### 3.3 Long strings and symmetric orbifolds

Given a set of Virasoro generators and a non-zero integer , 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
obey the Virasoro algebra with a larger central charge . Conversely, one might define
In general, the generators with , do not make sense because there are no
fractionalized Virasoro generators in the CFT. Such generators would be associated with multivalued modes
on the cylinder . However, in some cases, as we review below, the
Virasoro algebra (101) can be defined. The resulting central charge is smaller and given by
.
If a CFT with generators (101) can be defined such that it still captures the entropy of the original
CFT, the Cardy formula (90) applied in the original CFT could then be used outside of the usual Cardy
regime . Indeed, using the CFT with left-moving generators (101) and their right-moving
analogue, one has

which is valid when , . If is very large, Cardy’s formula (90) 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 [157].
The “long string CFT” can be made more explicit in the context of symmetric product orbifold
CFTs [186], which appear in the AdS_{3}/CFT_{2} correspondence [206, 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 AdS_{3}/CFT_{2} correspondence, as argued
in [157, 98, 23, 116, 31, 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 , one can construct the symmetric
product orbifold by considering the sigma-model with identical copies of the target space manifold
, identified up to permutations,

where is the permutation group on objects. The low energy (infrared) dynamics is a CFT with
central charge if the central charge of the low energy CFT of the original sigma model is .
The Virasoro generators of the resulting infrared CFT can then be formally constructed from the generators
of the original infrared CFT as (100). 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 (101) 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.