Supersymmetric warped AdS in extended topologically massive supergravity

We determine the most general form of off-shell <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math> supergravity field configurations in three dimensions by requiring that at least one off-shell Killing spinor exists. We then impose the field equations of the topologically massive off-shell supergravity and find a class of solutions whose properties crucially depend on the norm of the auxiliary vector field. These are spacelike-squashed and timelike-stretched <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">AdS</mi></mrow><mrow><mn>3</mn></mrow></msub></math> for the spacelike and timelike norms, respectively. At the transition point where the norm vanishes, the solution is null warped <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">AdS</mi></mrow><mrow><mn>3</mn></mrow></msub></math> . This occurs when the coefficient of the Lorentz–Chern–Simons term is related to the AdS radius by <math altimg="si3.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mi>ℓ</mi><mo>=</mo><mn>2</mn></math> . We find that the spacelike-squashed <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">AdS</mi></mrow><mrow><mn>3</mn></mrow></msub></math> can be modded out by a suitable discrete subgroup of the isometry group, yielding an extremal black hole solution which avoids closed timelike curves