# 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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**ABSTRACT:**Scalar fields minimally coupled to General Relativity in three dimensions are considered. For certain families of self-interaction potentials, new exact solutions describing solitons and hairy black holes are found. It is shown that they fit within a relaxed set of asymptotically AdS boundary conditions, whose asymptotic symmetry group coincides with the one for pure gravity and its canonical realization possesses the standard central extension. Solitons are devoid of integration constants and their (negative) mass, fixed and determined by nontrivial functions of the self-interaction couplings, is shown to be bounded from below by the mass of AdS spacetime. Remarkably, assuming that a soliton corresponds to the ground state of the sector of the theory for which the scalar field is switched on, the semiclassical entropy of the corresponding hairy black hole is exactly reproduced from Cardy formula once nonvanishing lowest eigenvalues of the Virasoro operators are taking into account, being precisely given by the ones associated to the soliton. This provides further evidence about the robustness of previous results, for which the ground state energy instead of the central charge appears to play the leading role in order to reproduce the hairy black hole entropy from a microscopic counting.Journal of High Energy Physics 12/2011; 2012(2). · 5.62 Impact Factor - SourceAvailable from: Ergin Sezgin[Show abstract] [Hide abstract]

**ABSTRACT:**We construct a new off-shell invariant in supergravity whose leading term is the square of the Riemann tensor. It contains a gravitational Chern–Simons term involving the vector field that belongs to the supergravity multiplet. The action is obtained by mapping the transformation rules of a spin connection with bosonic torsion and a set of curvatures to the fields of the Yang–Mills multiplet with gauge group SO(4, 1). We also employ the circle reduction of an action that describes locally supersymmetric Yang–Mills theory in six dimensions.Classical and Quantum Gravity 01/2011; 28(22). · 3.56 Impact Factor - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We present a complete solution of the constraints for three-dimensional N=2 conformal supergravity in terms of unconstrained prepotentials. This allows us to develop a prepotential description of the off-shell versions of N=2 Poincare and anti-de Sitter supergravity theories constructed in arXiv:1109.0496.Journal of High Energy Physics 09/2012; 2012(12). · 5.62 Impact Factor

Page 1

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

Page 2

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

1

Page 3

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.

2

Page 4

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)

3

Page 5

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.

4

Page 6

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”

5

Page 7

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)

6

Page 8

• 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.

7

Page 9

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].

8

Page 10

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.

9

Page 11

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)

10

Page 12

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)

11

Page 13

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)

12

Page 14

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)

13

Page 15

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

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- Available from Ergin Sezgin · Jun 2, 2014
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