Group manifold approach to higher spin theory

0.4 MB·English

by  Shan Hu

Group manifold approach to higher spin theory
5
1 Shan Hu a1, and Tianjun Li a,b2
0
2
a State Key Laboratory of Theoretical Physics and Kavli Institute for Theoretical Physics

p
e China (KITPC), Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing
S
100190, P. R. China
8
b School of Physical Electronics, University of Electronic Science and Technology of China,

]
h Chengdu 610054, P. R. China
t
-
p
e
h
[

3 Abstract
v
2
We consider the group manifold approach to higher spin theory. The deformed
2
3 localhigherspintransformationisrealizedasthediffeomorphismtransformationinthe
2 group manifold M. With the suitable rheonomy condition and the torsion constraint
0
imposed, the unfolded equation can be obtained from the Bianchi identity, by solving
.
1 which, fields in M are determined by the multiplet at one point, or equivalently, by
0
5 (Wµ[a(s−1),b(0)],H) in AdS4 ⊂ M. Although the space is extended to M to get the
1 geometrical formulation, the dynamical degrees of freedom are still in AdS . The
4
:
v 4d equations of motion for (W[a(s−1),b(0)],H) are obtained by plugging the rheonomy
µ
i
X condition into the Bianchi identity. The proper rheonomy condition allowing for the
r maximum on-shell degrees of freedom is given by Vasiliev equation. We also discuss
a
the theory with the global higher spin symmetry, which is in parallel with the WZ
model in supersymmetry.
1E-mail:[email protected]
2E-mail:[email protected]
1 Introduction
Group manifold approach provides a natural geometrical formulation for supergravity [1,
2, 3, 4]. The starting point is the supergroup Osp(1/4) or Osp(1/4). Supergravity field
and matter field are vielbein 1-form νA and 0-form H on the group manifold M, A,M¯ =
M¯
1,··· ,dimOsp(1/4). Local super Poincare transformation is realized as the diffeomorphism
transformation on M. The curvature RA for the 1-form can be defined, on which, the
M¯N¯
rheonomy condition is imposed [1, 2, 3, 4]. The condition requires that RA can be alge-
M¯N¯
braically expressed in terms of its purely “inner” components RA with µ,ν = 1,2,3,4 the
µν
indices in a four-dimensional submanifold M . Namely,
4
RA = rA |µνRB , or RA = rA |abRB (1.1)
M¯N¯ M¯N¯ B µν CD CD B ab
where rA |µν and rA |ab are constant holonomic and anholonomic tensors. a,b = 1,2,3,4.
M¯N¯ B CD B
The rheonomy condition ensures that fields on the whole M are determined by fields on M .
4
So the final dynamics is still in M , where the diffeomorphism transformation reduces to the
4
on-shell super Poincare transformation of the 4d fields. The equations of motion in M are
4
obtained by plugging the rheonomy condition into the Bianchi identity. Instead of imposing
the rheonomy condition, one can also construct the extended action, which is the integration
of some 4-form on a 4d submanifold M . Variation of the action with respect to both fields
4
and M gives the rheonomy condition as well as the 4d equations of motion.
4
In this paper, we will reformulate the group manifold method, adding an infinite number
of auxiliary fields so that the final system is equivalent to the unfolded dynamics approach
which is convenient for higher spin theory [5]. For simplicity, we will consider the minimal
bosonic 4d HS algebra ho(1|2 : [3,2]) with spin s = 0,2,··· [6]. The corresponding group
manifold is denoted as M. Fields are 1-form Wα and 0-form H on M with the curvature
M¯
2-form
1 1
dWα = (fα +Rα )Wβ ∧Wγ = fˆα Wβ ∧Wγ (1.2)
2 βγ βγ 2 βγ
and the 1-form
dH = H Wα. (1.3)
α
M¯ = 1,2,··· ,dimho(1|2 : [3,2]). α ∼ [a(s − 1),b(t)] is in the adjoint representation of
ho(1|2 : [3,2]). fα is the structure constant of ho(1|2 : [3,2]). The deformed higher spin
βγ
transformation is the diffeomorphism transformation on M.
The rheonomy condition is
fˆα = fˆα (R[a(s−1),b(s−1)],R[a(s−1),b(s−1)],··· ,H,H ,···)
βγ βγ ab ab;c1 c1
[a(s−1),b(s−1)] [a(s−1),b(s−1)]
H = h (R ,R ,··· ,H,H ,···), (1.4)
α α ab ab;c1 c1
where
[a(s−1),b(s−1)] [a(s−1),b(s−1)]
R = ∂ ···∂ R , H = ∂ ···∂ H. (1.5)
ab;c1···cn cn c1 ab c1···cn cn c1
1
∂ = WM¯∂ , a,b,c = 1,2,3,4. a is the abbreviation for the [0,a] element of ho(1|2 : [3,2]).
c c M¯
Different from the supergravity situation, the curvature depends on the “inner” components
as well as their “inner” derivatives. This is the most generic rheonomy condition. (1.4)
together with the Bianchi identity gives the unfolded equation
1
dWα = fˆα Wβ ∧Wγ,
2 βγ
dR[a(s−1),b(s−1)] = r[a(s−1),b(s−1)]Wγ,
ab;c1···cn ab;c1···cnγ
dH = h Wγ, (1.6)
c1···cn c1···cnγ
from which, (Wα,R[a(s−1),b(s−1)],H ) on the whole M is determined by its value at one
ab;c1···cn c1···cn
point. On AdS ⊂ M, we have the further relation
4
[a(s−1),b(s−1)] [a(s−1),b(s−1)]
(R ,R ,··· ,H,H ,···)
ab ab;c1 c1
∼ (W[a(s−1),b(0)],∂ W[a(s−1),b(0)],··· ,H,∂ H,···), (1.7)
µ ν1 µ ν1
[a(s−1),b(0)]
where ∂ is the derivative on AdS . So equivalently, with (W ,H) given on AdS ,
µ 4 µ 4
(Wα,H) on the whole M can be determined up to a gauge transformation. The dynam-
M¯
ical 1-form fields are W[a(s−1),b(0)], which is because in (1.4), the torsion constraint is also
implicitly imposed: fˆα and H do not depend on R[a(s−1),b(t)] with t 6= s − 1. For 0-
βγ α ab;c1···cn
form, the deformed higher spin transformation is ξM¯∂ = ǫα∂ , under which, the multiplet
M¯ α
[a(s−1),b(s−1)] [a(s−1),b(s−1)]
(R ,R ,··· ,H,H ,···) forms the complete representation on-shell.
ab ab;c1 c1
The whole dynamics is encoded in functions (fˆα ,h ), which should satisfy the Bianchi
βγ α
identity and also give the correct free theory limit. With the unfolded equation plugged in,
the Bianchi identities are polynomials of (R[a(s−1),b(s−1)],H ), by solving which, (fˆα ,h )
ab;c1···cn c1···cn βγ α
[a(s−1),b(s−1)]
is determined with the rest constraints on (R ,H ) acting as the 4d equations
ab;c1···cn c1···cn
of motion. The procedure is simple in supergravity but is extremely complicated in higher
spin theory. Instead of fixing (fˆα ,h ) and getting the 4d equations of motion by solving
βγ α
the Bianchi identity, one can first identify the on-shell degrees of freedom, for example,
Φσ˜ ∼ Φ[a(s+n),b(s)] in the twisted-adjoint representation of the higher spin algebra, then find
the suitable (fˆα ,h ) so that the Bianchi identity is satisfied for the arbitrary Φσ˜.
βγ α
[a(s−1),b(s−1)]
{H ,n = 0,1,···} ∪ {R ,s = 2,4,··· ,n = 0,1,···} (1.8)
c1···cn ab;c1···cn
and {Φ[a(s+n),b(s)],s = 0,2,··· ,n = 0,1,···} have the same number of indices. With the 4d
equations of motion imposed on (1.8), the two may contain the same number of degrees of
freedom. Written in terms of Φσ˜, the unfolded equation becomes
1
dWα = f¯α (Φσ˜)Wβ ∧Wγ, dΦα˜ = Fα˜(Φσ˜)Wβ. (1.9)
2 βγ β
It remains to find (f¯α ,Fα˜) satisfying the Bianchi identity and also giving rise to the correct
βγ β
free theory limit3. Vasiliev theory gives the elegant solution to this problem [7, 8, 9]. By
3As is shown in appendix C, there are (f¯α ,Fα˜) satisfying the Bianchi identity but failing to give the
βγ β
correctfree theorylimit. Itisunclearwhether the tworequirementscanuniquely fix (f¯α ,Fα˜)(upto afield
βγ β
redefinition) or not.
2
solving the Z part of the Vasiliev equation order by order, one may finally get the required
(f¯α ,Fα˜) [10].
βγ β
For supersymmetry, it is also possible to study the dynamics of the 0-form matter on
group manifold with the fixed background such as the WZ model. The component expan-
sion of the 0-form on superspace gives the spin 0 and 1/2 fields in 4d. The allowed gauge
transformation is the global super Poincare transformation, which is the diffeomorphism
transformation on M preserving the background. For higher spin theory, one can similarly
consider the 0-form H on M with
1
dWα = fα Wβ ∧Wγ, dH = H Wα. (1.10)
0 2 βγ 0 0 α 0
Wα describes the background with the vanishing curvature. The system has the global
0
HS symmetry. The component expansion of H on M gives the spin s = 0,2,··· fields
Rs . On the other hand, the linearized Vasiliev equation for the 0-forms on back-
a1···as,b1···bs
ground Wα is
0
dΦα˜ = kα˜ Φγ˜Wβ, (1.11)
βγ˜ 0
which is also invariant under the global HS transformation. kα˜ is the constant. With
βγ˜
Φ ≡ Φ[a(0),b(0)] = H, from dΦ = k Φγ˜Wβ, we have H = k Φγ˜. Rs can then be
βγ˜ 0 β βγ˜ a1···as,b1···bs
taken as the Weyl tensor of the linearized HS theory. With the space extended from AdS
4
to M, 0-forms in the linearized Vasiliev theory get the interpretation as the derivatives of a
single 0-form H on M.
The rest of the paper is organized as follows. In Section 2, we construct a symmetric
space M with the higher spin transformation group the isometry group. In Section 3, we
consider the theory with the local higher spin symmetry. The discussion and conclusion are
given in Section 4.
2 Symmetric space from the higher spin algebra
We will consider the minimal bosonic higher spin theory in AdS with the coordinate uµ,
4
µ = 1,2,3,4. The related HS algebra is ho(1|2 : [3,2]) with the basis {t ∼ t }
α A1···As−1,B1···Bs−1
[6]. t is in irreducible representations of SO(3,2) characterized by two row
A1···As−1,B1···Bs−1
rectangular Young tableaux, A ,B = 0,1,2,3,4, s = 2,4,···
i i
t = t = t ,
A1···As−1,B1···Bs−1 {A1···As−1},B1···Bs−1 A1···As−1,{B1···Bs−1}
t = 0, t B1···Bs−1 = 0. (2.1)
{A1···As−1,As}B2···Bs−1 A1···As−3CC,
With a ,b = 1,2,3,4, basis of ho(1|2 : [3,2]) can be rewritten as
i i
{t } = {t }
α A1···As−1,B1···Bs−1
= {t ,t ,t ,··· ,t }. (2.2)
0···0,b1···bs−1 0···0a1,b1···bs−1 0···0a1a2,b1···bs−1 a1···as−1,b1···bs−1
3
Let
{t } = {t ,t ,··· ,t } (2.3)
Q 0···0a1,b1···bs−1 0···0a1a2a3,b1···bs−1 a1···as−1,b1···bs−1
be the basis of a[E],
{t } = {t ,t ,··· ,t } (2.4)
A 0···0,b1···bs−1 0···0a1a2,b1···bs−1 0a1···as−2,b1···bs−1
be the basis of K, ho(1|2 : [3,2]) = a[E]⊕K.
[a[E],a[E]] ⊂ a[E], [a[E],K] ⊂ K, [K,K] ⊂ a[E]. (2.5)
a[E] is a subalgebra of ho(1|2 : [3,2]) generating a subgroup E. The coset space G[ho(1|2 :
[3,2])]/E is a symmetric space according to (2.5). With the group given, it is a standard
procedure in mathematics to construct the group manifold M for G[ho(1|2 : [3,2])]4 and the
symmetric space M for G[ho(1|2 : [3,2])]/E. In the following, we will give a construction
based on the operators and the conserved charges of the quantum higher spin theory in
AdS . For earlier work on space with the tensor coordinates, see [11, 12].
4
In quantum higher spin theory, there are conserved charges {Q } in one-
A1···As−1,B1···Bs−1
to-one correspondence with {t }. In particular, {Q } are generators of
A1···As−1,B1···Bs−1 A1,B1
SO(3,2). Suppose 0 is a point in the bulk of AdS , for example, (1,0,0,0,0) in x2 −x2 −
4 0 1
x2−x2+x2 = 1, and O(0) is the operator for the spin 0 field at 0, then the orbit generated
2 3 4
by SO(3,2) gives operators for the spin 0 field in the entire AdS .
4
{O(u)|u ∈ AdS } = {gO(0)g−1|g ∈ SO(3,2)}, (2.6)
4
where g = eiωA1,B1QA1,B1. Aside from AdS4, the orbit generated by G[ho(1|2 : [3,2])] gives
operators in an enlarged space M.
{O(z)|z ∈ M} = {gO(0)g−1|g ∈ G[ho(1|2 : [3,2])]}, (2.7)
where g = eiωA1···As−1,B1···Bs−1QA1···As−1,B1···Bs−1. In G[ho(1|2 : [3,2])], there is a subgroup E(z),
∀ e ∈ E(z), eO(z)e−1 = O(z). The higher spin algebra is decomposed as
ho(1|2 : [3,2]) = K(z)⊕a[E(z)] = g(z)K(0)g(z)−1 ⊕g(z)a[E(0)]g(z)−1 (2.8)
with K(z) the tangent space of M at z. M is the coset space G[ho(1|2 : [3,2])]/E. In
particular, SO(3,1) ⊂ E, SO(3,2) ⊂ G[ho(1|2 : [3,2])], AdS = SO(3,2)/SO(3,1), so M
4
has a fiber bundle structure with the fiber AdS attached at each point of the base manifold.
4
It remains to determine the subalgebra a[E]. Although the direct quantization of the
higher spin theory in AdS is still not available, its CFT dual is quite simple. In appendix
4
A, the CFT realization of O(0), or more accurately, O+(0), is given. It is shown that the
4HS transformation group is the global symmetry group of the 3d O(N) vector model and the dual
minimalbosonicHStheoryinAdS . Therelatedalgebraisho(1|2:[3,2]). Thegroupcanbenon-connected,
4
just as SO(3,1). Here G[ho(1|2:[3,2])] refers to the the connected piece containing the identity, which is a
simple group. So the related group manifold M is also connected.
4
charge Q corresponding to (2.3) commutes with O(0). So a[E] constructed
0···0a1···a2k−1,b1···bs−1
here is the same as (2.3).
The metric on the coset space M = G[ho(1|2 : [3,2])]/E is defined in group theory.
Alternatively, we can use the operator O(z) to get the same result. There is a one-to-one
correspondence between T (M) = {vM∂ |M = 1,··· ,dimM} and K(z). For the given ∂ ,
z M M
¯
∃ k (z) ∈ K(z) satisfying
M
∂ O(z) = i[k (z),O(z)]. (2.9)
M M
{k (z)}composethebasisforK(z), fromwhich, onecandefineaspecialsetofthecoordinate
M
on M
O(z) = eikM(0)zMO(0)e−ikM(0)zM. (2.10)
The metric on T (M) can be induced from K(z), i.e.
z
g (z) = hk (z)|k (z)i, (2.11)
MN M N
where hk (z)|k (z)i is the killing form. g is G[ho(1|2 : [3,2])] invariant. Under the
M N MN
G[ho(1|2 : [3,2])] transformation,
O(z) → gO(z)g−1 = O(z′). (2.12)
G[ho(1|2 : [3,2])] generates the isometric transformation z → z′ on M.
The tangent space on the coset space M is {t }. The group manifold of ho(1|2 : [3,2]) is
A
the manifold M with the tangent space {t }, dimM = dimho(1|2 : [3,2]). The coordinate
α
on M is Z , ik (Z) = ∂ g(Z)g(Z)−1, G (Z) = hk (Z)|k (Z)i.
M¯ M¯ M¯ M¯N¯ M¯ N¯
∂ O(Z) = i[k (Z),O(Z)]. (2.13)
M¯ M¯
When k (Z) ∈ E(Z), ∂ O(Z) = 0. Let {t } be a set of the orthogonal normalized basis
M¯ M¯ α
of ho(1|2 : [3,2]), one may assume k (Z) = g(Z)t g(Z)−1. k (Z) = Wα(Z)k (Z) and
α α M¯ M¯ α
k (Z) = WM¯(Z)k (Z) gives the vielbein on M:
α α M¯
WαWM¯ = δα, WαWN¯ = δN¯, η WαWβ = G . (2.14)
M¯ β β M¯ α M¯ αβ M¯ N¯ M¯N¯
η = fρ fσ = ht |t i is the killing metric for ho(1|2 : [3,2]) with the suitable normalization
αβ ασ βρ α β
assumed.5 Suppose ∂ k (Z) = ΓL¯ k (Z), ∂ k (Z) = φβ k (Z), there will be
N¯ M¯ N¯M¯ L¯ N¯ α N¯α β
∂ WM¯ +ΓM¯ WL¯ −φβ WM¯ = 0. (2.15)
N¯ α N¯L¯ α N¯α β
With the covariant derivative defined as D = ∂ −Γ and D = WM¯(∂ −φ ) = ∂ −φ ,
M¯ M¯ M¯ α α M¯ M¯ α α
we have
D ···D D O(Z) = in[k (Z),[k (Z),···[k (Z),O(Z)]···]], (2.16)
M¯n M¯2 M¯1 M¯1 M¯2 M¯n
D ···D D O(Z) = in[k (Z),[k (Z),···[k (Z),O(Z)]···]]. (2.17)
αn α2 α1 α1 α2 αn
5Notice that the killing metric η is indefinite having one sign for compact directions and the opposite
αβ
for non-compact directions. G(ho(1|2 : [3,2]) is obviously not a compact group as one can see from its
subgroup SO(3,2).
5
As is shown in Appendix A, for [Q ,O(0)] with k = 1,3,···
0···0a1···as,b1···bs+k
[Q ,O(0)]
0···0a1···as,b1···bs+k
t=1,2,···,2s+k−2r
= gc1···c2r+t [Q ,···[Q ,[Q ,O(0)]]···].
0···0a1···as,b1···bs+k 0c1···cr,cr+1···c2r+1 0,c2r+t−1 0,c2r+t
X
r=0,2,···,s
(2.18)
At the point Z, O(Z) = g(Z)O(0)g(Z)−1, Q (Z) = g(Z)Q g(Z)−1,
A A
[Q (Z),O(Z)]
0···0a1···as,b1···bs+k
t=1,2,···,2s+k−2r
= gc1···c2r+t
0···0a1···as,b1···bs+k
X
r=0,2,···,s
[Q (Z),···[Q (Z),[Q (Z),O(Z)]]···]. (2.19)
0c1···cr,cr+1···c2r+1 0,c2r+t−1 0,c2r+t
Since
D D ···D O(Z)
0,bs+k 0,bs+k−1 0a1···as,b1···bs+1
= ik[Q (Z),···[Q (Z),[Q (Z),O(Z)]]···], (2.20)
0a1···as,b1···bs+1 0,bs+k−1 0,bs+k
there will be
∂ O(Z)
0···0a1···as,b1···bs+k
t=1,2,···,2s+k−2r
= i1−tgc1···c2r+t D D ···D O(Z).
0···0a1···as,b1···bs+k 0,c2r+t 0,c2r+t−1 0c1···cr,cr+1···c2r+1
X
r=0,2,···,s
(2.21)
According to the definition, φβ and WM¯ are invariant under the global higher spin trans-
M¯γ α
formation, so is their contraction φ . φ is a scalar, so it must be a constant on M. (2.21)
α α
can be further rewritten as
∂ O(Z)
0···0a1···as,b1···bs+k
t=1,2,···,2s+k−2r
= Gc1···c2r+t ∂ ∂ ···∂ O(Z)
0···0a1···as,b1···bs+k 0,c2r+t 0,c2r+t−1 0c1···cr,cr+1···c2r+1
X
r=0,2,···,s
(2.22)
for some constant Gc1···c2r+t .
0···0a1···as,b1···bs+k
Just as the chiral constraint relates ∂ with ∂ , here, ∂ is determined by
θ¯ µ 0···0a1···as,b1···bs+k
∂ ∂ ···∂ . This is because [Q (Z),O(Z)]|0i are all
0,c2r+t 0,c2r+t−1 0c1···cr,cr+1···c2r+1 A1···As−1,B1···Bs−1
in the 1-particle Hilbert space of the higher spin theory, for which
{[Q (Z),···[Q (Z),[Q (Z),O(Z)]]···]}
0,bs+k 0,bs+2 0a1···as,b1···bs+1
∼ {[Q (Z),[Q (Z),··· ,[Q (Z),O(Z)]···]]} (2.23)
0a1···as,b1···bs+1 0,bs+2 0,bs+k
6
compose thecomplete basis.6 In[12], by considering the zerothlevel ofthe unfoldedequation
for the 0-form Φ in M, the similar result is also obtained. Φ = Φ[a(0),b(0)] is the lowest
component of Φ[a(s+t),b(s)]. Generically, one may expect
[Q0···0ap1···apsp,bp1···bpsp+kp(Z),···[Q0···0a11···a1s1,b11···b1s1+k1(Z),O(Z)]···]
∼ α(a ···a ,b ···b )[Q (Z),···[Q (Z),[Q (Z),O(Z)]]···],
1 s 1 s+k 0a1···as,b1···bs+1 0,bs+k−1 0,bs+k
X
(2.24)
where α(a ···a ,b ···b ) are constants to be determined.
1 s 1 s+k
∂0···0a11···a1s1,b11···b1s1+k1 ···∂0···0ap1···apsp,bp1···bpsp+kpO(Z)
∼ Λ(a ···a ,b ···b )∂ ∂ ···∂ O(Z). (2.25)
1 s 1 s+k 0,bs+k 0,bs+k−1 0a1···as,b1···bs+1
X
(2.25) is the G[ho(1|2 : [3,2])]-invariant differential operators on M, which will be useful in
section 3.6 when we try to construct the theory with the global higher spin symmetry. .
3 Theory with the local higher spin symmetry
In section 2, the background in M is fixed to be the intrinsic geometry with dWα−1fα Wβ∧
0 2 βγ 0
Wγ = 0, which is invariant under the global higher spin transformation preserving Wα. To
0 0
have the local higher spin symmetry, the 1-form Wα in M should be dynamical. We will
study the dynamics of the 1-form Wα and the 0-form H in M. With the suitable rheonomy
condition and the torsion constraint imposed, (Wα,H) in the whole M is determined by
[a(s−1),b(0)]
(W ,H) in AdS . We then discuss the relation between the unfolded equation in
µ 4
group manifold approach and the unfolded equation in Vasiliev theory. We will also make a
comment on theory with the global higher spin symmetry.
3.1 Higher spin theory on group manifold and the rheonomy con-
dition
The 1-form Wα is the vielbein on M. WαWM¯ = δα, WαWN¯ = δN¯, η WαWβ = G .7
M¯ M¯ β β M¯ α M¯ αβ M¯ N¯ M¯N¯
The curvature 2-form is defined as
1
Rα = dWα − fα Wβ ∧Wγ. (3.1)
2 βγ
It is convenient to use the 0-form Rα to parameterize Rα , Rα = Rα Wβ Wγ, Rα =
βγ M¯N¯ M¯N¯ βγ M¯ N¯ βγ
Rα WM¯WN¯.
M¯N¯ β γ
1 1
dWα = (fα +Rα )Wβ ∧Wγ = fˆα Wβ ∧Wγ, (3.2)
2 βγ βγ 2 βγ
6More precisely, it is {[Q (Z),···[Q (Z),Os (Z)]···]} that forms the complete basis,
0,bs+k 0,bs+2 a1···as,b1···bs
but (2.23) is enough to generate [QA1···As−1,B1···Bs−1(Z),O(Z)] since [QQ(Z),O(Z)]=0 for tQ in (2.3).
7Here, Wα is invertible, which is general enough to account for the 4d HS theory, in which, the relevant
M¯
field is Wα. Let {α}={α˜}∪{a}, {M¯}={M˜}∪{µ}, Wa ∼ea is usually required to be invertible, one can
µ µ µ
also suitably select Wα˜ to make the whole Wα invertible.
M˜ M¯
7
where fˆα is the deformed structure constant. The Bianchi identity is
βγ
∂ fˆα +fˆα fˆβ = 0, (3.3)
[γ ρσ] β[γ ρσ]
where ∂ = WM¯∂ . In addition, we can add the 0-form matter field H on M,
γ γ M¯
dH = H Wα ⇔ ∂ H = H , (3.4)
α α α
∂ H +H fˆα = 0. (3.5)
[ρ σ] α ρσ
The group manifold M is necessarily involved in the definition of Rα and H . (3.3) and
βγ α
(3.5) are defined in M as well.
The definition (3.2) and (3.4) is invariant under the diffeomorphism transformation gen-
erated by ξM¯,
δ Wα = ξN¯∂ Wα +∂ ξN¯Wα, δ fˆα = ξN¯∂ fˆα , δ H = ξN¯∂ H, δ H = ξN¯∂ H .
ξ M¯ N¯ M¯ M¯ N¯ ξ βγ N¯ βγ ξ N¯ ξ α N¯ α
(3.6)
With
ǫα = ξM¯Wα, ξM¯ = ǫαWM¯, (3.7)
M¯ α
(3.6) can be rewritten as
δ Wα = dǫα +fˆα ǫβWγ, δ fˆα = ǫβ∂ fˆα , δ H = ǫβH , δ H = ǫβ∂ H , (3.8)
ǫ βγ ǫ ρσ β ρσ ǫ β ǫ α β α
which is the deformed local higher spin transformation.
δ δ −δ δ = δ , [ǫ ,ǫ ]α = fˆα ǫγǫβ. (3.9)
ǫ2 ǫ1 ǫ1 ǫ2 [ǫ2,ǫ1] 2 1 βγ 2 1
The algebra is closed with the deformed structure constant fˆα .
βγ
If for some Λ, Rα = 0, fˆα = fα , the local gauge transformation generated by ǫΛ is
Λγ Λγ Λγ
undeformed. It is necessary to require Rα ≡ Rα = 0 to make the local Lorentz
[a(1),b(1)]γ (ab)γ
transformation undeformed. Also, since H is a scalar, H = 0 should hold so that δ H =
(ab) ǫab
ǫabH = 0. From (3.3) and (3.5),
(ab)
δ Rα = ǫab∂ Rα = ǫab[fα Rβ +fβ Rα ], δ H = ǫab∂ H = −ǫabfβ H .
ǫab ρσ (ab) ρσ (ab)β ρσ (ab)[ρ σ]β ǫab α (ab) α (ab)α β
(3.10)
The evolution along the (ab) direction is a local Lorentz transformation, so the group man-
ifold M effectively reduces to the coset space M = G[ho(1|2 : [3,2])]/SO(3,1). Recall that
in Section 2, we have discussed the coset space M = G[ho(1|2 : [3,2])]/E. For M to reduce
to M, there must be Rα = 0 so that the local gauge transformation generated by ǫQ is
Qγ
undeformed. However, at least in Vasiliev theory, Rα = 0 is valid but Rα = 0 does not
(ab)γ Qγ
necessarily hold.
When β 6= (ab), ∂ fˆα and ∂ H cannot be uniquely determined by (3.3) and (3.5).
β ρσ β α
Nevertheless, from (3.3) and (3.5), we have
∂ Rα = ∂ Rα +fˆα fˆβ (3.11)
γ ab [b a]γ β[γ ba]
∂ H = ∂ H +H fˆα (3.12)
γ a a γ α aγ
8
with H = ∂ H. a represents the [0,a] element of ho(1|2 : [3,2]). Let
a a
Rα = ∂ ···∂ Rα , H = ∂ ···∂ H, (3.13)
ab;c1···cn cn c1 ab c1···cn cn c1
if
Rα = rα (Rσ ,Rσ ,··· ,H,H ,···)
βγ βγ ab ab;c1 c1
H = h (Rσ ,Rσ ,··· ,H,H ,···) (3.14)
γ γ ab ab;c1 c1
with rα and h the polynomials of Rσ ,Rσ ,··· ,H,H ,··· with the constant coefficients,
βγ γ ab ab;c1 c1
then
∂ Rα = ∂ Rα +fˆα fˆβ = rα (Rσ ,Rσ ,··· ,H,H ,···) (3.15)
γ ab [b a]γ β[γ ba] ab;γ ab ab;c1 c1
∂ H = ∂ H +H fˆα = h (Rσ ,Rσ ,··· ,H,H ,···) (3.16)
γ a a γ α aγ a;γ ab ab;c1 c1
are also polynomials. Moreover, since
(∂ ∂ −∂ ∂ )F = fˆα ∂ F, (3.17)
β γ γ β γβ α
(∂ ∂ −∂ ∂ )Rα = fˆσ∂ Rα = fˆσrα (Rβ ,Rβ ,··· ,H,H ,···), (3.18)
c γ γ c ab γc σ ab γc ab;σ ab ab;c1 c1
(∂ ∂ −∂ ∂ )H = fˆσ∂ H = fˆσh (Rβ ,Rβ ,··· ,H,H ,···), (3.19)
c γ γ c a γc σ a γc a;σ ab ab;c1 c1
so
∂ Rα = ∂ ∂ Rα = ∂ rα −fˆσrα = rα (3.20)
γ ab;c γ c ab c ab;γ γc ab;σ ab;cγ
and
∂ H = ∂ H = ∂ ∂ H = ∂ h −fˆσh = h = h (3.21)
γ ac γ a;c γ c a c a;γ γc a;σ a;cγ acγ
are again polynomials. Subsequently, one can prove for n = 0,1,···, we have
∂ Rα = rα (Rβ ,Rβ ,··· ,H,H ,···),
γ ab;c1···cn ab;c1···cnγ ab ab;c1 c1
∂ H = h (Rβ ,Rβ ,··· ,H,H ,···), (3.22)
γ c1···cn c1···cnγ ab ab;c1 c1
or equivalently,
dRα = rα (Rβ ,Rβ ,··· ,H,H ,···)Wγ,
ab;c1···cn ab;c1···cnγ ab ab;c1 c1
dH = h (Rβ ,Rβ ,··· ,H,H ,···)Wγ, (3.23)
c1···cn c1···cnγ ab ab;c1 c1
where rα and h are polynomials of Rβ ,Rβ ,··· ,H,H ,···. Finally, we get
ab;c1···cnγ c1···cnγ ab ab;c1 c1
the unfolded equation
1
dWα = (fα +rα )Wβ ∧Wγ,
2 βγ βγ
dRα = rα Wγ,
ab;c1···cn ab;c1···cnγ
dH = h Wγ, (3.24)
c1···cn c1···cnγ
9