arXiv:1005.3952v2 [hep-th] 10 Dec 2010
More on Massive 3D Supergravity
Eric A. Bergshoeff1, Olaf Hohm2, Jan Rosseel1
Ergin Sezgin3and Paul K. Townsend4
1Centre for Theoretical Physics, University of Groningen,
Nijenborgh 4, 9747 AG Groningen, The Netherlands
email: E.A.Bergshoeff@rug.nl, email@example.com
2Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
3George and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, Texas A& M University, College Station, TX 77843, USA
4Department of Applied Mathematics and Theoretical Physics,
Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge, CB3 0WA, U.K.
Completing earlier work on three dimensional (3D) N
curvature-squared terms, we construct the general supergravity extension of ‘cosmo-
logical’ massive gravity theories. In particular, we show that all adS vacua of “new
massive gravity” (NMG) correspond to supersymmetric adS vacua of a “super-NMG”
theory that is perturbatively unitary whenever the corresponding NMG theory is per-
= 1 supergravity with
2 3D supergravity invariants
N = 1 superconformal tensor calculus . . . . . . . . . . . . . . . . . .
2.2A supersymmetric curvature squared action
2.3A new supersymmetric Snaction . . . . . . . . . . . . . . . . . . . . .
9 . . . . . . . . . . . . . . .
3 The general ‘curvature-squared’ model
3.1 Some notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Maximally symmetric vacua . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Review of supersymmetry-preservation conditions . . . . . . . . . . . .
3.5The pp-wave solution revisited . . . . . . . . . . . . . . . . . . . . . . .
4 Models with auxiliary S
4.1 Super-GMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Field equations and vacua . . . . . . . . . . . . . . . . . . . . .
4.1.2Other solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Generalized super-GMG . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Perturbative unitarity of generalized super-NMG
5.1 Quadratic approximation . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Review of Proca and Fierz-Pauli in adS . . . . . . . . . . . . . . . . . .
5.3 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1a = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2a ?= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The local dynamics of Einstein’s general relativity for a three-dimensional spacetime
is trivial because Einstein’s equations imply that the spacetime curvature is zero in
the absence of sources [1–3]. The addition to the standard Einstein-Hilbert (EH)
action of curvature-squared terms leads to non-trivial dynamics but, typically, some
propagated modes have negative energy, implying ghost particles in the quantum theory
and a corresponding loss of unitarity. This is an inevitable feature in four spacetime
dimensions  but it was recently discovered  that ghosts can be avoided in three
dimensions (3D) if (i) the EH term has the ‘wrong’ sign and (ii) the curvature-squared
invariant is constructed from the scalar1
K = RµνRµν−3
where Rµν is the Ricci tensor, and R its trace, for a metric g which we take to have
‘mostly plus’ signature. An equivalent expression is K = GµνSµν, where Gµν is the
Einstein tensor and Sµνthe Schouten tensor (the 2nd order ‘potential’ for the 3rd order
Cotton tensor, which is the 3D analog of the Weyl tensor). The inclusion of this K-
term in the action introduces a mass parameter m and linearizing about the Minkowski
vacuum one finds that two modes of helicities2±2 are propagated, unitarily, with mass
m. This model is now generally referred to as “new massive gravity” (NMG). The
addition of a (parity violating) Lorentz Chern-Simons (LCS) term leads to a model
that propagates the helicity ±2 modes with different masses m± ; this has been
called “general massive gravity” (GMG). The limit of GMG in which m− → ∞ for
fixed m+yields the well-known “topological massive gravity” (TMG) .
All these models have ‘cosmological’ extensions in which a cosmological constant
term is added to the Lagrangian density; we may take this to be −2m2λ times the
volume density, where λ is a dimensionless cosmological parameter. In this context it
is convenient to allow for an arbitrary coefficient σ of the EH term, so the Lagrangian
density for cosmological GMG is
−2λm2+ σR +
where LLCSis the Lorentz-Chern-Simons density. When λ = 0 there is a Minkowski
vacuum in which are propagated two modes, of helicities +2 and −2, and these are
propagated unitarily as long as σ < 0 and m2> 0; for σ = −1 this is the GMG model
described above, with masses m±such that m2= m+m−and µ = m+m−/(m−−m+).
More generally, it is convenient to allow for either sign of m2, in addition to either sign
of σ, because one does not know, a priori, what unitarity will permit in non-Minkowski
vacua. Note, however, that a change in sign of both σ and m2is equivalent to a change
1See also the discussion in [6,7].
2We use “helicity” to mean “relativistic helicity”, i.e. the scalar product of the relativistic 3-
momentum with the Lorentz rotation 3-vector, divided by the mass.
in the overall sign of the µ-independent terms in the action, from which it follows that
the dependence of the field equations on the signs of σ and m2is entirely through
the sign of the product m2σ. The same is true of the space of solutions, in particular
vacuum solutions, although conclusions concerning the unitarity of modes propagated
in a given vacuum will depend on the individual signs of both σ and m2.
All maximally-symmetric vacua of GMG were found in . By definition, such
vacua have the property that
where Λ is the cosmological constant, which is positive for de Sitter (dS) vacua and
negative for anti-de Sitter (adS) vacua, and zero for Minkowski vacua. When curvature-
squared terms are present it is important to distinguish the cosmological constant Λ
from the cosmological parameter λ, which becomes a quadratic function of Λ:
4m4λ = Λ?Λ + 4m2σ?. (1.4)
Observe that zero cosmological term allows non-zero cosmological constant; this is a
typical feature of higher-derivative gravity theories first pointed out in . Of particular
interest in the present context are the adS vacua because of their possible association
with a holographically dual conformal field theory (CFT) via the adS3/CFT2corre-
spondence [10,11]. In this connection, it was shown for NMG in  (completing earlier
partial results ) that the boundary CFT is non-unitary whenever the ‘bulk’ gravity
theory is unitary, and vice-versa, although there is a special case (recently analyzed
in more detail [14–16,43]) in which the central charge vanishes and the bulk massive
gravitons are replaced by bulk massive ‘photons’. This result was disappointing, but
perhaps to be expected in light of the similar difficulty afflicting cosmological TMG (we
refer the reader to [17–20] for up-to-date accounts). An obvious question is whether
this situation is any different in the context of a supergravity extension of GMG.
The off-shell N = 1 ‘graviton’ supermultiplet [21,22] comprises the dreibein (from
which one constructs the metric), the 3D Rarita-Schwinger potential and a scalar
field S. The off-shell supersymmetry transformations are independent of the choice
of action and it is possible to determine the general supersymmetric field configuration
without reference to the action . In particular, a maximally symmetric vacuum is
supersymmetric provided that
which is, of course, possible only when Λ ≤ 0, i.e. for Minkowski or adS vacua. In the
absence of the supergravity cosmological term, which is proportional to S, one does
not need the details of the non-linear theory to see that S = 0 is a solution of the
field equation for S, and hence that there exists a supersymmetric Minkowski vacuum.
The general conditions for unitarity of the linear theory in this vacuum were obtained
in , extending an analysis applied earlier to NMG . Generically, the scalar field
S has a kinetic term, and there is one unitary model of this type: the supersymmetric
extension of the R+R2model. Otherwise, unitarity in the Minkowski vacuum requires
that S be “auxiliary”, in the sense that there is no (∂S)2term, and this is indeed the
case for any supersymmetric extension of GMG, as was established already in  by
adapting earlier general results .
A fully non-linear N = 1 3D supergravity model with generic curvature-squared
terms was constructed in . This was partly motivated by the fact that the non-
linear details are crucial to an understanding of the physics in adS vacua. One question
of obvious interest is whether a given adS vacuum of GMG is supersymmetric in the
context of a supergravity extension of GMG. However, this question was not answered
by the construction of . For the question to make sense one needs a supergravity
model that has (cosmological) GMG as its bosonic truncation after elimination of any
auxiliary fields, and it is implicit in the results of  that, apparently, there is no
such model! There is no difficulty in the absence of curvature-squared terms; the EH
invariant includes an S2term and eliminating S converts the supergravity cosmological
term proportional to S into a standard cosmological term allowing (supersymmetric)
adS vacua. However, the supersymmetric extension of the NMG curvature-squared
scalar K presented in  includes both an S4and an RS2term, so the S equation of
motion is now cubic with R-dependent coefficients. Elimination of S then leads to an
infinite power series in R (irrespective of the ambiguity in the choice of solution to a
cubic equation). This means that none of the supergravity models constructed in 
can really be considered to be a “super-GMG” model, except in the super-TMG limit
(which has been known for some time [26–28]).
This state of affairs suggests that there was some ingredient missing from the anal-
ysis of . In this paper we supply the missing ingredient, and this allows an analysis
of unitarity for massive supergravity theories in adS vacua. The crucial observation is
that there is an additional super-invariant that includes both RS2and S4terms but
no curvature-squared term. This was missed in  because that paper only aimed to
construct a supersymmetric extension of the K and R2invariants; this was achieved
but without the appreciation that the result is not unique. Taking into account the new
super-invariant, one can find a supersymmetrization of the K invariant that includes
an S4term but not an RS2term3. There is a similar new invariant that can contribute
at the same dimension as the LCS term; although it includes an apparently undesir-
able RS term, its effects may cancel against those of the RS2term for special values
of S. This possibility motivates us to start with the most general model containing
no terms of dimension higher than R2but all terms of this dimension or less. This
general supergravity model contains two additional mass parameters as compared with
the model constructed in .
Of most interest are those special cases of the general model for which S can be elim-
inated by an algebraic equation with constant coefficients; in such cases, the bosonic
truncation yields a model of precisely GMG type. As will become clear, there is a
simple subclass of such models, which we refer to collectively as “super-GMG”, that
3Or vice versa. As already observed in , one of the two must be present because S can be entirely
absent only from super-conformal invariants.
is parametrized by the same two mass parameters (m,µ) as GMG itself. It turns out
that not all maximally symmetric vacua of GMG are solutions of super-GMG; some
dS vacua are excluded. In contrast, all adS vacua of GMG continue to be solutions of
super-GMG, although some map to two adS vacua of super-GMG because the latter
are distinguished by their dependence on a cosmological mass parameter M that differs
from (and is non-linearly related to) the cosmological parameter λ of GMG. This result
allows us to address the question of which adS vacua of GMG are supersymmetric so-
lutions of super-GMG. What we find can be summarized by saying that all adS vacua
of GMG are supersymmetric vacua of super-GMG but super-GMG has additional adS
vacua that are not supersymmetric.
Given a vacuum solution, the next step is to determine the quadratic approximation
to the action linearized about it, and thence the nature of the modes propagated, in
particular whether they are physical or ghosts. This settles the issue of perturbative
unitarity. Perturbative unitarity is a necessary condition for unitarity, and may be
sufficient in Minkowski vacua, but it is not sufficient in adS vacua because there are
then non-perturbative excitations to take into account; viz. BTZ black holes. In the
context of TMG there is the, by now well-known, problem that the ‘wrong-sign’ of
the EH term needed for perturbative unitarity implies a negative mass for BTZ black
holes, which translates to a negative central charge of the boundary CFT, although it
has been suggested that a superselection principle may allow the consistent exclusion
of BTZ black holes . In any case, we limit ourselves in this paper to a discussion
of perturbative unitarity.
In the supergravity context an analysis of perturbative unitarity generally requires
an analysis of fermionic field fluctuations, as well as bosonic field fluctuations, but
supersymmetric vacua are exceptional because perturbative unitarity of the bosonic
fluctuations implies perturbative unitarity of the fermionic fluctuations. This feature
of supersymmetric vacua greatly simplifies the analysis, and for this reason we consider
here only supersymmetric vacua. The results of  for the supersymmetric Minkowski
vacuum are still valid for the larger class of supergravity models found here, for reasons
already explained, so that leaves the supersymmetric adS vacua. A complete analysis
of perturbative unitarity for the adS vacua of NMG was presented in . No analogous
analysis for supergravity was attempted in , mainly because of the problems already
mentioned with the model constructed there. Here we shall show how the analysis
of  for perturbative unitarity of NMG extends to the supersymmetric adS vacua of
super-NMG. In particular, we shall show that the super-NMG model is perturbatively
unitary in a supersymmetric adS vacuum whenever the corresponding NMG model is
This paper is organized as follows. In section 2 we determine the new super-
invariants by means of the superconformal approach. These are then used in section 3 to
construct the bosonic truncation of the general curvature-squared supergravity model,
in which context we determine all maximally-symmetric vacua and revisit pp-wave
solutions. In section 4 we specialize to models in which the scalar field S is “auxiliary”
in the sense explained above. It turns out that this condition still allows propagating
fluctuations of S; we refer to those cases in which this does not happen as “generalized
super-GMG” and it is in this context that we find the“super-GMG” models that have
GMG as a bosonic truncation. In section 5 we further specialize to super-NMG, and
its “generalized” extension, determining the conditions for perturbative unitarity in
supersymmetric adS vacua. We present our conclusions, with some further discussion,
in section 6.
2 3D supergravity invariants
In order to determine the bosonic terms of 3D supergravity actions involving curvature
squared terms, it is convenient to combine global supersymmetry with local conformal
symmetry. In the conformal approach one first constructs a superconformal gauge
invariant action involving one or more compensating multiplets, which are then used to
gauge fix the superfluous superconformal symmetries to arrive at a standard Poincar´ e
supergravity invariant. For our purposes, we do not need to perform the complete
conformal programme. We only need to construct globally supersymmetric actions that
can be made invariant under local conformal transformations. This is because global
supersymmetry connects the S-dependent terms in the action to the (possibly higher-
derivative) kinetic terms for the compensating supermultiplet, and local conformal
invariance connects these kinetic terms to the R-dependent terms. After fixing the
compensating fields one ends up with an action containing all relevant R2and S-
dependent terms. The results are consistent with the bosonic truncations of the super-
invariants found in  but, surprisingly, we also find the bosonic truncation of a new
super-invariant. We will begin by recalling the essentials of the conformal procedure
and then show how the bosonic truncations of all relevant super-invariants may be
N = 1 superconformal tensor calculus
One starts with a (globally) supersymmetric action, involving one or more compen-
sating multiplets. These can then be coupled to the conformal supergravity multiplet,
that consists of the dreibein eµaand the gravitino ψµ, with the following transformation
rules under fermionic symmetries:
where ǫ is the ordinary Q-supersymmetry parameter and η is the parameter of the
In the following we will be mainly interested in the bosonic part of the action.
Restricting our attention to the bosonic level, conformal invariance means invariance
under dilatations D and special conformal transformations Ka. Invariance of a La-
grangian under these transformations can be achieved in three steps:
δψµ= Dµ(ω)ǫ + γµη,(2.1)
• In a first step, one ensures that all terms in the Lagrangian have the correct
behavior under global dilatations. Under these scale transformations, a field φ
transforms with a certain weight wφ:
δDφ = wφζφ, (2.2)
where ζ denotes the parameter of the dilatations. Invariance of the action under
global scale transformations is then accomplished when the sum of the weights of
all fields in each term adds up to the space-time dimension d (where derivatives
∂µhave weight one).
• In a second step, one takes care of the invariance of the action under local dilata-
tions by introducing a gauge field bµthat transforms as follows:
δDbµ= ∂µζ . (2.3)
All derivatives can then be turned into dilatation-covariant derivatives. E.g. for
a field φ with weight wφthis implies the following substitution:
∂µφ → Dµφ = (∂µ− wφbµ)φ. (2.4)
In a similar manner one can replace 2φ by a dilatation-covariant expression 2Cφ:
2Cφ = ηabDaDbφ = eaµ?∂µDaφ − (wφ+ 1)bµDaφ + ωµabDbφ?.
• In the last step, one takes care of the invariance under special conformal trans-
formations Ka. This can be achieved by adding terms involving the Ricci tensor
and scalar and by taking into account the following transformation rules under
δKbµ = 2ΛKµ,
δKDaφ = −2wφΛKaφ,
δK2Cφ = −2wφ(DcΛKc)φ + 2(d − 2 − 2wφ)Λc
δKRab = −2ηabDcΛc
δKR = −4(d − 1)DcΛKc,
K− 2(d − 2)DaΛKb,
where ΛKa are the parameters of the special conformal transformations. The
fact that bµtransforms with a shift under the special conformal transformations
means that, writing out all covariant derivatives, one finds that the dilatation
gauge field drops out in any conformal action.
These three steps are enough to ensure invariance under conformal transformations.
In particular, the last step allows one to extract the dependence of the conformal
Lagrangian on the curvatures. By employing a suitable gauge fixing, the (bosonic)
Lagrangian invariant under local super-Poincar´ e transformations can then be extracted.
In order to discuss this gauge fixing in more detail, let us note that in the following
we will always use an off-shell N = 1 scalar multiplet as compensating multiplet. This
consists of a real scalar φ, a Majorana fermion λ and a real auxiliary scalar S. The
transformation rules under ordinary and special supersymmetry are then given by
4¯ ǫλ,δS = −¯ ǫD /λ − 2(wφ− 1)¯λη,
δλ = D /φǫ −1
4Sǫ − 2wφφη. (2.7)
We choose the following gauge fixing conditions:
D − gauge
S − gauge
:φ = φ0= constant,
:λ = 0. (2.8)
As the S-gauge is not invariant under supersymmetry, the super-Poincar´ e rules will
involve a compensating S-transformation, with parameter
η = −1
In the following, we will always choose φ0such that4
Let us illustrate this procedure by constructing the ordinary two-derivative N = 1,
3D super-Poincar´ e action. We start from the (globally supersymmetric) action
EH= φ2φ −1
From now on, we will concentrate on the bosonic terms only. The action corresponding
to the Lagrangian (2.11) is not yet invariant under local conformal transformations. In
order to render it conformally invariant, we first note that it is invariant under global
scale transformations. These transformations consist of a scaling of the coordinates
and a scaling of the fields according to the following weights:
2= 1,wS= wφ+ 1 =3
One then has to replace the derivatives by covariant ones and add extra terms involving
curvatures. Using the rules (2.6), one can check that the action corresponding to
EH= −32φ2Cφ − 2S2+ 4Rφ2
4This convention is such that according to (2.1) the final supersymmetry rule of the gravitino is
given by : δψµ= Dµ(ω)ǫ +1
2Sγµǫ, as used in .
is conformally invariant, provided the metric transforms as usual with weight −2. The
super-Poincar´ e theory can now easily be recovered by using the gauge fixing conditions
(2.8) with, as a consequence of (2.10),
One thus finds the following Lagrangian
LEH= R − 2S2+ (fermionic terms), (2.15)
which is a standard result . We next consider a curvature squared term.
2.2 A supersymmetric curvature squared action
One can employ a similar reasoning as above starting from the higher-derivative su-
Ric= 2φ2φ −1
To ensure conformal invariance, one now has to choose different weights:
2= 0,wS= wφ+ 1 =1
One can again replace all derivatives by covariant ones and add terms involving the
curvatures to obtain a conformally invariant action. Focusing on the bosonic terms,
one obtains the following result:
4S2CS + 4φ2
Note that we have only written the relevant bosonic terms in this Lagrangian. The full
result contains extra terms5that vanish upon using the gauge fixing condition (2.8).
The third term cancels the Ka-variation of the (2Cφ)2term, while the last term cancels
the S2CS variation. Upon using the gauge fixing condition
one finds that
32S2R + (fermionic terms). (2.20)
5Of the form RabDaφDbφ, Rφ2Cφ and R(Dφ)2.
2.3 A new supersymmetric Snaction
An indication for the existence of a new supersymmetric invariant can be obtained by
comparing LRicconstructed above with the following two supersymmetric invariants
constructed in :
= K −1
= R2+ 16S2S + 12S2R + 36S4+ (fermionic terms).
2S4+ (fermionic terms),
If these were the only two invariants then LRicwould have to be a linear combination
of LK and LR2, but this is not the case! In particular, the RS2terms do not fit.
This means that there must exist a third invariant containing RS2but no curvature-
squared terms. To construct this invariant we need a globally supersymmetric invariant
not containing a quartic term in the compensating scalar φ. Starting from a superfield
Φ = φ + θαλα+ θ2S, one finds that there are two independent superspace actions of
These yield the component Lagrangians
= S4+ 48S2φ2φ − 12S2¯λ∂ /λ − 48Sφ(∂µ¯λ)γµγν(∂νλ) + ··· ,
= S4− 16S2(∂φ)2− 12S2¯λ∂ /λ − 32S2φ¯λλ − 16(∂S · ∂φ)¯λλ
+ 32S∂µφ¯λγµν∂νλ + ··· ,
where the dots indicate terms quartic in fermions. The next step consists in construct-
ing a conformally invariant Lagrangian out of Lrigid
not possible to make them conformally invariant separately; only the combination
. It turns out that it is
= 10S4+ 48S2φ2φ − 144S2(∂φ)2+ (fermionic terms)
can be made conformally invariant. This follows from the observation that
S2φ2Cφ − 3S2(Dφ)2+1
= 0. (2.25)
The combination Lrigid
can thus be made conformally invariant by taking the
2=14,wS= wφ+ 1 =3
by turning all derivatives into covariant ones and then adding the curvature-dependent
term 3RS2φ2. Upon using the gauge fixing condition φ0= 1, one ends up with the
LS4 = S4+3
10RS2+ (fermionic terms), (2.27)
which was not considered in .
The new S4invariant presented above can be generalized by noting that the fol-
lowing component Lagrangians are also invariant under rigid supersymmetry:
= Sn+ 16(n − 1)Sn−2φ2φ − 4(n − 1)Sn−2¯λγµ∂µλ
−8(n − 1)(n − 2)Sn−3φ(∂µ¯λ)γµγν(∂νλ) + ··· ,
= Sn− 16Sn−2(∂φ)2− 4(n − 1)Sn−2¯λγµ∂µλ − 16(n − 2)Sn−32φ¯λλ
−8(n − 2)(n − 3)Sn−4(∂S · ∂φ)¯λλ
+16(n − 2)Sn−3∂µφ¯λγµν∂νλ + ··· .
Again, only one linear combination of L(n)
This conformal combination leads to the following generalization of (2.27):
can be made conformally invariant.
LSn = Sn+
n − 1
6n − 14RSn−2+ (fermionic terms). (2.29)
Choosing n = 1 we recover the supergravity cosmological term
LS≡ LC= S + (fermionic terms). (2.30)
Choosing n = 2 we recover the standard EH terms
LS2 ≡ −1
where LEH is given in (2.15). Choosing n = 3 we arrive at a new invariant with
LS3 = S3+1
2RS + (fermionic terms).
Finally, we recover LS4 of (2.27) by choosing n = 4.
3 The general ‘curvature-squared’ model
We have now shown that there exist three locally supersymmetric actions with La-
grangians that have the same dimension as R2. The three Lagrangians are
= K −1
= R2+ 16S2S + 12S2R + 36S4+ (fermionic terms),
10RS2+ (fermionic terms).
2S4+ (fermionic terms),
We also found a fourth Lagrangian LRicof the same dimension but
16LS4 . (3.2)
In fact, all Lagrangians at this dimension are linear combinations of LK, LR2 and
LS4. Similarly, at one lower dimension we will have a linear combination of the scalar
density√−detg LS3 and the supersymmetric extension Ltop of the Lorentz-Chern-
Simons Lagrangian density LLCS.
Introducing the gravitational coupling constant κ, and the notation e =√−detg
for the volume density, we may now write the action for the most general 3D super-
gravity with no terms of dimension higher than R2as
8˜ m2LR2 +
ˇ m2LS4 +1
where (M,m, ˜ m, ˇ m) are mass parameters, as are (µ, ˇ µ) although the action depends
only on the dimensionless combinations (κ2µ,κ2ˇ µ), and
The bosonic Lagrangian density is
Lbos = eMS + σ?R − 2S2?+
This has six independent mass parameters (M,m, ˇ m, ˜ m, ˇ µ,µ), not counting the overall
gravitational coupling constant κ, and one dimensionless constant σ. In all, there are
therefore seven dimensionless parameters. We recall that we allow m2to be negative
as well as positive, and we will similarly allow ˜ m2and ˇ m2to take either sign.
3.1 Some notation
Before proceeding, we gather together here some useful definitions. First we recall the
definition of ˆ m2from :
Three new definitions are
In the case that ˜ m2= ∞, we drop the hats; for example
5ˇ m2. (3.8)
3.2 Field equations
We now turn to the field equations of the general model with Lagrangian density (3.5).
The S field equation is
M − 4σS −
˜ m2D2S .(3.9)
The metric field equation may be written as
2MS + σS2−S3
?GµνS −?DµDν− gµνD2?S?
?GµνS2−?DµDν− gµνD2?S2?, (3.10)
where (as in )
= εµτρDτSρν,Sµν= Rµν−1
2RRµν− 8RµλRλν+ 3gµν(RρσRρσ) ,
The trace of the metric field equation can be written as
M − 4σS −
3ˇ µD2S −
3.3Maximally symmetric vacua
The field equations simplify considerably for maximally-symmetric vacua, which are
characterized by the cosmological constant Λ. The S equation simplifies to
M − 4σS −
+ 3?S2+ Λ??1
For maximally symmetric spacetimes, the metric equation is implied by its trace. Using
the fact that
R = 6Λ,K = −3
for maximally symmetric metrics, the trace of the metric equation can be seen to reduce
M − 4σS −
S +?S2+ Λ??
= 0. (3.17)
Combining this with the S equation, we deduce that
S +(ˆ m′)2
= 0. (3.18)
There are therefore two classes of maximally symmetric vacua, as found for the less
general model of  but the present analysis is slightly simpler and better adapted to
the more general case now under consideration. We consider these two classes in turn.
• Supersymmetric vacua with
S2= −Λ ≥ 0. (3.19)
In this case both S and metric equation are solved when S solves the cubic
M − 4σS +
5ˇ m2S3= 0. (3.20)
Using the fact that S2= −Λ, we can rewrite this cubic equation as
S . (3.21)
Squaring both sides we then deduce that
16M2= 0. (3.22)
This is a cubic function of Λ that can be plotted as a curve in the (Λ,M2)
plane. In the limit that ˇ m2→ ∞ this curve reduces to the straight line of 
representing supersymmetric vacua.
• The remaining maximally symmetric vacua are generically non-supersymmetric,
and correspond to solutions of the quadratic equation
S +(ˆ m′)2
= 0. (3.23)
Using this in (3.15), we deduce that
M −4(ˆ m′)2
(ˆ m′′′)2−(ˆ m′)2
where S is a solution to (3.23). In the limit that |ˇ µ| → ∞, we have the following
cubic equation for Λ in terms of M2:
ˆ m2(ˆ m′′′)4
= −(ˆ m′)2?Λ + 4ˆ m2σ??
As expected, the sign of M is relevant only when ˇ µ is finite because otherwise the
field redefinition S → −S flips the sign of M without causing any other change.
In the further limit that ˇ m2→ ∞, the cubic reduces to the cubic found in 
and plotted there in the (Λ,M2) plane.
3.4 Review of supersymmetry-preservation conditions
The necessary and sufficient conditions for any bosonic field configuration of 3D su-
pergravity to be supersymmetric were found in . We shall review the result here as
we will want to know whether the solutions of the field equations that we consider are
supersymmetric solutions. A useful necessary condition for supersymmetry is that
16(∂S)2=?R + 6S2?2. (3.26)
When S is constant this implies that R + 6S2= 0, and this reduces to the condition
(1.5) for maximally symmetric vacua, defined by the condition (1.3). In this case,
one can show (by constructing the Killing spinors) that maximally symmetric vacua
satisfying (1.5) are also maximally supersymmetric.
More generally, a bosonic configuration of 3D supergravity is supersymmetric if the
metric and scalar field S take the form
ds2= dx2+ 2f(u,x)dudv + h(u,x)du2,S = −∂xlog
where the functions f and h are arbitrary, except that f is nowhere vanishing. This
4(R + 6S2),
which is obviously compatible with (3.26) but is a stronger condition.
All cases that we will consider here have constant S; in this case the configuration
(3.27) can be put into the form
ds2= dx2+ 2e∓2x/ℓdudv + h(u,x)du2,S = ±ℓ−1, (3.29)
for constant ℓ (with dimensions of inverse mass). Introducing the new coordinate
r = e∓x/ℓ, (3.30)
we see that the supersymmetric configurations for constant S can be put into the
r2+ 2r2dudv + h(u,r)du2,S = ±ℓ−1. (3.31)
When h = 0 we have an adS spacetime with adS radius ℓ.
3.5The pp-wave solution revisited
We know from  that there are supersymmetric pp-wave configurations, of the type
first discussed in , that solve the equations of motion of the curvature-squared
supergravity model constructed there. We now investigate this issue in the context of
the more general model. To this end, we first rewrite the metric of (3.31) as
ds2= 2e+e−+ e∗e∗,S = ±ℓ−1, (3.32)
e+= rdv +h(u,r)
du ,e−= rdu ,e∗=ℓ
The non-vanishing components of the Ricci and Cotton tensors are
R+− = R∗∗ = −2ℓ−2,
C−− = ℓ−1(r∂r+ 1)R−−.
The Ricci scalar is then given by R = −6/ℓ2.
Using these results, we find that the S field equation (3.9) reduces to
M ∓ 4σℓ−1±2ℓ−3
5ˇ m2= 0 . (3.35)
We also find that all components of the metric equation (3.10) are satisfied trivially
except the −− component, which gives
ˆ σ = σ ±ℓ−1
Trying a solution of the form h ∝ rn, we find that it solves the fourth-order ODE
as long as the power n satisfies the quartic characteristic equation
ˆ σ +1
h = 0, (3.36)
10ˇ m2. (3.37)
n(n − 2)
m2n(n − 2) +ℓ
µ(n − 1) + ℓ2ˆ σ
= 0 , (3.38)
which has roots 0,2,n+,n−, where
n±= 1 −ℓm2
4µ2− m2ℓ2ˆ σ. (3.39)
Thus, the generic supersymmetric pp-wave solution has
h(u,r) = h+(u)ℓ2−n+rn++ h−(u)ℓ2−n−rn−+ r2f2(u) + ℓ2f3(u), (3.40)
where h±,f2,f3 are arbitrary dimensionless functions of u. One can arrange for f2
and f3 to vanish by local coordinate transformations, so the solution is essentially
determined by the two dimensionless functions h±(u).
The solution (3.40) assumes that the four roots 0, 2, n+and n−are all different.
Several critical points can be identified, where some of these roots become degenerate.
We can distinguish the following cases:
• n+= n−, n±?= 0,2
In this case the characteristic equation has a doubly degenerate root; this arises
ˆ σ ±
in which case the generic solution (after setting f2= f3= 0) is
h(r,u) = ℓ2−k±rk±[h1(u)log(r/ℓ) + h2(u)] ,(3.42)
k±≡ 1 − (ℓm2
±/2µ) = 1 − ℓµˆ σ ∓
ℓ2µ2ˆ σ2− 1, (3.43)
and h1(u),h2(u) are arbitrary dimensionless functions of u.
• n−= 0 or n−= 2, n+?= 0,2
This case occurs when ℓµˆ σ = +1 (for n−= 0) or ℓµˆ σ = −1 (for n−= 2). In case
the root 0 becomes doubly degenerate, the generic solution (with f2= f3= 0) is
h(r,u) = ℓ2−k1rk1h1(u) + ℓ2h2(u)log(r/ℓ), (3.44)
k1= 2 −ℓm2
and h1(u),h2(u) are arbitrary dimensionless functions of u. For n− = 2, the
generic solution is given by
h(r,u) = ℓ2−k2rk2h1(u) + r2log(r/ℓ)h2(u), (3.46)
• n+= 0 or n+= 2, n−?= 0,2
This case is analogous to the previous one, with n−and n+interchanged. It thus
occurs when ℓµˆ σ = +1 (for n+= 0) or ℓµˆ σ = −1 (for n+= 2). The generic
solutions are given by (3.44) (for n+= 0) and (3.46) (for n+= 2).
• We can also consider the case for which the roots n = 0 and n = 2 become triply
degenerate. The conditions n+= n−= 0 occur for ℓµˆ σ = 1 and ℓm2= 2µ, while
n+= n−= 2 is obtained by taking ℓµˆ σ = −1 and ℓm2= −2µ. At these critical
points, the pp-wave solutions disappear and become diffeomorphic to adS3. New
doubly logarithmic solutions arise given by
ℓµˆ σ = +1 :h(r,u) = ℓ2log(r/ℓ)[h1(u)log(r/ℓ) + h2(u)] ,
h(r,u) = r2log(r/ℓ)[h1(u)log(r/ℓ) + h2(u)] , (3.48) ℓµˆ σ = −1 :
where, again, h1(u),h2(u) are arbitrary dimensionless functions of u.
All of the pp-wave solutions presented above reduce to those found in  in the
limit ˇ µ → ∞ and ˇ m2→ ∞, and for h ?= 0 they all preserve half the supersymmetry of
the adS3vacuum with Λ = −1/ℓ2; in the conventions of  the Killing spinor is
where ψ0is an arbitrary constant. For h = 0 the solution degenerates to the supersym-
metric adS3vacuum, which preserves both supersymmetries; the generic Killing spinor
now takes the form
for arbitrary constants ψ0and χ0.
4 Models with auxiliary S
In this section we will study special cases of the model defined by (3.5) for which
˜ m2= ∞. (4.1)
This defines a six-parameter subclass of models, all with the feature that the equation
for S is algebraic, in fact a cubic equation. However, the coefficients are not necessarily
constant and this will generically lead to a propagating scalar mode. This can be
avoided by imposing additional conditions on the parameters that define the following
classes of models:
Super − GMG :
Super − NMG :
˜ m2= ∞,
˜ m2= ∞,
ˇ µ2= ∞
ˇ µ2= ∞,
|µ| = ∞
(m′′)2= ∞⇔ˇ m2=3
We shall see that there are other “generalized” cases, with finite ˇ µ, in which a prop-
agating scalar can be avoided, but these arise as a consequence of a relation between
the parameters of the model and the vacuum value of S; see eq. (4.34) below.
We begin with the super-GMG model. In this case the Lagrangian density (3.5) sim-
L = e
MS − 2σS2+
+ σR +
which contains the four independent parameters M ,σ ,m and µ. The S equation of
motion is the cubic equation
M − 4σS +2S3
3m2= 0. (4.5)
The special feature of super-GMG is that the coefficients of this cubic equation are
constants, which means that S is constant. There is always at least one solution, and
it is unique when
9M2> 128m2σ3. (4.6)
This is satisfied automatically when m2σ < 0.
Given a solution S =¯S of (4.5), back-substitution into the Lagrangian density
where λ is related to¯S via the quartic equation
−2λm2+ σR +
4m4λ =¯S4− 4m2σ¯S2. (4.8)
This is just the cosmological GMG Lagrangian density of , hence the terminology
“super-GMG” for the model with bosonic Lagrangian density (4.4). The special case
in which |µ| = ∞ is then “super-NMG”.
4.1.1 Field equations and vacua
The metric equation of the general model simplifies enormously for super-GMG:
2MS + σS2−
2m2Kµν= 0. (4.9)
The trace of this equation can be written as
M − 4σS +2S3
Remarkably, the first-parenthesis terms vanish on using the S field equation (4.5).
Given that S =¯S solves that cubic equation, we see that the trace of the metric
equation further simplifies to
6As the equation for S is cubic rather than quadratic, this back-substitution is not equivalent to
Gaussian integration over S in the path integral. However, substitution into the field equations rather
than the action I[g,S] yields equations that are equivalent to those found from the action I[g,¯S], so
the back-substitution is still justified classically.
For maximally symmetric vacua, for which
24R2= 0, (4.12)
this equation reduces to
?¯S2+ Λ??4m2σ + Λ −¯S2?= 0, (4.13)
which also follows from a comparison of (4.8) with (1.4). There are therefore two classes
of vacua of super-GMG:
• Supersymmetric vacua with¯S2= −Λ. In this case
9m4M2= −4Λ?Λ + 6m2σ?2, (4.14)
with Λ < 0, so these vacua are either Minkowski or adS.
• Non-supersymmetric vacua with¯S2= 4m2σ + Λ ?= −Λ. In this case
9m4M2= 4?Λ + 4m2σ??Λ − 2m2σ?2,
with Λ > −4m2σ; for m2σ < 0 this implies that all non-supersymmetric vacua
are dS, but there are also non-supersymmetric adS vacua (with λ < 0) when
m2σ > 0.
A consequence of the restriction on Λ in each of these cases is that λ ≥ 0 when m2σ < 0.
Thus, not all of the vacua of GMG are vacua of super-GMG; the dS vacua for λ < 0
and m2σ < 0 are excluded.
As a simple illustration of the fact that there exist supersymmetric adS vacua,
m2σ < 0,M2≪??m2σ??. (4.16)
In this case there is a unique solution¯S of the cubic equation (4.5), and it takes the
The cosmological constant is therefore
Λ = −M2
≤ 0. (4.18)
It follows that¯S2= −Λ to the approximation at which we are working, whereas
¯S2?= Λ+4m2σ within the same approximation. We thus deduce that these adS vacua
are supersymmetric. The limit M → 0 yields the supersymmetric Minkowski vacuum.
To proceed further, it is convenient to define the two dimensionless parameters
32m2σ3,x = 1 +
m2Σ ? 0
m2Σ ? 0
m2Σ ? 0 or m2Σ ? 0
Figure 1: Graphical representation of the maximally-symmetric vacua of super-GMG in the
(x,y)-plane, with x and y defined in (4.19). Supersymmetric vacua correspond to
points on the solid curve y = 4−3x2−x3; all are adS except for the special point
g on this curve, which is Minkowski. The thick part of this curve corresponds to
supersymmetric adS vacua in which NMG is perturbatively unitary, as discussed
in section 5. All other vacua correspond to points on the dashed/dotted curve
y = 4 − 3x2+ x3. Those on the (thick) dashed line are dS, while those on the
(thin) dotted line are adS. The points a and h are dS vacua, while b, d, e and i are
supersymmetric adS. The point f is a non-supersymmetric Minkowski vacuum.
The point c can be dS or non-supersymmetric adS depending on the sign of m2σ.
Note that y ≥ 0 when m2σ > 0 and y ≤ 0 when m2σ < 0, and hence that m2σ may
have either sign when y = 0.
All maximally-symmetric vacua correspond to points in the (x,y)-plane that lie on
one of the two cubic curves
y = 4 − 3x2∓ x3,
where the upper sign yields the supersymmetric vacua. Taken together, these two cubic
curves yield a figure in the (x,y) plane, as shown in Fig 1. This figure is symmetric
under (x,y) → (−x,y), although this transformation exchanges a supersymmetric with
a non-supersymmetric vacuum, except at the fixed point (x,y) = (0,4) where the two
cubic curves cross. This crossing point corresponds to a supersymmetric adS vacuum
with λ = −σ2, as follows from
λ + σ2= σ2x2.
This is the unique vacuum on the y-axis, from which we deduce that the dS vacuum
of cosmological GMG with λ = −σ2and m2σ < 0 is not a solution of super-GMG.
As pointed out in , the adS vacuum at λ = −σ2and m2σ > 0 has very special
properties; in particular it admits a class of asymptotically adS black hole solutions,
with the extremal black hole solution interpolating between the adS vacuum and a
Kaluza-Klein solution with adS2× S1spacetime (see also [31,32]).
Let us now consider the possible vacua on each of the two cubic curves separately.
All points on the ‘supersymmetric’ cubic curve correspond to adS vacua except, of
course, the point at which this curve crosses the x-axis; at this point x = 1, so Λ = 0.
This is the supersymmetric Minkowski vacuum with M = 0, and λ = 0, although we
could consider this point as representing two vacua since it is valid for either choice of
sign of m2σ. There is also a supersymmetric adS vacuum for m2σ > 0 when M = 0;
this corresponds to the point (x,y) = (−2,0) at which the curve just touches the x-axis.
This has Λ = −6m2σ, and λ = 3σ2.
The analogous analysis for points on the ‘non-supersymmetric’ cubic curve is a
little more complex. Points on this curve with |x| > 1 correspond to dS vacua, either
with m2σ > 0 (for y > 0) or m2σ < 0 (for y < 0). The limiting point (x,y) = (1,2)
corresponds to a non-supersymmetric Minkowski vacuum with m2σ > 0 and λ = 0.
The other limiting point (x,y) = (−1,0) corresponds to a dS vacuum with m2σ < 0
and λ = 0 if it is approached from the y < 0 side. However, it can also be approached
from the y > 0 side, in which case it corresponds to an adS vacuum with m2σ > 0 and
λ = 0. Elsewhere on this cubic curve, i.e. for y > 0 and x < 1, points on the curve
correspond to adS vacua that are not supersymmetric except at the crossing point
(x,y) = (0,4).
To make contact with the analysis in  of the maximally-symmetric vacua of
GMG, we first recall that (1.4) has the solution
Λ = −2m2?
which shows that there are two possible vacua for each λ > −σ2. However, this becomes
4 vacua for each λ if one allows either sign of m2σ. This result is manifest from Fig.
1 since each value of λ > −σ2corresponds to two (vertical) lines in the (x,y) plane
that are parallel to, but not coincident with, the y-axis, and each of these vertical
lines cuts each of the two cubics curves once. Actually, this is not quite right for
λ = 0, but let us postpone consideration of this special case, and illustrate the generic
case with λ = 3σ2, which corresponds to x = ±2. The choice x = 2 yields a non-
supersymmetric dS vacuum at (x,y) = (2,0) (and hence Λ = 2m2σ > 0 and M = 0)
and a supersymmetric adS vacuum at (x,y) = (2,−16) (and hence Λ = 2m2σ < 0 and
M ?= 0). As shown in , the latter vacuum has very special properties; in particular,
linearization about it yields a quadratic model describing massive particles of spin 1
rather than spin 2. The other choice x = −2 yields a supersymmetric adS vacuum
at (x,y) = (−2,0) (and hence Λ = −6m2σ < 0 and M = 0) and a dS vacuum at
(x,y) = (−2,−16) (and hence Λ = −6m2σ > 0 and M ?= 0). There is complete
agreement with  and we now learn that the two adS vacua are supersymmetric in
the context of GMG.
The λ = 0 case, which corresponds to |x| = 1, is special because the point (x,y) =
(−1,0) represents two possible non-supersymmetric vacua, either dS or adS, depending
on the sign of m2σ, as we already observed above, and the same can be said of the
point (x,y) = (1,0) although both vacua are Minkowski. Taking this into account,
we have six vacua for λ = 0. One may ask how this is compatible with our earlier
conclusion that each value of λ > −σ2corresponds to four distinct vacua, allowing for
either sign of m2σ. The answer to this question is that two vacua may be equivalent
in the context of GMG but distinct in the context of super-GMG. For example, in
the GMG context the adS vacuum at (x,y) = (−1,2) would have to be considered
equivalent to the adS vacuum at (x,y) = (−1,0) because both have the same value of
Λ and λ. But these two vacua have different values of M2in the super-GMG context;
moreover, one is supersymmetric and the other is not. Similarly, the Minkowski vacuum
at (x,y) = (1,2) is equivalent to the m2σ > 0 Minkowski vacuum at (x,y) = (1,0)
in the GMG context, but they differ as vacua of super-GMG because they again have
different values of M2and one is supersymmetric and the other not.
4.1.2 Other solutions
Let us now turn to solutions of super-GMG that are not maximally symmetric. Of
particular interest are solutions that preserve some fraction of the supersymmetry of
a supersymmetric vacuum solution; this fraction is necessarily either 1/2 or 1. Let us
begin with the observation that since S =¯S, a constant, in any solution of super-GMG
(in contrast to the general model) all supersymmetric solutions have
R = −6¯S2. (4.23)
Using this to eliminate R from (4.12), we deduce that
K = −1
In other words, both R and K must be constants, such that the vacuum relation
(4.24) holds. This is a very strong condition that eliminates some otherwise plausible
For example, for the special case of λ = −1 and m2σ > 0, for which there is a
unique adS vacuum, there is also an adS2× S1‘Kaluza-Klein’ vacuum . In this
R = −4m2σ,
Since the relation (4.24) does not hold, this vacuum is not supersymmetric. It follows
immediately that the static extreme black hole that interpolates between the adS vac-
uum (at infinity) and the ‘Kaluza-Klein’ vacuum (near the horizon)  is also not
K = 2m4σ2. (4.25)
GMG has extremal BTZ black holes that are supersymmetric solutions of super-
GMG. This is because, firstly, the BTZ black holes are isometric to an adS vacuum
and hence solutions of super-GMG (because all adS vacua of GMG are solutions)
and, secondly, because the analysis of whether global identifications of adS preserve
some fraction of supersymmetry is independent of the choice of action. This argument
actually applies to the general curvature-squared model, but we concentrate on super-
GMG. Are there any other supersymmetric black holes?
To be supersymmetric a black hole must have zero Hawking temperature. This
immediately excludes the class of stationary black hole solutions of NMG found in .
It does not exclude the class found in , which all have zero Hawking temperature,
but we have not attempted to determine whether any of these are supersymmetric; it
would be a surprise if they were given the absence of non-BTZ supersymmetric static
4.2 Generalized super-GMG
We turn now to the more general models for which S is auxiliary. Given only the
condition (4.1), the bosonic truncation of the general action (3.3) is
2ˇ µRS −
MS − 2σS2+S3
+ σR +
where m′and m′′are as defined in (3.8). The S-equation of motion is algebraic:
M − 4σS +3S2
and it can be solved as a power series in R as long as
0 ?= A ≡ 2σ −3¯S
To see this, we set
S =¯S + αR +1
2βR2+ O?R3?, (4.29)
where¯S is a constant solution of the cubic equation
M − 4σ¯S +3¯S2
(m′)2= 0. (4.30)
Substitution into (4.27) yields
There is no solution of the assumed form if A = 0; in this case the series must involve
fractional powers of R. Assuming A ?= 0, elimination of S yields a Lagrangian density
of the form
L = e
−2¯λm2+ ¯ σR +
− 2¯λm2= M¯S − 2σ¯S2+
2(m′)2,¯ σ = σ +
We now have a model that involves, generically, an additional R2term as compared
with GMG, as well as higher powers of R. This leads to a loss of perturbative unitarity
in a Minkowski vacuum and we shall see in the following section that the same is true
for an adS vacuum. However, the additional R2term in the action is absent in the
special case that
and it is then obvious from (4.27) that all higher powers of R are also absent. The
Lagrangian density (4.32) is therefore precisely of GMG form in this case, with coeffi-
− 2¯λm2= M¯S − 2σ¯S2−
(m′′)2,¯ σ = σ +
For the analysis of the following section, it is convenient to introduce the new dimen-
The condition (4.34) can then be written more simply as a = 0. This condition defines
what we shall call the “generalized super-GMG” case. We say “case” rather than
“model” because the condition (4.34) is not just a relation between the parameters of
the general ‘auxiliary-S’ model but also involves¯S.
Observe that one way to achieve a = 0 is to set (m′′)2= ∞ and |ˇ µ| = ∞. We
can view this as the special case in which both a = 0 and |ˇ µ| = ∞ since these two
conditions imply (m′′)2= ∞. What is special about it is that no condition is imposed
on¯S, so we have a relation between the parameters of the general ‘auxiliary-S’ model
that define a subclass of models. This is precisely the “super-GMG” subclass, which
therefore arises as the |ˇ µ| = ∞ subcase of the a = 0 “generalized super-GMG” case.
Except for this special subcase,¯S is constrained by the relation
a = 2m2ℓ
¯S = (m′′)2/2ˇ µ. (4.37)
Consistency with (4.30) then requires that
ˇ µM = (m′′)2
20ˇ µ2ˇ m2
If the various mass parameters of the model defined by (4.26) satisfy this equation then
there exists a (constant) solution¯S of the equation for S for which I[g,¯S] is a GMG
action. One simple way in which this condition on the parameters can be satisfied is
to take ˇ m2= ∞ and ˇ µM = 2σm2.
5Perturbative unitarity of generalized super-NMG
We now turn to the issue of linearized perturbations about supersymmetric adS vacua.
One of our purposes is to make contact with the results of  on linearized pertur-
bations of NMG about adS vacua. The auxiliary tensor field method used there was
covariant, off-shell, and led to complete results that were easy to interpret. Here we
show how this method applies to super-NMG, and extend it to deal with the generalized
super-NMG case. However, we take as our starting point the generic parity-preserving
‘auxiliary-S’ model for which the Lagrangian density is obtained by taking the |µ| → ∞
L = e
MS − 2σS2+S3
+ σR +RS
As explained in the previous section, elimination of S leads generically to an infinite
series in powers of R. As each term could contribute to the quadratic approximation
in an expansion about an adS vacuum, it is simpler to retain S as an independent field
for the purposes of computing the quadratic action. It is also simpler to replace the
curvature-squared term K by an equivalent Lagrangian involving an auxiliary symmet-
ric tensor field fµν; the resulting action is
MS − 2σS2+S3
+ σR +RS
We wish to find the quadratic approximation to this action in a supersymmetric adS
vacuum with cosmological constant Λ = −1/ℓ2.
5.1 Quadratic approximation
We now set
= ¯ gµν+ κhµν,
S = ±ℓ−1+ κs,
?¯ gµν+ κhµν+ ℓ2κkµν
where hµν, kµν and s are independent fluctuation fields7, and ¯ gµν is the background
adS metric. We shall use the notation¯D to indicate a covariant derivative with respect
7The mass dimensions of these fluctuation fields are: [h] =1
2, [s] =3
2and [k] =5
to the standard Levi-Civita connection for the background metric. Expanding the full
Ricci tensor about the adS background we find that
Rµν = −2ℓ−2¯ gµν+ κR(1)
µν+ O(κ3) , (5.4)
µν = −1
We will need only the trace of the κ2term, which is
µν in the background metric.
?+ total derivative, (5.6)
where R(1)is the trace of R(1)
At this point it is useful to recall the gauge symmetries at the linearized level
and what the gauge-invariant objects are. The metric fluctuation transforms in the
standard way under linearized diffeomorphisms,
δξhµν = ¯Dµξν+¯Dνξµ, (5.7)
while kµνand s have been defined such that they are gauge-invariant. The invariant
curvature of hµνis given by the linearized Einstein tensor modified by the cosmological
Gµν(h) ≡ G(1)
µν(h) + Λhµν
2R(1)¯ gµν− 2Λhµν+ Λh¯ gµν,
which is the tensor that defines the linearized field equations of pure Einstein gravity
with cosmological constant.
Expanding the action about the vacuum, we find that all terms linear in the fluc-
tuations cancel provided
M = ±4σ
which is the S field equation in a supersymmetric vacuum with¯S = ±ℓ−1; this confirms
the existence of these vacua. For the quadratic terms in the Lagrangian we find the
manifestly gauge-invariant expression
2ˆ σhµνGµν(h) +
ℓm2s¯ gµνGµν(h) −
?kµνkµν− k2?−2σ ∓3
where a, the parameter defined in (4.36), is now given by
a = −m2
In the present context, the condition a = 0 yields the quadratic approximation for the
“generalized super-NMG” case, and the two conditions a = 0 and |ˇ µ| = ∞ yield the
quadratic approximation to super-NMG. As the analysis of propagating modes will
depend crucially on whether a is zero or non-zero, and as the a = 0 case is of more
relevance to “massive gravity”. The parameter ˆ σ introduced in (3.37) will also play a
significant role in what follows; it is useful to note that this parameter may be rewritten
ˆ σ = σ +
Our next goal is to analyze the modes propagated by the Lagrangian (5.10). After
some field redefinitions, we will be able to do this by comparison with Proca and Fierz-
Pauli theory in anti de Sitter space. For the convenience of the reader, we first review
this topic; one of our aims will be to determine the bounds on the masses of spin-1 and
spin-2 particles in adS that are implied by the absence of tachyons.
5.2 Review of Proca and Fierz-Pauli in adS
For a vector field Aµthe massive Proca Lagrangian in an adS background is given by
LProca = −1
It propagates massive spin-1 modes; in 3D this means that there are two modes, one
of helicity +1 and one of helicity −1. The existence of two modes can be seen by
inspecting the field equations. Variation with respect to Aµyields
¯DµFµν− M2Aν = 0⇒
¯DµAµ = 0 ,(5.14)
where the second equation (the subsidiary condition) follows by taking the divergence
of the first equation. The dynamical equation can then be written as
?¯D2− 2Λ − M2?Aµ = 0 .(5.15)
The subsidiary condition yields one constraint, which implies that there are in total
two propagating degrees of freedom.
For a symmetric tensor field ϕ and mass parameter M, the FP Lagrangian in an
adS background is
LFP(ϕ;M2) = −1
2M2¯ gµν¯ gρσϕµ[ρϕν]σ. (5.16)
For M2?= Λ this Lagrangian propagates, massive spin-2 modes; in 3D this means that
there are two modes, one of helicity +2 and one of helicity −2. The presence of two
propagating degrees of freedom can be seen by inspecting the field equation
2M2(ϕµν− ¯ gµν¯ ϕ) = 0, (¯ ϕ ≡ ¯ gµνϕµν).(5.17)
Taking the divergence of this equation and using the Bianchi identity¯DµGµν= 0, we
¯Dµϕµν−¯Dν¯ ϕ = 0
On the other hand, taking the trace of (5.17) and using the explicit form of R(1)in
(5.5), we get
¯Dµ¯Dνϕµν−¯D2¯ ϕ = 2?Λ − M2?¯ ϕ,
where Λ < 0 is the cosmological constant. Combining this with (5.18) we conclude that
¯ ϕ = 0 provided that M2?= Λ and hence that the symmetric tensor field ϕ is subject
to the subsidiary conditions
¯Dµ¯Dνϕµν−¯D2¯ ϕ = 0 . (5.18)
¯Dµϕµν = 0 ,¯ ϕ = 0 .(5.20)
The remaining, dynamical, equation is
?¯D2− 2Λ − M2?ϕµν= 0. (5.21)
The subsidiary conditions impose 3+1 constraints, so just two degrees of freedom are
propagated, and these can be shown to have helicities ±2. Observe that the specific
Fierz-Pauli mass term is crucial to this result because with a different relative coefficient
it would not be possible to derive (5.18), and the subsidiary condition ¯ ϕ = 0, needed
to eliminate scalar modes, would not be a consequence of the field equations.
In the special case of M2= Λ, the FP field equation does not imply that ¯ ϕ = 0.
In this case there is a ‘hidden’ gauge invariance,
δζ¯kµν = ¯Dµ¯Dνζ + Λ¯ gµνζ , (5.22)
with scalar gauge parameter ζ. This allows the trace ¯ ϕ to be set to zero by a gauge-
fixing condition. Theories of this type are known as partially massless [34,35], and in
3D they propagate a single mode without a well-defined helicity.
There is obviously a need for some lower bound on M2, in order to avoid tachyons.
Let us consider the generalization of (5.21) to arbitrary integer spin |s| 
?¯D2+ |s|(3 − |s|) − M2?ϕ(s)= 0 ,
where ϕ(s)denotes a traceless totally symmetric rank-|s| tensor satisfying the ‘divergence-
is derived is gauge invariant when M2= 0. Expanding the field ϕ(s)in terms of the uni-
tary irreducible representations (UIRs) of the adS3isometry group Sl(2;R)×Sl(2;R),
we find 
¯D2ϕ(s)= E0(E0− 2) − |s| ,
where (E0,s) denotes the lowest weight UIR with lowest energy E0 and helicity s.
These UIRs are nonsingular at the origin and normalizable with respect to the SO(2,2)
invariant measure [38,39]. Using the above formula in (5.23), we find
µν1...νs−1= 0. For |s| > 0, the action from which this field equation
M2= (E0− |s|)(E0+ |s| − 2) .(5.25)
Now, it is well known that the unitarity of the representation with lowest weight (E0,s)
is given by 
E0≥ |s| .
For s = 0 we deduce that M2≥ −1, which is the 3D version of the 4D Breitenlohner-
Freedman bound [39, 41, 42]. For s ≥ 1 we deduce that M2≥ 0, as claimed for
s = 1,2.
We are now ready to continue with our analysis of the quadratic Lagrangian (5.10).
The results depend crucially on whether the parameter a, defined in (5.11), is zero or
non-zero, so we consider these cases separately.
a = 0
When a = 0 the field s may be trivially eliminated and the quadratic Lagrangian (5.10)
2ˆ σhµνGµν(h) −
?kµνkµν− k2?, (5.27)
where ˆ σ is the parameter of (5.12). This is precisely eq. (4.17) of  when |ˇ µ| = ∞,
which corresponds to the super-NMG model; this was to be expected because super-
NMG has NMG as its bosonic truncation. The only difference between super-NMG and
generalized super-NMG in the context of a quadratic approximation is in the definition
of the parameter ˆ σ. How we now proceed depends on whether or not ˆ σ vanishes. We
shall consider these two subcases separately.
• ˆ σ ?= 0 : In this case we define a new symmetric tensor fluctuation field¯h by
m2ˆ σkµν. (5.28)
The quadratic Lagrangian then takes the diagonal form
2ˆ σ¯hµνGµν(¯h) −
m4ˆ σLFP(k;−m2ˆ σ), (5.29)
where LFPwas defined in (5.16). We see from this result that ˆ σ has the interpreta-
tion as the effective EH coefficient in a non-Minkowski vacuum. Because this term
propagates no modes, we effectively have an FP Lagrangian with M2= −m2ˆ σ.
As we explained earlier, the absence of tachyons requires M2≥ 0 (which is a
stronger condition that used in ) and hence m2ˆ σ < 0. We also require ˆ σ < 0
for positive kinetic energy (no ghosts) so we deduce that the combined conditions
for no ghosts and no tachyons are
m2> 0,σ +
4ℓˇ µ< 0 .(5.30)
Note that these conditions imply that σ < 0 in the NMG limit |ˇ µ| → ∞, but
σ > 0 is possible in the “generalized” case.
We should recall here that the case M2= Λ is special because it corresponds to
a partially massless mode . It is not clear to us whether our earlier conclusion
that M2≥ 0 is required for the absence of tachyons also applies in this special
• ˆ σ = 0 :
In this special case, we see from (5.27) that the fluctuation field hµνis a Lagrange
multiplier; the constraint it imposes has the general solution
kµν= 2¯D(µAν), (5.31)
for arbitrary vector field Aµ. Using this solution we arrive at the equivalent
where we have discarded a total derivative. This is a Proca Lagrangian for Aµ,
with positive kinetic energy provided m2> 0 and a specific value for the mass.
Alternatively, the Proca equations may be deduced from the equations of motion
of (5.27). The k field equation is
2(kµν− k¯ gµν) = 0. (5.33)
When combined with the Bianchi identity¯DµGµν = 0 and the h field equation
(5.31), this implies the Proca equations that follow from (5.32). Provided m2> 0
these equations propagate non-tachyonic modes of helicity ±1. This is consistent
with the corresponding result for NMG ; however, whereas ˆ σ = 0 was there
found to imply σ < 0, this is not true in the “generalized” case since it follows
from (5.12) that ˆ σ = 0 and a = 0 imply
σ = −
and this allows σ < 0 when ˇ µ is finite.
Finally, we remark that the equation (5.33) does not propagate any modes if one
adopts the standard Brown-Henneaux boundary conditions for the metric 
but weaker boundary conditions allow well known logarithmic bulk modes . It
may be verified that the Proca modes mentioned above are mapped by (5.31) into
the first descendants of the logarithmic modes; see  for a detailed description
of precisely such a descendant mode.
The occurrence of different formulations at the linearized level is similar to what
happens in TMG, in which case there exists a map of the linearized field equation
at the chiral point to that of a topologically massive photon . Alternatively,
the linearized theory can be mapped, non-covariantly, to a scalar field satisfying
the Breitenlohner-Freedman bound . The linearized solution of these equa-
tions are related to the logarithmic solutions of the metric formulation, as has
been shown in  for the scalar parametrization.
a ?= 0
When a ?= 0 we must return to the quadratic Lagrangian (5.10). Again we must
distinguish between the ˆ σ ?= 0 and the ˆ σ = 0 cases, so we consider them in turn.
• ˆ σ ?= 0 : When ˆ σ ?= 0 the Lagrangian becomes diagonal in terms of the new sym-
metric tensor fluctuation fields (¯h,¯k), defined by
¯kµν,kµν =¯kµν+ aℓ−1s ¯ gµν, (5.35)
where a is the mass parameter defined in (5.11). The quadratic Lagrangian then
takes the form
2ˆ σ¯hµνGµν(¯h) −
m4ˆ σLFP(¯k;−m2ˆ σ) +
ℓm2¯ks − bs2, (5.36)
2ˆ σ −(3a ∓ 2)(a ∓ 2)
If b ?= 0 then s may be trivially eliminated; this will give rise to an additional¯k2
mass term which will lead to a non-unitary theory (since the specific FP mass
term is crucial for unitarity).
If b = 0 then the field s becomes a Lagrange multiplier for the constraint¯k = 0,
which is one of the subsidiary conditions of the FP equations. However, the¯kµν
field equation now reads
Gµν(¯k) = m2ˆ σ?1
2¯kµν− aℓ−1s ¯ gµν
Taking the divergence we deduce that
Taking the trace, one finds
¯Dµ¯Dν¯kµν= 6ℓ−1m2ˆ σas ,(5.40)
and in combination with (5.39) this gives
?¯D2− 3m2ˆ σ?s = 0.(5.41)
In other words, the fluctuation s about the vacuum value of S is now a propagat-
ing mode! Whether the theory is ghost-free in presence of this mode, however,
remains to be investigated.
The fact that the ‘auxiliary’ field propagates is surprising in view of the fact that
the field equation for S is algebraic, a cubic equation in fact, but the coefficients
of this cubic equation are not constants when a ?= 0. We earlier argued that one
may solve for S as a power series in R in this case, but all orders of this series
are relevant to an expansion about adS, so it is not guaranteed that the solutions
for fluctuations of S will be local functions of the coefficients. We now see that
an ‘auxiliary S’, in the generalized sense that we have permitted in this section,
is not equivalent to ‘non-propagating S’.
• ˆ σ = 0 : Setting ˆ σ = 0 in the Lagrangian (5.10), but now allowing for a ?= 0, we
find that the fluctuation field hµνbecomes a Lagrange multiplier, as before. The
constraint it imposes has the general solution
kµν= aℓ−1s¯ gµν+ 2¯D(µAν), (5.42)
for arbitrary vector field Aµ. Using this solution we arrive at the equivalent
ℓm2s¯DµAµ− bs2, (5.43)
where we now have
b = −(3a ∓ 2)(a ∓ 2)
If b ?= 0, then the field s can be trivially eliminated, as before, and this will give
rise to additional (¯D · A)2terms which will lead to a non-unitary theory (since
the standard Proca form of the action is needed for unitarity).
If b = 0, then the field s becomes a Lagrange multiplier for the constraint¯D·A =
0, which is the Proca subsidiary condition. Furthermore, the Proca equation is
now modified to
¯DµFµν− 4ℓ−2Aν= 2aℓ−1∂νs .
Taking the divergence of this equation we deduce that
¯D2s = 0 .(5.46)
So the fluctuation field s propagates a scalar mode. The unitarity of the model
in presence of this mode remains to be investigated.
A curiosity that our analysis has uncovered is that a scalar field may be “auxiliary”
in the sense of having no kinetic term but still propagate modes in a non-Minkowski
vacuum if it is coupled to scalar products of propagating fields. The distinction between
“auxiliary” and “non-propagating” boils down, in the cases analysed, to whether a
dimensionless parameter a is non-zero (the generic case) or zero (the “non-propagating”
case). The latter option yields the cases that we have referred to as those of “generalized
super-NMG”. The more general “auxiliary S” models, with a ?= 0, propagate scalar
modes and are generically non-unitary although there may be special subcases that are
Within “generalized super-GMG” we find the “super-NMG” models. Since these
have NMG as a bosonic truncation (albeit with a restricted range of the NMG pa-
rameters) we should expect agreement with the results found for NMG in  . We
do, except for the stricter condition on perturbative unitarity that follows from the
stronger bound on the spin-2 Fierz-Pauli mass in adS vacua that we have justified
We have also shown that the super-NMG results extend to “generalized super-
NMG”, the only difference being that the “effective” EH coefficient ˆ σ now depends on
an additional parameter. This allows perturbative unitarity to be made consistent with
σ > 0, i.e. with “right-sign” EH term in the action. However, it should be recalled
that “generalized super-NMG” is not actually a class of “models” because its definition
depends on a choice of adS vacuum; in particular, the conclusion that σ < 0 is needed
for perturbative unitarity in Minkowski vacua is unchanged.
In this paper we have completed a study of three-dimensional (3D) N = 1 supergravity
theories with generic curvature-squared terms that was begun in . That paper was
titled “Massive 3D supergravity” but contact was made with the massive gravity models
introduced in  only in the context of an expansion about Minkowski spacetime,
where non-linear features are not crucial. The space of non-Minkowski vacua found
in  had no obvious relation to the space of non-Minkowski vacua found in , and
neither did there appear to be any supergravity model with a bosonic truncation that
could be identified with a massive gravity model. As we said in the introduction,
these unsatisfactory features suggest that there is some ingredient missing from the
analysis of , and we have shown here that this is indeed the case. The supergravity
results of  are correct but incomplete because there is an additional super-invariant
involving the auxiliary scalar field S of N = 1 supergravity that contributes to the
terms with the dimension of curvature-squared terms but not to the curvature-squared
terms themselves. Incorporating this invariant into a more general action allows the
choice of a special case in which S can be eliminated, at least classically, to yield a
model that is identical to the ‘cosmological’ extension of the “general massive gravity”
(GMG) model introduced in , and this includes as a special case the ‘cosmological’
extension of the parity-preserving “new massive gravity” (NMG) model studied in
detail in .
Actually, it is overstating the case to say that the new results of this paper are
suggested by complications for non-Minkowski found in  because it is far from
obvious, a priori, that a higher-derivative gravity model should arise as the truncation of
a supergravity model. In fact, the results of this paper confirm the contrary conclusion
for generic curvature-squared models, since the supergravity extension of the generic
model necessarily involves a kinetic term for the ‘auxiliary’ scalar, thus propagating a
field that was not present initially. The special feature of the NMG and GMG models,
already noted in , is that this term is absent, so that the ‘auxiliary’ field S really does
remain auxiliary in the sense that its field equation is algebraic, in fact cubic. However,
the incomplete results of  led to the conclusion that this cubic equation necessarily
has coefficients that are not all constant but depend upon the scalar curvature R.
Elimination of S then leads to an additional power series in R contribution to the
action; in particular, it leads to an additional unwanted R2term. Had this been the
last word on the matter, it would have encouraged the view that that massive 3D
gravities are mere curiosities. Conversely, the fact that one can recover NMG or GMG
as bosonic truncations of a 3D supergravity model, as shown in this paper, paves the
way to a further study of extended super-GMG models and encourages the belief that
these models should have a role to play in some ‘bigger picture’.
The main point of interest in the new massive gravity models such as NMG and
GMG is the fact that the higher-derivative terms are consistent with unitarity, at least
in the Minkowski vacuum. This result was shown in  to extend to the spin-3
of the supergravity models, as is of course implied by supersymmetry. The issue of
unitarity in adS vacua was studied in detail in  for NMG and we have here ex-
tended this analysis to super-NMG and some variants of it that also preserve parity.
As the bosonic truncation of super-NMG is equivalent to NMG after elimination of the
supergravity auxiliary field S, and as all adS vacua of NMG correspond to a supersym-
metric adS vacuum of super-NMG, the results of  for linearization about an adS
vacuum extend immediately to linearization of super-NMG about a supersymmetric
adS vacuum; in particular, there is no need to consider the spin-3
is determined by supersymmetry in a supersymmetric vacuum.
There is one caveat: we have shown here that the Fierz-Pauli mass M for a spin-2
field in adS3must satisfy M2≥ 0 in order that the associated spin-2 particle not be
a tachyon8whereas we allowed (provisionally) for a weaker bound in . This means
that the range of parameters for which the linearized theory is perturbatively unitary
is more restricted than stated in . Another subtlety is that although super-NMG
has been defined as the model as for which the bosonic truncation yields NMG after
elimination of the auxiliary field S, there is a larger class of models for which the field
linearized equations coincide with those of NMG if the parameters are tuned to the
choice of vacuum; specifically, we can tune the parameters so that the field equation
2sector because this
8We presume that this result is known but there are suggestions in the literature of a “Breitenlohner-
Freedman bound for spin 2” that allows M2≥ −1, as for spin zero.
for the fluctuation of S is algebraic. In this way, we slightly enlarge the class of models
that are perturbatively unitary in an adS vacuum. Within this larger class perturbative
unitarity is consistent with either sign of the Einstein-Hilbert term provided the new
parameter ˇ µ introduced in (3.3) is chosen appropriately.
Finally, we briefly consider the two-dimensional CFTs that might be holographically
related to the massive gravity models above when expanded about a supersymmetric
adS vacuum. Actually, we should expect a holographically dual superconformal field
theory, i.e. an SCFT, but it is unclear to us how the fermions may be taken into
account in a semi-classical approximation to the bulk supergravity theory, so we instead
consider only the bosonic truncations. According to the Brown-Henneaux analysis, for
generic adS3gravity theories the asymptotic symmetry group consists of two copies of
the Virasoro algebra corresponding to the two-dimensional conformal symmetry .
Their central charges encode important information about unitarity and the entropy of
BTZ black holes. In the case of parity-preserving gravity theories that contain higher
powers of the curvature tensor the (left and right) central charges are given by [47,48]
In the presence of the Lorentz Chern-Simons term, we need to add the contributions
±3/(2G3µ) to cL,R. Taking all these considerations into account, and starting from
the general model (3.1), we obtain the following values for the central charges
ˆ σ ±1
where we used Λ = −1/ℓ2. In the special case of super-GMG we have
ˆ σ = σ +
and hence agreement with the results of . Perhaps the most significant feature
of the formula (6.2) is that ˆ σ is the parameter determining the sign of the effective
linearized EH term in the chosen adS background, which must be negative for pertur-
bative unitarity. This means that the difficulty encountered in all previous massive 3D
gravity models, that one must choose between non-unitary gravitons or negative mass
BTZ black holes, is a rather general one that is not resolved in supergravity, no matter
how one adjusts the parameters.
We acknowledge helpful discussions and correspondence with Steve Carlip, Geoffrey
Comp` ere, Daniel Grumiller and Paul Howe. We also thank each others home institu-
tions for the hospitality extended during visits. PKT is supported by an EPSRC Senior
Fellowship. The research of ES is supported in part by NSF grants PHY-0555575 and
PHY-0906222. The work of O.H. is supported by the DFG–The German Science Foun-
dation and in part by funds provided by the U.S. Department of Energy (DOE) under
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