The classification of near-horizon geometries in 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 .
The method parallels the vacuum case in Section 4.2 closely. As in that case the near-horizon metric data reads and . Since cross sections of the horizon are one-dimensional, the Maxwell 2-form induced on must vanish identically. Hence, the most general near-horizon Maxwell field (23) in three dimensions is . It is straightforward to show that the 3D Maxwell equation , where is the Hodge dual with respect to the spacetime metric, is equivalent to the following equation on :17) and (18) are simply
Theorem 6.1. Consider a near-horizon geometry with a compact horizon cross section in Einstein–Maxwell- theory. If the near-horizon geometry must be either with a constant AdS2 Maxwell field, or the quotient of AdS3 Eq. (73) with a vanishing Maxwell field. If the only solution is the trivial flat near-horizon geometry . If there are no solutions.
Proof: Rather that solving the above ODEs we may use a global argument. Compactness requires to be a periodic coordinate on and since are globally defined they must be periodic functions of . For simply integrate Eq. (100) over to find that the only solution is the trivial flat one and . For we may argue as follows. Multiply Eq. (100) by and integrate over to obtain99). Hence must be a constant and . Equation (100) then implies is also a constant. We deduce the only possible solutions are , or . The former gives a near-horizon geometry 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- theory either has vanishing charge or angular momentum. The 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 . Charged rotating black holes were first obtained within a wide class of stationary and axisymmetric solutions to Einstein–Maxwell- theory , 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 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 .
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 to the spacetime action, resulting in a topologically-massive gauge theory. This only modifies the Maxwell equation:
Theorem 6.2. Consider a near-horizon geometry with a cross section in Einstein–Maxwell- theory with a Chern–Simons term and mass . If the functions and are constant and the near-horizon geometry is the homogeneous -bundle over AdS 2 (106). If the only solution is the trivial flat near-horizon geometry . If there are no solutions.
For the proof of this is identical to the Einstein–Maxwell case above. For one can also use the same argument as the Einstein–Maxwell case. Using the horizon equation (100) and Maxwell equation (104) one can show3 solution. If then and . The rest of the near-horizon data is given by and hence the near-horizon geometry is 3 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 (106), which is sometimes referred to as “warped AdS3”. Examples of charged rotating black-hole solutions in this theory have been found .
The spacetime Einstein–Maxwell equations are Eqs. (15), (22) with and , where is the Hodge dual with respect to the spacetime metric, and the Bianchi identity . The near-horizon Maxwell field is given by (23). The near-horizon geometry Einstein–Maxwell equations are given by Eqs. (17) and (18), where
Theorem 6.3 ([43, 154]). Consider a static near-horizon geometry in Einstein–Maxwell- theory, with compact horizon cross section . For it must be . For it must be where is one of the constant curvature surfaces .
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
Non-static near-horizon geometries are not fully classified, except under the additional assumption of axisymmetry.
Theorem 6.4 ([163, 154]). Any axisymmetric, non-static near-horizon geometry in Einstein–Maxwell- theory, with a compact horizon cross section, must be given by the near-horizon geometry of an extremal Kerr–Newman- black hole.
 solved the in the context of isolated degenerate horizons, where the same equations on arise.  solved the case with . Note that the horizon topology theorem excludes the possibility of toroidal horizon cross sections for .  also excluded the possibility of a toroidal horizon cross section if 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.
23). The corresponding near-horizon geometry equations are given by Eqs. (17) and (18) and
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 and , are qualitatively different to purely magnetic solutions, which correspond to and .
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 .
As in other five dimensional near-horizon geometry classifications, the assumption of rotational symmetry proves to be useful. Static near-horizon geometries in this class in general are either warped products of AdS2 and , 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 .
Proposition 6.1. Any static AdS3 near-horizon geometry with a -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 .
This classifies a subset of purely magnetic geometries. By combining the results of  together with Proposition 6.4 of , it can be deduced that for there are no purely magnetic AdS2 geometries; therefore, with these symmetries, one has a complete classification of purely magnetic geometries.
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 -rotational symmetry and compact cross section. It must be given by either , or a warped product of AdS 2 and an inhomogeneous .
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 , in Einstein–Maxwell–Chern–Simons theory with coupling , with non-trivial electric and magnetic fields, is a direct product of , where the metric on is not Einstein.
Explicit examples for were also found, which all have with -rotational symmetry. However, we should emphasise that no examples are known for minimal supergravity . 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 -rotational symmetry. For 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 and it is conjectured there are no others.
The classification of homogeneous near-horizon geometries can be completely solved, even including a cosmological constant . 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 is a homogeneous space, i.e., admits a transitive isometry group , such that the rest of the near-horizon data are invariant under . 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 and are Killing and the functions , must be constants. Thus, the horizon Einstein equations (17), (18), (113), (114) and Maxwell equation (115) simplify.
Firstly, note that if and vanish identically then is Einstein , so is a constant curvature space (the latter two can only occur if ). This family includes the static near-horizon geometry .
Now consider the case where at least one of and is non-vanishing. By contracting the Maxwell equation (115) with one can show that and hence by the Cauchy–Schwarz inequality and must be parallel as long as they are both non-zero. Thus, if one of is non-vanishing, we can write and for some constants where is a unit normalised Killing vector field. The near-horizon equations now reduce to
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 . The above result contains a number of special cases of interest, which we now elaborate upon. Firstly, note that reduces to the vacuum case, see Theorem 4.4.
Before discussing the general case consider , so the near-horizon geometry is static, which connects to Section 6.3.1. The constraint on the parameters is and hence, if one must have . For the connection is trivial and hence the near-horizon geometry is locally isometric to the dyonic solution. For we get examples of the geometries in Proposition (6.3), where with its standard homogeneous metric.
Now we consider in generality so at least one of is non-zero, in which case and so is the round . The horizon is then either with its homogeneous metric or , depending on whether the connection is non-trivial or not, respectively. Notably we have:
Corollary 6.2. Any homogeneous near-horizon geometry of minimal supergravity is locally isometric to , or the near-horizon limit of either (i) the BMPV black hole (including ), or (ii) an extremal nonsupersymmetric charged black hole with rotational symmetry.
The proof of this follows immediately from Theorem 6.5 with and . In this case the constraint on the parameters factorises to give two branches of possible solutions (a) or (b) . Case (a) gives two solutions. If it must have and corresponds to the BMPV solution (i) (for this reduces to ), whereas if it is the solution with . Case (b) also gives two solutions. If then , which gives with , otherwise we get solution (ii) with . Note that solution (ii) reduces to the vacuum case as .
The black-hole solution (ii) may be constructed as follows. A charged generalisation of the MP black hole can be generated in minimal supergravity . Generically, the extremal limit depends on 3-parameters with two independent angular momenta and symmetry. Setting gives two distinct branches of 2-parameter extremal black-hole solutions with enhanced rotational symmetry corresponding to the BMPV solution (i) (which reduces to the RN solution if ) 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:
This also follows from application of Theorem 6.5 with and . Solution (i) for reduces to the vacuum solution of Theorem 4.4, whereas for , it gives the static dyonic solution. Solution (ii) for gives , whereas for it gives a near-horizon geometry with , which for is . 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 symmetry.
For there are even more possibilities, since may be positive, zero, or negative. One then gets near-horizon geometries that generically have , respectively, equipped with their standard homogeneous metrics. Each of these may have a degenerate limit with , 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  correspond to and with . We will not investigate the full space of solutions in detail here.
The classification of near-horizon geometries in Einstein–Maxwell–CS theory, under the assumption of 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 .
In the special case of minimal supergravity one can show:
Proposition 6.4 (). Any near-horizon geometry of minimal supergravity with -rotational symmetry takes the form of Eqs. (58) and (62), where the functional form of can be fully determined in terms of rational functions of . In particular, 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 rotational symmetry and hence fall into the class of solutions covered by Theorem 3.5, so the near-horizon metric and field strength take the form of Eqs. (58) and (62) respectively. We will divide them by horizon topology.
Charged Myers–Perry black holes: This asymptotically-flat solution is known explicitly only for minimal supergravity (in particular it is not known in pure Einstein–Maxwell ), since it can be constructed by a solution-generating procedure starting with the vacuum MP solution. It depends on four parameters , and 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 .
Charged Kaluza–Klein black holes: The most general known solution to date was found in  (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 .
Supersymmetric black rings and strings: The asymptotically-flat supersymmetric black ring  and the supersymmetric Taub–NUT black ring  both have a near-horizon geometry that is locally . There are also supersymmetric black string solutions with such near-horizon geometries [93, 19].
Dipole black rings: The singly-spinning dipole black ring  is a solution to Einstein–Maxwell–CS for all . 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 is parameterised by 4-parameters 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
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  (the removal of Dirac-Misner string singularities imposes an additional constraint, so this solution has the same number of parameters as that of ). 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 .
Electro-magnetic Kerr black strings: Black string solutions have been constructed carrying five independent charges: mass , linear momentum along the of the string, angular momentum along the internal , as well as electric and magnetic charge . 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 at spatial infinity) .
For simplicity we will restrict our attention to the solutions with . The resulting near-horizon solution is parameterised by with and corresponds to an extremal string with non-zero magnetic charge and internal angular momentum:
Although the two solutions (124) and (125) 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.  discusses the possibility that the tensionless Kerr-string solution is the near-horizon geometry of these yet-to-be-constructed black rings.
Consider -invariant solutions of a general theory of the form (59). One may represent such solutions by a three-dimensional metric and a set of scalar fields (potentials) all defined on a three-dimensional manifold, see, e.g., . 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 with metric determined by the specific theory. In certain theories of special interest, the scalar manifold is a symmetric space equipped with the bi-invariant metric
Consider near-horizon geometries with isometry in such theories. It can be shown [162, 135, 155] that the classification problem reduces to an ODE on the orbit space for the , while 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 parameterised by the coordinate . The ODE is the equation of motion for a non-linear sigma model defined on this interval:
The most notable example which can be treated in the above formalism is vacuum -dimensional gravity for which . The classification problem for near-horizon geometries has been completely solved using this approach , as discussed earlier in Section 4.5.1.
Four-dimensional Einstein–Maxwell also possesses such a structure where the coset is now , although the near-horizon classification discussed earlier in Section 6.2 does not exploit this fact.
It turns out that minimal supergravity also has a non-linear sigma model structure with ( refers to the split real form of the exceptional Lie group ). The classification of near-horizon geometries in this case was analysed in  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 , with boundary conditions chosen such that the corresponding near-horizon data is smooth. Explicitly, this functional is given by it can be deduced this vanishes on such extrema. For vacuum gravity, for which , it was proved that with equality if and only if corresponds to a near-horizon geometry, i.e., near-horizon geometries are global minimisers of this functional [1, 132]. This result has also been demonstrated for four dimensional Einstein–Maxwell theory  and Einstein–Maxwell-dilaton theory [210, 211]. It would be interesting if this result could be generalised to other theories with hidden symmetry such as minimal supergravity.
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 . 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 (). Consider Einstein–Yang–Mills- theory with a compact semi-simple gauge group . Any static near-horizon geometry with compact horizon cross section is given by: if ; where is one of if .
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 , or on the higher genus surface as appropriate, with a gauge group (if there is a non-zero electric field the gauge group is broken to the centraliser of ). The moduli space of such connections have been previously classified . Hence, unless the gauge group is , 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 .
Non-static near-horizon geometries have also been classified under the assumption of axisymmetry.
Theorem 6.7 (). Any axisymmetric non-static near-horizon geometry with compact horizon cross section, in Einstein–Yang–Mills- with a compact semi-simple gauge group, must be given by the near-horizon geometry of an Abelian embedded extreme Kerr–Newman- 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 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 -symmetry.
Interestingly, Einstein–Yang–Mills theory with a negative cosmological constant is a consistent truncation of the bosonic sector of supergravity on a squashed . 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 , , -gauged supergravity that arises as a truncation of supergravity on .