The Maxwell group in 2+1 dimensions and its infinite-dimensional enhancements

The Maxwell group in 2+1 dimensions is given by a particular extension of a semi-direct product. This mathematical structure provides a sound framework to study different generalizations of the Maxwell symmetry in three space-time dimensions. By giving a general definition of extended semi-direct products, we construct infinite-dimensional enhancements of the Maxwell group that enlarge the ${\rm ISL}(2,\mathbb{R})$ Kac-Moody group and the ${\rm BMS}_3$ group by including non-commutative supertranslations. The coadjoint representation in each case is defined, and the corresponding geometric actions on coadjoint orbits are presented. These actions lead to novel Wess-Zumino terms that naturally realize the aforementioned infinite-dimensional symmetries. We briefly elaborate on potential applications in the contexts of three-dimensional gravity, higher-spin symmetries, and quantum Hall systems.


Introduction
The Maxwell algebra is an extension of the Poincaré symmetry that naturally arises in the study of particle systems in the presence of a constant electromagnetic field. It includes, apart from translations and Lorentz transformations, an Abelian ideal spanned by a second rank antisymmetric Lorentz tensor F µν , associated the electromagnetic field strength F µν " B µ A ν´Bν A µ . This enlargement stems from the fact that minimal coupling of a particle to the electromagnetic field requires to modify the canonical momentum in the form p µ Ñ π µ " p µ`Aµ . When the electromagnetic field is constant (i.e. homogeneous and static), it can be thought of as an extension of Minkowski space, whose isometries include non-commutative translations. This leads to the Maxwell algebra: rJ µν , J ρσ s " η µρ J νσ`ηνσ J µρ´ηµσ J νρ´ηνρ J µσ , rΠ µ , Π ν s " F µν , rJ µν , Π ρ s " η µρ Π ν´ηνρ Π µ , rΠ µ , F νρ s " 0 , rJ µν , F ρσ s " η µρ F νσ`ηνσ F µρ´ηµσ F νρ´ηνρ F µσ , rF µν , F ρσ s " 0 . (1.1) The Maxwell group is given by the semi-direct product of the Lorentz group and a centrally extended space of translations. It was first studied in [1] as the covariance group of the (3+1)-dimensional Klein Gordon and Dirac equations associated to a particle moving in a constant electromagnetic field 1 . However, previous indications of this symmetry were found in [5], where the kinematic symmetry of this kind of particle systems was shown to include a central extension of the translation group. This extension was characterized by the electric charge in pretty much the same way as the mass appears in the central extension of the Bargmann algebra.
Later on, the Maxwell algebra was used in the context of gravity. By gauging it, one can define a modified gravitational theory that extends General Relativity by including a generalized cosmological term [6]. In this case, the components of the antisymmetric tensor F µν are not related to an electromagnetic field, but reinterpreted as extra generators associated to the new field content of the theory. This provides a geometric framework to introduce vector inflatons in cosmological models [7] In the last years, different generalizations to higher dimensions, supersymmetric extensions, and deformations of the Maxwell algebra have been formulated 2 [12][13][14][15][16][17][18][19]. By means of algebra expansion methods [20,21], different families of Lie algebras that further extend the commutation relations (1.1) have been constructed, leading to novel (super)gravity theories in diverse dimensions [22][23][24][25][26][27][28][29]. Moreover, the Maxwell algebra can be understood as the simplest Poincaré extension within an infinitedimensional free algebra, which describes the motion of a charge distribution in an generic electro-magnetic field through multipole expansion [30,31]. 1 For a detailed analysis including the non-relativistic case, see [2], and for a geometric derivation of wave equations for particles coupled to constant electromagnetic field based on local representations of the Maxwell group, see [3,4]. 2 The supersymmetric extension of the Maxwell algebra has been used to construct supergravity models [8][9][10][11], as well as in the description of the superparticle on constant backgrounds [12].
The (2+1)-dimensional version of the Maxwell symmetry naturally arises when studying anyons coupled to a constant external electromagnetic field [32,33]. This is of particular interest in the study of quantum Hall systems, where magnetic translations associated to an electric charge in presence of a constant magnetic field B have non-vanishing Poisson brackets [34,35] tΠ x , Π y u " B .
(1. 2) In fact, in [36] it is shown that the Maxwell algebra properly describes the Girvin-Plazmann-MacDonald algebra for magnetic translations in quantum Hall systems [37]. Furthermore, a relativistic version of the Wen-Zee term was constructed by considering a Chern-Simons action invariant under the Maxwell algebra as an effective theory for the boundary dynamics of a topological insulator.
On the other hand, Chern-Simons forms are interesting in the context of gravitational theories, as they provide a gauge formulation of three-dimensional Einstein gravity [38,39]. Along these lines, a Maxwell-invariant Chern-Simons action defines an extended theory for gravity in 2+1 dimensions 3,4 [47][48][49]. In [49], it was shown that, given suitable boundary conditions, the asymptotic symmetry of such extended theory is given by an infinite-dimensional enhancement of the Maxwell algebra in 2+1 dimensions. This algebra can be understood as an Maxwell-like extension of the x bms 3 algebra 5 , known from the analysis of asymptotically flat three-dimensional Einstein gravity [51]. Moreover, the same extended x bms 3 symmetry was previously obtained as a particular expansion of the Virasoro algebra, and can also be recovered by means of a generalized Sugawara construction of a Maxwell-Kac-Moody algebra [52]. The Maxwell-Kac-Moody algebra, in turn, is by definition the symmetry of the WZW model that follows from a Maxwell-invariant Chern-Simons action after Hamiltonian reduction. The fact that both infinite-dimensional symmetries can be obtained from a Chern-Simons theory invariant under the maxwell 3 algebra suggests that they could also be relevant in the context of quantum Hall systems. Indeed, the analogy between three-dimensional gravity and the quantum Hall effect has been already considered in the literature (see, for instance, [53,54]). As, so far, only the algebras are known, a natural question is: what is the group structure behind these infinite-dimensional symmetries? symmetry correspond to semi-direct product groups [55][56][57]. Therefore, the group structure behind the aforementioned Maxwell-like symmetries must be given by a general mathematical construction that extends semi-direct products. In this article we show that this is indeed the case.
In this paper, we study the group structure of the Maxwell symmetry in 2+1 dimensions and its infinite-dimensional generalizations. The construction is based on a non-central extension of a semi-direct product group. Following the general construction of semi-direct products [58,59], the adjoint and the coadjoint representations of the extension here presented are given. The Kirillov-Kostant-Souriau (KKS) symplectic structure on coadjoint orbits is defined, and the Hamiltonian geometric action associated to an extended semi-direct product is computed along the lines of [60][61][62][63] (for a recent discussion in the context of three-dimensional gravity see [57]). Based on these definitions, we derive the Maxwell group in D dimensions as a particular case of extended semi-direct product and the action relativistic particle in a constant electromagnetic field as the corresponding geometric action. Subsequently, we focus on the (2+1)-dimensional case. Following the results found in [55,56], we show that the Maxwell group in three space-time dimensions can be formulated as a special type of extended semi-direct product based on the SL p2, Rq group. This construction can be generalized to define higher-spin extensions of the Maxwell symmetry in 2+1 dimensions, as well as infinite-dimensional enhancements of the BMS 3 group and the loop group LISL p2, Rq. By including central extensions, we construct Maxwell-like extensions of the z L ISL p2, Rq Kac-Moody group and the z BMS 3 group, whose associated Lie algebras are exactly the ones found in [52]. The geometric actions associated to these extended semi-direct products are shown to lead to novel Wess-Zumino terms. They are naturally globally invariant under the considered symmetry groups, and generalize the actions previously obtained in [60][61][62](for a recent discussion in the context of three-dimensional gravity see [57,64]).
The paper is organized as follows: In Section 2 we introduce extended semi-direct groups, define the adjoint and coadjoint representations, and construct the geometric action on coadjoint orbits. In Section 3 we show how the Maxwell group fits in this description. In the 2+1-dimensional case, we generalize the results to obtain Maxwell extensions of higher-spin symmetries, loop groups and the BMS 3 group. In section 4, we define central extensions of extended semi-direct products, and present two novel Wezz-Zumino terms based on the Maxwell extension of the Kac-Moody and the z BMS 3 group. In section 5 we conclude with a discussion and possible applications of our results.

Extending semi-direct products
In this section we define a particular extension of semi-direct products, which will be used later to formulate the Maxwell group in 2+1 dimensions and its generalizations.
Naturally, the starting point in our construction is a semi-direct product group K˙σV , where K is a Lie group, V is a vector space and σ : K Ñ GLpV q is a representation of K on V . The elements of a semi-direct product group are denoted by pairs pU, αq, where U P K and α P V , and the group operation is given by [58,59] pU, αq ‚ pV , βq " pU¨V , α`σ U βq , where U¨V denotes the product in K . Groups with this structure are ubiquitous in physics. Some examples are the Euclidean, the Galilei and the Poincaré groups [65], as well as the BMS group in three and four dimensions 6 [55,68].

Extended semi-direct product groups
In order to extend the semi-direct product group (2.1), we introduce a second vector spaceṼ , together with a skew-symmetric bilinear mapˆ: VˆV ÑṼ , and a representation ρ : K Ñ GLpṼ q of K onṼ compatible with σ in the sense that Then, we define the extended group V ext " V bṼ with elements pα, aq, α P V , a PṼ , and the following product Note that this operation is non-commutative, and therefore V ext endowed with the productˆdoes not define an Abelian group. Indeed, V ext defines a central extension of V , as the function Cpα, βq " 1 2 αˆβ is aṼ -valued two-cocycle on V , i.e. Cpα`β, γq`Cpα, βq " Cpα, β`γq`Cpβ, γq . (2.4) Since V andṼ are Abelian groups, they are respectively isomorphic to their corresponding Lie algebras. Thus, the Lie algebra associated to V ext is given by the direct sum V 'Ṽ with Lie bracket rpα, aq , pβ, bqs " p0, αˆβq.
Now we define an extended semi-direct product as the group where σˆρ is the direct product representation It is important to note that, even though K acts on a central extension of V , the group G defines a non-central extension of the original semi-direct product K˙σ V . The elements of G are given by triplets pU, α, aq, where U P K , α P V , and a PṼ . The group operation is the natural extension of (2.1), where now the term α`σ U β must be replaced by the pair pα, aqˆpσˆρq U pβ, bq. Using equation (2.3), this leads to pU, α, aq ‚ pV , β, bq "ˆU¨V , α`σ U β, a`ρ U b`1 2 αˆσ U β˙.
(2.7) 6 For a detailed analysis of the rigidity and stability of the bms algebra in three and four dimensions based on their semi-direct sum structure, see [66,67].

Extended semi-direct sum algebra
The Lie algebra associated to G is the extended semi-direct sum where k denotes de Lie algebra of K , and d σ and d ρ stand for the derivative representations of σ and ρ, respectively. Denoting the elements of g by triplets of the form pX , α, aq, with X P k, the adjoint action of G on g can be found by evaluating pU, β, bq ‚`e λX , λα, λa˘‚ pU, β, bq´1 The identity of G is pe, 0, 0q, where e is the identity in K . Thus, the inverse of an element pU, α, aq is given by pU, α, aq´1 " pU´1,´σ U´1 α,´ρ U´1 aq. This leads to where Ad U denotes the adjoint representation of K .
The bracket on g can be found by looking at the infinitesimal adjoint action, i.e.
This defines the adjoint representation of the Lie algebra g in terms of the adjoint representation of k, ad X , and the derivative representation d σ, The elements of the first two slots on the right-hand side of equations (2.10) and (2.12) reproduce the form of the adjoint representation of a semi-direct product group and its Lie algebra bracket, respectively (see [55,56]). The third slot, however, implies the presence of more Lie algebra generators and modifies the structure of the usual semi-direct sum algebra.

Coadjoint orbits
Let us consider now the dual space g˚. Its elements will be denoted by pJ, Π, F q where J is an element of the dual space K˚, while Π and F are elements of the dual vector spaces V˚andṼ˚, respectively. The pairing between g and it's dual space can be naturally constructed out of the corresponding pairings defined in k, V andṼ as xpJ, Π, F q , pX , α, aqy g " xJ, X y k`x Π, αy V`x F , ayṼ . (2.13) Given a point pJ 0 , Π 0 , F 0 q P g˚, a coadjoint orbit of the group G can be defined as where Adp U,α,aq denotes de coadjoint action of G on g˚. The space O pJ 0 ,Π 0 ,F 0 q Ă g˚is isomorphic to G { G pJ 0 ,Π 0 ,F 0 q , where G pJ 0 ,Π 0 ,F 0 q is the isotropy group of the orbit representative
(2. 16) In order to explicitly write down the coadjoint action of G on its Lie algebra dual, it is necessary to define three extra maps. The first two are the bilinear products d : VˆV˚Ñ g˚, and e :ṼˆṼ˚Ñ g˚, such that Note that the definition of d is familiar from the well-known semi-direct product construction [56,58,59], while e is simply the analogue definition forṼ . The third map necessary to construct the coadjoint representation of an extended semi-directproduct group G is a product _ : VˆṼ˚Ñ V˚, dual to the mapˆin the sense that xF , αˆβyṼ "´xα _ F , βy V . (2.18) These definitions, together with equation (2.10), lead to the following form for the coadjoint representation of an extended semi-direct product group: Adp U,α,aq pJ, Π, F q " "ˆAdŮJ`α d σŮ Π`a e ρŮ F`1 2 α d pα _ ρŮF q , σŮ Π`α _ ρŮF , ρŮF˙, Here AdŮJ stands for the coadjoint action of the group K on its Lie algebra dual k˚, while σ˚and ρ˚denote the dual representations of K on V˚andṼ˚, respectively. The definition of the infinitesimal adjoint action follows from differentiating equation (2.16) and yields @ adp X ,α,aq pJ, Π, F q , pY , β, bq D g`@ pJ, Π, F q , ad pX ,α,aq pY , β, bq D g " 0 . (2.20) Since the Lie bracket of the extended semi-direct sum algebra g is given by (2.12), equation (2.20) can be solved to give where d σ˚and d ρ˚stand for the dual derivative representations.

Geometric action
Coadjoint orbits are naturally endowed with a symplectic structure [69][70][71], which allows one to define a geometric Hamiltonian action with a well-defined Poisson structure [60][61][62][63]. In our notation, the Kirillov-Kostant-Souriau symplectic form on a coadjoint orbit O pJ 0 ,Π 0 ,F 0 q is given by where v pX ,α,aq denotes the vector tangent to the orbit at pJ 0 , Π 0 , F 0 q v pX ,α,aq " adp X ,α,aq`J 0 , Π 0 , F 0˘. (2.23) It can be shown that the the pull-back of Ω on G by the projection map is given by a locally exact two-form [62] P˚Ω " dA , where d is the exterior derivative on the group manifold G and the triplet pΘ U , Θ α , Θ a q corresponds to the left-invariant Maurer-Cartan form on G defined by Here wedge product between differential forms is assumed. The corresponding geometric action then reads [72,73] I G "´ż Γ A˚, (2.27) where Γ is a path in G , and A˚denotes the pull-back of the one form A to Γ. As remarked in [57,60], one can define a Hamiltonian H a gauge invariant function on the orbit, and include it in the geometric action by adding the term´ş dt H in (2.27). The Hamiltonian can be chosen in such a way that the global symmetries of the action are preserved 7 .
Using (2.12), the Maurer-Cartan equation (2.26) can be written as (2.29) 7 The preservation of the symmetries might imply a redefinition of the gauge parameters of the global symmetries. In [57], Hamiltonians for geometric actions are defined in different cases as particular Noether charges or combinations of them. In [33], it is shown how the different Hamiltonians can be found by looking at the invariant functions that label the coadjoint orbits.
Note that Θ U corresponds to the left-invariant Maurer-Cartan form on K . Its pullback to Γ reads ΘŮ " Θ U´9 Uptq¯dt , (2.30) where 9 Uptq is the tangent vector to the path Uptq in K (with overdot denoting derivative with respect to t). Following [74], the action Θ U on a vector is given by Extending this definition to the full path in G given by (2.28), with tangent vector p 9 Uptq, 9 αptq, 9 aptqq, we can use the product law (2.7) to find the general expression (2.32) Using (2.32) in equation (2.25), the action (2.27) can be written as The geometric action is naturally globally invariant under left transformations of the form pU, α, aq Ñ pV¨U, β`σ V α, b`σ V a`1 2 βˆσ V αq, while gauge symmetries are given by right transformations pU, α, aq Ñ pU¨V ptq, α`σ U βptq, a`σ U bptq`1 2 αˆσ U βptqq, provided pV ptq, βptq, bptqq P G pJ 0 ,Π 0 ,F 0 q , as shown in [57] 8 .

Extended semi-direct products under the adjoint action
Several semi-direct product groups that define symmetries of 2`1 dimensional systems have the particular structure K˙A d k pabq , where k pabq denotes the Lie algebra of K seen as a vector space, and K acts on it's Lie algebra under the adjoint action. Examples are the ISOp2, 1q group, the z BMS 3 group, and higher-spin extensions thereof [55,75] (for a detailed analysis, see also [76]). Here we show how to generalize this special kind of symmetries to the case of extended semi-direct product groups, which will be the key construction to define in a unified manner the different Maxwell-like symmetries in 2+1 dimensions that will be treated in the subsequent sections.
Let us consider a particular case of extended semi-direct product (2.7), where V andṼ are given by the Lie algebra associated to Lie group K seen as an Abelian vector 8 Note that the form of the geometric action (2.27) is equivalent to the general definition given in [57] when setting g´1 " pU, α, aq. When written in terms of g , the role of right and left transformations are interchanged. group, and the representations σ and ρ are given by the adjoint representation of K , i.e.
V "Ṽ " k pabq , σ " ρ " Ad . (2.34) In this case, the requirement (2.2) is trivially satisfied provided the anti-symmetric bilinear mapˆis chosen as With these considerations, we can define the extended vector space k pabq ext , given by the set k pabqˆkpabq , and endowed with a group law of the form (2.3). Thus, we can define the special extended semi-direct product where the group operation follows from equation (2.7), and is given by As in this case the derivative representations d σ and d ρ are both given by the adjoint representation of k, the general form for the adjoint representation of an extended semi-direct product (2.10) can be easily evaluated to give Differentiating this expression leads to the Lie algebra bracket associated to the corresponding Lie algebra which can also be directly obtained from the general expression (2.12): Coadjoint orbits are defined according to equation (2.14), where the form of the coadjoint representation follows from (2.16). In the case of a special extended semidirect products, the pairing (2.13) for the algebra h is written solely in terms of the paring in k, Putting this together allows one to write down the coadjoint representation of H directly from (2.19), yielding Adp U,α,aq pJ, Π, F q " "ˆAdŮJ`adα`AdŮΠ`1 2 adαAdŮF˘`adå AdŮF , AdŮΠ`adαAdŮF , AdŮF˙, while its infinitesimal form is given by adp X ,a,αq pJ, Π, F q "´adX J`adαΠ`adå F , adX Π`adαF , adX F¯. (2.44) The construction of the geometric action follows from equations (2.25) and (2.27) by means of means of (2.34) and (2.35). The kinetic terms reads 3 Maxwell groups

The Maxwell group in D dimensions
The Maxwell group in D dimensions, which will be denoted by Maxwell D , can be constructed by generalizing the original definition for D " 4, given in [1]. It can be written as an extended semi-direct product to the second exterior power of R D . Therefore, the Maxwell group is given by where R D ext a central extension of the translation group R D by Ź 2`R D˘. As seen in Section 2.1, this extension requires a skew-symmetric bilinear productˆ: R DˆRD Ñ Ź 2`R D˘. By choosing an orthonormal basis e µ (µ " 1, ... , D) on R D , the productĉ an be constructed by means of the usual cross product which defines the natural basis for Ź 2`R D˘. Then, the extended group of translations R D ext is the centrally extended group with elements pα, aq, α P R D , a P Ź 2`R D˘, endowed with a productˆof the form (2.3) with The action of a Lorentz transformation Λ P SOpD´1, 1q Ò on the elements of R D and Ź 2`R D˘a re given by the standard vector and tensor representations , respect- Note that, in this case, ρ corresponds to the exterior power representation ρ " σ Ź 2 pR D q . Therefore, it automatically satisfies equation (2.2). Putting all this together, the product law of the Maxwell D group follows directly from equation (2.7). Denoting its elements by triads of form pΛ, α, aq, we get The Maxwell algebra is given by the extended semi-direct sum As both R D and Ź 2`R D˘a re Abelian vector groups, their corresponding Lie algebras are isomorphic to themselves. This means we can define the generators so that the elements of the maxwell D algebra can be defined from the following exponential map where J µν stands for the generators of Lorentz transformations and J µν stands for the the usual matrix representation for so pD´1, 1q elements: The derivative representations d σ and d ρ are given by the infinitesimal limit of equation (3.4) (3.11) The definitions (3.3), (3.4) and (3.11), plus the fact that the adjoint action of the Lorentz group on its Lie algebra is given by Ad Λ X " ΛX Λ T , enables one to write down the adjoint representation of the Maxwell group using expression (2.10). Similarly, the bracket for the maxwell D algebra follows from (2.12). When applied to the generators (3.7) and (3.9), it leads directly to the commutation relations (1.1).

Particle action
The maxwell D algebra and its dual space are isomorphic. The isomorphism can be explicitly constructed by endowing R D with the D-dimensional Minkowski metric η µν " diag p´1, 1, ... , 1q, which maps contravariant vectors and tensors into covariant ones. The pairing (2.13) can be constructed considering (3.12) The coadjoint representation of the Maxwell group and the Maxwell algebra can be obtained from (2.43) and (2.44), respectively, where the dual representations σå nd ρ˚as well as their corresponding derivative representations are given by the covariant versions of equations (3.4) and (3.11). The products d and e in this case can be worked out using (2.17), and lead to elements in the Lie algebra dual so pD´1, 1qẘ ith components Once the coadjoint representation is known, coadjoint orbits can be defined according to equation (2.14). Then, one can construct a geometric action that is naturally invariant under the Maxwell group. The key ingredient in this construction is the left-invariant Maurer-Cartan form (2.32), which in this case reads The Maxwell group satisfies Mackey's theorem [1,4], and therefore its irreducible representations can be obtained from the irreducible representations of V ext . As these representations correspond to Poincaré particles in the presence of a constant electromagnetic field, the Hamiltonian geometric action (2.27) for the Maxwell group is expected to describe world lines of such particles. This is indeed the case. In fact, evaluating (2.33) for a representative of the form p0, Π 0 , F 0 q PṼ˚, we get The orbit inṼ˚with representative pΠ 0 , F 0 q is defined as O pΠ 0 ,F 0 q : pπ, f q " pσ Λ Π 0 , ρ Λ F 0 q, therefore we define the momentum and the electromagnetic field on the orbit as The path Γ Ă Maxwell D defines a path on the orbit O p0,Π 0 ,F 0 q through the projection map (2.24). We can consider an orbit of the Maxwell group describing a massive particle of mass M. The mass shell condition π µ π ν "´M 2 can be implemented in (3.16) as a Hamiltonian by means of a Lagrange multiplier e. Finally, by identifying the translation components α µ with space-time coordinates x µ and the elements a µν with extra coordinates y µν "´y νµ , the geometric action takes the form This is the action massive particle coupled to a constant electromagnetic field in Hamiltonian form, and it has been previously studied from the point of view of non-linear realizations [9,15,30]. Here, we have shown that it can obtained from the Kirillov-Kostant-Souriau symplectic structure on coadjoint orbits of the Maxwell group. The elements of the orbit O p0,Π 0 ,F 0 q can be computed using equation (2.19) together with (3.13) and (3.14). Their components read where we have defined p µ " δI Maxwell {δ 9 x " π µ`1 2 f µν x ν as the canonical momenta associated to x µ . We see that f µν corresponds to the momentum associated to the extra coordinates y µν , which plays the role of Lagrange multiplier for the constraint 9 f µν " 0. Therefore, π µ is the momenta that accounts for the minimal coupling to a constant electromagnetic potential The quantities J µν , Π µ , and F µν define a basis for the Noether charges (3.19). By definition, they satisfy a Poisson algebra isomorphic to glogal symmetry action of the algebra [57], i.e. the maxwell D algebra (1.1), As p µ and f µν are the canonical momenta associated to the coordinates x µ and y µν , respectively, one can define a coordinate representation for them  and fulfill equation (1.1) when Poisson brackets are replaced by commutators in the form t , u Ñ i r , s. As stated in [15], the differential operators (3.23) define the Killing vectors fields of a pD`DpD´1q{2q-dimensional superspace with metric ds 2 "ˆη µν` 1 4 px ρ x ρ η µν´xµ x ν q˙dx µ dx ν`η µν x ρ dx µ dy νρ`1 2 η µρ η νσ dy µν dy ρσ . (3.24) and whose isometries are given by (3.25) Note that the Hamiltonian used in equation (3.18) is not the most general one. In fact, it is possible to add more constraints to the action constructed out of the Casimir invariants of the system, which can fix components of the representative element F 0 , associated to the constant electromagnetic field [15].

The Maxwell group in 2+1 dimensions
The Maxwell group in 2+1 dimensions is obtained from equation (3.1) for D " 3, i.e.
However, we will show now that, unlike its higher-dimensional versions, the Maxwell 3 group can be alternatively described by the special kind of extended semi-direct product defined in Section 2.3. This construction will allow us to define different generalizations of the Maxwell symmetry in 2+1 dimensions.
In order to formulate the Maxwell 3 group as a special extended semi-direct product of the form (2.36), we will follow the same strategy used in [55] for the ISOp2, 1q group. First, we will consider the double covering of (3.26): which satisfy the commutation relations rt µ , t ν s " ǫ ρ µν t ρ . (3.30) The same can be done with the space Ź 2 pR 3 q. In fact, any sop3q matrix can be identified with a vector in R 3 . In other words, the spaces Ź 2 pR 3 q and R 3 can be shown to be isomorphic by means of Hodge duality. Thus, Ź 2 pR 3 q can also be identified with sl p2, Rq pabq , a " 1 2 a µν e µˆeν ÝÑ a " a ρ t ρ , a ρ " With this identification, inner product of vectors in R 3 is reproduced by the Killing form on sl p2, Rq 2Tr rαβs " η µν α µ β ν , (3.32) while the cross product (3.2) is translated into matrix commutation, αˆβ " rα, βs. This means that we can make the identification where sl p2, Rq pabq ext is endowed with a product of the form (2.3) given by pα, aqˆpβ, bq "ˆα`β , a`b`1 2 rα, βs˙. Λα ÝÑ UαU´1 , Λ µ ν " 2η µρ Tr Therefore, the elements of the Maxwell group can be considered as triplets pU, α, aq, where U is an SLp2, Rq matrix and u, µ are sl p2, Rq matrices. Hence, the group operation (3.5) turns into which matches the structure of (2.37). This means that the double covering of the Maxwell group in 2+1 dimensions corresponds to a special extended semi-direct product (2.36), where K " SL p2, Rq When one copy of the sl p2, Rq pabq algebras in (3.36) is eliminated, (3.37) reduces to the double covering of the Poincaré group in three dimensions, SL p2, Rq˙A d sl p2, Rq pabq , which has been studied in [55,56].
Following the construction developed in Section 2.3, the Maxwell algebra in 2+1 dimensions has the form (2.39) for k " sl p2, Rq, i.e.
This definition is isomorphic to (3.6) for D " 2`1 due to the isomorphism sop2, 1q » sl p2, Rq and the identification (3.33). Its bracket follows from equation (2.40) and the commutation relations of the sl p2, Rq algebra given in (3.30). As SL p2, Rq is a matrix group, the Lie algebra sl p2, Rq and its dual space can be identified by means of the Killing form (3.32). A base for sl p2, Rq˚is given by t µ " η µν t ν and therefore any element in sl p2, Rq can be written as an element of sl p2, Rq˚and vice-versa. Thus, the pairing between sl p2, Rq and its dual space is naturally given by xJ, X y slp2,Rq " J µ X ν Tr rt µ t ν s " The adjoint and the coadjoint action are equivalent and given by matrix conjugation, and their infinitesimal limits correspond to matrix commutation

The Maxwell algebra in 2+1 dimensions
The explicit form of the Maxwell algebra in 2+1 dimensions can be written down by defining the Following generators where t µ are the generators of the sl p2, Rq algebra and satisfy (3.30). Then, any element pX , α, aq P maxwell 3 can be written in the form In term of these generators, the bracket (2.40) leads to which can be put in the general form (1.1) for D " 2`1 by defining the Hodge duals J µν "´ǫ ρ µν J ρ , F µν "´ǫ ρ µν F ρ . It is important to note that the commutation relations (3.43) have recently been obtained in the context of non-relativistic symmetries as a particular conformal extension of the Galilean algebra [77]. This extension, in turn, can be thought as the simplest example of a Hiertarinta algebra, when the role of the generators Π µ and F µ are interchanged [78,79].

Generalizations
The algebraic structure (3.43) can be generalized to the case where K is an arbitrary matrix Lie group M. Thus, we can consider k as a general matrix Lie algebra m of the form where f c ab are the structure constants. In that case the natural generalization of (3.41) is given by J a " pT a , 0, 0q, Π a " p0, T a , 0q, F a " p0, 0, T a q, which applied to (2.40) leads to (3.45)

(3.48)
This corresponds exactly to the commutation relations of the higher-spin extension of the maxwell 3 algebra found in [45]. Therefore, one can define the spin-3 extension of the Maxwell 3 group as SL p3, Rq˙A d sl p3, Rq pabq ext . (3.49) In a similar way, more general higher-spin extensions of the Maxwell group can be defined considering the extended semi-direct product SL pN, Rq˙A d sl pN, Rq pabq ext .

Infinite-dimensional Maxwell groups in 2+1 dimensions
In this section, we will construct two infinite-dimensional groups whose underlying structure is given by a special extended semi-direct product K˙A d k pabq ext developed in Section 2.3. They correspond to extensions of the loop group LISLp2, Rq and the BMS 3 group. Their Lie algebras are given by infinite-dimensional enhancements of the Maxwell algebra in 2+1 dimensions (3.38). These groups are interesting, as they allow for central extensions, which will be considered in section 4.

Maxwell loop group
Consider the loop group LSL p2, Rq, whose elements are the continuous maps from the unit circle S 1 to the group SL p2, Rq and where the product of two group elements U, V P K is given by point-wise multiplication, i.e. U¨V " U pφq V pφq " pUV q pφq . (3.51) The corresponding Lie algebra L sl p2, Rq is the algebra of continuous maps from S 1 to sl p2, Rq. As functions on the unit circle can be expanded in Fourier series, the elements of L sl p2, Rq can be written as X " X µ pφqt µ " ř 8 m"´8 X µ m t m µ , where the generators of the loop algebra L sl p2, Rq can be written as and satisfy the commutation relations " Similarly to what happens with the Maxwell algebra, the dual space L sl p2, Rq˚is isomorphic to L sl p2, Rq. Its elements can be written as J " J µ pφqt µ " ř 8 m"´8 J m µ t µ m , where the dual basis is constructed out of the invariant sl p2, Rq metric η µν as t µ m " e imφ η µν t ν . The corresponding pairing is then given by xJ, X y L slp2,Rq " 1 2π Therefore, the adjoint and the coadjoint representations of LSL p2, Rq are given by expressions similar to (3.40), where now the group elements as well as the the elements of the corresponding Lie algebra and its dual space, are functions on S 1 .
We can define the Maxwell 3 loop group as the extended semi-direct product (2.36), where K " LSL p2, Rq. i.e. Here L sl p2, Rq pabq ext is given by L sl p2, Rq pabqˆL sl p2, Rq pabq as a set, and its product is defined by equation (3.34), with (3.30) replaced by (3.53).
Following the construction developed in Section 2.3, the Maxwell loop algebra in 2+1 dimensions, { Lmaxwell 3 , is given by the semi-direct sum L sl p2, Rq i ad L sl p2, Rq pabq ext .
Its elements can be written in a form analogue to (3.42) with φ-dependent components, which can be expanded in Fourier modes as pX , α, aq " X µ pφqJ µ`α µ pφqΠ µ`a µ pφqF µ " where J m µ " e imφ J µ , Π m µ " e imφ Π µ , and F m µ " e imφ F µ form a base for the Maxwell loop algebra. Using the definitions (3.41) and (3.52), these generators can be written as and its commutation relations follow from (2.40) and (3.53), which leads to (3.59) The elements of the dual space are naturally constructed as triples pJ, Π, F q, where J P L sl p2, Rq˚and Π, F P L sl p2, Rq pabq˚. The pairing between the L maxwell 3 algebra and it's dual follows directly from (2.41) and (3.54), and the adjoint and coadjoint representations together with the infinitesimal forms can be constructed in a straightforward way. The geometric action equation (2.45) in this case is given by a sigma model of the form This will lead to algebras of the form Lm i ad Lm pabq ext whose generators are given by J m a "`e imφ T a , 0, 0˘, Π m a "`0, e imφ T a , 0˘, F m a "`0, 0, e imφ T a˘, (3.62) and T a stands for the generators of m defined in equation (3.44). The base (3.62) will automatically satisfy an algebra of the form (3.59) with ǫ ρ µν Ñ f c ab . The geometric action in this case takes a form analogue to equation (3.60) where the pairing (3.54) is to be replaced by integration on S 1 of the corresponding the Killing form on M.

Extended BMS 3 group
The second infinite-dimensional Maxwell group we will be interested in is based on the orientation-preserving diffeomorphism group of the unit circle Diff`pS 1 q. In this case, an element U P Diff`pS 1 q is a reparametrization of S 1 : where prime denotes derivative with respect to φ and the group operation is given by function composition U¨V " U˝V . (3.64) The corresponding Lie algebra is the algebra of vector fields on the circle vect pS 1 q, whose elements will be denoted by X " X pφqB φ . The elements of the dual space vect pS 1 q˚correspond to quadratic differentials on S 1 , J " J pφq pd φq 2 , and the paring is given by where here α and a denote their components αpφq and apφq, and similarly for the elements pJ 0 , Π 0 , F 0 q in the dual space vect pS 1 q˚.
The corresponding Lie algebra, defines an extension of the bms 3 algebra with Maxwell structure. The Lie bracket has the form (2.40), which can be written in terms of the generators J m " pℓ m , 0, 0q , P m " p0, ℓ m , 0q , F m " p0, 0, ℓ m q , (3.73) and yields the following commutation relations rJ m , J n s " pm´nq J m`n , rP m , P n s " pm´nq F m`n , rJ m , P n s " pm´nq P m`n , rP m , F n s " 0 , rJ m , F n s " pm´nq F m`n , rF m , F n s " 0 . (3.74)

Centrally extended Maxwell-like groups in 2+1 dimensions
The special structure the extended semi-direct product K˙A d k pabq ext defined in Section 2.3 allows one to introduce a central extension in a straightforward way, provided a central extension of K is known. This can be achieved by generalizing the construction of centrally extended semi-direct products [55,56].
A central extensionK of a group K consists in a direct product p K " KˆR with elements pU, mq, where U P K and m P R, and group operation pU, mq pV , nq " pU¨V , m`n`C pU, V qq , (4.1) where C pU, V q defines a two-cocycle on K , i.e.
CpU¨V , W q`C pU, V q " CpU, V¨W q`CpV , W q . The corresponding centrally extended algebrak has elements pX , mq and its Lie bracket has the form rpX , mq , pY , nqs " prX , Y s , c pX , Y qq , where c is a Lie algebra two-cocycle, c P H 2 pk, Rq, given by

Central extensions of extended semi-direct products
Let us consider a special extended semi-direct product H " K˙A d k This yields to the following product law: pU, m 1 ; α, m 2 ; a, m 3 q ‚ pV , n 1 ; β, n 2 ; b, n 3 q " pW , q 1 ; ω, q 2 ; w , q 3 q , (4.7) where pW , q 1 q " pU, m 1 q pV , n 1 q , pω, q 2 q " pα, m 2 q`Ad pU,m 1 q pβ, n 2 q , pw , q 3 q " pa, m 3 q`Ad pU,m 1 q pb, n 3 q`1 2 ad pα,m 2 q Ad pU,m 1 q pβ, n 2 q . (4.8) Here the expressions Ad pU,m 1 q and ad pα,m 2 q denote the adjoint representation of the centrally extended groupK and the centrally extended algebrak , respectively. The former can be obtained in the usual way by differentiating the conjugation operation inK with the product law (4.1). This yields [81] Ad pU,mq pX , nq " pAd U X , n´xS pUq , X yq , where S is the Souriau cocyle on K , which is defined in terms of the two-cocycle C as xS pUq , X y "´d d λ

"
CpU, e λX U´1q`Cpe λX , U´1q ‰ˇˇˇλ "0 . (4.10) The later is the infinitesimal form of Ad pU, mq , which leads to the bracket (4.3), and has the form ad pA,mq pB, nq " pad A B,´xs pAq , Byq , (4.11) where s corresponds to the differential of S evaluated at the identity, and it is related with the two-cocycle c defined in (4.4) by cpA, Bq "´xs pAq , By , (4.12) Replacing these expressions in equation (4.8) leads to w " a`Ad U b`1 2 ad α Ad U β , q 3 " m 3`n3´x SpUq, by´1 2 xspαq, Ad U βy , (4.13) which determines the group operation (4.7).
The corresponding centrally extended algebra is given byĥ "k i adk pabq ext . Denoting its elements by pX , m 1 , α, m 2 , a, m 3 q, the associated bracket can be obtained by computing the adjoint representation ofĤ, which follows from the definition equation (2.10) and the prescription (4.6), Ad pU,m 1 ;β,m 2 ;b,m 3 q pX , n 1 ; α, n 2 ; a, n 3 q "´X ,ñ 1 ;α,ñ 2 ;ã,ñ 3¯, (4.14) whereX " Ad U X , (4.15) The infinitesimal limit of the expression (4.15) leads to the following bracket rpX , m 1 ; α, m 2 ; a, m 3 q , pY , n 1 ; β, n 2 ; b, n 3 qs " " prX , Y s , f 1 ; rX , βs´rY , αs , f 2 ; rX , bs´rY , as`rα, βs , f 3 q , (4. 16) where the central are written in terms of the Lie algebra cocycle (4.4) as (4.17) In the following will see how to apply this construction to the infinite-dimensional Maxwell-like groups in 2+1 dimensions studied in Section 3.4. This will lead to novel centrally extended groups, whose associated infinite-dimensional algebras have been recently discussed in [49,52] from the point of view of Lie algebra expansions and asymptotic symmetries in three-dimensional gravity theories.

Maxwell-Kac-Moody group
Let us construct now the central extension of the Maxwell loop group in 2+1 dimensions defined in (3.56). Following (4.5), first we need to the define the central extension of the loop group LSL p2, Rq. This corresponds to the Kac-Moody group y L SL p2, Rq, defined by the the following two-cocycle where the exteriord is defined on the unit disc D 1 . This two-cocyle extends the product (3.51) in LSL p2, Rq in the form (4.1). The definition (4.3) leads to the Kac-Moody algebra x L sl p2, Rq, where the two-cocycle (4.4) reads The corresponding Souriau cocyle and its differential can be computed using (4.10) and have the form S pUq " 1 2π Now we define the Maxwell-Kac-Moody group in 2+1 dimensions as the extended semi-direct product whose product law has the form given in equations (4.8) and (4.13) and its written in terms of the adjoint representation of L sl p2, Rq, given in (3.55), and the cocycles (4.18) and (4.20) previously defined.
The Maxwell-Kac-Moody algebra in 2+1 dimensions is therefore given by The adjoint representation of the { LMaxwell 3 group as well as the commutation relations of the { Lmaxwell 3 algebra can be directly read off from (4.15) and (4.16) for k " L sl p2, Rq. The commutation relations can be expressed in terms of a base of the form J m µ "`e imφ t µ , 0; 0, 0; 0, 0˘, k 1 " p0, 1; 0, 0; 0, 0q , Π m µ "`0, 0; e imφ t µ , 0; 0, 0˘, k 2 " p0, 0; 0, 1; 0, 0q , F m µ "`0, 0; 0, 0; e imφ t µ , 0˘, k 3 " p0, 0; 0, 0; 0, 1q , where t µ is the generator of the sl p2, Rq algebra (3.53). In terms of these generators, equation (4.16) leads to a extended version of the algebra (3.59) " (4.24) This structure can be generalized to more general Kac-Moody groups with Maxwell structure by centrally extending groups of the form (3.61). This can be done by first extending the loop group LM by means of a two-cocycle of the form (4.18) where the trace over sl p2, Rq elements is to be replaced by the Killing form on m. The resulting Lie algebra will have the form x Lm i ad x Lm pabq ext and its commutation relations will have the structure equation (4.24) with more general structure constants, defining a Kac-Moody version of (3.45). This type of algebras have been found in [52] as expansions of a Kac-Moody algebra x Lm.

Maxwell-like extension of the z BMS 3 group
In this section we will define and extension of the z BMS 3 group using the group structure (4.5). The starting point is the Virasoro group z Diff`pS 1 q, which is the central extension of the diffeomorphism group of the circle defined in equations (3.63) and (3.64). It is defined by the Thurston-Bott cocycle [81], which extends the function composition product (3.64) in the form (4.1). The corresponding Souriau cocycle is given by the Schwarzian derivative (4.26) and the bracket of the Virasoro algebra x vect pS 1 q follows from equation (4.3) by either using the definition (4.4) or (4.12), where (4.27) Now we consider the following extended semi-direct product based on the Virasoro group This is the natural Maxwell-like extension of the centrally extended BMS group in 2+1 dimensions, which can be defined as the semi-direct product of the Virasoro group and its Lie algebra [55]. These definitions, together with the adjoint representation of the Diff`pS 1 q group given in (3.66) and (3.67), define the group operation of the Maxwell-z BMS 3 group through equation (4.7). The Lie algebra defines a Maxwell extensions of the x bms 3 algebra, whose bracket is defined by equation (4.16). Considering a base of the form J m " pℓ m , 0; 0, 0; 0, 0q , c 1 " p0, 1; 0, 0; 0, 0q , P m " p0, 0; ℓ m , 0; 0, 0q , c 2 " p0, 0; 0, 1; 0, 0q , F m " p0, 0; 0, 0; ℓ m , 0q , c 3 " p0, 0; 0, 0; 0, 1q ,   (4.27). These commutation relations define a Maxwelllike extension of the x bms 3 algebra, which has been found in [52] as an expansion of the Virasoro algebra. Furthermore, it can be obtained from the Maxwell-Kac-Moody algebra (4.24) by means of a generalized Sugawara construction . On the other hand, the algebra (4.31) has been found in [49] as the asymptotic symmetry of a threedimensional Chern-Simons gravity theory invariant under the maxwell 3 algebra.

Wess-Zumino terms
In this section we construct the geometric actions for the { LMaxwell 3 group and the Maxwell-z BMS 3 group studied in the previous section by generalizing the construction shown in Section 2.3 to centrally extended groups of the form (4.5). In order to do this, we need to define the dual spaceĥ˚. Its elements will be denoted by sextuplets of the form pJ, c 1 , Π, c 2 , F , c 3 q, and the pairing betweenĥ and it's dual space can be defined as the natural extension of equation (2.41), i.e. xpJ, c 1 ; Π, c 2 ; F , c 3 q , pX , m 1 ; α, m 2 ; a, m 3 qyĥ " xpX , Π, Jq , pX , α, aqy h`c 1 m 1`c2 m 2`c3 m 3 .

Maxwell WZW model
In the case of the { LMaxwell 3 group (4.21), the Maurer-Cartan forms (4.38) can be evaluated using equations (2.32), (4.18) and (4.20). This leads to the following geometric action: where I LMaxwell 3 is the action (3.60) and I WZW takes the form (4.41) Here the exterior derivatived is defined on D 1 , as in (4.18). This correspond to a chiral Wess-Zumino-Witten (WZW) model, and it is globally invariant under the { LMaxwell 3 group. The first term in equation (4.41) corresponds to the usual chiral WZW model that defines the geometric action associated to the x L sl p2, Rq Kac-Moody group [60,63,72]. The second term corresponds to the flat chiral WZW, which has been studied in [57,82,83] in the context of three-dimensional gravity 9 . The third term is a novel chiral WZW model by itself and defines a Maxwell extension of the known WZW models previously discussed.

Maxwell gravitational WZ model
Let us consider again the Maxwell-z BMS 3 group defined in (4.28). In this case the Maurer-Cartan forms (4.38), and the geometric action (4.39) can be evaluated using (2.32), (4.25) and (4.26), which yields where I Maxwell-BMS 3 corresponds to the geometric action associated to the centerless Maxwell-BMS 3 group, given by (3.71), and I gWZ reads (4.43) 9 In order to make this term match the model found in [57,82,83], one has to replace the pair pU, αq by it's inverse pU, αq´1 " pU´1,´Ad U´1 αq, and integrate by parts.
The action I gWZ defines a Maxwell extension of the gravitational Wess-Zumino action [60,63,72,73], found as the geometric action describing world lines on the Virasoro group. In fact, by defining the change of variables F " U´1, ϕ " Upφq , (4.44) the first term in (4.43) (supplemented with the first term in (3.71)) takes exactly the same form as the Polyakov action found in [84] in the context of induced two-dimensional quantum gravity: (4.45) The second term in (4.43) corresponds to the BMS 3 extension of the gravitational WZ action, found in [57] as the geometric action on the BMS 3 group. It is related to the flat limit of Liouville theory, which defines the classical dual of asymptotically flat threedimensional Einstein gravity [83]. The third term in (4.43) defines a novel Extension of the Polyakov action for two-dimensional gravity, which is by definition invariant under the Maxwell-BMS 3 group.

Discussion and possible applications
In this paper, we have analysed the Maxwell group in 2+1 dimensions and it's infinitedimensional generalizations. This have been made by extending the notion a semidirect product group in a way that captures the underlying structure of the Maxwell symmetry. The definition of extended semi-direct products groups allowed us to define the adjoint and the coadjoint representation of Maxwell groups in a general fashion, and construct geometric actions on their coadjoint orbits . Subsequently, we have explicitly shown how the Maxwell D group fits in this description. The geometric action in this case leads to the description of a relativistic particle coupled to constant electromagnetic field. Subsequently, we have considered the semi-direct product of a Lie group and a central extension of its Lie algebra. When based on the SL p2, Rq symmetry, this group structure matches the double cover of the Maxwell group in 2+1 dimensions. Similarly, higher-spin extensions of the Maxwell group can be constructed by considering these special kind of extended semi-direct products.
Special extended semi-direct products admit central extensions, for which a general description has been given. Using this construction, two infinite-dimensional enhancements of the Maxwell symmetry have been constructed, namely, the { LMaxwell 3 Kac-Moody group and the Maxwell-z BMS 3 group. These symmetries define Maxwelllike extensions of the ISL p2, Rq Kac-Moody group and the z BMS 3 group, respectively. The corresponding Lie algebras of these infinite-dimensional groups have been previously studied in the context of algebra expansions and three-dimensional gravity [49,52]. The geometric actions associated to these infinite-dimensional groups lead to novel Wess-Zumino terms that generalize the known results presented, for instance, in [57,63]. The first geometric action is a novel Wess-Zumino-Witten model, invariant under the Maxwell-Kac-Moody group { LMaxwell 3 . The second case that we considered is the geometric action associated to the Maxwell-z BMS 3 group, which defines a Maxwell extension of gravitational Wess-Zumino action first found by Polyakov as an action for two-dimensional gravity [84]. In the following we briefly elaborate on the application of these geometric actions and comment on possible future directions.

Chern-Simons theory and quantum Hall systems
A (2+1)-dimensional field theory that naturally realizes the { LMaxwell 3 Kac-Moody symmetry at the level of its conserved charges is a Chern-Simons theory invariant under the maxwell 3 algebra (3.43): where A is a maxwell 3 -valued connection one-form on a manifold M, and x y denotes an invariant metric on maxwell 3 . The conserved charges in a Chern-Simons theory are known to satisfy a Kac-Moody algebra that enhances the original symmetry of the action [85]. It is therefore evident that, as in the case the gauge group is given by Maxwell 3 , the conserved charges of the theory will satisfy a Poisson algebra isomorphic to Maxwell Kac-Moody algebra (4.24). Hence, the { LMaxwell 3 Kac-Moody group can be relevant in any context involving a Maxwell 3 -invariant Chern-Simons action.
Chern-Simons forms provide effective theories for the the large scale dynamics of quantum Hall systems [86][87][88][89] (for a recent review see [90]). In [36], a Chern-Simons action invariant under the maxwell 3 algebra was used to construct an effective theory for a topological insulator. This leads to a relativistic version of the Wen-Zee term, which couples the background geometry to the Chern-Simons effective field in the quantum Hall fuilds [91]. The motivation to use the Maxwell symmetry in this description stems from the fact that it includes non-commuting translations, allowing to reproduce the Girvin-Plazmann-MacDonald algebra for magnetic translations [37]. As stated before, the current algebra of such Chern-Simons theory is naturally given by the Maxwell-Kac-Moody algebra { Lmaxwell 3 . This symmetry can be made explicit at the level of the action when studying its edge modes. In fact, the boundary dynamics of a Chern-Simons theory is described by WZW model, whose kinetic term has the general form where t and φ are coordinates on the boundary BM. It is easy to show that, considering and element of the Maxwell 3 group (3.37) in the form g " pU, α, aq´1, and the invariant metric on maxwell 3 xJ µ J ν y " γ 1 η µν , xJ µ Π ν y " γ 2 η µν , xΠ µ Π ν y " γ 3 η µν " xJ µ F ν y , this action is exactly the WZW model (4.41), obtained in Section 4.4 (where the Chern-Simons level k is related to the central charges through the constants entering in the invariant metric on the maxwell 3 algebra). Thus, the geometric action (4.41) describes the boundary dynamics of a Maxwell 3 -invariant Chern-Simons theory, and naturally provides an action for the edge modes of the topological insulator model introduced in [36]. One way to describe edge modes in such a system is to consider a dislocation at the boundary of a topological insulator in 3+1 dimensions as considered, for instance, in [92].

Three-dimensional gravity
Chern-Simons theories also provide a gauge theoretical formulation for gravity in 2+1 dimensions [38]. Along these lines, a Chern-Simons theory invariant under the Maxwell algebra (3.43) has been studied as an extension of three-dimensional Einstein gravity [47,48]. In [49], it was shown that the asymptotic symmetry of such a gravity theory is given by an extension of the bms 3 algebra, given precisely by the maxwell-x bms 3 algebra (4.31). This is a generalization of the known results for asymptotically AdS and and flat gravity in 2+1 dimensions, where the asymptotic symmetry group is known to be given by the conformal algebra in two dimensions and the BMS 3 group, respectively [51,93]. When the role of generators of the extended space of translations (3.33) are interchanged, this leads to an extended gravity theory invariant under the simplest example of a Hietarinta algebra [79]. This algebra is isomorphic to a particular conformal extension of the Galilean algebra, and its asymptotic symmetry is also given by the extended semidirect sum (4.29), where the generators P m and F m in equation (4.30) are interchanged [77]. In this article, we have succeeded in finding a group structure that leads to this infinite-dimensional algebra, namely, the Maxwell-z BMS 3 group (3.70). The relation of this group with Chern-Simons gravity can be analysed in the following way. The solution for the gauge connection used in [49] is given, up to a gauge transformation depending on the radial coordinate, by M2˙d φ¯F 0 , and depends only on the arbitrary functions of the boundary coordinates, given by M, N and F . Under the action of an improper gauge transformation, they change as which defines the solution space of the theory. It is an straightforward computation to show that this expression matches the infinitesimal form of coadjoint action Ad pY ,m 1 ;T ,m 2 ;R,m 3 q pM, c 1 ; N , c 2 ; F , c 3 q given in (4.14), when the Souriau cocycle on the Virasoro algebra (4.26), and the paring (3.65) are implemented. This shows that, when adopting the boundary conditions introduced in [49] , the solution space of the Maxwell 3 -invariant Chern-Simons theory matches the coadjoint representation of the Maxwell-z BMS 3 group presented here. On the other hand, a coadjoint orbit of the Maxwell-z BMS 3 group with representative pJ 0 , Π 0 , F 0 q defines a phase space for the geometric action (4.43). Therefore, the Maxwell extension gravitational WZ action can be conjectured to control the solution space of this Chern-Simons theory, when the boundary conditions found in [49] are adopted. This is an interesting subject for future research.
On the other hand, chiral WZW models appear in three-dimensional gravity as an intermediate point in the Hamiltonian reduction that leads to the corresponding classical dual defined at the boundary [94]. In our case, the Maxwell WZW model plays this role. This suggest there should be way to connect the Maxwell WZW model (4.41) with the gravitational Maxwell WZ action (4.43) by a Drinfeld-Sokolov reduction [95]. This is a work in progress.
Kac-Moody algebras can also provide asymptotic symmetries for gravity theories when the most general boundary conditions are adopted [96]. In this context, allowing the most general boundary conditions for the maxwell 3 Chern-Simons connection, would lead to the Maxwell-Kac-Moody algebra { Lmaxwell 3 as the asymptotic symmetry.

Higher-spin symmetries
In Section 3.3, we have shown how to construct higher-spin extensions of the Maxwell 3 group, which are given by extended semi-direct products of the form SL pN, Rq˙A d sl pN, Rq pabq ext . In the same way, higher-spin extensions of the maxwell-x bms 3 algebra can be constructed defining extended semi-direct sums based on the W N algebra.
This is interesting in the context of gravity, where W N algebras are known to describe asymptotic dynamics of higher-spin extensions of Einstein gravity in 2+1 dimensions [80,97,98]. In our case, these algebras define Maxwell extensions of the asymptotic symmetries found in flat higher-spin gravity [99][100][101], given by semidirect sums based on the W N algebra [75]. Thus, infinite-dimensional algebras of this kind are likely to define asymptotic symmetries of higher-spin gravity theories based on the Maxwell 3 group, like the one presented in [45]. On the other hand, W N algebras have also been found in the description of quantum Hall systems [102,103]. Based on the fact that a Maxwell 3 -invariant Chern-Simons theory captures features of the magnetic translations at the boundary of a topological insulator, it would be interesting to evaluate if Maxwell extensions of the W N symmetry could somehow emerge in this type of systems.