6 Solutions with Gauge Fields

In this section we will consider general near-horizon geometries coupled to non-trivial gauge fields. We will mostly focus on theories that are in the bosonic sector of minimal supergravity theories (since these are the best understood cases). Extremal, non-supersymmetric, near-horizon geometries may be thought of as interpolating between vacuum and supersymmetric solutions. They consist of a much larger class of solutions, which, at least in higher dimensions, are much more difficult to classify. In particular, we consider D = 3,4,5 Einstein–Maxwell theory, possibly coupled to a Chern–Simons term in odd dimensions, and D = 4 Einstein–Yang–Mills theory.

6.1 Three dimensional Einstein–Maxwell–Chern–Simons theory

The classification of near-horizon geometries in D = 3 Einstein–Maxwell theory with a cosmological constant can be completely solved. To the best of our knowledge this has not been presented before, so for completeness we include it here. It should be noted though that partial results which capture the main result were previously shown in [175].

The method parallels the vacuum case in Section 4.2 closely. As in that case the near-horizon metric data reads h = h (x)dx and 2 γ = dx. Since cross sections H of the horizon are one-dimensional, the Maxwell 2-form induced on H must vanish identically. Hence, the most general near-horizon Maxwell field (23View Equation) in three dimensions is ℱNH = d(rΔ (x)dv ). It is straightforward to show that the 3D Maxwell equation d ⋆ ℱ = 0, where ⋆ is the Hodge dual with respect to the spacetime metric, is equivalent to the following equation on H:

Δ ′ = hΔ. (99 )
The near-horizon Einstein equations (17View Equation) and (18View Equation) are simply
′ 1 2 2 h = 2h + 2Δ + Λ, (100 ) F = 12h2 − 12h ′ + Λ. (101 )

Theorem 6.1. Consider a near-horizon geometry with a compact horizon cross section H ∼= S1 in Einstein–Maxwell-Λ theory. If Λ < 0 the near-horizon geometry must be either AdS2 × S1 with a constant AdS2 Maxwell field, or the quotient of AdS3 Eq. (73View Equation) with a vanishing Maxwell field. If Λ = 0 the only solution is the trivial flat near-horizon geometry 1,1 1 ℝ × S. If Λ > 0 there are no solutions.

Proof: Rather that solving the above ODEs we may use a global argument. Compactness requires x to be a periodic coordinate on H ∼= S1 and since h,Δ are globally defined they must be periodic functions of x. For Λ ≥ 0 simply integrate Eq. (100View Equation) over H to find that the only solution is the trivial flat one h ≡ 0,Δ ≡ 0 and Λ = 0. For 2 Λ ≡ − ℓ2 < 0 we may argue as follows. Multiply Eq. (100View Equation) by ′ h and integrate over H to obtain

∫ ∫ 0 = (h ′2 − 2Δ2h ′)dx = (h′2 + 4 Δ2h2)dx, (102 ) S1 S1
where in the second equality we have integrated by parts and used Eq. (99View Equation). Hence h must be a constant and hΔ ≡ 0. Equation (100View Equation) then implies Δ is also a constant. We deduce the only possible solutions are h = 0,Δ = ± 1 ℓ, or h = ± 2,Δ = 0 ℓ. The former gives a near-horizon geometry AdS × S1 2 and the latter is the vacuum solution locally isometric to AdS3.

This result implies that the near-horizon limit of any charged rotating black-hole solution to 3D Einstein–Maxwell-Λ theory either has vanishing charge or angular momentum. The AdS2 × S1 solution is the near-horizon limit of the non-rotating extremal charged BTZ black hole, whereas the AdS3 solution is the near-horizon limit of the vacuum rotating extremal BTZ [15]. Charged rotating black holes were first obtained within a wide class of stationary and axisymmetric solutions to Einstein–Maxwell-Λ theory [45], and later by applying a solution generating technique to the charged non-rotating black hole [46, 174]. We have checked that in the extremal limit their near-horizon geometry is the AdS × S1 2 solution, so the angular momentum is lost in the near-horizon limit, in agreement with the above analysis. It would be interesting to investigate whether charged rotating black holes exist that instead possess a locally AdS3 near-horizon geometry. In this case, charge would not be captured by the near-horizon geometry, a phenomenon that is known to occur for five-dimensional supersymmetric black rings whose near-horizon geometry is locally AdS × S2 3.

In 2 + 1 dimensions an Abelian gauge field monopole is not isolated. Electric charge can be “screened” by adding a mass term to the gauge field. A natural way to do this is to add a Chern–Simons term ∫ μ 𝒜 ∧ ℱ to the spacetime action, resulting in a topologically-massive gauge theory. This only modifies the Maxwell equation:

d ⋆ ℱ + μℱ = 0, (103 )
where μ is the mass parameter of the gauge field. For the near-horizon Maxwell field it can be shown that this is equivalent to
Δ′ = (h + μ)Δ. (104 )
As in the pure Einstein–Maxwell case, a complete classification of near-horizon geometries to this theory is possible. To the best of our knowledge this has not been presented before.

Theorem 6.2. Consider a near-horizon geometry with a cross section H ∼= S1 in Einstein–Maxwell-Λ theory with a Chern–Simons term and mass μ. If Λ < 0 the functions Δ and h are constant and the near-horizon geometry is the homogeneous S1-bundle over AdS 2 (106View Equation). If Λ = 0 the only solution is the trivial flat near-horizon geometry 1,1 1 ℝ × S. If Λ > 0 there are no solutions.

For Λ ≥ 0 the proof of this is identical to the Einstein–Maxwell case above. For Λ = − 2ℓ2- one can also use the same argument as the Einstein–Maxwell case. Using the horizon equation (100View Equation) and Maxwell equation (104View Equation) one can show

∫ ∫ ′2 2 ′ ′2 2 2 0 = 1(h − 2Δ h )dx = 1(h + 4 Δ (h + μ ))dx, (105 ) S S
which implies h must be a constant and Δ (h + μ) ≡ 0. The horizon equation then implies Δ is a constant. If Δ = 0 one gets the vacuum AdS3 solution. If Δ ⁄= 0 then h = − μ and 12μ2 + 2Δ2 = 2ℓ2-. The rest of the near-horizon data is given by F = 1μ2 + Λ 2 and hence the near-horizon geometry is
2 2 2 2 g = − 2Δ( r dv +) 2dvdr − 2μrdvdx + dx = − 1μ2 + -22 r2dv2 + 2dvdr + (dx − μrdv )2 (106 ) 2 ℓ
and ℱ = Δdr ∧ dv. Note that if μ = 0 we recover the AdS2 × S1 solution, whereas if Δ → 0 we recover the vacuum AdS3 solution.

This implies that the near-horizon geometry of any charged rotating black-hole solution to this theory is either the vacuum AdS3 solution, or the non-trivial solution (106View Equation), which is sometimes referred to as “warped AdS3”. Examples of charged rotating black-hole solutions in this theory have been found [2].

6.2 Four dimensional Einstein–Maxwell theory

The spacetime Einstein–Maxwell equations are Eqs. (15View Equation), (22View Equation) with n = 2 and d ⋆ ℱ = 0, where ⋆ is the Hodge dual with respect to the spacetime metric, and the Bianchi identity d ℱ = 0. The near-horizon Maxwell field is given by (23View Equation). The near-horizon geometry Einstein–Maxwell equations are given by Eqs. (17View Equation) and (18View Equation), where

cd 2 γab 2 Pab = 2BacBbd γ + Δ γab − 2 B , (107 ) 2 E = Δ2 + B-, (108 ) 2 d ⋆2 B = ⋆2ihB + ⋆2(dΔ − Δh ), (109 )
where ⋆2 is the Hodge dual with respect to the horizon metric γab. Observe that ⋆2B is a function on H.

Static near-horizon geometries have been completely classified. For Λ = 0 this was first derived in [43Jump To The Next Citation Point], and generalised to Λ ⁄= 0 in [154Jump To The Next Citation Point].

Theorem 6.3 ([43, 154Jump To The Next Citation Point]). Consider a static near-horizon geometry in D = 4 Einstein–Maxwell-Λ theory, with compact horizon cross section H. For Λ ≥ 0 it must be AdS2 × S2. For Λ < 0 it must be AdS2 × H where H is one of the constant curvature surfaces S2,T 2,Σg.

It is worth remarking that if one removes the assumption of compactness one can still completely classify near-horizon geometries. The extra solution one obtains can be written as a warped product

2 g = ψ2(A0r2dv2 + 2dvdr ) + -dψ---+ P (ψ)dϕ2, (110 ) P (ψ) ℱ = edr ∧ dv + bψ−2dψ ∧ dϕ, (111 )
where P (ψ ) = A0 − c(2ψ )−1 − (e2 + b2)ψ− 2 − Λ ψ2∕3, which is an analyticaly continued Reissner–Nordström-Λ solution.

Non-static near-horizon geometries are not fully classified, except under the additional assumption of axisymmetry.

Theorem 6.4 ([163Jump To The Next Citation Point, 154Jump To The Next Citation Point]). Any axisymmetric, non-static near-horizon geometry in D = 4 Einstein–Maxwell-Λ theory, with a compact horizon cross section, must be given by the near-horizon geometry of an extremal Kerr–Newman-Λ black hole.

[163] solved the Λ = 0 in the context of isolated degenerate horizons, where the same equations on H arise. [154Jump To The Next Citation Point] solved the case with Λ ⁄= 0. Note that the horizon topology theorem excludes the possibility of toroidal horizon cross sections for Λ ≥ 0. [164Jump To The Next Citation Point] also excluded the possibility of a toroidal horizon cross section if Λ < 0 under the assumptions of the above theorem.

It is worth noting that the results presented in this section, as well as the techniques used to establish them, are entirely analogous to the vacuum case presented in Section 4.3.

6.3 Five dimensional Einstein–Maxwell–Chern–Simons theory

The field equations of D = 5 Einstein–Maxwell theory coupled to a Chern–Simons term are given by Eqs. (15View Equation) and (22View Equation) with n = 3 and

-2ξ- d ⋆ ℱ + √3--ℱ ∧ ℱ = 0 (112 )
where ℱ is the Maxwell two form and d ℱ = 0. The cases of most interest are ξ = 0 and ξ = 1, which correspond to pure Einstein–Maxwell theory and the bosonic sector of minimal supergravity respectively. The near-horizon Maxwell field is given by (23View Equation). The corresponding near-horizon geometry equations are given by Eqs. (17View Equation) and (18View Equation) and
( 2Δ2 1 ) Pab = 2BacBbd γcd + ---- − --B2 γab, (113 ) 3 3 4Δ2 B2 E = ---- + ---, (114 ) 3 3 d ⋆3 B = − ⋆3 ihB − ⋆3(dΔ − Δh ) + 4√ξ-ΔB, (115 ) 3
where ⋆3 is the Hodge dual with respect to the 3D horizon metric γab. In this section we summarise what is known about solutions to these equations, which is mostly restricted to the Λ = 0 case. Hence, unless otherwise stated, we assume Λ = 0 in this section.

A number of new complications arise that render the classification problem more difficult, most obviously the lack of electro-magnetic duality. Therefore, purely electric solutions, which correspond to Δ ⁄= 0 and B ≡ 0, are qualitatively different to purely magnetic solutions, which correspond to Δ ≡ 0 and B ⁄= 0.

6.3.1 Static

Perhaps somewhat surprisingly a complete classification of static near-horizon geometries in this theory has not yet been achieved. Nevertheless, a number of results have been proved under various extra assumptions. All the results summarised in this section were proved in [153Jump To The Next Citation Point].

As in other five dimensional near-horizon geometry classifications, the assumption of U (1)2 rotational symmetry proves to be useful. Static near-horizon geometries in this class in general are either warped products of AdS2 and H, or AdS3 and a 2D manifold, see the Corollary 3.1. The AdS3 near-horizon geometries are necessarily purely magnetic and can be classified for any ξ.

Proposition 6.1. Any static AdS3 near-horizon geometry with a 2 U (1 )-rotational symmetry, in Einstein–Maxwell–Chern–Simons theory, with a compact horizon cross section, is the direct product of a quotient of AdS3 and a round S2.

This classifies a subset of purely magnetic geometries. By combining the results of [153Jump To The Next Citation Point] together with Proposition 6.4 of [155Jump To The Next Citation Point], it can be deduced that for ξ = 1 there are no purely magnetic AdS2 geometries; therefore, with these symmetries, one has a complete classification of purely magnetic geometries.

Corollary 6.1. Any static, purely magnetic, near-horizon geometry in minimal supergravity, possessing 2 U (1)-rotational symmetry and compact cross sections H, must be locally isometric to 2 AdS3 × S with 1 2 H = S × S.

We now turn to purely electric geometries.

Proposition 6.2. Consider a static, purely electric, near-horizon geometry in Einstein–Maxwell–Chern–Simons theory, with a U(1)2-rotational symmetry and compact cross section. It must be given by either 3 AdS2 × S, or a warped product of AdS 2 and an inhomogeneous 3 S.

The latter non-trivial solution is in fact the near-horizon limit of an extremal RN black hole immersed in a background electric field (this can be generated via a Harrison type transformation).

Finally, we turn to the case where the near-horizon geometry possess both electric and magnetic fields. In fact one can prove a general result in this case, i.e., without the assumption of rotational symmetries.

Proposition 6.3. Any static near-horizon geometry with compact cross sections H, in Einstein–Maxwell–Chern–Simons theory with coupling ξ ⁄= 0, with non-trivial electric and magnetic fields, is a direct product of AdS2 × H, where the metric on H is not Einstein.

Explicit examples for 2 0 < ξ < 1∕4 were also found, which all have ∼ 3 H = S with 2 U (1)-rotational symmetry. However, we should emphasise that no examples are known for minimal supergravity (ξ = 1). Hence there is the possibility that in this case static near-horizon geometries with non-trivial electric and magnetic fields do not exist, although this has not yet been shown. If this is the case, then the above results fully classify static near-horizon geometries with 2 U (1)-rotational symmetry. For ξ = 0 the analysis of electro-magnetic geometries is analogous to the purely magnetic case above; in fact there exists a dyonic AdS2 geometry that is a direct product AdS2 × S2 × S1 and it is conjectured there are no others.

6.3.2 Homogeneous

The classification of homogeneous near-horizon geometries can be completely solved, even including a cosmological constant Λ. This does not appear to have been presented explicitly before, so for completeness we include it here with a brief derivation. We may define a homogeneous near-horizon geometry as follows. The Riemannian manifold (H, γab) is a homogeneous space, i.e., admits a transitive isometry group K, such that the rest of the near-horizon data (F, ha,Δ, Bab) are invariant under K. As discussed in Section (4.4), this is equivalent to the near-horizon geometry being a homogeneous spacetime with a Maxwell field invariant under its isometry group.

An immediate consequence of homogeneity is that an invariant function must be a constant and any invariant 1-form must be a Killing field. Hence the 1-forms h and j ≡ ⋆3B are Killing and the functions F, Δ,h2, j2 must be constants. Thus, the horizon Einstein equations (17View Equation), (18View Equation), (113View Equation), (114View Equation) and Maxwell equation (115View Equation) simplify.

Firstly, note that if h and j vanish identically then H is Einstein 1 2 Rab = 2(Δ + 2Λ )γab, so H is a constant curvature space S3, ℝ3,ℍ3 (the latter two can only occur if Λ < 0). This family includes the static near-horizon geometry AdS2 × H.

Now consider the case where at least one of h and j is non-vanishing. By contracting the Maxwell equation (115View Equation) with a b j j one can show that 2 2 2 (j ⋅ h) = j h and hence by the Cauchy–Schwarz inequality j and h must be parallel as long as they are both non-zero. Thus, if one of h,j is non-vanishing, we can write ha = kua and ja = qua for some constants k,q where ua is a unit normalised Killing vector field. The near-horizon equations now reduce to

( ) ( ) Rab = 1 k2 − 4q2 uaub + 4 q2 + 1 Δ2 + Λ γab, (116 ) 21 2 2 2 2 3 2 F = 2(k − 3q −)Δ + Λ, (117 ) √3- qdu = 2 k + 2ξq Δ ⋆ u. (118 )
This allows one to prove:

Theorem 6.5. Any homogeneous near-horizon geometry in Einstein–Maxwell–CS-Λ theory for which one of the constants k,q is non-zero, must be locally isometric to

( ) g = − 12k2 − 23q2 − Δ2 + Λ r2dv2 + 2dvdr + (ˆω + krdv )2 + ˆg, (119 ) ℱ = Δdr ∧ dv + qˆ𝜖, (120 )
where ˆω is a U(1)-connection over a 2D base with metric ˆg satisfying Ric (ˆg ) = λˆˆg, with ˆλ = 12k2 + 23q2 + Δ2 + 2Λ, and ˆ𝜖 is the volume form of the 2D base. The curvature of the connection is dωˆ = [k2 − 4q2 + Δ2 + 2Λ ]1∕2ˆ𝜖 3 and k2 − 4 q2 + Δ2 + 2Λ ≥ 0 3. If q ⁄= 0 the constants must satisfy
2 4 2 2 1 −2 2( √ -- )2 k − 3q + Δ + 2Λ = 4q Δ 3k + 4ξq , (121 )
whereas if q = 0 one must have kΔ = 0.

The proof of this proceeds as in the vacuum case, by reducing the horizon equations to the 2D orbit space defined by the Killing field u. The above result contains a number of special cases of interest, which we now elaborate upon. Firstly, note that q = Δ = 0 reduces to the vacuum case, see Theorem 4.4.

Before discussing the general case consider k = 0, so the near-horizon geometry is static, which connects to Section 6.3.1. The constraint on the parameters is (1 − 4ξ2)Δ2 = 43q2 − 2Λ and hence, if Λ ≤ 0 one must have 0 ≤ ξ2 < 1 4. For ξ = 0 the connection is trivial and hence the near-horizon geometry is locally isometric to the dyonic AdS × S1 × S2 2 solution. For 0 < ξ2 < 1 4 we get examples of the geometries in Proposition (6.3), where ∼ 3 H = S with its standard homogeneous metric.

Now we consider Λ = 0 in generality so at least one of k,q, Δ is non-zero, in which case ˆλ > 0 and so ˆg is the round S2. The horizon is then either H ∼= S3 with its homogeneous metric or H ∼= S1 × S2, depending on whether the connection ˆω is non-trivial or not, respectively. Notably we have:

Corollary 6.2. Any homogeneous near-horizon geometry of minimal supergravity is locally isometric to AdS3 × S2, or the near-horizon limit of either (i) the BMPV black hole (including AdS2 × S3), or (ii) an extremal nonsupersymmetric charged black hole with SU (2) × U (1) rotational symmetry.

The proof of this follows immediately from Theorem 6.5 with Λ = 0 and ξ = 1. In this case the constraint on the parameters factorises to give two branches of possible solutions (a) √ -- k = − 2q ∕ 3 or (b) -- -- Δ2 (k + 2√ 3q) = 4q2(k − 2q∕√ 3)∕3. Case (a) gives two solutions. If Δ ⁄= 0 it must have ∼ 3 H = S and corresponds to the BMPV solution (i) (for q → 0 this reduces to 3 AdS2 × S), whereas if Δ = 0 it is the AdS3 × S2 solution with H ∼= S1 × S2. Case (b) also gives two solutions. If k = 2q∕√3-- then Δ = 0, which gives AdS × S2 3 with H ∼ S1 × S2 =, otherwise we get solution (ii) with ∼ 3 H = S. Note that solution (ii) reduces to the vacuum case as Δ, q → 0.

The black-hole solution (ii) may be constructed as follows. A charged generalisation of the MP black hole can be generated in minimal supergravity [50]. Generically, the extremal limit depends on 3-parameters with two independent angular momenta J1,J2 and 2 ℝ × U (1) symmetry. Setting |J1| = |J2| gives two distinct branches of 2-parameter extremal black-hole solutions with enhanced SU (2) × U (1) rotational symmetry corresponding to the BMPV solution (i) (which reduces to the RN solution if J = 0) or solution (ii) (which reduces to the vacuum extremal MP black hole in the neutral limit).

It is interesting to note the analogous result for pure Einstein–Maxwell theory:

Corollary 6.3. Any homogeneous near-horizon geometry of Einstein–Maxwell theory is locally isometric to either (i)

( ) ( k cos𝜃d ϕ )2 d 𝜃2 + sin2𝜃d ϕ2 g = − 12k2 + 2q2 r2dv2 + 2dvdr + dψ + 1-2-----2-+ krdv + ---1--2----2---, 2k + 2q 2k + 2q -2- qsin𝜃d-𝜃 ∧-d-ϕ ℱ = ± √3qdr ∧ dv + 1k2 + 2q2 , (122 ) 2
or (ii)
( ) ( )2 2 2 2 g = − Δ2 + 4q2 r2dv2 + 2dvdr + dψ + Δ--cos𝜃dϕ-± √2qrdv + d-𝜃-+-sin-𝜃d-ϕ-, 3 Δ2 + 43q2 3 Δ2 + 43q2 q sin 𝜃d𝜃 ∧ dϕ ℱ = Δdr ∧ dv + ----2---4-2--. (123 ) Δ + 3q

This also follows from application of Theorem 6.5 with Λ = 0 and ξ = 0. Solution (i) for q → 0 reduces to the vacuum solution of Theorem 4.4, whereas for k = 0, it gives the static dyonic 1 2 AdS2 × S × S solution. Solution (ii) for Δ = 0 gives 2 AdS3 × S, whereas for Δ ⁄= 0 it gives a near-horizon geometry with 3 H ∼= S, which for q → 0 is 3 AdS2 × S. A charged rotating black-hole solution to Einstein–Maxwell theory generalising MP is not known explicitly. Hence this corollary could be of use for constructing such an extremal charged rotating black hole with ℝ × SU (2) × U(1) symmetry.

For 2 Λ = − 4∕ℓ < 0 there are even more possibilities, since ˆ λ may be positive, zero, or negative. One then gets near-horizon geometries that generically have 3 H ∼= S ,Nil,SL (2,ℝ ), respectively, equipped with their standard homogeneous metrics. Each of these may have a degenerate limit with H ∼= S1 × S2,T 3,S1 × ℍ2, as occurs in the vacuum and supersymmetric cases. The full space of solutions interpolates between the vacuum case given in Corollary 4.2, and the supersymmetric near-horizon geometries of gauged supergravity of Proposition 5.3. For example, the supersymmetric horizons [120Jump To The Next Citation Point] correspond to √ -- k = − 2 3q and k2 = 9∕ℓ2 with ˆλ = Δ2 − 3ℓ−2. We will not investigate the full space of solutions in detail here.

6.3.3 U (1 )2-rotational symmetry

The classification of near-horizon geometries in D = 5 Einstein–Maxwell–CS theory, under the assumption of 2 U(1) symmetry, turns out to be significantly more complicated than the vacuum case. This is unsurprising; solutions may carry two independent angular momenta, electric charge and dipole/magnetic charge (depending on the spacetime asymptotics). As a result, there are several ways for a black hole to achieve extremality. Furthermore, such horizons may be deformed by background electric fields [153].

In the special case of minimal supergravity one can show:

Proposition 6.4 ([155Jump To The Next Citation Point]). Any near-horizon geometry of minimal supergravity with 2 U (1)-rotational symmetry takes the form of Eqs. (58View Equation) and (62View Equation), where the functional form of Γ (x),Bij(x ),bi(x) can be fully determined in terms of rational functions of x. In particular, Γ (x) is a quadratic function.

The method of proof is discussed in Section 6.4. Although this solves the problem in principle, it turns out that in practice it is very complicated to disentangle the constraints on the constants that specify the solution. Hence an explicit classification of all possible solutions has not yet been obtained, although in principle it is contained in the above result.

We now summarise all known examples of five dimensional non-static near-horizon geometries with non-trivial gauge fields, which arise as near-horizon limits of known black hole or black string solutions. All these examples possess U (1)2 rotational symmetry and hence fall into the class of solutions covered by Theorem 3.5, so the near-horizon metric and field strength (g,ℱ ) take the form of Eqs. (58View Equation) and (62View Equation) respectively. We will divide them by horizon topology.

Spherical topology

Charged Myers–Perry black holes: This asymptotically-flat solution is known explicitly only for minimal supergravity ξ = 1 (in particular it is not known in pure Einstein–Maxwell ξ = 0), since it can be constructed by a solution-generating procedure starting with the vacuum MP solution. It depends on four parameters M, J1,J2, and Q corresponding to the ADM mass, two independent angular momenta and an electric charge. The extremal limit generically depends on three parameters and gives a near-horizon geometry with ∼ 3 H = S.

Charged Kaluza–Klein black holes: The most general known solution to date was found in [204] (see references therein for a list of previously known solutions) and carries a mass, two independent angular momenta, a KK monopole charge, an electric charge and a ‘magnetic charge’20. The extremal limit will generically depend on five-parameters, however, as for the vacuum case the extremal locus must have more than one connected component. These solutions give a large family of near-horizon geometries with 3 H ∼= S.

1 2 S × S topology

Supersymmetric black rings and strings: The asymptotically-flat supersymmetric black ring [64] and the supersymmetric Taub–NUT black ring [65] both have a near-horizon geometry that is locally 2 AdS3 × S. There are also supersymmetric black string solutions with such near-horizon geometries [93, 19].

Dipole black rings: The singly-spinning dipole black ring [69Jump To The Next Citation Point] is a solution to Einstein–Maxwell–CS for all ξ. It is a 3-parameter family with a single angular momentum and dipole charge possessing a 2-parameter extremal limit. The resulting near-horizon geometry with H ∼= S1 × S2 is parameterised by 4-parameters (q,λ, R1,R2 ) with one constraint relating them. Asymptotic flatness of the full black-hole solution imposes one further constraint, although from the viewpoint of the near-horizon geometry, this is strictly an external condition and we will deal with the general case here. The solution is explicitly given by

[ 2 2 ] 2 2 g = Γ (x) − r-dv--+ 2dvdr + ℓΓ-(x)dx-- ℓ2 (1 − x2 ) 2 ( ∘ ------ )2 2 2 2 2 + R1λ(1-+-λ-)H-(x-) dϕ1 + (1-−-λ) 1-−-λrdv + R2q-ω-0(1 −-x-)(dϕ2)2, q(1 − λ)F (x) λR1R2 1 + λ H (x)2 √-- [ ∘ ------ ] -3-- 1-−-q ω0qR2-(1-+-x-) 2 ℱ = 2 d 1 + q H (x ) dϕ , (124 )
where ∘ -q(1−λ) Γ (x) = λ(1+λ)F (x)H (x) with F (x) = 1 + λx, H (x) = 1 − qx and we have also defined the length scale ∘ λ(1+λ)q3- ℓ2 = R22 --1−λ--. The parameters satisfy 0 < λ,q < 1. The local metric induced on spatial cross sections H extends smoothly to a metric on S1 × S2 provided ∘ ---------3- ∘ -------------3 ω0 = F (1 )H (1) = F(− 1)H (− 1).

Charged non-supersymmetric black rings: The dipole ring solution admits a three-parameter charged generalisation with one independent angular momentum and electric and dipole charges [63] (the removal of Dirac-Misner string singularities imposes an additional constraint, so this solution has the same number of parameters as that of [69]). The charged black ring has a two-parameter extremal limit with a corresponding two-parameter near-horizon geometry. As in the above case, at the level of near-horizon geometries there is an additional independent parameter corresponding to the arbitrary size of the radius of the 1 S.

Electro-magnetic Kerr black strings: Black string solutions have been constructed carrying five independent charges: mass M, linear momentum P along the S1 of the string, angular momentum J along the internal 2 S, as well as electric Qe and magnetic charge Qm [49]. These solutions admit a four parameter extremal limit, which in turn give rise to a five-parameter family of non-static near horizon geometries (the additional parameter is the radius of the S1 at spatial infinity) [155Jump To The Next Citation Point].

For simplicity we will restrict our attention to the solutions with P = Qe = 0. The resulting near-horizon solution is parameterised by (a,cβ,sβ ) with 2 2 cβ − sβ = 1 and corresponds to an extremal string with non-zero magnetic charge and internal angular momentum:

[ 2 2 ] 2 2 4 4 2 2 2 r-dv-- ℓ-Γ (x-)dx- 4a-(cβ-+-sβ)-(1 −-x-) 2 g = Γ (x) − ℓ2 + 2dvdr + (1 − x2) + ℓ2Γ (x ) ω [ 1 23 3 2 2 2 2 ]2 2 Rd-ϕ-- 8a-cβs-β 2a-cβs-β(cβ-+-sβ)(1-−-x-) + a a − ℓ4 rdv + ℓ2Γ (x) ω , √ --3 4 4 [ ] 2--3a-sβcβ-(cβ-+-sβ)- --x-- ℱ = ℓ2 d Γ (x)ω , (125 )
where we have defined the one-form 2 -(c2β+s2β)- ω = d ϕ − ℓ2(c4β+s4β)rdv, the function a2 2 2 2 Γ (x ) = ℓ2 (1 + x + 4cβsβ ) and the length scale ℓ2 = 2a2(c4 + s4) β β. The induced metric on cross sections of the horizon H extends smoothly to a cohomogeneity-1 metric on 1 2 S × S.

Although the two solutions (124View Equation) and (125View Equation) share many features, it is important to emphasise that only the former is known to correspond to the near-horizon geometry of an asymptotically-flat black ring. It is conjectured that there exists a general black-ring solution to minimal supergravity that carries mass, two angular momenta, electric and dipole charges all independently. Hence there should exist corresponding 4-parameter families of extremal black rings. [155Jump To The Next Citation Point] discusses the possibility that the tensionless Kerr-string solution is the near-horizon geometry of these yet-to-be-constructed black rings.

6.4 Theories with hidden symmetry

Consider U(1)D −3-invariant solutions of a general theory of the form (59View Equation). One may represent such solutions by a three-dimensional metric h μν and a set of scalar fields ΦM (potentials) all defined on a three-dimensional manifold, see, e.g., [155Jump To The Next Citation Point]. Equivalently, such solutions can be derived from the field equations of a three-dimensional theory of gravity coupled to a scalar harmonic map whose target manifold is parameterised by the ΦM with metric GMN (Φ ) determined by the specific theory. In certain theories of special interest, the scalar manifold is a symmetric space G ∕K equipped with the bi-invariant metric

1 GMN (Φ)dΦM dΦN = ----Tr(Φ −1dΦ )2, (126 ) 4m
where Φ is a coset representative of G∕K and m is a normalisation constant dependent on the theory. Then the theory is equivalent to a three-dimensional theory of gravity coupled to a non-linear sigma model with target space G ∕K.

Consider near-horizon geometries with D−3 U (1) isometry in such theories. It can be shown [162Jump To The Next Citation Point, 135Jump To The Next Citation Point, 155Jump To The Next Citation Point] that the classification problem reduces to an ODE on the orbit space D−3 H ∕U (1) for the ΦM, while hμν is completely determined. We will assume non-toroidal horizon topology so the orbit space is an interval, which without loss of generality we take to be [− 1,1] parameterised by the coordinate x. The ODE is the equation of motion for a non-linear sigma model defined on this interval:

d [ dΦ ] --- (1 − x2 )Φ −1--- = 0, (127 ) dx dx
where the a coset representative Φ depends only on x. It is straightforward to integrate this matrix equation and completely solve for the scalar fields M Φ, which can then be used to reconstruct the full D-dimensional solution. Hence in principle one has the full functional form of the solution. However, in practice, reconstructing the near-horizon data is hindered by the non-linearity of the scalar metric.

The most notable example which can be treated in the above formalism is vacuum D-dimensional gravity for which G∕K = SL (D − 2,ℝ )∕SO (D − 2). The classification problem for near-horizon geometries has been completely solved using this approach [135], as discussed earlier in Section 4.5.1.

Four-dimensional Einstein–Maxwell also possesses such a structure where the coset is now G ∕K = SU (2,1 )∕SU (2), although the near-horizon classification discussed earlier in Section 6.2 does not exploit this fact.

It turns out that D = 5 minimal supergravity also has a non-linear sigma model structure with G ∕K = G ∕SO (4 ) 2,2 (G 2,2 refers to the split real form of the exceptional Lie group G 2). The classification of near-horizon geometries in this case was analysed in [155Jump To The Next Citation Point] using the hidden symmetry and some partial results were obtained, see Proposition 6.4.

It is clear that this method has wider applicability. It would be interesting to use it to classify near-horizon geometries in other theories possessing such hidden symmetry.

We note that near-horizon geometries in this class extremise the energy functional of the harmonic map D −3 Φ : H ∕U (1 ) → G ∕K, with boundary conditions chosen such that the corresponding near-horizon data is smooth. Explicitly, this functional is given by

∫ [ ] 1 2 -1-- −1 2 --2---- E [Φ ] = (1 − x )4m Tr(Φ ∂xΦ ) − 1 − x2 dx (128 ) −1
and from [155] it can be deduced this vanishes on such extrema. For vacuum gravity, for which m = 1, it was proved that E[Φ ] ≥ 0 with equality if and only if Φ corresponds to a near-horizon geometry, i.e., near-horizon geometries are global minimisers of this functional [1, 132Jump To The Next Citation Point]. This result has also been demonstrated for four dimensional Einstein–Maxwell theory [87Jump To The Next Citation Point] and D = 4, 5 Einstein–Maxwell-dilaton theory [210Jump To The Next Citation Point, 211Jump To The Next Citation Point]. It would be interesting if this result could be generalised to other theories with hidden symmetry such as minimal supergravity.

6.5 Non-Abelian gauge fields

Much less work has been done on classifying extremal black holes and their near-horizon geometries coupled to non-Abelian gauge fields.

The simplest setup for this is four-dimensional Einstein–Yang–Mills theory. As is well known, the standard four dimensional black-hole uniqueness theorems fail in this case (at least in the non-extremal case), for a review, see [205]. Nevertheless, near-horizon geometry uniqueness theorems analogous to the Einstein–Maxwell case have recently been established for this theory.

Static near-horizon geometries in this theory have been completely classified.

Theorem 6.6 ([164Jump To The Next Citation Point]). Consider D = 4 Einstein–Yang–Mills-Λ theory with a compact semi-simple gauge group G. Any static near-horizon geometry with compact horizon cross section is given by: 2 AdS2 × S if Λ ≥ 0; AdS2 × H where H is one of 2 2 S ,T ,Σg if Λ < 0.

The proof employs the same method as in Einstein–Maxwell theory. However, it should be noted that the Yang–Mills field need not be that of the Abelian embedded solution. The horizon gauge field may be any Yang–Mills connection on 2 S, or on the higher genus surface as appropriate, with a gauge group G (if there is a non-zero electric field E the gauge group G is broken to the centraliser of E). The moduli space of such connections have been previously classified [14]. Hence, unless the gauge group is SU (2 ), one may have genuinely non-Abelian solutions. It should be noted that static near-horizon geometries have been previously considered in Einstein–Yang–Mills–Higgs under certain restrictive assumptions [25].

Non-static near-horizon geometries have also been classified under the assumption of axisymmetry.

Theorem 6.7 ([164]). Any axisymmetric non-static near-horizon geometry with compact horizon cross section, in D = 4 Einstein–Yang–Mills-Λ with a compact semi-simple gauge group, must be given by the near-horizon geometry of an Abelian embedded extreme Kerr–Newman-Λ black hole.

The proof of this actually requires new ingredients as compared to the Einstein–Maxwell theory. The AdS2-symmetry enhancement theorems discussed in Section 3.2 do not apply in the presence of a non-Abelian gauge field. Nevertheless, assuming the horizon cross sections are of S2 topology allows one to use a global argument to show the symmetry enhancement phenomenon still occurs. This implies the solution is effectively Abelian and allows one to avoid finding the general solution to the ODEs that result from the reduction of the Einstein–Yang–Mills equations. One can also rule out toroidal horizon cross sections, hence giving a complete classification of horizons with a U (1)-symmetry.

Interestingly, Einstein–Yang–Mills theory with a negative cosmological constant is a consistent truncation of the bosonic sector of D = 11 supergravity on a squashed 7 S [188]. It would be interesting if near-horizon classification results could be obtained in more general theories with non-Abelian Yang–Mills fields, such as the full 𝒩 = 8, D = 4, SO (8)-gauged supergravity that arises as a truncation of D = 11 supergravity on S7.

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