Gravitational free energy in topological AdS/CFT

TL;DR

This paper investigates the holographic duals of topologically twisted gauge theories, showing that their gravitational free energy is independent of boundary metrics and is always zero for smooth fillings, with implications for large N gauge theories.

Contribution

It establishes that the gravitational free energy in topological AdS/CFT is metric-independent and always zero for smooth fillings, extending to four-dimensional cases.

Findings

Renormalized gravitational free energy is metric-independent.

Gravitational free energy of smooth fillings is always zero.

Results apply to topological AdS$_5$/CFT$_4$ and large N gauge theories.

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.