JHEP10(2019)039 Springer July 4, 2019 xtension of : October 4, 2019 , Sigma Models : September 3, 2019 : Received e coadjoint representa- Published Accepted actions on coadjoint orbits s that naturally realize the ) Kac-Moody and the R pace-time dimensions. By giv- , es a sound framework to study y elaborate on potential appli- e construct infinite-dimensional er-spin symmetries, and quantum alpara´ıso, Published for SISSA by https://doi.org/10.1007/JHEP10(2019)039 . 3 1905.09421 The Authors. c Chern-Simons Theories, Gauge , Global Symmetries

The Maxwell group in 2+1 dimensions is given by a particular e , [email protected] group by including non-commutative supertranslations. Th 3 Instituto de Pontificia F´ısica, Universidad de Cat´olica Casilla V 4059, Chile Valpara´ıso, E-mail: [ Open Access Article funded by SCOAP BMS Abstract: Patricio Salgado-Rebolledo Keywords: tion in each caseare is presented. defined, These and actions theaforementioned lead corresponding infinite-dimensional to symmetries. geometric novel Wecations Wess-Zumino briefl in term the contexts ofHall three-dimensional systems. gravity, high The Maxwell group ininfinite-dimensional 2+1 enhancements dimensions and its a semi-direct product. Thisdifferent mathematical generalizations structure of provid the Maxwelling symmetry a in general three definition s ofenhancements extended of semi-direct the products, Maxwell w group that enlarge the ISL(2 ArXiv ePrint: JHEP10(2019)039 4 6 9 1 4 = 25 10 14 16 17 21 23 24 → 27 10 21 µ µν (1.1) p F , of a particle µν , . us and static), F ] = 0 ] = 0 ] = ν νρ ρσ Π F F , isometries include non- , , µ µ µν Π [ Π [ orentz group and a centrally F [ y that naturally arises in the romagnetic field. It includes, a: l momentum in the form elian ideal spanned by a second ] as the covariance group of the , 1 , µσ µσ F J νρ νρ η η − − group νρ 3 νρ – 1 – F J µσ d µσ , associated the electromagnetic field strength BMS η η µν − − F dimensions , µρ µρ µ F D J Π νσ νσ νρ η η η + + − ν νσ νσ Π F J µρ µρ µρ η η η ] = ] = ] = . When the electromagnetic field is constant (i.e. homogeneo ρ . This enlargement stems from the fact that minimal coupling µ ρσ ρσ µ Π A J F , A , , ν + µν ∂ µν µν µ J [ J J − p [ [ ν 4.1 Central extensions4.2 of extended semi-direct Maxwell-Kac-Moody products 4.3 group Maxwell-like extension4.4 of the Wess-Zumino terms 3.2 The Maxwell3.3 group in 2+1 The dimensions Maxwell3.4 algebra in 2+1 Infinite-dimensional dimensions Maxwell groups in 2+1 dimensions 2.1 Extended semi-direct2.2 product groups Coadjoint orbits 2.3 Extended semi-direct products under the adjoint action 3.1 The Maxwell group in = A µ µ The Maxwell group is given byextended the semi-direct space product of of translations. the L It was first studied in [ ∂ The Maxwell algebra is anstudy extension of of the particle Poincar´esymmetr systemsapart in from the translations and presence Lorentz of transformations,rank antisymmetric an a Lorentz Ab tensor constant elect 1 Introduction 5 Discussion and possible applications 4 Centrally extended Maxwell-like groups in 2+1 dimensions 3 Maxwell groups Contents 1 Introduction 2 Extending semi-direct products it can be thoughtcommutative of translations. as This leads an to extension the of Maxwell Minkowski algebr space, whose π to the electromagnetic field requires to modify the canonica JHEP10(2019)039 ], ]. e- 41 39 , (1.2) , 40 38 ] by means of ]. By means 43 , 19 ]. Furthermore, – ]. ems was shown 42 12 37 12 Girvin-Plazmann- [ 2 ppears in the central ]. This is of particular ed as extra generators ty. By gauging it, one context of gravitational rally arises when study- 33 ], respectively. , al Relativity by including 46 – 32 c translations associated to have non-vanishing Poisson l systems [ es a geometric framework to mulated ated to a particle moving in sions, supersymmetric exten- 44 ed to construct supergravity mod- s extension was characterized y considering a Chern-Simons gnetic field through multipole s previously considered in [ B ], and for a geometric derivation of eory for the boundary dynamics eld [ 2 nsional Einstein gravity [ n defines an extended theory for ant backgrounds [ ional free algebra, which describes y in 1+1 dimensions [ relativistic limits of Chern-Simons gravity see [ ructed in [ ] 7 netic field based on local representations of the B. = ], it was shown that, given suitable bound- } y 49 Π – 2 – , x Π { ]. In [ ], different families of Lie algebras that further extend However, previous indications of this symmetry were ]. In this case, the components of the antisymmetric 49 21 6 ]. Moreover, the Maxwell algebra can be understood as – 1 , 29 47 20 – ) have been constructed, leading to novel (super)gravity th [ 4 22 , 1.1 3 ]. 4 , 3 ]. ] 31 , 35 ] it is shown that the Maxwell algebra properly describes the , are not related to an electromagnetic field, but reinterpret ], where the kinematic symmetry of this kind of particle syst 30 36 5 34 µν ], as well as in the description of the superparticle on const F 11 – On the other hand, Chern-Simons forms are interesting in the The (2+1)-dimensional version of the Maxwell symmetry natu In the last years, different generalizations to higher dimen Later on, the Maxwell algebra was used in the context of gravi Supersymmetric extensions, higher-spin extensions, and non- The supersymmetric extension of the Maxwell algebra has been us For a detailed analysis including the non-relativistic case, A Chern-Simons theory invariant under the Maxwell algebra wa 8 4 2 1 3 associated to the newintroduce field vector content of inflatons the in theory. cosmological models This [ provid brackets [ theories invariant under the Maxwell algebra have been const els [ In fact, in [ Maxwell group, see [ where it was used todimensional obtain reduction. the Cangemi-Jackiw action for gravit Along these lines, a Maxwell-invariant Chern-Simonsgravity actio in 2+1 dimensions wave equations for particles coupled to constant electromag MacDonald algebra for magnetic translationsa in relativistic quantum version Hal of theaction invariant Wen-Zee under term the Maxwell was algebra constructed asof b an effective a th topological insulator. interest in the studyan of electric quantum Hall charge systems, in where presence magneti of a constant magnetic field the simplest within Poincar´eextension an infinite-dimens the motion of aexpansion charge distribution [ in an generic electro-ma ing anyons coupled to a constant external electromagnetic fi sions, and deformations of theof algebra Maxwell expansion algebra methods have [ beenthe commutation for relations ( tensor found in [ a constant electromagnetic field. (3+1)-dimensional Klein Gordon and Dirac equations associ theories, as they provide a gauge formulation of three-dime can define a modifieda gravitational generalized theory cosmological that term extends [ Gener ories in diverse dimensions [ to include a centralby extension the of electric the translation chargeextension group. in of the pretty Thi Bargmann much algebra. the same way as the mass a JHEP10(2019)039 ], D he 56 on¨u- , 55 ] extend their ike extensions nsion here pre- 52 group and the 3 symmetry correspond 3 both the Maxwell-Kac- ) group. This construc- of the , metry in 2+1 dimensions R s can be formulated as a onstruction of a Maxwell- , sions. This algebra can be terms. They are naturally known from the analysis of a Maxwell-invariant Chern- [ eory is given by an infinite- l mathematical construction BMS (2 th infinite-dimensional sym- en already considered in the the results found in [ 5 ons theory suggests that they al construction of semi-direct . Indeed, the analogy between an extended semi-direct prod- he Maxwell symmetry in 2+1 e-dimensional symmetries? hat this is indeed the case. ive supertranslations. On the duct and the action relativistic nding geometric action. Subse- f the BMS ]. Moreover, the same extended algebra found in [ c structure on coadjoint orbits is ction is based on a non-central ex- 3 51 algebra, group, whose associated Lie algebras c bms 3 3 c bms [ BMS – 3 – ] (for a recent discussion in the context of three- 63 – ]. Therefore, the group structure behind the afore- ]. ]). As, so far, only the algebras are known, a natural 60 57 50 ]. The geometric actions associated to these extended – 54 , related with the above-mentioned Maxwell extension by an In¨ 52 55 , 3 53 c bms ]). Based on these definitions, we derive the Maxwell group in ) Kac-Moody symmetry as well as the 57 ]. The Maxwell-Kac-Moody algebra, in turn, is by definition t R , 52 ) Kac-Moody group and the ). By including central extensions, we construct Maxwell-l ], the adjoint and the coadjoint representations of the exte R R , , 59 , 58 ISL (2 ISL (2 \ L L symmetry was previously obtained as a particular expansion In this paper, we study the group structure of the Maxwell sym In to answer this question, it is important to note that A semi-simple enlargement of 3 5 c are exactly the ones found in [ of the to semi-direct product groups [ group tion can be generalizeddimensions, to as define well higher-spin as extensions infinite-dimensional of enhancements t o semi-direct products are shown to lead to novel Wess-Zumino we show that thespecial Maxwell type group of in extended three semi-direct product space-time based dimension on the SL dimensional gravity see [ dimensions as a particular caseparticle of in extended a semi-direct constant pro electromagneticquently, we field focus as the on correspo the (2+1)-dimensional case. Following other hand, the ISL(2 mentioned Maxwell-like symmetries must bethat given by extends a semi-direct genera products. In this article we show t literature (see, for instance,question [ is: what is the group structure behind theseMoody infinit algebra and the Maxwell-like partners extensionPoincar´e-like of in the that they include non-commutat symmetry of the Wess-Zumino-Witten model thatSimons follows action from after Hamiltonianmetries reduction. can be The obtained fact from thatcould a also bo Maxwell-invariant be Chern-Sim relevant in thethree-dimensional context of gravity quantum and Hall the systems quantum Hall effect has be asymptotically flat three-dimensional Einstein gravity [ Wigner contraction, was also studied in [ bms dimensional enhancement of the Maxwellunderstood algebra as in an 2+1 Maxwell-like dimen extension of the ary conditions, the asymptotic symmetry of such extended th Kac-Moody algebra [ and can also be recovered by means of a generalized Sugawara c uct is computed along the lines of [ and its infinite-dimensional generalizations. Thetension constru of a semi-directproducts product [ group. Following thesented are gener given. Thedefined, Kirillov-Kostant-Souriau symplecti and the Hamiltonian geometric action associated to JHEP10(2019)039 , is ˜ V we and K 5 (2.2) (2.1) (2.4) (2.3) ∈ . The K a , as the V , ∈ V al gravity V on U , where ∈ K V α ], as well as the σ ), ⋉ 65 , we define central 4 K α, a ), where , and a representation group. In section . . ) ˜ V 3 . U, α  endowed with the product ( β tain Maxwell extensions of V → , β, γ ´egroups [ s generalizations. Naturally, ( ext [ ) BMS × ∈ C wo novel Wezz-Zumino terms results. V β V uct group nd generalize the actions previ- ct products, which will be used α , i.e. U × nd construct the geometric action 1 2 σ ) + V α, β γ V we introduce extended semi-direct + + algebra in three and four dimensions based ∀ : ), we introduce a second vector space with elements ( b + ) is a representation of 2 group. In section bms defines a central extension of V ] ˜ 3 × + V 2.1 ]. 59 V , α α, β β ⊗ ext ( · 68 , in the sense that GL( U C V , V σ β , a U 58 σ → 55 × + = [ – 4 – ) = α α 6 K ) = ( : U ext  ]. σ α, β V σ ( 67 . Groups with this structure are ubiquitous in physics. V, β C , ) = ( K ) = 66 • we show how the Maxwell group fits in this description. -valued two-cocycle on β ) + ) β, b 3 ˜ V × ˆ + ( α β, γ U, α ( compatible with ) ( is a + U ˜ V β ρ α ](for a recent discussion in the context of three-dimension α, a ( ( × C on 62 – α 1 2 K 60 is a vector space and ) = ) of ˜ V V denotes the product in α, β ]). ( V C GL( · 64 , U , and the group operation is given by [ → 57 V The paper is organized as follows: in section For a detailed analysis of the rigidity and stability of the K 6 ∈ , together with a skew-symmetric bilinear map : ˜ ˆ Then, we define the extended group and the following product: function a , on their semi-direct sum structure, see [ BMS group in three and four dimensions 2.1 Extended semi-directIn product order groups to extend the semi-direct product group ( 2 Extending semi-direct products In this section we definelater a to particular formulate the extension Maxwell ofthe group semi-dire in starting 2+1 point dimensions in and our it construction is a semi-direct prod In the 2+1-dimensional case,higher-spin symmetries, we loop generalize groups theextensions and results of the BMS extended to semi-direct ob based products, on the and Maxwell presentconclude extension t with of a the discussion Kac-Moody and and possible the applications of our see [ groups, define the adjoint and coadjointon representations, a coadjoint orbits. In section where Some examples are the Euclidean, the Galilei and the Poincar + does not define an . Indeed, ρ Note that this operation is non-commutative, and therefore ously obtained in [ globally invariant under the considered symmetry groups, a V α elements of a semi-direct product group are denoted by pairs JHEP10(2019)039 ˜ ), G V ⊕ (2.8) (2.7) (2.9) (2.6) (2.5) (2.10) (2.11) (2.12) V .e. X, α, a , extended  ) reproduce . β . . . , the group A  is the =0 U . The elements V 2.12  =0 λ β ir corresponding

λ V β Ad G

i × ) U σ 1 dσ σ − α ⋉ ) × × ) and ( . The group operation + K ˜ β α V a Y, β, b , must be replaced by the ( 2 1 Y 1 2 2.10 ∈ ) U, β, b β dρ ( − dρ a + U l semi-direct sum algebra. s. ( • in terms of the adjoint repre- uct group and its b α σ . −  U U g and product ,λα,λa b ) by triplets of the form ( . This leads to + ρ σ . Thus, the inverse of an element a  X g , and is given by the direct sum λX a α U + dσ e × stand for the derivative representa- K 1 , V ( , − β ext U ∈ as the group dρ . ext , λα, λa Ad ρ + α , ρ ext V α , dρ β, a α V K b U − d Y V U , λX dλ ) σ σ A e and ρ α, dσ K dρ U  × 1 + × σ acts on a central extension of − • ∈ − ( dσ ) = Ad ) = ( U dσ ) ), this leads to ). – 5 – ⋉ The Lie algebra associated to β U σ K dρ i β can be found by evaluating X − k K 2.3 α, a V , α g − , × ( · Y, β, b 1 is the identity in = a = ( U, β, b , while U − on ( ) U U ) e g , α h G U K ρ Y , dσ  G ), where ]). The third slot, however, implies the presence of more X d × dλ ), where now the term β, ρ X,α,a = ( 56 σ ) = ad A 1 ( )] = (0 , 2.1 U  − ) ) = U, α, a 55 ). Using eq. ( Ad 0), where β, b , ( V, β, b )] = dσ )] = ad , 0 ( β, b extended semi-direct product ) can be found by looking at the infinitesimal adjoint action, i ( • − e, U, α, a X, α, a ) U g ( α ) , and the derivative representations ) α, a ) = is ( ρ U Y, β, b Y, β, b X ( ( on , , , respectively. Denoting the elements of × G are Abelian groups, they are respectively isomorphic to the ) ρ U,β,b ) U, α, a ( σ , ad A, σ ( ˜ is the direct product representation V k U denotes the of X, α, a Ad , the adjoint action of ˆ ρ ( + ( and k U ) ) bracket Ad × and denotes the Lie algebra of σ X, α, a X, α, a ∈  ) is given by ( σ k [( [( V α, a U,β,b X are given by triplets ( ( = The Now we define an G Ad U, α, a with semi-direct sum tions of where Extended semi-direct sum algebra. pair ( the form of thebracket, adjoint respectively representation (see of [ aLie semi-direct algebra prod generators and modifies the structure of the usua The identity of sentation of This defines the adjoint representation of the Lie algebra where Ad Since The elements of the first two slots on the right-hand side of eq ( Lie algebras. Thus, the Lie algebra associated to defines a non-central extension of the original semi-direct It is important to note that, even though with Lie bracket [( where of is the natural extension of ( JHEP10(2019)039 is → uc- is a ∗ ∗ g V (2.17) (2.18) (2.15) (2.19) (2.16) (2.13) (2.14) G ⊂ ) where × , ) V 0 ,F  : ,F Π F . 0 , ⊙ . ∗ U g Π J, , i o 0 ) denote the dual o . J  G ( ∗ 0 ˜ V F , ρ ρ . O i ∈ stands for the coad- ∗ U a ,F ) as ˜ V ρ 0 X i J X, α, a must be such that the ˜ ( Π ∨ V and ∗ U can be defined as , , ∗ 0 α ∗ ) F, a . The third map necessary g G h σ J F, dρ U, α, a ˜ h ( near products and V ,F on its Lie algebra dual, it is , + on Π + Π =  = V . mi-direct-product group V 0 G ∗ U i ,  ˜ are elements of the dual vector V J, G . The space V k 0 i ( i ∗ . Here Ad ,F h , α , σ , while and its dual space can be naturally g F 0 X ∗ ,F ) Π k g = Π 0 U h F, β , F F,X g on Π 0 Ad + ∗ U , E ∨ ⊚ J ) ρ 0 k G i dσ α a J is the isotropy group of the orbit represen- ) ∨ h in the sense that ) = ) 0 α − h ( × J, X 1 , ,F h U,α,a − X, α, a = 0 – 6 – ), lead to the following form for the coadjoint ∗ ( ⊙ ( V U U,α,a Π ˜ , = i ) ∗ ( V σ α 0 i . Its elements will be denoted by ( , while Π and α g J 1 2 X 2.10 ∗ ∗ β i ( Ad X ) g U,α,a G K + × dσ ( ) = Ad U , dσ F G/ σ Ad ,F ∗ U Π The coadjoint action of , , a coadjoint orbit of the group F, α h ∈ ρ Π X, α, a ∗ h ) ( ) g is familiar from the well-known semi-direct product constr on its Lie algebra dual ⊚ = , J, , where ( ,F ) ) ∈ a V ⊙ 0 n K , dual to the map i Π  is simply the analogue definition for ∗ , such that ,F ,F 0 0 ∗ = J, U, α, a V Π ,X ⊚ g ( Π + Π ( ) , ,F ) 0 ∗ 0 U Π n 0 J, → J σ ( → ,F ( Π h ) = ⊙ 0 = ∗ ∗ , , respectively. The pairing between ⊙ Π U,α,a ) 0 ˜ , α ∗ ˜ V denotes de coadjoint action of ∗ ( V 0 ,F 0 h α J ˜ J V ], while ) ,F × Π G/ G ( × Ad 0 + ) remains invariant under transformations on the orbit 59 O Π D J, ˜ V , V J , ( 0 : U,α,a and : ) ∗ ( ∗ U J ( 58 ∗ 2.13 ∨ ⊚ , G V Ad U,α,a 56  ( ∗ = is an element of the dual space , and Ad ∗ tion [ Note that the definition of pairing ( to construct the coadjoint representation of an extended se These definitions, together withrepresentation of eq. an ( extended semi-direct product group: g product tative isomorphic to Coadjoint representation. necessary to define three extra maps. The first two are the bili Let us consider now the dual space 2.2 Coadjoint orbits where we have used the identity In order to explicitly write down the coadjoint action of Given a point joint action of the group where Ad constructed out of the corresponding pairings defined in spaces J JHEP10(2019)039 ) ) a Θ ruc- , 2.20 (2.28) (2.25) (2.27) (2.26) (2.22) (2.21) (2.20) (2.23) (2.24) α ic form Θ , ], one can U ), eq. ( 60 , ,  . 2.12 , F 57 ith a well-defined g ∗ X = 0 E  , 0 . g )]

g ) )]

) ,F a ) 0 0 F , dρ a Θ Π ,F , Θ ∨ Y, β, b e corresponding geometric , 0 , ( α is given by ( 0 Y, β, b α , α Π , ( and the triplet (Θ J g Θ . ) , ) , Θ 0 , defined by a  ) , )) G U J t 0 α G Π + ( (Θ ∗ X X,α,a ,F (Θ ( , U,α,a , a 0 , ∗ ( X, α, a ) )  ad a [( Π t , 0 ( , , , Ad Θ 0 to Γ. As remarked in [ ∗ )  , 0 F , dσ J ,F , α A α 7→ 0 A ,F ) Γ ⊚ ) Θ t ) by the projection map ,F Π Π ( , Z 0 , a U 0 G U Π J, − ) and yields – 7 – ( G ( J , X,α,a

0 ∗ ( = [(Θ Π + U, α, a J ] 2.16 + 1 2 G 7−→ −→ − ⊙ :( I g 62 D − t ) = ad E α R = 0 ) ) , respectively. The definition of the infinitesimal adjoint = ,F + ∗ A ) = 0  ˜ ) is given by a V J Π I ⊂ , X,α,a ) 0 Θ ∗ X , ( 0 Y, β, b , J v ( ( Y,β,b Γ: α ,F ad and , ( A 0 O  ) Θ ∗ a gauge invariant function on the orbit, and include it in the Π , v , , ] ]. In our notation, the Kirillov-Kostant-Souriau symplect ) 0 V → U ,F Coadjoint orbits are naturally endowed with a symplectic st J , H 73 63 ) = ( Π Ω = d G – G , on O ∗ X,α,a ,F stand for the dual derivative representations. J, ( d (Θ P 60 72 ( K v P: Π ∗ ) J, dρ Ω ( denotes the vector tangent to the orbit at ( ) ) X,α,a ∗ ( ], which allows one to define a geometric Hamiltonian action w and ad ∗ X,α,a 71 X,α,a D denotes the pull-back of the one form ∗ ( ( – v dσ ∗ ad 69 A and corresponds to the left-invariant Maurer-Cartan form on where d is the exterior derivative on the group where Γ is a path in where define a Hamiltonian Poisson structure [ on a coadjoint orbit can be solved to give Here wedge product betweenaction differential then forms reads is [ assumed. Th action follows from differentiating eq. ( where representations of ture [ Geometric action. Since the Lie bracket of the extended semi-direct sum algebr It can be shown that the pull-back of Ω on is given by a locally exact two-form [ JHEP10(2019)039 ] 57 (2.29) (2.31) (2.32) (2.33) (2.30) )), provided t . (  . Its pull-back β ˜ V U K σ  × ˙ 7 α α . 2 1 × ), with tangent vector t cases as particular Noether  α ) t ) + erved. 1 2 t ( . 2.28 ( ˙ α t b − . = U ˙ × a λ α left transformations of the form σ

of the gauge parameters of the global )   t Θ (with overdot denoting derivative , + ) ( 0 ). The Hamiltonian can be chosen λ , α × ( , a F K 1 2 α ) dt ) can be written as ∗ U U t given by ( 1 ( Θ 2.27 ρ − rbits.  , the role of right and left transformations are ), while gauge symmetries are given by − ) to find the general expression β 1 2 ) on a vector is given by  ) g ) in , , , ) t α t t G U 2.26 t ( ) ( 8 ( in ( , U V U ( − +  σ 2.7 t ˙ ˙ a ]. U ) ( σ U α a Θ ) is equivalent to the general definition given in [ U t V ˙ +   H α ( U 57

Θ Θ × 1 1 ˙ ˙ U Θ – 8 – ) can be written as U U α U − dt − 2.27 , α d ) β , )  Θ dλ ) Θ t t 0 2 1 t ad R ( ( ( U 2 1 U Π U = Θ dσ dρ 2.27 = − + V ρ σ ∗ U − − − · ∗ U  σ a ], it is shown how the different Hamiltonians can be found by ) = = Θ =

Θ U t = = = V 33 ( (    σ a + α ], the action Θ ˙ U ˙ ˙ ˙ a a a U k → +  74 E ˙ ˙ ˙ dΘ , as shown in [ dΘ α, α, α, dΘ ) ) ) U ˙ 0 U ˙ ˙ ˙ Θ α, b U, U, U, ( ,F    0 V U ), the action ( ). When written in terms of Π a σ α U , Θ U, α, a 0 , Θ Θ Θ + J 0 ( 2.25 J G U, α, a D  ). Following [ U, β ∈ t ], Hamiltonians for geometric actions are defined in differen ), the Maurer-Cartan equation ( = ( · corresponds to the left-invariant Maurer-Cartan form on dt )), we can use the product law ( )) 1 57 t t V ( − U ( ( Z ˙ g 2.12 ) in eq. ( a , , b ) is the tangent vector to the path ) ) t = → t t ( ( ( 2.32 ˙ G ) ˙ U α I , β , ) ) Using ( t t Note that the form of the geometric action ( The preservation of the symmetries might imply a redefinition ( ( 8 7 ˙ V U U, α, a interchanged. ( charges or combinations oflooking at them. the invariant In functions that [ label thewhen coadjoint setting o ( in such a way that the global symmetries of the action are pres Extending this definition to the full path in Note that Θ symmetries. In [ geometric action by adding the term right transformations ( The geometric action is naturally( globally invariant under Using ( to Γ reads with respect to where JHEP10(2019)039 e ar ries and seen (2.34) (2.35) (2.40) (2.37) (2.38) (2.39) (2.36) V K . ded semi-   , i.e. β . . K X ] (for a detailed ), where  U  β , given by the set 75 α β 2.7 Ad , U ): ext (ab) ad 55 k 1 2 Ad + ad α 2.12 a − oduct ( Y ad es in 2+1 dimensions that α ). Thus, we can define the space 2 1 U f 2 + 1 dimensional systems ll be the key construction to ad ion ( cket associated to the corre- + . are both given by the adjoint 2.3 − Ad b . b seen as an Abelian vector group,  k U X dρ β denotes the Lie algebra of K ∈ = Ad ad , ρ , + Ad and (ab) + ad α, k α, β = (ab) ext b Y ∀ k (ab) ext ), and is given by dσ X k ad β , a U Ad ad 2.7 U , σ − Ad ⋉ β , i – 9 – α β , where k ad K X (ab) k + Ad = − = (ab) ad k = ad a h = H U β ˜ Y, V Ad V, α X × · Ad ⋉ are given by the adjoint representation of ) is trivially satisfied provided the anti-symmetric biline = α U ad ρ β, K 2.2 V   X U group, and higher-spin extensions thereof [ and acts on its Lie algebra by the adjoint action. Examples are th 3 ) = )] = Ad σ K ad ]). Here we show how to generalize this special kind of symmet [ BMS − ) can be easily evaluated to give V, β, b Y, β, b , the general form for the adjoint representation of an exten 76 ( α ( k ) = , • U 2.10 ) ) Ad , and endowed with a group law of the form ( X, α, a X, ( X, α, a U, α, a U ) ( (ab) [( is chosen as extended semi-direct product k 1) group, the , Ad × × U,β,b Let us consider a particular case of extended semi-direct pr  ( are given by the Lie algebra associated to Lie group = Ad (ab) ˜ and the representations In this case, the requirement ( V With these considerations, we can define the extended vector k representation of direct product ( As in this case the derivative representations special map to the case ofdefine extended semi-direct in product a groups, unifiedwill which manner be wi the treated different in Maxwell-like the symmetri subsequent sections. analysis, see also [ sponding Lie algebra Differentiating this expression leads to the Lie algebra bra Several semi-direct product groupshave that the define particular symmetries structure o 2.3 Extended semi-direct products under the adjoint action which can also be directly obtained from the general express as a vector space, and where the group operation follows from eq. ( ISO(2 JHEP10(2019)039 → can (3.1) (3.2) D (2.44) (2.43) (2.41) (2.45) (2.42) × R directly , × is the or-  H , D ) by means F . k ↑ R . As seen in ∗ U . , can be con-   : k  D 1) 2.27 D Ad , F × R 1  ]. It can be writ- ∗ X ) lead to ˙ F, t 1 α 2 , , the product − ∗ U α ad k D V i 2.18 ad D R ) and ( Ad 2 1 ∗ α F, by F, a ∗ α h − 2.25 D , representation of ) on ˙ a + R ) and ( = SO(  k D ext Π+ad i , β . , 0 ∗ U ∗ α µ Π + ad K , α 2.17 e F ⋉R ), where the form of the coadjoint = 4, given in [ ∗ X ,...,D is the vector space of translations, Π Ad ∗ U ↑ ⊗ h ad = ad D ν D F, 2.14 Ad = 1 + 1) e anti-symmetric matrices, isomorphic to β R ∗ U , k  µ i 1 F, − ∨ ( D ∗ a = + Ad ν − α ), where µ ∗ a e k × e

J, X V D = ˙ h ⊗ 2.5 α D . Therefore ( , µ ∗ β – 10 – +ad = 0 e Π + ad h  Π ⊚ ∗ α i = cross product ∗ U ) F = SO( α = ad ν ∗ U e is written solely in terms of the paring in D dimensions ∗ = Ad + ad ρ

h × Ad π J ). In the case of a special extended semi-direct products, D . Therefore, the Maxwell group is given by α ∗ = dimensions, µ + X, α, a ∗ X ⊙ e D k ( ∗ ad , E R D α σ ad 2.16 2 1 ) ) is the space of  Maxwell ˙ dimensions, which will be denoted by Maxwell U  ( ,F D Π+ U D Π ) = R U ∗ Θ J, , ,F 2 ( 0 Ad h Π ). This gives J V ) =  D J, ) for the algebra α ∗ ∼ =  ( ,F ) 2.35 ) ) implies that dt Π D 2.13 +ad is a central extension of the translation group ( J, Z , this extension requires a skew-symmetric bilinear produc ( 2.34 ), yielding J X,a,α ) ∗ ( so ∗ U = . By choosing an orthonormal basis ext D 2.1 ) and (  ad = R H 2.19 D Ad I U,α,a ˜ ∗ ( V R Coadjoint orbits are defined according to eq. (  2.34 = 2 Ad structed by generalizing the original definition for section be constructed by means of the usual where The construction of the geometric action follows from eqs. ( tochronus in the second exterior power of while its infinitesimal form is given by The Maxwell group in V and while eq. ( ten as an extended semi-direct product ( of ( representation follows from ( the pairing ( 3.1 The Maxwell group in 3 Maxwell groups Putting this together allows one to write down the coadjoint from ( JHEP10(2019)039 . ) ) D ext D and ), it R R (3.7) (3.9) (3.8) (3.5) (3.6) (3.3) (3.4) s are ( 3.4 2 (3.10) (3.11) D , 3.9 V . R σ σ =0 e λ =

. × ρ  ν ρ ν ) and ( e . e . , endowed with e ν  i ×  e 3.7 × D ρλ µ β µ × R e a Λ e ) , respectively λ µ 2 µ σ e × µν ) β a V 1): α ν αβ µν 2 , λ , a α ∈ 2 1 tors 1 on the elements of , J β ) a ν ν µ − ↑ + − e e , Λ . ν + ( µ T α D 1) D β ). Similarly, the bracket for the × µ ( , Λ µ R λσ D ext . b µ 1 Λ a α R so , λ α ∈ 1 2 ( , e λ µσ 2.10 − ρ 0 η 1 2 i α + Λ ). Denoting its elements by triads of ) λX = , ρ ν D , e , enables one to write down the adjoint δ ), 1) = µν T T 2.7  , 0) J − Λ β 1 Λ , SO( ( β, a = (0 ). a 0 α, a d h dλ × νσ − X , Λ ∈ µν η 1.1 µν ). Putting all this together, the product law + Λ ρ µ µν D = ≡ F δ ( . Then, the extended group of translations X J = Λ are given by the infinitesimal limit of eq. (  – 11 – a ( 2.2 , α 4 1 µν , α = Λ so D , ν X ≡ F σ dρ R e = Λ = ρ ΛΣ 0) ) µν ), plus the fact that the adjoint action of the Lorentz a algebra can be defined from the following exponential µν × 2 , , ρ D  a ). When applied to the generators ( µ X µν J µ µ and V 2 1 D e e J , e 3.11 ( ν ) = + 2.12 µν dσ α a µ ν , dρ maxwell 2 1 µ ) with = (0 Π ρ , β, b maxwell µ e µ are Abelian vector groups, their corresponding Lie algebra corresponds to the exterior power representation = 2.3 ) and ( σ α (Σ  = Λ Π ρ α D • + 3.4 α σ ) ρ R Λ , a ) µν ), ( µ 2 J ≡ µν e group follows directly from eq. ( V µ , α, a J µν 3.3 α ( α Λ D X (Λ σ µν = 1 2 and ), we get X α stands for the generators of Lorentz transformations ˆ + of the form ( D 2 1 are given by the standard vector and tensor representations algebra follows from ( ) = stands for the usual matrix representation for R µν , α, a  = D J µν D α J R The Maxwell algebra is given by the extended semi-direct sum The action of a Lorentz transformation Λ X X, α, a 2 ( dσ The derivative representations so that the elements ofmap the and The definitions ( isomorphic to themselves. This means we can define the genera where As both Therefore, it automatically satisfies eq. ( V group on its Lie algebra is given by Ad maxwell leads directly to the commutation relations ( of the Maxwell form (Λ a product which defines the natural basis for is the centrally extended group with elements ( representation of the Maxwell group using expression ( Note that, in this case, JHEP10(2019)039 , i- ),  as 0 ∗ ) for F ρ 2.17 Λ (3.17) (3.16) (3.14) (3.13) (3.12) (3.15) , ρ 2.33 0 and Π ∗ . Λ σ σ  ) , . As these repre- µ ) = ˙ ) α ext ν . µρ xwell algebra can be ly invariant under the α V π, f F µν − -dimensional Minkowski :( νσ on the orbit as F η ν , ) D ˙  0 . α ) − ρσ µ ,F ρ . 0 ρσ α α e left-invariant Maurer-Cartan X tions of νσ . In fact, evaluating ( ( ts can be defined according to a 0 with components ρσ d (Π F 2 1 σ µν F ce of a constant electromagnetic ∗ µν O σ α J µρ − F e given by the covariant versions of 1) µ η with the , . νσ − Λ , νσ 1 µν η ], and therefore its irreducible repre- ρ ν − σ η D ˙ ν ρ µ 4 a α ρ − µ α R α , µ  µσ , η d µ µν 1 η D ρ F 2 1 ) = Λ ( 2 1 F Π µν α ν ) νρ in this case can be worked out using ( ) for the Maxwell group is expected to de- ( = µν η Π T = so , we get µν = is defined as 2 1 η ∗ electromagnetic field Λ ⊚ µ µρ +  µν 0 ˜ ) V µν 2.27 – 12 – η 0 = − a F X F , f νρ ∈ µ − ρσ ,F and F 0 µ µν ∨ µν α  (Λ 0 µ a 0 µ J F Π , 2 1 µσ d α ⊙ Π Π ν 1 1 2 2 ( and the µ η  µ ( ,F + νρ algebra and its dual space are isomorphic. The isomor- σ 0 Π , = = Π = η ν µ ν ( ρσ Π k ˙ D ˜ Λ α = Λ V V α a i , ), respectively, where the dual representations ρ ρ i i µ dΛ d 0 1 2 α µ µ 1), which maps contravariant vectors and tensors into covar ) 1 ν 0 π , α Λ µ − ) can be constructed considering = = 2.44 J, X F, a 1 2 Π h h h maxwell µν µν ,..., (ΛΠ = = Λ = Λ momentum 2.13 ) 1  a , α Λ Π) F 1 dt Θ Θ The Θ ) and ( ). The products ⊙ ⊚ − Γ Z a α ( ( with representative 2.43 3.11 = ∗ D ˜ V = diag ( ) takes the form ), which in this case reads ). Then, one can construct a geometric action that is natural ) and ( µν η 2.18 Maxwell 2.32 3.4 I 2.14 The coadjoint representation of the Maxwell group and the Ma therefore we define the sentations correspond to Poincar´eparticles infield, the the presen Hamiltonian geometric action ( well as their corresponding derivative representationseqs. ar ( The orbit in eq. ( Once the coadjoint representation is known, coadjoint orbi while ( scribe world lines ofa such representative particles. of This the form is indeed the case sentations can be obtained from the irreducible representa Maxwell group. The key ingredientform in ( this construction is th and lead to elements in the Lie algebra dual obtained from ( phism can be explicitly constructed by endowing metric Particle action. The Maxwell group satisfies Mackey’s theorem [ ant ones. The pairing ( JHEP10(2019)039 ], 57 , re- = 0. (3.20) (3.22) (3.23) (3.18) (3.21) (3.19) ) as a µν µν y ˙ ). Their f Kostant- . 3.16 stant elec-  3.14 , and  ints 2 assive particle µ , . µν x ). By definition, M F + = = 0 = 0 ) and ( , 3.19 µ } } }  ν π with extra coordinates νρ ρσ µ 3.13 Π through the projection µρ π , ∂ µν ,F ,F ) µ , 0 µ a ∂y e 2 Π µν ρ ρ t of view of non-linear real- ,F Π { y action of the algebra [ 0 F ctromagnetic field in Hamil- ν well group. The elements of { . µ − , { y Π f as the canonical momenta as- , µ , hem  x (0 − νρ ν µν ) ν y ∂ x µ O µν x ˙ , νρ ∂y x f can be implemented in ( − , ∂ µν i ν µν 2 1 f ρ 2 f ∂y x µσ 2 1 µσ − ρ 1 2 ν F ) together with ( J − µ M f . Finally, by identifying the translation → − + y νρ = e , ν + νρ − µρ η ˙ µ x η µ µ y + 2.19 and the elements µν  µ π p − = − µ µ x + µν ( x ν ∂ = = ∂ νρ µ 1 2 νρ ∂x π ,A , f – 13 – p ν F ˙ ∂y J x µ ν ν µ x µ ν − π x µσ x µσ /δ A x ∂ η µν η ∂x 1 2 µν − f − ˙ + i y − − , ν ν µ + +  p corresponds to the momentum associated to the extra , ∂ p , µρ µ µ µρ µ µν ∂x µ define a basis for the Noether charges ( µν F → − π x f Maxwell J ∂ µ = µν Π f µν ∂x x f µ ∂ νσ δI νσ µ 1 2 = = = µν νρ p η ), η   ∂y π F η µ i i i defines a path on the orbit = + µν µν + + Π 1.1 − J µ F µ D ˙ x ν p νσ νσ µ → − → − → − , and Π F J π µ µ  µρ µρ µρ µν µν can be computed using eq. ( Π η η η J F ,Π dt ]. Here, we have shown that it can obtained from the Kirillov- ) 0 algebra ( = = = µν Maxwell , which play the role of Lagrange multipliers for the constra with space-time coordinates are the canonical momenta associated to the coordinates 30 ,F Z J } } } . We see that is the momenta that accounts for the minimal coupling to a con D 0 , µ ρ µν µ ⊂ Π = ρσ ρσ , µν µ α y Π x 15 , the geometric action takes the form f π . The mass shell condition , (0 D ,J , ,F ). We can consider an orbit of the Maxwell group describing a m νµ O 9 µν M µν y µν J maxwell J J and { − 2.24 { { µ Maxwell = p I µν y components The quantities which leads to i.e. the they satisfy a Poisson algebra isomorphic to glogal symmetr Souriau symplectic structure onthe coadjoint orbit orbits ofcomponents the read Max of mass Hamiltonian by means of a Lagrange multiplier This is the actiontonian massive form, particle and coupled it toizations has [ been a previously constant studied ele from the poin map ( coordinates The path Γ tromagnetic potential As Therefore, spectively, one can define a coordinate representation for t where we have defined sociated to JHEP10(2019)039 , (ab) } → ) group (3.30) (3.28) (3.31) (3.29) (3.27) (3.25) (3.24) (3.26) , R 3 , { , 1) group. (2 , ρσ sl rm dy zations of the µν dy = 3, i.e. , associated to the , 0 . νσ ) algebra, seen as an D ! η . F 1 R µν µρ , a µν − can be shown to be iso- η a (2 ] for the ISO(2 2 1 ) for 1 0 0 + 3 µν sl 

R (3) matrix can be identified 55 ement ρ + ρ 3.1 ǫ 1 2 x so ut of the Casimir invariants of 2 1 ed semi-direct product defined νρ ρ al versions, the Maxwell . . µ = = dy , Λ 1 3 ext 3 ext ) and µ ν ρ t µ 3 ) define the Killing vectors fields of t α R dx µ . ( ρ ⋉ R ⋉ R , − α ρ 2 ): x , a ) 3.23 t ρ ) can also be identified with V 1) ρ ! = R x 3 µν t , µν , ρ η ρ ρ R α ν 3.26 ǫ ( 0 1 1 0 a ) is not the most general one. In fact, it is 2 Λ + ). In fact, any

µ 3 = ν V ] = −→ α 2 1 R ), a – 14 – ν ( 3.18 group as a special extended semi-direct product dx µ t 2 R = SO (2 = SL (2 = µ 1 2 e , , 3 3 3 V µ µ 1 , −→ t (2 dx t + α [ µ ν sl  α e = ρσ ) , ]. y ν + α × σ x ! ν 15 ν µ µ Maxwell Maxwell x e Λ x ν ρ 1 0 µ µ µν 0 1 − − a Λ Λ 1 2

µν can be identified with elements of the η → → 1 2 ρ = 3 ], the differential operators ( µ x R a µν = ρ x . In other words, the spaces 15 y x 3 1 2)-dimensional superspace with metric ( t ), we will follow the same strategy used in [ / ) when Poisson brackets are replaced by commutators in the fo R 1 4 1) 1.1 . This construction will allow us to define different generali + 2.36 − µν 2.3 D η ( denotes the generators of  µ D t = + In order to formulate the Maxwell 2 ]. As stated in [ D , ds [ morphic by means of Hodge duality. Thus, The same can be done with the space where with a vector in constant electromagnetic field [ Abelian vector group which satisfy the commutation relations of the form ( Secondly, vectors in Maxwell symmetry in 2+1 dimensions. the system, which can fix components of the representative el 3.2 The MaxwellThe group Maxwell in group 2+1 in dimensions 2+1 dimensions is obtained from eq. ( possible to add more constraints to the action constructed o i a ( First, we will consider the double covering of ( Note that the Hamiltonian used in eq. ( and whose isometries are given by and fulfill eq. ( can be alternatively described byin the section special kind of extend However, we will show now that, unlike its higher-dimension JHEP10(2019)039 ) ), ). ≃ 3.5 1) ) and 2.36 3.32 , (3.37) (3.39) (3.35) (3.34) (3.36) (3.38) (3.32) (3.33) ]. This R , which (2 , ), where ) can be so (2 R rm ( (ab) α, β , ) sl , R (2 = [ , U, α, a  sl  β (2 ) reduces to the 1 − sl × . ) and the commuta- α 3.37 3 Ad , R . ⋉ i . ) 2.40  1 R mi-direct product ( α, UβU (ab)  . ,  ] − ) ) is a matrix group, the Lie J, X h . 1 2 R U R ) given by is reproduced by the Killing ed as triplets ( , ν , 2 1 α, β t (ab) ext 3 + [ ) (2 2.3 SL (2 (ab) ext U = 1 R ) R 1 2 ρ sl , the Maxwell algebra in 2+1 di- − , µ t R  , , × + (2 X 2.3 ν b µ (2 ) is eliminated, ( Tr sl β J U bU µ + (ab) sl µρ 1 2 ). As SL (2 ) α + Ad η 3.36 ad R ⋉ µν , i ] = ) η , a 3.30 = 2 + 1 due to the isomorphism β , a = 2 ν (2 ) ) matrices. Hence, the group operation ( and therefore any element in 1 R ), i.e. t ν , R R + − µ sl ν D µ R ] = , , – 15 – t t , α Λ ) are realized as the adjoint action of elements ] = (2 (2 µν  (2 αβ Tr [ η 55 3.4 sl sl UβU , ν sl ) for (ab) ext 1 ). This means that the double covering of the Maxwell = = SL (2 + = algebras in ( ) ) = X ). Its bracket follows from eq. ( = − 2Tr [ and vice-versa. Thus, the pairing between µ are 3.6 3 3 R µ k J ∗ , t 2.37 β, b ) (ab) 3.33 (2 = R ) UαU u, µ , ˆ ) + ( sl R UV, α ) for ) is translated into matrix commutation, ) algebra [ R ) , , (2 maxwell  ) algebra given in ( ]. R → (2 Maxwell (2 −→ sl , R 3.2 sl 56 , 2.39 α, a i sl α (2 , ( ) = 3 ext (2 Λ sl R is given by 55 sl ) J, X ∗ h is endowed with a product of the form ( ) R V, β, b , R ( ) matrix and , ) ) and its dual space can be identified by means of the Killing fo • (ab) ext R R (2 ) ) R , , ) on the , R sl R , (2 (2 , = SL (2 (2 sl sl U, α, a ) and the identification ( K sl ( R SL (2 Following the construction developed in section , is an SL(2 ∈ (2 its dual space is naturally given by When one copy of the Therefore, the elements of the Maxwell group can be consider written as an element of U Lorentz transformations on vectors ( A base for tion relations of the double covering of the Poincar´egroup in three dimensions, mensions has the form ( group in 2+1 dimensionswhere corresponds to a special extended se turns into U algebra which matches the structure of ( This definition is isomorphic to ( has been studied in [ With this identification, the inner product of vectors in while the cross product ( form on means that we can make the identification where sl JHEP10(2019)039 ]. . ρ ) is 79 t , ν (3.43) (3.42) (3.44) (3.41) (3.40) (3.45) J 78 mple of 3.41 µ ) leads to X ρµ ν 2.40 ǫ ), respectively, = J 2.43 lization of ( , ∗ X ) ). Then, any element µ ad t as a general matrix Lie , , ) have recently been ob- are interchanged [ ) and ( , 0 3.30 , k ρ c , µ ρ F . F t (0 3.43 F presentations of the Maxwell 2.38 n be written down by defining µ ν n by matrix conjugation, and , , µν ( ab , . c ρ ≡ rticular conformal extension of Y F ǫ n, f µ µ µ and a ), which applied to ( X F µ ] = 0 ] = ] = 0 ). a ] = 0 ] = ] = 0 + ν ν ν b b b , µν Π T ρ µ c , , F F = 2 + 1 by defining the Hodge duals Π F F Π ǫ , , 2.44 0 , , , , ) can be generalized to the case where T Π , µ a µ a µ a 0) µ = D ab c , F F (0 Π Π α Π Π [ [ µ [ [ f [ [ Y 3.43 t ) leads to + ≡ , X ) and ( µ a ) for ] = (0 ad b J ) algebra and satisfy ( , , – 16 – , , F , , 2.40 µ c ≡ T c ρ ρ 1.1 c R ρ 2.40 , , , µ X Π J F a F J Π 0), 1 , (2 ab ab ab Π T c c c − µν µν µν a [ ρ ρ ρ sl f f f ǫ ǫ ǫ ) = T , , ] = ] = ] = (0 UJU ] = ] = ] = . 0) b b b ν ν ν ρ , J ≡ = F Π 0 J F X, α, a Π , F , , , , ( a , , a J a a µ µ µ µ ρ µν t ∗ U J Π J J ǫ [ J ( [ J [ J [ [ [ ]. This extension, in turn, can be thought as the simplest exa − Ad ≡ 0), 77 , The ( = µ can be written in the form 0 , , J 3 µν 1 a − F T ( , ρ ≡ J maxwell a UXU of the form are the structure constants. In that case the natural genera are the generators of the ρ µν J ∈ ǫ = c m ab µ ) − t f X = It is important to note that the commutation relations ( U is an arbitrary matrix Lie group M. Thus, we can consider µν Ad X, α, a algebra which can be put in the general form ( given by Generalizations. K tained in the context ofthe Galilean non-relativistic algebra symmetries [ as aa pa Hiertarinta algebra, when the role of the generators J the following generators ( together with their infinitesimal forms ( 3.3 The Maxwell algebraThe explicit in form 2+1 of the dimensions Maxwell algebra in 2+1 dimensions ca In term of these generators, the bracket ( where where their infinitesimal limits correspond to matrix commutatio The adjoint and the coadjoint action are equivalent and give This allows one to constructgroup in the 2+1 adjoint dimensions and and its the Lie algebra coadjoint from re expressions JHEP10(2019)039 . , , σ σ 2.3 ) ) ρ ρ (3.49) (3.50) (3.46) (3.48) (3.47) F F ). Now ν ν ( ( R ). In fact σ µ µ σ , R ǫ ǫ , (3 . group. Their , (3 sl ] = ] = ) 3 tensions, which ) sl νρ νρ µν µ . F J = T T , , , , ices. One can define k µ µ 0 0 (ab) ext ] , , ) J F [ , [ R 80 (0 (0 ) developed in section , . φ ≡ ≡ N, e higher-spin extension of ( α ( α ) when µ , whose generators are given ext U µν Π F sl (ab) , Maxwell group can be defined F ), whose elements are the con- ) and the BMS k α α F ) ) α ) s of the Maxwell algebra in 2+1 (ab) ext 3.45 R R ν ν Ad . ) | | , , ) = , and have the form R T Ad ) ) , , groups whose underlying structure ⋉ } , R π α σ σ σ σ ⋉ ) , ) ) ) ǫ ǫ , satisfying [ ν , µν ρ ρ | (ab) ext } R ) (3 ρ ρ , ) , + 2 K SL (2 ( ( F σ ISL(2 | | P F 0) µν R σ sl L ǫ µ µ φ , , , ) L N, ν ν 0) , ( ( ( ( ( ρ µ T , ρ ρ ad σ µ µ σ µν ( (3 µ , | ǫ ǫ ε η ε η F T T i µ µ { T T sl ( ν ) , , ( µν T ] = ] = ] = ] = ρ σ µ = { R Ad ǫ ǫ ε η (0 (0 , νρ νρ – 17 – ρσ ρσ a ] as a particular expansion of the ) ⋉ = ≡ ≡ (3 F P P P P ) R ,U a ] = ] = ] = , , , , 45 µ , , sl ν R µ µ ) µν to the group SL(2 T } νρ ρσ , µν µν Π φ J T Π 1 [ ( P µν [ , T T P J S [ µ , [ , U SL (2 ]. Therefore, one can define the spin-3 extension of the P µ T SL (3 , µν [ , T 45 , µ , [ . α T α , One example of such generalization is given by spin-3 exten- 7−→ −→ [ 4 Π ) one can see that these generators satisfy the commutation J F 0) { 1 φ , α α 0) S ) ) Consider the loop group 0 , = ν ν : , | | 0 3.46 , ) ) , a , σ σ µν σ σ U µ ) ) ǫ ǫ Π ρ ρ T T , ρ ρ ( ( ( ( } | | J P ) can be defined as µ µ ≡ ≡ ν ν ( ( µν ). These groups are interesting, as they allow for central ex ) and ( ( ( R σ µ σ µ µ , J ) plus ǫ ǫ ε η ε η µν J (3 , algebra found in [ 3.38 2.40 J µ ] = ] = ] = ] = sl 3 0 and parentheses denote symmetrization of the enclosed ind 3.43 J group as νρ νρ ρσ ρσ { 3 F J J J = ε < , , , , a µ µ maxwell µν µν J J Π [ [ J J [ [ is given by a special extended semi-direct product 3.4 Infinite-dimensional MaxwellIn groups this in section, we 2+1 will construct dimensions two infinite-dimensional Lie algebras are given bydimensions infinite-dimensional ( enhancement In a similar way, moreconsidering general the higher-spin extended extensions semi-direct of product the SL( a basis for the extended semi-direct sum algebra where Maxwell loop group. They correspond to extensions of the loop group will be considered in section Maxwell relations ( tinuous maps from the unit circle by sion of the Maxwell algebra found in [ Higher-spin extensions. This corresponds exactlythe to the commutation relations of th Using eqs. ( we will see that this symmetry can be obtained from eq. ( JHEP10(2019)039 . ), ν to t 1 µν . Its S 2.36 is iso- (3.57) (3.55) (3.54) (3.58) (3.56) (3.52) (3.53) (3.51) η , where ∗ , ext ) (ab) µ m ) imφ  t R , e R , m µ ] m µ , J (2 F = (2 µ m sl µ m X,J a sl . t −∞ L , = L m + = [  µ as ∞ m − m µ ad m µ eries, the elements of J ) are given by expres- X t P i µν Π ∗ X , space R m µ η . 0 ) µ m , = J , ad α R 0 µ , responding Lie algebra and t (ab) ext + ) as a set, and its product is −∞ , (2 ) ∞ , where the generators of the . SL (2 φ ≡ = X m µ ] ( R -dependent components, which m µ sl ) L m , µ t J m µ φ φ f (ab) ) metric L J 1 2 form a base for the Maxwell loop µ is given by point-wise multiplica- µ m m ) (2 F R )( , the Maxwell in 2+1 X,Y µ R = . X , X sl = K , n F ) , J 2.3 (2 L = [ UV + (2 R ∈ , ) with −∞  ). , sl m ρ −∞ 0 sl = imφ (2 Ad Y µ t ∞ , = e X t sl ∞ m L ⋉ X i m µ 3.42 µν ) = ( m 3.53 ρ U, V ) ), these generators can be written as t = φ ǫ P × ad , imφ ( R = 0 e m µ , = J, X = µ V 3.52 h , – 18 – F  = ) (ab) µ ) is the algebra of continuous maps from F 1 ≡ n ν ) t ) φ t ) R m µ − ( dφ R φ SL (2 , π t , m µ φ , ( 2 ( L U m µ µ 0 (2 Π , and µ t (2 a ) and ( Z  µ = ≡ . sl UJU X loop group as the extended semi-direct product ( sl + 1 π 3 , Π 1 L µ 2 ) replaced by ( V 3 S = L = 3.41  · Π 0 J imφ = ) , X U e ∗ U 3.30 ) φ 0 ( R , , = µ Ad m µ (2 Maxwell α t m µ sl L + L , is given by the semi-direct sum , ) can be written as Π i ). i.e. µ 3 ), ≡ , 1 ). Its elements can be written as R ), with ( is given by R J µ , − ) R , m µ , J φ (2 3.40 J, X J ( h 3.34 (2 (ab) ext µ sl ) maxwell \ imφ sl SL (2 UXU L X e L R L , L = = (2 = ) = ) can be written as m µ X sl R ). As functions on the unit circle can be expanded in Fourier s K J U , L R We can define the Maxwell Following the construction developed in section Similarly to what happens with the Maxwell algebra, the dual , (2 Ad X, α, a (2 sl ( algebra. Using the definitions ( where elements can be written in a form analogue to ( can be expanded in Fourier modes as Therefore, the adjoint and the coadjoint representations o where now the group elementsits as well dual as space, the are elements functions of on the cor sions similar to ( dimensions, where The corresponding pairing is then given by Here defined by eq. ( and where the product of two group elements morphic to the dual basis is constructed out of the invariant tion, i.e. loop algebra and satisfy the commutation relations L sl The corresponding Lie algebra JHEP10(2019)039 ∗ ∈  1 . J S (3.62) (3.65) (3.61) (3.63) (3.64) (3.60) (3.59) , whose  ]  ˙ vect 1 α S : ) will auto- α, 1 , [ ), where S  1 2 algebra and its a 3.62 vect , ,F is the loop group 3 + T 0 Π ˙ a K > J, ,  imφ ) ) is to be replaced by n 1 φ , e + − maxwell ( 0 ′ m ρ , U 3.54 L . The geometric action in loop groups with Maxwell 0 straightforward way. The 0 F riples (  ) where ab . . c , . µν on the circle ρ ) f ). The base ( ≡ ǫ UF , and the paring is given by M ), which leads to φ π , U 2 2.36 ( m a + = = 0 ) → morphism group of the unit circle ] = 0 ˙ 3.44   F α X ν dφ 3.53 n ν n ν µν 1 ) ) + 2 whose generators are given by ρ F − φ F )( . Π φ , ǫ and the group operation is given by , ( , ( , φ U µ (  m µ 0 m µ φ V. U (ab) ext 0 ) is a reparametrization of F (ab) ext J [ Π , 1 Π ◦ Π ) and ( m dφJ m a   S U = L π ( L U ) = ) with T 2 + + π J . The elements of the dual space 0 2.40 ad ) where the pairing ( = φ Z Ad , ˙ imφ , U – 19 – , i 1 ∂ , 3.59 ), while the adjoint and coadjoint representations ⋉ 1 ) V + 2 = n defined in eq. ( Diff n S n , e · − m φ 3.60 + ) φ + 0 + ( ∈ M 1 L ( U m m ρ  m ρ U 3.54 . The pairing between the m ρ S X LM 0 ( ∗ U J Π F ≡ J vect  = µν µν µν i (ab) ρ ρ ρ m a ,U ) ǫ ǫ ǫ Tr ) ) and ( Π R X The second infinite-dimensional Maxwell group we will be φ = = = , J, X ( dt 1    h , (2 n ν n ν n ν 2.41 U S Z sl  J F Π 0 , , , L , dφ π m µ ) in this case is given by a sigma model of the form 7−→ −→ m µ m µ 0 2 ∈ , J 0 1 J group. J φ a   Z  S 2.45 of the corresponding the on T 3 : ,F π 1 1 2 U S imφ e =  3 and Π ≡ ∗ ). In this case, an element ) stands for the generators of 1 m a R a S , J Maxwell ( associated to a matrix Lie group T This construction can be easily generalized to more general L (2 + I sl and This will lead to algebras of the form interested in is based onDiff the orientation-preserving diffeo LM where prime denotes derivative with respect to this case takes a form analogue to eq. ( function composition Extended BMS matically satisfy an algebra of the form ( integration on together with the infinitesimalgeometric forms action ( can be constructed in a The corresponding Lie algebra is the algebra of vector fields and its commutation relations follow from ( dual follows directly from ( structure by considering an extended semi-direct product ( elements will be denoted by correspond to quadratic differentials on L The elements of the dual space are naturally constructed as t JHEP10(2019)039 ) , the (3.67) (3.68) (3.70) (3.66) (3.71) (3.73) (3.69) (3.74) (3.72) m U, α, a s follow ), using ℓ , m  X 2.37 U ◦ , ]  −∞ , 2  = ′ 74 n ˙ , read ∞ m α dφ , + ) α  ) m P 1 57 m φ − ( ℓ S π F 1 , 2 , . ˙ 1 ) α 0 ′ − n 2 , , = α ) U vect ext (ab) − (0 ,  ◦ dφ X 2 1 φ , . 1 ( are defined by the action of (ab) ext ≡ m are given by triplets ( ∂ J S   +   3 1 ) m ), and similarly for the elements 2 ) 1 ˙ endowed with an extended sum a S i . φ ons of φ φ F S ) ( ] = 0 ] = ( ] = 0 group as the extended semi-direct (  ( ′ n , respectively [ vect n n n φ a ) can be evaluated to give 3 + ∗ ( Y 0 X (ab)  ′ , ) P F F ) m , vect  Ad 1 F  , , , φ vect ℓ φ 1 φ 1 ( S 2.45 ( ⋉ ) m 0) m m ′ ad ∂ + S − , ) J ) and n P i P X F 1 U [ m φ [ [ U (  − ℓ imφ S ◦ − vect h 1 ( , α ˙ vect ie α ) S m + ) + 2 (0 = φ − of the BMS φ × , , 0 , ( – 20 – ( ′ and J in Fourier modes as ≡ n n n = and its group law can be read off from ( X Y  + ] = ( + vect + ∗ U .  ) ) m 1 (ab) n + Π ∗ 1  m m m m = Diff φ φ  ℓ  1 S ℓ = S P ˙ ( ( 1 Ad 1 , 3 ′ U P J F S 3 0 S S m J X ) ) ) J ℓ ,   n n n , [ algebra with Maxwell structure. The Lie bracket has the vect  bms vect - φ − − ′ vect 3 − − − 0) ∂ vect vect 1 , U ) = = ∈ 0 m m m ) bms φ , ) leads to the Witt algebra ). Similarly, the adjoint and the coadjoint representation J ( dt φ Y ( m 1 ∗ X ′ X ). The elements of Maxwell-BMS maxwell ℓ α, a ) − Z ( ] = ( ] = ( ] = ( 3.67 1 3.67 ad n n n ad U Maxwell-BMS − ), and the geometric action ( ≡ dφ 2.35 ◦ π P F J U denote their components is the set 2 , , Maxwell-like extension , ( m ) and 0 a X m m 1 m 2.43 Z J ) and ( S J J J = (ab) ext ( [ = [ [  and + ) and ( 3 1 X 3.66 S α U ), which can be written in terms of the generators ) and ( ) in the dual space 2.3 Diff 0 ), ( Ad ∈ ,F vect 2.40 2.38 0 3.64 U The corresponding Lie algebra, Π , Maxwell-BMS 0 I J Defining the generators and expanding the elements in where given by ( product while the infinitesimal adjoint and coadjoint representati where here and yields the following commutation relations Now we define the eqs. ( from ( bracket defined in eq. ( diffeomorphisms on elements of The adjoint and the coadjoint representations of form ( ( with defines an extension of the JHEP10(2019)039 ). H 2.3 with (4.3) (4.2) (4.8) (4.1) (4.5) (4.6) (4.4) (4.7) 2.36 R × K . , ion of centrally = ) . ) 2 3 b are central terms, K =0 as defined in ( 2 3 β, n λ w, q ( ) and its Lie bracket . defined in section ; m = ) ) 2 . 1 1 ext (ab) λ ) k

, 3 (ab) ext i ω, q X, m U,m and k ) , respectively. The former Ad ( ; , )) V,W k 1 ˆ 2 X ( ⋉ 1 Ad a, m C )) ward way, provided a central Ad λ m ( ) K ⋉ 2 , U, V , e W, q ( 1 ) by implementing the following → = ) + Y K C X,Y 2 m α,m ( ( λ W 2.7 H + e c . · ) = ( ( , ad 3 , a n ] 2 1 ) ext (ab) + has elements ( , 2 − C ), given by b, n k ˆ U, V ; ) ) k ˆ ( R ) where ) + 2 2 X,Y , 3 Y C 3 Ad k 2 denote the adjoint representation of the cen- α, m ( V, m λ ⋉ ( ) · 2 β, n consists in a direct product β, n can be constructed, the central extension of b, n 2 , i.e. , and group operation ) = – 21 – , e ˆ ( ; a, m ( K H 1 U → ; ) R ) )] = ([ ˆ X K 1 K K 2 1 1 α,m = ∈ ( ∈ λ , ]. U, V e c V, n ˆ ( U,m Y, n ( U,m ) H ( ) = ( ( m ( ( , α 1 56 α, m C C , • ; h , ) ) 1 ) 1 2 V, n 3 55 and ad V, n and ) + d ) )( dλ )( ) + Ad 1 ) + Ad U, m X, m K 1 1 2 of a group U, m 3 a, m ( d [( ; ∈ V,W and the centrally extended algebra U,m 2 dλ ˆ U, m ( K · ( → U α, m U, m a, m ˆ K U ( U ) = α, m C ; 1 ) = ( ) = ( ) = ( 3 1 2 X,Y ( is known. This can be achieved by generalizing the construct ), where c U, m ω, q w, q ( W, q ) defines a two-cocycle on ( ( ( K U, m U, V ( is a Lie algebra two-cocycle, c C A central extension extended semi-direct products [ where has the form can be defined as Here, the expressions Ad and the group operation can be obtained from eq. ( where trally extended group generalization If a non-trivial central extended group The corresponding centrally extended algebra Its elements are sextuplets ( The special structure theallows extended one semi-direct product to introduceextension a of central extension in a straightfor 4 Centrally extended Maxwell-like groups in 2+1 dimensions This yields to the following product law: elements ( 4.1 Central extensions ofLet us extended consider semi-direct a special products extended semi-direct product where JHEP10(2019)039 with (4.9) (4.10) (4.11) (4.12) (4.14) (4.13) (4.16) (4.15) as ˆ K C , ) and the ), and has i . Denoting β 4.3 2.10 U . , (ab) ext , k ˆ )  Ad 3 .  , ad 3 X ) , f ˜ n i ] U α =0 ( k ˆ λ s a,

h Ad ; ˜ i α, β 2 = , β ) 1 2 1 ˜ n ) , ˆ h ation operation in , ad − i ) ] + [ α, 1 2 , i , i − i ; ˜ ,U ) ,X 1 + , b , β  ) ˜ n ) ) ,B Y, a λX [ α X U ) e U U . ( ˜ U U, V ( ( ( U X, − A ( i ( C ]  C s Ad Ad , ,B β − hS ) − hS − hS + ) + ) X, b ) = − h 1 β 3 1 2 3 ad A ( ;[ − n n n ( s 2 2 1 )] = B, 3 U s , which leads to the bracket ( X, n  + + + ) leads to the following bracket A a, n ) , f + U , ; ] 3 1 2 λX 2 evaluated at the identity, and it is related with − h b, n α i ). i − m m m ; U, m – 22 – 4.15 U ( S 2 , β Y, α , b U, e [ = = = 4.7 α, n ) ) = ) ( ; Ad ) leads to ) = (ad 3 1 2 C 1 ] − X β, n X  ) = (Ad h ] ; U , which follows from the definition eq. ( U , which is defined in terms of the two-cocycle ), the associated bracket can be obtained by comput- 4.8 1 81 β A, B 3 ˆ d B, n ( H dλ K X, n ( c X, n ( (Ad ) by ) X, β (Ad ( Y, n ) s − s ( + ad ) ;[ 3 β , q h h , , , a, m 1 β , 4.4 b U A,m = 2 ) + ( i + X b,m X 3 , f U,m ; i i ( U i ] 2 U Ad ad ,X α , a , α Ad ,X Ad ) Ad ) ) a, m β,m , α, m ) ; ad ; 1 U 1 ad X,Y U U ad 2 ( U 2 1 ( ( ). This yields [ ( − − U,m + defined in ( ( 4.1 = ([ X, α a hS β , q X, m b α, m ), c U U U − hS − hS − hS ; U U Ad 1 3 1 2 4.6 n n n V , q · = Ad = = = Ad = = Ad + Ad + Ad X, m is the Souriau cocyle on 1 3 2 ˜ a ˜ ˜ α U a α corresponds to the differential of [( X ˜ ˜ ˜ n n n S s = = = The corresponding centrally extended algebra is given by ω w W The later is the infinitesimal form of Ad Replacing these expressions in eq. ( where ing the adjoint representation of where prescription ( the form which determines the group operation ( its elements by ( can be obtained in the usual way by differentiating the conjug The infinitesimal limit of the expression ( the product law ( where the two-cocycle JHEP10(2019)039 ) ). tic R , 4.20 ) and (4.23) (4.22) (4.19) (4.20) (4.21) (4.17) (4.18) (2 sl 4.10 L ) as ) and ( = ), defined by k 4.4 R , 4.18 ) for , , , SL (2 [ L 1) 0) 0) 4.16 , the infinite-dimensional , , , . . , of the form ) leads to the Kac-Moody ) p group in 2+1 dimensions 0; 0 1; 0 0; 0  computed using ( (ab) ext φ ebra cocycle ( , , , 1 ) and its written in terms of ) . This will lead to novel cen- ( . (ab) ext . 4.3 ′ − ) R ) ) and ( ) , 0; 0 0; 0 1; 0 3.4 R X φ 4.13 , , , onal algebras have been recently ( , (2 ′ π VV 1 2 4.15 α, β (2 ¯ d Y sl ( ), and the cocycles ( ) in 2+1 dimensions is therefore given sl c = (0 = (0 = (0 d . This two-cocyle extends the prod- L U φ 1 , ¯ d d 3 2 1 ( ) = L 1 ) and ( ) k k k D Ad 3.55 ) + X − ad ( ⋉ in 2+1 dimensions as the extended semi- 4.8 U , , , i s )  dφ X group as well as the commutation relations Y, α π Y, a ) reads ) (    R 2 ( 3 c , R 0 0 0 0 , Tr c , , , , 4.4 Z , 1 1 – 23 – µ − ). The definition ( − (2 − t ) D π ) 1 ) 0; 0 0; 0 Z 2 U SL (2 sl , , ′ 4.1 ), given in ( π imφ µ [ d 1 L L \ U t e 4 R Maxwell X, b X,Y X, β 0; 0 , ) = ( ( ( π 1 L = = , 0; 2 c c c (2 , imφ µ 3 3 ), first we need to the define the central extension of the t ) = e = = = sl X,Y 0; 0 0; 4.5 3 1 2 ) = ( L , , imφ c f f f U 0 0 e U, V ( (    maxwell \ \ C S Maxwell-Kac-Moody algebra Maxwell L = = = ) in the form ( L m µ m µ m µ R ). This corresponds to the Kac-Moody group , ] from the point of view of Lie algebra expansions and asympto Maxwell-Kac-Moody group J ¯ d is defined on the unit disc F Π R , 52 algebra can be directly read off from ( ), where the two-cocycle ( , ). Following ( SL (2 R 3 , 49 L SL (2 (2 L 3.56 sl ) in maxwell d \ L L 3.51 by of the The adjoint representation of the The corresponding Souriau cocyle and its differential can be direct product Now we define the previously defined. The the adjoint representation of have the form algebra Let us construct nowdefined the in central ( extension ofloop the group Maxwell loo trally extended groups, whosediscussed associated infinite-dimensi in [ symmetries in three-dimensional gravity theories. 4.2 Maxwell-Kac-Moody group whose product law has the form given in eqs. ( The commutation relations can be expressed in terms of a base the following two-cocycle uct ( where the central elements are written in terms of the Lie alg In the followingMaxwell-like we groups will in 2+1 see dimensions how studied to in section apply this construction to where the exterior JHEP10(2019)039 ) . ). 1 m S , ( d L n 3.45 group + (4.25) (4.27) (4.28) (4.29) (4.26) (4.24) − 3 m, δ µν [ BMS ). It is defined m δ . 3 ) k 3.64 φ ). The correspond- y version of ( ( ion of the Diff + ) by either using the ′′′ 4.1 , n . Y ) ) where the trace over ′ 4.3 + )  ′ ) and ( m ρ φ ( (ab) ext V 3.59 F 4.18 groups with Maxwell struc-   f the Virasoro group and its . The resulting Lie algebra 1 based on the Virasoro group ), which is the central exten- , . (ab) ext 3.63 µν ρ m 1 group using the group struc-  S ǫ 1 dφ X S π Log 3 ( , 2 S = = 0  ). In terms of these generators, 0 + 2 ] = 0 c   Z vect V ν n ν n ν ! ) in the form ( [ ◦ c π ′ BMS ′′ F vect 3.53 \ Diff F ′ Π 1 Ad ). This can be done by first extending , , U , U 24 µ U ⋉ ad 3.64 m µ m µ  follows from eq. (

) F group i [ 3.61 1 Π Π  2 3   ) = 3  1 S 1 ( Log S ] as expansions of a Kac-Moody algebra − S of the centrally extended BMS group in 2+1 + ′ , , dφ 52 ′′′ , – 24 – π X,Y ) algebra ( n 2 \ ( U n BMS c n U vect c \ Diff c 0 R − vect − − , Z = m, m, m, = , (2 π δ δ ) = δ 3 1 3 and its commutation relations will have the structure φ sl U 48 µν µν ∂ µν ( ], ) c bms − φ - S ), define the group operation of the Maxwell- [ (ab) ext BMS 81 m δ ( m δ m δ 2 3 1 m ′′′ k ) = k k d L 3.67 X + + + π ad maxwell 1 n n n U, V i ), where 24 + ( + + Maxwell- m ρ m ρ C by means of a two-cocycle of the form ( m ρ m ) and ( Maxwell-like extension d L J Π F 4.12 ) = µν µν µν X ρ ρ ρ 3.66 LM ). The Lie algebra ( ǫ ǫ ǫ s ]. These definitions, together with the adjoint representat ) or ( = = = 4.7 55    4.4 n ν n ν n ν is the generator of the J F ). The starting point is the Virasoro group Π ) with more general structure constants, defining a Kac-Mood ) leads to a centrally extended version of the algebra ( , µ , , t ) elements is to be replaced by the Killing form on m µ m µ m µ 4.5 R J 4.24 4.16 J J ,    (2 dimensions, which can be definedLie algebra as [ the semi-direct product o eq. ( ture ( which extends the function composition product ( through eq. ( Now we consider the following extended semi-direct product In this section, we will define an extension of the This is the natural will have the form sion of the diffeomorphism group of the circle defined in eqs. ( definition ( ing Souriau cocycle is given by the Schwarzian derivative by the Thurston-Bott cocycle [ This type of algebras have been found in [ the loop group sl eq. ( 4.3 Maxwell-like extension of the and the bracket of the Virasoro algebra where group given in ( This structure can be generalizedture to by more centrally general extending Kac-Moody groups of the form ( JHEP10(2019)039 . , ). , 3 i n ) m − 3 α 4.16 (4.30) (4.33) (4.35) (4.32) (4.34) (4.31) ( c m, s t prod- δ + 3 algebra. 3 2 c 3 m m onstruction + 2 3 c  c 12 group and the  + 1 + , 1 3 maxwell − n  m U + 3 1 c m can be obtained from S , , , ). In order to do this, + ˜ F 3 F , c , h c ) ; \ i ) 2 1) 0) 0) Maxwell 4.5 n ) 3 , , , − L , , c − ) F ˜ ) Π 0; 0 1; 0 0; 0 F, c ] as the asymptotic symmetry , , ), leads to α ; ∗ U α m ( , , , ( 1 ( ant under the 49 s X, α, a s Ad ( 2 → 3 0; 0 0; 0 1; 0 ˜ 3.69 , c J, c  c , , , ] = ( ] = 0 ] = 0 ) ∗ α  ν n n + + ,J ad F P F  , h , ,F , and its dual space can be defined as the = (0 = (0 = (0  Π ) =  µ 1 ∗ α 3 1 ) m m 3 2 1 ˆ h − 2 c c c − F X, ad P P [ ( [ U [ , c U 1 2 F, c h , , , ; 2 , , (Π , S = S 3 0) 0) 0) algebra, whose bracket is defined by eq. ( n – 25 – 2 n n ) + , c h ˆ c , , , c − i a → − − 3 ) ( m − ;Π 3 − ). These commutation relations have been found m, ) by means of a generalized Sugawara construction. ℓ m, m, s Π 0; 0 0; 0 1 δ c δ δ 3 . Its elements will be denoted by sextuplets of the ) has been found in [ bms , , F 3 Π 3 3 0; c ∗ , ∗ U m can be defined by using the prescription ˆ h ∗ U , J, c m 4.27 4.24 a, m m m ℓ + ( 0; 0 ; ) 4.31 ˆ 1 3 2 , ) 2 H Ad 1  Ad 0; 0 0; 3 c 12 c c 12 12 , , m   ℓ 1 ∗ α ), i.e. ∗ α + J, c + + a,m − ; ( α, m n 2 n n ; ) and ( = (0 = (0 = ( U + 1 + + 2.41 . → + ad + ad m m m m m m α,m ; S    J 3.68 1 3 1 1 J P F P 1 F J ), and the pairing between to centrally extended groups of the form ( c − − X, m ) ) ) 3 − ( n n n U,m ) in the general expression for special extended semi-direc U − U , U ∗ ( ) 2.3 − − − F 3 4.6 , F, c group studied in the previous section, by generalizing the c S S S Ad ∗ U 2 3 2 1 m m m c c 3 c , c F, c Ad ; − − − Π 2  , stands for the generator of the Witt algebra ( ] = ( ] = ( ] = ( ∗ a ). This leads to J Π F 1 [ , c BMS n n n U ∗ ∗ U ∗ U m P F J ℓ ;Π J, c , , 2.43 , ] as an expansion of the Virasoro algebra. Furthermore, they 1 + ad m m m 52 = Ad = Ad J, c J J J [ [ ( [ ˜ ˜ ˜ J h Π = Ad F where 4.4 Wess-Zumino terms In this section, we construct the geometric actions for the together with ( shown in section Maxwell- natural extension of eq. ( ucts ( we need to define the dual space of a three-dimensional Chern-Simons gravity theory invari in [ the Maxwell-Kac-Moody algebra ( On the other hand, the algebra ( Considering a base of the form where The coadjoint representation of where we have used ( defines a Maxwell extensions of the form ( JHEP10(2019)039 3 . .  ] in ) by ). and ˙ α )] ′ 83 ˆ (4.40) (4.39) (4.41) (4.36) (4.37) (4.38) 3 H α U, α , 2.45 Θ 1 2 ) can be \ , 82 Maxwell , a + , L   4.34 ˙ ). Then, the ;Θ ] a 57 ]. The second 2 ˙ i α ˙ ˙ α, Θ 72 U , , 1 α, 2.14 , [ ˙ α h ˆ − . U, i 1 2 63  ) k U , ;Θ i 3 3 ), the Maurer-Cartan 2 − 1 ˙ α  ˙ Θ 60 a Θ previously discussed. , Θ tudied in [ , U ) ,  a nt under the 4.21 dt ¯ ′ d α U 1 ( U ;Θ − Z 1 s ). This leads to the following 2 (Θ , h 3 − U , , , given by expression ( Θ c , the expression ( t 1 2 ) ). This corresponds to a chiral according to ( , U ˆ h h = rom the direct generalization of 3 H ], one has to replace the pair ( α 4.20 group (  + λ , + ˆ

Θ ∈ H 3 83 Tr  k k , WZW  4.18 Tr ;Θ ˙ , 1 a ) a I 1  3 of D dt ) E 82 E ˙ Θ α, + ˙ Z ˙ ) a , ;Θ λ a , c , ) and ( ), plus the central terms 3 ( 2 3 Z 0 ˙ 57 ˙ U ,c ˙ dt α, U, α, F Θ on its dual space can be generalized from 0 \ takes the form ;  , Maxwell dφ F ,U , as in ( π ; ˙ 4.18 (Θ 2 2.32 ˙ Z ˆ h ) Kac-Moody group [ ) α 2 U, 2 1 2 U, L , t 0 R 1  ) takes the form ,c , c π (  Θ ) D 0 Maxwell Z c , ), ( 0 1 ;Θ 4 3 a α L 1 − The third term is a novel chiral WZW model by WZW 3 dt ;Π π – 26 – (2 I , c Θ Θ + I 1 c ;Π 2 Θ 4.36 U 0 , , 9 1 i ,c , sl 2.32 Z = ) ) 0 F ˙ + U U J 2 , c ; d L C 3 (  U U 1 c 0 2 ( (  ˙ α ) to the case of centrally extended groups, the solution − J O ′ ) and [(Θ , c + S S d 0 U dλ U ′ 1 2 D D  ) corresponds to the usual chiral WZW model that defines 1 ), and integrate by parts. ˙ \ 2.31 a U ;Π − Maxwell α − 3.60 = = = 1 1 1 L ) stands for the left-invariant Maurer-Cartan form on ˙ U − α, − 3 I    4.41 ¯ d is defined on  , c In the case of the U ˙ ˙ ˙ a a a U 0 ) = Θ ˙ U, h 3 , J Tr ˙ ˙ ˙ Ad ( a  α, α, α, Θ h − Tr 1 , dt ), the action eq. ( , ˙ ˙ ˙ Γ a ;Θ 1 U, U, U, Θ dt Z 2 − Z    ;Θ Θ dt U 3 2 1 4.32 = Z 2 , dφ π Θ Θ Θ α ˆ is the action ( Θ 2 Z = ( H dφ , 0 π I 1 3 1 2 α Z ;Θ c − 0 1 ) 2 π Z + ;Θ c Θ 2 ) can be evaluated using eq. ( 1 1 , π is the geometric action associated to the group H U, α c 4 U + Θ I Maxwell ). Given a fixed element ( ), , L H 4.38 = I I U = 2.44 ˆ 2.27 H (Θ In order to make this term match the model found in [ I 9 d WZW I satisfies where group. The first term in eq. ( where term corresponds to thethe flat context chiral of WZW three-dimensional model, gravity. itself which has and been defines a s Maxwell extension of the knownits WZW inverse ( models Wess-Zumino-Witten (WZW) model, and it isthe globally geometric action invaria associated to the Maxwell WZW model. forms ( geometric action: of this equation can be shown to be given by ( used to define theexpression coadjoint of orbits the correspondingeq. geometric ( action follows f By generalizing the expression ( where (Θ Using the pairing ( Similarly, the infinitesimal coadjoint action of eq. ( Here the exterior derivative JHEP10(2019)039 , . 60 rless  group (4.42) (4.43) (4.45) (4.44) 2 3 3 ′ ′′ . F F 2 U ′ o action [ [ BMS − ◦ U ˙ 2 ase leads to the α ′′′ ′ F ′′′ F ′   ˙ e Virasoro group. In )) takes exactly the U U U π sions and its infinite- ′ 1 ′′′ ◦ c U 48 3.71  α a central extension of its ), and the geometric ac- ral fashion, and construct − U ,  ′ group. It is related to the e of the Maxwell symmetry. ) ing the notion a semi-direct ◦ U wed us to define the adjoint 4.38 F dt 3 ) defines a novel Extension of α ( al of asymptotically flat three- gWZ y definition invariant under the , we have explicitly shown how 0 ), which yields I  Z J ′ , 4.43 + extension of the gravitational WZ dt F dφ ) electromagnetic field. Subsequently, 4.26 3 π  φ 2 3 ( Z 0 ˙ F Z U reads dφ dt π π 3 = 2 c ) and ( 0 48 ] in the context of induced two-dimensional Z Z gWZ 84 + I π dϕ Maxwell-BMS 2 4.25 ) symmetry, this group structure matches the ′ π , ϕ I c – 27 – ˙ 2 Let us consider again the Maxwell- 1 24 U R U 0 − , ), ( = Z + ′′′ U 3 ′ ), and  → ˙ = U 2.32 U U ′  [ ′ BMS ′ F ◦ 3.71 U ]. The third term in (   ′ a ′ ′′ ′′ 83 U  U U U   Maxwell- dt I ) corresponds to the BMS π 1 dt Z c ) (supplemented with the first term in ( 48 Z 4.43 corresponds to the geometric action associated to the cente dφ π + ] as the geometric action on the BMS 2 dφ 3 4.43 0 0 π 57 2 J Z defines a Maxwell extension of the gravitational Wess-Zumin 0  ). In this case the Maurer-Cartan forms ( group, given by ( group. π Z ′ 3 group fits in this description. The geometric action in this c ˙ 3 3 c U U π 24 1 gWZ D 4.28 c I 48 dt + ) can be evaluated using ( ], found as the geometric action describing world lines on th = Maxwell-BMS Z 73 I , 4.39 dφ π gWZ 2 72 I 0 , Z defined in ( action, found in [ Maxwell-BMS where description of a relativistic particle coupledwe to constant have considered theLie semi-direct algebra. product of When a based Lie group on and the SL(2 geometric actions on theirthe coadjoint Maxwell orbits . Subsequently In this paper,dimensional we generalizations. have analysed This the haveproduct been Maxwell group made in group a by inThe way extend 2+1 that definition dimen captures of the extendedand underlying structur semi-direct the products coadjoint groups representation of allo Maxwell groups in a gene 5 Discussion and possible applications the Polyakov action for two-dimensionalMaxwell-BMS gravity, which is b same form as the Polyakovquantum action gravity: found in [ tion ( The action dimensional Einstein gravity [ The second term in ( flat limit of Liouville theory, which defines the classical du 63 Maxwell gravitational WZ model. fact, by defining the change of variables the first term in ( JHEP10(2019)039 s 3 he [ BMS ]. It is . The 3 85 denotes an gical insula- ): \ h i , the conserved Maxwell 3 L . This symmetry ]. The motivation Kac-Moody group 3.43 3 3 91 special kind of extended maxwell \ ZW model, whose kinetic , ly, higher-spin extensions sions, for which a general uilds [ L e fact that it includes non- scale dynamics of quantum ts edge modes. In fact, the dy group algebra ( \ Maxwell  iated to the Maxwell- hic to Maxwell Kac-Moody comment on possible future ss-Zumino action first found 3 ino terms that generalize the L A mmetry of the action [ Plazmann-MacDonald algebra en by Maxwell ]. In the following we briefly A (2+1)-dimensional field the- , which couples the background ∧ viously studied in the context of finite-dimensional enhancements 84 A ], a Chern-Simons action invariant ]. The geometric actions associated maxwell ∧ 36 52 , A 2 3 49 + ]). In [ ]. The first geometric action is a novel Wess- group, respectively. The corresponding Lie A 90 d 63 3 , ∧ – 28 – Kac-Moody symmetry at the level of its conserved 57 A Kac-Moody group can be relevant in any context 3 [ BMS  3 M Z π k \ 4 Maxwell \ L Maxwell = ]. As stated before, the current algebra of such Chern-Simon L . The conserved charges in a Chern-Simons theory are known 3 37 CS group. These symmetries define Maxwell-like extensions of t I 3 -valued connection one-form on a manifold M, and -invariant Chern-Simons action. 3 3 maxwell ] (for a recent review see [ algebra was used to construct an effective theory for a topolo [ BMS 89 3 – maxwell 86 ). Hence, the maxwell is a 4.24 ) Kac-Moody group and the R A , Chern-Simons forms provide effective theories for the large Special extended semi-direct products admit central exten to satisfy a Kac-Moody algebra that enhances the original sy invariant metric on Hall systems [ where tor. This leads to ageometry relativistic to version the of the Chern-Simons Wen-Zee effective term field in quantum Hall f can be made explicitboundary at dynamics the of level a of Chern-Simons the theory action is when described studying by i W commuting translations, allowing tofor reproduce magnetic translations the [ Girvin- theory is naturally given by the Maxwell-Kac-Moody algebra under the to use the Maxwell symmetry in this description stems from th therefore evident that, as in this case the gauge group is giv directions. Chern-Simons theory and quantumory Hall that naturally systems. realizes the Zumino-Witten model, invariant under the Maxwell-Kac-Moo elaborate on the application of these geometric actions and to these infinite-dimensional groupsknown lead results to presented, for novel instance, Wess-Zum in [ involving a Maxwell charges is a Chern-Simons theory invariant under the of the Maxwell groupsemi-direct products. can be constructed by considering these double cover of the Maxwell group in 2+1 dimensions. Similar charges of the theoryalgebra ( will satisfy a Poisson algebra isomorp second case that we considered is the geometric action assoc algebras of these infinite-dimensional groupsalgebra have expansions been and pre three-dimensional gravity [ group, which defines a Maxwellby extension Polyakov of as gravitational an We action for two-dimensional gravity [ ISL (2 description has been given. Usingof this the construction, Maxwell two symmetry in have been constructed, namely, the and the Maxwell- JHEP10(2019)039 N , nsla- ) are M group, ]. One , 0 3 4.30 36 F ring in the , ].  i ure that leads 92 ν dφ ) describes the ). The relation , , F in eq. (  ′ µ E ′′ 3 Y (where the Chern- 4.41 3.70 J m , and the invariant invariant under the  h 1 M Z g change as F − 2 d 2 = ) 4.4 t 1 + 2 s and the BMS − + µν location at the boundary g and ′ group ( Y η ymptotic symmetry group el introduced in [ ′ algebra, given precisely by 3 oordinates, given by 3 D J m γ U, α, a Z e following way. The solution 3 t M otic symmetry is also given by P for instance, in [ = + , + Z ′ ′ bms = ( i [ BMS ν π Z T Y g k  Π J J 12 µ . It is easy to show that, considering + − Π + 2 + 2 h

′ du ∂M T Y g ′  ′ 1 ′ 1 , J − J Π ), obtained in section ]. This algebra is isomorphic to a particular M µν ]. Along these lines, a Chern-Simons theory , ˙ + ], it was shown that the asymptotic symmetry t ) has been studied as an extension of three- + gg η dt 79 1 2 ′′′ 38 – 29 – ′′′ ) in the form 49 + ′′′ γ − 4.41 + Y R g T 3.43 0 2 J

2 = 2 3.37 Π i − − −  ν ] is given, up to a gauge transformation depending on dt ]. In [ ′  ′ ′ ), where the generators 2 1 -invariant Chern-Simons theory, and naturally provides Π 49 Y R T 48 Z 3 Chern-Simons theories provide a gauge theoretical for- dφ µ algebra). Thus, the geometric action ( + ,  J M M ′ 4.29 M 1 3 dφ h group ( π 47 ). This is a generalization of the known results for asymp- J 2 3 M 0 + 2 + 2 + 2 t , Z dφ 4.31 Y R T + π ′ ′ ′ µν k maxwell + 4 η J 0 1 M M M γ = J = = = ]. When the role of the generators of the extended space of tra + = dφ i 93 Z 3 J dt algebra ( ]. In this article, we have succeeded in finding a group struct is related to the central charges through the constants ente M ν δ δ , WZW δ M are coordinates on the boundary 3 I J 77 M k µ 51 1 2 φ  c J bms 1 2 - h + ) are interchanged, this leads to an extended gravity theory maxwell = and t 3.33 . Under the action of an improper gauge transformation, they A F maxwell respectively [ invariant metric on the and depends only on the arbitrary functions of the boundary c the radial coordinate, by the extended semidirect suminterchanged [ ( to this infinite-dimensional algebra, namely,of the this Maxwell- group with Chern-Simons gravityfor can the be gauge analysed in connection th used in [ is known to be given by thetions conformal ( algebra in twosimplest dimension example of aconformal Hietarinta extension algebra of [ the Galilean algebra, and its asympt mulation for gravity ininvariant 2+1 under dimensions the [ Maxwelldimensional Einstein algebra gravity ( [ of such a gravity theory is giventotically by AdS an and extension flat of gravity the in 2+1 dimensions, where the as way to describe edge modesof a in topological such a insulator system in is 3+1 toThree-dimensional dimensions consider as gravity. considered, a dis Simons level boundary dynamics of a Maxwell an action for the edge modes of the topological insulator mod and this action is exactly the WZW model ( where term has the general form metric on the and element of the Maxwell JHEP10(2019)039 ] - 3 3 N 3 he 49 c W bms - [ group BMS 3 algebra. ). There- -invariant ) with the 3 N group, like asymptotic ly to define maxwell ]. This is a [ 3 BMS W 4.43 95 4.41 onal gravity as an del ( tforward computa- bo, D. Sorokin and J. for gravity theories when orm of coadjoint action the Maxwell- ledgements are given to J. ein gravity in 2+1 dimen- an be conjectured to control slations at the boundary of a ndary conditions found in [ to the corresponding classical ms based on the sentation of the Maxwell- rch. ed on the Maxwell algebras are known to describe if Maxwell extensions of the N ]. In this context, allowing the most . ), when the Souriau cocycle on the W ], given by semi-direct sums based on ] , the solution space of the Maxwell 96 algebras have also been found in the de- (ab) ext 49 4.14 101 ) N – N ) are implemented. This shows that, when W 99 Chern-Simons connection, would lead to the W ( , we have shown how to construct higher-spin ]. Based on the fact that a Maxwell 3 3.65 – 30 – ad 3.3 103 i ) by a Drinfeld-Sokolov reduction [ as the asymptotic symmetry. , 3 N ) given in ( maxwell 3 4.43 102 W , c ]. In our case, the Maxwell WZW model plays this role. F ; maxwell \ . In the same way, higher-spin extensions of the 94 In section 2 ) defines a phase space for the geometric action ( L 0 , c (ab) ext group, which are given by extended semi-direct products of t ) ,F N 3 0 R ; 1 ]. On the other hand, Π , ), and the paring ( N, , c 0 45 ( J ]. Thus, infinite-dimensional algebras of this kind are like sl M 4.26 ( 75 ) Ad 3 ]. In our case, these algebras define Maxwell extensions of the ⋉ ) 98 R,m ; , R 2 97 N, algebra [ , T,m ; 1 N 80 Kac-Moody algebras can also provide asymptotic symmetries On the other hand, chiral WZW models appear in three-dimensi W Y,m ( group presented here. On the other hand, a coadjoint orbit of with representative ( the one presented in [ Maxwell-Kac-Moody algebra Higher-spin symmetries. symmetry could somehow emerge in this type of systems. Acknowledgments The author would likeTosiek to for thank useful P. suggestions Brzykcy, N. and Merino, comments. G. Special Palum acknow asymptotic symmetries of higher-spin gravity theories bas Chern-Simons theory captures featurestopological of insulator, the it magnetic would tran be interesting to evaluate symmetries found in flatthe higher-spin gravity [ extensions of the Maxwell general boundary conditions for the intermediate point in thedual Hamiltonian defined reduction at that the leads This boundary suggest [ there shouldgravitational be Maxwell way WZ to actionwork connect ( in progress. the Maxwell WZW mo fore, the Maxwell extensionthe of solution the space of gravitational this WZ Chern-Simonsare action theory, adopted. when c the This bou is an interesting subject for future resea Virasoro algebra ( adopting the boundary conditions introduced in [ invariant Chern-Simons theory matches the coadjoint repre tion to showAd that this expression matches the infinitesimal f which defines the solution space of the theory. It is a straigh scription of quantum Hall systems [ the most general boundary conditions are adopted [ This is interesting in the context of gravity, where algebra can be constructed defining extended semi-direct su form SL ( the asymptotic dynamicssions of [ higher-spin extensions of Einst JHEP10(2019)039 , ]. ] ]. = 4 SPIRE ic and ommons ]. D SPIRE IN = 4 ]. IN ][ D ]. ][ SPIRE sical 3160581 is also chanics: a group ]. ◦ IN . ]. SPIRE ][ , ]. IN SPIRE arXiv:1005.3714 [ l algebras of ][ IN ant N SPIRE SPIRE ][ IN (2010) 090401 [ IN manuscript. P. Salgado- SPIRE (1990) 568 ][ IN [ 104 redited. arXiv:1201.2850 ugh Proyecto Postdoctorado [ 31 bout different aspects of the ]. arXiv:1801.08860 ]. [ (2010) 102301 Deformations of Maxwell Maxwell superalgebra and (1972) 701 arXiv:1012.4402 [ 51 SPIRE Generalized cosmological term from Maxwell symmetries and some 20 SPIRE (1983) 1295 arXiv:1809.07871 IN [ IN [ arXiv:1701.04234 ][ [ 24 (2013) 01160 (2018) 211 arXiv:0906.4464 [ Phys. Rev. Lett. 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