Gauging the higher-spin-like symmetries by the Moyal product
Maro Cvitan, Predrag Dominis Prester, Stefano Giaccari, Mateo Pauliši?, Ivan Vukovi?
PPreprint typeset in JHEP style - HYPER VERSION
ZTF-EP-21-03
Gauging the higher-spin-like symmetries by the Moyalproduct
M. Cvitan a , P. Dominis Prester b , S. Giaccari c , M. Pauliˇsi´c b , I. Vukovi´c b a Department of Physics, Faculty of Science, University of Zagreb,Bijeniˇcka cesta 32, 10000 Zagreb, Croatia b Department of Physics, University of Rijeka,Radmile Matejˇci´c 2, 51000 Rijeka, Croatia c Department of Sciences, Holon Institute of Technology (HIT),52 Golomb St., Holon 5810201, IsraelE-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We analyze a novel approach to gauging rigid higher derivative (higher spin)symmetries of free relativistic actions defined on flat spacetime, building on the formalismoriginally developed by Bonora et al. and Bekaert et al. in their studies of linear couplingof matter fields to an infinite tower of higher spin fields. The off-shell definition is basedon fields defined on a 2 d -dimensional master space equipped with a symplectic structure,where the infinite dimensional Lie algebra of gauge transformations is given by the Moyalcommutator. Using this algebra we construct well-defined weakly non-local actions, bothin the gauge and the matter sector, by mimicking the Yang-Mills procedure. The theoryallows for a description in terms of an infinite tower of higher spin spacetime fields onlyon-shell. Interestingly, Euclidean theory allows for such a description off-shell, which weused to calculate some basic 4-point tree-level amplitudes for matter fields. The ordinaryspacetime matter does not scatter into non-trivial angles, while amplitudes for master mat-ter have exponentially softer UV behaviour when compared to QED. Owing to its formalsimilarity to non-commutative field theories, the formalism allows for the introduction ofa covariant potential which plays the role of the generalised vielbein. This covariant for-mulation uncovers the existence of other phases and shows that the theory can be writtenin a matrix model form. The symmetries of the theory are analyzed and conserved cur-rents are explicitly constructed. By studying the spin-2 sector we show that the emergentgeometry is teleparallel under disguise, with the induced linear connection being oppositeto Weitzenb¨ock’s. a r X i v : . [ h e p - t h ] F e b ontents
1. Introduction 22. Moyal-higher-spin formalism in the Yang-Mills formulation 4
3. MHS formalism in the covariant formulation 15
4. Model building: MHS gauge sector 21
5. Model building: matter sector 31
6. Some explicit calculations 37 . Low-spin sector 42 s ≤
8. Conclusions 49A. Properties of the Moyal product 51B. Proof of the theorem on triviality 53C. Weyl-Wigner formalism 55
C.1 Wigner function 56C.2 Second order/metric formalism 57C.3 First order/frame formalism 57
1. Introduction
In the recent period we have witnessed a revival of interest in studying properties of gaugequantum field theories. Taking into account that it is a mathematical framework underlyingour current fundamental description of Nature (via the Standard Model of elementaryparticles), the interest is hardly surprising. A contributive factor to this shift of interests isthat frameworks invented to solve long standing issues present in the high energy regime, inparticular to quantize gravity (such as string theory), are after many years of developmentstill far from fulfilling their promise. There is a growing feeling that fresh new ideas areneeded and these may arise from efforts to improve our understanding of quantum fieldtheories by relaxing some of the standard postulates.The well-known but still poorly understood property of all free relativistic field theorieson Minkowski spacetime, is that they possess an infinite tower of abelian rigid symmetrieswith a corresponding tower of conserved currents represented by Lorentz tensor fields ofall ranks s ≥ s currents). As the rigid symmetryvariations and corresponding conserved currents contain a number of derivatives growingwith s , one can speak of higher derivative or higher spin (HS) symmetries. The low spinsector ( s ≤
2) of these symmetries plays an immensely important role, as spin-1 describesinternal symmetries and gives conservation of electric or non-Abelian charges while spin-2describes translations in spacetime and leads to conservation of energy and momentum.Completing the theories by making the rigid low spin symmetries local one obtains two Likewise, in fermionic free field theories there is also an infinite tower of half-integer spin symmetriesand corresponding currents. We shall not discuss half-integer spin symmetries in this paper. We shall use both terms interchangeably, depending on the context. – 2 –illars of our current description of the fundamental forces in Nature, Maxwell and Yang-Mills theory in the spin-1 case and theories intimately related to General Relativity inthe spin-2 case. While it is easy to dismiss HS symmetries ( s >
2) due to their higher-derivative nature, it is somewhat surprising that they do not have any role in the currentdescription. This may be an indication that we are missing something related to higherderivative symmetries in our understanding of the QFT formalism and maybe even inour fundamental model of Nature. It is not inconceivable that the degrees of freedomassociated with these symmetries may account for dark matter, a missing puzzle in thestandard cosmological model.The idea of gauging higher derivative symmetries is not new, and is at the core ofthe higher spin theory program (e.g., see reviews [2, 3, 4, 5] and references therein). Thestandard approach is to introduce higher spin massless fields and construct higher spingauge QFTs. While it is easy to write equations for free higher spin fields, constructionof consistent interacting QFTs containing fundamental fields of spin higher than two inMinkowski spacetime is an unsolved problem of theoretical physics. There are severalno-go theorems and obstructions which put strong restrictions on such constructions. In d ≥ To this we add that in flat spacetimeeven the free HS theory has strongly non-local or higher-derivative field equations whenthe HS symmetry is unconstrained [17, 18, 19, 20], a property further studied by inducedaction methods in [21, 22]. All in all, it appears that gauging higher derivative symmetriesin general requires abandoning the condition of the strong locality of the action.In [23, 24] a different formalism for gauging higher derivative symmetries in Minkowskispacetime was proposed. It was based on the observation from [25, 26] that if one introducesinteractions in otherwise free Klein-Gordon theory by coupling an infinite tower of externalHS fields linearly to the conserved currents of the free theory, the resulting theory haslocal higher derivative symmetries whose algebra can be elegantly written by introducingan auxiliary space and using Moyal product acting on master fields defined on spacetimeand the auxiliary space. The key ingredient was provided in [27] where it was shown thata similar construction can be done starting from the free Dirac theory, resulting in thesame algebra of local higher derivative symmetries. In this case one directly obtains themaster field gauge potential, allowing for the construction of a Yang-Mills-like action forthe gauge potential master field by mimicking standard Yang-Mills prescription, including See [1] for recent review and developments. Though there were some interesting developments recently, see e.g. [6, 7, 8, 9]. See [10, 9, 11, 12] and reviews [13, 5]. One can avoid traditional no-go theorems by taking AdS spacetime as a background, allowing forVasiliev higher spin program [15, 2] to be developed, but so far only with partial success as no off-shellformulation of a consistent interacting HS QFT in d ≥ – 3 –he BRST quantization rules (at least on the formal level) [24]. The outstanding featureof the construction is that one obtains a theory defined off-shell, with the local symmetryobtained by gauging rigid higher spin symmetries of matter, in Minkowski spacetime. Weshall refer to this framework as Moyal-higher-spin (MHS). In [24] it was shown that anattempt to write the MHS Yang-Mills (MHSYM) theory as an off-shell theory of a towerof spacetime HS fields, by making reductions of the configuration space and scaling thecoupling constant with an infinite volume factor, faces challenges of projecting out ghostsand preserving unitarity. We stress here that reductions considered in [24] break theoriginal form of the MHS symmetry, without assurance that enough of symmetry survivesto guarantee unitarity, which means that the troubles should not be ascribed to the MHStheory in its original form, but to the procedure which basically changes the original theory.Instead of trying to achieve the standard HS description in terms of finite spin space-time fields, in this paper we propose to take the MHS theory as it is naturally defined, whichis a theory fundamentally described by master fields, and explore its properties. In Sec. 2we review the MHS construction and state some basic properties of the MHSYM theory.In particular, we show that MHSYM theory has a perturbatively stable vacuum and doesnot possess Ostrogradsky ghosts, and make connection with standard non-commutativeYM theory. We explain why the on-shell description in terms of an infinite tower of finitespin spacetime fields does not have a non-singular off-shell formulation which keeps man-ifest symmetry in the Lorentzian case, but does have such a description in the Euclideancase. In Sec. 3 we show that the introduction of the MHS covariant potential allows fora more general description of the MHS formalism which bears resemblance to teleparallelgeometry. In Sec. 4 we study in more detail MHSYM theory and its generalisations, anal-yse its symmetries, conserved currents and phases. In Sec. 5 we analyse coupling to thematter sector and show that beside standard spacetime matter fields one can introducematter in the form of master fields. In Sec. 6 we calculate a few basic tree level 4-pointamplitudes for different types of matter in the Euclidean formulation where one can use arepresentation in terms of an infinite tower of finite spin spacetime fields. We show that forthe ”minimal” matter (described by spacetime fields) amplitudes vanish when the set ofoutgoing momenta is different from the set of ingoing momenta while for matter describedby master fields amplitudes have an exponential fall-off. Both results are promising inthe sense that the amplitudes in the MHS theory possess much softer UV behaviour whencompared to the Maxwell theory. In Sec. 7 we study the low spin sector of the expansionin HS fields and show that the spin-2 component has a differential geometric interpretationwith the linear connection being opposite to Weitzenb¨ock’s connection, meaning that it isteleparallel in disguise. Finally, a few technical appendices are added at the end.
2. Moyal-higher-spin formalism in the Yang-Mills formulation
It is known that actions of free field theories in flat d -dimensional Minkowski spacetime M – 4 –re invariant under infinitesimal rigid transformations of the form δ ε φ ( x ) = ∞ (cid:88) n =0 ( − i ) n +1 ε µ ...µ n ∂ µ · · · ∂ µ n φ ( x ) (2.1)where symmetric Lorentz tensors ε µ ...µ n are infinitesimal rigid (constant in spacetime)real parameters. This is an infinite dimensional Abelian symmetry, which gives the rigid U (1) symmetry for n = 0, and spacetime translations for n = 1. For n > s = n + 1. Inthe case when φ is real, one has only terms with odd n (even spin). With a slight abuse oflanguage we shall refer to the whole tower as HS transformations and HS symmetries, andrefer to its s ≤ d -dimensional auxiliary space U , withcoordinates denoted by u µ , and borrowing from the Wigner-Weyl phase space formalism,we write free field actions in the master space as M × U S [ φ ] = (cid:90) d d x d d u W φ ( x, u ) (cid:63) K ( u ) (2.2)where K ( u ) is the kinetic function and (cid:63) is the Moyal product defined by a ( x, u ) (cid:63) b ( x, u ) = a ( x, u ) exp (cid:20) i (cid:16) ← ∂ x · → ∂ u − → ∂ x · ← ∂ u (cid:17)(cid:21) b ( x, u ) . (2.3)The rigid HS variation (2.1) of the Wigner function then takes the following form δ ε W φ ( x, u ) = − i [ ε ( u ) (cid:63) , W φ ( x, u )] (2.4)where ε ( u ) = ∞ (cid:88) n =0 ε µ ...µ n u µ . . . u µ n . (2.5)The details of the construction can be found in Append. C and Sec. 5.1.The advantage of the master space description comes from the properties of the Moyalproduct: associativity, Leibniz identity, Jacobi identity, hermicity, and cyclicity of trace. The most important properties of the Moyal product relevant for us are summarized in the Append. A. – 5 –his allows us to gauge the rigid HS symmetry in the spirit of the standard Yang-Mills(YM) procedure, by promoting the HS parameter to a master field ε ( u ) → ε ( x, u ) (2.6)while keeping the Moyal product structure by which local HS symmetry is represented. Toavoid confusion with the existing literature we shall refer to this construction as the MHSsymmetry. There are two obvious representations, adjoint and fundamental. In the adjointrepresentation MHS variations act as δ ε A ( x, u ) = − i [ ε ( x, u ) (cid:63) , A ( x, u )] (2.7)where A ( x, u ) can be either a composite master field, as in the case of matter describedby standard spacetime fields, in which case A ( x, u ) is the Wigner function W φ ( x, u ),or an elementary master field. We shall refer to an object transforming in the adjointrepresentation as an MHS tensor. In the fundamental representation MHS variations actas δ ε χ ( x, u ) = − i ε ( x, u ) (cid:63) χ ( x, u ) (2.8)which can be applied only to complex master fields.Finite (large) MHS transformations in the fundamental representation are φ E ( x, u ) = e − i E ( x,u ) (cid:63) (cid:63) χ ( x, u ) (2.9)and in the adjoint representation are A g ( x, u ) ≡ A E ( x, u ) = e − i E ( x,u ) (cid:63) (cid:63) A ( x, u ) (cid:63) e i E ( x,u ) (cid:63) . (2.10)The group multiplication is represented by the Moyal product. The Baker-Campbell-Hausdorff lemma guarantees that a real function E ( x, u ) exists such that e − i E ( x,u ) (cid:63) = e − i E ( x,u ) (cid:63) (cid:63) e − i E ( x,u ) (cid:63) (2.11)for any real E ( x, u ) and E ( x, u ), which means that the group of local MHS transformationsis closed. The inverse is given by (cid:16) e − i E ( x,u ) (cid:63) (cid:17) − = e i E ( x,u ) (cid:63) . (2.12)By making transformations local we have changed the nature of the symmetry group,from abelian to non-abelian. The Lie algebra structure of local MHS transformations isgiven by the Moyal bracket [ δ ε , δ ε ] = δ i [ ε (cid:63) ,ε ] . (2.13)Taking into account that the Moyal bracket satisfies the Jacobi identity, it follows that theMHS transformations possess an infinite dimensional non-abelian Lie algebra structure.We shall return to the detailed study of the matter sector in Sec. 5. We now turn ourfocus to the gauge sector. In this case matter field φ ( x ) transforms in the Hilbert space representation, see Sec. 5.1. – 6 – .2 The MHS potential and field strength Requiring a gauge symmetry necessitates the introduction of a respective connection. Thealgebraic properties of the Moyal product allow us to repeat the standard steps of the Yang-Mills (YM) construction. The first step is to introduce an MHS (master) gauge potential h ( x ), which is a Lie algebra valued 1-form on spacetime M . The potential represents theconnection, which generally transforms under gauge transformations as h g ( x ) = g ( x ) h ( x ) g ( x ) − − ig ( x ) d g ( x ) − (2.14)where d is the exterior derivative. In the case of MHS transformations this implies h E a ( x, u ) = e − i E ( x,u ) (cid:63) (cid:63) h a ( x, u ) (cid:63) e i E ( x,u ) (cid:63) − ie − i E ( x,u ) (cid:63) (cid:63) ∂ xa e i E ( x,u ) (cid:63) . (2.15)In the YM-like description on the Minkowski spacetime there is no difference betweenLatin indices a, b, . . . and the Greek indices µ, ν, . . . , so, e.g., ∂ xa = δ µa ∂ xµ , h a = δ µa h µ and u a = δ µa u µ . However, we shall later show that the Taylor expansion of master fields around u = 0 leads to an induced differential geometry interpretation in which Latin indices behaveas frame indices.For (infinitesimal) MHS variations, parameterized by E ( x, u ) = ε ( x, u ), (2.15) becomes δ ε h a ( x, u ) = ∂ xa ε ( x, u ) + i [ h a ( x, u ) (cid:63) , ε ( x, u )] (2.16)which is the variation obtained in [27] by studying a Dirac field linearly coupled to aninfinite tower of HS fields.Having the potential we can introduce the MHS covariant exterior derivative D (cid:63) = d + i h ˆ (cid:63) (2.17)where ˆ (cid:63) stands for the Moyal-wedge product. We can write (2.16) as δ ε h a ( x, u ) = D (cid:63)a ε ( x, u ) (2.18)where by D (cid:63)a we denote the MHS covariant derivative acting on the MHS tensors, D (cid:63)a ≡ ∂ xa + i [ h a ( x, u ) (cid:63) , ] . (2.19)Using the covariant exterior derivative we define the MHS field strength in the standardway F = D (cid:63) h = dh + i h ˆ (cid:63) h (2.20)which in our case gives F ab ( x, u ) = ∂ xa h b ( x, u ) − ∂ xb h a ( x, u ) + i [ h a ( x, u ) (cid:63) , h b ( x, u )] . (2.21)By construction, the MHS field strength transforms covariantly, in the adjoint representa-tion F g ( x ) = g ( x ) F ( x ) g ( x ) − (2.22)– 7 –hich in our case of MHS transformations is F E ab ( x, u ) = e − i E ( x,u ) (cid:63) (cid:63) F ab ( x, u ) (cid:63) e i E ( x,u ) (cid:63) . (2.23)For infinitesimal MHS variations this becomes δ ε F ab ( x, u ) = i [ F ab ( x, u ) (cid:63) , ε ( x, u )] . (2.24)It is straightforward to show that the MHS field strength obeys the Bianchi identity D (cid:63) F = 0 . (2.25)In YM theory field strength measures the triviality of the configuration. The sameapplies here, which means that the MHS potential is trivial on an open subset of M d if andonly if the MHS field strength vanishes there h is pure gauge ⇐⇒ F = 0 . (2.26)This can be proven in the similar fashion as it is usually done in standard YM theory, usingthe fact that the Moyal product satisfies the algebraic properties of matrix multiplication.The proof of (2.26) is presented in Append. B.Before moving to construction of explicit MHS theories, let us briefly comment on thestructure of the rigid MHS symmetry in the gauge sector. From (2.16) follows δ ε h a ( x, u ) = 2 ε ( u ) sin (cid:18) ← ∂ u · → ∂ x (cid:19) h a ( x, u ) . (2.27)If we expand the MHS parameter as in (2.5) one realizes that the MHS potential (asany MHS tensor) in general transforms differently than the spacetime matter fields whosetransformation is given in (2.1). The only exceptions are low-spin transformations with s ≤ Let us demonstrate this on the case of a pure spin-3 variation ε ( u ) = ζ µν u µ u ν forwhich (2.27) gives δ (3) ζ h a ( x, u ) = u µ ζ µν ∂ xν h a ( x, u ) . (2.28)A real spacetime matter field, in comparison, is neutral under the rigid spin-3 variation.A similar departure from the standard form for rigid HS transformations, given in(2.1), appears for matter described by elementary master fields. This is a price to be paidfor a simple and consistent formalism. It is also one of the places on which we deviate fromstandard Noether constructions based on HS symmetries. There is a restriction on the space of configurations for which this is valid. See Append. B for moredetails. MHS potential is real and so is neutral under spin-1 U (1) transformations, so the only non-trivial caseis n = 2. – 8 – .3 MHS Yang-Mills model Before we continue with the formal mathematical development of the idea, let us examinethe physical aspects encoded in it. Following the standard YM construction the naturalcandidate for an action in the MHS gauge sector is the MHS Yang-Mills (MHSYM) actiondefined by S ym = − g (cid:90) d d x d d u F ab ( x, u ) (cid:63) F ab ( x, u )= − g (cid:90) d d x d d u F ab ( x, u ) F ab ( x, u ) + (boundary terms) . (2.29)Frame (Latin) indices are contracted using the Minkowski metric η ab . In many applica-tions, such as finding equations of motion (EoM) or analyzing proper gauge symmetries,boundary terms in the action can be discarded. However, there are instances in which onehas to be more careful, e.g., analyses of asymptotic symmetries and non-trivial topologicalstructures. The action is weakly non-local, but has at most quartic interaction terms. EoMare (cid:50) x h a − ∂ xa ∂ xb h b + i (cid:16) h b (cid:63) , ∂ xb h a ] − [ h b (cid:63) , ∂ xa h b ] + [ ∂ xb h b (cid:63) , h a ] (cid:17) + (cid:2) h b (cid:63) , [ h a (cid:63) , h b ] (cid:3) = 0 . (2.30)The MHSYM theory was proposed in this form in [24]. Note that it is based on thesimplest symmetry group for the spin-1 sector, which is U (1). Had we started by assuminga non-abelian symmetry group G (e.g., SU ( N )), the MHS construction would have beennaturally generalized by lumping together the ”standard” internal YM structure with theMHS structure [24]. In this case the MHS potential would be valued in the Lie algebra g of the group G . To keep the presentation as simple as possible we shall constrain ourselvesin this paper to the U (1) spin-1 group.In Sec. 3 we shall generalise the YM-like formalism which will allow us to see that thisform describes just one phase of the theory. Before that, let us mention some of its basicproperties. • Noncommutative structure. The MHSYM model generally falls into the class ofnon-commutative (NC) theories in the broad sense. However, it differs from thestandardly studied NC-YM theories. In particular, here the spacetime by itself iscommutative and non-commutativity is present only between the spacetime and theauxiliary space. It is however possible to interpret MHSYM model as a singularlimit of a particular NC-YM theory in 2 d dimensions. It is known that NC-YM fieldtheory defined in a background with a constant metric tensor G ij can be rewrittenas S nc = − g (cid:90) d d (cid:98) x √ G G ik G jl F ij ( (cid:98) x ) (cid:63) F kl ( (cid:98) x ) , (2.31) We use the convention η ab = diag( − , , . . . , In [29] a somewhat similar model was constructed by taking the associative limit of a gauge theorydefined on non-associative fuzzy spaces, the difference being that in [29] the star product which does notsatisfy the hermicity property (A.4) appeared instead of the Moyal product. For a review of NC field theory see [30]. – 9 –here (cid:98) x = { (cid:98) x i } are coordinates and i, j, . . . are indices in the 2 d -dimensional space.NC gauge field strength is F ij ( (cid:98) x ) = ∂ i h j ( (cid:98) x ) − ∂ j h i ( (cid:98) x ) + i [ h i ( (cid:98) x ) (cid:63) , h j ( (cid:98) x )] . (2.32)The star product encodes NC structure and can be written as a ( (cid:98) x ) (cid:63) b ( (cid:98) x ) = e i θ ij ∂∂ξi ∂∂ζj a ( (cid:98) x + (cid:98) ξ ) b ( (cid:98) x + (cid:98) ζ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:98) ξ = (cid:98) ζ =0 , (2.33)where θ ij is a constant antisymmetric matrix. The action is invariant under gaugetransformations δ ε h i ( (cid:98) x ) = ∂ i ε ( (cid:98) x ) + i [ h i ( (cid:98) x ) (cid:63) , ε ( (cid:98) x )] . (2.34)Let us now consider a special case in which G and θ are given by G ≡ (cid:32) η (cid:15) η (cid:33) , θ ≡ (cid:32) η − η (cid:33) (2.35)where η is d × d Minkowski metric tensor. If we write (cid:98) x = ( x, u ) we see that the starproduct defined in this way is identical to the Moyal product (2.3). Defining g ≡ g (cid:15) d (2.36)we obtain S nc = S ym + O ( (cid:15) ) . (2.37)In other words, by taking the scaling limit (cid:15) → g ym fixed and finite, theNC field theory defined by (2.31)-(2.35) becomes formally identical to the MHSYMaction (2.29). In this limit g nc → ∞ , so it is a strong coupling limit from the NC-YMfield theory perspective. Note that the limit is singular from the perspective of NCfield theory because the background metric tensor (2.35) is singular when (cid:15) = 0. • Symmetries. The MHSYM action is off-shell invariant under the MHS transforma-tions up to boundary terms. This will be important in the study of conservation lawsand corresponding conserved charges. There is also a symmetry h (cid:48) a ( x (cid:48) , u (cid:48) ) = Λ ab h b ( x, u ) , x (cid:48) µ = Λ µν x ν + ξ µ , u (cid:48) µ = Λ µν u ν + τ µ (2.38)where Λ are Lorentz matrices and ξ µ and τ µ are arbitrary constant vectors. TheLorentz transformations acting in spacetime and the auxiliary space must be thesame in order to keep the Moyal product invariant. We see that besides the standardPoincar´e group (Λ , ξ ) of spacetime isometries, there is also an independent group oftranslations in the auxiliary space. Its existence is a consequence of the fact that theMHSYM action is only weakly non-local on the master space. We show in Sec. 5 thata matter sector may break the auxiliary space translation symmetry.– 10 – Restricted MHS symmetry (even-spin only). The MHSYM action is also symmetricunder the auxiliary space reflection u (cid:48) µ = − u µ , h (cid:48) a ( x, u (cid:48) ) = − h a ( x, u ) (2.39)since the HS field strength transforms as F (cid:48) ab ( x, u (cid:48) ) = − F ab ( x, u ) . (2.40)The MHS variation of the transformed MHS potential keeps the form (2.16), with ε (cid:48) ( x, u (cid:48) ) = − ε ( x, u ) . (2.41)It follows that the MHS potential can consistently be restricted to be odd in theauxiliary space h a ( x, − u ) = − h a ( x, u ) . (2.42)Indeed, it is easy to show that HS transformations are compatible with this if theMHS parameter is also restricted to be odd ε ( x, − u ) = − ε ( x, u ) . (2.43)Using the expansion (2.5) it follows that this restriction corresponds to gauging onlyspin-even rigid HS transformations. • Coupling constant. The dimension of the MHS potential is (length) − so the dimen-sion of the coupling constant g ym is (length) − for all d . It appears that the MHSYMtheory has a scale, which we denote by (cid:96) h , already at the classical level. However,we shall show in Sec. 4.2 that the theory in the classical regime does not have anintrinsic scale and that the scale symmetry is spontaneously broken by the choice ofthe vacuum. To make contact with the canonical formalism it is natural to pass tothe dimensionless auxiliary coordinates ¯ u and a rescaled coupling constant ¯ g h andthe MHS potential ¯ h a , defined by¯ u = (cid:96) h u , ¯ g ym = (cid:96) d/ h g ym , ¯ h a = h a / ¯ g ym . (2.44)The dimension of ¯ g ym is (length) d − , the same as in the standard Maxwell or Yang-Mills theories, and in d = 4 it is zero. In the canonical normalization cubic terms andquartic terms in the action have the coupling given by ¯ g ym (cid:96) D − h and ¯ g (cid:96) Dh , respec-tively, where D is the total number of spacetime derivatives in a given monomial. • Stability of the vacuum. The configuration h a ( x, u ) = 0 is obviously a solution ofEoM. In the weak-field regime, in which one can ignore terms in the action higherthen quadratic, the action is formally similar to the one for the Maxwell theory, sowe can immediately obtain the expression for the spatial energy density U ≈ g (cid:90) d d − x (cid:90) d d u (cid:18) (cid:88) j F j ( x, u ) + (cid:88) j In the MHS formalism a natural description is in terms of fields defined on the master space.The classical configuration space is restricted by the requirement that master fields havewell-defined Moyal products. Furthermore, due to the infinite volume of the auxiliary spacethe MHS potential must satisfy proper fall-off conditions, so as to guarantee convergenceof integrals over auxiliary space in the expressions for observables such as energy andmomentum (see, e.g., (2.45)).The natural objective would be to understand the content of the theory in terms ofWigner’s classification of irreducible representations of the Poincar´e group. In particular,is there a purely spacetime description in terms of the standard (finite spin) spacetimefields? Now, by looking at the Eq. (2.5) the simplest assumption would be to assume thatsuch a description is provided by the Taylor expansion in the auxiliary space h a ( x, u ) = ∞ (cid:88) n =0 h ( n ) µ ··· µ n a ( x ) u µ . . . u µ n . (2.46)There are several reasons that make this assumption attractive. This expansion is man-ifestly Lorentz covariant. The coefficients in the expansion are spacetime fields that areLorentz tensors of rank n + 1 symmetric in their n (Greek) indices, and which by (2.30)satisfy Maxwell-like linearized EoM. Such Lorentz tensor fields are reducible and gener-ally break into irreducible representations with integer spins s ≤ n + 1. We shall refer tothe spacetime field h ( n ) a as the spin-( n + 1) field. The expansion (2.46) leads us to the Using an auxiliary space to formally pack a tower of HS spacetime fields into some generator masterfield, as in (2.46), is a standard trick used in conventional HS constructions and analyses of EoM’s. A totally symmetric Lorentz tensor field of rank n satisfying Maxwell-like EoM contains irreduciblerepresentations with spins s = n, n − , n − , . . . , – 12 –pacetime description in terms of infinite tower(s) of HS spacetime fields with unboundedspin. If we restrict the HS potential to an odd function in the auxiliary space (2.42) thetower will only contain spacetime fields with even spin. In [27, 24] it was shown that HSspacetime fields defined by (2.46) linearly couple to the corresponding HS currents whenspacetime matter fields are minimally coupled to the MHS potential (see Sec. 5.1 and inparticular Eq. (5.22)). Also, it is straightforward to show that the truncation to n = 0and n = 1 sectors is consistent both with the HS transformations and the MHSYM EoM,meaning that on the level of EoM the low-spin sectors ( s = 1 or 2) may be decoupled fromthe true HS sector ( s > h ( n ) µ ...µ n a ( x ) are non-vanishing, in particular low-spin ( s ≤ spacetime fields generated by theexpansion (2.46) do not reflect the spectrum (particle content) of the MHSYM theory .No expansion in a discrete basis of functions in the auxiliary space exists which providesat the same time (i) convergence of integrals over auxiliary space inside the action needed forthe purely spacetime off-shell description, and (ii) manifest Lorentz covariance. One maytry to bypass the requirement (i) and regularize integrals over master space by absorbingthe (infinite) volume through a (classical) renormalization of the coupling constant. Suchan approach was proposed in [24] with partial success. The problem with this strategy ingeneral is that the theory is essentially redefined, which introduces the danger of losingsome of the crucial qualities present in the classically regular master space description, suchas perturbative stability of the vacuum. Alternatively, one can ignore the requirement (ii)and consider expansions in an (orthonormal) discrete basis of functions on { f r ( u ) } whichbreaks manifest Lorentz covariance. Formally, the Lorentz symmetry of the theory is stillthere and from a purely classical viewpoint there is nothing wrong with this. It would beinteresting to see whether one can eventually reconstruct manifest Lorentz covariance onthe level of observables. We leave this question for our future work.In summary, we conclude that the MHSYM theory apparently does not have a La-grangian (off-shell) description in terms of (a tower of ) HS spacetime fields carrying stan-dard massless finite spin irreducible representations of the Poincar´e group. On the onehand this may seem strange considering our starting point was a tower of global HS sym-metries, but on the other hand it is not surprising considering that attempts based onstandard HS fields in Minkowski spacetime background not only failed but also produceda number of no-go theorems. – 13 – .5 Euclidean theory It is interesting that a viable purely spacetime off-shell description is possible in the Eu-clidean version of the theory. In this case it is easy to find regular expansions of theHS master potential leading to convergent integrals over the auxiliary space, which aremanifestly covariant under the isometries of the flat Euclidean spacetime.The simplest choice closely resembling (2.46) would be h a ( x, u ) = ∞ (cid:88) n =0 h ( n ) µ ··· µ n a ( x ) u µ . . . u µ n e − ( (cid:96) h u ) / (2.47)where u = δ µν u µ u ν . Comparing with (2.46) we see that the difference is the exponentialfactor. This factor guarantees convergence of all integrations over auxiliary space afterusing (2.47) in the MHSYM action or in the energy-momentum spacetime tensor. In thisway we obtain a regular Euclidean spacetime action and other observables as functionalsof the infinite tower of spacetime potentials { h ( n ) µ ··· µ n a ( x ) } . A downside of this approachis the lack of a consistent on-shell truncation to a finite subset of spacetime fields.Let us apply (2.47) to the quadratic (free field) part of the MHSYM action in thecanonical normalization (2.44) S (0)ym [ h ] = 14 g (cid:90) d d x (cid:90) d d u (cid:0) ∂ a h b ( x, u ) − ∂ b h a ( x, u ) (cid:1) . (2.48)Integrating over the auxiliary space we obtain the quadratic part of the action in the(purely) spacetime description S (0)ym [ h ] = (cid:90) d d x L ( x ) (2.49)with the spacetime Lagrangian density given by L = ∞ (cid:88) n,m =0 n + m even (cid:0) ∂ a h ( n ) µ ...µ n b − ∂ b h ( n ) µ ...µ n a (cid:1) K ( n + m ) µ ...µ n ν ...ν m (cid:0) ∂ a h ( n ) ν ...ν m b − ∂ b h ( n ) ν ...ν m a (cid:1) . (2.50)The kinetic matrix is totally symmetric and given by K (2 r ) µ ...µ r = c d (cid:96) − ( d +2 r ) h g Γ (cid:16) r + d (cid:17) (2 r − d + 2 r − δ ( µ µ · · · δ µ r − µ r ) (2.51)where c d is a (finite) numerical coefficient depending on d (but not on m or n ) whose exactexpression is not of relevance here. We see that the quadratic term is not diagonal in thespacetime HS fields, but strongly mixed. In fact, it is as non-diagonal as it gets, becauserotation invariance prevents mixing between spin-odd (even n ) and spin even (odd n ) fieldsin the quadratic Lagrangian. This complicates the extraction of the physical spectrum andcalculation of the propagator, as for this one first has to invert the kinetic matrix (2.51). That such mixing of HS fields in the linearised EoM is a generic property was already shown by usinginduced action methods [22]. – 14 –he problem of diagonalizing the quadratic part of the action can be avoided using anorthonormal basis of functions in the auxiliary space { f r (¯ u ) } , (cid:90) d d ¯ u f r (¯ u ) f s (¯ u ) = δ rs (2.52)to expand master fields as ¯ h a ( x, ¯ u ) = (cid:88) r ¯ h ( r ) a ( x ) f r (¯ u ) . (2.53)Here we used the canonical normalization introduced in (2.44). Examples of bases whichare covariant under SO ( d ) rotations are those based on the spherical harmonics. Note thata basis of this type cannot be naturally Wick rotated to the Minkowski space.This makes the quadratic part of spacetime action in MHSYM theory diagonal, everyterm being of the Maxwell type S [¯ h ] = − (cid:88) r (cid:90) d d x (cid:0) ∂ a ¯ h ( r ) b − ∂ b ¯ h ( r ) a (cid:1) . (2.54)In a similar way one can integrate the interacting part of the MHSYM action over theauxiliary space to obtain a purely spacetime action which is a weakly non-local functionalof spacetime fields { ¯ h ( r ) a ( x ) } .Note that after using an orthonormal basis to diagonalize the kinetic term, we can inprinciple express all results in terms of the manifestly covariant (under isometries) space-time fields h ( n ) µ ··· µ n a ( x ) by expressing the orthonormal basis { f r (¯ u ) } in terms of the non-orthogonal basis used in (2.47). This allows us to bypass the problem of diagonalization ofthe kinetic matrix (2.51). The expansion (2.47) has the advantage that the spacetime fieldscarry representations of both Euclid and Minkowski isometries ( SO ( d ) and SO (1 , d − 3. MHS formalism in the covariant formulation Here we explore the MHS structure introduced in Sec. 2.2 in more depth. We use a formalsimilarity with non-commutative field theories to borrow some of the techniques, and showthat there exists a covariant frame-like formulation. This formulation will be important inunderstanding the emergent geometrical description of the spin-2 sector. As we will show,it also offers a better starting point for a background independent formulation in terms ofmatrix models.In the YM-like approach the basic object that covariantly transforms, in the adjointrepresentation, under MHS transformations is the MHS master field strength, see (2.23).Any master field A ( x, u ) transforming in the same way, which is A E ab ··· ( x, u ) = e − i E ( x,u ) (cid:63) (cid:63) A ab ··· ( x, u ) (cid:63) e i E ( x,u ) (cid:63) (3.1)– 15 –e call an MHS tensor. An MHS tensor in general can have any number of frame-likeindices (denoted by Latin letters a, b, . . . ), on which MHS transformations do not act. Forthe moment we assume flat background and trivial frames, so frame indices are raised andlowered with the Minkowski metric tensor. For infinitesimal MHS transformations thisgives the MHS variation δ ε A a ··· ( x, u ) = i [ A a ··· ( x, u ) (cid:63) , ε ( x, u )] . (3.2)The important property of MHS tensors is that the Moyal product of MHS tensors is againan MHS tensor. This is a trivial consequence of (3.1) and (2.12). In standard YM gauge theories, to construct a covariant object from the gauge potentialwe need to take a derivative. This object is the gauge field strength. Non-commutativityof the MHS structure allows us to construct an MHS tensor without using derivatives, inthe following way e a ( x, u ) ≡ u a + h a ( x, u ) . (3.3)Using (2.16) it is easy to show that e a ( x, u ) transforms under MHS variations as δ ε e a ( x, u ) = i [ e a ( x, u ) (cid:63) , ε ( x, u )] . (3.4)which is exactly (3.2). The presence of such an object is not unexpected from the viewpointof NC field theories. By using it instead of the MHS potential h a ( x, u ) we can write allequations in the MHS gauge sector in a manifestly MHS covariant way (i.e., by usingexclusively MHS tensors), a feat not possible in the standard YM theories.We refer to e a ( x, u ) as the MHS vielbein. The motivation comes from analysing itsTaylor expansion in the auxiliary coordinates e a ( x, u ) = ∞ (cid:88) n =0 e ( n ) µ ...µ n a ( x ) u µ · · · u µ n . (3.5)As we show in Sec. 7 the spin-2 ( n = 1) spacetime component transforms under the spin-2part of the MHS transformations as a vector frame under diffeomorphisms. Moreover, if weassume (2.42), which is consistent with the MHS symmetry, this component is the lowestterm in the expansion. When coupled to spacetime matter, this vector frame plays the roleof the vielbein, as shown in Sec. 5.1.This expansion also illuminates the meaning of (3.3). Performing Taylor expansions(3.5) and (2.46) one obtains that the corresponding spacetime fields are connected through e ( n ) µ ...µ n a ( x ) = h ( n ) µ ...µ n a ( x ) , n (cid:54) = 1 (3.6)and e (1) µa ( x ) = δ aµ + h (1) µa ( x ) . (3.7) Note that e a ( x, u ) can be understood as MHS covariant coordinates on the auxiliary space. – 16 –e see that (3.3) defines the MHS potential with respect to the empty Minkowski back-ground e (0) a = u a ≡ δ µa u µ . (3.8)This is not surprising, but it shows the limits of practical usability of (3.3). We can againsee that the MHS vielbein is the fundamental object in the theory, and that (3.3) is sensibleonly if we are interested in expansions around the empty Minkowski vacuum.Strictly speaking, to identify e (1) µa ( x ) as a spacetime vielbein, an invertibility conditionshould be imposed. This condition is apparently not required in the MHS formalism, whichopens the possibility of accommodating configurations and phases with non-geometric in-terpretations. In Sec. 2.2 we have noted that D (cid:63)a = ∂ xa + i [ h a ( x, u ) (cid:63) , ] (3.9)fulfills al the requirements usually requested from a covariant derivative when acted onan MHS tensor. Using (3.3) we obtain that the background independent formulation of(3.9) is given by D (cid:63)a = i [ e a ( x, u ) (cid:63) , ] . (3.11)As we will see soon, this form is not only more generic but also usually more convenient forperforming calculations. Note that we can write the MHS variation of the MHS vielbeinin a manifestly covariant form as δ ε e a ( x, u ) = D (cid:63)a ε ( x, u ) . (3.12)Having defined the MHS vielbein and covariant derivative a natural object to constructis T ab ( x, u ) ≡ D (cid:63)a e b ( x, u ) = i [ e a ( x, u ) (cid:63) , e b ( x, u )] (3.13)which is an antisymmetric MHS tensor T ab ( x, u ) = − T ba ( x, u ) . (3.14) The requirements are that (i) it is gradient linear, (ii) it maps tensors into tensors, (iii) obeys theLeibniz rule and (iv) it is the inverse of the integral, which in our context means (cid:90) d d x d d u D (cid:63)a A a... ( x, u ) = (boundary terms) . (3.10)Using the properties of the Moyal product listed in Sec. A it is easy to show that the definition (3.11)satisfies all four properties. – 17 –s we show in Sec. 7.1, the Moyal bracket in the spin-2 sector behaves as the Lie bracketof vector fields. It then follows from (3.13) that T ab can be interpreted both as the gener-alized anholonomy and the generalized torsion. We shall refer to it as the MHS torsion.Expanding around flat background (3.3), we get T ab ( x, u ) = ∂ xa h b ( x, u ) − ∂ xb h a ( x, u ) + i [ h a ( x, u ) (cid:63) , h b ( x, u )] (3.16)which is the MHS field strength obtained in the YM-like construction and defined in (2.21).Our convention is to use the symbol T ab in generic situations, and the symbol F ab when(3.3) is meaningful.One could ask what is the HS generalization of the Riemann tensor. In differentialgeometry the Riemann tensor is extracted from the commutator of covariant derivatives.The commutator of MHS covariant derivatives, acting on an arbitrary MHS tensor, gives[ D (cid:63)a , D (cid:63)b ] A c... ( x, u ) = i [ T ab ( x, u ) (cid:63) , A c... ( x, u )] (3.17)thus it is defined by the MHS torsion. As a special case,[ D (cid:63)a , D (cid:63)b ] e c ( x, u ) = D (cid:63)c T ba ( x, u ) (3.18)from which we see that there is no extra independent structure in our formalism correspond-ing to the generalized Riemann tensor. Using the Jacobi identity (A.8) it is straightforwardto show that the MHS torsion satisfies the MHS Bianchi identity, D (cid:63)a T bc ( x, u ) + D (cid:63)b T ca ( x, u ) + D (cid:63)c T ab ( x, u ) = 0 (3.19)which is the same as (2.25).Note that by putitng A c... ( x, u ) = ε ( x, u ) in (3.17) we can write the MHS variation ofthe MHS torsion as δ ε T ab ( x, u ) = [ D (cid:63)a , D (cid:63)b ] ε ( x, u ) (3.20)= D (cid:63)a δ ε e b ( x, u ) − D (cid:63)b δ ε e a ( x, u ) . (3.21) The simplest HS tensor without frame indices in our formalism is g ( x, u ) ≡ e a ( x, u ) (cid:63) e a ( x, u ) . (3.22)For the obvious reason we call it the MHS metric. If we use (3.5) and a similar Taylorexpansion for the MHS metric g ( x, u ) = ∞ (cid:88) s =0 g µ ...µ s ( s ) ( x ) u µ · · · u µ s (3.23) The latter is obvious when we write T ab in the following form T ab = D (cid:63)a e b ( x, u ) − D (cid:63)b e a ( x, u ) − i [ e a ( x, u ) (cid:63) , e b ( x, u )] . (3.15) – 18 –t follows from (3.22) that the s = 2 component is given by g µν (2) ( x ) = η ab e (1) µa ( x ) e (1) νb ( x ) + (HS contributions) (3.24)where every monomial in ”(HS contributions)” contains field(s) e ( n ) µ ...µ n a ( x ) with n ≥ ≥ 3. Up to spin s > g ( x, u ) ≡ u + h ( x, u ) . (3.25)Taylor expanding both sides around u = 0, we get for s (cid:54) = 2 g µ ...µ s ( s ) ( x ) = h µ ...µ s ( s ) ( x ) , s (cid:54) = 2 (3.26)and for s = 2 g µν (2) ( x ) = η µν + h µν (2) ( x ) . (3.27)We see that the MHS field h ( x, u ) measures the deviation from the flat background. Using(3.22), (3.25) and (3.4) we get the MHS variation of h ( x, u ) δ ε h ( x, u ) = 2( u · ∂ x ) ε ( x, u ) + i [ h ( x, u ) (cid:63) , ε ( x, u )] (3.28)which is exactly the variation found in [25, 26] in the analysis of MHS symmetries of the freeKlein-Gordon field linearly coupled to the infinite tower of spacetime HS fields. In [25, 26]it was argued that h ( x, u ) should be a composite field, and here we made it explicit. In Sec.5.1 we show that h ( x, u ) indeed is the field which couples minimally to the Klein-Gordonfield in the MHS formalism. Using (3.22), (3.25) and (3.3) we obtain h ( x, u ) = 2 u a h a ( x, u ) + h a ( x, u ) (cid:63) h a ( x, u ) . (3.29)In particular the s = 0 component of h ( x, u ), which provides seagull vertices for s ≥ h (0) ( x ) = h a ( x, u ) (cid:63) h a ( x, u ) (cid:12)(cid:12)(cid:12) u =0 . (3.30)The MHS covariant derivative is not metric-compatible since Q a ( x, u ) ≡ D (cid:63)a g ( x, u ) = i [ e a ( x, u ) (cid:63) , g ( x, u )] (3.31)is generally not vanishing. We refer to the HS tensor Q a ( x, u ) as the MHS nonmetricitytensor. The underlying geometry in our construction appears not to be of the Riemann-Cartan type. Note that the MHS nonmetricity tensor (3.31) can be written as Q a ( x, u ) = { e b ( x, u ) (cid:63) , T ab ( x, u ) } (3.32)– 19 –.e., it is completely determined by the MHS torsion.To summarize, the geometry emerging in the MHS theory has all fundamental tensors(torsion, Riemann tensor and nonmetricity) non-vanishing. While the geometry may lookexotic at a first glance, it is in fact closely related to the teleparallel geometry. In Sec.7 we study in detail the spin-2 sector, and show that the emergent linear connection isopposite to the Weitzenb¨ock connection. In the construction above we have drawn analogies with differential geometry. It is thennatural to ask can the MHS formalism accommodate the covariance under local Lorentztransformations of the vielbein E aµ ( x ) → Λ ab ( x ) E bµ ( x ) . (3.33)If we try to generalize this to the MHS vielbein e a ( x, u ) → Λ ab ( x ) e b ( x, u ) (3.34)we obtain that none of the HS objects we constructed (metric, torsion, MHSYM action)transform covariantly. Covariance is present only for rigid transformations, Λ ab ( x ) = Λ ab .The apparent failure of local Lorentz covariance is a consequence of not taking properlyinto account the fact that our construction so far was based on a special choice of theinertial frame. Local Lorentz transformations turn inertial frames into non-inertial, so wehave to extend our formalism to non-inertial frames. To accommodate for this we have tointroduce the inertial spin connection A abµ ( x ) in the ambient Minkowski spacetime anduse the covariant derivative instead of ordinary one ∂ x → D = ∂ x + A (3.35)inside the Moyal products. Note that the inertial spin connection is flat, which means that A abµ = Λ ac ∂ µ Λ cb and [ D µ , D ν ] = 0 . (3.36)This guarantees that Moyal products of the MHS tensors transform covariantly under localLorentz transformations, e.g., T ab ( x, u ) → Λ ac ( x )Λ bd ( x ) T cd ( x, u ) (3.37)from which follows that the MHSYM theory is locally Lorentz covariant when properlyformulated. It is, though, not locally Lorentz invariant because one has to introduce theinertial connection to achieve covariance. To use the terminology of [32]. One can say that the geometry induced by MHS formalism is teleparallel in disguise. – 20 – . Model building: MHS gauge sector Having developed the MHS formalism and having it cast into a more general form, letus turn back to the question of constructing candidates for the theories based on theMHS symmetry. As for degrees of freedom, we expect to have an MHS gauge sectordescribed by the MHS vielbein e a ( x, u ), which is an MHS tensor (transforms in the adjointrepresentation), and a matter sector spanned by a set of matter fields collectively denotedby ψ which can be in different representations of the MHS symmetry (discussed in moredetail in Sec. 5). Correspondingly, the action is a sum of two parts, S [ e, ψ ] = S hs [ e ] + S m [ ψ, e ] (4.1)one describing the pure MHS gauge theory, and the other the matter sector and its couplingto the MHS vielbein. We assume that the action is weakly non-local in the master space,by which we mean that both parts in (4.1) can be written in terms of a master spaceLagrangian S hs , m = (cid:90) d d x d d u L hs , m ( x, u ) . (4.2)To keep it simple here we only consider Lagrangian terms which are (Moyal) polynomialsmanifestly covariant under the MHS symmetry. This means that L hs ( x, u ) is an MHStensor, while L m ( x, u ) may be an MHS scalar or an MHS tensor, depending on the mattercontent. In both cases MHS invariance of the action δ ε S [ e ] = 0 (4.3)is manifestly guaranteed. In case of a master space Lagrangian being an MHS tensor, in-variance of the action is a consequence of the trace property of the Moyal product (A.12). In addition, we assume that the Lagrangian is a Lorentz scalar.From (4.1) it follows that the EoM in the MHS gauge sector is0 = F a hs ( x, u ) + J a m ( x, u ) (4.4)where F a hs ( x, u ) = δS hs [ e ] δe a ( x, u ) , J a m ( x, u ) = δS m [ ψ, e ] δe a ( x, u ) (4.5)To obtain the EoM, the trace property of the Moyal product can be used, since variationsof the fields must vanish at the boundary. This makes it equal whether the functionalderivative is ”left” or ”right”. One should in general distinguish between δ L A [ e, ψ ] = (cid:90) d d x d d u δe a (cid:63) δ L Aδe a , δ R A [ e, ψ ] = (cid:90) d d x d d u δ R Aδe a (cid:63) δe a (4.6) Assuming proper gauge transformations for which boundary terms vanish. – 21 –ut on places where it does not make a difference, we will omit an explicit subscript.The EoM in matter sector are given by0 = δS m [ ψ, e ] δψ . (4.7)Using (3.12), (3.10) and (A.12) the MHS variation of S hs can be written as δ ε S hs [ e ] = (cid:90) d d x d d u F a hs ( x, u ) δ ε e a ( x, u )= (cid:90) d d x d d u F a hs ( x, u ) D (cid:63)a ε ( x, u )= − (cid:90) d d x d d u D (cid:63)a F a hs ( x, u ) ε ( x, u ) . (4.8)Then, from (4.3) we get the off-shell identity D (cid:63)a F a hs ( x, u ) = 0 . (4.9)Applying D (cid:63)a on EoM (4.4) and using (4.9) we get D (cid:63)a J a m ( x, u ) = 0 (4.10)which states that the matter master current is (on-shell) covariantly conserved. Let us first analyze the MHS gauge sector. The matter sector is studied in Sec. 5. Thesimplest acceptable Lagrangian term that is dynamical and has the flat vacuum e a ( x, u ) = u a as a solution of the EoM it generates is L hs ( x, u ) = 14 g T ab ( x, u ) (cid:63) T ba ( x, u ) ≡ L ym ( x, u ) . (4.11)This is the Lagrangian of the MHSYM theory already introduced in Sec. 2.3. The corre-sponding action is S ym [ e ] = 14 g (cid:90) d d x d d u T ab ( x, u ) (cid:63) T ba ( x, u )= 14 g (cid:90) d d x d d u [ e a (cid:63) , e b ] (cid:63) [ e a (cid:63) , e b ] . (4.12)Under a generic variation δe a ( x, u ) that vanishes on the boundary of the integration volume,the MHSYM action behaves as δS ym [ e ] = 12 g (cid:90) d d x d d u {D (cid:63)b T ba ( x, u ) (cid:63) , δe a ( x, u ) } = 1 g (cid:90) d d x d d u D (cid:63)b T ba ( x, u ) δe a ( x, u ) (4.13) It is assumed here that the MHS variation is a proper gauge transformation, in which case boundaryterms in (3.10) and (A.12) vanish. – 22 –hich means that its contribution to the EoM is F a ym ( x, u ) = 1 g D (cid:63)b T ba ( x, u )= 1 g (cid:2) e b ( x, u ) (cid:63) , [ e a ( x, u ) (cid:63) , e b ( x, u )] (cid:3) . (4.14)The EoM of the pure MHSYM theory is then D (cid:63)b T ba ( x, u ) = 0 . (4.15)It is important to observe that the MHSYM theory is classically a scale-free theory fromthe master space perspective. If one takes the MHS vielbein to be dimensionless, thenthe pure MHSYM coupling constant g ym is also dimensionless. Moreover, as the couplingconstant can be absorbed by rescaling the MHS vielbein, it is a theory without an intrinsiccoupling constant. We have seen that in the YM formulation the theory was not scale-free,so how is this possible? The answer is that the scale was introduced by a choice of theempty flat vacuum. In this normalization, it is given by e a ( x, u ) vac = (cid:96) h u a . (4.16)Note that the scale (cid:96) h can be changed by ”canonical” scale transformations e (cid:48) a ( x, u ) = e a ( λx, u/λ ) (4.17)which form a subgroup of MHS transformations. The Minkowski vacuum spontaneouslybreaks a part of the MHS symmetry. A few words are in order about vacua in MHSYM theory. They are solutions of theEoM which satisfy the condition T ab ( x, u ) = 0 (4.18)We have already mentioned that the flat configuration e a ( x, u ) = u a is, at least from theclassical viewpoint, a well-defined Lorentz-invariant vacuum. However, it is not the casethat all configurations satisfying (4.18) are MHS gauge equivalent to the flat vacuum. Anobvious example is an ”empty” configuration e a ( x, u ) = 0 which is a fixed point of MHSgauge transformations. To obtain some insight into the structure of vacua, let us examinethe vacua that are of the form e a ( x, u ) = M aµ u µ (4.19)where M are arbitrary constant real d × d matrices. MHS transformations preserving theshape of these configurations have the gauge parameter master field of the form E Λ ( x, u ) = x µ Λ µν u ν (4.20) There are other mechanisms which might introduce a scale in the MHSYM theory. In Sec. 2.5 we haveseen that this may happen when the theory is written in a purely spacetime form. In Sec. 4.4 we show that all conserved charges, including the energy-momentum tensor, vanish forconfigurations satisfying (4.18). – 23 –here Λ is again an arbitrary constant real d × d matrix. Using (B.21) it is easy to showthat such an MHS transformation when acting on a vacuum (4.19) produces the same typeof vacuum with matrix M Λ given by M Λ = M e Λ . (4.21)where matrix multiplication is assumed. A corollary is that two vacua of the type (4.19),which are defined with matrices of different rank, are MHS gauge inequivalent.This analysis sugests that the MHSYM theory contains different phases. The flatvacuum e a ( x, u ) = u a describes empty flat (Minkowski) background and defines a geometricphase. When expanded around the flat vacuum solution as in (3.3) the linear part ofthe EoM is second-order in spacetime derivatives, and in this phase the theory has aperturbative regime (in the coupling constant). In contrast, the configuration e a ( x, u ) = 0does not have an emergent regular geometric description and defines a non-perturbativestrongly-coupled unbroken phase (it is the only vacuum with a trivial orbit with respectto the MHS transformations). Let us now analyze possible generalizations of the MHSYM theory by adding additionalterms to the action. There is just one lower-dimensional term allowed by the MHS sym-metry for a generic number of spacetime dimensions d , L ( x, u ) = λ g ( x, u ) = λ e a ( x, u ) (cid:63) e a ( x, u ) (4.22)which produces the following EoM contribution F a ( x, u ) = λ e a ( x, u ) . (4.23)This term is not dynamical and has the appearance of a mass term, but in the geometricphase it behaves as a generalized cosmological constant term. When added to the MHSYMaction, the flat configuration e a ( x, u ) = δ µa u µ is no longer solution of EoM, so this termchanges the vacuum in the geometric phase.For general d there is one additional independent term, which is of the same dimensionas F , L ( x, u ) = λ g ( x, u ) (cid:63) g ( x, u ) (4.24)which would contribute the following EoM term F a ( x, u ) = λ { g ( x, u ) (cid:63) , e a ( x, u ) } . (4.25)Adding this term also removes the flat configuration e a ( x, u ) = u a from the solution space,but there is more. This term is dynamical and so it is interesting to see how it contributesto the linearized EoM in the geometric phase. If we write e a ( x, u ) = e (0) a ( x, u ) + h a ( x, u ) (4.26) By a regular geometric description we mean a description based on the non-degenerate emergent metrictensor or vielbein. – 24 –here e (0) a ( x, u ) is the solution of the EoM (a background), it is straightforward to show F a ( x, u ) = λ (cid:16) { g (0) ( x, u ) (cid:63) , h a ( x, u ) } + (cid:8) { e (0) b ( x, u ) (cid:63) , h b ( x, u ) } (cid:63) , e a (0) ( x, u ) (cid:9) + O ( h ) (cid:17) . In case of the simplest type of background belonging to the geometric phase, e (0) a ( x, u ) = e (0) µa ( x ) u µ (4.27)the contribution to the linearized EoM is at most second-order in spacetime derivatives. Ifthe background is not of this type, it must have an infinite Taylor expansion in u and as aconsequence its contribution to the linearized EoM have an infinite number of terms withno bounds on order in spacetime derivatives.Similarly, we can construct potential Lagrangian terms by taking higher polynomialsin the MHS vielbein, all of them having the higher dimension in the geometric phase in d > d = 2 r , there exists a Lorentz-scalarMHS tensor P r ( x, u ) = (cid:15) a b ...a r b r T a b ( x, u ) (cid:63) · · · (cid:63) T a r b r ( x, u ) (4.28)where (cid:15) a ...a d is the Levi-Civita symbol. As it is a generalization of the Chern term we referto it as the MHS Chern tensor. It can be used to construct Lagrangian terms, the simplestbeing L P ( x, u ) = λ P P r ( x, u ) . (4.29)Note that this term is parity-odd. In d = 4 it has the same dimension as the MHSYMterm. It is not hard to show that it is a topological term, P r ( x, u ) = D (cid:63)a (cid:16) (cid:15) a b ...a r b r e b ( x, u ) (cid:63) T a b ( x, u ) (cid:63) · · · (cid:63) T a r b r ( x, u ) (cid:17) (4.30)where we used (3.13) and (3.19). As a consequence it does not contribute to the (bulk)EoM, but may possibly lead to non-perturbative effects if the theory contains topologicallynon-trivial configurations analogous to instantons.If the number of spacetime dimensions is odd, d = 2 r +1, we can construct the followingLagrangian term L CS ( x, u ) = (cid:15) ab c ...b r c r { e a ( x, u ) (cid:63) , T b c ( x, u ) (cid:63) · · · (cid:63) T b r c r ( x, u ) } . (4.31)This tensor is parity-odd as well. From (4.30) it follows that it can be obtained as aboundary term from the MHS Chern term. It is thus natural to call it the MHS Chern-Simons tensor. It produces the following contribution to the EoM F a CS ( x, u ) = d (cid:15) ab c ...b r c r T b c ( x, u ) (cid:63) · · · (cid:63) T b n c n ( x, u ) (4.32)which we call the MHS Cotton tensor. In d = 3 the MHS Chern-Simons term has a lowerdimension than the MHSYM term so it dominates in the IR regime. As the EoM of thepure MHS Chern-Simons theory in d = 3 is (cid:15) abc T bc ( x, u ) = 0 ⇒ T bc ( x, u ) = 0 (4.33)which shows that the MHSCS theory is topological.– 25 – .4 Conservation laws and conserved charges in MHSYM theory4.4.1 Covariant vs. non-covariant conservation laws Here we would like to examine in more detail the question of conservation laws and con-served charges in MHS theories, taking MHSYM theory as an example. As is well-known ina Lorentz covariant theory a current satisfying the continuity equation on-shell (i.e., withEoM applied) ∂ xµ J µ ( x ) . = 0 (4.34)encodes a local conservation of charge defined by Q V ( t ) = (cid:90) V d d − x J ( x ) . (4.35)We refer to (4.34) as the conservation law. There are situations, especially when localsymmetries are present, in which the total charge identically vanishes for all physical con-figurations. In the case of local symmetries such trivial charges are connected to theproper gauge transformations. Gauge transformations usually contain a subgroup of im-proper gauge transformations that are connected to non-trivial conserved charges. As theconserved charges in gauge theories can be written as asymptotic integrals, improper gaugesymmetries are recognized by their ”soft” fall-off in the limit r → ∞ . Here we are inter-ested in extracting conservation laws and non-trivial charges in the framework of the MHSsymmetry.In theories with local symmetries covariant conservation laws appear naturally. In thecase of MHS symmetry they are defined in the master space and are of the form D (cid:63)a J a ( x, u ) . = 0 (4.36)where D (cid:63)a is the MHS covariant derivative and J a ( x, u ) is an MHS (tensor) current. Anexample is the matter current J m ( x, u ), defined in (4.5), which is the source in the MHSvielbein EoM. In ordinary theories with non-commutative local symmetries, such as YMtheory and GR, a covariant conservation law does not directly imply conserved charges.Let us take as an example the matter energy-momentum tensor in GR, which is covariantlyconserved. This does not imply conservation of the matter energy and momentum. Wecan construct the corresponding energy-momentum pseudotensor which is conserved inthe sense of (4.34), but which also contains a contribution from the spin-2 sector. In thecase of the MHS symmetry, the covariant conservation (4.36) automatically generates theconservation law (4.34). This is because the MHS covariant derivative is by the definition aMoyal commutator and every Moyal commutator is a total divergence in the master space0 . = D (cid:63)a J a ( x, u ) = ∂ xµ A µ J ( x, u ) + ∂ µu B J µ ( x, u ) . (4.37) For equalities valid on-shell we use the symbol ” . =”. The equalities valid for generic field configurationssatisfying proper boundary conditions are denoted by the simple equality sign ”=”. – 26 –ntegrating both sides of (4.37) over the auxiliary space and assuming that boundary termsin the auxiliary space are zero, we conclude J µ ( x ) ≡ (cid:90) d d u A µ J ( x, u ) (4.38)is conserved. While covariantly conserved master field currents can usually be writtenin closed and compact expressions, we see from (4.37) and the structure of the Moyalproduct that physically conserved spacetime currents J µ ( x ) may have a rather involvedand cumbersome form when written explicitly.One example is the presence of the conserved matter charge in the MHS theory basedon U (1) spin-1 subgroup that is generated by the constant improper MHS transformation ε ( x, u ) = ε = const. The MHS vielbein is neutral (invariant) under its action and so doesnot contribute to the U (1) charge. It is the only conserved charge with such propertiesthat is generated by the MHS symmetry. We now pass to a detailed study of conservationlaws in the case of the MHSYM theory. In the geometric phase conserved charges are directly obtained from the EoM following thestandard procedure used in ordinary YM theory and GR. Let us demonstrate this in thecase of MHSYM theory coupled to the matter whose EoM is1 g D (cid:63)b F ba ( x, u ) . = J a m ( x, u ) . (4.39)We now use (3.3) and move all nonlinear terms in h a ( x, u ) to the right hand side, obtaining1 g ∂ xb F ba (1) ( x, u ) . = ˜ J a ( x, u ) (4.40)where F ab (1) ( x, u ) = ∂ xa h b ( x, u ) − ∂ xb h a ( x, u ) (4.41)and ˜ J a ( x, u ) = J a m ( x, u ) − ig (cid:16) h b (cid:63) , ∂ xb h a ] − [ h b (cid:63) , ∂ ax h b ] + [ ∂ xb h b (cid:63) , h a ] (cid:17) −− g (cid:2) h b (cid:63) , [ h a (cid:63) , h b ] (cid:3) . (4.42)Taking a = 0 in (4.40) we get ˜ J ( x, u ) . = 1 g ∂ xj F j ( x, u ) (4.43)which is the MHS Gauss’s law, while taking the spacetime divergence of Eq. (4.40) yieldsthe continuity equation ∂ xa ˜ J a ( x, u ) . = 0 (4.44)– 27 –howing that the current ˜ J ( x, u ) is conserved in the master space (before integratingover auxiliary space). From Gauss’s law (4.43) it follows that the corresponding locallyconserved charge can be written as a surface space integral˜ Q V ( t, u ) = (cid:90) V d d − x ˜ J ( x, u ) (4.45) . = (cid:90) V d d − x ∂ xj F j ( x, u ) . = (cid:73) S ( V ) d d − a j F j ( x, u ) (4.46)which is Gauss’s law in the integral form. Equation (4.44) encodes a tower of conservedcharges. To see this, we Taylor expand both sides in auxiliary coordinates around u = 0to obtain an infinite set of conserved charges˜ Q µ ··· µ n . = (cid:73) d d − a j F j µ ··· µ n (1) ( x ) (4.47)where F j ( x, u ) = ∞ (cid:88) n =0 F j µ ··· µ n (1) ( x ) u µ · · · u µ n . (4.48) Let us now construct conservation laws by applying the Noether method. For simplicity,we restrict ourselves to the pure MHSYM theory. First, using the MHSYM EoM D (cid:63)a T ab ( x, u ) . = 0 (4.49)we conclude that a generic on-shell variation of the MHSYM master Lagrangian can bewritten as δL ym ( x, u ) . = − g D (cid:63)a { T ab ( x, u ) (cid:63) , δe b ( x, u ) } . (4.50)If we take the variation to be an MHS variation δ ε e a ( x, u ) = D (cid:63)a ε ( x, u ) (4.51)we see that the Lagrangian transforms as an MHS tensor, δ ε L ( x, u ) = i [ L ( x, u ) (cid:63) , ε ( x, u )] . (4.52)We now use (4.50) and (4.52) to write0 . = 12 g (cid:18) D (cid:63)a (cid:8) T ab ( x, u ) (cid:63) , D (cid:63)b ε ( x, u ) (cid:9) − i (cid:2) T ab ( x, u ) (cid:63) T ab ( x, u ) (cid:63) , ε ( x, u ) (cid:3)(cid:19) . (4.53)As both terms on the right hand side are Moyal commutators the equation has the form0 . = ∂ xµ A µε ( x, u ) + ∂ µu B εµ ( x, u ) . (4.54)– 28 –gain, integrating over the auxiliary space and assuming that all boundary terms vanish,we obtain a standard conservation law (in the form of the continuity equation) ∂ xµ J µε ( x ) . = 0 , J µε ( x ) ≡ (cid:90) d d u A µε ( x, u ) . (4.55)The corresponding conserved charges Q ε = (cid:90) d d − x J ε ( x ) (4.56)are non-trivial only for a small class of MHS parameters corresponding to improper gaugetransformations. It is expected that rigid variations ε ( x, u ) = ε ( u ), which we can expandas ε ( u ) = ∞ (cid:88) n =0 ξ µ ··· µ n u µ . . . u µ n (4.57)with ξ µ ··· µ n constant and completely symmetric, fall into this class. Let us now focus onthem. We have already analyzed the n = 0 case, which does not affect MHS vielbein andso (4.53) becomes trivial (0 . = 0). Let us now consider n ≥ u a → e a ( x, u ), ε ( x, u ) = ∞ (cid:88) n =0 ξ a ··· a n e a ( x, u ) (cid:63) . . . (cid:63) e a n ( x, u ) (4.58)where ξ a ··· a n is a constant tensor with symmetries guaranteeing reality of the MHS pa-rameter ε ( x, u ). Now we use this in (4.53) where we want to write the second term onthe right hand side as a covariant divergence (the first term is already in this form). Wedo this by using the identity[ A (cid:63) . . . (cid:63) A n (cid:63) , X ] = n (cid:88) j =1 [ A j (cid:63) , A j +1 (cid:63) . . . (cid:63) A n (cid:63) X (cid:63) A (cid:63) . . . (cid:63) A j − ] (4.59)valid for generic master space functions A j ( x, u ) and X ( x, u ), to write i (cid:104) e a (cid:63) . . . (cid:63) e a n (cid:63) , T ab (cid:63) T ab (cid:105) = n (cid:88) j =1 D (cid:63)a j (cid:16) e a j +1 (cid:63) . . . (cid:63) e a n (cid:63) T ab (cid:63) T ab (cid:63) e a (cid:63) . . . (cid:63) e a j − (cid:17) . (4.60) We do not claim that (4.57) is a complete set of improper MHS transformations. Quite the contrary,the experience from Maxwell’s theory and GR teaches us that there should be a much larger set of conservedcharges. We leave the more complete analysis of asymptotic symmetries to future work. We are motivated by Jackiw’s covariantisation trick [33]. For ε ( x, u ) in (4.58) to be real it has to be expressible purely in terms of Moyal commutators and/oranticommutators. If the number of anticommutators is odd, the parameter ξ a ··· a n is imaginary. When ξ a ··· a n is completely symmetric the expression (4.58) is a covariantization of (4.57). – 29 –sing this in (4.53) we obtain D (cid:63)a T aξ ( x, u ) . = 0 (4.61)with the covariantly conserved currents given by T aξ = ξ b ··· b n (cid:18) { T ac (cid:63) , D (cid:63)c ( e b (cid:63) . . . (cid:63) e b n ) } ++ 12 n (cid:88) j =1 δ ab j e b j +1 (cid:63) . . . (cid:63) e b n (cid:63) T cd (cid:63) T cd (cid:63) e b (cid:63) . . . (cid:63) e b j − (cid:19) . (4.62)The currents related to totally symmetric ξ b ··· b n T ab ··· b n = { T ac (cid:63) , D (cid:63)c (cid:0) e ( b (cid:63) . . . (cid:63) e b n ) (cid:1) } ++ 12 n (cid:88) j =1 δ a ( b j e b j +1 (cid:63) . . . (cid:63) e b n (cid:63) T | cd | (cid:63) T cd (cid:63) e b (cid:63) . . . (cid:63) e b j − ) (4.63)play a special role as they are obtained by covariantizing the rigid MHS symmetries. An-other reason for its special status is that they have the softest behavior at spatial infinity( r → ∞ ) in the geometric phase, which means that they are main candidates for produc-ing non-trivial charges. Since the corresponding conserved charges are described by totallysymmetric tensors, they should be related to the charges (4.47) obtained by the previousmethod.The n = 1 case in (4.57) corresponds to spacetime translations, leading to energy-momentum conservation, and is therefore of special importance. Fixing n = 1 in (4.63) weget the covariant master energy-momentum tensor T ab ( x, u ) = { T ac (cid:63) , T bc } − η ab T cd (cid:63) T cd (4.64)which is symmetric, and in d = 4 traceless. As expected, the obtained expression has thesame form as in non-commutative field theories [34, 35, 36, 37]. There is another way to represent MHS theories discussed above, which uses the connectionbetween the Moyal product and the Weyl-ordered operator product well known from thephase space formulation of a quantized particle. If we define the Hilbert space H with thecomplete set of operators ˆ x µ , ˆ u µ satisfying commutation relations[ˆ x µ , ˆ u ν ] = iδ µν , [ˆ x µ , ˆ x ν ] = 0 = [ˆ u µ , ˆ u ν ] (4.65)there is a bijective map (for a fixed ordering scheme) between the set of linear operatorson H and the set of functions on the master space, i.e., End( H ) (cid:51) ˆ O ←→ O ( x, u ) ∈ C ∞ ( M × U ) . (4.66) One usually refers to O ( x, u ) as the symbol of operator ˆ O . – 30 –f one defines a product of two operators with the (symmetric) Weyl ordering of x and u ,its pull-back to the master space (through the map (4.66)) defines the Moyal product ofcorresponding master space functions (symbols)ˆ O ˆ O ←→ O ( x, u ) (cid:63) O ( x, u ) . (4.67)This map is such that the trace of an operator is given by the integral of the correspondingfunction over the master spacetr( ˆ O ) = (cid:90) d d x d d u (2 π ) d O ( x, u ) . (4.68)Using this map it is now evident that all models for MHS theories can be written in thisoperator language, and therefore as a type of matrix models. For example, the MHSYMtheory can be written as S ym = − (2 π ) d g ym tr (cid:16) [ˆ e a , ˆ e b ][ˆ e a , ˆ e b ] (cid:17) (4.69)where ˆ e a are operators on H , components of a vector in the SO (1 , d − 1) representation.The MHS symmetry is now represented by unitary linear operatorsˆ U E = exp( − i ˆ E ) (4.70)which act on the MHS vielbein operator asˆ e a → ˆ U E ˆ e a ˆ U †E (4.71)with all operator products defined with symmetric Weyl ordering. 5. Model building: matter sector Now we turn to the question of consistent coupling of matter fields to the MHS vielbein.We investigate three possible routes for achieving this. There is a way to couple matter fields to the MHS vielbein that naturally accommodatesthe operator (matrix model) formulation introduced in Sec. 4.5 and has the additionalbonus of describing matter by purely spacetime fields. The latter means that we do nothave to increase the number of degrees of freedom in the matter sector. This approach wasoriginally introduced in [25, 26] for the case of the Klein-Gordon field and in [27, 24] forthe case of the Dirac field.The starting point in the construction is the observation, already noted in Sec. 4.5, thatthe MHS vielbein can be represented by a linear operator acting on a particular Hilbertspace on which ”position operators” ˆ x µ and their conjugate momenta ˆ u µ are represented. For an explicit proof that the Moyal product is a matrix product see [38]. – 31 –he matter configuration φ is represented by a state vector | α φ (cid:105) in this Hilbert space.Matter fields are then simply wave functions in the x -representation φ r ( x ) = (cid:104) x, r | α φ (cid:105) (5.1)where the index r allows for a non-trivial finite representation of the Lorentz group andpossibly also of some internal group of symmetries. The classical action for a free matterfield can simply be written as a particular expectation value S (0) m [ φ ] = (cid:104) α φ | ˆ K (ˆ u ) | α φ (cid:105) (5.2)where ˆ K (ˆ u ) is a linear operator depending on spins and masses of matter fields. For bosonic(fermionic) fields it is quadratic (linear) in u . In particular, for the complex Klein-Gordonfield defined on flat spacetime, ˆ K s (ˆ u ) = η ab ˆ u a ˆ u b − m (5.3)while for the Dirac field, ˆ K D (ˆ u ) = − γ ( γ a ˆ u a + m ) (5.4)where γ a are Dirac matrices and where again the flat inertial vielbein ˆ E µa = δ µa is used.To prove this, one uses the fact that ˆ u µ act on wave functions in the x -representation asˆ u µ → − i∂ µ .As described in Sec. 4.5, the MHS transformations are represented by unitary linearoperators ˆ U E = exp( − i ˆ E (ˆ x, ˆ u )), | α φ (cid:105) E = ˆ U E | α φ (cid:105) . (5.5)where ˆ E † = ˆ E , and where a symmetric ordering of ˆ x µ and ˆ u µ is assumed. The free fieldactions are invariant under the rigid MHS transformations for which ˆ E = ˆ E (ˆ u ). Theminimal way to have matter actions symmetric under the local MHS transformations is tomake the substitution ˆ u a → ˆ e a = ⇒ ˆ K (ˆ u ) → ˆ K (ˆ e ) (5.6)in (5.2) to obtain S m [ φ, e ] = (cid:104) α φ | ˆ K (ˆ e ) | α φ (cid:105) (5.7)which is the action for matter minimally coupled to the MHS vielbein. The (local) MHSsymmetry is a straightforward consequence of (4.71) and (5.5). As before, we can write matter actions in the Moyal product language. As reviewedin Append. C, using the phase space formalism we can write minimally coupled matteractions in the form S m [ φ, e ] = (cid:90) d d x d d u Tr( W φ ( x, u ) (cid:63) K ( e ( x, u )) (5.8) Again, putting aside subtleties that appear for improper gauge transformations. – 32 –here the trace is performed over Lorentz and internal indices carried by matter fields andthe Wigner function can be written as( W φ ( x, u )) rs = φ r ( x ) (cid:63) δ d ( u ) (cid:63) φ s ( x ) ∗ . (5.9)While it follows from (5.5) that matter fields transform under MHS symmetry as φ r ( x ) E = (cid:104) x, r | ˆ U E | α φ (cid:105) (5.10)the Wigner function transforms as (3.1), i.e., it is an MHS tensor. It then follows thatLagrangians for minimally coupled matter, defined by (5.8), are also MHS tensors. Notethat coupling the matter this way explicitly breaks the translational symmetry in theauxiliary space. The matter action is also formally defined for non-geometric configurations.To understand the nature of the minimal coupling in MHS theory in the geometricphase, we first use (3.3) to separate free and interacting parts of the action by writing K ( e ( x, u )) = K ( u ) + K int ( h ( x, u ); u ) (5.11)where the explicit dependence of K int on u is present only for bosonic fields. Couplingto the MHS potential is linear for fermionic matter and quadratic for bosonic matter.Substituting this into (5.8) we get S m [ φ, h ] = S (0) m [ φ ] + S (int) m [ φ, h ] (5.12)where by definition S (0) m is the action for the free field and the interaction term can bewritten as S (int) m [ φ, h ] = (cid:90) d d x d d u Tr (cid:0) W φ ( x, u ) (cid:63) K int ( x, u ) (cid:1) = (cid:90) d d x d d u (cid:0) φ ∗ r ( x ) (cid:63) K rs int ( x, u ) (cid:63) φ s ( x ) (cid:1) δ d ( u ) . (5.13)This form suggests that Taylor expanding in the auxiliary space K rs int ( x, u ) = Γ rs · ∞ (cid:88) n =0 K µ ··· µ n ( n ) ( x ) u µ . . . u µ n (5.14)would be the natural thing to do. Indeed, using this we get S (int) m [ φ, h ] = ∞ (cid:88) n =0 (cid:90) d d x J ( n ) ( x ) · K ( n ) ( x ) (5.15)where J ( n ) ( x ) are the conserved currents in the free theory, related to the rigid HS sym-metries (2.1), in a form we refer to as ”simple”. In the equations above the simbol ” · ”denotes a contraction of all indices that are not explicitly written.Let us demonstrate the above construction on two important examples of matter, Diracand Klein-Gordon fields. In case of the complex Klein-Gordon field ϕ ( x ) we have K ( x, u ) = g ( x, u ) − m (5.16)– 33 –here g ( x, u ) is the MHS metric defined in (3.22). Using (3.25) and (5.3) we obtain K int ( x, u ) = h ( x, u ) (5.17)where h ( x, u ) is a composite object obtained from the MHS potential by (3.29). In thiscase Γ rs = 1, therefore the interacting part of the action (5.15) is given by S (int) m [ ϕ, h ] = ∞ (cid:88) s =0 (cid:90) d d x J ( s ) µ ··· µ s ( x ) h µ ··· µ s ( s ) ( x ) . (5.18)It can be shown (see [25, 26] and Append. C) that the spin- s simple current is J ( s ) µ ··· µ s ( x ) = i s s ϕ ( x ) ∗ ↔ ∂ µ · · · ↔ ∂ µ s ϕ ( x ) . (5.19)In case of the Dirac field ψ ( x ) we have K ( x, u ) = − γ (cid:0) γ a e a ( x, u ) + M (cid:1) (5.20)which means that K int ( x, u ) = − γ γ a h a ( x, u ) . (5.21)We see that (Γ a ) rs = − ( γ γ a ) rs . Taylor expanding the MHS potential h a ( x, u ) as in (3.3)we get S (int) m [ ψ, h ] = ∞ (cid:88) n =0 (cid:90) d d x J a ( n ) µ ··· µ n ( x ) h ( n ) µ ··· µ n a ( x ) (5.22)where the HS currents ([27, 24] and Append. C) are given by J a ( n ) µ ··· µ n ( x ) = i n n ¯ ψ ( x ) γ a ↔ ∂ µ · · · ↔ ∂ µ n ψ ( x ) . (5.23)Since we can write the Wigner function for off-diagonal matrix elements (cid:104) α | ( · · · ) | β (cid:105) ,this formalism can accommodate Lagrangian terms quadratic and non-diagonal in space-time matter fields. Another way to couple matter in an MHS symmetric way is to describe it by master fields, φ ( x, u ). This type of matter is necessary if one wants to introduce supersymmetry. Thesimplest representations are adjoint and fundamental. Matter in the adjoint representation is described by MHS tensors, which means that theMHS covariant derivative is given by D (cid:63)a φ ( x, u ) = i [ e a ( x, u ) (cid:63) , φ ( x, u )] . (5.24)– 34 –n case of minimal coupling the action is then constructed in the standard way, by substi-tuting ∂ xa → D (cid:63)a . This type of matter shares some properties with the MHS gauge sectoraction: the master fields are real, actions are also defined in the non-geometric phases, andthey can be written in the form of matrix models.Let us apply this to the free Majorana spin-1/2 field ψ ( x, u ). The MHS action forminimal coupling is S M [ ψ, e ] = 12 (cid:90) d d x d d u ¯ ψ ( x, u ) (cid:63) ( iγ a D (cid:63)a − M ) ψ ( x, u ) . (5.25)In the operator formulation this is S M [ ψ, e ] = − (2 π ) d (cid:16) ¯ˆ ψ (cid:0) γ a [ˆ e a , ˆ ψ ] + M ˆ ψ (cid:1)(cid:17) . (5.26)Putting together the MHSYM action and the action for one minimally coupled masslessMajorana field one gets the simplest supersymmetric MHS theory.In case of the real scalar field the minimal coupling is described by the following action S s [ ϕ, h ] = (cid:90) d d x d d u (cid:2) η ab ( D (cid:63)a ϕ ) ∗ (cid:63) D (cid:63)b ϕ − m ϕ ∗ (cid:63) ϕ − V (cid:63) ( ϕ ∗ (cid:63) ϕ ) (cid:3) . (5.27) Matter in the fundamental representation of MHS symmetry transforms as φ E ( x, u ) = e − i E ( x,u ) (cid:63) (cid:63) φ ( x, u ) (5.28)from which it follows φ E ( x, u ) ∗ = φ ( x, u ) ∗ (cid:63) e i E ( x,u ) (cid:63) (5.29)In the YM-like formalism the MHS covariant derivative in the fundamental representationis D (cid:63)a φ = ∂ xa φ + i h a (cid:63) φ . (5.30)It is simple to check the MHS covariance( D (cid:63)a φ ) E = e − i E (cid:63) (cid:63) D (cid:63)a φ . (5.31)MHS invariants are constructed by Moyal-sandwiching MHS tensors between φ ∗ or( D (cid:63)a φ ) ∗ from the left and φ or D (cid:63)a φ from the right. Using these invariants we can producecandidates for Lagrangian terms, with minimal coupling defined in the usual manner bya substitution of MHS covariant derivative for partial spacetime derivative in free fieldactions. When discussing Majorana spinors, d is assumed to be such that it allows for their existence. – 35 –owever, minimal prescription based on (5.30) can be defined only in the geometricphase. In addition, it is not natural from the the perspective of a matrix model formulation.These shortfalls can be avoided by using the prescription ∂ xa φ ( x, u ) → i e a ( x, u ) (cid:63) φ ( x, u ) . (5.32)From the relation i e a ( x, u ) (cid:63) φ ( x, u ) = iu a φ ( x, u ) + 12 ∂ xa φ ( x, u ) + i h ( x, u ) (cid:63) φ ( x, u ) (5.33)it is obvious that it differs from (5.30). To understand the origin of this degeneracy ofminimal prescriptions, let us consider an example of the master Dirac field ψ ( x, u ). In thiscase it is easy to show that − ¯ ψγ a (cid:63) e a (cid:63) ψ = i ψγ a (cid:63) D (cid:63)a ψ − i D (cid:63)a ψ (cid:63) γ a ψ + u a ¯ ψγ a (cid:63) ψ . (5.34)The first two terms on the right hand side produce the master Lagrangian kinetic termwhich one would obtain by the minimal coupling prescription based on (5.30), leading tothe action S D [ ψ, e ] = (cid:90) d d x d d u ¯ ψ ( x, u ) (cid:63) (cid:0) iγ a D (cid:63)a − M (cid:1) ψ ( x, u ) . (5.35)On the left hand side of (5.34) is the expression which takes natural matrix model formwhen used in the action S D [ ˆ ψ, ˆ e ] = − Tr (cid:16) ˆ¯ ψ ( γ a ˆ e a + M ) ˆ ψ (cid:17) (5.36)and is formally defined for all phases of the MHS theory (it also takes care of hermicity byautomatism). The two actions differ already at the free field level, i.e., for h a ( x, u ) = 0. Wenow see that the difference between Lagrangians in two prescriptions is the second term onthe right hand side of (5.34) which is an MHS scalar. Its existence is a consequence of thefact that Lagrangian terms for matter in the fundamental representation are MHS scalars,which means that they can be multiplied by functions of the auxiliary coordinates withoutbreaking any of the important symmetries. The last observation opens up a new and exciting possibility for constructing theoriesin the matter sector. Let us demonstrate this on the example of the Dirac master field, bytaking the action to be S D [ ψ, e ] = − (cid:90) d d x d d u (cid:0) ¯ ψ (cid:63) ( γ a e a + M ) (cid:63) ψ (cid:1) δ d ( u ) (5.37)Note that it preserves Poincare symmetry of spacetime and explicitly breaks the auxiliaryspace translations. Observe that now there is no difference between the two forms of the Of course we should be careful not to break symmetries which we would like to preserve, such as Lorentzsymmetry and translations in spacetime. The symmetry under translations in the auxiliary space is brokenby such multiplications, but it is not obvious why we should try to protect this symmetry. – 36 –inimal prescription defined above. What is outstanding is that if one Taylor-expandsboth ψ and e a around u = 0, as in (3.5), then one can integrate over u in (5.37) to obtainpurely spacetime action, which is to say off-shell description in terms of the infinite towerof spacetime fields in Minkowski spacetime. The quadratic part of the action is non-diagonal in spacetime fields. An additional bonus is that one can construct new types ofLagrangian terms, e.g. in the case of the Dirac master field one can add the MHS Lorentzterm L L ( x, u ) = λ L (cid:16) ¯ ψ ( x, u ) (cid:63) γ a Σ bc T bc ( x, u ) (cid:63) ψ ( x, u ) (cid:17) ∂ au δ d ( u ) , Σ bc = i γ b , γ c ] (5.38)We leave more detailed analysis for the future work.Let us mention that mater fields in the fundamental representation have an additionalpeculiarity that the rigid MHS variations with n = 1 ( s = 2) act differently than in thecase of the MHS vielbein and previously discussed realizations of matter, δ ε (1) φ = − i ε µ u µ (cid:63) φ = − i ε µ u µ φ − ε µ ∂ xµ φ . (5.39)We see that it does not describe spacetime translations. One consequence is that the MHStransformations in this case can be consistently truncated only to the lowest spin sector( n = 0) when master fields are Taylor-expanded around u = 0. 6. Some explicit calculations The idea here is to perform a few sample perturbative calculations in the realm of theMHSYM theory coupled to matter, to see if the MHSYM theory produces mathematicallysensible results when quantized. First, we must first calculate the Feynman rules. We willwork in the Euclidean formulation, where the spacetime description is possible withoutbreaking manifest isometries. We expand the master fields in an orthonormal basis inauxiliary space, as described in (2.52)-(2.53). A benefit of this representation is that theMHS propagator is simple and obtained immediately from (2.54), D ( r,s ) µν ( k ) = i δ µν k δ rs = D (QED) µν ( k ) δ rs (6.1)where we use the Feynman gauge.As for the propagators and vertices which include matter, they depend on the type ofthe matter. In Sec. 5 we introduced three types, of which we analyze minimal (spacetime)matter and master field matter in the fundamental representation. For simplicity, weassume a single field of spin-1/2 type minimally coupled to the MHS potential. We remind the reader that in the case of the MHS gauge sector such description exists in the Euclideantheory, but not in Minkowski spacetime. – 37 – .1 Minimal spacetime matter6.1.1 Feynman rules Here the matter is described by standard spacetime fields, as described in Sec. 5.1, so thepropagators are of the usual form. To make the demonstration as simple as possible, wetake matter to be a single Dirac field ψ ( x ) minimally coupled to the MHS vielbein.To calculate vertices we use (5.13) with (5.21), which after using (2.53) and the nor-malization (2.44), becomes S D, int [ ψ, ¯ h ] = − ¯ g ym (cid:88) r (cid:90) d d x ¯ h ( r ) a ( x ) (cid:90) d d ¯ u ¯ ψ ( x ) (cid:63) γ a f r (¯ u ) (cid:63) ψ ( x ) δ d (¯ u ) (6.2)which leads to the following expression for the interacting part of the (spacetime) La-grangian density L int ( x ) = − ¯ g ym (cid:88) r ¯ h ( r ) a ( x ) J a ( r ) ( x ) (6.3)where the currents are given by J a ( r ) = ∞ (cid:88) n =0 ∞ (cid:88) m =0 i n ( − i ) m (cid:96) n + mh n + m n ! m ! ∂ µ . . . ∂ µ n ¯ ψγ a ∂ ν . . . ∂ ν m ψ ∂ µ . . . ∂ µ n ∂ ν . . . ∂ ν m f r (¯ u ) (cid:12)(cid:12)(cid:12) ¯ u =0 . Figure 1: Interaction vertex for the minimal coupling. There is only one type of the vertex, depicted in Fig. 1. If we denote by q (cid:48) the outgoingfermion momentum and with q the ingoing fermion momentum, then the vertex contribu-tion is V βα ( rµ ) ( q (cid:48) , q ) = i ( γ µ ) βα V r ( q (cid:48) , q ) = V (QED) βαµ V r ( q (cid:48) , q ) (6.4)where the vertex factor V r ( q (cid:48) , q ) is the MHS contribution to the standard QED result, andis given by V r ( q (cid:48) , q ) = ∞ (cid:88) n =0 ∞ (cid:88) m =0 (cid:96) n + mh n + m n ! m ! q (cid:48) µ . . . q (cid:48) µ n q ν . . . q ν m ∂ µ . . . ∂ µ n ∂ ν . . . ∂ ν m f r (¯ u ) (cid:12)(cid:12)(cid:12) ¯ u =0 = exp (cid:18) (cid:96) h q + q (cid:48) ) · ∂ ¯ u (cid:19) f r (¯ u ) (cid:12)(cid:12)(cid:12) ¯ u =0 = f r (cid:18) ¯ u + (cid:96) h q + q (cid:48) ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ¯ u =0 = f r (cid:18) (cid:96) h q + q (cid:48) ) (cid:19) . (6.5)– 38 –omparing with QED, we see that the difference is in the vertex functions given bythe basis functions in the auxiliary space f r (¯ u ) evaluated on¯ u → (cid:96) h q + q (cid:48) ) . (6.6)The basis functions vanish in the limit | ¯ u | → ∞ faster than any power, making the UVlimit soft. The formula for the MHS vertex factor (6.5) suggests that the UV behavior ofthe MHSYM should be better than in QED. As an example, we take the tree-level 4-point amplitude for the Dirac field. There iseffectively one Feynman diagram, shown in Figure 2, which has two channels: t and u . (cid:1) Figure 2: Tree level 4-point amplitude Feynman diagram. Let us start with the t -channel. Using the vertex (6.5) and the propagator (6.1) whosemomentum is k = p − p (cid:48) = p (cid:48) − p , the t -channel contribution reads M t = M (QED) t A t (6.7)where M (QED) t is the QED result and A t is given by A t = − (cid:88) r V r ( p , p (cid:48) ) V r ( p , p (cid:48) )= (cid:88) r f r (cid:18) (cid:96) h p + p (cid:48) ) (cid:19) f r (cid:18) (cid:96) h p + p (cid:48) ) (cid:19) = δ d (cid:18) (cid:96) h p + p (cid:48) − p − p (cid:48) ) (cid:19) (6.8)In the last line we used the completeness relation (cid:88) r f r (¯ u ) f r (¯ v ) = δ d (¯ u − ¯ v ) (6.9)Finally, using the energy-momentum conservation condition p + p = p (cid:48) + p (cid:48) = ⇒ p (cid:48) − p = p − p (cid:48) (6.10)we obtain A t = δ d (cid:0) (cid:96) h ( p − p (cid:48) ) (cid:1) (6.11)– 39 –he u -channel Feynman diagram is obtained simply by exchanging 1 (cid:48) ↔ (cid:48) in theresults for the t -channel, which means that the momentum of the propagator is k = p − p (cid:48) = p − p (cid:48) , and M u = M (QED) u A u (6.12)where A u = δ d (cid:0) (cid:96) h ( p − p (cid:48) ) (cid:1) . (6.13)The full tree-level amplitude is then M = 12 ( M t − M u )= 12 M (QED) t δ d (cid:0) (cid:96) h ( p − p (cid:48) ) (cid:1) − M (QED) u δ d (cid:0) (cid:96) h ( p − p (cid:48) ) (cid:1) . (6.14)It is vanishing unless the set of momenta in the final state is the same as the set ofmomenta in the initial state. This result is expected from the viewpoint of the Coleman-Mandula theorem, despite the fact that MHS theory does not fulfill all assumptions ofthe theorem. The result is also interesting from the perspective of the search for the darkmatter candidates in cosmology. We mentioned that the MHSYM theory can be consistently truncated if we require e a ( x, − u ) = − e a ( x, u ) (”truncation to spin-even sector”). In this case the basis functions f r (¯ u ) are also odd, f r ( − ¯ u ) = − f r (¯ u ), and the completeness relation becomes (cid:88) r f r (¯ u ) f r (¯ v ) = 12 (cid:16) δ d (¯ u − ¯ v ) − δ d (¯ u + ¯ v ) (cid:17) . (6.15)The t -channel amplitude is now M t = 12 M (QED) t (cid:16) δ d (cid:0) (cid:96) h ( p − p (cid:48) ) (cid:1) − δ d (cid:0) (cid:96) h ( p + p ) (cid:1)(cid:17) . (6.16)The main conclusion, that the amplitude is ultralocal in momentum space, remains thesame. Now we take that matter is represented by a single master Dirac field ψ ( x, ¯ u ) in thefundamental representation of the MHS symmetry (see Sec. 5.2). The free action is S D [ ψ ] = (cid:90) d d x d d ¯ u ¯ ψ ( x, ¯ u ) γ a ∂ xa ψ ( x, ¯ u ) (6.17)while the interaction term is S D , int [ ψ, ¯ h ] = − ¯ g ym (cid:90) d d x d d ¯ u ¯ ψ ( x, ¯ u ) (cid:63) (cid:0) γ a ¯ h a ( x, ¯ u ) (cid:1) (cid:63) ψ ( x, ¯ u )= − ¯ g ym (cid:90) d d x d d ¯ u Tr (cid:0) ψ ( x, ¯ u ) (cid:63) ¯ ψ ( x, ¯ u ) γ a (cid:1) ¯ h a ( x, ¯ u ) (6.18) For recent speculations that higher-spin particles may describe dark matter see [39]. – 40 –here Tr denotes the trace over spinor indices. Now we also expand the master Dirac fieldin the orthonormal basis in the auxiliary space ψ β ( x, ¯ u ) = (cid:88) j ψ ( j ) β ( x ) f j (¯ u ) (6.19)where β is a spinor index. The basis used for the matter field does not have to be the sameas the one used for the MHS potential. Using (6.19) in (6.17) gives us the propagator forthe Dirac spacetime fields D ( ij ) αβ ( p ) = D αβ ( p ) δ ij (6.20)where the first term on the RHS is the usual propagator for the Dirac field. Using (6.19)and (2.53) in (6.18) we obtain that the vertex is V ( iα )( jβ )( ra ) ( p (cid:48) , p ) = V (QED) αβa V (hs) ijr ( p (cid:48) , p ) (6.21)where the QED vertex is V (QED) αβa = − i ¯ g ym ( γ a ) αβ (6.22)and the MHS vertex factor is given by V (hs) ijr ( p (cid:48) , p ) = (cid:90) d d ¯ u f ∗ i (cid:16) ¯ u + (cid:96) h p (cid:17) f j (cid:16) ¯ u + (cid:96) h p (cid:48) (cid:17) f r (¯ u ) . (6.23)This completes the knowledge of Feynman rules in this theory. Again, comparing withQED the difference is the presence of a vertex factor that softens the UV behavior. We want to calculate the tree-level 4-point amplitude with matter fields on all legs. Again,there is essentially one diagram, depicted in Fig. 2, which has two channels. Let us firstcalculate the t -channel diagram. From Feynman rules calculated above we obtain theamplitude M t = M (QED) t A ( ij ) t (6.24)where A ( ij ) t = (cid:88) r V (hs) i j r ( p (cid:48) , p ) V (hs) i j r ( p (cid:48) , p )= (cid:90) d d ¯ u f ∗ i (¯ u + (cid:96) h p ) f j (¯ u + (cid:96) h p (cid:48) ) (cid:90) d d ¯ v f ∗ i (¯ v + (cid:96) h p ) f j (¯ v + (cid:96) h p (cid:48) ) (cid:88) r f r (¯ u ) f r (¯ v )= (cid:90) d d ¯ u f ∗ i (¯ u + (cid:96) h p ) f j (¯ u + (cid:96) h p (cid:48) ) f ∗ i (¯ u + (cid:96) h p ) f j (¯ u + (cid:96) h p (cid:48) ) . (6.25)In passing from second to third line the completeness relation was used. The u -channelcontribution to the amplitude is obtained from (6.24) and (6.25) by exchanging p (cid:48) ↔ p (cid:48) and i ↔ i . The total tree-level amplitude is M tree = M t − M u . (6.26)– 41 –here are no Dirac-delta functions which are present in the case of simple spacetime mat-ter. The integral in (6.25) is convergent and, due to the asymptotic fall-off of functions f j (¯ u ) when | ¯ u | → ∞ , the MHS contribution certainly makes the UV behavior softer whencompared to the standard spinor QED. The basis functions are of the form f r (¯ u ) = P r (¯ u ) e − ( (cid:96) h ¯ u ) . (6.27)where P r are polynomials. Using this in (6.25) we obtain A ( ij ) t = P ( ij ) ( (cid:96) h p ) exp − (cid:96) h (cid:88) i,j =1 ( p i − p j ) (6.28)where p ≡ p (cid:48) , p ≡ p (cid:48) and P ( ij ) are polynomial functions. The exponential factor makesthe UV behaviour much softer when compared with QED. 7. Low-spin sector We have seen in Sec. 5.1 that the minimal way to incorporate interacting matter inside theMHS framework leads to the picture in which matter perceives spacetime fields obtainedby Taylor expanding the MHS vielbein in the auxiliary space e a ( x, u ) = ∞ (cid:88) n =0 e ( n ) µ ...µ n a ( x ) u µ · · · u µ n (7.1)as a HS background. From this viewpoint the lowest two components, e (1) a ( x ) and e (1) µa ( x ),play the roles of the U (1) potential and the emergent spacetime vielbein, respectively. Inthis section we focus on the low-spin sector n ≤ s ≤ e a ( x, u ) = A a ( x ) + E aµ ( x ) u µ + . . . (7.2)and similarly for the MHS variation parameter ε ( x, u ) = (cid:15) ( x ) + ε µ ( x ) u µ + . . . (7.3)and in all expressions ignore higher spin components (with n > 1) denoted above by ellipses.We noted before that the truncation to the low-spin sector is formally consistent at thelevel of MHS symmetry and EoM. However, we should keep in mind that such truncatedconfigurations are not physical (see Sec 2.4), so our findings based on this truncation areof limited importance.If B ( x, u ) and C ( x, u ) are generic master fields, their Moyal bracket truncated to lowspin sector, i [ B ( x, u ) (cid:63) , C ( x, u )] = ∂C ( x, u ) ∂x µ ∂B ( x, u ) ∂u µ − ∂B ( x, u ) ∂x µ ∂C ( x, u ) ∂u µ + . . . = −{ B ( x, u ) , C ( x, u ) } PB + . . . (7.4)– 42 –s given by the Poisson bracket, where the master space plays the role of the phase space.From (7.4) it follows that the set of spin-2 truncated master fields is closed under the Moyalbracket. The Taylor expansion of (7.4) is given by i [ B ( x, u ) (cid:63) , C ( x, u )] = B ν (1) ( x ) ∂ ν C (0) ( x ) − C ν (1) ( x ) ∂ ν B (0) ( x )+ (cid:16) B ν (1) ( x ) ∂ ν C µ (1) ( x ) − C ν (1) ( x ) ∂ ν B µ (1) ( x ) (cid:17) u µ + . . . (7.5)= £ B (1) C (0) − £ C (1) B (0) + (cid:0) £ B (1) C (1) (cid:1) µ u µ + . . . (7.6)The last line is obtained by recognizing the differential-geometric structure, with Lie deriva-tives treating B (0) ( x ) and C (0) ( x ) as scalar fields and A µ (1) ( x ) and B µ (1) ( x ) as vector fieldson the spacetime manifold. We will see below that this is generally true in our construction– all expressions truncated to the low-spin sector ( s ≤ 2) are going to be diff-covariant.Let us apply this to the MHS variation of the MHS vielbein (3.4). Using (7.2), (7.3)and (7.6) we obtain that the low-spin spacetime fields transform as δ (cid:15) A a ( x ) = £ E a (cid:15) ( x ) (7.7) δ ε A a ( x ) = − £ ε A a ( x ) (7.8) δ (cid:15) E aµ ( x ) = 0 (7.9) δ ε E aµ ( x ) = (cid:0) £ E a ε (cid:1) µ ( x ) (7.10)We now see that the MHS variation with n = 1 acts as an infinitesimal diffeomorphismdefined by x (cid:48) µ = x µ + ε µ ( x ) (7.11)under which E aµ ( x ) behaves as a set of vector fields, while A a ( x ) behaves as a set of scalars.Assuming that the frame E aµ ( x ) is regular, i.e., there exists a co-frame E aµ ( x ) satisfying E aµ ( x ) E aν ( x ) = δ νµ , E bµ ( x ) E aµ ( x ) = δ ba (7.12)an MHS variation with n = 0 acts on A µ ( x ) = E aµ ( x ) A a ( x ) as δ (cid:15) A µ ( x ) = − ∂ µ (cid:15) ( x ) (7.13)while the frame E aµ ( x ) is invariant. Taken all together, E aµ ( x ) can be identified as the(inverse) vielbein, while A a ( x ) can be identified as a U (1) gauge potential vector field inthe non-coordinate basis of the vielbein. The n = 0 MHS variations are infinitesimal U (1)gauge transformations, while n = 1 MHS variations are infinitesimal diffeomorphisms. Notethat this interpretation is only valid in the geometric phase, in which the frame E aµ ( x )invertible, to which we now turn our attention.A word of caution is necessary here. If we keep higher spin contributions, the diff-covariant structure, at least as defined in the standard way, is apparently lost. This canbe traced to the mixing (or twisting) of the HS transformations noted already at the level– 43 –f the rigid transformations in Sec. 2.2. The effects of twisting can be seen by analysing n = 1 finite (large) MHS transformations of MHS tensors, E ( x, u ) = E µ ( x ) u µ (7.14)where for the sake of simplicity we assume that the n = 0 component of the MHS tensoris vanishing X a... ( x, u ) = X (1) µa... ( x ) u µ + O ( u ) (7.15)From the definition (3.1) and the Baker-Campbell-Hausdorff formula it follows that thespacetime vector field X (1) µa... ( x ) transforms as (cid:0) X (1) E a... (cid:1) µ = (cid:0) exp( £ E ) X (1) a... (cid:1) µ + . . . . (7.16)This should be compared with the diff-transformation of a vector field V (cid:48) µ ( x (cid:48) ) = ∂ ν ζ µ ( x ) V ν ( x ) , x (cid:48) µ = ζ µ ( x ) (7.17)The large MHS transformation (7.16) is a diffeomorphism, where the connection betweenparameter fields seems to be given by ζ µ ( x ) − x µ = ∞ (cid:88) r =0 ( E ( x ) · ∂ ) r ( r + 1)! E µ ( x )= e E ( x ) · ∂ − E ( x ) · ∂ E µ ( x ) . (7.18)We have checked this relation up to quartic order [40, 41]. We see that large MHS transfor-mations in spin-2 sector are indeed finite diffeomorphisms, but that the naturally definedparameters of the two descriptions are related in a complicated way given by (7.18).Let us now analyze the metric, whose low-spin components in the MHS framework arenaturally obtained from (3.22). The result is g (0) ( x ) = 12 A a ( x ) A a ( x ) + 12 ∂ ν E aµ ( x ) ∂ µ E aν ( x ) + . . . (7.19) g µ (1) ( x ) = E aµ ( x ) A a ( x ) + . . . (7.20) g µν (2) ( x ) = E aµ ( x ) E aν ( x ) + . . . . (7.21)Relations (7.20)-(7.21) confirm the identification of A µ ( x ) ≡ E aµ ( x ) A a ( x ) (7.22)as a U (1) vector potential, E aµ ( x ) as a vielbein and g µν ( x ) ≡ E aµ ( x ) E aν ( x ) (7.23)as the (inverse) metric tensor. The terms on the right hand side of (7.19) are responsiblefor producing the seagull interaction terms when a Klein-Gordon matter field is minimallycoupled to the MHS field. – 44 –o understand the induced geometry, we have to find the induced linear connection.The natural way to obtain it is by analysing the n = 1 component of the MHS covariantderivative of an MHS tensor. Using (3.13) and (7.6) we get( D (cid:63)a B b... ) µ (1) ( x ) = (cid:0) £ E a B (1) b... (cid:1) µ ( x ) + . . . (7.24)from which it follows that the induced covariant derivative of a vector field should be givenby ( ∇ E a V ) µ ≡ E aν ∇ ν V µ = (cid:0) £ E a V (cid:1) µ . (7.25)Multiplying by E aµ ( x ) we finally obtain ∇ ν V µ = E aν (cid:0) £ E a V (cid:1) µ = ∂ ν V µ + E aµ ∂ ρ E aν V ρ . (7.26)This means that the MHS symmetry induces the following linear connectionΓ µρν = E aµ ∂ ρ E aν = − E aν ∂ ρ E aµ . (7.27)The obtained linear connection is very much different from the Levi-Civita connection. Forone, the torsion tensor is generally non-vanishing, as it can explicitly be checked T µρν = Γ µνρ − Γ µρν = ξ µρν ≡ E aρ E bν ξ µab (7.28)where ξ µab ( x ) is ξ µab = (cid:0) £ E a E b (cid:1) µ = ξ cab E cµ . (7.29) ξ cab ( x ) are known as coefficients of anholonomy. As a consistency check, let us calculatethe n = 1 component of the HS torsion. It is easy to show that it is given by T (1) µab ( x ) = ξ µab ( x ) + . . . (7.30)which is consistent with (7.28).Also, the linear connection (7.27) is not metric compatible, the nonmetricity tensorbeing Q ρµν ≡ ∇ ρ g µν = T µρσ g ρν + T ν ρσ g ρµ = T µν σ + T νµσ (7.31)which is generally non-vanishing. Note that the nonmetricity tensor is not independent butis fully (algebraically) expressible in terms of torsion. The same is true for the Riemanntensor for which it can be shown that it can be expressed in terms of the torsion tensorand its covariant derivatives.Let us also calculate the spin connection in the geometry induced by the MHS con-struction. The simplest way to find it is to use A abµ = E aν ∇ µ E bν . (7.32)– 45 –sing (7.27) we get A abµ = E aν E bρ T ν µρ = T aµb . (7.33)We see that A abµ is not antisymmetric in its first two indices, which is a manifestation ofmetric incompatibility. Again, we see that the induced spin connection is fully determinedby the torsion. We have seen that the induced spacetime geometry found in the s = 2 ( n = 1) sector of MHStheory seems rather unusual. The linear connection is metric-incompatible, and both thetorsion and the Riemann tensor are non-vanishing. However, it is just teleparallel geometryin disguise. The key observation is that there is only one independent fundamental tensor,the torsion, and all others are expressible in terms of it.Let us first briefly review the concept of distant parallelism or teleparallelism. Letus assume that a differentiable manifold is equipped with a linear connection Γ + , which isnot symmetric. Teleparallelism is a requirement on the linear connection that there existsa frame of vector fields (an inertial frame) E aµ ( x ) that globally satisfies ∇ + µ E aσ ≡ ∂ µ E aσ + Γ σ + ρµ E aρ = 0 . (7.34)From the definition of the covariant derivative it follows that the linear connection is givenby Γ σ + ρµ = E aσ ∂ µ E aρ = − E aρ ∂ µ E aσ (7.35)which is known as the Weitzenb¨ock connection. If the metric is naturally defined by takingthe inertial frame as the vielbein g µν = η ab E aµ E bν (7.36)then from (7.34) it obviously follows that the Weitzenb¨ock connection is metric compatible ∇ + ρ g µν = 0 (7.37)An outstanding property of the Weitzenb¨ock connection is that its corresponding spin(Lorentz) connection is vanishing A a + bµ = E bσ ∇ µ E aσ = 0 (7.38)for inertial frames. Inertial frames are related to one another through global Lorentztransformations, E aµ ( x ) → Λ ab E bµ ( x ) . (7.39) For a detailed exposition of teleparallel geometry and gravity see the book [42]. – 46 –ote that by performing a local Lorentz transformation E aµ ( x ) → Λ ab ( x ) E bµ ( x ) , ∂ µ Λ ab (cid:54) = 0 (7.40)one passes to a non-inertial frame which does not satisfy (7.34). As a consequence the spinconnection in the transformed frame is non-vanishing but still trivial (i.e. flat), A a + bµ → Λ bc ∂ µ Λ ca . (7.41)It follows directly from (7.38) that the Riemann tensor also vanishes R a + bµν = 0 (7.42)which, as a consistency check, one could also show using the Weitzenb¨ock linear connection.This means that, beside the metric, the only nontrivial fundamental tensor in teleparallelgeometry is torsion, which is given (using the inertial frame) by T µ + νρ ≡ Γ µ + ρν − Γ µ + νρ = − ξ µνρ (7.43)where the anholonomy ξ was defined in (7.29).The simplest Lagrangians of teleparallel gravity theories are of the form [43] S tg = (cid:90) d d x E (cid:0) c T ρ + µν T + µνρ + c T ρ + µν T νµ + ρ + c T ρ + µρ T + νµν (cid:1) (7.44)It can be shown that if one takes c = 14 , c = 12 , c = − µνρ = Γ µ + ρν (7.46)i.e., our linear connection is the opposite of the teleparallel one. It is then not strange thattorsions are related by T µνρ = − T µ + νρ . (7.47)This means that the covariant derivative induced by MHS symmetry can be written interms of the covariant derivative of teleparallel geometry ∇ = ∇ + − T + . (7.48)– 47 –sing (7.47) and (7.48) we can express any covariant expression in the teleparallel geometryas a covariant expression in the opposite of teleparallel geometry. As a special case, it meansthat a manifestly covariant EoM in the emergent MHS geometry can be expressed as amanifestly covariant EoM in teleparallel geometry, and vice versa. This will be importantbelow in the discussion of s = 2 sector of EoM in the MHSYM model.Teleparallel gravity can be obtained by gauging the group isometric to the groupof spacetime translations [45, 46, 47]. As the global MHS transformations have such asubgroup ( n = 1 sector), it is not surprising that there is a connection between the MHStheory and teleparallel gravity. s ≤ sector In view of the preceding discussion on the induced spacetime geometry in the MHS con-struction, it is interesting to study the s ≤ s ≤ s = 1 component of the EoM is given by0 = E bν ∂ ν F ba − ξ νba ∂ ν A b + . . . (7.49)where F is the 2-form field strength of the spin-1 U (1) spacetime vector potential 1-form A , i.e. F ab = E aµ E bν F µν = E aµ E bν ( ∂ µ A ν − ∂ ν A µ )= E aν ∂ ν A b − E bν ∂ ν A a + A c ξ cab . (7.50)The s = 2 component of the EoM is0 = E bν ∂ ν ξ µba − ξ νba ∂ ν E bµ + . . . . (7.51)After some manipulation we can rewrite this equation in the equivalent form0 = E cν ∂ ν ξ abc − ξ acd ξ cdb + . . . . (7.52)Also, using (7.52) and (7.50) we can write the s = 1 EoM component (7.49) as0 = E bν ∂ ν F ba − ξ bca F bc + . . . (7.53)which is manifestly U (1)-gauge invariant.Terms denoted by ellipses, and also complete s > s > s ≤ 2) sector at the levelof EoM. Let us rewrite the low-spin EoM (7.52)-(7.53) within the framework of teleparallelgeometry by using (7.47) and the fact that in teleparallel gravity Lorentz connection is We remind the reader that such truncated configurations are not physically acceptable in our formalism. – 48 –anishing in inertial frames so the Lorentz covariant derivative is simply the coordinatederivative D + µ = ∂ µ = ⇒ D + a = E aµ ∂ µ . (7.54)Using all of this we can write (7.53) as D + b F ba + T bca + F bc = 0 (7.55)which is now fully diff- and U (1) covariant. Similarly, (7.52) becomes D + c T abc + + T a + cd T cdb + = 0 (7.56)which is manifestly diff-covariant. We stress that in this form all objects in (7.55) and(7.56) should be calculated using the Weitzenb¨ock connection. In other words, they areformally written within the realm of teleparallel gravity, and not the geometry induced bythe MHS symmetry.There is an interesting piece of history here. The equation (7.56) was first written byAlbert Einstein in 1929, with a motivation to unify electromagnetism with gravity [48]. .Einstein observed that it is not possible to write the diff-covariant action which produces(7.56) as its EoM inside the realm of the teleparallel geometry, because the left hand side isnot covariantly conserved (in particular it does not belong to the class of theories defined in(7.44)). It is amusing that we obtained EoM (7.56) from an action principle by truncatingMHSYM theory. 8. Conclusions We have investigated a novel way of gauging higher derivative (or higher spin) rigid sym-metries present in all free field theories in flat spacetime, originally proposed in [23, 24].The proposal is based on the symmetries observed for relativistic matter fields coupledlinearly to an infinite tower of higher-spin fields [25, 26, 27]. The important feature ofthis proposal, which we refer to as Moyal-higher-spin (MHS), is that the gauge potentialis intrinsically (off-shell) defined on the master space, which is a direct product of theordinary spacetime and an auxiliary space of the same dimensionality. The Moyal productintroduces a particular type of non-commutativity which acts between the spacetime andthe auxiliary space, whereas spacetime commutativity is kept. We have shown that one canconstruct actions for the MHS gauge potential and matter coupled to it by mimicking thestandard Yang-Mills procedure, providing in this way such desired properties as Poincaresymmetry, a perturbatively stable vacuum, the formal application of BRST quantizationrules, mass generation by spontaneous symmetry breaking, supersymmetrization and L ∞ structure. It appears that the construction is well defined and consistent not only classi-cally, but possibly also on the quantum level. We have shown that the gauge potential can For this reason he added by hand one more equation to EoM in an attempt to project out unwanteddegrees of freedom. Already in 1930 he abandoned this attempt. – 49 –e manifestly covariantized, in a way which has resemblances to the teleparallel gravityconstruction based on gauging the group of translations. A consequence of this generali-sation, which we call geometry-like formulation is an appearance of new phases, includingthe strongly coupled with an unbroken vacuum. It also provides a way of casting the MHStheory into an (infinite dimensional) matrix model form.An obvious question is how to connect the MHS construction to the standard Wignerclassification of irreducible representations of the Poincare group. On the level of equationsof motion, the MHS Yang-Mills theory can be written in terms of an infinite tower of tensorspacetime fields with unbounded rank, thus making contact with higher spin field theories.The spin-2 sector has a differential geometric interpretation, being essentially teleparallelunder disguise, with a linear connection opposite to Weizenb¨ock’s. The MHS theory for-mally allows for more phases, even with a singular emergent tangent frame (vielbein). TheYM phase corresponds to a regular emergent geometry , while the unbroken phase presentsthe maximally singular (i.e. vanishing) frame. However, when comparing with the stan-dard higher spin program there are important differences. Higher spin fields obtained inthe spacetime MHS decomposition are not independent in the usual sense, one consequencebeing that there is no regular off-shell purely spacetime and manifestly Lorentz covariantdescription of the MHS theory. Even on the level of the linearized equations, which aresecond-order in derivatives and of the Maxwell-type, there is a mixing between differentirreducible components of the higher spin spacetime fields. All of this points at the possi-bility that the MHSYM theory is not compatible with the Fock space composed of weaklyinteracting particles with finite spins. One can now understand how the intrinsic masterspace nature of the MHS construction surpasses the barriers raised by no-go theorems.Interestingly, a manifestly covariant off-shell description of MHS theory in terms ofan infinite tower of higher-spin spacetime fields is possible in the Euclidean regime. Wehave used this description to calculate tree-level four-point amplitudes for different types ofmatter. In the case of the ordinary spacetime matter, the amplitude vanishes for nontrivialoutgoing momenta, a result which is interesting from the perspective of the dark matterproblem. In the case of the master space matter, the amplitudes have factors which areexponentially suppressed at high energies, leading to a much softer UV behaviour in com-parison to QED. This and the fact that the theory formally allows for the BRST proceduregives us a hope that MHS theory may be consistently quantised, while its soft high energybehaviour will avoid such issues as the Landau pole.There are some obvious questions arising from our analysis. The fact that the MHStheory is based on gauging rigid higher spin symmetries and allows for higher-spin de-scriptions under some circumstances raises questions as to its possible connection to thetensionless limit of string field theory. That it has a natural matrix model formulation sug-gests possible connections with ideas on higher-spin constructions proposed by Steinacker[49, 50]. Another question is whether the MHS theory, in the present or modified form,has any connections to the infinite (or continuous) spin representations of the Poincaregroup, which also contain a continuous (but compact) ”index” [51, 52]. The more detailedstudy of the matter sector and the generation of mass through the Higgs mechanism maylead to more surprises. We leave these and other questions to our future work. It is our– 50 –pinion that results so far already suggest that MHS formalism is of enough interest towarrant attention and further study. At least it shows that relativistic field theories stilloffer surprises, especially when one is ready to relax some of the standard assumptionsusually taken for granted. Acknowledgments P.D.P. would like to thank Erwin Schroedinger Institute for Mathematics and Physics (Uni-versity of Vienna) for support during the visit under the framework of ESI Programme andWorkshop ”Higher Spins and Holography”, and Evgeny Skvortsov and Per Sundell for stim-ulating discussions. S.G is also grateful to Dario Francia, Per Sundell, Massimo Taronna,Konstantin Alkalaev, Alexey Sharapov, Evgeny Skvortsov, Xavier Bekaert, Euihun Joungand to the organizers and participants of the workshop “Higher Spin Gravity: Chaotic,Conformal and Algebraic Aspects” at the Asia Pacific Center for Theoretical Physics fordiscussions and comments on topics relevant to the paper. We especially thank LorianoBonora as this work evolved from our mutual collaboration and for useful discussions andcomments on early versions of the manuscript.This research has been supported by theUniversity of Rijeka under the project uniri-prirod-18-256. The research of S.G. has beensupported by the Israel Science Foundation (ISF), grant No. 244/17. A. Properties of the Moyal product We use the following definition for the Moyal product of functions on the master space M × U with coordinates ( x, u ): a ( x, u ) (cid:63) b ( x, u ) = a ( x, u ) exp (cid:20) i (cid:16) ← ∂ x · → ∂ u − → ∂ x · ← ∂ u (cid:17)(cid:21) b ( x, u )= exp (cid:20) i ∂ x · ∂ w − ∂ y · ∂ u ) (cid:21) a ( x, u ) b ( y, w ) (cid:12)(cid:12)(cid:12) y = x , w = u . (A.1)The partial derivatives are defined in the usual way, i.e., ∂ xµ = ∂∂x µ , ∂ µu = ∂∂u µ . (A.2)The Moyal product is associative (cid:0) a ( x, u ) (cid:63) b ( x, u ) (cid:1) (cid:63) c ( x, u ) = a ( x, u ) (cid:63) (cid:0) b ( x, u ) (cid:63) c ( x, u ) (cid:1) (A.3)and Hermitian under the complex conjugation( a ( x, u ) (cid:63) b ( x, u )) ∗ = b ( x, u ) ∗ (cid:63) a ( x, u ) ∗ . (A.4)The Moyal product can, for a class of functions with well behaved fall of conditions,be calculated in the integral form a ( x, u ) (cid:63) b ( x, u ) = (cid:90) d d y d d z d d v (2 π ) d d d w (2 π ) d e i ( yw − zu ) a ( x + y , u + v ) b ( x + z , u + w ) (A.5)– 51 –nd a very useful and convenient way to make calculations with the Moyal product is bypromoting coordinates to operators, of which we note one possible way: a ( x, u ) (cid:63) b ( x, u ) = a ( x, u ) b ( x, u † ) (A.6)with u = u − i (cid:126)∂ x , u † = u + i ←− ∂ x (A.7)Note that the Moyal commutator of real functions is purely imaginary, while the Moyalanticommutator of real functions is real. It is important to note that the Moyal commutatorobeys the Jacobi identity[ a (cid:63) , [ b (cid:63) , c ]] + [ c (cid:63) , [ a (cid:63) , b ]] + [ b (cid:63) , [ c (cid:63) , a ]] = 0 (A.8)and the derivation (or Leibniz) property[ a (cid:63) , b (cid:63) c ] = [ a (cid:63) , b ] (cid:63) c + b (cid:63) [ a (cid:63) , c ] . (A.9)The same properties are obeyed by the ordinary (matrix) commutator. From (A.9) itfollows { [ a (cid:63) , b ] (cid:63) , a } = [ a (cid:63) a (cid:63) , b ] . (A.10)The Moyal product satisfies the adjoint property under integration (cid:90) d d x d d u (cid:0) a ( x, u ) (cid:63) b ( x, u ) (cid:1) c ( x, u ) = (cid:90) d d x d d u a ( x, u ) (cid:0) b ( x, u ) (cid:63) c ( x, u ) (cid:1) = (cid:90) d d x d d u b ( x, u ) (cid:0) c ( x, u ) (cid:63) a ( x, u ) (cid:1) (A.11)where a , b and c are square-integrable functions on the master space. If we put c ( x, u ) = 1we obtain (cid:90) d d x d d u a ( x, u ) (cid:63) b ( x, u ) = (cid:90) d d x d d u a ( x, u ) b ( x, u ) + (boundary terms) . (A.12)It is convenient to define deformations on standard functions on the master space byusing Moyal instead of ordinary product in the Taylor expansion. We denote such functionswith the (cid:63) subscript, e.g., e a ( x,u ) (cid:63) = ∞ (cid:88) n =0 n ! a ( x, u ) (cid:63)n (A.13)where a ( x, u ) (cid:63)n is the Moyal product with n factors of a ( x, u ) a ( x, u ) (cid:63)n = a ( x, u ) (cid:63) a ( x, p ) (cid:63) . . . (cid:63) a ( x, u ) . (A.14)– 52 – . Proof of the theorem on triviality Here we prove that the HS field strength measures the triviality of HS configurations, i.e.HS master space field is pure gauge ⇐⇒ F ab ( x, u ) = 0 (B.1)in a domain of configurations containing h a ( x, u ) = 0.Proof. From (2.21) it follows directly that the theorem is valid in the linear approximation,since the linear term in (3.16) can be interpreted as the exterior derivative of a form h µ ...µ n a ( x ) if Greek indices µ j are treated as internal. It means that in the linearizedtheory if and only if F ab ( x, u ) = 0 in a ball, there exists a master space function ε ( x, u )such that h a ( x, u ) = ∂ a ε ( x, u ) (B.2)in the same ball. But this is just the linearised pure gauge condition for the MHS potential h a ( x, u ).Let us extend this to large fields. First we prove the left-to-right arrow in (B.1). If theMHS vielbein field is pure gauge, then by (3.1) we can write it as e a ( x, u ) = e − i E ( x,u ) (cid:63) (cid:63) u a (cid:63) e i E ( x,u ) (cid:63) = u a − ie − i E ( x,u ) (cid:63) (cid:63) ∂ xa e i E ( x,u ) (cid:63) (B.3)so a pure gauge MHS master field h a ( x, u ) is of the form h a ( x, u ) = − ie − i E ( x,u ) (cid:63) (cid:63) ∂ xa e i E ( x,u ) (cid:63) . (B.4)Plugging this into (2.21), and using the identities e − i E ( x,u ) (cid:63) (cid:63) e i E ( x,u ) (cid:63) = 1 , e − i E ( x,u ) (cid:63) (cid:63) ∂ xa e i E ( x,u ) (cid:63) = − ∂ xa e − i E ( x,u ) (cid:63) (cid:63) e i E ( x,u ) (cid:63) (B.5)we conclude that MHS field strength F ab ( x, u ) vanishes for pure gauge HS fields. Therefore,HS phase space field is pure gauge = ⇒ F ab ( x, u ) = 0 . (B.6)Proving the opposite direction of (B.1) happens to be more involved. We want to findthe general solution of the equation F ab ( x, u ) = 0 . (B.7)To do this, let us start from the linearized solution (B.2) and build a full solution by aformal perturbative series h a ( x, u ) = ∞ (cid:88) n =1 ∆ ( n ) a ( x, u ) . (B.8) In this construction it is not assumed that the HS potential h a ( x, u ) is small. We can introduce a formalparameter θ , and consider (B.8) as an expansion in θ . Eventually we put θ → – 53 –ntroducing (B.8) into (B.7), using (2.21), and collecting the terms of the same order, weobtain ∂ xa ∆ ( n ) b ( x, u ) − ∂ xb ∆ ( n ) a ( x, u ) = − i n − (cid:88) r =1 [∆ ( r ) a ( x, u ) (cid:63) , ∆ ( n − r ) b ( x, u )] . (B.9)We see that it has a form which can be attacked by mathematical induction. For n = 1 itbecomes ∂ xa ∆ (1) b ( x, u ) − ∂ xb ∆ (1) a ( x, u ) = 0 (B.10)for which the general solution is ∆ (1) a ( x, u ) = ∂ xa E ( x, u ) (B.11)where E ( x, p ) is an arbitrary function. For n = 2 we get ∂ xa ∆ (2) b − ∂ xb ∆ (2) a = − i [∆ (1) a (cid:63) , ∆ (1) b ]= − i [ ∂ xa E (cid:63) , ∂ xb E ]= − i ∂ xa [ E (cid:63) , ∂ xb E ] − ∂ xb [ E (cid:63) , ∂ xa E ]) (B.12)for which the solution is∆ (2) b ( x, u ) = − i E ( x, u ) (cid:63) , ∂ xa E ( x, u )] + ∂ xa E (cid:48) ( x, u ) . (B.13)The trivial exact part of the solution, which appears at every order, is of the same formas the first-order solution – it introduces nothing new and can therefore be ignored in theconstruction of the general solution. Now we conjecture that the generic solution is givenby ∆ ( n ) a = ( − i ) n − n ! [ E (cid:63) , [ E (cid:63) , . . . [ E (cid:63) , ∂ a E ]] . . . ] ( n − − i ) n n ! [ E (cid:63) , [ E (cid:63) , . . . [ E (cid:63) , u a ]] . . . ] ( n Moyal brackets) . (B.14)This can be proved by induction. Using (B.14) in (B.8) gives us finally h a ( x, u ) = ∞ (cid:88) n =1 ( − i ) n n ! [ E ( x, u ) (cid:63) , [ E ( x, u ) (cid:63) , . . . [ E ( x, u ) (cid:63) , u a ]] . . . ]= e − i E ( x,u ) (cid:63) (cid:63) u a (cid:63) e i E ( x,u ) (cid:63) − u a = − i e − i E ( x,u ) (cid:63) (cid:63) ∂ xa e i E ( x,u ) (cid:63) (B.15)where during the passage from the first to the second line we used the Baker-Campbell-Hausdorff lemma. We have obtained (B.4), therefore we proved that F ab ( x, u ) = 0 = ⇒ HS potential is a pure gauge (B.16)– 54 –n some neighbourhood of h a ( x, u ) = 0. (cid:4) The perturbative construction outlined above has some limitations, in the sense thatone cannot obtain all configurations with vanishing field strength by gauge transformingthe configuration h a ( x, u ) = 0, which is e (0) a ( x, u ) = u a = δ µa u µ . (B.17)To see this let us consider configurations of the form e a ( x, u ) = M aµ u µ (B.18)with M a constant real d × d matrix, for which (B.7) is also true. If we want to connectsuch configurations with the vacuum (B.17) by an MHS transformation, the MHS gaugeparameter must be of the form E Λ ( x, u ) = x µ Λ µν u ν (B.19)with Λ again a constant real d × d matrix. Now one has[ E Λ ( x, u ) (cid:63) , u µ ] = i Λ µν u ν (B.20)from which follows e Λ a ( x, u ) ≡ e − i E Λ ( x,u ) (cid:63) (cid:63) u a (cid:63) e i E Λ ( x,u ) (cid:63) = ∞ (cid:88) n =0 ( − i ) n n ! [ E Λ ( x, u ) (cid:63) , [ E Λ ( x, u ) (cid:63) , . . . [ E Λ ( x, u ) (cid:63) , u a ]] . . . ]= δ µa ( e Λ ) µν u ν (B.21)If M cannot be written as an exponential of some matrix, the corresponding configuration(B.18) is not MHS gauge equivalent to the vacuum (B.17).There is more than one gauge orbit consisting of solutions of EoM satisfying T ab ( x, u ) =0, which is to say there are many vacua in the (pure) MHSYM theory. The above construc-tion of the orbit of h a ( x, u ) = 0 was limited by taking e (0) a ( x, u ) = δ µa u µ as the 0 th -orderterm. If we had started with e (0) a ( x, u ) = M aµ u µ , with matrix M of rank < d ,we wouldinstead cover a different gauge orbit. C. Weyl-Wigner formalism Borrowing the language from the phase space formulation of a quantized particle, we canrewrite free field actions using the Wigner function, and show how a gauging of the MHSsymmetry leads to the existence and transformation properties of the MHS potential. In asimilar way, this was originally realized in [25, 26], and in [27] for the frame formalism.– 55 – .1 Wigner function One can rewrite the free field action for a (for simplicity massless) complex scalar field inthe following way (as displayed in sect. (5.1) and using ˆ K s (ˆ u ) = η ab ˆ u a ˆ u b ) S (0) [ φ ] = (cid:90) d d x ∂ µ φ ∂ µ φ † = (cid:104) α φ | ˆ K s (ˆ u ) | α φ (cid:105) = Tr (cid:16) ˆ K s (ˆ u ) | α φ (cid:105)(cid:104) α φ | (cid:17) . (C.1)Under the Weyl-Wigner map, a trace of a product of two operators is mapped to an integralover the phase space of a Moyal product of their respective symbols. The necessary symbolsare calculated using the Wigner map W [ ˆ F ] = f ( x, u ) = (cid:90) d d q (cid:104) x − q | ˆ F | x + q (cid:105) e iq · u (C.2)providing the symbol of the kernel W [ ˆ K ] = (cid:90) d d q (cid:104) x − q | η ab ˆ u a ˆ u b | x + q (cid:105) e iq · u = u (C.3)and the symbol of the projector (which turns out to be the Wigner function [53, 54]) W [ | α φ (cid:105)(cid:104) α φ | ] = (cid:90) d d q (cid:104) x − q | α φ (cid:105)(cid:104) α φ | x + q (cid:105) e iq · u = (cid:90) d d q φ ( x − q/ φ † ( x + q/ e iq · u = (2 π ) d φ ( x ) (cid:63) δ d ( u ) (cid:63) φ † ( x ) (C.4)where (cid:63) is the Moyal product. The expression in the last line can be proved in the followingway (2 π ) d φ ( x ) (cid:63) δ d ( u ) (cid:63) φ † ( x ) = (cid:90) d d q φ ( x ) (cid:63) e iqu (cid:63) φ † ( x )= (cid:90) d d q φ ( x ) (cid:63) (cid:104) e iqu φ † ( x + q (cid:105) = (cid:90) d d q φ ( x − q e iqu φ † ( x + q . (C.5)To avoid having factors of (2 π ) in expressions, we redefine the Wigner function as W φ ≡ (2 π ) − d W [ | α φ (cid:105)(cid:104) α φ | ] = φ ( x ) (cid:63) δ d ( u ) (cid:63) φ † ( x ) . (C.6)By using (5.5), it can be shown that the Wigner function transforms as an MHS tensor inthe adjoint representation δ ε W φ ( x, u ) = i [ W φ ( x, u ) (cid:63) , ε ( x, u )] . (C.7)Alternatively, the Wigner function can be thought of as the fundamental object used todescribe matter, and its transformation properties can be postulated as above.– 56 – .2 Second order/metric formalism Using the Wigner function, the action for a free complex scalar field can be written out inthe phase space as S = (cid:90) d d x ∂ µ φ ∂ µ φ † = (cid:90) d d x d d u u (cid:63) W φ (C.8)and we can see that for a rigid gauge parameter ε ( u ) we have a symmetry of the action δS = i (cid:90) d d x d d u (cid:0) [ ε ( u ) (cid:63) , W φ ( x, u ) (cid:63) u ] + W φ ( x, u ) (cid:63) [ u (cid:63) , ε ( u )] (cid:1) = 0 . (C.9)The first term is a boundary term and is discarded. The second term has a Moyal com-mutator of only u -dependent quantities, so it vanishes.Considering a local symmetry ε = ε ( x, u ), the variation of free field action is non-vanishing δS = i (cid:90) d d x d d u W φ ( x, u ) (cid:63) [ u (cid:63) , ε ( x, u )] . (C.10)In the spirit of Yang Mills theory, this calls for a compensating field to obtain local invari-ance u → u − h ( x, u ) . (C.11)To first order in changes in W φ and h , by neglecting total derivatives under the integral,we have δS = (cid:90) d d x d d u W φ (cid:63) ( − δh + i [( u − h ) (cid:63) , ε ]) . (C.12)To keep the action symmetric under local transformations, we can infer that h ( x, u ) musttransform as δh ( x, u ) = 2 u · ∂ x ε ( x, u ) − i [ h ( x, u ) (cid:63) , ε ( x, u )] (C.13)reproducing the result in [26]. C.3 First order/frame formalism The free scalar field action can be written in an equivalent way (due to the cyclicity of theMoyal product under integration) leading to the first order or frame representation. S = (cid:90) d d x ∂ µ φ ∂ µ φ † = (cid:90) d d x d d u u a (cid:63) W φ (cid:63) u a . (C.14)Again, for a rigid ε ( x, u ) = ε ( u ) we have a symmetry since δS = i (cid:90) d d x d d u ([ u a (cid:63) , W φ (cid:63) ε (cid:63) u a ] + [ ε (cid:63) , u a ] (cid:63) { W φ (cid:63) , u a } ) . To see this, add and subtract W φ ( x, u ) (cid:63) u (cid:63) ε ( u ) under the integral. To see this add and subtract + W φ (cid:63) ε (cid:63) u a (cid:63) u a − W φ (cid:63) ε (cid:63) u a (cid:63) u a + ε (cid:63) u a (cid:63) W φ (cid:63) u a − ε (cid:63) u a (cid:63) W φ (cid:63) u a and use cyclicity of the Moyal product under integration. – 57 –he first term is discarded as it is a boundary term, and the second term vanishes since ε = ε ( u ).In case where the gauge parameter ε = ε ( x, u ) is also a function of x , the second termdoes not vanish, so again a compensating field is introduced, however this time it is aLorentz vector field h a ( x, u ). S = (cid:90) d d x d d u ( u a + h a ) (cid:63) W φ (cid:63) ( u a + h a ) . (C.15)The variation of the action becomes: δS = (cid:90) d d x d d u (cid:16) δh a (cid:63) { W φ (cid:63) , ( u a + h a ) } + i ( u a + h a ) (cid:63) W φ (cid:63) ε (cid:63) ( u a + h a ) (C.16) − i ( u a + h a ) (cid:63) ε (cid:63) W φ (cid:63) ( u a + h a ) (cid:17) . (C.17)Following the logic above and discarding the boundary terms we obtain: δS = (cid:90) d d x d d u ( δh a + i [ ε (cid:63) , ( u a + h a )]) (cid:63) { W φ (cid:63) , ( u a + h a ) } from which we conclude that the action is locally invariant if the compensating field hasthe following infinitesimal transformation law: δh a ( x, u ) = ∂ a ε ( x, u ) + i [ h a ( x, u ) (cid:63) , ε ( x, u )] . (C.18)This reproduces the known result obtained using a free fermion action [27]. References [1] F. W. Hehl and Y. N. Obukhov, Conservation of Energy-Momentum of Matter as the Basisfor the Gauge Theory of Gravitation , Fundam. Theor. Phys. (2020), 217-252.[arXiv:1909.01791 [gr-qc]].[2] X. Bekaert, S. Cnockaert, C. Iazeolla and M. A. Vasiliev, Nonlinear higher spin theories invarious dimensions , Proceedings of the First Solvay Workshop (Brussels, May 12-14, 2004),eds. R. Argurio, G. Barnich, G. Bonelli and M. Grigoriev (Int. Solvay Institutes, 2006)[arXiv:hep-th/0503128 [hep-th]].[3] A. K. H. Bengtsson, Towards Unifying Structures in Higher Spin Gauge Symmetry , SIGMA (2008) 013. [arXiv:0802.0479 [hep-th]].[4] V. E. Didenko and E. D. Skvortsov, Elements of Vasiliev theory , [arXiv:1401.2975 [hep-th]].[5] R. Rahman and M. Taronna, From Higher Spins to Strings: A Primer , [arXiv:1512.07932[hep-th]].[6] E. Skvortsov and T. Tran, One-loop Finiteness of Chiral Higher Spin Gravity , JHEP (2020), 021. [arXiv:2004.10797 [hep-th]].[7] E. Skvortsov, T. Tran and M. Tsulaia, More on Quantum Chiral Higher Spin Gravity , Phys.Rev. D (2020) no.10, 106001. [arXiv:2002.08487 [hep-th]].[8] D. Ponomarev, Chiral Higher Spin Theories and Self-Duality , JHEP (2017), 141.[arXiv:1710.00270 [hep-th]]. – 58 – 9] D. Ponomarev and E. D. Skvortsov, Light-Front Higher-Spin Theories in Flat Space , J. Phys.A (2017) no.9, 095401. [arXiv:1609.04655 [hep-th]].[10] R. R. Metsaev, Poincare invariant dynamics of massless higher spins: Fourth order analysison mass shell, Mod. Phys. Lett. A , 359-367 (1991).[11] M. Taronna, On the Non-Local Obstruction to Interacting Higher Spins in Flat Space, JHEP , 026 (2017). [arXiv:1701.05772 [hep-th]].[12] R. Roiban and A. A. Tseytlin, On four-point interactions in massless higher spin theory inflat space, JHEP , 139 (2017). [arXiv:1701.05773 [hep-th]].[13] X. Bekaert, N. Boulanger and P. Sundell, How higher-spin gravity surpasses the spin twobarrier: no-go theorems versus yes-go examples, Rev. Mod. Phys. (2012) 987.[arXiv:1007.0435 [hep-th]].[14] F. A. Berends, G. J. H. Burgers and H. Van Dam, On the theoretical problems in constructinginteractions involving higher-spin massless particles , Nucl. Phys. B260 (1985) 295.[15] M. A. Vasiliev, Consistent equation for interacting gauge fields of all spins in(3+1)-dimensions , Phys. Lett. B (1990) 378; Properties of equations of motion ofinteracting gauge fields of all spins in (3+1)-dimensions, Class. Quant. Grav. (1991) 1387; Algebraic aspects of the higher spin problem, Phys. Lett. B (1991) 111; More onequations of motion for interacting massless fields of all spins in (3+1)-dimensions, Phys.Lett. B (1992) 225.[16] C. Sleight and M. Taronna, Higher-Spin Gauge Theories and Bulk Locality , Phys. Rev. Lett. (2018) no.17, 171604. [arXiv:1704.07859 [hep-th]].[17] D. Francia, J. Mourad and A. Sagnotti, Current Exchanges and Unconstrained Higher Spins ,Nucl. Phys. B (2007), 203. [arXiv:hep-th/0701163 [hep-th]].[18] D. Francia, Geometric Lagrangians for massive higher-spin fields , Nucl. Phys. B (2008),77. [arXiv:0710.5378 [hep-th]].[19] A. Sagnotti, Higher Spins and Current Exchanges , PoS CORFU2011 (2011), 106.[arXiv:1002.3388 [hep-th]].[20] A. Campoleoni and D. Francia, Maxwell-like Lagrangians for higher spins , JHEP (2013),168. [arXiv:1206.5877 [hep-th]].[21] L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari, B. Lima de Souza and T. ˇStemberga, One-loop effective actions and higher spins , JHEP (2016) 084. [arXiv:1609.02088[hep-th]].[22] L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari and T. ˇStemberga, One-loop effectiveactions and higher spins. Part II , JHEP (2018) 080. [arXiv:1709.01738 [hep-th]].[23] L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari and T. ˇStemberga, HS in flatspacetime. The effective action method , Eur. Phys. J. C (2019) no.3, 258.[arXiv:1811.04847 [hep-th]].[24] L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari and T. ˇStemberga, HS in flatspacetime. YM-like models , [arXiv:1812.05030 [hep-th]].[25] X. Bekaert, E. Joung and J. Mourad, On higher spin interactions with matter , JHEP (2009) 126. [arXiv:0903.3338 [hep-th]]. – 59 – 26] X. Bekaert, E. Joung and J. Mourad, Effective action in a higher-spin background , JHEP (2011) 048. [arXiv:1012.2103 [hep-th]].[27] L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari, M. Pauliˇsi´c and T. ˇStemberga, Worldline quantization of field theory, effective actions and L ∞ structure , JHEP (2018)095. [arXiv:1802.02968 [hep-th]].[28] X. Bekaert, Higher spin algebras as higher symmetries , Ann. U. Craiova Phys. (2006)no.II, 58. [arXiv:0704.0898 [hep-th]].[29] P. de Medeiros and S. Ramgoolam, Non-associative gauge theory and higher spininteractions , JHEP (2005) 072. [arXiv:hep-th/0412027].[30] M. R. Douglas and N. A. Nekrasov, Noncommutative field theory , Rev. Mod. Phys. (2001), 977. [arXiv:hep-th/0106048].[31] L. Bonora and S. Giaccari, Supersymmetric HS Yang-Mills-like models , Universe (2020)no.12, 245. [arXiv:2011.00734 [hep-th]].[32] R. L. Bishop and R. J. Crittenden, Geometry of Manifolds , (Academic Press, Elsevier, 1964)[33] R. Jackiw, Gauge-Covariant Conformal Transformations , Phys. Rev. Lett. (1978) 1635.[34] M. Abou-Zied, H. Dorn, Comments on the energy momentum tensor in non-commutativefield theories , Phys. Lett. B (2001) 183. [arXiv:hep-th/0104244].[35] A. Das, J. Frenkel, Energy-momentum tensor in noncommutative gauge theories , Phys. Rev.D (2003) 067701. [arXiv:hep-th/0212122].[36] J. M. Grimstrup, B. Kloib¨ock, L. Popp, M. Schweda, M. Wickenhauser, V. Putz, Theenergy-momentum tensor in nocommutative gauge field models , Int. Jour. Mod. Phys. A (2004) 5615. [arXiv:hep-th/0210288].[37] H. Balasin, D. N. Blaschke, F. Gieres, M. Schweda, On the energy-momentum tensor inMoyal space , Eur. Phys. J. C (2015) 284. [arXiv:1502.03765 [hep-th]].[38] S. A. Merkulov, The Moyal product is the matrix product , [arXiv:math-ph/0001039][39] S. Alexander, L. Jenks and E. McDonough, Higher Spin Dark Matter , [arXiv:2010.15125[hep-ph]].[40] “xAct: Efficient tensor computer algebra for Mathematica.” http://xact.es/index.html.[41] T. Nutma, xTras: A field-theory inspired xAct package for mathematica , Comp. Phys. Comm. (2014) 1719. [arXiv: 1308.3493 [cs.SC]].[42] R. Aldrovandi and J. G. Pereira, Teleparallel Gravity: An Introduction (Springer, Dordrecht,2012).[43] K. Hayashi and T. Shirafuji, New general relativity , Phys. Rev. D .12 (1979) 3524.[44] A. Einstein, Riemann-Geometrie mit Aufrechterhaltung des Begriffes des Fernparallelismus ,Sitzber. Preuss. Akad. Wiss. (1928) 217; Neue M¨oglichkeit f¨ur eine einheitlicheFeldtheorie von Gravitation und Elektrizit¨at , Sitzber. Preuss. Akad. Wiss. (1928) 224.[45] K. Hayashi and T. Nakano, Extended translation invariance and associated gauge fields , Prog.Theor. Phys. (1967) 491.[46] J. G. Pereira and Y. N. Obukhov, Gauge Structure of Teleparallel Gravity , Universe (2019)139. [arXiv:1906.06287 [gr-qc]]. – 60 – 47] M. Le Delliou, E. Huguet and M. Fontanini, Teleparallel theory as a gauge theory oftranslations: Remarks and issues , Phys. Rev. D (2020) 024059. [arXiv:1910.08471[gr-qc]].[48] A. Einstein, Auf die Riemann-Metrik und den Fern-Parallelismus gegr¨undete einheitlicheFeldtheorie , Math. Ann. (1929) 685.[49] H. C. Steinacker, On the quantum structure of space-time, gravity, and higher spin in matrixmodels , Class. Quant. Grav. (2020) no.11, 113001. [arXiv:1911.03162 [hep-th]].[50] H. C. Steinacker, Higher-spin gravity and torsion on quantized space-time in matrix models ,JHEP (2020), 111. [arXiv:2002.02742 [hep-th]].[51] X. Bekaert and E. D. Skvortsov, Elementary particles with continuous spin , Int. J. Mod.Phys. A (2017) no.23n24, 1730019. [arXiv:1708.01030 [hep-th]].[52] P. Schuster and N. Toro, A Gauge Field Theory of Continuous-Spin Particles , JHEP (2013) 06. [arXiv:1302.3225 [hep-th]].[53] E. P. Wigner, On the quantum correction for thermodynamic equilibrium , Phys. Rev. (1932) (5): 749.[54] D. B. Fairlie, Moyal brackets, star products and the generalized Wigner function , Chaos,Solitons Fractals 10.2-3 (1999) 365. [arXiv:hep-th/9806198]., Chaos,Solitons Fractals 10.2-3 (1999) 365. [arXiv:hep-th/9806198].