Higher spin algebras as higher symmetries
Xavier Bekaert
Laboratoire de Mathématiques et Physique Théorique
Unité Mixte de Recherche 6083 du CNRS, Fédération Denis Poisson
Université François Rabelais, Parc de Grandmount
37200 Tours, France
Abstract
The exhaustive study of the rigid symmetries of arbitrary free field
theories is motivated, along several lines, as a preliminary step in the
completion of the higher-spin interaction problem in full generality. Some
results for the simplest example (a scalar field) are reviewed and com-
mented along these lines.
Expanded version of the lectures presented at the “5th international school
and workshop on QFT & Hamiltonian systems” (Calimanesti, May 2006).
1 Higher-spin interaction problem
Whereas covariant gauge theories describing arbitrary free massless fields
on constant-curvature spacetimes of dimension n are firmly established by
means of the unitary representation theory of their isometry groups, it is
still open to question whether non-trivial consistent self-couplings and/or
cross-couplings among those fields may exist for n > 2 , such that the
deformed gauge algebra is non-Abelian. The goal of the present paper is
to advocate that a lot of information on the interactions can be extracted
from the symmetries of the free field theory.
The conventional local free field theories corresponding to unitary irre-
ducible representations of the helicity group SO(n− 2) , that are spanned
by completely symmetric tensors, have been constructed a while ago (for
some introductory reviews, see [1]). In order to have Lorentz invariance
manifest and second order local field equations with minimal field content,
the theory is expressed in terms of completely symmetric double-traceless
tensor gauge fields hµ1... µs of rank s > 0, the gauge transformation of
which reads
δξ hµ1µ2... µs =
∇ µ1 ξµ2...µs + cyclic , (1)
E-mail address: Xavier.Bekaert@lmpt.univ-tours.fr
http://arxiv.org/abs/0704.0898v2
where
∇ is the covariant derivative with respect to the background Levi–
Civita connection and “cyclic” stands for the sum of terms necessary to
have symmetry of the right-hand-side under permutations of the indices.
The gauge parameter ξ is a completely symmetric traceless tensor field
of rank s − 1. In this relativistic field theory, the “spin” is equal to the
rank s. For spin s = 1 the gauge field hµ represents the photon with U(1)
gauge symmetry while for spin s = 2 the gauge field hµν represents the
graviton with linearized diffeomorphism invariance. The gauge algebra of
field independent gauge transformations such as (1) is of course Abelian.
Non-Abelian gauge theories for “lower spin” s 6 2 are well known and
essentially correspond to Yang-Mills (s = 1) and Einstein (s = 2) theories
for which the underlying geometries (principal bundles and Riemannian
manifolds) were familiar to mathematicians before the construction of the
physical theory. In contrast, the situation is rather different for “higher
spin” s > 2 for which the underlying geometry (if any!) remains obscure.
Due to this lack of information, it is natural to look for inspiration in
the perturbative “reconstruction” of Einstein gravity as the non-Abelian
gauge theory of a spin-two particle propagating on a constant-curvature
spacetime (see e.g. [2] for a comprehensive review).
Let us denote by
S [hµ1... µs ] the Poincaré-invariant, local, second-
order, quadratic, ghost-free, gauge-invariant action of a spin-s symmetric
tensor gauge field. In order to perform a perturbative analysis via the
Noether method [3], the non-Abelian interaction problem for a collection
of higher (and possibly lower) spin gauge fields is formulated as a defor-
mation problem.
Higher-spin interaction problem: List all Poincaré-invariant local
deformations
S[h] =
S [h] + ε
S [h] + O(ε
of a positive sum, with at least one s > 2,
S [h] =
S [hµ1... µs ]
of quadratic actions such that the deformed local gauge symmetries
δξh =
δξ h + ε
δξ h + O(ε
are already non-Abelian at first order, in the deformation parameters ε
and do not arise from local redefinitions
h → h + ε φ(h) + O(ε
) , ξ → ξ + ε ζ(h, ξ) + O(ε
of the gauge fields and parameters.
This well-posed mathematical problem is expected to possess non-
trivial solutions including higher-spin fields, as strongly indicated by Vasiliev’s
works (for some reviews, see [4] and references therein) and deserves to
be investigated further along systematic lines.
2 The Noether method
The assumption that the deformations are formal power series in some
deformation parameters ε enables to investigate the problem order by
order. The crucial observation of any perturbation theory is that the first
order deformations are constrained by the symmetries of the undeformed
system. In the present case, the Noether method scrutinizes the gauge
symmetry of the action, δξS = 0 . At zeroth order, the latter equation is
satisfied by hypothesis. At first order, it reads
S = 0 . (2)
This equation may be used to constrain the possible deformations by
reinterpreting them as familiar objects of the undeformed gauge theory.
By definition, an observable of a gauge theory is a functional which is
gauge-invariant on-shell, while a reducibility parameter of a gauge theory
is a gauge parameter such that the corresponding gauge variation vanishes
off-shell.
First-order deformations in terms of the undeformed theory:
• First-order deformations of the action are observables of the undeformed
theory.
• First-order deformations of the gauge symmetries evaluated at reducibil-
ity parameters of the undeformed gauge theory define symmetries of the
undeformed theory.
Proof: In (2) the infinitesimal variation
S of the undeformed action
is proportional to the undeformed Euler–Lagrange equations. This proves
the fist part of the theorem. Reducibility parameters ξ of the undeformed
gauge theory verify
h = 0 by definition. Inserting this fact into (2)
with ξ = ξ gives
S = 0 , which is precisely the translation of the second
part of the theorem.
In the mathematical litterature, a (conformal) Killing tensor of a
pseudo-Riemannian manifold is a symmetric tensor field ξ such that its
symmetrized covariant derivative with respect to the Levi–Civita connec-
tion, ∇µ1 ξµ2...µs + cyclic, vanishes (modulo a term proportional to the
metric for conformal Killing tensors). Therefore, any reducibility parame-
ter ξ of the spin-s symmetric gauge field theory on the constant-curvature
spacetime M is identified with a Killing tensor of rank s− 1 of the mani-
fold M. The space of Killing tensors on any constant-curvature spacetime
is known to be finite-dimensional [5], thus the linear gauge symmetries (1)
are irreducible.
These results suggest two strategies for addressing the higher-spin in-
teraction problem. The most ambitious one is the computation of all lo-
cal observables of the free gauge theory associated to deformations of the
gauge algebra. This result would provide the exhaustive list of algebra-
deforming first order vertices, but this computation is technically demand-
ing and seems out of reach in the completely general case. Nevertheless,
the BRST reformulation of the problem [6] allowed the complete classi-
fication of non-Abelian deformations in various particular cases (see e.g.
the review [7] and references therein). Actually, a more humble strategy
is the computation of all rigid symmetries of the free irreducible gauge
theory. It is of interest because the knowledge of these rigid symmetries
would strongly constrain the candidates for gauge symmetry deforma-
tions. Indeed, the constant tensors appearing in the rigid symmetries
could be compared with the complete list [5] of constant-curvature space-
time Killing tensors.
3 Free theory symmetries
Bosonic fields are usually described in terms of their components living in
some subspace V of the space ⊗(Rn) of tensors on Rn (e.g. V = ⊙(Rn)
for symmetric tensor fields). The background metric of the constant-
curvature spacetime induces some non-degenerate bilinear form on V .
This defines a non-degenerate sesquilinear form 〈 | 〉 on the space L2(Rn)⊗
V of square-integrable fields taking values in the countable space V (the
components). Let † stands for the adjoint with respect to the sesquilinear
form 〈 | 〉 .
Any quadratic action for bosonic fields ψ can be expressed as a quadratic
S [ψ] =
〈ψ | K | ψ 〉 , (3)
where the kinetic operatorK is self-adjoint, K† = K. Because the sesquilin-
ear form 〈 | 〉 is non-degenerate, the Euler-Lagrange equation extremizing
the quadratic action is the linear equation
δ〈ψ |
= K|ψ 〉 = 0 . (4)
Moreover, the quadratic form 〈ψ | K | ψ 〉 is degenerate if and only if the
kinetic operator K is degenerate. This happens if and only if there exists
a linear operator P (on L2(Rn) ⊗ V ) such that KP = 0. Infinitesimal
gauge symmetries then read
δχ | ψ 〉 = P | χ 〉 ,
with gauge parameters χ . The Noether identity is P†K = (KP)† = 0 .
A symmetry of the quadratic action (3) is an invertible linear pseudo-
differential operator U preserving the quadratic form 〈 | K | 〉. In other
words,
KU = K .
The group of off-shell symmetries is the group of symmetries of the quadratic
action endowed with the composition ◦ as product. A symmetry genera-
tor of the quadratic action (3) is a linear differential operator T which is
self-adjoint with respect to the quadratic form 〈 | K | 〉. More concretely,
KT = T
Any symmetry generator T defines a symmetry U = eiT of the quadratic
action (3). If T = T† then the linear operator T is a symmetry generator
of the quadratic action if and only it commutes with K. The real Lie
algebra of off-shell symmetries is the algebra of symmetry generators of
the quadratic action endowed with i times the commutator as Lie bracket,
{ , } := i [ , ].
A symmetry of the linear equation (4) is a linear differential operator
T obeying
KT = SK , (5)
for some linear operator S. Such a symmetry T preserves the space KerK
of solutions to the equations of motion. Any symmetry generator T of
the action (3) is always a symmetry of the equation of motion (4) with
S = T† in (5). A symmetry T is trivial on-shell if T = RK for some
linear operator R. Such an on-shell-trivial symmetry is always a sym-
metry of the field equation (4), since it obeys (5) with S = KR. The
algebra of on-shell-trivial symmetries obviously forms a left ideal in the
algebra of linear differential operators endowed with the composition ◦
as multiplication. Furthermore, it is also a right ideal in the algebra of
symmetries of the linear equation (4). The complex associative algebra of
on-shell symmetries is the associative algebra of symmetries of the linear
equation quotiented by the two-sided ideal of on-shell-trivial symmetries.
The complex Lie algebra of on-shell symmetries is the algebra of on-shell
symmetries endowed with the commutator as Lie bracket.
Notice that when K is non-degenerate, a linear operator T = RK is
a symmetry generator of the quadratic action (3) if and only if R is self-
adjoint. Moreover, the Lie subalgebra of such on-shell-trivial symmetry
generators is an ideal in the Lie algebra of off-shell symmetries.
4 Higher-spin algebras
Let g be the Lie algebra corresponding to the finite-dimensional (confor-
mal) isometry group G of the constant-curvature spacetime of dimension
n > 2. For n = 2 , the spacetime may be arbitrary and the conformal
algebra is of course infinite-dimensional. If the free field theory is rela-
tivistic, then g is linearly realized on the space L2(Rn)⊗ V (respectively,
KerK) of off-shell (resp. on-shell) fields. This induces a linear realiza-
tion of the universal enveloping algebra U(g) over C. The real form of
this realization corresponding to the self-adjoint operators, endowed with
i times the commutator as Lie bracket, is nowadays referred to as (confor-
mal) on/off-shell higher-spin algebra of the constant-curvature spacetime
(see e.g. [8] for an elementary introduction to such algebraic structures).
The name comes from the fact that its generators are in “higher-spin”
representations of the Lorentz group, and the algebra is said to be “on”
or “off” shell whether the algebra is realized on the space of solutions of
the Euler-Lagrange equations or not.
The isometry algebra g of a constant-curvature spacetime is a module
of the Lorentz subalgebra o(n − 1, 1) ⊂ g for the adjoint representation.
This module decomposes as the sum of two irreducible o(n−1, 1)-modules:
the “translations” are in the vector module ∼= R
n while the boosts and ro-
tations are in the antisymmetric module ∼= ∧
2(Rn). These representations
are labelled by one-column Young diagrams of, respectively, one and two
cells. The number of columns is associated with the spin. The fact that
the generators of U(g) are in higher-spin representations is summarized in
the following result.
Universal enveloping algebra of isometries: The universal envelop-
ing algebra U(g) of the isometry algebra g of an n-dimensional constant-
curvature spacetime is an infinite-dimensional module of the general linear
Lie algebra gl(n), decomposing as an infinite sum of finite-dimensional ir-
reducible gl(n)-modules labelled by the set of all Young diagrams, with
multiplicity one, the first column of which has length 6 n.
Proof: The Poincaré-Birkhoff-Witt theorem states that the universal en-
veloping algebra U(g) is isomorphic to the symmetric algebra ⊙(g) as a
vector space. As a gl(n)-module, the vector space g is isomorphic to the
sum Rn ⊕ ∧2(Rn) of irreducible modules. This leads to the following
isomorphism of modules:
⊙ (g) ∼=
. (6)
The idea is to evaluate the right-hand-side of (6) using the available tech-
nology on Kronecker products of irreducible representations [9]. The mod-
ule ⊙(Rn) decomposes as the infinite sum of irreducible modules labelled
by all one-row Young diagrams with multiplicity one. A formula of Lit-
tlewood for symmetric plethsyms implies that the module ⊙
∧2 (Rn)
decomposes as the infinite sum of irreducible modules, with multiplic-
ity one, labelled by all Young diagrams with columns of even lengths.
The Kronecker product in (6) decomposes as the infinite sum of all the
Kronecker products between a one-row Young diagram and a Young dia-
gram with columns of even lengths, each with multiplicity one. Using the
Littlewood–Richardson rule, one may show that the result of this compu-
tation is the infinite sum of irreducible modules labelled with all possible
Young diagrams, each with multiplicity one. The Young diagrams whose
first column has length greater than n lead to vanishing modules, hence
they do not appear in the series.
The higher-spin algebras are important in relativistic field theories
because they always appear as spacetime symmetry algebras in the free
limit.
Spacetime symmetries of relativistic free field theories: If the
Lie algebra of off/on-shell symmetries contains the (conformal) isometry
algebra g of some constant-curvature spacetime M, then it also contains
the (conformal) off/on-shell higher-spin algebra of M.
Proof: The Poincaré-Birkhoff-Witt theorem states that one can realize
the universal enveloping U(g) as Weyl-ordered polynomials in the elements
of the Lie algebra g. The above theorem is proved by observing that
any Weyl-ordered polynomial in on-shell symmetries is itself an on-shell
symmetry. As observed in [10], the same is true for symmetry generators.
As an important corollary, the theorem implies that any relativistic
free field theory has an infinite number of rigid symmetries, and therefore
it possesses an infinite number of conserved currents via the Noether the-
orem, as it is well known. Notice that relativistic integrable models are
precisely such that they possess an infinite set of commuting rigid symme-
tries corresponding to an infinite set of conserved charges in involution.
The infinite-dimensional subalgebra of symmetries of the free field theory
generated by the translations only is, of course, Abelian. Actually, the fac-
torization property is deeply related to the preservation of this subalgebra
of symmetries at the interacting level [11]. Thus the relationship between
higher-spin algebras and integrable models appears to be very intimate
(see also [12] and references therein). The strong form of the Maldacena
conjecture (in the large N limit) and the integrability properties recently
enlightened in this context are further indications of such a relationship.
Symmetries may be characterized by their action on the spacetime co-
ordinates. A smooth change of coordinates is generated by a first-order
linear differential operator. Therefore, a higher-order linear differential
operator does not generate coordinate transformations. For instance, an
isometry generator is a first-order linear differential operator correspond-
ing to a Killing vector field, but the spacetime higher-symmetries are
powers of such isometry generators, hence they are higher-order linear
differential operators. They do not generate coordinate transformations
and this explains why spacetime higher-symmetries are usually not con-
sidered in textbooks.
Let us focus on the first non-trivial example of free field theory: the
quadratic action of a complex scalar field on an n-dimensional spacetime
M. In such case, the space V = C and the kinetic operator K can be
taken to be a constant mass term plus the Laplacian on M,
A scalar field is said to be conformal if its kinetic operator is the conformal
Laplacian
4 (n− 1)
R , (7)
where R denotes the scalar curvature. The quadratic action and the linear
equation are symmetric under the full conformal algebra o(n, 2) if and only
if the scalar field is conformal and has conformal weight 1− n/2.
Higher symmetries of the conformal scalar field: For the quadratic
action of a complex conformal scalar field on a constant-curvature space-
time M of dimension n > 2, the following spaces over R are isomorphic:
• The Lie algebra of off-shell symmetries quotiented by the ideal of on-
shell-trivial symmetry generators,
• A real form of the associative algebra of on-shell symmetries.
• The conformal on-shell higher-spin algebra,
• The real algebra of Weyl-ordered polynomials in the conformal Killing
vector fields quotiented by the ideal generated by the conformal Laplacian,
endowed with i times the commutator as Lie bracket. The symbols of these
differential operators,
T = (−i)
µ1...µr
∇µ1 . . .
∇µr + lower + on-shell-trivial ,
may be represented by real traceless symmetric tensor fields ξ which are
conformal Killing tensors.
Moreover, in n = 2 dimensions the theorem is valid for an arbitrary space-
time manifold.
Proof: The theorem can be extracted from the results of [13] on flat
spacetime of dimension n > 2 by taking into account that any constant-
curvature spacetime M can be seen as a conic in the projective null cone
of the ambient space Rn,2 . The two-dimensional case is addressed by
using the left/right-moving coordinates.
Notice that the on-shell higher-spin algebra of a non-conformal scalar
field on a constant-curvature spacetime is a proper subalgebra of the uni-
versal enveloping algebra of the isometry algebra g: it decomposes as
the infinite sum of irreducible o(n− 1, 1)-modules labelled by all two-row
Young diagrams with multiplicity one, as reviewed in [4, 7]. This algebra
is in one-to-one correspondence with the space of reducibility parameters
of the infinite tower of symmetric tensor gauge fields where each field ap-
pears once and only once for each given spin s > 0. Moreover, notice that
the AdSn+1/CFTn correspondence for n > 2 in the weak tension/coupling
limit also makes use of the isomorphism between the on-shell higher-spin
algebra of a non-conformal scalar field on AdSn+1 and the on-shell sym-
metry algebra of a conformal scalar field on Rn−1,1 (see [14] for the cor-
respondence at the level of conserved currents). Remark also that the
conformal on-shell higher-spin algebra of a two-dimensional spacetime for
a massless scalar field is isomorphic to the direct sum of u(1) and the two
Lie algebras of differential operators for the left and right moving sectors
respectively. Each of such algebras of differential operators is isomorphic
to the algebra W∞ with zero central charge [15].
The deep connection between higher-spin algebras and integrable mod-
els is exhibited by the following example in n = 2 dimensions.
Higher symmetries of the interacting scalar field: A non-linear
action of a real scalar field on the two-dimensional Minkowski spacetime,
without derivative interaction term, of the form
S[φ] =
〈φ | � | φ 〉+
xV (φ) , V (φ) = O(φ
is invariant under an infinite number of local infinitesimal rigid symmetry
transformations, independent of the coordinate xµ, if and only if
V (φ) = ±
cos (αφ)− 1
, m ∈ R ,
the parameter α is either purely real or imaginary. In such case, the field
φ either corresponds to a free massless scalar field (m = 0), a free massive
scalar field (m 6= 0 , α = 0) or sine-Gordon theory (m 6= 0 , α 6= 0).
Moreover, via linearisation, there is a one-to-one correspondence be-
tween:
• The set of on-shell non-trivial, polynomial in the field derivatives, coordinate-
independent, symmetry transformations of the sine-Gordon Lagrangian
• The Lie algebra of coordinate-independent off-shell symmetries of a free
real scalar field quotiented by the ideal of on-shell-trivial symmetry gener-
ators,
• A proper Abelian Lie subalgebra of the on-shell higher-spin algebra of
the Minkowski plane,
• The space of harmonic odd polynomials in the momenta Pµ = −i∂µ .
These differential operators T may be represented by real traceless sym-
metric constant tensors λ:
T = i λ
µ1...µ2q+1∂µ1 . . . ∂µ2q+1 + on-shell-trivial .
Proof: The first part of the theorem is a straightforward consequence
of the results of [16] in the case when V (φ) is at least quadratic in φ
(by hypothesis). The second part is proven by selecting all coordinate-
independent symmetries of a free real scalar field and comparing them
with the conserved currents of [16]. In both cases, the Noether correspon-
dence between non-trivial conserved currents and non-trivial symmetries
(see e.g. [17] for a precise statement of this isomorphism) is performed
via the Hamiltonian formulation of a two-dimensional scalar field where
one of the light-cone coordinate plays the role of “time.”
5 A gauge principle for higher-spins ?
The analogy with lower-spins suggests to guess the full non-Abelian gauge
theory by making use of the “gauge principle.” Moreover, this point of
view actually provides a concrete motivation for using the higher-spin
algebras in the interaction problem.
The idea is to consider some “matter” system described by a quadratic
action (3) with some algebra of rigid symmetries. The rigid symmetries U
of this system are by definition in the “fundamental” representation of the
algebra of off-shell symmetries of the action (3). Connections are usually
introduced in order to “gauge” these rigid symmetries by allowing U to be
a smooth function on Rn taking values in the group of off-shell symmetries
of the action (3). In order to construct a covariant derivative D = ∂ + Γ,
one introduces a connection defined as a covariant vector field Γµ taking
values in the Lie algebra of off-shell symmetries and transforming as
| ψ 〉 −→ U | ψ 〉 , Γ −→ UDU
, (8)
in such a way that
D | ψ 〉 −→ UD | ψ 〉 .
The minimal coupling is the replacement of all partial derivatives ∂ in the
kinetic operator K(∂) by covariant derivatives D which should ensure that
the quadratic action 〈ψ | K(D) | ψ 〉 is preserved by gauge symmetries
(8). The connection transforms in the “adjoint” representation of the
rigid symmetries while the matter field transforms in the “fundamental.”
(More precisely, the covariant derivative transforms in the adjoint while
the matter field belongs to a module of the gauge algebra.)
The introduction of a connection requires the introduction of some new
dynamical fields: the “gauge” sector. In Yang-Mills gauge theories, the
rigid symmetry is internal and the connection is itself made of spin-1 gauge
fields. For spacetime symmetries, the relation between the connection
and the gauge field is more complicated. For instance, in Einstein gravity
the Levi-Civita connection is expressed in terms of the first derivative of
the metric via the torsionlessness and metricity constraints. In general,
the spin-s tensor field propagating on a constant-curvature spacetime is
expected to be the perturbation of some background field
gµ1...µs =
g µ1...µs + ε hµ1...µs ,
so that the deformed gauge symmetries would be of the form
δξgµ1µ2... µs = ε (Dξ)µ1µ2... µs , (9)
where the covariant derivative D = ∇ + O(ε) starts as the covariant
derivative with respect to the Levi–Civita connection for the spacetime
metric plus non-minimal corrections. Thus the background connection
is identified with the Levi-Civita connection for the background metric,
and the linearization of (9) reproduces (1). Furthermore, the reducibility
parameters of (1) exactly correspond to the gauge symmetries (9) leaving
the background geometry invariant. In the present case, this group of
rigid symmetries contains the isometry group g of the constant-curvature
spacetime. The classical theory of (in)homogeneous pseudo-orthogonal
groups tells us that completely symmetric tensor fields which are invariant
under g are constructed from products of the background metric:
g (µ1µ2 . . .
µ2m−1µ2m)
Thus, along these lines, only even-spin symmetric tensor fields can be
perturbations of a non-vanishing higher-spin background in a constant-
curvature spacetime. The first-order deformation of the gauge symmetries
(1) following from (9) would be of the schematic form
δξ hµ1µ2... µs = (
Γ · ξ )µ1µ2... µs , (10)
where
Γ stands for the linearized connection (including the linearized
Levi-Civita connection) and the dot stands for the action on the gauge
parameter ξ. The transformations (10) evaluated on Killing tensors ξ
of the background spacetime would be rigid symmetry transformations
of the free gauge theory. This property highly constrains the possible
expressions for the linearized connection.
Let us now consider the expansion of the minimally coupled action for
the “matter” sector in power series of ε :
〈ψ | K(D) | ψ 〉 =
〈ψ | K(∂) | ψ 〉 + ε 〈h | J 〉 + O(ε
where J denotes a set of symmetric tensors which are bilinear in ψ and
their derivatives. Assuming that the “matter” sector is strictly distinct
from the “gauge” sector, the gauge invariance of the complete action at
first order in ε requires the symmetric tensors Jµ1µ2... µs to be conserved
up to terms proportional to the “matter” free field equations (and deriva-
tives thereof) corresponding to first-order deformations
δξ | ψ 〉 =
U | ψ 〉 (11)
of the gauge transformations of the “matter” sector, where
U is a lin-
ear differential operator depending linearly on ξ and its derivatives. At
zeroth order in ǫ , the “gauge” group does not act on the matter. There-
fore, at leading order, the transformation law (8) reads as (11). Via the
Noether correspondence, the space of all rigid symmetries of the “matter”
quadratic action determines the space of all on-shell-conserved currents
bilinear in the “matter” fields. The latter ones determine, at first order,
the “fundamental” representation of the “gauge” group. The transforma-
tions (11) evaluated on Killing tensors ξ must define off-shell symmetries
of the “matter” quadratic action. Their algebra algebra is non-Abelian,
hence the “gauge” algebra is already non-Abelian at first order.
As a suggestive example, one may consider a “matter” sector contain-
ing only a single scalar field.
Noether cubic couplings of a scalar field: The minimally coupled
action of a complex scalar field on flat spacetime, given by
S[φ] =
〈φ | � −m
| φ 〉 − ε
x hµ1µ2... µsJ
µ1µ2... µs + O(ε
is invariant at first order in ε , for any symmetric tensor field ξµ1µ2... µs−1 ,
under infinitesimal symmetry transformations
δξhµ1µ2... µs =
δξ hµ1µ2... µs + O(ε) ,
δξ | φ 〉 = εT | φ 〉+ O(ε
) , (12)
where the symbol of the differential operator T is represented by ξ and the
lower order terms depend on derivatives of ξ,
T = (−i)
µ1...µs−1∂µ1 . . . ∂µs−1 + lower+ on-shell-trivial ,
if and only if the on-shell-conserved current J is equivalent to a Noether
current associated to the coordinate-independent off-shell symmetries of
the free scalar field. This defines a one-to-one correspondence between
equivalence classes of such symmetric Noether currents J , bilinear in φ
and its derivatives, and equivalence classes of such deformations δξφ at
first order.
Proof: The explicit equation expressing the gauge invariance of the min-
imaly coupled action for any symmetric tensor field ξ(x) of rank s − 1
precisely states that the symmetric tensor J of rank s is conserved mod-
ulo terms proportional to field equation of the scalar field φ. The one-
to-one correspondence, precisely explained in [17], between equivalence
classes of on-shell conserved currents and equivalence classes of off-shell
symmetry transformations shows explicitly that J is necessarily related to
a coordinate-independent transformation of the form (12). In turn, these
transformations are obtained by evaluating the transformation (12), at
lowest order in ε and on gauge parameters ξ equal to constant Killing
tensors. The sufficiency is proven by making use of the symmetric con-
served currents of [18]. The second part of the theorem follows from the
fact that trivial currents define trivial deformations and conversely, as it
can be seen explicitly.
In the lower-spin case, one recovers the standard minimal coupling
procedure. For s = 1 , the minimal coupling stops at second order in ε
since Jµ is the U(1) current and hµ is the Abelian vector gauge field. For
s = 2 , the minimal coupling at first order is the usual coupling between
a spin-two gauge field and the energy-momentum tensor Jµν leading to
the coordinate transformations of the scalar field, generated by the vector
fields T = −i ξµ(x) ∂µ . The commutators of such infinitesimal transfor-
mations close and define the Lie bracket of vector fields, so the underlying
gauge symmetry algebra may already be guessed at first order for gravity:
it is the Lie agebra of smooth vector fields, i.e. the Lie algebra for the
group of diffeomorphisms. The minimally coupled action is obtained to
all orders by introducing the Levi-Civita connection.
In the higher-spin case, it should be stressed that the trace condi-
tions on the gauge field and parameter have not been stated in the former
proposition because they may indeed be relaxed in order to simplify its
formulation. (Nevertheless, these constraints may be included by consis-
tently imposing weaker conservation laws on double-traceless currents.)
Moreover, it is convenient to remove trace constraints for searching a
Non-Abelian higher-spin gauge symmetry algebra. Actually, the trace
constraints may be removed for free field theories in several ways (see [19]
for some reviews, and [20] for the latest developments). The Lie algebra of
gauge transformations (12) for the infinite tower of all gauge parameters
(1 6 s < ∞) is a real form of the algebra of linear differential opera-
tors on Rn endowed with i times the commutator as Lie bracket. Notice
also that the unital associative algebra of linear differential operators on
n is isomorphic to the universal enveloping algebra of vector fields on
n . (Strictly speaking, this is true only for polynomial vector fields and
differential operators, more sophisticated mathematical statements may
be required for smooth functions, but this point is only technical.) More
concretely, the symbol of a differential operator of order r is represented
by a symmetric tensor field of rank r. In the light of these remarks, it is
tempting to conjecture that, for higher-spin gauge theories, the algebra of
Hermitian differential operators,
µ1...µr (x) ∂µ1 . . . ∂µr + Hermitian conjugate
generalizes the algebra of infinitesimal diffeomorphisms for gravity. An-
other argument in favour of this conjecture may be presented in the
“gauge” sector by looking at the metric-like formulation of higher-spins
arising from the frame-like formulation of Vasiliev, at first order in the
coupling constant [21].
6 Conclusion
The conclusion is that there are two complementary but distinct ways
of using rigid symmetries of the free theory in order to guess the proper
gauge symmetry principle of higher-spin gauge theories.
On the one hand, the infinite set of rigid symmetries of the free (or,
maybe, even integrable) “matter” sector, might be gauged by the intro-
duction of a connection via a minimal coupling prescription. The idea of
using a massive scalar field as free matter sector and an infinite tower of
massless symmetric tensor fields as interacting gauge sector is in agree-
ment with the isomorphism between the off-shell higher-spin algebra and
the space of reducibility parameters. (If tensor fields are used as free
“matter” sector, then the symmetry algebra could be larger. Following
the lines of the Vasiliev construction in such case, the structure of the uni-
versal enveloping algebra points towards a larger infinite tower of gauge
fields including mixed-symmetry tensors.)
On the other hand, in the free “gauge” sector, rigid symmetries linked
to reducibility parameters may arise from the linearization of the gauge
symmetries of some non-linear action. Thus the complete knowledge of the
rigid symmetries of free higher-spin gauge theories would indicate what
can be the linearized connection.
Acknowledgments
I. Bakas, G. Barnich, N. Boulanger, T. Damour and J. Remmel are
thanked for very useful exchanges. The author is grateful to the orga-
nizers for their invitation to this enjoyable meeting and the opportunity
to present his lecture. The Institut des Hautes Études Scientifiques de
Bures-sur-Yvette is acknowledged for its hospitality.
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	Higher-spin interaction problem
	The Noether method
	Free theory symmetries
	Higher-spin algebras
	A gauge principle for higher-spins ?
	Conclusion

  The exhaustive study of the rigid symmetries of arbitrary free field theories
is motivated, along several lines, as a preliminary step in the completion of
the higher-spin interaction problem in full generality. Some results for the
simplest example (a scalar field) are reviewed and commented along these lines.

Higher spin algebras as higher symmetries
Xavier Bekaert
Laboratoire de Mathématiques et Physique Théorique
Unité Mixte de Recherche 6083 du CNRS, Fédération Denis Poisson
Université François Rabelais, Parc de Grandmount
37200 Tours, France
Abstract
The exhaustive study of the rigid symmetries of arbitrary free field
theories is motivated, along several lines, as a preliminary step in the
completion of the higher-spin interaction problem in full generality. Some
results for the simplest example (a scalar field) are reviewed and com-
mented along these lines.
Expanded version of the lectures presented at the “5th international school
and workshop on QFT & Hamiltonian systems” (Calimanesti, May 2006).
1 Higher-spin interaction problem
Whereas covariant gauge theories describing arbitrary free massless fields
on constant-curvature spacetimes of dimension n are firmly established by
means of the unitary representation theory of their isometry groups, it is
still open to question whether non-trivial consistent self-couplings and/or
cross-couplings among those fields may exist for n > 2 , such that the
deformed gauge algebra is non-Abelian. The goal of the present paper is
to advocate that a lot of information on the interactions can be extracted
from the symmetries of the free field theory.
The conventional local free field theories corresponding to unitary irre-
ducible representations of the helicity group SO(n− 2) , that are spanned
by completely symmetric tensors, have been constructed a while ago (for
some introductory reviews, see [1]). In order to have Lorentz invariance
manifest and second order local field equations with minimal field content,
the theory is expressed in terms of completely symmetric double-traceless
tensor gauge fields hµ1... µs of rank s > 0, the gauge transformation of
which reads
δξ hµ1µ2... µs =
∇ µ1 ξµ2...µs + cyclic , (1)
E-mail address: Xavier.Bekaert@lmpt.univ-tours.fr
http://arxiv.org/abs/0704.0898v2
where
∇ is the covariant derivative with respect to the background Levi–
Civita connection and “cyclic” stands for the sum of terms necessary to
have symmetry of the right-hand-side under permutations of the indices.
The gauge parameter ξ is a completely symmetric traceless tensor field
of rank s − 1. In this relativistic field theory, the “spin” is equal to the
rank s. For spin s = 1 the gauge field hµ represents the photon with U(1)
gauge symmetry while for spin s = 2 the gauge field hµν represents the
graviton with linearized diffeomorphism invariance. The gauge algebra of
field independent gauge transformations such as (1) is of course Abelian.
Non-Abelian gauge theories for “lower spin” s 6 2 are well known and
essentially correspond to Yang-Mills (s = 1) and Einstein (s = 2) theories
for which the underlying geometries (principal bundles and Riemannian
manifolds) were familiar to mathematicians before the construction of the
physical theory. In contrast, the situation is rather different for “higher
spin” s > 2 for which the underlying geometry (if any!) remains obscure.
Due to this lack of information, it is natural to look for inspiration in
the perturbative “reconstruction” of Einstein gravity as the non-Abelian
gauge theory of a spin-two particle propagating on a constant-curvature
spacetime (see e.g. [2] for a comprehensive review).
Let us denote by
S [hµ1... µs ] the Poincaré-invariant, local, second-
order, quadratic, ghost-free, gauge-invariant action of a spin-s symmetric
tensor gauge field. In order to perform a perturbative analysis via the
Noether method [3], the non-Abelian interaction problem for a collection
of higher (and possibly lower) spin gauge fields is formulated as a defor-
mation problem.
Higher-spin interaction problem: List all Poincaré-invariant local
deformations
S[h] =
S [h] + ε
S [h] + O(ε
of a positive sum, with at least one s > 2,
S [h] =
S [hµ1... µs ]
of quadratic actions such that the deformed local gauge symmetries
δξh =
δξ h + ε
δξ h + O(ε
are already non-Abelian at first order, in the deformation parameters ε
and do not arise from local redefinitions
h → h + ε φ(h) + O(ε
) , ξ → ξ + ε ζ(h, ξ) + O(ε
of the gauge fields and parameters.
This well-posed mathematical problem is expected to possess non-
trivial solutions including higher-spin fields, as strongly indicated by Vasiliev’s
works (for some reviews, see [4] and references therein) and deserves to
be investigated further along systematic lines.
2 The Noether method
The assumption that the deformations are formal power series in some
deformation parameters ε enables to investigate the problem order by
order. The crucial observation of any perturbation theory is that the first
order deformations are constrained by the symmetries of the undeformed
system. In the present case, the Noether method scrutinizes the gauge
symmetry of the action, δξS = 0 . At zeroth order, the latter equation is
satisfied by hypothesis. At first order, it reads
S = 0 . (2)
This equation may be used to constrain the possible deformations by
reinterpreting them as familiar objects of the undeformed gauge theory.
By definition, an observable of a gauge theory is a functional which is
gauge-invariant on-shell, while a reducibility parameter of a gauge theory
is a gauge parameter such that the corresponding gauge variation vanishes
off-shell.
First-order deformations in terms of the undeformed theory:
• First-order deformations of the action are observables of the undeformed
theory.
• First-order deformations of the gauge symmetries evaluated at reducibil-
ity parameters of the undeformed gauge theory define symmetries of the
undeformed theory.
Proof: In (2) the infinitesimal variation
S of the undeformed action
is proportional to the undeformed Euler–Lagrange equations. This proves
the fist part of the theorem. Reducibility parameters ξ of the undeformed
gauge theory verify
h = 0 by definition. Inserting this fact into (2)
with ξ = ξ gives
S = 0 , which is precisely the translation of the second
part of the theorem.
In the mathematical litterature, a (conformal) Killing tensor of a
pseudo-Riemannian manifold is a symmetric tensor field ξ such that its
symmetrized covariant derivative with respect to the Levi–Civita connec-
tion, ∇µ1 ξµ2...µs + cyclic, vanishes (modulo a term proportional to the
metric for conformal Killing tensors). Therefore, any reducibility parame-
ter ξ of the spin-s symmetric gauge field theory on the constant-curvature
spacetime M is identified with a Killing tensor of rank s− 1 of the mani-
fold M. The space of Killing tensors on any constant-curvature spacetime
is known to be finite-dimensional [5], thus the linear gauge symmetries (1)
are irreducible.
These results suggest two strategies for addressing the higher-spin in-
teraction problem. The most ambitious one is the computation of all lo-
cal observables of the free gauge theory associated to deformations of the
gauge algebra. This result would provide the exhaustive list of algebra-
deforming first order vertices, but this computation is technically demand-
ing and seems out of reach in the completely general case. Nevertheless,
the BRST reformulation of the problem [6] allowed the complete classi-
fication of non-Abelian deformations in various particular cases (see e.g.
the review [7] and references therein). Actually, a more humble strategy
is the computation of all rigid symmetries of the free irreducible gauge
theory. It is of interest because the knowledge of these rigid symmetries
would strongly constrain the candidates for gauge symmetry deforma-
tions. Indeed, the constant tensors appearing in the rigid symmetries
could be compared with the complete list [5] of constant-curvature space-
time Killing tensors.
3 Free theory symmetries
Bosonic fields are usually described in terms of their components living in
some subspace V of the space ⊗(Rn) of tensors on Rn (e.g. V = ⊙(Rn)
for symmetric tensor fields). The background metric of the constant-
curvature spacetime induces some non-degenerate bilinear form on V .
This defines a non-degenerate sesquilinear form 〈 | 〉 on the space L2(Rn)⊗
V of square-integrable fields taking values in the countable space V (the
components). Let † stands for the adjoint with respect to the sesquilinear
form 〈 | 〉 .
Any quadratic action for bosonic fields ψ can be expressed as a quadratic
S [ψ] =
〈ψ | K | ψ 〉 , (3)
where the kinetic operatorK is self-adjoint, K† = K. Because the sesquilin-
ear form 〈 | 〉 is non-degenerate, the Euler-Lagrange equation extremizing
the quadratic action is the linear equation
δ〈ψ |
= K|ψ 〉 = 0 . (4)
Moreover, the quadratic form 〈ψ | K | ψ 〉 is degenerate if and only if the
kinetic operator K is degenerate. This happens if and only if there exists
a linear operator P (on L2(Rn) ⊗ V ) such that KP = 0. Infinitesimal
gauge symmetries then read
δχ | ψ 〉 = P | χ 〉 ,
with gauge parameters χ . The Noether identity is P†K = (KP)† = 0 .
A symmetry of the quadratic action (3) is an invertible linear pseudo-
differential operator U preserving the quadratic form 〈 | K | 〉. In other
words,
KU = K .
The group of off-shell symmetries is the group of symmetries of the quadratic
action endowed with the composition ◦ as product. A symmetry genera-
tor of the quadratic action (3) is a linear differential operator T which is
self-adjoint with respect to the quadratic form 〈 | K | 〉. More concretely,
KT = T
Any symmetry generator T defines a symmetry U = eiT of the quadratic
action (3). If T = T† then the linear operator T is a symmetry generator
of the quadratic action if and only it commutes with K. The real Lie
algebra of off-shell symmetries is the algebra of symmetry generators of
the quadratic action endowed with i times the commutator as Lie bracket,
{ , } := i [ , ].
A symmetry of the linear equation (4) is a linear differential operator
T obeying
KT = SK , (5)
for some linear operator S. Such a symmetry T preserves the space KerK
of solutions to the equations of motion. Any symmetry generator T of
the action (3) is always a symmetry of the equation of motion (4) with
S = T† in (5). A symmetry T is trivial on-shell if T = RK for some
linear operator R. Such an on-shell-trivial symmetry is always a sym-
metry of the field equation (4), since it obeys (5) with S = KR. The
algebra of on-shell-trivial symmetries obviously forms a left ideal in the
algebra of linear differential operators endowed with the composition ◦
as multiplication. Furthermore, it is also a right ideal in the algebra of
symmetries of the linear equation (4). The complex associative algebra of
on-shell symmetries is the associative algebra of symmetries of the linear
equation quotiented by the two-sided ideal of on-shell-trivial symmetries.
The complex Lie algebra of on-shell symmetries is the algebra of on-shell
symmetries endowed with the commutator as Lie bracket.
Notice that when K is non-degenerate, a linear operator T = RK is
a symmetry generator of the quadratic action (3) if and only if R is self-
adjoint. Moreover, the Lie subalgebra of such on-shell-trivial symmetry
generators is an ideal in the Lie algebra of off-shell symmetries.
4 Higher-spin algebras
Let g be the Lie algebra corresponding to the finite-dimensional (confor-
mal) isometry group G of the constant-curvature spacetime of dimension
n > 2. For n = 2 , the spacetime may be arbitrary and the conformal
algebra is of course infinite-dimensional. If the free field theory is rela-
tivistic, then g is linearly realized on the space L2(Rn)⊗ V (respectively,
KerK) of off-shell (resp. on-shell) fields. This induces a linear realiza-
tion of the universal enveloping algebra U(g) over C. The real form of
this realization corresponding to the self-adjoint operators, endowed with
i times the commutator as Lie bracket, is nowadays referred to as (confor-
mal) on/off-shell higher-spin algebra of the constant-curvature spacetime
(see e.g. [8] for an elementary introduction to such algebraic structures).
The name comes from the fact that its generators are in “higher-spin”
representations of the Lorentz group, and the algebra is said to be “on”
or “off” shell whether the algebra is realized on the space of solutions of
the Euler-Lagrange equations or not.
The isometry algebra g of a constant-curvature spacetime is a module
of the Lorentz subalgebra o(n − 1, 1) ⊂ g for the adjoint representation.
This module decomposes as the sum of two irreducible o(n−1, 1)-modules:
the “translations” are in the vector module ∼= R
n while the boosts and ro-
tations are in the antisymmetric module ∼= ∧
2(Rn). These representations
are labelled by one-column Young diagrams of, respectively, one and two
cells. The number of columns is associated with the spin. The fact that
the generators of U(g) are in higher-spin representations is summarized in
the following result.
Universal enveloping algebra of isometries: The universal envelop-
ing algebra U(g) of the isometry algebra g of an n-dimensional constant-
curvature spacetime is an infinite-dimensional module of the general linear
Lie algebra gl(n), decomposing as an infinite sum of finite-dimensional ir-
reducible gl(n)-modules labelled by the set of all Young diagrams, with
multiplicity one, the first column of which has length 6 n.
Proof: The Poincaré-Birkhoff-Witt theorem states that the universal en-
veloping algebra U(g) is isomorphic to the symmetric algebra ⊙(g) as a
vector space. As a gl(n)-module, the vector space g is isomorphic to the
sum Rn ⊕ ∧2(Rn) of irreducible modules. This leads to the following
isomorphism of modules:
⊙ (g) ∼=
. (6)
The idea is to evaluate the right-hand-side of (6) using the available tech-
nology on Kronecker products of irreducible representations [9]. The mod-
ule ⊙(Rn) decomposes as the infinite sum of irreducible modules labelled
by all one-row Young diagrams with multiplicity one. A formula of Lit-
tlewood for symmetric plethsyms implies that the module ⊙
∧2 (Rn)
decomposes as the infinite sum of irreducible modules, with multiplic-
ity one, labelled by all Young diagrams with columns of even lengths.
The Kronecker product in (6) decomposes as the infinite sum of all the
Kronecker products between a one-row Young diagram and a Young dia-
gram with columns of even lengths, each with multiplicity one. Using the
Littlewood–Richardson rule, one may show that the result of this compu-
tation is the infinite sum of irreducible modules labelled with all possible
Young diagrams, each with multiplicity one. The Young diagrams whose
first column has length greater than n lead to vanishing modules, hence
they do not appear in the series.
The higher-spin algebras are important in relativistic field theories
because they always appear as spacetime symmetry algebras in the free
limit.
Spacetime symmetries of relativistic free field theories: If the
Lie algebra of off/on-shell symmetries contains the (conformal) isometry
algebra g of some constant-curvature spacetime M, then it also contains
the (conformal) off/on-shell higher-spin algebra of M.
Proof: The Poincaré-Birkhoff-Witt theorem states that one can realize
the universal enveloping U(g) as Weyl-ordered polynomials in the elements
of the Lie algebra g. The above theorem is proved by observing that
any Weyl-ordered polynomial in on-shell symmetries is itself an on-shell
symmetry. As observed in [10], the same is true for symmetry generators.
As an important corollary, the theorem implies that any relativistic
free field theory has an infinite number of rigid symmetries, and therefore
it possesses an infinite number of conserved currents via the Noether the-
orem, as it is well known. Notice that relativistic integrable models are
precisely such that they possess an infinite set of commuting rigid symme-
tries corresponding to an infinite set of conserved charges in involution.
The infinite-dimensional subalgebra of symmetries of the free field theory
generated by the translations only is, of course, Abelian. Actually, the fac-
torization property is deeply related to the preservation of this subalgebra
of symmetries at the interacting level [11]. Thus the relationship between
higher-spin algebras and integrable models appears to be very intimate
(see also [12] and references therein). The strong form of the Maldacena
conjecture (in the large N limit) and the integrability properties recently
enlightened in this context are further indications of such a relationship.
Symmetries may be characterized by their action on the spacetime co-
ordinates. A smooth change of coordinates is generated by a first-order
linear differential operator. Therefore, a higher-order linear differential
operator does not generate coordinate transformations. For instance, an
isometry generator is a first-order linear differential operator correspond-
ing to a Killing vector field, but the spacetime higher-symmetries are
powers of such isometry generators, hence they are higher-order linear
differential operators. They do not generate coordinate transformations
and this explains why spacetime higher-symmetries are usually not con-
sidered in textbooks.
Let us focus on the first non-trivial example of free field theory: the
quadratic action of a complex scalar field on an n-dimensional spacetime
M. In such case, the space V = C and the kinetic operator K can be
taken to be a constant mass term plus the Laplacian on M,
A scalar field is said to be conformal if its kinetic operator is the conformal
Laplacian
4 (n− 1)
R , (7)
where R denotes the scalar curvature. The quadratic action and the linear
equation are symmetric under the full conformal algebra o(n, 2) if and only
if the scalar field is conformal and has conformal weight 1− n/2.
Higher symmetries of the conformal scalar field: For the quadratic
action of a complex conformal scalar field on a constant-curvature space-
time M of dimension n > 2, the following spaces over R are isomorphic:
• The Lie algebra of off-shell symmetries quotiented by the ideal of on-
shell-trivial symmetry generators,
• A real form of the associative algebra of on-shell symmetries.
• The conformal on-shell higher-spin algebra,
• The real algebra of Weyl-ordered polynomials in the conformal Killing
vector fields quotiented by the ideal generated by the conformal Laplacian,
endowed with i times the commutator as Lie bracket. The symbols of these
differential operators,
T = (−i)
µ1...µr
∇µ1 . . .
∇µr + lower + on-shell-trivial ,
may be represented by real traceless symmetric tensor fields ξ which are
conformal Killing tensors.
Moreover, in n = 2 dimensions the theorem is valid for an arbitrary space-
time manifold.
Proof: The theorem can be extracted from the results of [13] on flat
spacetime of dimension n > 2 by taking into account that any constant-
curvature spacetime M can be seen as a conic in the projective null cone
of the ambient space Rn,2 . The two-dimensional case is addressed by
using the left/right-moving coordinates.
Notice that the on-shell higher-spin algebra of a non-conformal scalar
field on a constant-curvature spacetime is a proper subalgebra of the uni-
versal enveloping algebra of the isometry algebra g: it decomposes as
the infinite sum of irreducible o(n− 1, 1)-modules labelled by all two-row
Young diagrams with multiplicity one, as reviewed in [4, 7]. This algebra
is in one-to-one correspondence with the space of reducibility parameters
of the infinite tower of symmetric tensor gauge fields where each field ap-
pears once and only once for each given spin s > 0. Moreover, notice that
the AdSn+1/CFTn correspondence for n > 2 in the weak tension/coupling
limit also makes use of the isomorphism between the on-shell higher-spin
algebra of a non-conformal scalar field on AdSn+1 and the on-shell sym-
metry algebra of a conformal scalar field on Rn−1,1 (see [14] for the cor-
respondence at the level of conserved currents). Remark also that the
conformal on-shell higher-spin algebra of a two-dimensional spacetime for
a massless scalar field is isomorphic to the direct sum of u(1) and the two
Lie algebras of differential operators for the left and right moving sectors
respectively. Each of such algebras of differential operators is isomorphic
to the algebra W∞ with zero central charge [15].
The deep connection between higher-spin algebras and integrable mod-
els is exhibited by the following example in n = 2 dimensions.
Higher symmetries of the interacting scalar field: A non-linear
action of a real scalar field on the two-dimensional Minkowski spacetime,
without derivative interaction term, of the form
S[φ] =
〈φ | � | φ 〉+
xV (φ) , V (φ) = O(φ
is invariant under an infinite number of local infinitesimal rigid symmetry
transformations, independent of the coordinate xµ, if and only if
V (φ) = ±
cos (αφ)− 1
, m ∈ R ,
the parameter α is either purely real or imaginary. In such case, the field
φ either corresponds to a free massless scalar field (m = 0), a free massive
scalar field (m 6= 0 , α = 0) or sine-Gordon theory (m 6= 0 , α 6= 0).
Moreover, via linearisation, there is a one-to-one correspondence be-
tween:
• The set of on-shell non-trivial, polynomial in the field derivatives, coordinate-
independent, symmetry transformations of the sine-Gordon Lagrangian
• The Lie algebra of coordinate-independent off-shell symmetries of a free
real scalar field quotiented by the ideal of on-shell-trivial symmetry gener-
ators,
• A proper Abelian Lie subalgebra of the on-shell higher-spin algebra of
the Minkowski plane,
• The space of harmonic odd polynomials in the momenta Pµ = −i∂µ .
These differential operators T may be represented by real traceless sym-
metric constant tensors λ:
T = i λ
µ1...µ2q+1∂µ1 . . . ∂µ2q+1 + on-shell-trivial .
Proof: The first part of the theorem is a straightforward consequence
of the results of [16] in the case when V (φ) is at least quadratic in φ
(by hypothesis). The second part is proven by selecting all coordinate-
independent symmetries of a free real scalar field and comparing them
with the conserved currents of [16]. In both cases, the Noether correspon-
dence between non-trivial conserved currents and non-trivial symmetries
(see e.g. [17] for a precise statement of this isomorphism) is performed
via the Hamiltonian formulation of a two-dimensional scalar field where
one of the light-cone coordinate plays the role of “time.”
5 A gauge principle for higher-spins ?
The analogy with lower-spins suggests to guess the full non-Abelian gauge
theory by making use of the “gauge principle.” Moreover, this point of
view actually provides a concrete motivation for using the higher-spin
algebras in the interaction problem.
The idea is to consider some “matter” system described by a quadratic
action (3) with some algebra of rigid symmetries. The rigid symmetries U
of this system are by definition in the “fundamental” representation of the
algebra of off-shell symmetries of the action (3). Connections are usually
introduced in order to “gauge” these rigid symmetries by allowing U to be
a smooth function on Rn taking values in the group of off-shell symmetries
of the action (3). In order to construct a covariant derivative D = ∂ + Γ,
one introduces a connection defined as a covariant vector field Γµ taking
values in the Lie algebra of off-shell symmetries and transforming as
| ψ 〉 −→ U | ψ 〉 , Γ −→ UDU
, (8)
in such a way that
D | ψ 〉 −→ UD | ψ 〉 .
The minimal coupling is the replacement of all partial derivatives ∂ in the
kinetic operator K(∂) by covariant derivatives D which should ensure that
the quadratic action 〈ψ | K(D) | ψ 〉 is preserved by gauge symmetries
(8). The connection transforms in the “adjoint” representation of the
rigid symmetries while the matter field transforms in the “fundamental.”
(More precisely, the covariant derivative transforms in the adjoint while
the matter field belongs to a module of the gauge algebra.)
The introduction of a connection requires the introduction of some new
dynamical fields: the “gauge” sector. In Yang-Mills gauge theories, the
rigid symmetry is internal and the connection is itself made of spin-1 gauge
fields. For spacetime symmetries, the relation between the connection
and the gauge field is more complicated. For instance, in Einstein gravity
the Levi-Civita connection is expressed in terms of the first derivative of
the metric via the torsionlessness and metricity constraints. In general,
the spin-s tensor field propagating on a constant-curvature spacetime is
expected to be the perturbation of some background field
gµ1...µs =
g µ1...µs + ε hµ1...µs ,
so that the deformed gauge symmetries would be of the form
δξgµ1µ2... µs = ε (Dξ)µ1µ2... µs , (9)
where the covariant derivative D = ∇ + O(ε) starts as the covariant
derivative with respect to the Levi–Civita connection for the spacetime
metric plus non-minimal corrections. Thus the background connection
is identified with the Levi-Civita connection for the background metric,
and the linearization of (9) reproduces (1). Furthermore, the reducibility
parameters of (1) exactly correspond to the gauge symmetries (9) leaving
the background geometry invariant. In the present case, this group of
rigid symmetries contains the isometry group g of the constant-curvature
spacetime. The classical theory of (in)homogeneous pseudo-orthogonal
groups tells us that completely symmetric tensor fields which are invariant
under g are constructed from products of the background metric:
g (µ1µ2 . . .
µ2m−1µ2m)
Thus, along these lines, only even-spin symmetric tensor fields can be
perturbations of a non-vanishing higher-spin background in a constant-
curvature spacetime. The first-order deformation of the gauge symmetries
(1) following from (9) would be of the schematic form
δξ hµ1µ2... µs = (
Γ · ξ )µ1µ2... µs , (10)
where
Γ stands for the linearized connection (including the linearized
Levi-Civita connection) and the dot stands for the action on the gauge
parameter ξ. The transformations (10) evaluated on Killing tensors ξ
of the background spacetime would be rigid symmetry transformations
of the free gauge theory. This property highly constrains the possible
expressions for the linearized connection.
Let us now consider the expansion of the minimally coupled action for
the “matter” sector in power series of ε :
〈ψ | K(D) | ψ 〉 =
〈ψ | K(∂) | ψ 〉 + ε 〈h | J 〉 + O(ε
where J denotes a set of symmetric tensors which are bilinear in ψ and
their derivatives. Assuming that the “matter” sector is strictly distinct
from the “gauge” sector, the gauge invariance of the complete action at
first order in ε requires the symmetric tensors Jµ1µ2... µs to be conserved
up to terms proportional to the “matter” free field equations (and deriva-
tives thereof) corresponding to first-order deformations
δξ | ψ 〉 =
U | ψ 〉 (11)
of the gauge transformations of the “matter” sector, where
U is a lin-
ear differential operator depending linearly on ξ and its derivatives. At
zeroth order in ǫ , the “gauge” group does not act on the matter. There-
fore, at leading order, the transformation law (8) reads as (11). Via the
Noether correspondence, the space of all rigid symmetries of the “matter”
quadratic action determines the space of all on-shell-conserved currents
bilinear in the “matter” fields. The latter ones determine, at first order,
the “fundamental” representation of the “gauge” group. The transforma-
tions (11) evaluated on Killing tensors ξ must define off-shell symmetries
of the “matter” quadratic action. Their algebra algebra is non-Abelian,
hence the “gauge” algebra is already non-Abelian at first order.
As a suggestive example, one may consider a “matter” sector contain-
ing only a single scalar field.
Noether cubic couplings of a scalar field: The minimally coupled
action of a complex scalar field on flat spacetime, given by
S[φ] =
〈φ | � −m
| φ 〉 − ε
x hµ1µ2... µsJ
µ1µ2... µs + O(ε
is invariant at first order in ε , for any symmetric tensor field ξµ1µ2... µs−1 ,
under infinitesimal symmetry transformations
δξhµ1µ2... µs =
δξ hµ1µ2... µs + O(ε) ,
δξ | φ 〉 = εT | φ 〉+ O(ε
) , (12)
where the symbol of the differential operator T is represented by ξ and the
lower order terms depend on derivatives of ξ,
T = (−i)
µ1...µs−1∂µ1 . . . ∂µs−1 + lower+ on-shell-trivial ,
if and only if the on-shell-conserved current J is equivalent to a Noether
current associated to the coordinate-independent off-shell symmetries of
the free scalar field. This defines a one-to-one correspondence between
equivalence classes of such symmetric Noether currents J , bilinear in φ
and its derivatives, and equivalence classes of such deformations δξφ at
first order.
Proof: The explicit equation expressing the gauge invariance of the min-
imaly coupled action for any symmetric tensor field ξ(x) of rank s − 1
precisely states that the symmetric tensor J of rank s is conserved mod-
ulo terms proportional to field equation of the scalar field φ. The one-
to-one correspondence, precisely explained in [17], between equivalence
classes of on-shell conserved currents and equivalence classes of off-shell
symmetry transformations shows explicitly that J is necessarily related to
a coordinate-independent transformation of the form (12). In turn, these
transformations are obtained by evaluating the transformation (12), at
lowest order in ε and on gauge parameters ξ equal to constant Killing
tensors. The sufficiency is proven by making use of the symmetric con-
served currents of [18]. The second part of the theorem follows from the
fact that trivial currents define trivial deformations and conversely, as it
can be seen explicitly.
In the lower-spin case, one recovers the standard minimal coupling
procedure. For s = 1 , the minimal coupling stops at second order in ε
since Jµ is the U(1) current and hµ is the Abelian vector gauge field. For
s = 2 , the minimal coupling at first order is the usual coupling between
a spin-two gauge field and the energy-momentum tensor Jµν leading to
the coordinate transformations of the scalar field, generated by the vector
fields T = −i ξµ(x) ∂µ . The commutators of such infinitesimal transfor-
mations close and define the Lie bracket of vector fields, so the underlying
gauge symmetry algebra may already be guessed at first order for gravity:
it is the Lie agebra of smooth vector fields, i.e. the Lie algebra for the
group of diffeomorphisms. The minimally coupled action is obtained to
all orders by introducing the Levi-Civita connection.
In the higher-spin case, it should be stressed that the trace condi-
tions on the gauge field and parameter have not been stated in the former
proposition because they may indeed be relaxed in order to simplify its
formulation. (Nevertheless, these constraints may be included by consis-
tently imposing weaker conservation laws on double-traceless currents.)
Moreover, it is convenient to remove trace constraints for searching a
Non-Abelian higher-spin gauge symmetry algebra. Actually, the trace
constraints may be removed for free field theories in several ways (see [19]
for some reviews, and [20] for the latest developments). The Lie algebra of
gauge transformations (12) for the infinite tower of all gauge parameters
(1 6 s < ∞) is a real form of the algebra of linear differential opera-
tors on Rn endowed with i times the commutator as Lie bracket. Notice
also that the unital associative algebra of linear differential operators on
n is isomorphic to the universal enveloping algebra of vector fields on
n . (Strictly speaking, this is true only for polynomial vector fields and
differential operators, more sophisticated mathematical statements may
be required for smooth functions, but this point is only technical.) More
concretely, the symbol of a differential operator of order r is represented
by a symmetric tensor field of rank r. In the light of these remarks, it is
tempting to conjecture that, for higher-spin gauge theories, the algebra of
Hermitian differential operators,
µ1...µr (x) ∂µ1 . . . ∂µr + Hermitian conjugate
generalizes the algebra of infinitesimal diffeomorphisms for gravity. An-
other argument in favour of this conjecture may be presented in the
“gauge” sector by looking at the metric-like formulation of higher-spins
arising from the frame-like formulation of Vasiliev, at first order in the
coupling constant [21].
6 Conclusion
The conclusion is that there are two complementary but distinct ways
of using rigid symmetries of the free theory in order to guess the proper
gauge symmetry principle of higher-spin gauge theories.
On the one hand, the infinite set of rigid symmetries of the free (or,
maybe, even integrable) “matter” sector, might be gauged by the intro-
duction of a connection via a minimal coupling prescription. The idea of
using a massive scalar field as free matter sector and an infinite tower of
massless symmetric tensor fields as interacting gauge sector is in agree-
ment with the isomorphism between the off-shell higher-spin algebra and
the space of reducibility parameters. (If tensor fields are used as free
“matter” sector, then the symmetry algebra could be larger. Following
the lines of the Vasiliev construction in such case, the structure of the uni-
versal enveloping algebra points towards a larger infinite tower of gauge
fields including mixed-symmetry tensors.)
On the other hand, in the free “gauge” sector, rigid symmetries linked
to reducibility parameters may arise from the linearization of the gauge
symmetries of some non-linear action. Thus the complete knowledge of the
rigid symmetries of free higher-spin gauge theories would indicate what
can be the linearized connection.
Acknowledgments
I. Bakas, G. Barnich, N. Boulanger, T. Damour and J. Remmel are
thanked for very useful exchanges. The author is grateful to the orga-
nizers for their invitation to this enjoyable meeting and the opportunity
to present his lecture. The Institut des Hautes Études Scientifiques de
Bures-sur-Yvette is acknowledged for its hospitality.
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	Higher-spin interaction problem
	The Noether method
	Free theory symmetries
	Higher-spin algebras
	A gauge principle for higher-spins ?
	Conclusion
