Super Virasoro Algebras from Chiral Supergravity

1. Super Virasoro Algebras from Chiral Supergravity Ibaraki Univ. Yoshifumi Hyakutake Based on arXiv:1211xxxx + work in progress

2. 1. Introduction 3D gravity with negative cosmological constant has been one of the interesting testing grounds to uncover quantum natures of gravity. The vacuum solution is global AdS3 geometry. This theory also contains black hole solution (BTZ black hole) which have mass and angular momentum. Banados, Teitelboim, Zanelli

3. BTZ black hole has inner and outer horizons : By using the area formula, the entropy of the BTZ black hole is evaluated as This is thermodynamic entropy. Then can we derive this from the statistical viewpoint? Brown and Henneaux showed that there exist Virasoro algebras at the boundary . Central charges for left and right moving modes are evaluated as Brown, Henneaux Statistical entropy can be calculated by Cardy formula, and the result coincides with thermodynamic one.

4. Chiral SUGRA AdS3 Chiral CFT2 We want to show that super Virasoro×Virasoro appears at the boundary. cf. Banados et al.

5. 2. Supergravity Lagrangian and Local SUSY Let us consider the Lagrangian in three dimensions which is written by vielbein and gravitino . Action is invariant under general coordinate transformation.

6. Action is also invariant under local supersymmetry. : Majorana spinor By using the relation we can show EOM:

7. 3. Current for General Covariance Let us apply Noether’s method to derive current for general coordinate invariance in a covariant way. Variation of Lagrangian becomes Up to EOM, is evaluated as Current for general coordinate invariance is given by EOM used

8. is added to make Hamiltonian well-defined. To determine this, we evaluate variation of the current. where This is called symplectic current and gives variation of Hamiltonian. should cancel total divergent term. Therefore variation of current is given by EOM used

9. 4. Current for Local Supersymmetry Let us apply Noether method to derive current for local supersymmetry in a covariant way. Up to EOM, is evaluated as Current for local supersymmetry is defined as EOM used Variation of current under local supersymmetry is given by EOM used

10. 5. Asymptotically Symmetry Group for AdS3 So far, we have constructed currents and their variations. We want to evaluate these at the boundary of AdS3. We need to know isometries which preserve this metric at the boundary. So called ASG. Let us consider coordinate transformation which satisfy following boundary condition.

11. Then is solved as Now we expand by Fourier modes Commutation relations become

12. Next let us consider local supersymmetry which satisfies boundary condition. is solved as Product of two solutions and become Therefore we identify normalization is fixed here

13. Now we expand by Fourier modes Then we find Since should be integer Global AdS3 :NSsector Massless BTZ black hole :Rsector

14. Remark on vielbein and spin connection This is not well-defined because this does not go to zero faster than original value of vielbein as goes to infinity. So we use local Lorentz transformation. Now variation of vielbein is well-defined.

15. In a similar way

16. 6. Super Virasoro Algebra from Supergravity Now we are ready to investigate algebras at the boundary. Hamiltonian is defined as Variation of Hamiltonian is given by Let us evaluate this in the background of massless BTZ (ground state of R sector). The energy is zero, so

17. Explicit calculation is done as follows. Substituting Fourier mode expansion, we obtain Algebras at the boundary of AdS3 become

18. Supercharge is defined as Variation of the supercharge is given by Let us evaluate this in the background of massless BTZ (ground state of R sector). The energy is zero, so Explicit calculation shows

19. Substituting Fourier mode expansion, we obtain Algebras at the boundary of AdS3 become Consistency check

20. 7. Summary Currents for general coordinate invariance and local supersymmetry are constructed in a covariant way. Asymptotically supersymmetry group is derived. By using local Lorentz symmetry, ASG is formulated in terms of vielbein and spin connection. Super Virasoro × Virasoro is constructed at the boundary of AdS3.

21. Generalization to supergravity with Lorentz Chern-Simons term is possible. Super Virasoro algebra is modified as

22. Effective central chargeand black hole entropy In order to determine , we need to shift to make Virasoro algebra in a canonical form. Then energy of global AdS3 becomes zero, hence