Since we aim at drawing parallels between black holes and two-dimensional CFTs ( CFTs), it is useful to describe some key properties of 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 CFT. Only future research will tell if 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 [130, 169].
A 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 on the plane or on the cylinder. It is generated by two sets of vector fields , obeying the Lie bracket algebra
A CFT can be uniquely characterized by a list of (primary) operators , the conformal dimensions of these operators (their eigenvalue under and ) 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.
Living Rev. Relativity 15, (2012), 11
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