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

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

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

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.

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

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

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)

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

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

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

R2+L3 transformation Since the transformed matrix then becomes

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

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

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