Abstract
Since Euclidean global AdS2 space represented as a strip has two boundaries, the state-operator correspondence in the dual CFT1 reduces to the standard map from the operators acting on a single copy of the Hilbert space to states in the tensor product of two copies of the Hilbert space. Using this picture we argue that the corresponding states in the dual string theory living on AdS2 × K are described by the twisted version of the Hartle–Hawking states, the twists being generated by a large unitary group of symmetries that this string theory must possess. This formalism makes natural the dual interpretation of the black hole entropy—as the logarithm of the degeneracy of ground states of the quantum mechanics describing the low energy dynamics of the black hole, and also as an entanglement entropy between the two copies of the same quantum theory living on the two boundaries of global AdS2 separated by the event horizon.
Highlights
Introduction and SummaryWith the help of radial quantization, local operators in a conformal field theory in d dimensions (CFTd ) can be mapped in a one-to-one fashion to states in the same CFT on S d−1 × R, with R labelling the time direction
Since S 0 is a collection of two points, the states live in the Hilbert space H ⊗ H of two copies of the CFT1
On the other hand the absence of spatial separation makes all operators in the Hilbert space H of a single copy of the CFT1 local
Summary
With the help of radial quantization, local operators in a conformal field theory in d dimensions (CFTd ) can be mapped in a one-to-one fashion to states in the same CFT on S d−1 × R , with R labelling the time direction. The dominant contribution to this amplitude comes from the configuration where the cut extends all the way across the disk, representing the usual AdS2 × K geometry without any twist While this gives an abstract prescription for associating the states living in two copies of the Hilbert space of CFT1 to states of string theory on AdS2 × K, in general we cannot explicitly construct these states since the action of W on the bulk fields is not known. For the special cases when W can be associated with some known ZZk symmetry generator of string theory on AdS2 × K, we can explicitly construct the corresponding state; the path integral will be over all field configurations whose boundary values jump by the action of this discrete symmetry transformation W at some point on the boundary Even though this is a special case, this at least demonstrates that it is possible for global AdS2 to admit non-trivial quantum states. Symmetry, in that this symmetry is present in the near horizon geometry but not necessarily present away from the horizon
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