Maximally supersymmetric AdS solutions and their moduli spaces

Abstract

We study maximally supersymmetric AdS D solutions of gauged supergravities in dimensions D ≥ 4. We show that such solutions can only exist if the gauge group after spontaneous symmetry breaking is a product of two reductive groups H R × H mat , where H R is uniquely determined by the dimension D and the number of supersymmetries N while H mat is unconstrained. This resembles the structure of the global symmetry groups of the holographically dual SCFTs, where H R is interpreted as the R-symmetry and H mat as the flavor symmetry. Moreover, we discuss possible supersymmetry preserving continuous deformations, which correspond to the conformal manifolds of the dual SCFTs. Under the assumption that the scalar manifold of the supergravity is a symmetric space we derive general group theoretical conditions on these moduli. Using these results we determine the AdS solutions of all gauged supergravities with more than 16 real supercharges. We find that almost all of them do not have supersymmetry preserving deformations with the only exception being the maximal supergravity in five dimensions with a moduli space given by SU(1, 1)/U(1). Furthermore, we determine the AdS solutions of four-dimensional N = 3 supergravities and show that they similarly do not admit supersymmetric moduli.1 If we denote the conserved current of a global symmetry of the boundary CFT by J it couples via ∂AdS A ∧ * J to the gauge field A of a local symmetry in the bulk.2 Note, however, that there are SCFTs without an R-symmetry, as for example three-dimensional N = 1 theories. In this case also the gauge group factor H g R of the dual supergravity solution is trivial.

Holographic RG flows for four-dimensional $$ \mathcal{N}=2 $$ SCFTs

“…This "bottom-up" approach to studying general properties of SCFTs with holographic duals by employing gauged supergravity has recently received some attention in the context of understanding the space of exactly marginal couplings of SCFTs in various dimensions, see [3][4][5][6][7][8][9][10][11][12] and references thereof. While these results present interesting holographic predictions for strongly interacting SCFTs it should be noted that some of the supergravity results can also be shown more generally, without a reference to holography, using purely algebraic techniques [13].…”

“…The necessary and sufficient conditions for such domain walls in half-maximal supergravity were given in [28]. 11 Here we summarize them. The conditions use the same vector X and the same superpotential W defined in section 5.1, however now the scalars are non-constant and depend on the radial coordinate.…”

“…12 The superpotential (5.10) can also be written as W = 2 P mn + P +mn . One can then use (5.26) to show that ∂ Σ P + is proportional to P + , and is therefore analogous to (5.28).…”

“…One can see that the latter are in fact equivalent to the constraints (5.25)-(5.28), together with the condition that the superpotential is extremized.We now proceed to discuss two explicit examples which display all these features. 11 The analysis of [28] was restricted to an embedding tensor satisfying ξM = 0. Recall that we are also assuming this condition here as it is necessary for a fully supersymmetric AdS5 vacuum.…”

“…The supersymmetry constraints on the embedding tensor reflect the discussion of the gauge group and are given by 12) where Λ has the same form as in (3.12) and therefore describes the embedding of SO (3) into H . Similarly, Ξ is a transformation in SO(1, n − m) whose non-vanishing components are Ξ 0 a and Ξ a 0 given by…”