Supersymmetric Solutions

5 Supersymmetric Solutions

By definition, a supersymmetric solution of a supergravity theory is a solution that also admits a Killing spinor, i.e., a spinor field $ψ$ that satisfies $Dμ ψ = 0$ , where $D μ$ is a spinorial covariant derivative that depends on the matter fields of the theory. Given such a Killing spinor $ψ$ , the bilinear $μ μ K = ¯ψΓ ψ$ is a non-spacelike Killing field. By definition, a supersymmetric horizon is invariant under the Killing field $K μ$ and thus $K μ$ must be tangent to the horizon. Hence $K μ$ must be null on the horizon; in other words the horizon is a Killing horizon of $K μ$ . Furthermore, since $K2 ≤ 0$ both outside and inside the horizon, it follows that on the horizon $2 dK = 0$ , i.e., it must be a degenerate horizon. Hence any supersymmetric horizon is necessarily a degenerate Killing horizon.

5.1 Four dimensions

The simplest supergravity theory in four dimensions admitting supersymmetric black holes is minimal $𝒩 = 2$ supergravity, whose bosonic sector is simply standard Einstein–Maxwell theory. The general supersymmetric solution in this theory is given by the Israel–Wilson–Perjés metrics. Using this fact, the following near-horizon uniqueness theorem has been proved:

Theorem 5.1 ([41 ]). Any supersymmetric near-horizon geometry in $𝒩 = 2$ minimal supergravity is one of the maximally supersymmetric solutions $ℝ1,1 × T 2$ or $AdS2 × S2$ .

Notice that staticity here follows from supersymmetry. In Section 7 we will discuss the implications of this result for uniqueness of supersymmetric black holes in four dimensions.

For $𝒩 = 2$ gauged supergravity, whose bosonic sector is Einstein–Maxwell theory with a negative cosmological constant, an analogous classification of supersymmetric near-horizon geometries has not been performed. Nevertheless, one may deduce the following result, from a classification of all near-horizon geometries of this theory under the additional assumption of axisymmetry:

Proposition 5.1 ([154 ]). Any supersymmetric, axisymmetric, near-horizon geometry in $𝒩 = 2$ gauged supergravity, is given by the near-horizon limit of the 1-parameter family of supersymmetric Kerr–Newman-AdS₄ black holes [150 ].

Note that the above near-horizon geometry is non-static. This is related to the fact that supersymmetric AdS black holes must carry angular momentum. It would be interesting to remove the assumption of axisymmetry. Some related work has been done in the context of supersymmetric isolated horizons [29 ].

Supersymmetric black holes are not expected to exist in $𝒩 = 1$ supergravity. For the general $𝒩 = 1$ supergravity the following result, supporting this expectation, has been established.

Proposition 5.2 ([115 ]). A supersymmetric near-horizon geometry of $𝒩 = 1$ supergravity is either the trivial solution $ℝ1,1 × T 2$ , or $ℝ1,1 × S2$ where $S2$ may be a non-round sphere.

An example of a supersymmetric near-horizon geometry of the form $1,1 2 ℝ × S$ , with a round $2 S$ , was given in [176 ].

5.2 Five dimensions

The simplest $D = 5$ supergravity theory admitting supersymmetric black holes is $𝒩 = 1$ minimal supergravity. The bosonic sector is Einstein–Maxwell theory with a Chern–Simons term given by Eq. (112 ) with a specific coupling $ξ = 1$ . Supersymmetric solutions to this theory were classified in [93 ]. This was used to obtain a complete classification of supersymmetric near-horizon geometries in this theory.

Theorem 5.2 ([191 ]). Any supersymmetric near-horizon geometry of minimal supergravity is locally isometric to one of the following maximally supersymmetric solutions: $AdS3 × S2$ , $ℝ1,1 × T 3$ , or the near-horizon geometry of the Breckenridge–Myers–Peet–Vafa (BMPV) black hole (of which $AdS2 × S3$ is a special case).

Note that here supersymmetry implies homogeneity. The $2 AdS3 × S$ near-horizon geometry has cross sections $H ∼= S1 × S2$ and arises as the near-horizon limit of supersymmetric black rings [64 , 65 ] and supersymmetric black strings [93 , 19 ]. Analogous results have been obtained in $D = 5$ minimal supergravity coupled to an arbitrary number of vector multiplets [111 ]. As discussed in Section 7, the above theorem can be used to prove a uniqueness theorem for topologically spherical supersymmetric black holes.

The corresponding problem for minimal gauged supergravity has proved to be more difficult. The bosonic sector of this theory is Einstein–Maxwell–Chern–Simons theory with a negative cosmological constant. The theory admits asymptotically AdS₅ black-hole solutions that are relevant in the context of the AdS/CFT correspondence [120 , 38 ]. The following partial results have been shown.

Proposition 5.3 ([120 ]). Consider a supersymmetric, homogeneous near-horizon geometry of minimal gauged supergravity. Cross sections of the horizon must be one of the following: a homogeneously squashed $S3$ , $Nil$ or $SL (2,ℝ )$ manifold.

The near-horizon geometry of the $3 S$ case was used to construct the first example of an asymptotically AdS₅ supersymmetric black hole [120 ]. Analogous results in gauged supergravity coupled to an arbitrary number of vector multiplets (this includes $U (1 )3$ gauged supergravity) were obtained in [151 ]. Unlike the ungauged theory, homogeneity is not implied by supersymmetry, and indeed there are more general solutions.

Proposition 5.4 ([161 ]). The most general supersymmetric near-horizon geometry in minimal gauged supergravity, admitting a $U (1)2$ -rotational symmetry and a compact horizon section, is the near-horizon limit of the topologically-spherical supersymmetric black holes of [38 ].

The motivation for assuming this isometry group is that all known black-hole solutions in five dimensions possess this. Interestingly, this result implies the non-existence of supersymmetric AdS₅ black rings with $ℝ × U (1)2$ isometry.

In fact, recent results allow one to remove all assumptions and obtain a complete classification. Generic supersymmetric solutions of minimal gauged supergravity preserve $1 4$ -supersymmetry.

Proposition 5.5 ([121 , 106 ]). Any $1 2$ -supersymmetric near-horizon geometry in minimal gauged supergravity must be invariant under a local $U (1 )2$ -rotational isometry.

Furthermore, the following has also been shown.

Proposition 5.6 ([105 ]). Any supersymmetric near-horizon geometry in minimal gauged supergravity with a compact horizon section must preserve $1 2$ -supersymmetry.

This latter result is proved using a Lichnerowicz type identity to establish a correspondence between Killing spinors and solutions to a horizon Dirac equation, and then applying an index theorem. Therefore, combining the previous three propositions gives a complete classification theorem for near-horizon geometries in minimal gauged supergravity.

Theorem 5.3 ([161 , 105 , 106 ]). A supersymmetric near-horizon geometry in minimal gauged supergravity, with a compact horizon section, must be locally isometric to the near-horizon limit of the topologically-spherical supersymmetric black holes [38 ], or the homogeneous near-horizon geometries with the $Nil$ or $SL (2,ℝ )$ horizons.

This theorem establishes a striking corollary for the corresponding black hole classification theorem.

Corollary 5.1. Supersymmetric black rings in minimal gauged supergravity do not exist.

We emphasise that the absence of supersymmetric AdS₅ black rings is rather suprising, given asymptotically-flat counterparts are known to exist [64 ].

Parts of the above analysis have been generalised by $U (1)n$ gauged supergravity, although the results are slightly different.

Proposition 5.7 ([151 ]). Consider a supersymmetric near-horizon geometry in $n U(1)$ minimal gauged supergravity with $U (1)2$ -rotational symmetry and a compact horizon section. It must be either: (i) the near-horizon limit of the topologically spherical black holes of [160 ]; or (ii) $AdS3 × S2$ with $H ∼= S1 × S2$ or (iii) $AdS3 × T2$ with $H ∼= T 3$ . These latter two cases have constant scalars and only exist in certain regions of the scalar moduli space (not including the minimal theory).

Therefore, in this theory one cannot rule out the existence of supersymmetric AdS₅ black rings (although as argued in [151 ] they would not be connected to the asymptotically-flat black rings [66 ]). It would be interesting to complete the classification of near-horizon geometries in this more general theory, along the lines of the minimal theory.

5.3 Six dimensions

The simplest supergravity in six dimensions is minimal supergravity. The bosonic field context of this theory is a metric and a 2-form potential with self-dual field strength. The classification of supersymmetric solutions to this theory was given in [112 ]. This was used to work out a complete classification of supersymmetric near-horizon geometries.

Theorem 5.4 ([112 ]). Any supersymmetric near-horizon geometry of $D = 6$ minimal supergravity, with a compact horizon cross section, is either $1,1 4 ℝ × T$ , $1,1 ℝ × K3$ or locally $3 AdS3 × S$ .

The $AdS3 × S3$ solution has $H ∼= S1 × S3$ and arises as the near-horizon limit of a supersymmetric rotating black string.

Analogous results have been obtained for minimal supergravity coupled to an arbitrary number of scalar and tensor multiplets [3 ].

5.4 Ten dimensions

Various results have been derived for heterotic supergravity and type IIB supergravity.

The bosonic field content of $D = 10$ heterotic supergravity consist of the metric, a 2-form gauge potential and a scalar field (dilaton). The full theory is invariant under 16 supersymmetries. There are two classes of supersymmetric near-horizon geometries [113 ]. One is the direct product $ℝ1,1 × H$ , with vanishing flux and constant dilaton, where $H$ is Spin(7) holonomy manifold, which generically preserves one supersymmetry (there are solutions in this class which preserve more supersymmetry provided $H$ has certain special holonomy). In the second class, the near-horizon geometry is a fibration of AdS₃ over a base $B7$ (with a $U (1 )$ -connection) with a $G2$ structure, which must preserve $2$ supersymmetries. This class may preserve $4,6,8$ supersymmetries if $B7$ is further restricted. In particular, an explicit classification for $12$ -supersymmetric near-horizon geometries is possible.

Proposition 5.8 ([113 ]). Any supersymmetric near-horizon geometry of heterotic supergravity invariant under 8 supersymmetries, with a compact horizon cross section, must be locally isometric to one of $3 4 AdS3 × S × T$ , $3 AdS3 × S × K3$ or $1,1 4 ℝ × T × K3$ (with constant dilaton).

A large number of heterotic horizons preserving $4$ supersymmetries have been constructed, including explicit examples where $H$ is $SU (3)$ and $S3 × S3 × T 2$ [114 ].

The bosonic field content of type IIB supergravity consists of a metric, a complex scalar, a complex 2-form potential and a self-dual 5-form field strength. The theory is invariant under 32 supersymmetries. A variety of results concerning the classification of supersymmetric near-horizon geometries in this theory have been derived [102 , 101 , 103 ]. Certain explicit classification results are known for near-horizon geometries with just a 5-form flux preserving more than two supersymmetries, albeit under certain restrictive assumptions [101 ]. More generally, the existence of one supersymmetry places rather weak geometric constraints on the horizon cross sections $H$ : generically $H$ may be any almost Hermitian spin $c$ manifold [103 ]. There are also special cases for which $H$ has a $Spin (7)$ structure and those for which it has an $SU (4)$ structure (where the Killing spinor is pure). More recently, it has been shown that any supersymmetric near-horizon geometry in type IIB supergravity must preserve an even number of supersymmetries, and furthermore, if a certain horizon Dirac operator has non-trivial kernel the bosonic symmetry group must contain $SO (2, 1)$ [104 ].

5.5 Eleven dimensions

The bosonic field content of $D = 11$ supergravity consists of a metric $gμν$ and a 3-form potential $C μνρ$ . The near-horizon limit of the 4-form field strength $ℱ = dC$ can be written as Eq. (27 ), where $Y$ and $X$ are a 2-form and closed 4-form respectively on the 9-dimensional horizon cross sections $H$ . The near-horizon Einstein equations are Eqs. (17 ) and (18 ) with $Λ = 0$ and the matter field terms are given by [116 ]

We note that the dominant and strong energy conditions are satisfied: $ab 1 2 1- 2 Pabγ = 4Y + 48X ≥ 0$ and $E ≥ 0$ . Therefore, the general results established under these assumptions, discussed in Section 3, are all valid, including most notably the horizon topology theorem.

Various classification results have been derived for supersymmetric near-horizon geometry solutions under the assumption that cross sections of the horizon are compact. Static supersymmetric near-horizon geometries are warped products of either $1,1 ℝ$ or AdS₂ with $M9$ , where $M9$ admits a particular $G$ -structure [117 ]. We note that these warped product forms are guaranteed by the general analysis of static near-horizon geometries in Section 3.2.1. Supersymmetric near-horizon geometries have been studied more generally in [116 ]. Most interestingly, a near-horizon (super)symmetry enhancement theorem has been established.

Theorem 5.5 ([118 ]). Any supersymmetric near-horizon geometry solution to eleven dimensional supergravity, with compact horizon cross sections, must preserve an even number of supersymmetries. Furthermore, the bosonic symmetry group must contain $SO (2, 1)$ .

The proof of this follows by first establishing a Lichnerowicz type identity for certain horizon Dirac operators and then application of an index theorem. As far as bosonic symmetry is concerned, the above result is a direct analogue of the various near-horizon symmetry theorems discussed in Section 3.2, which are instead established under various assumptions of rotational symmetry.

	Abstract
1	Introduction
	1.1	Black holes in string theory
	1.2	Gauge/gravity duality
	1.3	Black hole classification
	1.4	This review
2	Degenerate Horizons and Near-Horizon Geometry
	2.1	Coordinate systems and near-horizon limit
	2.2	Curvature of near-horizon geometry
	2.3	Einstein equations and energy conditions
	2.4	Physical charges
3	General Results
	3.1	Horizon topology theorem
	3.2	AdS₂-structure theorems
4	Vacuum Solutions
	4.1	Static: all dimensions
	4.2	Three dimensions
	4.3	Four dimensions
	4.4	Five dimensions
	4.5	Higher dimensions
5	Supersymmetric Solutions
	5.1	Four dimensions
	5.2	Five dimensions
	5.3	Six dimensions
	5.4	Ten dimensions
	5.5	Eleven dimensions
6	Solutions with Gauge Fields
	6.1	Three dimensional Einstein–Maxwell–Chern–Simons theory
	6.2	Four dimensional Einstein–Maxwell theory
	6.3	Five dimensional Einstein–Maxwell–Chern–Simons theory
	6.4	Theories with hidden symmetry
	6.5	Non-Abelian gauge fields
7	Applications and Related Topics
	7.1	Black-hole uniqueness theorems
	7.2	Stability of near-horizon geometries and extremal black holes
	7.3	Geometric inequalities
	7.4	Analytic continuation
	7.5	Extremal branes
	Acknowledgements
	References
	Footnotes