Super Virasoro Algebras from Chiral Supergravity

Ibaraki Univ.Yoshifumi Hyakutake

Based on arXiv:1211xxxx + work in progress

1. Introduction

3D gravity with negative cosmological constant has been one of the interesting testing grounds to uncover quantum natures of gravity.

This theory also contains black hole solution (BTZ black hole) which have mass and angular momentum.

The vacuum solution is global AdS3 geometry.

Banados, Teitelboim, Zanelli

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

Statistical entropy can be calculated by Cardy formula, and the result coincides with thermodynamic one.

Brown, Henneaux

AdS3

Chiral CFT2

We want to show that super Virasoro × Virasoro appears at the boundary.

Chiral SUGRA

cf. Banados et al.

Let us consider the Lagrangian in three dimensions which is written by vielbein and gravitino .

2. Supergravity Lagrangian and Local SUSY

Action is invariant under general coordinate transformation.

Action is also invariant under local supersymmetry.

: Majorana spinor

By using the relation

we can show

EOM:

3. Current for General CovarianceLet 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

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

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

Variation of current under local supersymmetry is given by

EOM used

EOM used

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.

which satisfy following boundary condition.

Let us consider coordinate transformation

Then is solved as

Now we expand by Fourier modes

Commutation relations become

Next let us consider local supersymmetry which satisfies boundary condition.

is solved as

Product of two solutions and become

Therefore we identifynormalization is fixed here

: NS sector

: R sector

Now we expand by Fourier modes

Then we find

Since should be integer

Global AdS3

Massless BTZ black hole

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.

In a similar way

6. Super Virasoro Algebra from Supergravity

Hamiltonian is defined as

Now we are ready to investigate algebras at the boundary.

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

Explicit calculation is done as follows.

Substituting Fourier mode expansion, we obtain

Algebras at the boundary of AdS3 become

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

Substituting Fourier mode expansion, we obtain

Algebras at the boundary of AdS3 become

Consistency check

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.

Generalization to supergravity with Lorentz Chern-Simons term is possible.

Super Virasoro algebra is modified as

Effective central charge and 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