3 Two-Dimensional Conformal Field Theories

Since we aim at drawing parallels between black holes and two-dimensional CFTs (2d CFTs), it is useful to describe some key properties of 2d CFTs. Background material can be found, e.g., in [120, 149, 234]. An important caveat to keep in mind is that there are only sparse results in gravity that can be interpreted in terms of a 2d CFT. Only future research will tell if 2d CFTs are the right theories to be considered (if a holographic correspondence can be precisely formulated at all) or if generalized field theories with conformal invariance are needed. For progress in this direction, see [130Jump To The Next Citation Point, 169].

A 2d CFT is defined as a local quantum field theory with local conformal invariance. In two-dimensions, the local conformal group is an infinite-dimensional extension of the globally-defined conformal group SL (2,ℝ ) × SL (2,ℝ ) on the plane or on the cylinder. It is generated by two sets of vector fields L , ¯L n n, n ∈ ℤ obeying the Lie bracket algebra

[Lm, Ln] = (m − n)Lm+n, [Lm, ¯Ln] = 0, (85 ) ¯ ¯ ¯ [Lm, Ln] = (m − n)Lm+n.
From Noether’s theorem, each symmetry is associated to a quantum operator. The local conformal symmetry is associated with the conserved and traceless stress-energy tensor operator, which can be decomposed into left and right moving modes ℒn and ¯ ℒn, n ∈ ℤ. The operators ¯ ℒn, ℒn form two copies of the Virasoro algebra
[ℒ ,ℒ ] = (m − n)ℒ + cLm (m2 − A )δ , m n m+n 12 L m+n,0 [ℒ , ¯ℒ ] = 0, (86 ) [ m n ] c ¯ℒm, ¯ℒn = (m − n)ℒ¯m+n + -Rm (m2 − AR )δm+n,0, 12
where ℒ− 1,ℒ0, ℒ1 (and ¯ℒ−1,ℒ¯0, ¯ℒ1) span a SL (2,ℝ ) subalgebra. The pure numbers cL and cR are the left and right-moving central charges of the CFT. The auxiliary parameters AL, AR depend if the CFT is defined on the plane or on the cylinder. They correspond to shifts of the background value of the zero eigenmodes ¯ ℒ0,ℒ0. In many examples of CFTs, additional symmetries are present in addition to the two sets of Virasoro algebras.

A 2d CFT can be uniquely characterized by a list of (primary) operators 𝒪, the conformal dimensions of these operators (their eigenvalue under ℒ 0 and ¯ℒ 0) and the operator product expansions between all operators. Since we will only be concerned with universal properties of CFTs here, such detailed data of individual CFTs will not be important for our considerations.

We will describe in the next short Sections 3.1, 3.2 and 3.3 some properties of CFTs that are conjectured to be relevant to the Kerr/CFT correspondence and its extensions: the Cardy formula, some properties of the discrete light-cone quantization (DLCQ) and some properties of symmetric product orbifold CFTs.

 3.1 Cardy’s formula
 3.2 DLCQ and chiral limit of CFTs
 3.3 Long strings and symmetric orbifolds

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