1. SL(4,R) sigma model & Charged black rings
Da-Sheng Kung
Chiang-Mei Chen
National Central University, Taiwan
8 January 2008

2. Neutral black ring
The black ring is a five-dimensional black hole with an event horizon of topology S1×S2

3. SL(4,R) σ-model
The starting point is a five-dimensional action with gravity, dilaton field, and three-form field
We perform a two-step reduction to three-dimensional theory in which all physical variables can be transformed to scalar fields

4. SL(4,R) σ-model The three-dimensional action is rewritten as
This action is invariant under the 15-parametric SL(4,R) transformations
R-transformation: pure gauge.
S-transformation: two gauge, three different scale, and two electric Harrison transformations.
L-transformation: two magnetic Harrison transformations and Ehlers-like part of S-duality.

5. Asymptotic flatness preserving
The general SL(4,R) transformation can be represented by generator
Define to be the asymptotic form of matrix Suppose it’s symmetric, then asymptotic flatness preserving requires that This will give the condition

6. Asymptotic flatness preserving
The flat geometry of black ring is
The corresponding scalar potentials of target space are

7. Asymptotic flatness preserving
According to the asymptotic condition, only four transformations keep asymptotic flatness:
Pure gauge:
Electric Harrison: (generating Kaluza-Klein vector)
Gauge & magnetic Harrison: (generating form field)

8. Charged black ring
For constructing charged ring solution, it’s natural to choose neutral black ring as the seed solution
where
Define a useful function
After reduction to three-dimensional theory, vector fields can be parameterized by

9. S3 transformation where
The electric charge is produced by the electric Harrison transformation with the parameter So that
where

10. S3 transformation Black ring with Kaluza-Klein charge takes the form
The Kaluza-Klein vector is
with components

11. R2+L3 transformation
Since
the transformed matrix then becomes

12. R2+L3 transformation
By the relationship between matrix elements and target-space potentials, one can obtain
where

13. R2+L3 transformation Black ring with form-field charge reads
The corresponding gauge potentials are

14. Thank You