To gain further information about the supergravity model we study it is instructive to calculate the spectrum of linearized fluctuations of all supergravity fields around the maxi-mally supersymmetric AdS4vacuum solution in the theory. In appendixCwe show in some detail how to calculate the masses for all bosonic fluctuations of the supergravity fields.
Similar spectrum calculations in AdS4 for non-supersymmetric higher-derivative gravita-tional theories were done in [19, 56]. It is important to note that in the presence of the four-derivative terms in the Lagrangian some of the auxiliary fields used in the conformal supergravity construction outlined in section2 acquire kinetic term and lead to dynamical excitations. Since the AdS4 background is supersymmetric, the linearized bosonic fluctua-tions in our model should combine with the linearized fluctuafluctua-tions for the fermionic fields to form supergravity multiplets. While we have not explicitly computed the masses for all fermionic fluctuations, we have indeed confirmed that the bosonic fluctuations nicely fit into two supersymmetric multiplets. To discuss these results further it is convenient to use AdS/CFT and map all linearized supergravity modes around AdS4 to operators in the dual 3d N = 2 SCFT. In the language of 3d N = 2 superconformal representation theory, see [57] for the notation we use and a comprehensive review, we find that the supergravity modes organize into one short and one long multiplet.
The short multiplet is denoted by A1A¯1[1](0)2 in [57] and is simply the stress-energy multiplet of the SCFT, see table 1. The supergravity modes in this multiplet are the massless metric and graviphoton excitations as well as the corresponding gravitino modes.
The long multiplet is denoted by LL¯[1](0)δ in [57], where [j] indicates the Lorentz spin of the superconformal primary, the subscript is its conformal dimension and the superscript
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s 2 32 32 1 1 1 1 1 12 12 0
∆ δ+ 1 δ+32 δ+12 δ+ 2 δ+ 1 δ+ 1 δ+ 1 δ δ+32 δ+12 δ+ 1
r 0 ±1 ±1 0 ±2 0 0 0 ±1 ±1 0
Table 2. Spin s, conformal dimension ∆, and R-charge r of the operators in the long LL¯[1](0)δ multiplet. The superconformal primary operator is indicated in red.
is its R-charge. The operators comprising this multiplet are summarized in table 2. The conformal dimensions of the operators in this multiplet are not determined in terms of theirr charge and are given in terms of the quantity
δ= 1 2 +1
2 s
1 + L2
8πGN(c1−c2). (4.13) It is well-known that gravitational theories with higher-curvature corrections often have ghosts [58], see [59] for a review and further references. These ghosts are manifested by the fact that the massive spin-2 mode arising from the metric has negative energy due to a wrong sign kinetic term. Our model is no exception to this general feature of four-derivative gravitational theories and also suffers from this ghost problem. As can be seen from (C.22), massive spin-2 modes with negative energy will be present for any value of the HD coefficients as long as c16=c2. In addition to this, our four-derivative supergravity model can suffer from instabilities associated to the violations of the BF bound by the massive scalar mode. Violations of the BF bound are associated with complex conformal dimensions in the dual CFT. Using the form of the conformal dimension in (4.13) we find that the BF bound is violated when
L2
8πGN(c1−c2) <−1. (4.14) In a specific microscopic model given by the ABJM theory arising from N M2-branes we find that in the large N limit the 4d supergravity parameters scale as L2/GN ∼ N3/2 and c1,2 ∼ N1/2. Moreover we find that c1 = 3c2 < 0, see the discussion around (7.50), which indeed leads to a violation of the BF bound according to (4.14). More generally, we expect that in any supergravity model with a consistent higher-derivative expansion L2/GN is parametrically larger than c1,2 and thus the BF bound will be violated for all models withc1 < c2.
Violations of the BF bound and the presence of propagating ghost modes will both generally lead to violations of unitarity in the theory. In the case at hand it is possible that the presence of both phenomena simultaneously could somehow be compatible with unitarity, though concluding this would require a much more thorough exploration of the dynamics of the theory. If unitarity is violated, it should be viewed as an artefact of the four-derivative approximation of the gravitational theory that we have used here; the UV completion of the 4d N = 2 supergravity theory by string or M-theory is expected to restore unitarity.
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5 Black hole thermodynamics
We now turn our attention to studying black holes. In particular, we study how the HD terms in our gravitational theory affect the thermodynamic properties of a wide class of AdS4 black holes. HD effects on black hole thermodynamics in anti-de Sitter spacetimes have been investigated previously in the literature, but this analysis can typically only be carried out perturbatively, since the black hole solutions themselves are generically modified by HD effects. The theory we presented in section2, by contrast, has the unique property that any solution to the two-derivative theory is also a solution to the four-derivative theory. This guarantees that black hole solutions will not be modified by HD effects, thus allowing us to do an in-depth and complete analysis of HD modifications to the black hole thermodynamic quantities.
In order to look at proper black hole solutions, we first need to translate the actions presented in section2from Euclidean into Lorentzian signature. In particular, as discussed in section 2.2.1, this is accomplished simply by a Wick-rotation of one of the Euclidean tangent space coordinates into a Lorentzian time direction. After implementing this, we can present the bulk Lorentzian gravitational action Sbulk succinctly as follows:12
Sbulk =−S2∂+ (c2−c1)SW2 −c2SGB, (5.1) whereS2∂,SW2, and SGB, are defined in (3.2), (3.3), and (3.4), respectively
As discussed in section 3.1, these bulk actions must be supplemented with particular boundary counterterm actions in order to regularize divergences. We studied these bound-ary counterterms in detail for the Euclidean theory, and a similar analysis holds for the Lorentzian bulk theory (5.1). The end result is that each of the three pieces of the action have their own associated counterterm, which read:
S2∂CT =− 1 8πGN
Z
d3x√
−h
K−L 2R − 2
L
, SGBCT = 4Z d3x√
−h (J −2GµνKµν), SWCT2 =SGBCT+64πGN
L2 S2∂CT .
(5.2)
The full counterterm action is therefore given by
SCT =−S2∂CT+ (c2−c1)SWCT2 −c2SGBCT =1 +64πGN(c2−c1) L2
S2∂CT−c1SGBCT . (5.3) The full Lorentzian action for our higher-derivative theory, including both the bulk pieces and the boundary counterterm pieces, is given simply by the sum of the bulk and boundary actions:
SHD =Sbulk+SCT . (5.4)
This action SHD is our starting point for analyzing how four-derivative terms affect black hole thermodynamics in our supergravity theory. We now consider an arbitrary
12Note that the overall minus sign on each term is due to the action picking up a sign when going from Euclidean to Lorentzian signature, see (2.48).
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stationary black hole solution to the original two-derivative theory, equipped with at least two Killing vectors: one time-like Killing vector associated with time-translations, and one space-like Killing vector associated with azimuthal rotations. Correspondingly, the two-derivative black hole will have conserved total mass and azimuthal angular momentum, which we denote by M0 and J0, respectively. Additionally, we allow the solution to be charged under the graviphoton gauge field with electric chargeQ0 and magnetic chargeP0. We emphasize that the zero subscript indicates that these conserved quantities are all read off in the original two-derivative theory.
In the following section, we show how these charges as well as other thermodynamic properties of the black hole are modified when we consider it as a solution to the four-derivative theory at hand. We then go on to show how the higher-four-derivative corrections to thermodynamic quantities are compatible with the quantum statistical relation, before moving on to discuss the relations of these results to some recent proposals related to the weak gravity conjecture.