1.2 Gauge fields as Kaluza–Klein vectors

Since the work of Kaluza and Klein, one can conceive that our U (1) electromagnetic gauge field could originate from a Kaluza–Klein vector of a higher-dimensional spacetime of the form ℳ4 × X, where ℳ4 is our spacetime, X is compact and contains at least a U (1 ) cycle (the total manifold might not necessarily be a direct product). Experimental constraints on such scenarios can be set from bounds on the deviation of Newton’s law at small scales [197, 2].

If our U (1) electromagnetic gauge field can be understood as a Kaluza–Klein vector, it turns out that it is possible to account for the entropy of the Reissner–Nordström black hole in essentially the same way as for the Kerr black hole [159Jump To The Next Citation Point]. This mainly follows from the fact that the electric charge becomes an angular momentum J2 = Q in the higher-dimensional spacetime, which is on the same footing as the four-dimensional angular momentum J = J 1 lifted in the higher-dimensional spacetime.

We will assume throughout this review that the U (1) electromagnetic gauge field can be promoted as a Kaluza–Klein vector.

As far as the logic goes, this assumption will not be required for any reasoning in Section 2, even though it will help to understand striking similarities between the effects of rotation and electric charge. The assumption will be a crucial input in order to formulate the Reissner–Nordström/CFT correspondence and its generalizations in Section 4 and further on. This assumption is not required for the Kerr/CFT correspondence and its (extremal or non-extremal) extensions, which are exclusively based on the U (1) axial symmetry of spinning black holes.

In order to make this idea more precise, it is important to study simple embeddings of the U (1) gauge field in higher-dimensional spacetimes as toy models for a realistic embedding. In asymptotically-flat spacetimes, let us introduce a fifth compact dimension χ ∼ χ + 2πR χ, where 2πR χ is the length of the U (1) Kaluza–Klein circle and let us define

ds2 = ds2 + (d χ + A)2. (2 ) (4)
The metric (2View Equation) does not obey five-dimensional Einstein’s equations unless the metric is complemented by matter fields. One simple choice consists of adding a U (1) gauge field A (5), whose field strength is defined as
√ -- --3- dA (5) = 2 ⋆(4) F, (3 )
where ⋆(4) is the four-dimensional Hodge dual. The five-dimensional metric and gauge field are then solutions to the five-dimensional Einstein–Maxwell–Chern–Simons theory, as reviewed, e.g., in [185].

These considerations can also be applied to black holes in anti-de Sitter spacetimes. However, the situation is more intricate because no consistent Kaluza–Klein reduction from five dimensions can give rise to the four-dimensional Einstein–Maxwell theory with cosmological constant [204Jump To The Next Citation Point]. As a consequence, the four-dimensional Kerr–Newman–AdS black hole cannot be lifted to a solution of any five-dimensional theory. Rather, embeddings in eleven-dimensional supergravity exist, which are obtained by adding a compact seven-sphere [69, 109].

Therefore, in order to review the arguments for the Reissner–Nordström/CFT correspondence and its generalizations, it is necessary to discuss five-dimensional gravity coupled to matter fields. We will limit our arguments to the action (1View Equation) possibly supplemented by the Chern–Simons terms

∫ ---1--- 5 I J K SCS = 16πG d xCIJK A ∧ F ∧ F , (4 ) 5
where CIJK = C(IJK) are constants. This theory will suffice to discuss in detail the embedding (2View Equation) – (3View Equation) since the five-dimensional Einstein–Maxwell–Chern–Simons theory falls into that class of theories. We will not discuss the supergravities required to embed AdS–Einstein–Maxwell theory.

Let us finally emphasize that even though the scale R χ of the Kaluza–Klein direction is arbitrary as far as it allows one to perform the uplift (2View Equation), it is constrained by matter field couplings. For example, let us consider the toy model of a probe charged massive scalar field ϕ (x) of charge qe in four dimensions, which is minimally coupled to the gauge field. The wave equation reads as

1 (√ --- αβ ) 2 √----D α − gg Dβϕ (x) + μ ϕ(x) = 0, (5 ) − g
where the derivative is defined as D α = ∂α − iqeA α. This wave equation is reproduced from a five-dimensional scalar field ϕ (x,χ ) (5d) probing the five-dimensional metric (2View Equation), if one takes
iqeχ ϕ(5d)(x,χ ) = ϕ(x)e , (6 )
and if the five-dimensional mass is equal to μ2 = μ2 + q2 (5d) e. However, the five-dimensional scalar is multivalued on the circle χ unless
qeR χ ∈ ℕ. (7 )
This toy model illustrates that the scale R χ can be constrained from consistent couplings with matter. We will use this quantization condition in Section 6.4.

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