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

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 (?)

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) 026011

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

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

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.

*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)]

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

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

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.

Computation:

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

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.

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

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

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). “

*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 !

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

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.

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

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

Noether charge for (b) case: PB algebras:

Here, the Lie bracket is On the constraint surface,

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

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

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.

Current Interest: Workshop Organization