Fracton gauge fields from higher-dimensional gravity

We show that the fractonic dipole-conserving algebra can be obtained as an Aristotelian (and pseudo-Carrollian) contraction of the Poincar\'e algebra in one dimension higher. Such contraction allows to obtain fracton electrodynamics from a relativistic higher-dimensional theory upon dimensional reduction. The contraction procedure produces several scenarios including the some of the theories already discussed in the literature. A curved space generalization is given, which is gauge invariant when the Riemann tensor of the background geometry is harmonic.


Introduction
Fracton phases of matter represent a remarkable class of quantum states with novel and intriguing properties, challenging conventional paradigms in condensed matter physics [1][2][3][4][5].These exotic phases are characterized by their restricted mobility of excitations, leading to unconventional patterns of long-range entanglement and topological order [6][7][8].Understanding and describing the nature of fracton phases has emerged as a forefront of research, first in the fields of condensed matter theory and more recently in high energy theory, promising groundbreaking insights into the behavior of quantum matter.
Gauge theories have been instrumental in describing a wide array of physical phenomena, from the fundamental forces of nature to condensed matter systems.Actually, to delve into the fascinating realm of the so-called gapless fracton phases, it is crucial to explore the role of symmetric gauge fields [5,[9][10][11][12][13][14].These gauge fields mediate the interactions between fractonic matter.Therefore, the inclusion of symmetric gauge fields enriches the theoretical framework, paving the way for an insightful analysis of fracton phases in various materials and scenarios.On the other hand, fractonic theories couple naturally to Aristotelian geometries [11][12][13][15][16][17][18], i.e., manifolds whose tangent space isometry group is given only by rotations and spacetime translations, but no boosts as in the more familiar case of Riemann-Cartan geometry.Recently, generalizations of fracton electrodynamics defined on curved space have been constructed by gauging the Monopole Dipole Momentum Algebra (MDMA) [11][12][13]18].
The main results of this paper are that the dipole conserving symmetry group can be embedded in Poincaré, and that symmetric gauge fields can be obtained from a relativistic non-Einstenian gravity theory in one dimension higher.The higher-dimensional theory is described by a Yang-Mills-like action with Poincaré as gauge group.Nonetheless, the theory is assumed to be in a Higgs phase with symmetry breaking pattern isopd `1, 1q Ñ sopd, 1q.The fracton gauge fields are obtained after a particular limit defined by a Lie algebra contraction that connects the Poincaré algebra with the MDMA.Such contraction is a suitable combination of pseudo-Carrollian and Aristotelian limits respectively.Additionally by dimensionally reducing the system we identify the extra dimension of the system with the internal U p1q associated to monopole transformations.Similarly, the dipole transformations in the dimensionally reduced theory is obtained from the spacetime boosts along the compactified spatial dimension.
The structure of this paper is as follows: In Section 2, we provide a brief overview of dipole conserving systems, highlighting their defining characteristics.In addition, we introduce symmetric gauge theories as the fields mediating the interactions between fractons, and discuss the incompatibility between the gauge principle and spacetime curvature.In Section 3 we define a novel contraction of the Poincaré algebra that leads to the MDMA algebra in one dimension less.Then, we define a higher-dimesional Riemann-squared gravitational theory, and after the algebra contraction we dimensionally reduce it.After doing so, we discuss the results and finish with some conclusions and comments on possible future developments in Section 4.

Symmetric gauge fields and dipole conservation
Before discussing dipole conserving systems, let us consider a spinless particle in d `1 spatial dimensions with momentum P A , and angular momentum J AB " x A P B ´xB P A .If the system is translational and rotational invariant P A and J AB will be constants of motion, which implies that 9 P A " 0, (2.1) Actually notice that the conservation of angular momentum does not introduce new constants of motion, instead it constraints the particles's motion by requiring the velocity to be parallel to the momentum.Now let us pick coordinates x A " px i , R zq, interpret x i as the coordinates of the physical space, z the coordinate of some extra dimension, and R a constant with dimensions of length.Then, we redefine the transverse momentum as Q " RP z , and the transverse angular momentum Q i " RJ i z .After doing so, we zoom out the z´direction by sending R Ñ 0. Therefore, the angular momenta become From the lower dimensional perspective the transverse momentum takes the form of an internal charge, and the angular momentum Q i corresponds to the dipole moment of that charge Q.For such a particle the conservation of all the charges imply x i " constant, (2.6) which is unusual since the particle is not allowed to move, however, its momentum is not constrained to be related to the velocity.Although a system with such properties seems to be dynamically trivial, we notice that motion is allowed once more than one particle are included.For example, let us consider two fractons with charges Q 1 " q, Q 2 " ´q such that Q " 0, and Q i " qpx i p1q ´xi p2q q for such a system the conservation of momentum and dipole imply Thus the actual dynamical variables for a dipole are the center of mass position X i ptq, and the relative momentum W i ptq between the particles forming the dipole.If in addition we impose conservation of angular momentum we obtain the condition which for d " 3 constraints the force 9 W i as 9 X i ptq " V i ptq, (2.11) (2.12) In fact, we can go beyond the single particle picture assuming locality and introducing the densities ̺, p i such that we can express the conserved charges as With these definitions and requiring 9 Q " 0, 9 P i " 0, we obtain the continuity equations On the other hand, dipole and angular momentum conservation 9 Q i " 0, and 9 J ij " 0 demands the constraints (2.17) with L ijk " ´Likj , and K ij the dipole current.However, without lost of generality the stress tensor can be improved such that the new stress tensor T ij is symmetric.With such improvement the conservation of angular momentum is automatic if momentum is conserved.Similarly, to guarantee conservation of both charge and dipole moment it is enough to satisfy the generalized continuity equation1

Fracton Gauge Theory
In fact, the charge (dipole) conservation can be derived once the scalar φ, and symmetric tensor A ij gauge fields are minimally coupled to the fracton current as with the gauge field transforming as Actually, in a system with dynamical gauge fields, the corresponding generalized electrodynamic theory with electric and magnetic fields defined as [9] with action S " 1 2 Notice that it is possible to construct a symmetric spacetime tensor A µν with A 0µ " ´Bµ φ, and transformation law δA µν " B µ B ν ε.The field strength for the enhanced model is defined as and the action (2.23) can be written as where indices are raised and lowered with the Minkowski metric η µν " diagp´`¨¨¨`q.
Notice, that such construction posses and accidental gauge symmetry δA µν " B µ β ν that leaves the field strength invariant but does not respect the symmetry property of the gauge field.Actually we notice that the alternative gauge field Ãµν " A µν `Bµ φ τ ν with τ ν " δ 0 ν has field strength Fµνλ " F µνλ , therefore, same equation of motions.It is important to emphasize that the constraint τ µ A µν " B µ φ explicitly breaks the apparent relativistic symmetry.

Fractons on curved manifolds
In order to illustrate the tension between the fractonic gauge principle and curved spaces, let us consider a Minkowskian manifold with metric g µν and a timelike vector τ µ .We can define a spatial metric by introducing a clock τ µ as In addition, we introduce an Aristotelian covariant derivative with the torsion-free connection and B µ τ ν ´Bν τ µ " 0. With these ingredients at hand, we consider the symmetric tensor gauge field A µν of the previous section satisfying the covariant condition under gauge transformations we postulate the field A µν transforms as and its corresponding field strength reads The issue this construction has is that the field strength is not gauge invariant, in fact it can be shown that δF µνρ " r∇ µ , ∇ ν sB ρ ǫ " ´Rα ρµν B α ǫ. (2.31) Therefore, a minimal extension of the action (2.25) can be However, this action is not invariant in arbitrary backgrounds.In fact, it preserves gauge invariance for background spacetime with curvature satisfying In addition, notice that the theory can be reformulated using frame fields e a µ such that h µν " δ ab e a µ e b ν , and satisfying ∇ µ e a ν `ωab µ e bν " 0 with ω ab µ the sopdq connection.Moreover, we introduce the inverse spatial vielbein e µ a and define A a µ " A µν e aν .Therefore the gauge field A µν can be split as where we have used the constraint (2.28).Besides, the field A a µ satisfies and thus can be put in the form (2.36) Using the fact that the Riemann tensor constructed out of the Aristotelian connection (2.27) satisfies R µν ρσ τ µ " 0, and the field strength of A µν satisfies F µνρ τ ρ " 0 the action (2.32) can be written in terms of A aµ as Alternatively to the action (2.32), it is possible to introduce a Higgs mechanism for the dipole symmetry as shown in Ref. [11], by doing so a gauge invariant action can be obtained after introducing a Stückelberg field ψ α transforming as δψ α " B α ǫ such that the new gauge field is gauge invariant and therefore its field strength is gauge invariant, and no constraints on the background spacetime are needed.Therefore, we define the action S " ´1 2 Since R α µνρ τ α " 0, the longitudinal component of the Stückelberg field ψ µ τ µ does not appear in the action, therefore, it will be completely undetermined.In fact, if we fix ψ µ to be of the form and use Eq.(2.34) we can write the invariant gauge field as Āµν " `Aa µ ´Dµ ψ a ˘eaν . (2.44) With that choice for the Stückelberg the action reduces to which corresponds to the action obtained in [11] after gauging the dipole conserving symmetry group.
In the next section, we will define the Monopole Dipole Momentum Algebra and embed it into the Poincare group in one extra dimension and show that the actions (2.32), (2.37), (2.45) can be obtained from a gravitational gauge theory upon dimensional reduction after applying an Aristotelian limit.
3 From Poincaré gauge theory to fracton gauge fields Following the intuition built in Sect. 2 we will embed the fracton algebra into the Poincaré algebra isopd `1, 1q with the purpose of deriving the fractonic gauge theory as some limit of a gauge theory of gravity in one dimension higher.

Fractonic Symmetry Algebra
In systems with dipole conservation the charge conservation cannot be described as a regular internal U p1q, since the total value of the dipole moment in the system changes once a space translation is applied (see Eq. (2.13)).Actually, this property is consequence of the generators of space translations and the transformation generated by the dipole charge not commuting, and forming a non-Abelian symmetry algebra [20,21] where P i , Q j , Q are the generators of translations, "dipole", and U p1q transformations.
In addition, if the system possesses rotational invariance, the bracket Eq. (3.1) has to be supplemented with where J ij is the generator of the sopdq rotation group.We will refer to this algebra as Monopole Dipole Momentum Algebra (MDMA).The commutation relation Eq. (3.1) has important implications on the properties of fractons systems on curved space.In fact, as we discussed in the previous section spacetime curvature is in tension with the fractonic gauge transformations, and in particular the origin of the symmetry breaking has been argue to have the same origin as the translational invariance breaking in gauge theories of spacetime symmetry groups [11].
We also notice that MDMA is isomorphic to the Carroll algebra [22,23] rP i , C j s " which requires to reinterpret the U p1q generator Q as the Hamiltonian H and the dipole generator Q i and the generator of Carrollian boosts C i .The analogy between the dipole conserving group and the Carroll symmetry has been extensively used to construct fractonic models of particles and gauge fields [23][24][25].A remarkable property of Carroll theories is that it corresponds to the vanishing speed of light limit of the Poincaré algebra [22], and are also characterized by immobile excitations [23,25,26] The Carroll algebra can be obtained from the Poincaré algebra rJ µν , P ρ s " η ρrν P µs , rJ µν , J ρσ s " η rµrρ J σsνs , where µ " p0, iq and η µν " diagp´, `, ¨¨¨, `q, by means of the following Lie algebra contraction [27] when the speed of light c vanishes.This suggests that the dipole conserving algebra can be obtained by means of a similar limiting procedure from Poincaré.In the next section we show that this is indeed the case.

Poincaré and the fracton symmetry algebra
In this section we will show how the MDMA in d `1 dimensions can be obtained after contracting Poincaré algebra in isopd `1, 1q in d `2 dimensions, given by where η Â B " diagp´`¨¨¨`q is the Minkowski metric and Â " 0, 1, . . ., d, d `1 " n.
The resemblance between the dipole conserving algebra and the Carroll algebra suggests that a rescaling analog to (3.4) will allow to obtain the later as a Lie algebra contraction of Poincaré.However, temporal translations are also present in fractonic systems, where the generator H commutes with all the generators of the dipole conserving algebra.Thus, instead of using time as the longitudinal direction in the contraction, we can use a spatial direction Â " n to define a pseudo-Carrollian contraction3 similar to (3.4), i.e.
The extra dimension will be interpreted as the internal direction that will be associated to the conservation of monopole charge.However, the resulting symmetry will still be relativistic, which contrast with the absence of boost symmetry in the dipole conserving algebra and the Aristotelian character of fractonic systems.Thus, one should supplement (3.6) with a contraction of Aristotelian nature that eliminates transformations that connect space and time translations.This can be achieved through the rescaling Combining the Aristotelian contraction (3.6) and the pseudo-Carrollian contraction (3.7) lead us to define the following rescaling of the elements of the higher-dimensional Poincaré algebra where the generators in the first line of Eqs.(3.9) are to be interpreted as spacetime translations, U p1q, dipole transformations, and rotations respectively.Taking the limit ε Ñ 0, we find that the set of generators tJ ab , H, P a , Q, Q a u close in a sub-algebra defined by the following non-vanishing commutators rJ ab , P c s " δ crb P as , (3.10b) ) rQ a , Qs " σ 2 P a , (3.10e) whereas the generators tG a , Ku form an ideal with commutation relations The sub-algebra (3.10) is an extension of the MDMA defined by (3.1) and (3.2), and reduces to it in the limit σ Ñ 0. Contrary to the standard contractions where the parameters are removed from the algebra, we will take the strict limit ε Ñ 0, but keep the leading terms in σ for reasons that will become clearer below.

Poincaré Gauge Theory
After showing that the MDMA can be obtained after contracting the Poincaré algebra, we will proceed constructing a relativistic gravity theory in five-dimensions with Poincaré as gauge group, and then dimensionally reduce it.This analysis has the purpose of understanding the puzzling relation between fracton phases of matter and gravity theories.As we will show, starting from a fully boost invariant action in higher dimensions is not enough to recover an invariant fracton gauge theory.This is related to the fact that in the strict σ Ñ 0 limit the lower dimensional theory is purely gravitational without fracton gauge fields.Therefore, the gauge group will be realized non-linearly by adding a Higgs field Ψ A associated to the transverse boosts J An , where un-hatted capital indices take values A " 0, . . ., d.Therefore, the non-linear connection one-form A " A μdx μ, μ " 0, . . ., d `1, taking values on the isopd `1, 1q algebra Eq. (3.5) reads where E Âµ and Ω Â B µ are the non-linear vielbein and spin-connection, respectively.On the other hand, the corresponding curvature reads where T Â and R Â B are the torsion and the curvature forms We choose these fields to be such that, under a Poincaré transformation with gauge parameter ε " Υ ÂP Â `1 2 Θ Â B J Â B , they transform only under the lower-dimensional Lorentz group sopd, 1q with parameters Θ AB , including as well diffeomorphisms with parameter Ξ μ, we obtain the set of transformations where L stands for the Lie derivative.Actually, A is related to the isopd `1, 1q gauge fields by a gauge transformation with an element e Ψ A J An e Φ ÂP Â belonging to the coset isopd 1, 1q{sopd, 1q (for details, see [29]), A " e Ψ A J An e Φ ÂP Â ´Ã `d¯e ´Φ B P B e ´ΨB J Bn . (3.17) In fact, the vielbein and spin connection transforms under the isopd `1, 1q Poincaré transformations in the standard way, i.e.
In addition, the Stückelberg fields Φ Â and Ψ A obey the transformation rules In terms of these fields, the non-linear gauge fields can be expressed as where we have defined the translational-invariant vielbein Notice that by construction the fields E Â are invertible.In fact, we define the inverse vielbein In addition, we can define the spacetime metric and its inverse.
It is also convenient to introduce an affine connection Γ ρ μν by means of the vielbein postulate In terms of which the components of the torsion and the curvature read Finally, with these ingredients we define the volume form and a Hodge dual operation that we can use to define the Yang-Mills-like action where x¨¨¨y stands for an invariant metric on the isopd `1, 1q algebra in d `2 dimensions.However, note that the Higgsing of the theory allows us to define a bi-linear form only invariant under the action of the unbroken subgroup SOpd, 1q.With that criteria, the most general choice is (3.27)

Aristotelian contraction
As previously pointed out our goal is to implement the contraction (3.9) of the Poincaré algebra on the gravitational theory described by Eq. (3.26).This can be achieved after rescaling the gauge fields as ) ) ) E n " σ ρ. (3.28g) Actually, in the limit ε Ñ 0, the connection shown in Eq. (3.12) reduces to a non-Lorentzian one taking values on the elements of the algebra defined by (3.10) and (3.11), which can be written as where The corresponding contracted curvature is given by where F " dA `A ^A is the curvature associated to the connection A, which reads with R ab " dω ab `ωa c ^ωcb , (3.33b) f " dρ `ea ^va , (3.33d) and D is the covariant exterior derivative with respect to ω ab .The term D A B, on the other hand, is the covariant derivative of B with respect to A.
D A B " dB `rA, Bs " `Dµ a `σ2 v a ^u˘G a `pdu `µa ^va q K. (3.34)Notice also that in the limit ε Ñ 0, the non-vanishing components of the invariant tensor (3.27) are δρ " L Ξ ρ, (3.36c) where the gauge parameters are related to the ones in Eqs.(3.15) by Θ ab " θ ab .From the contracted algebra perspective these transformations can be obtained from a gauge transformation of the connection defined in Eq. (3.30) as, δA " L ξ A `dλ `rA, λs, with λ " 1 2 θ ab J ab .
In the Aristotelian limit ε Ñ 0, the linear connection (3.16) can also be decomposed in the form (3.29) with where we have defined the following rescaling for the field Similarly, using (3.20) and rescaling Ψ A in the form the Aristotelian non-linear fields can be expressed in terms of the linear ones as where fields λ, h a , a, ṽa and ωab transform as Notice that, apart from the parameter associated to local rotations, we have defined Θ an " σ b a .Since we are interested in expressing the action (3.26) in terms of the rescaled fields given in Eq. (3.28), we notice that the d `2 metric and its inverse can be decomposed as where the inverse vielbein has been rescaled as E μ Â " pτ μ, e μ a , σ ´1ρ μq and we have defined The fields satisfy the orthogonality relations and the square root metric determinant can be written as All this allows us to expand the higher-dimensional action in powers of σ, S " with the leading order terms taking the form (3.48)

Dimensional reduction
Before starting with the dimensional reduction procedure, we would like to point out that, as it generically happens with gravitational theories, local translations are "spontaneously broken" [30].Nonetheless, we are interested in a system where the fracton charge (momentum in the extra direction) is conserved.Therefore, we will require the existence of a spacelike (transverse) killing vector K, and to use coordinates x μ " px µ , zq such that K " B z .Moreover, from the perspective of this construction the fracton U p1q transformations can be understood as translations along the extra spacetime dimension (see Eq. (3.9)).Therefore, since the generator Q commutes with all the rest after the contraction it is natural to require the existence of a Killing vector K.That would guarantee that gauge fields remain invariant when a transverse diffeo with constant parameter is applied.Thus we require which implies all components of the fields to be z-independent.After doing so, we introduce the gauge fixing conditions From the orthogonality relations between the higher-dimensional vielbein and its inverse, it follows that setting (3.50) implies whereas the rest of the fields satisfy the relations τ µ e µ a " 0, familiar from non-Lorentzian geometry.The higher-dimensional metric tensor then takes the form and the determinant of the metric reduces to In order to make the dipole symmetry explicit, we express the non-linear fields ρ µ and v μ in terms of a µ , ṽa μ, ψ a , ω ab μ, τ µ and e a µ .At leading order in σ we can write ρ µ " a µ ´ea µ ψ a `Opσ 2 q, v a μ " ṽa μ ´Dμ ψ a `Opσ 2 q.
(3.58) Implementing all these conditions, the transformations of the pd`1q-dimensional fields take the form ´R µν ac ψ c ᾱ0 R abµν pṽ aµ ´Dµ ψ a q pṽ bν ´Dν ψ b q `α1 ´f µν ´T a µν ψ a ¯´f µν ´T bµν ψ b 2α 3 `ṽ a µ ´Dµ ψ a ˘pṽ µ a ´Dµ ψ a q  . (3.64) Notice that, in the strict limit σ " 0, the limit pocedure leads to S " S 1 .However, other limit can be defined that lead to the action S 2 instead.In the following we discuss a few interesting cases: (3.65)Indeed, one can see that integrating out the auxiliary fields yields the action (3.63) after properly rescaling the constants α n .Now, in the limit σ Ñ 0, λ ab µν , λ a µν and λ µν become Lagrange multipliers enforcing the constraints R ab µν " 0, T a µν " 0, B rµ τ νs " 0, (3.66) and thus leading to the action S " S 2 on flat space.For α 3 " 0 and α 2 " 1, the action boils down to S " ´1 2 Splitting the index µ into space and time components µ " p0, iq, choosing τ µ " p1, 0, . . ., 0q, e a µ " p0, δ a i ), ω ab µ " 0, and gauge fixing a µ " pφ, 0, . . ., 0q we can write where we have defined ṽµν " e aµ ṽa ν .We now split the field ṽa µ as where u a µ is the part of ṽa µ that solves fµν " 0 and thus has the form Notice that the curvature f µν is gauge invariant in the absence of torsion and the second term in (3.67) is now a mass term for the fields B 0i and B ij .In the low energy regime of the theory we can keep only the gapless modes and therefore neglect B a µ .In this case, defining leads to the action (2.23) proposed in [9].The gauge transformation for φ and A ij follow from (3.59) and the gauge invariance of the gauge condition a i " 0. This yields which matches (2.21).
‚ Proca extension -Turning on the constant α 3 in the previous example leads to a Proca extension of Fracton electrodynamics.Indeed, by renaming, and α 3 " m 2 , the action (2.23) reads where . . . in the action contains the kinetic term for the B field, and the interactions between the gauge field A a µ and B a µ .Notice again that the B field will be massive and gauge invariant.
‚ Curved space generalization -Implementing the rescaling α 1 Ñ σ ´2α 1 and α 2 Ñ σ ´2α 2 and taking the limit σ Ñ 0 in the action (3.63) we find ) which corresponds to a torsion-full generalization of the action proposed in [11].Indeed, the term proportional to α 2 is precisely the action (2.45).
‚ Constrained curved background -Another possibility is to treat the action (3.63) perturbatively in the parameter σ, by assuming the fields ṽa µ and a µ do not backreact on the geometry and considering the gravitational fields to be on-shell .We consider the case α 3 " 0 and α 0 {2 " α 2 " 1.The action S 1 then reduces to an Aristotelian version of the Stephenson-Kilmister-Yang model [31][32][33].The field equations coming from S 1 after varying with respect to the metric fields and the spin connection read ) We consider a solution with vanishing torsion T a µν " 0 " B rµ τ νs .One consequence of this is that the resulting action S 2 is gauge invariant even in absence of the Stückelbeg field ψ a thanks to the field equation (3.75).This means that on such gravitational backgrounds it is possible to describe a phase of the system with unbroken dipole symmetry where ψ a " 0.Moreover, as in the flat case, after decomposing ṽa µ in the form (3.69), the term fµν f µν is a mass for B a µ and therefore this field can be neglected in a low energy description of the system.Under all these considerations, the action S 2 takes the simple form where ∇ µ acts with ω ab µ on tangent space indices and with Γ σ µν on spacetime indices.Using these relations we can write the projections of the gauge field A a µ along the inverse tetrad τ µ and e µ a as A aµ τ µ " τ µ ∇ µ pe ν a a ν q ´e µ a ∇ µ pτ ν a ν q , A aµ e µ b " e µ b A ab ´1 2 ∇ rµ a νs e µ a e ν b . (

Conclusions and outlook
In this paper, we have studied the connection between symmetric gauge fields and gravity after understanding MDMA as a contraction of Poincaré algebra.This analysis gives a geometric interpretation to the fracton charge as the momentum of the matter field in a transverse (internal) spacetime dimension, and dipole charge as the angular momentum along that direction.
The main result of the paper is twofold: on one hand we have constructed a Lie algebra contraction that allows to obtain the MDMA from the Poincaré algebra in one dimension higher by putting together a combination of a pseudo-Carrollian contraction and an Aristotelian one.On the other hand, we have derived the action (3.63), which describes fracton gauge fields coupled to Aristotelian geometry.This action was obtained from a higher-dimensional Poincaré gauge theory in a symmetry-broken phase after applying a dimensional reduction and the pseudo-Carrollian-Aristotelian limit.
Different σ Ñ 0 limits of the resulting action were analized which led in particular to the original gauge theory of fracton electrodynamics proposed by Pretko on flat space, together with a flat space Proca extension of the theory, a spontaneously broken phase in curved space, and a symmetric phase in curved space with the harmonic condition D µ R abµν " 0.
As future directions, we envisage possible generalizations of our results.For instance, one could explore the inclusion of fractonic matter in our model by considering higherdimensional relativistic matter fields coupled to our Poincaré gauge theory.Additionally, one could generalize the Lie algebra contraction here considered to fractonic symmetries that generalize the MDMA by including higher moment charges.Indeed, due to the isomorphism between the Bargmann algebra and the extension of the MDMA that includes conservation of the trace of the cuadrupole moment, it would be interesting to understand the relation between Newton-Cartan gravity and fracton gauge theories with such a gauge group.Moreover, it would be of interest to explore supersymmetric generalizations of our results.By exploiting the relation between the Carroll algebra and the MDMA, supersymmetric extensions of Carroll could be of use in the study of spin 1{2 fractons.
xP a P b y " α3 δ ab , xHHy " ´α 3 , (3.35) whereas the components xG a G b y and xKKy vanish.This means that the gauge fields µ a and u entering B decouple from the rest of the gauge fields in the Aristotelian limit and do not appear in the action (3.26) after the contraction.Thus, for simplicity we remove the connection B from the analysis and consider only the connection A. The transformations in Eqs.(3.15) lead to the following symmetry transformations for the gauge fields in Eqs.(3.30) when ε Ñ 0 δτ " L Ξ τ, (3.36a) δe a " L Ξ e a ´θa b e b , (3.36b) .78) This is precisely the action (2.32) when the gauge fixing condition a µ " φ τ µ (3.79) is imposed.Indeed, the generalization of A a µ in the curved space case has the form τ µ ˘eaν `Aab e b µ , A ab " A ba .(3.80)When the conditions (3.49), (3.50), (3.51) and (3.58) are imposed, at leading order in the σexpansion the vielbein postulate (3.23) and its inverse relation lead to the lower-dimensional vielbein postulate and its inverse∇ µ τ ν " 0, ∇ µ e a ν " 0, ∇ µ τ ν " 0, ∇ µ e ν a " 0.(3.81) .7) Therefore, we select the time and spatial direction p0, nq and split the indices as Â " p0, a, nq, where a " 1, ..., d.The commutation relations (3.5) then take the form rJ 0n , P n s " P 0 , rJ ab , J 0c s " δ crb J 0as , (3.8h)rJ ab , J cd s " δ rarc J dsbs ,(3.8i)rJ 0a , J 0n s " J an , (3.8j)rJ 0a , J bn s " δ ab J 0n ,(3.8k)rJ ab , J cn s " δ crb J asn , ) Ξ μB μ " ξ µ B µ ´ǫB z " ξ ´ǫB z .(3.60)where we have remaned Ξ n " ´ǫ.Demaning that the gauge conditions (3.50) and (3.51) are invariant under gauge transformations (3.59), restricts the diffeomorphism parameter ξ µ and the gauge paramenters θ ab and b a to be z-independent.Due to the conditions (3.50),(3.51)and (3.58), the gauge connection (3.30) satisfies A z " Q.This together with the fact that the fields are z-independent imply that the field strength two-form (3.32) satisfies F zµ " B rz τ µs H `T a s theory -we can introduce the rescaling α n Ñ 1 σ 2 α n and introduce auxiliary fields λ ab µν , λ a µν and λ µν , which allow to rewrite (3.63) as µν λ µν ´2λ µν B rµ τ νs ˘`S 2 `Opσ 2 q.
.82) One can show the equivalence between the actions (2.37) and (3.78) by noticing that these relations reduce to (2.35) after the gauge fixing condition (3.79) is implemented.Similarly, one can define A µν " A aµ e a ν ´∇µ a ν , (3.83) which can be shown to be explicitly symmetric after replacing (3.80), and matches (2.36) after imposing the gauge condition (3.79).