving the bulk flatness constraint and substituting back, one obtains a chiral Maxwell-WZW model. Imposing the asymptotically flat boundary conditions
\[
e_u=\omega_\phi,\qquad \omega_u=0,\qquad \sigma_u=e_\phi
\]
and using a Gauss-like parametrization of the boundary group element leads, after field redefinitions,
\[
\varphi=\rho_2,\qquad \xi=2(\rho_0\beta_1-\beta_2),\qquad \chi=2(\rho_0\gamma_1-\gamma_2)-\left(\beta_0+\rho_0\beta_2-\frac12\rho_0^2\beta_1\right)\beta_1,
\]
to the reduced boundary action
\[
I_{\rm MBD} = \int du\,d\phi\, \left( \kappa_1\,\varphi'\dot\varphi +\kappa_2(\xi'\dot\varphi-\varphi'^2) +\kappa_3\left(\chi'\dot\varphi+\frac14\dot\xi\,\xi'-\xi'\varphi'\right) \right).
\]
The \(\kappa_2\) term is exactly the flat Liouville action of Barnich-Gomberoff-González, the \(\kappa_1\) term is the exotic extension, and the \(\kappa_3\) term is the genuinely Maxwell correction [2506.07651].

The same boundary theory also arises as the geometric action on coadjoint orbits of the Maxwell extension of the \(\mathfrak{bms}_3\) group. In that formulation, the orbit action reproduces the kinetic part of the boundary dynamics, and the full action is obtained by adding the Hamiltonian
\[
\mathcal H=Q[0,-1,0] = -\int d\phi\,\mathcal P = \frac{1}{48\pi}\int d\phi\, \left( c_2\,\varphi'^2+c_3\,\varphi'\xi' \right)
\]
[2506.07651]. The boundary theory is therefore not merely inspired by the asymptotic algebra; it is literally the corresponding coadjoint-orbit action.

A transgression-based antecedent of this boundary correspondence appears in the construction of a Maxwell gauged Wess-Zumino-Witten model. There, taking the Maxwell group element
\[
g=e^{-\frac12 h^{ab}\mathsf Z_{ab}}e^{-\phi^a \mathsf P_a}
\]
and comparing the Chern-Simons forms of \(\mathcal A\) and \(\mathcal A^g\), one obtains the induced two-dimensional action
\[
\mathrm{S}^{(2)}\left[\omega, \phi, h, e \right]=\kappa\, \int_{\partial \mathcal{M}} \big(\alpha _{1}\, \epsilon _{abc}\, R^{ab}\, \phi ^{c}+\alpha _{2}\,  R_{~c}^{a}\wedge h_{~a}^{c} \big)
\]
[1411.1780]. On shell, after a field redefinition, this reduces to the \(n=1\) equations of two-dimensional topological gravity, while still retaining a distinct off-shell boundary field content [1411.1780].

## 6. Torsion, Carrollian limits, hypersymmetry, and higher-spin generalizations

A torsional generalization is obtained by deforming the Maxwell algebra so that
\[
[P_a,P_b]=\epsilon_{ab}{}^{c}\left(M_c+\frac1\ell P_c\right)
\]
in the notation \(\{J_a,P_a,M_a\}\). The corresponding Chern-Simons action contains both cosmological and torsional terms and reduces, for \(f^a=0\), to a particular sector of the Mielke-Baekler model [2005.07690]. For \(\alpha_2\neq0\), the field equations imply
\[
R^a=0,\qquad K^a=0,\qquad W^a=0,
\]
hence
\[
T^a=-\frac1\ell \epsilon^{abc}e_b e_c.
\]
The geometry is therefore Riemann-Cartan or teleparallel rather than torsionless [2005.07690]. The asymptotic algebra can be written either in the \((J,P,M)\) basis or, after a change of basis, as \(\widehat{\mathfrak{bms}_3\oplus\widehat{\mathfrak{vir}}\) with three independent central charges [2005.07690]. The 2025 thermodynamic analysis of the torsional theory identifies BTZ-like black hole geometries in this sector [2510.25559].

The ultra-relativistic limit of Maxwell gravity leads to Maxwellian Carroll gravity. A direct Carroll contraction of the relativistic Maxwell Chern-Simons theory yields a finite action, but the associated invariant tensor is degenerate because the \(Z_a\) sector lacks a kinetic pairing [2107.05716]. To cure this, an extended Maxwellian Carroll algebra with additional generators \(S\), \(L_a\), and \(T\) is introduced; its nondegenerate invariant tensor supports a proper Chern-Simons theory, and the corresponding field equations set all curvatures to zero [2107.05716]. This shows that the Carrollian limit exists, but a well-defined nondegenerate Chern-Simons formulation requires algebraic extension.

Maxwell Chern-Simons gravity also admits hypersymmetric extensions. The simplest hyper-Maxwell algebra enlarges \(\{J_a,P_a,Z_a\}\) by a \(\Gamma\)-traceless vector-spinor \(Q_{\alpha a}\), with
\[
\{Q_{\alpha a},Q_{\beta b}\} = -\frac43 \eta_{ab} Z_c (C\Gamma^c)_{\alpha\beta} +\frac53 \epsilon_{abc} C_{\alpha\beta} Z^c +\frac23 Z_{(a|}(C\Gamma_{|b)})_{\alpha\beta}.
\]
The gauge connection becomes
\[
A=e^aP_a+\omega^aJ_a+k^aZ_a+\bar\psi^a Q_a,
\]
and the Chern-Simons action acquires the term \(2i\bar\psi_a D\psi^a\) in the \(\alpha_2\) sector [2105.12243]. A notable feature is
\[
\delta e^a=0,
\]
so hypersymmetry acts on the Maxwell field rather than on the dreibein in the standard supergravity fashion [2105.12243]. More elaborate contractions of \(\mathfrak{osp}(1|4)\) and \(\mathfrak{sp}(4)\) produce further hyper-Maxwell theories with non-propagating spin-4 gauge fields and one or two spin-\(\frac52\) fields [2105.12243].

Higher-spin generalizations begin with the spin-3 Maxwell algebra constructed by \(S\)-expansion of \(\mathfrak{sl}(3,\mathbb R)\). Besides \(\{M_a,P_a,Z_a\}\), one introduces symmetric traceless generators \(\{M_{ab},P_{ab},Z_{ab}\}\), so that the gauge connection is
\[
A = \omega^a M_a+e^a P_a+k^a Z_a+\omega^{ab}M_{ab}+e^{ab}P_{ab}+k^{ab}Z_{ab}.
\]
The corresponding Chern-Simons action defines a vanishing-cosmological-constant higher-spin gravity model coupled to higher-spin topological matter, and it arises as the flat limit of the higher-spin AdS-Lorentz theory [1712.09975]. The asymptotic analysis of the spin-3 Maxwell theory then yields a nonlinear higher-spin extension of the Maxwell-\(\mathfrak{bms}_3\) algebra, denoted
\[
\mathfrak{hs}_3\mathfrak{max}\text{-}\mathfrak{bms}_3,
\]
with six towers of generators and three independent central charges. The same algebra can be obtained as the \(\ell\to\infty\) limit of three copies of \(\mathcal W_3\) [2412.04992]. This places Maxwell gravity in a hierarchy where the spin-2 Maxwell deformation of \(\mathfrak{bms}_3\) extends naturally to nonlinear higher-spin asymptotic symmetries [2412.04992].

A common misconception is to conflate these Maxwell-gravity constructions with unrelated three-dimensional Chern-Simons systems that involve gauge sectors beyond gravity. By contrast, studies based on enlarged orthogonal groups unifying Lorentz and internal gauge Chern-Simons terms, or supersymmetric \(su(2,1|2)\) models with \(SU(2)\) gauge symmetry, do not employ the Maxwell algebra and do not introduce the characteristic \(Z_a\) generators with \([P_a,P_b]\sim Z_a\) [1703.08237; 1505.03834]. Likewise, static Maxwell-Chern-Simons electrodynamics mapped to two-dimensional Euclidean dilaton gravity is not a reformulation of three-dimensional gravity as a Chern-Simons theory for the Maxwell algebra, but an induced two-dimensional gravitational sector from abelian Maxwell-Chern-Simons electrodynamics [2503.19803].

In aggregate, the literature presents three-dimensional Maxwell Chern-Simons gravity as a topological but nontrivially enlarged gauge theory in which the extra Maxwell sector reshapes asymptotic symmetries, boundary dynamics, vacuum charges, and admissible extensions. The standard torsionless theory preserves the local metric content of flat Einstein gravity while adding a genuine Maxwell deformation of \(\mathfrak{bms}_3\) and a third central charge [1805.08834]. Boundary reduction produces a Maxwellian extension of flat Liouville theory [2506.07651]. Torsional, Carrollian, hypersymmetric, and higher-spin versions show that the Maxwell mechanism extends coherently across several deformation regimes, provided the invariant bilinear form remains nondegenerate and the enlarged algebra is chosen appropriately [2005.07690; 2107.05716; 2105.12243; 1712.09975; 2412.04992].

Source: https://www.emergentmind.com/topics/three-dimensional-maxwell-chern-simons-gravity