Gravitational free energy in topological AdS/CFT

Abstract

We define and study a holographic dual to the topological twist of $\mathcal{N}=4$ gauge theories on Riemannian three-manifolds. The gravity duals are solutions to four-dimensional $\mathcal{N}=4$ 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 AdS${}_5$/CFT${}_4$. 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 $\mathcal{N}=4$ $SU(N)$ super-Yang-Mills, respectively.

arXiv comment46 pages; v2: corrected typos, updated references

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• Topological Insights into Black Hole Thermodynamics: Non-Extensive Entropy in CFT framework
• The Supersymmetry of Higher-Derivative Supergravity in AdS${}_4$ Holography
• Supersymmetry anomalies in new minimal supergravity
• Twisted String Theory in Anti-de Sitter Space
• Topological AdS/CFT and the $\Omega$ deformation
• Boundary Conditions in Topological AdS${}_4$/CFT${}_3$