super virasoro algebras from chiral supergravity ibaraki univ. yoshifumi hyakutake based on...
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Super Virasoro Algebras from Chiral Supergravity
Ibaraki Univ.Yoshifumi Hyakutake
Based on arXiv:1211xxxx + work in progress
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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
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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
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AdS3
Chiral CFT2
We want to show that super Virasoro × Virasoro appears at the boundary.
Chiral SUGRA
cf. Banados et al.
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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.
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Action is also invariant under local supersymmetry.
: Majorana spinor
By using the relation
we can show
EOM:
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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
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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
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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
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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
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Then is solved as
Now we expand by Fourier modes
Commutation relations become
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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
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: NS sector
: R sector
Now we expand by Fourier modes
Then we find
Since should be integer
Global AdS3
Massless BTZ black hole
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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.
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In a similar way
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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
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Explicit calculation is done as follows.
Substituting Fourier mode expansion, we obtain
Algebras at the boundary of AdS3 become
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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
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Substituting Fourier mode expansion, we obtain
Algebras at the boundary of AdS3 become
Consistency check
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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.
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Generalization to supergravity with Lorentz Chern-Simons term is possible.
Super Virasoro algebra is modified as
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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