Master actions for linearized massive gravity models in 3-D

0.17 MB·English

by  Pio J. Arias

AEI-2012-003
SB/F/389-11
2 Master actions for linearized massive
1 gravity models in 3-D
0
2
Pio.J.Ariasa, Adel Khoudeirb and J. Stephanyc,d
n
a aCentro de F´ısica Te´orica y Computacional, Facultad de Ciencias,
J
Universidad Central de Venezuela, AP 20513,Caracas 1020-A, Venezuela

3
1 bCentro de F´ısica Fundamental, Departamento de F´ısica,

Facultad de Ciencias, Universidad de Los Andes, M´erida 5101, Venezuela
]
h
t cMax-Planck-Institut fu¨r Gravitationsphysik,Albert-Einstein-Institut
-
p Am Mu¨hlenberg 1, 14476 Golm, Germany
e
h dDepartamento de F´ısica, Universidad Sim´on Bol´ıvar,
[ Apartado Postal 89000, Caracas 1080-A, Venezuela

1
e-mail: [email protected], [email protected], [email protected]
v
7
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9
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Abstract
.
1
We present a unified analysis of the self-dual, second order, topologi-
0
cally massive andtherecentlyintroduced fourth ordermodels of massive
2
1 gravityin3D. Weshowthatthereisafamily offirstorderactionswhich
: interpolatebetweenthesedifferentsingleexcitationmodels. Weshowhow
v
the master actions are related by duality transformation. We construct
i
X bythesamemethodthemasteractionwhichrelatesthefourthordernew
massive model with two excitations and the usual second order model
r
a withFierz-Paulimass. Weshowthatthemoregeneralmodelobtainedby
adding a Chern-Simons term to the new massive model is equivalent off
shelltothesecondorderspontaneouslybrokenlinearizedmassivegravity.
Keywords:Massive gravity,self duality,topological mass
PACS numbers: 11.10 Kk, 04.60.Kz, 04.60.Rt
1 Introduction
Threedimensionalmassivetensormodelshavearichstructurewhichemergedin
thethreedecadessincethediscoveryofthetopologicallymassivemodel(TMM)
[1]. This model, in which the non propagating Einstein-Hilbert action is com-
plemented with a gravitational Chern Simons term, yields upon linearization
a third order gauge invariant model which propagates a single parity sensible
excitation. Other unitary models witha single excitationwerethenrecognized.
The self-dual model (SDM), which upgrades to the case of spin 2, the ideas
used to construct the vectorial self dual theory [2] equivalent to topologically
massiveelectrodynamics [3] was introduced in Ref. [4]. It provides a firstorder
equation for the tensor field . A second order model with the same spectrum
usually denominated the intermediate model (IM) was also formulated in the
samework. MasteractionsrelatedtheSDMtothe IMandtheIMtothe TMM
were presented. The IM allows for a curved space [5] formulation which con-
sists of the ordinary Einstein Hilbert action and a first order, vector like quasi
topologicaltermconstructedoutofthe dreibeinvariables. This modelisdiffeo-
morphism invariant but has the local Lorentz invariance spontaneously broken
[6]. More recently a Lorentz, gauge and conformal fourth order model (FOM)
which also describes a single massive mode has been presented [7]. This model
is not unrelated with the also recently introduced and very interesting system
dubbed new massive gravity (NMG)[8] [9] which describes in an elegant and
gauge invariant form the two physical excitations of a massive,parity invariant
spin 2 particle. These theories are ghost free , despite of the fact that their
field equations contain higher terms in the curvature in contrast to equivalent
situation in D 4 [10] [11]). This has motivated to review [12] in this con-
≥
text the standard relation between negative energy excitation and higher order
equations as was already proposed elsewhere [13].
The TMM and the NMG have in common that the Einstein-Hilbert term
appears with a sign opposite to the conventional one in D 4 gravity. Com-
≥
bining these two theories, the most general unitary action for massive spin 2
[14]isconstructedandshowntodescribetwophysicalexcitationswithdifferent
massesinagaugeinvariantway. Fromthis theory,appearsinaparticularlimit
theFOM[7],[14]. Allthesemodelsmaybe formulatedforgenericgravitational
background and when the background is AdS, new interestingly phenomena
arise [15]. The linearization on flat and AdS backgrounds have been studied in
references [9] and [12].
As already mentioned, in a form which is analogous to the case of vector
fields where the self dual theory [2] is equivalent to topologically massive elec-
trodynamics [3], at linearized level the self dual massive spin 2 [4] is equivalent
to the topological massive gravity. The self dual model do not have any lo-
cal invariance and their action is first order. Nevertheless, it may be viewed as
gaugefixedformulationoftheirrelatedhigherordermodels[16,17,18]. Classical
andquantumaspectsofthese dualequivalenceshavebeen studiedinreferences
[19, 20, 21, 22].
In section 2 we take a systematic approach to establish the equivalences,
1
at the linearized level in topological trivial manifolds, of the models mentioned
both for one and two physical excitations. For the family of parity sensible
models we display a network of master actions ultimately relating the SDM
action to the FOM. The process of increasing the order of the action is accom-
plished with a corresponding improvement of the symmetries in the chain of
master actions. Although the models obtained from the linearization of metric
curved gravity models may be written in terms of symmetric tensor fields the
structure discussed in this article is best shown using non symmetric tensors.
Thisisevidentinthediscussionofsection3wherethemasteractionsareshown
to emerge from the application of dual transformations to the different mode.
In section 4 we construct using the duality transformation, the master action
which forces the equivalence of the NMG with the usual massive Fierz Pauli
modeldiscussedin[8]. We alsoshowthe equivalenceofthe mostgeneralfourth
order model [14], [23] with the second order model considered in ref [6] which
propagates unitarily two different masses. Since all the actions presented here
arequadraticinthefields,thediscussionoftheseequivalencesintheframework
of path integral functional may also be pursued.
2 Models with one excitation
We start with the self-dual (SDM) first order action action [4]
m 1
I [w]= <w wνµ w µw ν >+ <w ǫµνρ∂ w λ >, (1)
0 − 2 µν − µ ν 2 µλ ν ρ
where <> denotes integration in 3D. This model is equivalent to the second
order intermediate model (IM) which can be written in the two alternative
forms,
1 1
I2th[h] = <ǫναβ∂ h ǫρµσ∂ h > <(ǫαβγ∂ h )2 >
2m α βρ µ σν −4m β γα
1
<h ǫµνρ∂ h λ > (2)
− 2 µλ ν ρ
1 1
= <h ρǫναβ∂ W [h]> <h ǫµνρ∂ h λ > .
2m ν α βρ −2 µλ ν ρ
with
W [h] (W )ρσh
µν µν ρσ
≡
1
= [δαδσ η ηασ]ǫ τρ∂ h . (3)
ν µ − 2 µν α τ ρσ
This is the linearized version of the massive vector Chern-Simons gravity [5]
whose action is given by the Einstein-Hilbert action and a vectorial Chern-
Simonstermconstructedoutofthedreibeinvariables. TheIMisinvariantunder
gaugetransformations,δh =∂ ζ . Equivalencewith the SDM maybe shown
µν µ ν
in different ways. For example a detailed analysis of the constraints structure
2
of the models lead to interpretthe self-dualmodel as a gauge fixed formulation
of the intermediate model [18]. These models may be also related by a duality
transformation[19,22]whichestablishestheequivalenceontopologicallytrivial
manifolds leaving room to a more subtle relation on general manifolds as it
happens with the vector models [24, 25, 26]. The dual relation between the
modelsissignalizedbytheexistenceofamasteractionwhichprovidesthemost
direct demonstration of their equivalence. This master action, which is the
analogousof the one used by Deser and Jackiw in their treatment of the vector
models [3] was given in [4]
m
I [w,h] = <w wνµ wµwν >+<w ǫµνρ∂ h λ >
1 − 2 µν − µ ν µλ ν ρ
1
<h ǫµνρ∂ h λ >. (4)
− 2 µλ ν ρ
intermsofgeneraltensorfields h andw neithersymmetric orantisymmet-
µν µν
ric. Note that the structure of this action, is given by a Fierz-Pauli mass term
for one of the fields plus a term involving the rotor of both fields ending with
a vector Chern Simons like term. This structure is exploited below to general-
ize this master action. Either the SDM or the IM may be derived covariantly
by substituting the equations of motions but this procedure should be supple-
mentedbyamorecarefulanalysistoguaranteethe canonicalequivalenceofthe
systems[4, 18]. To obtain the IM one takes variations in (4) respect to w and
obtains the identity
1
w (h)= W [h] . (5)
ρµ µν
m
This equation may be viewed as a kind of self dual change of variables and will
be used repeatedly in what follows. Substituting (5) in (4) leads to the second
order intermediate model (2). To recover the SDM one considers the second of
the equations of motions of (4),
ǫµνρ∂ w =ǫµνρ∂ h . (6)
ν ρσ ν ρσ
Substitution of this in I gives the SDM. Why this covariant treatment works
1
may be understood by noting that (6) forces the transversecomponents of w
µν
and h to be equal and then,
µν
w h =∂ λ . (7)
µν µν µ ν
−
Setting λ =0 using the gauge invariance of I ,
ν 1
δh =∂ ζ ; δw =0, (8)
µν µ ν µν
it follows that
w =h . (9)
µν µν
Combined with (5) this implies
1 1 1
h = [ǫρτσ∂ h η ǫλτσ∂ h ]= W [h] . (10)
µρ τ σµ ρµ τ σλ µρ
m − 2 m
3
Thesearetheequationsofmotionoftheselfdualtheory. Alternativelyonemay
note that I does not depend on the longitudinal part of h .
1 µν
We can iterate the mechanism which forces the equivalence between I and
1
I andintroduceasecondandathirdmasteractionsI andI givenrespectively
0 2 3
by
m
I [w,h,v] = <w wνµ wµwν >+<w ǫµνρ∂ h λ >
2 − 2 µν − µ ν µλ ν ρ
1
<h ǫµνρ∂ vλ >+ <v ǫµνρ∂ v λ >. (11)
− µλ ν ρ 2 µλ ν ρ
m
I [w,h,v,u] = <w wνµ wµwν >+<w ǫµνρ∂ h λ >
3 − 2 µν − µ ν µλ ν ρ
<h ǫµνρ∂ vλ >+<v ǫµνρ∂ vλ > (12)
− µλ ν ρ µλ ν ρ
1
<u ǫµνρ∂ u λ >.
− 2 µλ ν ρ
IneachcasethevectorChernSimonstermisre-expressedwithapparentredun-
dancy using an auxiliary field. One may also view the procedure as a duality
transformationaswillbe discussedbelow. The relationbetweenv andh in
µν µν
I havethesamestructurethattheonebetweenh andw inI justdiscussed.
2 µν µν 1
FollowingthesamestepsoneobtainsthatI [w,h,w(h)]=I [w,h]withw(h)de-
2 1
finedby(6). ThismeansthatI isequivalenttoI andforcefullytoI andI2th.
2 1 0
AnalogouslyitisstraightforwardtoshowthatI [w,h,w,u(w)]=I [w,h,u]with
3 2
u(w) defined now by the equation of motion with the same structure that (6)
andits correspondingequivalence with allthe previousactions. Belowwe show
how these master actions may be used to discuss the equivalence of all the sin-
gle excitation models of linearized gravity in 3D. Other actions I , written in
N
termsofmorefieldsmayalsobeconstructedbythesamemechanismandareall
equivalent, but when reduced to models with a single field they do not lead to
new unitary models different of the four discussed in this paper. In particular,
themodels oforderhigherthanfourinthederivativeswhichmaybeassociated
with them using covariant methods, propagate ghost excitations.
To continue our discussionlet us show how I may be relatedto thirdorder
2
TMM whose action is given by [1]
1 1
ITMM[h]= <h ǫµνρ∂ W λ[h]>+ <W [h]ǫµνρ∂ W λ[h]>.
−2m µλ ν ρ 2m2 µλ ν ρ
(13)
Here the first term is the linearized Einstein-Hilbert action with the opposite
sign and the second term is the linearized true Chern-Simons gravitational ac-
tion. This action can also be expressed alternatively in terms only of the sym-
metric part H = 1(h +h ) 1. The TMM may be obtained directly from
µν 2 µν νµ
1We use also the Einstein tensor Gµν =ǫρµσ∂ρWσν[h]=ǫρµσǫανβ∂ρ∂αHσβ, sothat Gµν =
(cid:3)Hµν +.... The Schouten tensor is Sµν = (Wµν)ρσWρσ[h] and the Cotton tensor, which
is symmetric, transverse and traceless is defined by Cνµ ≡ ǫµρσ∂ρSσν, with Sµν ≡ Gµν −
12ηµνG=Rµν− 14ηµνR
4
the SDM by performing the the self-dual change of variables w(h) of equation
(5) in (1). The connection with I goes through the following action
2
1
I2th[h,v] = <h ǫναβ∂ W [h]>
2 2m νρ α βρ
1
<h ǫµνρ∂ v λ >+ <v ǫµνρ∂ v λ > (14)
− µλ ν ρ 2 µλ ν ρ
Thisisobtainedsubstituting(5),whichisagainoneofthe equationsofmotion,
in (11).This procedure is the same used to obtain the second order model (2)
fromI . I2th[h,v]worksasamasteractionforI2th andITMM andisequivalent
1 2
to the slightly different one introduced in [4]. One of the equations of motion
states that the transverse parts of v and h are the same and using this in
the action we obtain that I2th[h(v),v)] = I2th[v]. On the other hand, making
2
independent variations respect to h , we obtain
µν
1
ǫµρσ∂ v ν = ǫµρσ∂ W ν[h] . (15)
ρ σ m ρ σ
Substituting (15) in (14) the third order action of the topologically massive
gravity emerges.
At this point we call the attention of the reader for the first time to Fig-
ure (1) where the relations between the different actions are summarized. The
curved arrow between I and ITMM indicates the self-dual change of variables
0
mentionedaboveandthestraightarrowsshowtheconnectionsamongthecanon-
ically equivalent actions.
I
0
I
1
I2 I2th
I I2th
3 2
I2th ITMM
3
ITMM ITMM ITMM
2 Quad 23
I4th
Figure 1: Connections tree between the self-duals models
Beforecontinuingwenotethatafamily I2th andinparticularI2th mayalso
N 3
be constructed using the mechanism of splitting the vector Chern Simons term
5
with the aidof auxiliary field. All these actions are equivalent. The action I2th
3
which we write in the form,
1 1
I2th[h,v,u] = <ǫναβ∂ h ǫρµσ∂ h > <(ǫαβγ∂ h )2 > (16)
3 2m α βρ µ σν −4m β γα
1
<h ǫµνρ∂ v λ >+<v ǫµνρ∂ u λ > <u ǫµνρ∂ u λ >
− µλ ν ρ µλ ν ρ −2 µλ ν ρ
isobtainedalsodirectlyfromI byeliminatingw. IfinI2th[h,v,u]weeliminate
3 3
the field v we obtain a second order master action for ITMM which we denote
by ITMM
2
1 1
ITMM[h,u] = <h ǫµνρ∂ W [h]>+ <u ǫµνρ∂ W [h]>
2 −2m µλ ν ρλ m µλ ν ρλ
1
<u ǫµνρ∂ u >. (17)
µλ ν ρλ
− 2
Now, making independent variations respect to u we obtain (15) with u in-
µν
stead of v, which substituted back in the action allows to recover the action
of TMM. We note that ITMM[h,u] may also be obtained making the self dual
2
changeofvariables (w w(h),h u) ofthe form(5) in I . All this relations
1
→ 7−→
are reflected in Figure (1).
Contrarytowhatonecouldhavebe expectedonecannotarrivefromITMM
2
toafourthorderactionandadifferentschemeshouldbedevisedtomakecontact
withthe fourthorderactionrecentlyintroducedinRef.[7,14]. This is givenby,
1
I4th[h] = <ǫναβ∂ W (h)ǫρµσ∂ W (h)> (18)
2m3 α βρ µ σν
1 1
<(ǫαβγ∂ W )2 > <W (h)ǫµνρ∂ W λ(h)>
− 4m3 β γα(h) −2 µλ ν ρ
Stressing the entangledstructure of these systems we note first that this fourth
orderaction is obtained after substituting the self-dualchange of variables(10)
into (2). This is indicated again with the curved arrow which appears between
I2th and I4th in Figure (1).
There exist two master actions which connect ITMM and I4th. The first is
denoted by ITMM because it use an auxiliary field u with a quadratic Fierz-
Quad µν
Pauli like mass term to express the term proportional to the Einstein action in
first order. It is given by
m
ITMM[h,u] = <u uνµ uµuν > <u ǫµνρ∂ hλ >
Quad 2 µν − µ ν − µλ ν ρ
1
+ <W [h]ǫµνρ∂ Wλ[h]>, (19)
2m2 µλ ν ρ
It is obviously equivalent to ITMM. Making independent variations on h
µσ
yields
1
ǫµνρ∂ [u (W )αβW [h]]=0. (20)
ν ρσ − m2 ρσ αβ
6
Locally,thesolutionis: u = 1 (W )αβW [h]+∂ ζ . Pluggingintothe
µν m2 µν αβ µ ν
action (after some algebraic manipulations), the fourth order action of Ref.[7,
14], is obtained.
The other master action is obtained introducing an auxiliary field v to
µν
split the Einstein action without changing its degree. It has the form,
1
ITMM[h,v] = <v ǫµρσ∂ W [v]> <h ǫµρσ∂ W [v]>
23 2 µν ρ σν − µν ρ σν
1
+ <W [h]ǫµνρ∂ Wλ[h]>. (21)
2m µλ ν ρ
Using the equations of motion we have
ǫµρσ∂ W [v]=ǫµρσ∂ W [h], (22)
ρ σν ρ σν
which establishes the equivalence with I .
TMM
On the other hand, independent variations on h lead to
µν
1
ǫµρσ∂ [W [v] (W )αβ[W [h]]=0 . (23)
ρ σν σν αβ
− m
This has as a solution
1
v = W [h] . (24)
µν µν
m
Using these results in ITMM the fourth order action is reached.
23
Here we reachthe limit to whichthe structure discussedin this sectionmay
be exploited. If we make the self-dual change of variables (5) into the action of
the topologically massive gravity (13) we obtain a 5th order action
1 1
I5th[h]= <W [h]ǫµνρ∂ Wλ[h]> <W [h]ǫµνρ(cid:3)∂ Wλ[h]>,
−2m2 µλ ν ρ −2m4 µλ ν ρ
(25)
which may be written in terms of symmetric variable H as
µν
1 1
I5th[H]= <H Cµν > <H (cid:3)Cµν >, (26)
−2m2 µν −2m4 µν
and was shown to propagate a ghost in Ref.[32].
3 Duality
Inthissectionwemaketheconnectionbetweenthemasteractionsconsideredin
the previous section and duality transformations. We use the gauge invariance
of some terms in the different models and an auxiliary field, which is restricted
to be a pure gauge by a constraint equation enforced by a Lagrange multiplier
to construct dual actions which are found to be the master actions. We start
considering the vector Chern-Simons term in the self dual action (1). This
term is invariant under δw = ∂ ζ We modify the self dual action with the
µν µ ν
7
introduction of a pair of variables B and h ) of which h is a Lagrange
µν µν µν
multiplier in the form,
1
IDual[w,B,h] = <(w +B )ǫµνρ∂ (w λ+B λ)>
0 2 µλ µλ ν ρ ρ
m
<w wνµ wµwν > <h ǫµνρ∂ B λ >. (27)
− 2 µν − µ ν − µλ ν ρ
The field equations which a
ǫ ρσ∂ [w +B ] m[w η w]=0, (28)
µ ρ σν σν µν µν
− −
ǫ ρσ∂ [w +B h ]=0 (29)
µ ρ σν σν σν
−
and
ǫ ρσ∂ B =0. (30)
µ ρ σν
The constraint(30) forces B =∂ ζ to be a pure gauge which can be gauged
µν µ ν
away to recover the self dual action. On the other hand to express the action
in terms of the Lagrange multiplier field, we note that Eq. (29) has the local
solution
B = w +h +l , (31)
µν µν µν µν
−
with ǫµνρ∂ l = 0. Substituting (31) into (27), we obtain the master action
ν ρσ
I [w,h](4)whichinterpolatesbetweenI andI2th[h]asdiscussedintheprevious
1 0
section.
InthesamewaywemayconstructthemasteractionI2th[h,v]whichconnects
2
the TMM with the IM starting from the latter. We modify the vector Chern
Simons term in the second expression of I2th[h] in Eq.(2) to write
1 1
I2th = <h ǫµαβ∂ W [h]> <[h +f ]ǫµνρ∂ [h +f ]>
Dual 2 µρ α βρ −2 µσ µσ ν ρσ ρσ
+ <v ǫµνρ∂ fλ > (32)
µλ ν ρ
Here, v is the multiplier which enforces the constraint
µν
ǫ ρσ∂ f =0, (33)
µ ρ σν
on the auxiliary field f insuring that it is a pure gauge and in consequence
µν
(32) is locally equivalent to (2). The other equations of motion are in this case
1
ǫ ρσ∂ [ W [h] (h +f )]=0, (34)
µ ρ σν σρ σρ
µ −
and
ǫ ρσ∂ [v (h +f )]=0, (35)
µ ρ σν σρ σρ
−
Solving f in (35) as
µν
f = h +v +l (36)
µν µν µν µν
−
with ǫ αβ∂ l =0 and substituting this solutioninto (32), we reachI2th[h,v].
µ α βν 2
8
Finally, followingthe procedureoutlinedabovewe canconstructthe master
actionITMM whichconnectstheTMMwiththefourthordermodel. Wemodify
23
ITMM[v] by introducing a pair of non symmetric fields, f and h in the
µν µν
following way:
1 1
I = <[v +f ]ǫµνρ∂ W [v+f]>+ <W [v]ǫµνρ∂ W [v]>
µσ µσ ν ρσ µσ ν ρσ
−2 2m
+ <h ǫµνρ∂ W λ[f]>. (37)
µλ ν ρ
In this case, h again plays the role of a multiplier enforcing the constraint
µν
that now takes the form
ǫ ρσ∂ W [f]=0. (38)
µ ρ σν
Ifinstead,weconsidertheequationofmotionobtainedtakingvariationsrespect
to f we get
ǫ ρσ∂ W [v+f h]=0. (39)
µ ρ σν
−
This equation can be solved as
f =h v +l , (40)
µν µν µν µν
−
where l satisfies ǫ ρσ∂ W =0 and substituting into (37) we reach ITMM
µν µ ρ σν[l] 23
It is interesting to note that it is also possible to construct ITMM out of
23
I2th in one step by modifying the first term in (2) instead of the second with
the auxiliary field restricted by a constraint of the form (35). The duality
transformations considered in this section may also be formulated in the path
integral approach in which case the elimination of one of the fields in favor of
the Lagrange multiplier is done by means of a quadratic functional integration
[22, 26, 27]
4 Models with two excitations
In 3D there exist also more than one model of linearized gravitywith two exci-
tations. The conventional Fierz-Pauliaction is given by
1 m2
I [h]= <h ǫµρσ∂ W [h]> <h hνµ h2 >. (41)
FP µν ρ σν µν
2 − 2 −
and the fourth order model
1
IG[h] = <W [v]Wνµ Wµ[v]Wν[v]> (42)
4 −2 µν [v] − µ ν
1
+ <W [v]ǫµνρ∂ W [W(v)]>,
2m2 µλ ν ρλ
which was proposed in [14].
In 3D describes the ghost free propagation of two physical excitations with
oppositehelicities. Thiscanbemadeexplicitbyobservingthat(41)isequivalent
9