1. 3D (Higher Spin) Gravity Black Holes and Statistical Entropy
APCTP topical program "String theory and Cosmology“ (30 Nov. 2012)
3D (Higher Spin) Gravity Black Holes and Statistical Entropy
Mu-In Park
Kunsan Nat’al Univ., Korea
In progress …
In collaboration with S. Nam

2. Contents (?)
1. Virasoro Algebras in 3D Chern-Simons Theories with Boundary: Classical Central Extensions
2. Black Hole Entropy from Classical Virasoro Algebras
3. Test 1: BTZ Black Hole
4. Test 2: Scalar-Haired BTZ Black Hole
5. Test 3: 3D Kerr-de Sitter Space
6. New Test (?): Higher-Spin Gravity Black Holes
7. Conclusion (?)

3. Refs. [1] Oh & Park, Mod. Phys. Lett. A14 (1999) 231
[2] Park, Phys. Lett. B440 (1998) 275
[3] Nucl. Phys. B544 (1999) 377
[4] Nucl. Phys. B634 (2002) 339
[5] Phys. Lett. B597 (2004) 237
[6] Phys. Rev. D77 (2008)

4. 1. Virasoro Algebras in 3D Chern-Simons Theories with Boundary: Classical Central Extensions
Refs:
[1] (Jackiw,) Oh & Park, Mod. Phys. Lett. A14 (1999) 231
[3] Park, Nucl. Phys. B544 (1999) 377

5. Kac-Moody algebra of gauge transformation
A. Noether charge
Let’s start with CS Lagrangian on a disc
up to a boundary term. Here,
Trace

6. Variation of the Lagrangian gives
up to the total time-derivative term.
In order to get the usual equation of motion, even with the boundary,
we choose the boundary conditions
for each time t.

7. *Symplectic reduction vs. Dirac method
Gauss law constraint:
(a) Symplectic reduction: by solving the Gauss law constraint. [Oh&Park (‘99)]
(b) Dirac method: Constraint is not solved but imposed after all the (Poisson bracket) analysis. [Park (’99)]

8. Remarks:
(1). It is believed that these two methods are equivalent, classically.
(2). Dirac method is unique when the constraints can not be solved.

9. Under the time-independent gauge transformations ( )
the Lagrangian transforms as
Noether charge is given by
Symmetry !

10. B. Poisson & Dirac bracket algebras of Noether charge
From the symplectic structure, we have
with the Poisson bracket,
Then, the Poisson algebras of Q are
Kac-Moody algebra
with a classical central
term: integrated form.

11. Computation:

12. Dirac bracket algebras
Gauss law constraint becomes
2nd class constraints when
does not vanish on the boundary

13. Dirac bracket
so that
i.e., Gauss law constraint can be
imposed consistently in the Hamiltonian dynamics.
Here,
Remark: Quantization is possible in the Dirac bracket algebra, not in the Poisson bracket algebra.

14. The DB algebras of Q:
The full Q= is the same as in PB algebras, but not and , separately.

15. Gauge transformations:
where is the gauge transf. only on the boundary.
The differentiable generator Q produces the usual bulk gauge transformation even with boundary.
DB algebras: is frozen but replaced by

16. Remarks
1. Full bulk gauge degrees of freedom are transferred completely into the boundary: A realization of
holographic principle, saying
“ Bulk world is an image of data on the Boundary, or vice versa “.
2. “From the connection of CS theory to diverse areas of physics, the principle can be more widely than currently limited cases of 3D AdS or (Kerr-) dS (‘99). “

17. *Re-examining the origin of the central terms
From the basic PB,
one obtains, for the Abelian case,
Q: ?
But this depends on the test function !

18. For the test function which does not vanish on the boundary,
from
Angular part is not modified
if the test function is single-valued

19. For the test function which is not constant on the boundary,
Then, one can find
, in agreement with the other computation, for the Abelian case.
So, the central term is the result of unusual delta-function formula due to boundary.

20. Virasoro algebras of diffeomorphism.
A. Noether charge
Under time-independent Diff.
the Lagrangian transforms as

21. Two possible BC for Diff. invariance
(a) : Not interesting ! No central charge (Witt algebra).
(b) : More interesting !
Non-zero central charge.
with

22. Noether charge for (b) case:
PB algebras:

23. Here, the Lie bracket is
On the constraint surface,

24. DB algebras:
Gauge transf:
, with the boundary transf. Diff
Holography still works !

25. Remarks 1. We have as a generalization of
2. One CS theory has one copy of Virasoro algebra on the boundary.
3. Two copies of CS theories=AdS;
Two copies of Virasoro algebras (Brown- Henneaux) 2

26. 6. New Test (?): Higher-Spin Gravity Black Holes
Higher-Spin Gravity can be considered a higher-rank (internal) group of CS.
Issue:
Are the known BC for CS valid in HS gravity ?
If not, we need another computation or formula for the Virasoro algebra.
Hope to present somewhere else.

27. Current Interest: Workshop Organization