Flat Space as a Limit from AdS
10.4 A Flat Space (Higher-Spin) Cardy Formula
In this section I will show how to obtain a flat space analogue of the Cardy formula3, which allows one to holographically compute the thermal entropy of a flat space cosmology. While taking the limit itself is simple, it is an essential and important point to note that one has to take theinner horizonlimit of the Cardy formula and not the well known result for the outer BTZ horizon. The reason for this is, as explained in the beginning of this chapter, that the limit from BTZ to FSCs can be geometrically interpreted as zooming in between the region between the inner and outer BTZ horizon. In order to make this point explicit I will first show that taking the flat space limit of the (standard) outer horizon Cardy formula does not yield the correct result for the microstate counting of FSCs.
The Cardy formula that determines the entropy of the outer horizon of a BTZ black hole is given by [151]:
SAout = Aout
4G = 2π s
cL 6 + 2π
s c¯L¯
6 =Souter. (10.39)
The left hand side is the Bekenstein–Hawking entropy associated with the outer horizonAout, while the right hand side is the Cardy formula for outer horizons. The central chargescand¯care given by
[Ln,Lm] =(n−m)Ln+m+ c
12 n3−nδn+m,0, (10.40a) [¯Ln,L¯m] =(n−m) ¯Ln+m+ c¯
12 n3−nδn+m,0, (10.40b)
3Originally the Cardy formula provided a way to count the number of states of a CFT at non-zero temperature [151]. However, it also coincides with the thermal entropy of BTZ black holes [13]
and thus provides a holographic way to count the microstates of the BTZ.
whereL ≡L0,L ≡¯ L¯0and their respective conformal weightsh,¯hwhen acting on a highest weight state|h,¯hiare related to the mass and angular momentum in the usual way
L=h= 1
2(M `−J), L¯= ¯h= 1
2(M `+J). (10.41) In order to perform the ˙Inönü–Wigner contraction I make the same identifications as in (10.22) which in turn also means that the eigenvalues,hL,hM, ofL0 andM0 when acting on a highest weight state|hL, hMiare given by
hL=h−¯h, hM = 1
` h+ ¯h. (10.42)
In addition, I define the quantitiesMandN in a similar fashion as in (10.14) M= 12 L which is obviously not the correct result as its ` → ∞ limit diverges with ` as expected.
As mentioned earlier the cosmological horizon of a FSC is obtained as a limit of the inner BTZ horizon. Thus, one has to consider a modified Cardy formula for the BTZ in order to take the limit. This modified Cardy formula should count the microstates of the inner BTZ horizon in order to be a valid starting point for a flat space contraction. Such a modified Cardy formula is given by [148,149]:
SAint = Aint
The modification in comparison to the (standard) Cardy formula (10.39) consists of a relative minus sign between the right-(L) and left-(L) moving contributions.¯ In order to perform the ˙Inönü–Wigner contraction I repeat the same steps as before but use now (10.45) instead of (10.39). This yields
Sinner= Taking the`→ ∞limit gives a prediction for the microscopic entropy of the dual quantum field theory:
This agrees precisely with the results obtained in [61,62,71].
Another sanity check is to compare the results obtained this way with the results for Einstein gravity wherec = ¯cand hence cL = 0. The expression (10.47) then simplifies to
The result (10.48) (after translating conventions forc-normalization) agrees per-fectly with the results in [61,62].
Flat Space Chiral Gravity: One can also use the contractions used previously to determine the microscopic entropy of flat space chiral gravity (FSχG), a theory that can be obtained as a limit [57] of Topologically Massive Gravity (TMG) [152]. The corresponding action is given by
ITMG= 1 term andµis the corresponding Chern-Simons coupling. Flat space chiral gravity arises in the limitGN → ∞ while keeping fixedµGN so that µGN = c3L remains finite. This is particularly interesting as the central charges of the dual field theory are of a form that allow for unitary highest-weight representations of thebms3∼gca2 algebra [III], i.e. cL6= 0, cM = 0, a topic I will also elaborate on in Chapter11.
In [71] it has been shown that the entropy formula for FSCs in TMG takes exactly the same form as (10.47) but with cL = µG3 entropy for FSCs in flat space chiral gravity is given by
SFSFSCχG = 2π s
cLhL
6 =SCFTChiral. (10.50)
The result (10.50) coincides precisely with what one would expect of one chiral half of a CFT [63]. This fits very nicely with the suggestion that flat space chiral gravity is indeed the chiral half of a CFT [57,153].
FSC with Spin-3 Charges: Even though I have not yet introduced higher-spin sym-metries in flat space, it is nevertheless instructive to consider a flat space limit from a rotating BTZ black hole with spin-3 chargesW andW¯ [150]. This will enable one to make a prediction for a Cardy-like formula for FSCs that carry spin-3 charges.
Following [135] one can write a Cardy-like formula for theouterhorizon entropy of a spin-3 charged BTZ as
Souter= 2π
whereCandC¯ are dimensionless constants defined via and theC→ ∞( ¯C→ ∞)limit corresponds to the limit of vanishing spin-3 charges.
In order to successfully perform a contraction that yields a Cardy-like formula for a spin-3 charged FSC one has again to determine theinnerhorizon spin-3 BTZ formula.
In addition one has to find an expression ofCandC¯in terms of flat space analogues of these constants, which I will callRandP. This is actually a non-trivial problem sinceCandC¯ are related to the canonical charges (L,L,¯ W,W¯) in a nonlinear way.
In the following I present a method to solve this problem. First I will introduce the flat space analogues of the spin-3 chargesW andW¯ as
V = 12 W
Using these relations and replacingW andW¯ byVandZ in (10.52) one can deduce a suitable ansatz forC andC¯ in terms ofRandP by demanding that up toO(`12) the l.h.s and the r.h.s of (10.52) have to agree. It turns out that a suitable ansatz for CandC¯ is given by
Again, as in the spin-2 case, the outer horizon limit does not yield the correct result as can be easily shown by replacing all AdS quantities in (10.51) by their flat space counterparts and taking the`→ ∞limit. This yields the following expression
Souter= π
which is divergent with`and thus, as expected, the outer horizon formula (10.51) is not the correct expression for a contraction to flat space. In close analogy to the spin-2 case I assume that the following formula provides an appropriate microstate counting of theinnerhorizon entropy of a spin-3 charged BTZ
Sinner= 2π
whose only difference, as compared to the outer horizon formula, is again a relative minus sign between the two left- and right-moving contributions. In addition the C → ∞andC¯→ ∞limit yields the correct expression for the spin-2 inner horizon formula (10.45) as it should be. Using (10.58) and performing the same steps as before one obtains the following result after taking the`→ ∞limit
SFSCSpin-3= π 6
cL
√ M
r 1− 3
4R+cM
N 4R −6 + 3P√ R 4√
M(R −3)q1− 4R3
. (10.59)
As expected theR → ∞limit yields again (10.47). It is also important to note that the part of (10.59) which is proportional to cM can alternatively also be derived by solving holonomy conditions (as in the AdS case) [VII], which I will also show explicitly in Chapter11. This shows that my assumption (10.58) is not only plausible but seems indeed to be the correct expression for the inner horizon entropy of a spin-3 charged BTZ black hole as its ˙Inönü–Wigner contraction leads to the correct flat space result.
With this derivation I want to close this chapter on flat space as a limit from AdS and proceed with the next chapter, in which I will study unitarity of highest-weight representations ofF W-algebras.