#
State Operator Correspondence and Entanglement in AdS_{2}/CFT_{1}

## Abstract

**:**

## 1. Introduction and Summary

## 2. CFT${}_{1}$ and Its State-Operator Correspondence

## 3. AdS${}_{2}$ Space in Different Coordinates

**Figure 1.**Global AdS${}_{2}$ and the location of the horizon(s). The two vertical solid lines label the two boundaries of AdS${}_{2}$ at $\sigma =-\pi $ (left) and $\sigma =0$ (right). The dashed lines label the locations of the event horizons of the original black hole.

**Figure 2.**Conformal map from unit disk to the strip. The left boundary of the strip is at $\sigma =-\pi $ and the right boundary is at $\sigma =0$.

## 4. AdS${}_{2}$/CFT${}_{1}$ Correspondence

## 5. States in String Theory on AdS${}_{2}$

**Figure 3.**Generating a state in string theory on AdS${}_{2}$ from a state ${W}_{ab}{|a\rangle}_{\left(1\right)}{|b\rangle}_{\left(2\right)}$ in the two copies of the Hilbert space of CFT${}_{1}$. The thick semi-circular line is the boundary of AdS${}_{2}$ whereas the thin diameter is the line on which the string fields, appearing in the argument of ${f}_{W}$, live. The dashed line reaching the boundary of AdS${}_{2}$ denotes a cut which relates the field configurations on the right of the cut to those on the left of the cut by a transformation by W.

## 6. Conformal Invariance of the Correlation Functions

- The elements of the group approach Diff$\left({S}^{1}\right)$ as we approach the boundary.
- They are well defined in the interior of AdS${}_{2}$.

## 7. Entanglement vs. Statistical Entropy

## 8. Information Loss Problem

## 9. Speculations on the Enhanced Symmetry

- In the classical limit the black hole entropy $lnN$ goes to infinity. Thus if the $U\left(N\right)$ symmetry is present in the classical limit, then it must appear as a $U\left(\infty \right)$ symmetry which is broken down to $U\left(N\right)$ by quantum effects. Since $U\left(\infty \right)$ is a symmetry of the infinite dimensional complex Grassmannians, one could wonder if Grassmannians might play a role in string theory on AdS${}_{2}\times \mathrm{K}$. Alternatively the $U\left(N\right)$ symmetry could arise only as a symmetry of the quantum theory with no classical analog.
- Typically for a BPS black hole in a supersymmetric theory carrying a fixed charge, some of the moduli scalar fields are fixed at the horizon due to the attractor mechanism, but the other moduli may remain free and label the moduli space of the near horizon geometry. As we move around in this moduli space, the discrete symmetries of the theory may change, being either non-existent or a small group of symmetries at a generic point but getting enhanced to bigger groups at special points. On the other hand the spectra of the black hole ground states at different points in this moduli space are expected to be isomorphic since they represent the BPS states carrying a given set of charges. It should in principle be possible to use these isomorphisms to represent the action of the discrete symmetries at different points in the moduli space on the same Hilbert space. Thus all these discrete symmetries must be embedded in the single $U\left(N\right)$ group that acts on the N degenerate ground states of the black hole.To take a concrete example, consider type IIA string theory compactified on $K3\times {T}^{2}$ and take a black hole that carries only fundamental string winding, momentum, Kaluza–Klein (KK) monopole and H-monopole charges associated with the two circles of ${T}^{2}$. Although the full moduli space of the theory is parametrized locally by the $SO(6,22)/SO\left(6\right)\times SO\left(22\right)$ coset space, some of the moduli are fixed in the near horizon geometry leaving behind only a locally $SO(4,22)/SO\left(4\right)\times SO\left(22\right)$ space. The moduli labelling this space include in particular the metric and the 2-form fields on $K3$. Now it has been recently speculated in [77,78,79,80,81,82,83,84] that the symmetry group of supersymmetric states in a sigma model with target space $K3$ includes the Matthew group ${M}_{24}$ even though there is no known point in the $K3$ moduli space at which the corresponding string theory has manifest ${M}_{24}$ symmetry. In this case this group must also have a natural action on the space of BPS states of the black hole described above and sit inside the $U\left(N\right)$ group acting on the N degenerate ground states of the black hole. A better understanding of why ${M}_{24}$ appears as a symmetry of supersymmetric states in the sigma model could help us realize this as an explicit symmetry of string theory in this particular near horizon geometry. This will still fall short of realizing the whole $U\left(N\right)$ group as a manifest symmetry, but will help us realize a large subgroup of $U\left(N\right)$.For special values of the charges the symmetry group may also include a subgroup of the duality group associated with compactification on ${T}^{2}$. Note however that none of these symmetries may be a symmetry of the asymptotic theory since they can be broken by the expectation values of the moduli fields at infinity.
- Supergravity theories reduced to two dimensions typically have a large group of continuous duality symmetries. Normally in the presence of charged particles this symmetry breaks down to a discrete subgroup. However since in the AdS${}_{2}$ geometry there are no charged excitations one could wonder if these continuous duality symmetries could play any role in building up the $U\left(N\right)$ group. In this context it is encouraging to note that the enhanced discrete symmetries at special points in the moduli space are naturally embedded in this continuous duality group.

## Acknowledgements

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**MDPI and ACS Style**

Sen, A.
State Operator Correspondence and Entanglement in AdS_{2}/CFT_{1}. *Entropy* **2011**, *13*, 1305-1323.
https://doi.org/10.3390/e13071305

**AMA Style**

Sen A.
State Operator Correspondence and Entanglement in AdS_{2}/CFT_{1}. *Entropy*. 2011; 13(7):1305-1323.
https://doi.org/10.3390/e13071305

**Chicago/Turabian Style**

Sen, Ashoke.
2011. "State Operator Correspondence and Entanglement in AdS_{2}/CFT_{1}" *Entropy* 13, no. 7: 1305-1323.
https://doi.org/10.3390/e13071305