De-Sitter space can be thought of as a 4 dimensional hyperboloid embedded in 5D Minkowski space. Hence, the symmetry group of dS is SO(1,4) whose generators are,
JAB=i(XA∂∂XB−XB∂∂XA)
where A,B=0,1,2,3,4. They follow the standard SO(1,4) commutation relations,
[JAB,JCD]=i(ηBCJAD+ηADJBC−ηACJBD−ηBDJAC)
where ηAB=(−1,1,1,1,1). The coordinates on the dS hyperboloid (whose equation is, ηABXAXB=1H2) are,
X0=−12H(Hη−1Hη)+x22η
Xi=xiHη
X4=−12H(Hη+1Hη)+x22η
Here i=1,2,3. (η,→x) are parameters on the hyperboloid. The 5D Minkowski metric restricted to the hyperboloid, in terms of these coordinates, becomes,
ds2=−dη2+d→x2H2η2.
Split the SO(1,4) generators as,
Lij=Jij
D=J04
Pi=J0i+J4i
Ki=J4i−J0i
It can be easily checked that in terms of the η,→x coordinates,
Lij=i(xi∂j−xj∂i)
D=−i(η∂η+xi∂i)
Pi=−iH∂i
Ki=i[−2Hxi(η∂η+xi∂i)+H(−η2+x2)∂i]
These generators follow the standard conformal algebra (check the one given in the big yellow book[1]). The point η,→x=0 is left invariant by D,Lij,Ki. Hence, if ϕ(η,→x) is a classical field,
Lijϕ(0)=Sijϕ(0)
Dϕ(0)=−iΔϕ(0)
Kiϕ(0)=0
Now, SO(1,4) has two Casimir operators,
C1=−12JABJAB
C2=−WAWA
where WA=18ϵABCDEJBCJDE. In terms of the conformal genertors,
C1=D2−12LijLij−12{Pi,Ki}
by making C1 act on ϕ(0), the eigenvalues turn out to be,
C1=−s(s+1)−Δ(Δ−3)
In terms of the conformal generators, the components of WA are,
W0=12ϵijkJijJk4=12Lk(Pk+Kk)
Wi=12ϵijkJjkJ04−ϵijkJj4J0k=−LiD+14ϵijk{Kj,Pk}
W4=−12ϵijkJijJ0k=12Lk(Pk−Kk)
where Lk=−12ϵkijLij and it has the following commutation rules,
[Li,Lj]=iϵijkLk
[Li,D]=0
[Li,Pj]=iϵijkPk
[Li,Kj]=iϵijkKk
Now, W2=−W20+W2i+W24=(W4+W0)(W4−W0)−[W0,W4]+W2i, so when it acts on a classical field, the first term does not contribute since it has a Ki on the right which annihilates the field. Also, it is easy to see that Lk commutes with W0 and W4. Therefore,
[W0,W4]=−Lk[Jk4,LiJ0i]=−iLkWk.
In the formula for Wk, commuting the Pi past the Ki gives a factor of iLk so that
Wkϕ(0)=−Lk(D−i)ϕ(0).
Thus, we get,
C2ϕ(0)=s(s+1)(Δ+1)(Δ+2)ϕ(0)
which is incorrect according to this[2] reference (their q is my Δ−1). The first Casimir is correct. The correct answer for C2 should be −s(s+1)(Δ−1)(Δ−2). Can somebody point out where I made a mistake?
[1]: https://www.springer.com/gp/book/9780387947853
[2]: https://arxiv.org/abs/hep-th/0309104