Flat Space Higher-Spin Gravity
12.1 Spin-3 Gravity in Flat Space
Flat Space Higher-Spin Gravity
„
Once upon a midnight dreary, while I pondered, weak and weary,Over many a quaint and curious volume of forgotten lore—
While I nodded, nearly napping, suddenly there came a tapping, As of some one gently rapping, rapping at my chamber door.
“Tis some visitor,” I muttered,“tapping at my chamber door”—
Only this and nothing more.”
– Edgar Allen Poe The Raven
I
n the previous chapter I looked for unitary highest-weight representations of F W-algebras that are the flat space analogues of W-algebras, which play an important role as the asymptotic symmetry algebras of (non-)AdS higher-spin gravity theories. In this chapter I review how theseF W-algebras arise as asymptotic symmetries of asymptotically flat spacetimes using spin-3 gravity in flat space as an explicit example. This will serve as a basis for Chapter13where I will extend the higher-spin description in flat space by including chemical potentials.On a technical level, especially in terms of a Chern-Simons description, higher-spin theories in three dimensional flat space are actually not very different from the description in AdS3. I have already shown how to describe spin-2 gravity in flat space using anisl(2,R)valued Chern-Simons gauge field previously in Section10.1.
The higher-spin extension of this formalism works in complete analogy to the AdS case by simply replacing the gauge algebraisl(2,R)→isl(N,R). Depending on the choice of embedding ofisl(2,R) ,→ isl(N,R) one again obtains different theories with different spectra of spins. The biggest subtlety one has to take care of is the invariant bilinear form which is different for isl(N,R) in comparison tosl(N,R).
Aside from that subtlety one can apply most of the techniques and intuition already known from AdS3 holography.
12.1 Spin-3 Gravity in Flat Space
Similar to the AdS case, spin-3 gravity in the principal embedding1is the simplest higher-spin extension one can consider in flat space and has been first described in [150].
1Here principal embedding refers to the principal embedding ofisl(2,R),→isl(3,R)
The starting point is again the Chern-Simons action (2.3) whereAtakes values in the principal embedding ofisl(3,R)which is generated by the generatorsLn, Mn, Un, Vn as
[Ln, Lm] = (n−m)Ln+m, (12.1a)
[Ln, Mm] = (n−m)Mn+m, (12.1b)
[Ln, Um] = (2n−m)Un+m, (12.1c) [Ln, Vm] = (2n−m)Vn+m, (12.1d) [Mn, Um] = (2n−m)Vn+m, (12.1e) [Un, Um] =σ(n−m)(2n2+ 2m2−nm−8)Ln+m, (12.1f) [Un, Vm] =σ(n−m)(2n2+ 2m2−nm−8)Mn+m, (12.1g) where the factorσ fixes the overall normalization of the spin-3 generatorsUnand Vnand can be chosen at will. I will fix this normalization constant toσ =−13. The algebra (12.1) naturally comes with aZ2 grading so that the generatorsLn,Un
are even andMn,Vnare odd, respectively. Then even with even gives even, even with odd gives odd and odd with odd vanishes. In terms of ˙Inönü–Wigner contractions, the-independent generators are the even generators and the generators linear in are the odd generators [70].
There are various ways of constructing matrix representations ofisl(N,R)algebras.
One particular neat way to do this is outlined in AppendixA.2. The basic idea is to combine fundamental matrix representations of sl(2,R) and the Grassmann parameterin a block diagonal matrix in a certain way and then use2= 0for all computations involving the matrix representation constructed this way.
Moreover, this construction can be used to define various traces that correspond to different bilinear forms on (subalgebras of)isl(N,R). For example the invariant bilinear form h·,·i, which is relevant for a flat space description in terms of the Chern-Simons action (2.3) can be obtained from the representation I described above by defining a (hatted) trace
hGaGbi=Trb GaGb:= d d
1
4Tr GaGbγ(D)∗
=0, (12.2)
with
γ?(D) = 1lD×D 0 0 −1lD×D
!
, (12.3)
and for two generators Ga,Gb ∈isl(N,R). The subscript inγ(D)? is the dimension of the matrix representation ofsl(N,R)used to construct theisl(N,R)matrix rep-resentation. Since the matrix representations used in this thesis are based on the fundamental representation ofsl(N,R)one has for all practical purposesD=N.
Another invariant bilinear form2can be obtained from the representation I described above by defining a (twisted) trace over a product ofnisl(N,R)generatorsGi as
Tre
n
Y
i=1
Gi
!
= 1 2Tr
n
Y
i=1
d dGiγ(D)?
!
. (12.4)
Thus, there are basically three notions of traces available, the normal, twisted and hatted trace. This allows one to use these traces as tools to pick out information which is related to even (odd) [mixed] contributions by using the trace (twisted trace) [hatted trace].
The twisted trace (12.4) for example can be used to determine the spin-2 and spin-3 fields since those correspond to symmetric products of powers of the zuvielbein which is a purely odd quantity. Hence the metric can equivalently be determined by
gµν = 1
2Tre AµAν=heµeνi . (12.5) The spin-3 field is similarly defined from the cubicsl(3,R)-Casimir or, equivalently, by using again the twisted trace
Φµνλ= 1
6Tre AµAνAλ=heµeνeλi. (12.6) Explicit expressions forisl(3,R) valued connections that obey asymptotically flat boundary conditions were established independently in [150] and [154] (see also [157]). As I have shown in Section10.1(see also [74]) one can, similar to the AdS case, gauge away the radial dependence of the action using
A=b−1db+b−1a(u, ϕ)b, b=er2M−1. (12.7) In the spin-3 gravity setup one can then choose the following boundary conditions fora(u, ϕ)[150,154]
a(u, ϕ) =aϕ(u, ϕ) dϕ+au(u, ϕ) du, (12.8) where
aϕ(u, ϕ) =L1−M
4 L−1−N
2 M−1+V
2U−2+ZV−2, (12.9a) au(u, ϕ) =M1−M
4 M−1+V
2V−2, (12.9b)
with
N =L(ϕ) +u
2M0(ϕ), Z =U(ϕ) +u
2V0(ϕ), (12.10)
2This is actually only an invariant bilinear form on the subalgebra of odd generators.
where all other components have to vanish at the asymptotic boundaryr→ ∞. This form of the state dependent functions is a result of the equations of motion which requireF = dA+A ∧ A= 0which is tantamount to
∂uM=∂uV = 0, ∂uN = 12∂ϕM, ∂uZ= 12∂ϕV. (12.11) The canonical analysis for the connection (12.7) in combination with the boundary conditions (12.8) follows exactly the same lines as already outlined in Chapter4.
Thus, the first step is to determine the gauge transformations which preserve the boundary conditions (12.8). In analogy to the AdS case one can write the gauge transformations as ε=b−1 P1
which only depend on the angular coordinateϕ. This leads to the following trans-formation behavior of the state dependent functions under infinitesimal gauge transformations
In this way, one obtains as the variation of the canonical boundary charge
which is finite and integrable in field space. After integration one obtains the canonical boundary charge
which is also conserved in retarded time, i.e.∂uQ= 0.
The conserved charge (12.15) in combination with (12.13) can again be used to first determine the Dirac bracket algebra of the asymptotic symmetries and in turn also the semiclassical asymptotic symmetry algebra as
[Ln, Lm] =(n−m)Ln−m+cL
This algebra can also be obtained as an ultrarelativistic limit of the asymptotic symmetries of spin-3 gravity in AdS3 and is the spin-3 extension of thebms3 algebra encountered in spin-2 gravity. As in the AdS case, however, this algebra is only valid for large central chargescL andcM and therefore needs further modifications in order to also be valid for small values ofcL and cM where products of operators have to be normal ordered.
12.2 Quantum Asymptotic Symmetries
SinceΛnandΘnare nonlinear operators one should introduce a notion of normal ordering for these operators. I will choose the same vacuum conditions as in (11.1) and (11.5) and define normal ordering accordingly as3
: Λ :n:= X
p≥−1
Ln−pMp+ X
p≥−1
MpLn−p. (12.19)
In order to be compatible with the Jacobi identities one again has to modify the semiclassical algebra (12.16) in a suitable way. After performing some algebraic gymnastics one obtains
[Ln, Lm] = (n−m)Ln+m+cL
12(n3−n)δn+m,0, (12.20a)
[Ln, Mm] = (n−m)Mn+m+cM
12 (n3−n)δn+m,0, (12.20b)
[Ln, Um] = (2n−m)Un+m, (12.20c)
[Ln, Vm] = (2n−m)Vn+m, (12.20d)
[Mn, Um] = (2n−m)Vn+m, (12.20e)
[Un, Um] = (n−m)(2n2+ 2m2−nm−8)Ln+m+192
cM(n−m) : ˜Λ :n+m (12.20f)
−96 cL+445
c2M (n−m)Θn+m+cL
12n(n2−1)(n2−4)δn+m,0,
(12.20g) [Un, Vm] = (n−m)(2n2+ 2m2−nm−8)Mn+m+ 96
cM
(n−m)Θn+m
+cM
12 n(n2−1)(n2−4)δn+m,0, (12.20h)
where
: ˜Λ :n= : Λ :n−3
10(n+ 2)(n+ 3)Mn, (12.21) which is nothing else than the contractedW3 algebra (10.30) already presented in Section10.3. This example shows explicitly how the quantumF W3 algebra arises as the asymptotic symmetry algebra of spin-3 gravity in flat space.
Other flat space higher-spin gravity theories can be analyzed in the exact same way by starting out with a different embedding ofisl(2,R) ,→ isl(N,R) and choosing appropriate boundary conditions for the gauge fieldA.
3SinceΘnis a sum of products of commuting operators it does not make a difference whether or not this operator is normal ordered, and thus one can omit the normal ordering altogether.