We construct rigid supersymmetric gauge theories on Riemannian five-manifolds. We follow a holographic approach, realizing the manifold as the conformal boundary of a six-dimensional bulk supergravity solution. This leads to a systematic classification of five-dimensional supersymmetric backgrounds with gravity duals. We show that the background metric is furnished with a conformal Killing vector, which generates a transversely holomorphic foliation with a transverse Hermitian structure. Moreover, we prove that any such metric defines a supersymmetric background. Finally, we construct supersymmetric Lagrangians for gauge theories coupled to arbitrary matter on such backgrounds.Comment: 35 pages: v2: minor corrections and references added. Published versio
We consider a general class of asymptotically locally AdS5 solutions of minimal gauged supergravity, which are dual to superconformal field theories on curved backgrounds S 1 × M3 preserving two supercharges. We demonstrate that standard holographic renormalization corresponds to a scheme that breaks supersymmetry. We propose new boundary terms that restore supersymmetry, and show that for smooth solutions with topology S 1 × R 4 the improved on-shell action reproduces both the supersymmetric Casimir energy and the field theory BPS relation between charges. I. THE SUPERSYMMETRIC CASIMIR ENERGYIn [1, 2] a new observable of d = 4 superconformal field theories has been introduced: the supersymmetric Casimir energy. This is defined by putting the theory on certain curved backgrounds M 4 = S 1 β × M 3 , where S 1 β is a circle of length β and M 3 is a compact threemanifold. These are rigid supersymmetric backgrounds, and the supersymmetric Casimir energy is defined asHere the partition function Z susy is computed with periodic boundary conditions for the fermions around S 1 β . A key point is that, unlike the vacuum energy of general d = 4 conformal field theories (CFTs), E susy is schemeindependent and thus an intrinsic observable.The rigid supersymmetric backgrounds of interest comprise a metric on M 4 of the formwhere τ ∼ τ + β is a coordinate on S 1 β . The vector ∂ ψ is Killing, and generates a transversely holomorphic foliation of M 3 , with local transverse complex coordinate z. The local one-form a satisfies da = iu e w dz ∧ dz, where w = w(z,z), u = u(z,z). In addition there is a non-dynamical Abelian gauge field, which couples to the R-symmetry current and arises when the field theory is coupled to background conformal supergravity, given byNotice that the second line is locally pure gauge; however, the constant γ will play an important role. The background geometry thus depends on the choice of the two functions w(z,z), u(z,z), and via (1) the supersymmetric Casimir energy also a priori depends on this choice. These backgrounds admit two supercharges of opposite R-charge, and associated to each of these is an integrable complex structure (i.e. they are ambiHermitian). In [3] it is argued that the supersymmetric partition function depends on the background only via the choice of complex structure(s). In the present set-up, this implies that Z susy depends only on the transversely holomorphic foliation generated by ∂ ψ . In particular, deformations of w(z,z) and u(z,z) that leave this foliation fixed should not change E susy .Later in this paper we will focus on the case that topologically M 3 ∼ = S 3 . Here we may embed S 3 ⊂ R 4 = R 2 ⊕ R 2 , and write, where ϕ 1 , ϕ 2 are standard 2π periodic azimuthal angles. In this case the above statements imply that E susy should depend only on b 1 , b 2 , and the explicit calculation in [1] givesHere a and c are the usual trace anomaly coefficients for a d = 4 CFT. For field theories admitting a large N gravity dual in type IIB supergravity, to leading order in the ...
We derive a simple formula for the action of any supersymmetric solution to minimal gauged supergravity in the AdS 4 /CFT 3 correspondence. Such solutions are equipped with a supersymmetric Killing vector, and we show that the holographically renormalized action may be expressed entirely in terms of the weights of this vector field at its fixed points, together with certain topological data. In this sense, the classical gravitational partition function localizes in the bulk. We illustrate our general formula with a number of explicit examples, in which exact dual field theory computations are also available, which include supersymmetric Taub-NUT and Taub-bolt type spacetimes, as well as black hole solutions. Our simple topological formula also allows us to write down the action of any solution, provided it exists.
Holographic renormalization is a systematic procedure for regulating divergences in observables in asymptotically locally AdS spacetimes. For dual boundary field theories which are supersymmetric it is natural to ask whether this defines a supersymmetric renormalization scheme. Recent results in localization have brought this question into sharp focus: rigid supersymmetry on a curved boundary requires specific geometric structures, and general arguments imply that BPS observables, such as the partition function, are invariant under certain deformations of these structures. One can then ask if the dual holographic observables are similarly invariant. We study this question in minimal N = 2 gauged supergravity in four and five dimensions. In four dimensions we show that holographic renormalization precisely reproduces the expected field theory results. In five dimensions we find that no choice of standard holographic counterterms is compatible with supersymmetry, which leads us to introduce novel finite boundary terms. For a class of solutions satisfying certain topological assumptions we provide some independent tests of these new boundary terms, in particular showing that they reproduce the expected VEVs of conserved charges.
We define a holographic dual to the Donaldson-Witten topological twist of N = 2 gauge theories on a Riemannian four-manifold. This is described by a class of asymptotically locally hyperbolic solutions to N = 4 gauged supergravity in five dimensions, with the four-manifold as conformal boundary. Under AdS/CFT, minus the logarithm of the partition function of the gauge theory is identified with the holographically renormalized supergravity action. We show that the latter is independent of the metric on the boundary four-manifold, as required for a topological theory. Supersymmetric solutions in the bulk satisfy first order differential equations for a twisted Sp(1) structure, which extends the quaternionic Kähler structure that exists on any Riemannian four-manifold boundary. We comment on applications and extensions, including generalizations to other topological twists.
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