State Operator Correspondence and Entanglement in AdSIn a saddle-like curved Flatland, the so-called AdS2 space, many concepts of theoretical physics simplify: the state-operator correspondence, entanglement between two regions, the Hartle-Hawking wave function, and others. This makes the AdS2 arena an excellent toy model to settle many questions about the structure of Hilbert spaces and operators; black hole entropy and initial conditions of the Universe; the information loss paradox._{2}/CFT_{1}

In two-dimensional conformal field theories (QFTs) that we know as a description of world sheets in string theory, we usually learn about the "state-operator correspondence" for the first time. The spectrum of possible local operators is in one-to-one correspondence with the states in the Hilbert space obtained by quantizing a 1-dimensional sphere (a circle). Similarly, operators on the boundary of the world sheet are mapped to states of an open string (a half-circle). The dimensions of the operators match the energies of the states in a simple way.

However, this correspondence works for any conformal field theory - radial quantization is enough to see why. The states live on a sphere. There exists one dimensionality in which a sphere becomes disconnected, namely 0 dimensions.

The zero-sphere, or "S^0", is the set of points on a 1-dimensional line that have a fixed distance from the origin. Clearly, they're two points - at "-R" and "+R". It follows that the state-operator correspondence in 1 dimension (CFT1) becomes kind of trivial: the operators are mapped to states on an "S^0" because "S^0" are two points, and indeed, the space of operators is isomorphic to the tensor product of two copies of the Hilbert space. The operators just tell you how you should connect states on the left side of Flatland to those on the right side.

Sen discusses lots of coordinate systems for AdS2 and its Euclidean cousin. The identity operator is mapped to the maximally entangled system that identifies the two Hilbert spaces on the boundary. However, before we identify them, we may apply any U(N) transformation where N is the number of degenerate massless states of a "black hole" (below a gap; all conceivable excited states are ignored in the paper). The CFT1 description is kind of trivial - all of us know N-dimensional Hilbert spaces.

However, Sen also addresses the description in terms of the AdS2 bulk, one that involves the Hartle-Hawking wave function and its twisted cousins. Well, they're twisted by any U(N) transformation. Sen conjectures that such a U(N) symmetry must therefore be well-defined even in the bulk - and unbroken near the horizon.

**Known chaps inside the U(N)**

He suggests that some particular elements of this U(N) are actually well-understood because they're some discrete symmetries of various backgrounds. He speculates that Mathieu groups, especially the M24 one, recently suggested to underlie any conformal field theory for a K3 surface (despite the fact that it never becomes an isometry for a particular K3 surface - not a shocking statement as M24 has about 245 million elements), are also a subgroup of this U(N).

Well, I am a bit uncertain whether Ashoke says a sufficiently sharp thing about the "existence" of this U(N) in the bulk. Whenever there are N degenerate states at some energy level, there is a U(N) group relating them that commutes with the energy (because the states are degenerate), but in most cases, we don't try to apply the group to the whole physical system. In particular, a spin-j representation of SU(2) has (2j+1) states but we rarely speak about the U(2j+1) symmetry. In some sense, it's there but it's broken at "such a low level", almost explicitly, that the symmetry doesn't imply anything away from the particular multiplet.

Also, Sen says that both the "counting of microstates" and "counting the entanglement entropy" are equally good and manifestly equivalent definitions of a black hole entropy in AdS2. That's very nice but kind of too trivial: if he could show that this equivalence may be extended to all black holes in all spacetimes, it would already be as interesting as one dreamed before he completed the reading of this excellent paper.

At any rate, I think that most quantum gravity experts who are into similar questions should try to write their version of the interpretation what's going on in AdS2 with all these concepts.

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