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 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 mathcal{N}=4 SU(N) super-Yang-Mills, respectively.
Highlights
In [4] we defined a holographic dual to the Donaldson-Witten topological twist of N = 2 gauge theories on a Riemannian four-manifold (M4, g)
We prove a similar result for topological AdS5/CFT4
In this paper we have defined and studied a holographic dual to the topological twist of N = 4 gauge theories on Riemannian three-manifolds and verified that the renormalized gravitational free energy is independent of the boundary three-metric, providing an additional construction of topological AdS/CFT beyond [4]
Summary
The AdS/CFT correspondence has been especially influential in the context of threedimensional field theories. The ABJM theory in flat spacetime R1,2 is conjectured to be holographically dual to M-theory on AdS4 × S7/Zk. In order to study the gravity dual of the field theory defined on different manifolds M3 in the large N limit, one may consider a consistent truncation of eleven-dimensional supergravity on S7, or S7/Zk, to an effective four-dimensional bulk supergravity theory. In particular the SU(2) × SU(2) isometry of the internal space, which becomes a Spin(4)R gauged R-symmetry of the consistently truncated four-dimensional theory, corresponds to the Spin(4)R R-symmetry of the field theory dual. In the rest of the paper we will work entirely within the Das-Fischler-Rocek fourdimensional N = 4 gauged supergravity theory Any solution to this theory, for a bulk asymptotically locally hyperbolic four-manifold Y4, automatically uplifts on S7/Zk to give a gravity dual to the ABJM theory defined on the conformal boundary M3 = ∂Y4. In particular we note that the effective four-dimensional Newton constant is
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