More on Massive 3D Supergravity

Abstract

Completing earlier work on three dimensional (3D) N=1 supergravity with curvature-squared terms, we construct the general supergravity extension of cosmological massive gravity theories. We expand about supersymmetric anti-de Sitter vacua, finding the conditions for bulk unitarity and the critical points in parameter space at which the spectrum changes. We discuss implications for the dual conformal field theory. Comment: 53 pages, 1 figure

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arXiv:1005.3952v2 [hep-th] 10 Dec 2010
UG-10-18
MIT-CTP-4129
DAMTP-2010-15
MIFP-10-8
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, j.rosseel@rug.nl
2Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
email: ohohm@mit.edu
3George and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, Texas A& M University, College Station, TX 77843, USA
email: sezgin@tamu.edu
4Department of Applied Mathematics and Theoretical Physics,
Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge, CB3 0WA, U.K.
email: P.K.Townsend@damtp.cam.ac.uk
ABSTRACT
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-
turbatively unitary.
= 1 supergravity with

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Contents
1Introduction2
23D supergravity invariants
2.1
N = 1 superconformal tensor calculus . . . . . . . . . . . . . . . . . .
2.2A supersymmetric curvature squared action
2.3A new supersymmetric Snaction . . . . . . . . . . . . . . . . . . . . .
6
6
9. . . . . . . . . . . . . . .
10
3The general ‘curvature-squared’ model
3.1Some notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3Maximally symmetric vacua . . . . . . . . . . . . . . . . . . . . . . . .
3.4Review of supersymmetry-preservation conditions . . . . . . . . . . . .
3.5The pp-wave solution revisited . . . . . . . . . . . . . . . . . . . . . . .
11
12
13
13
15
15
4Models with auxiliary S
4.1Super-GMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1Field equations and vacua . . . . . . . . . . . . . . . . . . . . .
4.1.2Other solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2Generalized super-GMG . . . . . . . . . . . . . . . . . . . . . . . . . .
18
18
19
23
24
5Perturbative unitarity of generalized super-NMG
5.1Quadratic approximation . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2Review of Proca and Fierz-Pauli in adS . . . . . . . . . . . . . . . . . .
5.3Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1a = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2a ?= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
26
28
30
30
32
33
6Discussion34
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1Introduction
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 [4] but it was recently discovered [5] 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
8R2,(1.1)
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± [5]; 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) [8].
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
LGMG=
?
−detg
?
−2λm2+ σR +
1
m2K
?
+1
µLLCS,(1.2)
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.
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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 [5]. By definition, such
vacua have the property that
Gµν= −Λgµν,
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:
(1.3)
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 [9]. 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 [12] (completing earlier
partial results [13]) 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 [23]. In particular, a maximally symmetric vacuum is
supersymmetric provided that
S2= −Λ,
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 [23], extending an analysis applied earlier to NMG [24]. 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
(1.5)
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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 [5] by
adapting earlier general results [25].
A fully non-linear N = 1 3D supergravity model with generic curvature-squared
terms was constructed in [23]. 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 [23]. 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 [23] 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 [23] 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 [23]
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 [23]. 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 [23] 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 [23].
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 [5], one of the two must be present because S can be entirely
absent only from super-conformal invariants.
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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 [29]. 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 [23] 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 [12]. No analogous
analysis for supergravity was attempted in [23], mainly because of the problems already
mentioned with the model constructed there. Here we shall show how the analysis
of [12] 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
perturbatively unitary.
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”
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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.
23D 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 [23] 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
determined.
2.1
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:
δeµa=1
2¯ ǫγaψµ,
where ǫ is the ordinary Q-supersymmetry parameter and η is the parameter of the
special S-supersymmetries.
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)
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• 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
Ka:
(2.5)
δKbµ = 2ΛKµ,
δKDaφ = −2wφΛKaφ,
δK2Cφ = −2wφ(DcΛKc)φ + 2(d − 2 − 2wφ)Λc
δKRab = −2ηabDcΛc
δKR = −4(d − 1)DcΛKc,
KDcφ,
K− 2(d − 2)DaΛKb,
(2.6)
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.
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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
δφ =
1
4¯ ǫλ,δS = −¯ ǫD /λ − 2(wφ− 1)¯λη,
δλ = D /φǫ −1
4Sǫ − 2wφφη.(2.7)
We choose the following gauge fixing conditions:
Ka− gauge
D − gauge
S − gauge
:bµ= 0,
: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
8
S
wφφ0ǫ.(2.9)
In the following, we will always choose φ0such that4
wφφ0= −1
4.(2.10)
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
Lrigid
EH= φ2φ −1
4
¯λγµ∂µλ +1
16S2. (2.11)
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:
wφ=1
2,wλ= wφ+1
2= 1,wS= wφ+ 1 =3
2.(2.12)
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
Lconf
EH= −32φ2Cφ − 2S2+ 4Rφ2
(2.13)
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 [23].
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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),
φ0= −1
2.(2.14)
One thus finds the following Lagrangian
LEH= R − 2S2+ (fermionic terms), (2.15)
which is a standard result [21]. We next consider a curvature squared term.
2.2A supersymmetric curvature squared action
One can employ a similar reasoning as above starting from the higher-derivative su-
persymmetric action
Lrigid
Ric= 2φ2φ −1
4
¯λγµ∂µ2λ +1
16S2S . (2.16)
To ensure conformal invariance, one now has to choose different weights:
wφ= −1
2,wλ= wφ+1
2= 0,wS= wφ+ 1 =1
2.(2.17)
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:
Lconf
Ric= 4?2Cφ?2+1
4S2CS + 4φ2
?
RµνRµν−23
64R2
?
−1
32S2R. (2.18)
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
φ0=1
2, (2.19)
one finds that
LRic= RµνRµν−23
64R2+1
4S2S −1
32S2R + (fermionic terms). (2.20)
5Of the form RabDaφDbφ, Rφ2Cφ and R(Dφ)2.
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2.3A 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 [23]:
LK
= K −1
= R2+ 16S2S + 12S2R + 36S4+ (fermionic terms).
2S2R −3
2S4+ (fermionic terms),
LR2
(2.21)
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
this type:
Irigid
1
=
?
d3xd2θ(D2Φ)3Φ,Irigid
2
=
?
d3xd2θ(D2Φ)2DαΦDαΦ.(2.22)
These yield the component Lagrangians
Lrigid
1
Lrigid
2
= 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
(2.23)
1
and Lrigid
2
. It turns out that it is
Lrigid
1
+ 9Lrigid
2
= 10S4+ 48S2φ2φ − 144S2(∂φ)2+ (fermionic terms)
can be made conformally invariant. This follows from the observation that
(2.24)
δK
?
+ 9Lrigid
S2φ2Cφ − 3S2(Dφ)2+1
16RS2φ2
?
= 0.(2.25)
The combination Lrigid
following weights:
12
can thus be made conformally invariant by taking the
wφ= −1
4,wλ= wφ+1
2=14,wS= wφ+ 1 =3
4,(2.26)
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
following Lagrangian:
LS4 = S4+3
10RS2+ (fermionic terms),(2.27)
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which was not considered in [23].
The new S4invariant presented above can be generalized by noting that the fol-
lowing component Lagrangians are also invariant under rigid supersymmetry:
L(n)
1
= 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)
1
This conformal combination leads to the following generalization of (2.27):
L(n)
2
(2.28)
and L(n)
2
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
2LEH,(2.31)
where LEH is given in (2.15). Choosing n = 3 we arrive at a new invariant with
Lagrangian
LS3 = S3+1
2RS + (fermionic terms).
Finally, we recover LS4 of (2.27) by choosing n = 4.
(2.32)
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
LK
= K −1
= R2+ 16S2S + 12S2R + 36S4+ (fermionic terms),
= S4+3
10RS2+ (fermionic terms).
2S2R −3
2S4+ (fermionic terms),
LR2
LS4
(3.1)
We also found a fourth Lagrangian LRicof the same dimension but
LRic≡ LK+1
64LR2 +15
16LS4 .(3.2)
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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
?
+1
µLLCS
I[g,S] =
1
κ2
?
d3xe
?
MLC+ σLEH+
1
m2LK+
1
8˜ m2LR2 +
1
ˇ m2LS4 +1
ˇ µLS3
?
(3.3)
?
,
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
LLCS=1
2ελµνΓρ
λσ
?
∂µΓσ
ρν+2
3Γσ
µτΓτ
νρ
?
.(3.4)
The bosonic Lagrangian density is
?
2
˜ m2
Lbos = eMS + σ?R − 2S2?+
?
1
m2
?
?2?
K −1
2RS2−3
2S4
?
+
1
ˇ m2
?
S4+3
10RS2
?
−
(∂S)2−9
4
?
S2+1
6R +1
ˇ µ
?
S3+1
2RS
??
+1
µLLCS.(3.5)
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.
3.1Some notation
Before proceeding, we gather together here some useful definitions. First we recall the
definition of ˆ m2from [23]:
1
ˆ m2=
Three new definitions are
1
(ˆ m′)2
1
(ˆ m′′)2
1
(ˆ m′′′)2
In the case that ˜ m2= ∞, we drop the hats; for example
1
(m′)2=
1
m2−
3
˜ m2.(3.6)
=
1
ˆ m2−
1
ˆ m2−
1
ˆ m2−
2
3ˇ m2,
3
5ˇ m2,
27
40ˇ m2.
=
=(3.7)
1
m2−
2
3ˇ m2,
1
(m′′)2=
1
m2−
3
5ˇ m2.(3.8)
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3.2Field equations
We now turn to the field equations of the general model with Lagrangian density (3.5).
The S field equation is
?
M − 4σS −
SR
15ˇ m2
?
+ 3
?
S2+1
6R
??1
ˇ µ−
2S
(ˆ m′)2
?
= −4
˜ m2D2S .(3.9)
The metric field equation may be written as
0 =
?
−
−1
2MS + σS2−S3
2
˜ m2
1
2(ˆ m′′)2
2ˇ µ+
3S4
4(ˆ m′)2
?
+
gµν+ σGµν+1
µCµν+
1
2m2Kµν+
1
2˜ m2Lµν
?
∂µS∂νS −1
2gµν(∂S)2
?
1
2ˇ µ
?GµνS −?DµDν− gµνD2?S?
−
?GµνS2−?DµDν− gµνD2?S2?,(3.10)
where (as in [23])
eCµν
= εµτρDτSρν,Sµν= Rµν−1
2DµDνR −1
2RRµν− 8RµλRλν+ 3gµν(RρσRρσ) ,
= −1
4gµνR,(3.11)
Kµν
= 2D2Rµν−1
+9
2gµνD2R −13
8gµνR2
(3.12)
Lµν
2DµDνR +1
2gµνD2R −1
8gµνR2+1
2RRµν.(3.13)
The trace of the metric field equation can be written as
?
−
M − 4σS −
1
3m2
1
15ˇ m2SR
K +1
24R2
?
=
S +
?
?2(∂S)2+ D2R?+
S2+1
6R
??
2σ +S
ˇ µ+
2
3ˇ µD2S −
R
12ˆ m2−
3S2
2(ˆ m′)2
2
3(ˆ m′′)2D2S2.
?
?
?
1
3˜ m2
(3.14)
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 −
2
5ˇ m2SΛ
?
+ 3?S2+ Λ??1
ˇ µ−
2S
(ˆ m′)2
?
= 0.(3.15)
For maximally symmetric spacetimes, the metric equation is implied by its trace. Using
the fact that
R = 6Λ,K = −3
2Λ2,(3.16)
for maximally symmetric metrics, the trace of the metric equation can be seen to reduce
to
?
M − 4σS −
2
5ˇ m2SΛ
?
S +?S2+ Λ??
2σ +
Λ
2ˆ m2+S
ˇ µ−
3S2
2(ˆ m′)2
?
= 0.(3.17)
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Combining this with the S equation, we deduce that
?S2+ Λ??
S2−4(ˆ m′)2
9ˇ µ
S +(ˆ m′)2
9
?
4σ +
Λ
ˆ m2
??
= 0. (3.18)
There are therefore two classes of maximally symmetric vacua, as found for the less
general model of [23] 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
equation
M − 4σS +
2
5ˇ m2S3= 0.(3.20)
Using the fact that S2= −Λ, we can rewrite this cubic equation as
M =
?
4σ +
2Λ
5ˇ m2
?
S . (3.21)
Squaring both sides we then deduce that
Λ
?
σ +
Λ
10ˇ m2
?2
+1
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 [23]
representing supersymmetric vacua.
• The remaining maximally symmetric vacua are generically non-supersymmetric,
and correspond to solutions of the quadratic equation
S2−4(ˆ m′)2
9ˇ µ
S +(ˆ m′)2
9
?
4σ +
Λ
ˆ m2
?
= 0. (3.23)
Using this in (3.15), we deduce that
M −4(ˆ m′)2
27ˇ µ
?
σ −
20Λ
(ˆ m′′′)2
?
=4S
3
?
σ +
4Λ
(ˆ m′′′)2−(ˆ m′)2
9ˇ µ2
?
,(3.24)
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
?9M
16
?2
= −(ˆ m′)2?Λ + 4ˆ m2σ??
Λ +1
4(ˆ m′′′)2σ
?2
.(3.25)
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 [23]
and plotted there in the (Λ,M2) plane.
14