We construct asymptotically AdS_5 solutions to 5-dimensional Einstein-Maxwell
theory with Chern-Simons term which are dual to 4-dimensional gauge theories,
including N=4 SYM theory, in the presence of a constant background magnetic
field B and a uniform electric charge density \rho. For the solutions
corresponding to supersymmetric gauge theories, we find numerically that a
small magnetic field causes a drastic decrease in the entropy at low
temperatures. The near-horizon AdS_2 \times R^3 geometry of the purely
electrically charged brane thus appears to be unstable under the addition of a
small magnetic field. Based on this observation, we propose a formulation of
the third law of thermodynamics (or Nernst theorem) that can be applied to
black holes in the AdS/CFT context.
We also find interesting behavior for smaller, non-supersymmetric, values of
the Chern-Simons coupling k. For k=1 we exhibit exact solutions corresponding
to warped AdS_3 black holes, and show that these can be connected to
asymptotically AdS_5 spacetime. For k\leq 1 the entropy appears to go to a
finite value at extremality, but the solutions still exhibit a mild singularity
at strictly zero temperature. In addition to our numerics, we carry out a
complete perturbative analysis valid to order B^2, and find that this
corroborates our numerical results insofar as they overlap.Comment: 45 pages v2: added note about subsequent results found in
arXiv:1003.130