3D Maxwell Chern-Simons Gravity

• Three-dimensional Maxwell Chern-Simons gravity is a topological gauge theory that extends flat Einstein gravity by replacing the Poincaré algebra with the Maxwell algebra.
• It employs a Chern-Simons formulation where the dreibein, spin connection, and an additional Maxwell gauge field together yield modified vacuum structures and asymptotic symmetries.
• The theory’s rich algebraic structure enables versatile extensions, including torsional, Carrollian, hypersymmetric, and higher-spin generalizations with corresponding boundary dynamics.

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 (Concha et al., 2018). 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 (Concha et al., 2018).

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 $\mathfrak{iso}(2,1)$0 span an Abelian ideal (Concha et al., 2018). In tensorial notation this same extension may be written as $\mathfrak{iso}(2,1)$1, with $\mathfrak{iso}(2,1)$2 in three dimensions [(Höfenstock et al., 9 Jun 2025); (Valdivia, 2014)].

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 $\mathfrak{iso}(2,1)$3 or, in another notation, $\mathfrak{iso}(2,1)$4, that modifies both the vacuum and the asymptotic charges [(Concha et al., 2018); (Valdivia, 2014)]. The Maxwell algebra can also be obtained from other parent structures. One route is as a contraction of the AdS-Lorentz algebra (Concha et al., 2018, Concha et al., 2021). Another is through $\mathfrak{iso}(2,1)$5-expansion of AdS or $\mathfrak{iso}(2,1)$6, which is especially important for higher-spin extensions because it simultaneously produces the invariant tensor needed for the Chern-Simons action [(Valdivia, 2014); (Caroca et al., 2017)].

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 [(Bergshoeff et al., 2014); (Afshar et al., 2014)]. By contrast, in Maxwell gravity the new one-form belongs directly to the gauge connection of the enlarged algebra (Concha et al., 2018).

2. Chern-Simons formulation and field equations

In the standard spin-2 Maxwell theory, the gauge connection is

$\mathfrak{iso}(2,1)$7

where $\mathfrak{iso}(2,1)$8 is the dreibein, $\mathfrak{iso}(2,1)$9 the dualized spin connection, and $J_a$0 the gravitational Maxwell gauge field (Concha et al., 2018). An equivalent $J_a$1-valued notation is

$J_a$2

with $J_a$3, $J_a$4, and $J_a$5 (Höfenstock et al., 9 Jun 2025).

The nonvanishing invariant bilinear form is

$J_a$6

$J_a$7

or, in the notation of the boundary-dynamics analysis,

$J_a$8

with all other independent pairings vanishing (Concha et al., 2018, Höfenstock et al., 9 Jun 2025). The three coefficients parameterize the Lorentz or exotic sector, the Einstein-Hilbert sector, and the genuinely Maxwell coupling.

The Chern-Simons action is

$J_a$9

and explicitly becomes

$P_a$0

Here

$P_a$1

(Concha et al., 2018). In equivalent notation,

$P_a$2

(Höfenstock et al., 9 Jun 2025).

Gauge transformations generated by

$P_a$3

act as

$P_a$4

$P_a$5

and on shell diffeomorphisms are gauge transformations with $P_a$6 (Concha et al., 2018).

For $P_a$7, the equations of motion reduce to

$P_a$8

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 (Concha et al., 2018). This is why all Einstein geometries remain embedded as particular solutions, even though the full field content is larger (Concha et al., 2018). The same conclusion is emphasized in the boundary-dynamics construction: the geometry determined by $P_a$9 and $Z_a$0 is the same as in the Poincaré theory, while the novel ingredient is the extra flatness equation for $Z_a$1, sourced by the $Z_a$2 term in $Z_a$3 (Höfenstock et al., 9 Jun 2025).

A parallel formulation appears in the transgression-based derivation of a Maxwell gauged Wess-Zumino-Witten model. There the connection is written as

$Z_a$4

and the field equations are

$Z_a$5

(Valdivia, 2014). This is the same geometric content in tensorial notation.

3. Asymptotic structure and Maxwell-$Z_a$6

The asymptotic analysis of the spin-2 theory is performed at null infinity in BMS gauge. The metric is written as

$Z_a$7

with

$Z_a$8

(Concha et al., 2018). A compatible asymptotic Maxwell branch is

$Z_a$9

where

$[P_a,P_b]\sim Z_c$0

The asymptotic data are therefore $[P_a,P_b]\sim Z_c$1, $[P_a,P_b]\sim Z_c$2, and $[P_a,P_b]\sim Z_c$3 (Concha et al., 2018).

In radial gauge,

$[P_a,P_b]\sim Z_c$4

the asymptotic connection becomes

$[P_a,P_b]\sim Z_c$5

Residual transformations are controlled by three arbitrary boundary functions $[P_a,P_b]\sim Z_c$6, $[P_a,P_b]\sim Z_c$7, and $[P_a,P_b]\sim Z_c$8, inducing

$[P_a,P_b]\sim Z_c$9

$\{J_a,P_a,Z_a\},\qquad a=0,1,2,$0

$\{J_a,P_a,Z_a\},\qquad a=0,1,2,$1

(Concha et al., 2018).

The canonical charge integrates to

$\{J_a,P_a,Z_a\},\qquad a=0,1,2,$2

The independent charges are

$\{J_a,P_a,Z_a\},\qquad a=0,1,2,$3

$\{J_a,P_a,Z_a\},\qquad a=0,1,2,$4

$\{J_a,P_a,Z_a\},\qquad a=0,1,2,$5

(Concha et al., 2018).

In Fourier modes $\{J_a,P_a,Z_a\},\qquad a=0,1,2,$6, $\{J_a,P_a,Z_a\},\qquad a=0,1,2,$7, $\{J_a,P_a,Z_a\},\qquad a=0,1,2,$8, the Poisson algebra is

$\{J_a,P_a,Z_a\},\qquad a=0,1,2,$9

$$[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,\qquad [J_a,Z_b]=\epsilon_{abc}Z^c,$$0

$$[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,\qquad [J_a,Z_b]=\epsilon_{abc}Z^c,$$1

$$[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,\qquad [J_a,Z_b]=\epsilon_{abc}Z^c,$$2

$$[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,\qquad [J_a,Z_b]=\epsilon_{abc}Z^c,$$3

with

$$[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,\qquad [J_a,Z_b]=\epsilon_{abc}Z^c,$$4

(Concha et al., 2018). This is the defining Maxwell deformation of $$[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,\qquad [J_a,Z_b]=\epsilon_{abc}Z^c,$$5: supertranslations no longer commute but close on the new Abelian generators $$[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,\qquad [J_a,Z_b]=\epsilon_{abc}Z^c,$$6, 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 $$[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,\qquad [J_a,Z_b]=\epsilon_{abc}Z^c,$$7, with $$[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,\qquad [J_a,Z_b]=\epsilon_{abc}Z^c,$$8 (Höfenstock et al., 9 Jun 2025).

The same algebraic pattern persists in generalized asymptotic conditions with chemical potentials. In the torsionless theory, the temporal component may be chosen as

$$[J_a,J_b]=\epsilon_{abc}J^c,\qquad [J_a,P_b]=\epsilon_{abc}P^c,\qquad [J_a,Z_b]=\epsilon_{abc}Z^c,$$9

and the canonical densities are

$$[P_a,P_b]=\epsilon_{abc}Z^c,\qquad [Z_a,Z_b]=0,\qquad [Z_a,P_b]=0.$$0

(Avilés et al., 29 Oct 2025). 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

$$[P_a,P_b]=\epsilon_{abc}Z^c,\qquad [Z_a,Z_b]=0,\qquad [Z_a,P_b]=0.$$1

with

$$[P_a,P_b]=\epsilon_{abc}Z^c,\qquad [Z_a,Z_b]=0,\qquad [Z_a,P_b]=0.$$2

A corresponding dreibein is

$$[P_a,P_b]=\epsilon_{abc}Z^c,\qquad [Z_a,Z_b]=0,\qquad [Z_a,P_b]=0.$$3

with torsionless spin connection

$$[P_a,P_b]=\epsilon_{abc}Z^c,\qquad [Z_a,Z_b]=0,\qquad [Z_a,P_b]=0.$$4

(Concha et al., 2018). The Maxwell field is nontrivial and depends on constants $$[P_a,P_b]=\epsilon_{abc}Z^c,\qquad [Z_a,Z_b]=0,\qquad [Z_a,P_b]=0.$$5 and gauge functions $$[P_a,P_b]=\epsilon_{abc}Z^c,\qquad [Z_a,Z_b]=0,\qquad [Z_a,P_b]=0.$$6, with a convenient gauge choice

$$[P_a,P_b]=\epsilon_{abc}Z^c,\qquad [Z_a,Z_b]=0,\qquad [Z_a,P_b]=0.$$7

(Concha et al., 2018).

The diffeomorphism or Noether charge is

$$[P_a,P_b]=\epsilon_{abc}Z^c,\qquad [Z_a,Z_b]=0,\qquad [Z_a,P_b]=0.$$8

For stationary configurations this yields

$$[P_a,P_b]=\epsilon_{abc}Z^c,\qquad [Z_a,Z_b]=0,\qquad [Z_a,P_b]=0.$$9

$[P_a,P_b]\sim Z_c$0

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 (Concha et al., 2018).

For Minkowski geometry,

$[P_a,P_b]\sim Z_c$1

the vacuum charges are

$[P_a,P_b]\sim Z_c$2

Hence the vacuum energy is shifted by the Maxwell parameter $[P_a,P_b]\sim Z_c$3, while the vacuum angular momentum is controlled by the gravitational Chern-Simons coupling $[P_a,P_b]\sim Z_c$4 (Concha et al., 2018).

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 (Avilés et al., 29 Oct 2025). 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 (Avilés et al., 29 Oct 2025). 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 (Avilés et al., 29 Oct 2025).

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 (Höfenstock et al., 9 Jun 2025). In radial gauge,

$[P_a,P_b]\sim Z_c$5

with reduced boundary connection

$[P_a,P_b]\sim Z_c$6

and

$[P_a,P_b]\sim Z_c$7

Residual gauge transformations act on $[P_a,P_b]\sim Z_c$8 by the Maxwellian extension of the usual $[P_a,P_b]\sim Z_c$9 laws (Höfenstock et al., 9 Jun 2025).

After solving the bulk flatness constraint and substituting back, one obtains a chiral Maxwell-WZW model. Imposing the asymptotically flat boundary conditions

$\mathfrak{iso}(2,1)$00

and using a Gauss-like parametrization of the boundary group element leads, after field redefinitions,

$\mathfrak{iso}(2,1)$01

to the reduced boundary action

$\mathfrak{iso}(2,1)$02

The $\mathfrak{iso}(2,1)$03 term is exactly the flat Liouville action of Barnich-Gomberoff-González, the $\mathfrak{iso}(2,1)$04 term is the exotic extension, and the $\mathfrak{iso}(2,1)$05 term is the genuinely Maxwell correction (Höfenstock et al., 9 Jun 2025).

The same boundary theory also arises as the geometric action on coadjoint orbits of the Maxwell extension of the $\mathfrak{iso}(2,1)$06 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

$\mathfrak{iso}(2,1)$07

(Höfenstock et al., 9 Jun 2025). 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

$\mathfrak{iso}(2,1)$08

and comparing the Chern-Simons forms of $\mathfrak{iso}(2,1)$09 and $\mathfrak{iso}(2,1)$10, one obtains the induced two-dimensional action

$\mathfrak{iso}(2,1)$11

(Valdivia, 2014). On shell, after a field redefinition, this reduces to the $\mathfrak{iso}(2,1)$12 equations of two-dimensional topological gravity, while still retaining a distinct off-shell boundary field content (Valdivia, 2014).

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

A torsional generalization is obtained by deforming the Maxwell algebra so that

$\mathfrak{iso}(2,1)$13

in the notation $\mathfrak{iso}(2,1)$14. The corresponding Chern-Simons action contains both cosmological and torsional terms and reduces, for $\mathfrak{iso}(2,1)$15, to a particular sector of the Mielke-Baekler model (Adami et al., 2020). For $\mathfrak{iso}(2,1)$16, the field equations imply

$\mathfrak{iso}(2,1)$17

hence

$\mathfrak{iso}(2,1)$18

The geometry is therefore Riemann-Cartan or teleparallel rather than torsionless (Adami et al., 2020). The asymptotic algebra can be written either in the $\mathfrak{iso}(2,1)$19 basis or, after a change of basis, as $\mathfrak{iso}(2,1)$20 with three independent central charges (Adami et al., 2020). The 2025 thermodynamic analysis of the torsional theory identifies BTZ-like black hole geometries in this sector (Avilés et al., 29 Oct 2025).

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 $\mathfrak{iso}(2,1)$21 sector lacks a kinetic pairing (Concha et al., 2021). To cure this, an extended Maxwellian Carroll algebra with additional generators $\mathfrak{iso}(2,1)$22, $\mathfrak{iso}(2,1)$23, and $\mathfrak{iso}(2,1)$24 is introduced; its nondegenerate invariant tensor supports a proper Chern-Simons theory, and the corresponding field equations set all curvatures to zero (Concha et al., 2021). 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 $\mathfrak{iso}(2,1)$25 by a $\mathfrak{iso}(2,1)$26-traceless vector-spinor $\mathfrak{iso}(2,1)$27, with

$\mathfrak{iso}(2,1)$28

The gauge connection becomes

$\mathfrak{iso}(2,1)$29

and the Chern-Simons action acquires the term $\mathfrak{iso}(2,1)$30 in the $\mathfrak{iso}(2,1)$31 sector (Caroca et al., 2021). A notable feature is

$\mathfrak{iso}(2,1)$32

so hypersymmetry acts on the Maxwell field rather than on the dreibein in the standard supergravity fashion (Caroca et al., 2021). More elaborate contractions of $\mathfrak{iso}(2,1)$33 and $\mathfrak{iso}(2,1)$34 produce further hyper-Maxwell theories with non-propagating spin-4 gauge fields and one or two spin-$\mathfrak{iso}(2,1)$35 fields (Caroca et al., 2021).

Higher-spin generalizations begin with the spin-3 Maxwell algebra constructed by $\mathfrak{iso}(2,1)$36-expansion of $\mathfrak{iso}(2,1)$37. Besides $\mathfrak{iso}(2,1)$38, one introduces symmetric traceless generators $\mathfrak{iso}(2,1)$39, so that the gauge connection is

$\mathfrak{iso}(2,1)$40

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 (Caroca et al., 2017). The asymptotic analysis of the spin-3 Maxwell theory then yields a nonlinear higher-spin extension of the Maxwell-$\mathfrak{iso}(2,1)$41 algebra, denoted

$\mathfrak{iso}(2,1)$42

with six towers of generators and three independent central charges. The same algebra can be obtained as the $\mathfrak{iso}(2,1)$43 limit of three copies of $\mathfrak{iso}(2,1)$44 (Concha et al., 2024). This places Maxwell gravity in a hierarchy where the spin-2 Maxwell deformation of $\mathfrak{iso}(2,1)$45 extends naturally to nonlinear higher-spin asymptotic symmetries (Concha et al., 2024).

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 $\mathfrak{iso}(2,1)$46 models with $\mathfrak{iso}(2,1)$47 gauge symmetry, do not employ the Maxwell algebra and do not introduce the characteristic $\mathfrak{iso}(2,1)$48 generators with $\mathfrak{iso}(2,1)$49 (Saghir et al., 2017, Alvarez et al., 2015). 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 (Bittencourt et al., 25 Mar 2025).

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{iso}(2,1)$50 and a third central charge (Concha et al., 2018). Boundary reduction produces a Maxwellian extension of flat Liouville theory (Höfenstock et al., 9 Jun 2025). 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 (Adami et al., 2020, Concha et al., 2021, Caroca et al., 2021, Caroca et al., 2017, Concha et al., 2024).