1.1 Classes of effective field theories

The Kerr/CFT correspondence is an effective description of rotating black holes with an “infrared” CFT. Embedding this correspondence in string theory has the potential to give important clues on the nature of the dual field theory. Efforts in that direction include [225Jump To The Next Citation Point, 22Jump To The Next Citation Point, 157Jump To The Next Citation Point, 98Jump To The Next Citation Point, 23Jump To The Next Citation Point, 116Jump To The Next Citation Point, 31Jump To The Next Citation Point, 243Jump To The Next Citation Point, 246Jump To The Next Citation Point, 115Jump To The Next Citation Point, 130Jump To The Next Citation Point]. However, the details of particular CFTs are irrelevant for the description of astrophysical black holes, as long as we don’t have a reasonable control of all realistic embeddings of the standard model of particle physics and cosmology in string theory. Despite active research in this area, see, e.g., [153, 117, 217], a precise description of how our universe fits in to the landscape of string theory is currently out-of-reach.

In an effective field theory approach, one concentrates on long-range interactions, which are described by the physical Einstein–Maxwell theory. However, it is instructive in testing ideas about quantum gravity models of black holes to embed our familiar Einstein–Maxwell theory into the larger framework of supergravity and study the generic properties of rotating black holes as toy models for a physical string embedding of the Kerr–Newman black hole.

Another independent motivation comes from the AdS/CFT correspondence [205, 265Jump To The Next Citation Point]. Black holes in anti-de Sitter (AdS) spacetime in d + 1 dimensions can be mapped to thermal states in a dual CFT or CFT in d dimensions. Studying AdS black holes then amounts to describing the dynamics of the dual strongly-coupled CFT in the thermal regime. Since this is an important topic, we will discuss in this review the AdS generalizations of the Kerr/CFT correspondence as well. How the Kerr/CFT correspondence fits precisely in the AdS/CFT correspondence in an important open question that will be discussed briefly in Section 7.2.

In this review we will consider the following class of four-dimensional theories,

∫ ( S = ---1-- dDx √ −-g R − 1-f (χ )∂ χA∂ μχB − V (χ) − k (χ )F I F Jμν 16 πG 2 AB μ IJ μν ) +hIJ (χ)𝜖μνρσFμIνF Jρσ , (1 )
possibly supplemented with Planck-suppressed higher-derivative corrections. We focus on the case where fAB (χ) and kIJ(χ) are positive definite and the scalar potential V (χ) is non-positive in (1View Equation). This ensures that matter obeys the usual energy conditions and it covers the case of zero and negative cosmological constant. Some theories of interest contained in the general class (1View Equation) are the Einstein–Maxwell gravity with negative or zero cosmological constant and the bosonic sector of 𝒩 = 8 supergravity. Note that the phenomenology described by the action (1View Equation) is limited by the absence of charged scalars, massive vectors, non-abelian gauge fields and fermions.

The explicit form of the most general single-center spinning–black-hole solution of the theory (1View Equation) is not known;however, see [270, 221] for general ansätze. For Einstein and Einstein–Maxwell theory, the solutions are, of course, the Kerr and Kerr–Newman geometries that were derived about 45 years after the birth of general relativity. For many theories of theoretical interest, e.g., 𝒩 = 8 supergravity, the explicit form of the spinning–black-hole solution is not known, even in a specific U-duality frame (see, e.g., [49] and references therein). However, as we will discuss in Section 2.3, the solution at extremality greatly simplifies in the near-horizon limit due to additional symmetries and takes a universal form for any theory in the class (1View Equation). It is for this reason mainly that we find convenient to discuss theory (1View Equation) in one swoop.

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