Notes on HigherSpin Diffeomorphisms
Abstract
:1. Introduction
2. HigherSpin Gauge Symmetries in the Unconstrained Metriclike Formulation
2.1. NonAbelian Deformations of HigherSpin Gauge Symmetries
2.2. Two Examples of Infinitesimal HigherSpin Gauge Symmetries
2.3. Problems with HigherSpin Diffeomorphisms
2.4. Notation and Terminology
2.5. NoGo Theorems
2.6. Definitions
2.7. Automorphisms
3. Symplectomorphisms of the Cotangent Bundle
3.1. Lagrangian Submanifolds
3.2. Flows of Hamiltonians of Degree Zero
3.3. Lagrangian Foliations
3.4. Flows of Hamiltonians of Degree One
 (a)
 preserves the tautological oneform,
 (b)
 is the lift of a flow on the base manifold M generated by a base vector field, $\widehat{X}={X}^{\mu}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{\partial}_{\mu}$,
 (c)
 is generated by a homogenous Hamiltonian of degree one in the momenta, $H(x,p)={X}^{\mu}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{p}_{\mu}$.
 (a)
 preserves the vertical polarisation, i.e., it maps cotangent spaces ${T}_{m}^{*}M$ to cotangent spaces ${T}_{{m}^{\prime}}^{*}M$,
 (b)
 is the composition of a vertical symplectomorphism and the lift of a diffeomorphism of the base M,
 (c)
 is generated by a Hamiltonian of degree one in the momenta, $H(x,p)={X}^{\mu}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{p}_{\mu}+f\left(x\right)$.
3.5. Flows of Hamiltonians of Higher Degree
3.6. Summary
4. Quantisation of the Cotangent Bundle: Differential Operators as Symbols and Vice Versa
4.1. Quantisation of the Cotangent Bundle
 (i)
 a collection of isomorphisms ${\tilde{q}}_{i}={\sigma}_{i}\circ {q}_{i}:{\mathcal{S}}_{i}\stackrel{\sim}{\to}{\mathrm{gr}}_{i}\mathcal{A}$ of vector spaces, and
 (ii)
 an isomorphism $\tilde{q}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}\mathcal{S}\stackrel{\sim}{\to}\mathrm{gr}\phantom{\rule{0.166667em}{0ex}}\mathcal{A}$ of Schouten algebras between $\mathcal{S}$ and the Poisson limit of $\mathcal{A}$,
4.2. Compatibility Condition
 (i)
 it is an algebra morphism, i.e., it relates the product ★ in $\mathcal{B}$ to the product ∘ in $End\left(\mathcal{A}\right)$ one has:$$\widehat{{b}_{1}\u2605{b}_{2}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\widehat{b}}_{1}\circ {\widehat{b}}_{2}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}\forall {b}_{1},{b}_{2}\in \mathcal{B}\phantom{\rule{0.166667em}{0ex}},$$In particular, an anchor (39) defines a representation of the $\mathcal{A}$ring $\mathcal{B}$ on its base algebra $\mathcal{A}$.
 (ii)
 it relates their unit maps, i.e., the anchor extends the canonical isomorphism (32) in the sense that$$\widehat{i\left(a\right)}=\widehat{a}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}\forall a\in \mathcal{A}\phantom{\rule{0.166667em}{0ex}},$$$${\widehat{1}}_{\mathcal{B}}=i{d}_{\mathcal{A}}\phantom{\rule{0.166667em}{0ex}}.$$
4.3. Examples of Quantisation of the Cotangent Bundle
5. Quantisation of the Cotangent Bundle: Going beyond Differential Operators
5.1. QuasiDifferential Operators
5.2. Criteria on the Strict Product
5.3. Strict HigherSpin Diffeomorphisms
 1.
 A strict higherspin diffeomorphism ${C}_{\u2605}^{\infty}\left({T}^{*}M\right)\stackrel{\sim}{\to}{C}_{\u2605}^{\infty}\left({T}^{*}N\right)$ induces a standard diffeomorphism $M\stackrel{\sim}{\to}N$,
 2.
 Its restriction to the subalgebra of differential operators of order zero is an isomorphism ${\mathcal{S}}^{0}\left(M\right)\stackrel{\sim}{\to}{\mathcal{S}}^{0}\left(N\right)$ of commutative algebras,
 3.
 Its restriction to the whole subalgebra of differential operators is an isomorphism $\mathcal{D}\left(M\right)\stackrel{\sim}{\to}\mathcal{D}\left(N\right)$ of associative algebras.
5.4. Formal Completion
6. Almost Differential Operators
6.1. Formal Power Series over an Algebra
6.2. AlmostDifferential Operators
7. Deformation Quantisation: Sample of Results
7.1. Star Products
7.2. Equivalences of Star Products and Automorphisms of Deformations
7.3. Derivations
8. Quasi Differential Operators
8.1. Star Products of Symbols of Differential Operators
8.2. Formal QuasiDifferential Operators
 (1)
 Is differential, and
 (2)
 Reduces to the star product on $\mathcal{S}\left(M\right)[\hslash ]$.
9. HigherSpin Diffeomorphisms
9.1. Looking for HigherSpin Diffeomorphisms
9.2. Formal HigherSpin Diffeomorphisms
 1.
 symplectomorphisms from the cotangent bundle ${T}^{*}{M}^{\prime}$ to the cotangent bundle ${T}^{*}M$,
 2.
 classical limit of formal higherspin diffeomorphisms between the manifolds M and ${M}^{\prime}$,
 3.
 equivalence classes of ℏ linear isomorphisms between the associative algebras ${C}_{\u2605}^{\infty}\left({T}^{*}M\right)\u301a\hslash \u301b$ and ${C}_{{\u2605}^{\prime}}^{\infty}\left({T}^{*}{M}^{\prime}\right)\u301a\hslash \u301b$ of formal quasidifferential operators with respect to the equivalence of star products,
 4.
 ℏ linear isomorphisms between the formal algebras $\mathcal{Q}\mathcal{D}\left(M\right)$ and $\mathcal{Q}\mathcal{D}\left({M}^{\prime}\right)$ of quasidifferential operators.
9.3. Formal HigherSpin Flows
10. Quotient Algebra of AlmostDifferential Operators
10.1. Subalgebra
10.2. Quotient Algebra
11. Conclusions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Proof of Technical Lemma
Notes
1  
2  A distinct but related issue is the degree of (non)locality of higherspin interactions. It has been a subject of intense scrutiny and debate over the last years. This open problem will not be adressed here, although one may hope that the unavoidable nonlocality of finite higherspin gauge symmetries might shed some light on this issue in the future. 
3  This difficulty was recognised immediately by Fronsdal, despite he actually found such a nonabelian deformation together with a Lie bracket over the space of traceless symmetric tensor fields [33]. 
4  In spacetime dimension four, it is wellknown [7,8,9,10,11] that these trace constraints can be taken into account in the spinorial framelike formulation (where the fields are base oneforms and fibre tensorspinors). At nonlinear level, the corresponding gauged algebra of infinitesimal symmetries is the Weyl algebra of polynomial differential operators on the tangent space [1,2,3,4,5,6]. Nogo theorems analogous to [29,30] have been established for the Weyl algebra [34,35]. 
5  
6  Let us repeat that, in the present paper, the focus is on finite gauge symmetries in the unconstrained metriclike formulation for technical simplicity. It is natural to expect that our main conclusions should apply to the original constrained metriclike formulation of Fronsdal without any qualitative change. 
7  
8  The prescription for (9) is as follows: one consider a covariantised Weyl map $\mathcal{W}:h(x,p)\mapsto \widehat{H}$ where the operators are obtained from their symbols (3) via (i) the quantisation rule $p\mapsto i\ell \nabla $ and (ii) the anticommutatorordering prescription with respect to the covariant derivative
$$\widehat{H}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\left(\right)}_{exp}h(x,p)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\sum _{s\u2a7e0}\frac{{(i\ell )}^{s}}{{2}^{s}\phantom{\rule{0.166667em}{0ex}}s!}{\left(\right)}_{{\nabla}_{{\mu}_{1}}}$$

9  Let us stress that the words “quantum”, “quantisation”, etc, throughout this paper should be taken in a mathematical technical sense, not in a physical literal sense. In deformation quantisation, “quantum” is synonymous of “associative” while “classical” is synonymous of “Poisson”. 
10  For the sake of simplicity, reality conditions will not be discussed in this paper. All algebras considered here will be complex, of which suitable reals forms (e.g., of Hermitian operators) can be extracted if necessary. From now on, factors of i will be dropped from all formulae. 
11  Note that the socalled “Lie’s third theorem” stating that every finitedimensional Lie algebra $\mathfrak{g}$ over the real numbers is associated to a Lie group G does not hold in general for infinitedimensional Lie algebras. 
12  Moreover, if the algebra $\mathcal{A}$ has a unit element, then one further requires that $1\in {\mathcal{A}}_{0}$. Here, all associative algebras are assumed to have a unit element and, accordingly, morphisms of associative algebras relate their unit elements. 
13  Such a Poisson algebra was called a “classical Poisson algebra” in [29]. Since this terminology may be confusing to people familiar with the vocabulary in deformation quantisation (where Poisson algebras are, by definition, classical), one chose to call them Schouten algebras (as a tribute to the Schouten bracket of symmetric multivector fields). Note that a Gerstenhaber algebras are the supercommutative analogues of Schouten algebras. 
14  There would have been a way out if the higherspin Lie derivative was locally nilpotent. However, it has been shown (cf. Section 3 of [29]) that $a{d}_{\widehat{X}}$ is locally nilpotent (more precisely: for any zerothorder differential operator $\widehat{f}$, there exists a positive integer n such that $a{d}_{\widehat{X}}^{n}\widehat{f}=0$) iff $\widehat{X}$ is a differential operator of order zero. This is the crucial technical lemma behind the nogo theorems. 
15  The Stone?Weierstrass theorem ensures that $\mathcal{S}\left(M\right)$ is dense inside ${C}^{\infty}\left({T}^{*}M\right)$. 
16  If the quantised Schouten algebra happens to be equal to the Poisson limit of the almostcommutative algebra (i.e., $\mathcal{S}=\mathrm{gr}\mathcal{A}$) then one also requires that the restriction ${q}_{i}$ of the quantisation is a section of the restriction ${\sigma}_{i}$ of the principal symbol map (${\sigma}_{i}\circ {q}_{i}=i{d}_{{\mathcal{S}}_{i}}$). 
17  Note that the injectivity can be assumed without loss of generality, in the sense that one can always focus on the quotient algebra $\mathcal{A}/ker\phantom{\rule{0.166667em}{0ex}}i$. The terminology originates from the fact that, in particular, the unit map relates the two units in the sense that $i\left({1}_{\mathcal{A}}\right)={1}_{\mathcal{B}}$). 
18  
19  This terminology is borrowed from [44] where the anchor is defined for bialgebroids (an extra compatibility condition with the coproduct is added). 
20  This quantisation is not canonical since, by construction, it depends explicitly on a choice of specific coordinate system. Nevertheless, this normaltype quantisation can be made geometrical (i.e., globally welldefined and coordinateindependent) for a generic manifold M by considering the following data: an affine connection on the base manifold M (cf. [45]). Retrospectively, the corresponding normal coordinates provide a privileged coordinate system. 
21  Usually, the term “strict quantisation” refers to one of the (many) mathematical approaches to the problem of quantisation and often refers to the axiomatisation by Rieffel (see, e.g., their book [48]). Here, the term is understood in a nontechnical sense. 
22  Since the idea behind pseudodifferential operators is that they behave asymptotically like differential operators (whose “order” can be any real number), they should face the same problem that was encountered for differential operators in Section 2.7. 
23  This can be shown as follows: First, the relation $\widehat{X}\left[f\right]=(\widehat{X}\circ \widehat{f})\left[1\right]$ is true by the very definition of the map (32). Second, the quantisation map is assumed to be an algebra morphism, hence $Q:X\u2605f\mapsto \widehat{X\u2605f}=\widehat{X}\circ \widehat{f}$. Third, the quantisation map is assumed compatible, thus $\widehat{X}\left[f\right]=\widehat{X\u2605f}\left[1\right]={(X\u2605f)}_{0}$. This ends the proof. In particular, for functions X on the cotangent bundle which are polynomial in the momenta, the relation (63) reproduces the action on functions f of differential operators $\widehat{X}$ with symbol X. 
24  
25  In the applications to higherspin gravity, this formal parameter should tentatively be identified with the parameter ℓ with the dimension of a length, mentioned below Equation (9). However, for the general mathematical considerations of Section 6, Section 7, Section 8 and Section 10, it is a purely formal deformation parameter. 
26  One speaks of strict deformations if the series are convergent for $0\u2a7d\hslash \u2a7d1$, plus some extra technical assumptions, cf. the celebrated definition by Rieffel [48]. 
27  This is true globally if the second Betti number of $\mathcal{M}$ vanishes. 
28  This remains true globally if the first Betti number of $\mathcal{M}$ vanishes. 
29  This is a slight abuse of terminology since, strictly speaking, they are not inner derivations. In fact, by definition ${}^{\u2605}a{d}_{X}$ is an inner derivation, but ${\mathcal{L}}_{X}^{\hslash}$ is not (since $1/\hslash \notin \mathbb{C}\u301a\hslash \u301b$). 
30  
31  As shown in [57,59], the analogue of the Gelfand–Naimark–Segal construction applies for such formal deformations ${C}_{\u2605}^{\infty}\left({T}^{*}M\right)\u301a\hslash \u301b$. This establishes on firm ground the quantum mechanical interpretation of these elements as operators acting on a Hilbert space. This is not explored here but could be important in the future for looking for a suitable real form of the algebra of formal quasidifferential operators considered here. 
32  For the sake of completeness, note that the corresponding star products for a noncanonical symplectic twoform on the cotangent bundle have been studied in [59]. 
33  This last property holds because the star product is natural. Detailed statements about the quantum correction to the Heisenbergpicture time evolution can be found in Appendix B of [57]. 
34  For instance, if the base manifold is the Euclidean space $M={\mathbb{R}}^{n}$, then the function $f\left(x\right)exp(1/\overrightarrow{p}\phantom{\rule{0.166667em}{0ex}}{}^{2})$ is such that its Taylor series along the direction of momenta vanishes on the zero section $\overrightarrow{p}=\overrightarrow{0}$. 
35  This can be checked explicitly for the normal star product in Darboux coordinates, via the Formula (115). Nevertheless, the conclusion is coordinatefree and remains valid for any equivalent differential star product. 
36  From the point of view of higherspin gravity, the status of such a strong condition on higherspin diffeomorphisms is unclear since it would remove the Maxwell gauge symmetries of the spinone sector (since they correspond to vertical automorphisms of the cotangent bundle that do not preserve the zero section). Nevertheless, they appear as a reasonable candidate subclass of higherspin symmetries which could be compatible with some weak notion of locality. 
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Lie Algebra  Dimension  Exact Symplectomorphisms  Basis of Hamiltonian Vector Fields  Basis of Hamiltonians 

${V}^{*}$  n  Vertical translations  $\frac{\partial}{\partial {p}_{a}}$  ${x}^{a}$ 
${V}^{*}\odot {V}^{*}$  $\frac{n(n+1)}{2}$  Linear vertical symplectom  ${x}^{b}\phantom{\rule{0.166667em}{0ex}}\frac{\partial}{\partial {p}_{c}}+{x}^{c}\phantom{\rule{0.166667em}{0ex}}\frac{\partial}{\partial {p}_{b}}$  $\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{x}^{b}\phantom{\rule{0.166667em}{0ex}}{x}^{c}$ 
${V}^{*}\oplus ({V}^{*}\odot {V}^{*})$  $\frac{n(n+3)}{2}$  Affine vertical symplectom  $\frac{\partial}{\partial {p}_{a}}$, ${x}^{b}\frac{\partial}{\partial {p}_{c}}+{x}^{c}\phantom{\rule{0.166667em}{0ex}}\frac{\partial}{\partial {p}_{b}}$  ${x}^{a}$, ${x}^{b}\phantom{\rule{0.166667em}{0ex}}{x}^{c}$ 
V  n  Horizontal translations  $\frac{\partial}{\partial {x}^{a}}$  ${p}_{a}$ 
$\mathfrak{gl}\left(V\right)$  ${n}^{2}$  Lift of linear transformations  ${x}^{b}\phantom{\rule{0.166667em}{0ex}}\frac{\partial}{\partial {x}^{c}}{p}_{c}\phantom{\rule{0.166667em}{0ex}}\frac{\partial}{\partial {p}_{b}}$  ${x}^{b}{p}_{c}$ 
$\mathfrak{igl}\left(V\right)$  ${n}^{2}+n$  Lift of affine transformations  $\frac{\partial}{\partial {x}^{a}}$, ${x}^{b}\frac{\partial}{\partial {x}^{c}}{p}_{c}\frac{\partial}{\partial {p}_{b}}$  ${p}_{a}$, ${x}^{b}{p}_{c}$ 
$V\odot V$  $\frac{n(n+1)}{2}$  Linear changes of polarisation  ${p}_{a}\frac{\partial}{\partial {x}^{b}}+{p}_{b}\frac{\partial}{\partial {x}^{a}}$  $\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{p}_{a}\phantom{\rule{0.166667em}{0ex}}{p}_{b}$ 
$\mathfrak{isp}\left({T}^{*}V\right)$  $2{n}^{2}+3n$  Affine symplectomorphisms  Affine vector fields  degree 2 
Classical  Quantum  

Algebra  Poisson algebra (symplectic)  Associative algebra (central) 
${C}^{\infty}\left({T}^{*}M\right)$  $\mathcal{Q}\mathcal{D}\left(M\right)$  
Elements  Functions on the cotangent bundle  Quasidifferential operators 
$X(x,p)$  $\widehat{X}(x,\partial )$  
Graded/Filtered  Schouten algebra  Almostcommutative algebra 
subalgebra  $\mathcal{S}\left(M\right)$  $\mathcal{D}\left(M\right)$ 
Elements  Symbols  Differential operators 
$X(x,p)={\displaystyle \sum _{r=0}^{k}}\frac{1}{r!}\phantom{\rule{0.166667em}{0ex}}{X}^{{\mu}_{1}\cdots {\mu}_{r}}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{p}_{{\mu}_{1}}\cdots {p}_{{\mu}_{r}}$  $\widehat{X}={\displaystyle \sum _{r=0}^{k}}\frac{1}{r!}\phantom{\rule{0.166667em}{0ex}}{X}^{{\mu}_{1}\cdots {\mu}_{r}}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{\partial}_{{\mu}_{1}}\cdots {\partial}_{{\mu}_{r}}$  
Commutative  Base algebra  Order zero subalgebra 
subalgebra  ${C}^{\infty}\left(M\right)\subset \mathcal{S}\left(M\right)$  ${\mathcal{D}}^{0}\left(M\right)\subset \mathcal{D}\left(M\right)$ 
Elements  Functions on the base  Differential operators of order zero 
$f\left(x\right)$  $\widehat{f}$ 
Mathematical Objects  Classical (Undeformed)  Quantum (Deformed) 

Un/deformed  Order ${\hslash}^{0}$  Formal power series in ℏ 
Algebra  Symplectic algebra  Associative algebra 
of functions  ${C}^{\infty}\left(\mathcal{M}\right)$  ${C}_{\u2605}^{\infty}\left(\mathcal{M}\right)\u301a\hslash \u301b$ 
Linear maps  $\mathbb{C}$linear  ℏlinear 
Associative algebra of  Endomorphism algebra  Algebra of ℏlinear endomorphisms 
endomorphisms  $\mathrm{End}\left(\right)open="("\; close=")">\phantom{\rule{0.166667em}{0ex}}{C}^{\infty}(\mathcal{M}\phantom{\rule{0.166667em}{0ex}}$  $\mathrm{End}\left(\right)open="("\; close=")">\phantom{\rule{0.166667em}{0ex}}{C}^{\infty}\left(\mathcal{M}\right)\phantom{\rule{0.166667em}{0ex}}$ 
Group of  Group of symplectomorphisms  Group of ℏlinear automorphisms 
finite automorphisms  $\mathrm{Aut}\left(\right)open="("\; close=")">\phantom{\rule{0.166667em}{0ex}}{C}^{\infty}\left(\mathcal{M}\right)\phantom{\rule{0.166667em}{0ex}}$  $\mathrm{Aut}\left(\right)open="("\; close=")">\phantom{\rule{0.166667em}{0ex}}{C}^{\infty}\left(\mathcal{M}\right)\u301a\hslash \u301b\phantom{\rule{0.166667em}{0ex}}\u301a\hslash \u301b$ 
Lie algebra of  Lie algebra of  Lie algebra of ℏlinear derivations 
infinitesimal automorphisms  symplectic vector fields  $\mathfrak{der}\left(\right)open="("\; close=")">\phantom{\rule{0.166667em}{0ex}}{C}^{\infty}\left(\mathcal{M}\right)\u301a\hslash \u301b\phantom{\rule{0.166667em}{0ex}}\u301a\hslash \u301b$ 
Group of finite  Hamiltonian  Flows of nontrivial 
inner automorphisms  flows $\in {C}^{\infty}\left(\mathcal{M}\right)$  inner automorphisms 
Lie algebra of infinitesimal  Lie algebra of  Lie algebra of nontrivial 
inner automorphisms  Hamiltonian vector fields  inner derivations 
Group of  Trivial group  Normal subgroup of pert. redefs 
finite redefinitions  (identity map)  $exp\left(\right)open="("\; close=")">\phantom{\rule{0.166667em}{0ex}}\hslash \phantom{\rule{0.166667em}{0ex}}\mathrm{End}\left(\right)open="("\; close=")">\phantom{\rule{0.166667em}{0ex}}{C}^{\infty}\left(\mathcal{M}\right)\phantom{\rule{0.166667em}{0ex}}$ 
Lie algebra of  Trivial algebra  Ideal of infinitesimal pert. redefs 
infinitesimal redefinitions  (zero map)  $\hslash \phantom{\rule{0.166667em}{0ex}}\mathrm{End}\left(\right)open="("\; close=")">\phantom{\rule{0.166667em}{0ex}}{C}^{\infty}(\mathcal{M}\phantom{\rule{0.166667em}{0ex}}$ 
Group of  Trivial group  Group of starproduct 
selfequivalences  (identity map)  selfequivalences 
Lie algebra of  Trivial algebra  Lie algebra of 
infinitesimal selfequivalences  (zero map)  trivial derivations 
Classical  Quantum  

Algebra  Poisson algebra (symplectic)  Associative algebra (central) 
${C}^{\infty}\left({T}^{*}M\right)$  $\mathcal{Q}\mathcal{D}\left(M\right)$  
Elements  Functions on the cotangent bundle  Quasidifferential operators 
$X(x,p)$  $\widehat{X}(x,\partial )$  
Finite  Symplectomorphisms  Higherspin 
automorphisms  of ${T}^{*}M$  diffeomorphisms of M 
Flow of inner  Hamiltonian flow  Higherspin flow 
automorphisms  on ${T}^{*}M$  on M 
Infinitesimal  Symplectic vector field  Infinitesimal higherspin 
automorphism  on ${T}^{*}M$  diffeomorphism of M 
Inner  Hamiltonian vector  Higherspin 
derivation  field on ${T}^{*}M$  Lie derivative on M 
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Bekaert, X. Notes on HigherSpin Diffeomorphisms. Universe 2021, 7, 508. https://doi.org/10.3390/universe7120508
Bekaert X. Notes on HigherSpin Diffeomorphisms. Universe. 2021; 7(12):508. https://doi.org/10.3390/universe7120508
Chicago/Turabian StyleBekaert, Xavier. 2021. "Notes on HigherSpin Diffeomorphisms" Universe 7, no. 12: 508. https://doi.org/10.3390/universe7120508