3.3 Long strings and symmetric orbifolds

Given a set of Virasoro generators ℒn and a non-zero integer N ∈ ℤ0, one can always redefine a subset or an extension of the generators, which results in a different central charge (see, e.g., [25]). One can easily check that the generators
short -1- ℒn = N ℒNn (100 )
obey the Virasoro algebra with a larger central charge short c = N c. Conversely, one might define
ℒlonng = N ℒn ∕N . (101 )
In general, the generators long ℒ n with n ⁄= N k, k ∈ ℤ do not make sense because there are no fractionalized Virasoro generators in the CFT. Such generators would be associated with multivalued modes einϕ∕N on the cylinder (t,ϕ ) ∼ (t,ϕ + 2π ). However, in some cases, as we review below, the Virasoro algebra (101View Equation) can be defined. The resulting central charge is smaller and given by long c = c∕N.

If a CFT with generators (101View Equation) can be defined such that it still captures the entropy of the original CFT, the Cardy formula (90View Equation) applied in the original CFT could then be used outside of the usual Cardy regime TL ≫ 1. Indeed, using the CFT with left-moving generators (101View Equation) and their right-moving analogue, one has

π2 cL cR 𝒮CFT = -3-(N-(N TL) + N--(N TR)), (102 )
which is valid when N TL ≫ 1, N TR ≫ 1. If N is very large, Cardy’s formula (90View Equation) would then always apply. We will use the assumption of the existence of such a “long string CFT” in Section 4.4 to justify the validity of Cardy’s formula outside the usual Cardy regime as done originally in [157Jump To The Next Citation Point].

The “long string CFT” can be made more explicit in the context of symmetric product orbifold CFTs [186], which appear in the AdS3/CFT2 correspondence [206Jump To The Next Citation Point, 114, 123] (see also [230] and references therein). These orbifold CFTs can be argued to be relevant in the present context, since the Kerr/CFT correspondence might be understood as a deformation of the AdS3/CFT2 correspondence, as argued in [157, 98, 23, 116, 31Jump To The Next Citation Point, 243, 246, 115, 130].

Let us then briefly review the construction of symmetric product orbifold CFTs. Given a conformally-invariant sigma-model with target space manifold ℳ, one can construct the symmetric product orbifold by considering the sigma-model with N identical copies of the target space manifold ℳ, identified up to permutations,

N ( N ) Sym (ℳ ) ≡ ⊗ ℳ ∕SN , (103 )
where SN is the permutation group on N objects. The low energy (infrared) dynamics is a CFT with central charge cSym = N c if the central charge of the low energy CFT of the original sigma model is c. The Virasoro generators of the resulting infrared CFT can then be formally constructed from the generators ℒm of the original infrared CFT as (100View Equation). Conversely, if one starts with a symmetric product orbifold, one can isolate the “long string” sector, which contains the “long” twisted operators. One can argue that such a sector can be effectively described in the infrared by a CFT, which has a Virasoro algebra expressed as (101View Equation) in terms of the Virasoro algebra of the low energy CFT of the symmetric product orbifold [211]. The role of these constructions for the Kerr/CFT correspondence remains to be fully understood.

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