Gravitational free energy in topological AdS/CFT

We define and study a holographic dual to the topological twist of N = 4 gauge theories on Riemannian three-manifolds. The gravity duals are solutions to four-dimensional N = 4 gauged supergravity, where the three-manifold arises as a conformal boundary. Following our previous work, we show that the renormalized gravitational free energy of such solutions is independent of the boundary three-metric, as required for a topological theory. We then go further, analyzing the geometry of supersymmetric bulk solutions. Remarkably, we are able to show that the gravitational free energy of any smooth four-manifold filling of any three-manifold is always zero. Aided by this analysis, we prove a similar result for topological AdS5/CFT4. We comment on the implications of these results for the large N limits of topologically twisted gauge theories in three and four dimensions, including the ABJM theory and N = 4 SU(N) super-Yang{Mills, respectively.