General relativity from three forms in seven dimensions

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

General

relativity

from

three-forms

in

seven

dimensions

Kirill Krasnov

SchoolofMathematicalSciences,UniversityofNottingham,UK

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received5May2017

Receivedinrevisedform8June2017 Accepted9June2017

Availableonline29June2017 Editor:M.Cvetiˇc

We consider acertaintheoryof3-forms in7dimensions,and studyitsdimensionalreductionto 4D, compactifyingthe7-dimensionalmanifoldonthe3-sphereofafixedradius.Weshowthattheresulting 4Dtheoryis(Riemannian)GeneralRelativity(GR)inPlebanskiformulation,modulocorrectionsthatare negligibleforcurvaturessmallerthanPlanckian.Possiblythemostinterestingpointofthisconstruction isthatthedimensionallyreducedtheoryisGRwithanon-zerocosmologicalconstant,andthevalueof thecosmologicalconstantisdirectlyrelatedtothesizeofS3_{.RealisticvaluesofcorrespondtoS}3_of Plancksize.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

It is almost universally agreed that General Relativity (GR) should be viewed asa low energyapproximation to some other theory.Theusualviewisthatthisissomequantumgravitytheory thatgivestheultra-violet(UV)completionofperturbatively quan-tisedGR.

ItisalsopossiblethatGRarisesasthelowenergylimitofsome otherclassicaltheory,anditisonlythisdistinctfromGRclassical theory thatis tobe UV completed quantummechanically. A sce-narioofthissortispartofstringtheory,where4D gravityarises bycompactificationfrom11Dsupergravity.Thelatteristobe com-pletedquantummechanicallybyM-theory.

Scenariosembedding4DGRintootherclassicaltheoriesare in-terestingforseveralreasons.First, whatcomes out fromsuch an embeddingisusuallymorethanjustGR.Therearetypicallymore degreesof freedom, and this is ofinterest both phenomenologi-callyaswellasforthequestionoftheUVcompletion.Indeed,UV incompletetheoriesmayariseasthelowenergyapproximationof UV completetheories,thelatterhaving moredegreesoffreedom thantheformer.

In the extensively studied Kaluza–Klein scenarios 4D gravity arisesfromahigherdimensionaltheorythatisagaingravity (pos-siblywithextra fields).The purposeof thisletteristo point out that(Riemannian signature)GeneralRelativityinfourdimensions may be obtained by compactification from a higher-dimensional theoryofaverydifferentnature.Thus,weshowthatacertain dy-namicaltheoryof3-formsin7dimensions,whencompactifiedon

E-mailaddress:[email protected].

the3-sphere(ofafixedsize)givesa4Dgravitytheorythatatlow energiesisindistinguishablefromGR.

Mathematically, the idea of our construction is rather simple and naturalandcan be explained already here. It iswell-known thatgravitycanbeusefullydescribedusingthelanguageofframe fields instead of the metric. A frame eI is a collection of one-forms that are declared orthonormal. Thisdefines the metricvia

ds2

=

η

I JeIeJ,where

η

I J istheflat metric.Whentheframe field

is used todescribe geometry, itis alsovery naturalto allow the spinconnection wI J _{(laterusedtoconstructthecurvature)to}

be-comeanindependentobject.Itsrelationtoderivativesoftheframe isthenfixedbyitsfieldequation.

The(spin)connectionwI J _{canlocallybeviewedasaone-form}

with values in the Lie algebra of Lorentz group, or orthogonal group SO

(

D

)

of appropriate dimension D if one wants to de-scribe metrics of Riemannian signature. On the other hand, the frameone-formsare not

so

(

D

)

Liealgebravalued(apartfromthe caseof 3D gravity wherethe frame indexcan be identifiedwith theLiealgebraindex).However,onecanconstructtwo-formsthat are Lie algebra valued, taking the wedge product of frame with itself BI J

_:=

_eI

_∧

_eJ_{. This gives rise to a formalism for gravity}

in which the basic fields are Lie algebra valued one-forms wI J

andtwo-formsBI J.Thetwo-formsmusthoweverbeappropriately constrainedinordertoguarantee thatthey comefromthe frame field. This formalism is particularly simple in 4 dimensions, see below,butcanalsobeusedtodescribegravityinanydimension, see[1].

The main idea behind the constructions of this letter is that it is natural to combine the one- and two-forms of such a

for-http://dx.doi.org/10.1016/j.physletb.2017.06.025

mulation of gravity into a single object — a 3-form in a higher dimensionalspace. The space inquestion is (locally)the product ofthe spacetimeandthe SO

(

D

)

groupmanifolds,i.e.itisthe to-talspaceofthe principalSO

(

D

)

bundleoverspace(time).LetmI J

beLiealgebravaluedone-formsontheSO

(

D

)

groupmanifold,i.e. Maurer–Cartan forms.To exhibit thefields wI J and BI J as com-ponentsofa 3-forminthetotalspaceof thebundle,we usethe matrixnotationinwhichmI J

_,

_wI J _{areone-formsandB}I J _are

two-formsvaluedinthespaceofanti-symmetric D

×

D matrices. We thenconsiderthefollowing3-form

C

=

Tr

(

m

∧

m

∧

m

+

m

∧

m

∧

w

+

m

∧

B

) .

(1)

Thus, thespin connection 1-forms wI J _{arise astwo vertical one}

horizontal componentsofthe3-formC above,andthefields BI J

ariseastheoneverticaltwohorizontalcomponents.Thisrecovers all fields of the 2-formformulation of gravity as components of a 3-formin the total spaceof the SO

(

D

)

group bundle over the spacetime.In[2,3]itwasexplainedhow3Dgravitycanbeusefully describedinthisfashion.Thepurposeofthisletteristooutlinethe constructionthatworksfor4Dgravity.

While the idea described is general enough to work in any dimension,the3-formsaremostinterestingobjectsin6and7 di-mensions,aswereviewbelow.Ontheotherhand,thetotalspace ofthe SO

(

4

)

group principal bundle over a 4-dimensional mani-fold is 10 dimensional. But instead of working with the full 4D rotationgroupone cantake oneofits chiralhalves,i.e.eitherits self-dual oranti-self-dual part, both ofwhich are 3-dimensional. Thereisaformalismfor4Dgravity[4,5]thatworkswithone- and two-formswithvaluesintheLie algebraofthechiral halfofthe rotationgroup. We will baseour construction on thisformalism. Inthiscasethetotalspaceofthebundleis7-dimensional,andwe are in the setting where 3-forms are mostnatural and interest-ing.

Thefundamentalfactaboutageneric(orusingtheterminology of[6]stable)3-formC ona7-dimensionalmanifold

M

isthatit definesametric gC.Themetricisexplicitlygivenbythefollowing formula

gC

(ξ,

η

)

volC

= −

1

6iξC

∧

iηC

∧

C

.

(2)

Here gC

(ξ,

η)

istheresultofthemetriccontractionoftwovector

fields

ξ,

η

,volC isthevolume formforgC,andiξ,η isthe

opera-tionofinsertion ofa vector fieldinto aform. The metric(2)has beenknownfor morethan acentury, seee.g. [7]fora historical perspective.It isultimatelyrelatedto thegeometryofspinors in 7and 8dimensions, see e.g. [8] forthe discussion ofthe spinor aspect,andtooctonions,seee.g.[9].

Generic3-formsin7dimensionsarerelatedtotheexceptional group G2.ThiscanbedefinedasthesubgroupofGL

(

7

)

that

sta-bilises ageneric3-form,see[10]andmorerecently[6].Thespace of generic 3-forms (at a point) can then be identified with the coset GL

(

7

)/

G2. It is easy to see that the spaces GL

(

7

)/

G2 and

3

R

7havethesamedimension35.ThefactthatC definesgC ex-plainswhyG2isasubgroupofSO

(

7

)

.

The volume form volC, playing an important role below, can

also be described explicitly asa homogeneitydegree 7

/

3 object builtfromC.Thus,let

˜

a1...a7 _{bethe densitiesedcompletely}

anti-symmetrictensortakingvalues

±

1 inanycoordinatesystem.Here

a

=

1

,

. . . ,

7. We can then construct the following degree 7 and weight3scalar:

˜

a1...a7

˜

b1...b7

˜

c1...c7_C

a1b1c1

. . .

Ca7b7c7

.

(3)

Thecuberootofthisexpressionisa multipleofvolC.Thisisnot

themostusefulinpractise wayofcomputingthevolume form—

itisusuallymuchmoreeffectivetocomputethevolumevolC from

thedeterminantofgC.

Let usnow make C dynamical. Consider the following action principle

S

[

C

] =

1

2

M

C

∧

dC

+

6

λ

volC

.

(4)

Thefirsttermhere(i.e.thecase

λ

=

0)describesatopologicalfield theoryconsideredin[11].TheEuler–Lagrangeequationsfollowing from(4)are

dC

=

λ

∗C

.

(5)

Here ∗C isthe Hodge dualof C computedusing gC.The

numer-icalcoefficient ontheright-hand-sidehereissimplestverifiedby noticingthatvolC

= −

(

1

/

7

)

C

∧

∗C,andthenusingthe

homogene-ityto compute thevariation of volC withrespect to C.We note

that,becausethetwotermsin(4)scaledifferently,byrescalingC

we canalways achieve

λ

=

1 attheexpenseofintroducing a pa-rameterinfrontoftheaction.We willdosofromnowon.Thus, therearenofreeparametersinthetheory(4).

Real 3-forms C are of two possible types, see e.g. [7]. Forms of one type give gC ofsignature

(

4

,

3

)

. Forms of the other type

givetheRiemanniansignaturemetrics gC.Such3-formssatisfying

(5)describe what[8]call nearly parallel G2 structures. Notethat (5)impliesthat∗C isclosed.However,thisequationalsosaysthat

dC

=

0.Thus,thecriticalpointsof(4)arenotthepossiblymore fa-miliarinthiscontexttorsion-free G2 structuressatisfyingdC

=

0, d∗C

=

0 anddescribingG2holonomymanifolds.A related

observa-tionisthattheequation(5)impliesthatthemetricgC isEinstein with non-zeroscalar curvature, see proposition3.10 from[8]. In contrast,G2-holonomymanifoldsareRicciflat,seee.g.[6].

Equations(5)havebeenstudiedintheliterature[8].The vari-ational principle (4) is a subcase of a more general action (29) in [12], with1- and 5-forms set to zero andthe auxiliary met-ric integratedout. It is alsosimilar to prepotentials appearing in the literature on G2 compactifications of M-theory,see e.g. [13].

Nevertheless,itseemsthatthetheoryof3-forms(4)hasnotbeen studiedintheliterature.Notethataction(4)isdifferentfromthe onesconsideredbyHitchin[6].ThesimplestHitchinactionisthe lasttermin(4),restrictedto3-formsinafixedcohomologyclass. Ouractionisthesumofthosein[11]and[6],withtheonly con-strainton C beingthat itisgeneric (orstable),whichisan open condition.

Additionalinformationaboutthetheory(4)isgiveninarecent paper[14]. Inparticular,thisreferencedemonstratesthat(4)isa 7D theorywith3propagatingdegreesoffreedom. Also,some so-lutionsoftheequations(5)aredescribed, inparticularthosethat canbeobtainedasS3 _{bundlesover S}4_.

We nowdescribe arelation to4D GeneralRelativity(GR). We claimthat(4)dimensionallyreducedon S3 _{(ofafixedradius)isa}

4Dtheoryofgravitythatisforallpracticalpurposes indistinguish-able from(Riemannian signature) GR.This means that whilethe reducedtheory is, strictly speaking,not GR,it coincides withGR forWeyl curvatures smallerthan Planckian,whichis anyway the regime wherewe cantrust GRasa classicaltheory.All thisis to beexplainedinmoredetailsbelow.

coupledtoMaxwell.Allowingthesizeofthecircletobecome dy-namicalgivesrisetoanadditionalmasslessscalarfieldin4D,and toavoidconflict withobservations thismustbe givenamass,or stabilisedinsomeotherway.

TheKaluza–Klein mechanismgivesa geometricallycompelling unificationofgravity withelectromagnetism. Also,aspointedout byKaluza,itrelatesthequantum ofelectriccharge tothesizeof thecompactextradimension.TheKaluza–Kleinmechanismcanbe generalisedtodescribenon-Abeliangaugefields.A comprehensive reviewonKKise.g.[15].

Letusreturntoourstory.Weclaimedthat4DGRarisesasthe dimensionalreduction ofthe theory (4)of3-forms in7D. Unlike theKK case,nounificationisachievedhere,thereducedtheoryis puregravity.AlsounlikeKK,gravityin4Darisesfroma7Dtheory ofaverydifferentsort—thetheory (4)isa dynamicaltheory of 3-forms,notmetrics.Whileitmaybeamusingthat4DGRadmits alifttoatheory ofsuch adifferentnatureas(4),isthisauseful perspectiveon4Dgravity?

Now comes what we believe is the most interesting physics pointaboutourconstruction.As wewillshow,thedimensionally reducedtheoryisGRwithnon-zerocosmologicalconstant,andthe valueofthecosmologicalconstantisdirectlyrelatedtothesizeof the S3_{.As the5D Kaluza–Kleinstory makesthe electriccharge a}

dynamicallydeterminablequantity,atleastinprinciple,via some “spontaneous compactification” mechanism, in our setup the 4D cosmological constant becomes in principle determinable by the dynamicsoftheextradimensions.

Thus, our 7D lift of 4D GR makes the 4D cosmological con-stant adynamical object.Havingsaid this, wemustalso saythat inthisletterwelimitourselvestojustdemonstratingtherelation betweentheradiusof S3 and

.No attemptatstudying the dy-namics of the extra dimensions (and thus predicting

) will be made.Still,thisshouldbe keptinmindasone ofthemotivations forourconstruction, inadditionto thosedescribed inthe begin-ningofthistext.

Afterthesemotivationalremarks,wearereadytodescribethe dimensional reduction. We phrase the discussion that follows in termsofrealobjects.Inthiscasethedimensionallyreducedtheory istheRiemanniansignatureGR.Allobjectscanalsobe complexi-fied,inthiscaseoneobtainscomplexifiedGR.Thesubtlerissueof realityconditionsrelevantfortheLorentziansignaturetheorywill betreatedelsewhere.

Tocarryoutthedimensionalreduction,weneedtoassumethat the3-formC in7Dis“independent”of3ofthe7coordinates.The appropriate forourpurposeswayofdoing thisistoassume that we have the group SU

(

2

)

that acts on

M

freely. This gives

M

the structure of an SU

(

2

)

principal bundle over a 4-dimensional base M. Our considerations here are local, over a region in M. We parametrise fibre points as g

∈

SU

(

2

)

, withthe group action on the fibre beingthe right action ofSU

(

2

)

on itself. Denote by

m

=

g−1dg the Maurer–Cartan one-forms on SU

(

2

)

. Theseforms transform covariantly under the right action of SU

(

2

)

on itself. To establish further notations, let A be an SU

(

2

)

connection on the base M, i.e. a 2

×

2 anti-Hermitian matrix valued one-form on M, andlet A

=

g−1_{Ag be its lift into the total space of the}

bundle. Then W

=

m

+

A is the connection one-form in the to-talspaceof thebundle.Simple standardcomputation showsthat

F

:=

dW

+

W W isa2-formthatispurely horizontalF

=

g−1Fg, whereF

=

dA

+

AAisthecurvatureoftheconnectionone-formon thebase.Hereandinwhatfollows,forbrevity,weomitthewedge productsymbol.

AgeneralSU

(

2

)

invariant 3-formon X canbe writtenas C

=

Tr

(φ

m3

+

Am2

+

Bm

)

+

c.Here

φ

∈

0

(

M

),

c

∈

3

(

M

)

,while A

,

B

are liftsto thetotal spaceofthebundle ofLie algebravalued 1-and2-formsonthebaseM respectively.Notethatnoneofthe35

components of C has been lost here, asa simplecount of com-ponents in

φ,

A

,

B

,

c shows. The above parametrisation of C is howevernot theonemostsuitedforcomputations.Wenote that thetermquadraticinmcanalwaysbeeliminatedbyshiftingmby aLiealgebravalued1-form.Thisalsoredefines B

,

c.Thissuggests weparametrise

C

= −

2 Tr

1 3

φ

3_W3

₊

_φ

_{W B}

+

c

.

(6)

Here W

=

m

+

A is a connection in the totalspace of the bun-dle.Geometrically, anSU

(

2

)

invariant 3-formC inthetotalspace ofan SU

(

2

)

bundleover M definesaconnectionby declaringthe horizontal vector fieldsto be those that are in the kernelof the 1-formiξViηVC forarbitraryverticalvectorfields

ξ

V

,

η

V.In(6)we

simplychosetoparametrisethe3-formC bythisconnection.The parametrisation(6)ismostsuitedforpractical computations. Nu-mericalprefactorsareforfutureconvenience.

Asimplecomputationthengives

dC

= −2 Tr

φ

2d

φ

W3

+

(φ

3F

+

φ

B

)

W2 (7)

+

(

d

φ

B

+

φ

dAB

)

W

+

φ

F B

+

dc

.

Here dAB

=

g−1

(

dB

+

AB

−

BA

)

g is the liftto the bundle of the

covariantderivativeofLiealgebra-valued2-formBwithrespectto the connection A.Another simple computation using some trace identitiesgives

1 2

M

CdC

=

SU(2)

−

2

3Tr

(

m

3

₎

₍₈₎

×

M

−

2 Tr

(φ

4BF

+

(φ

2

/

2

)

BB

)

+

φ

3dc

.

We learn that the dimensional reduction of the first, topological term in(4), modulo the prefactor equal to thevolume of SU

(

2

)

, is the so-calledBF theory witha

-term, coupled to the scalar and3-formfields.Wefindthisresultinteresting initsownright. Thedimensionalreductionofthetopologicaltheoryistopological. Thus, ifthereisno second termin(4), varyingwithrespectto c

gives

φ

=

const,andwerecovertheusual Lagrangianofthe topo-logicalBFtheorywiththe

term.

Letusnowunderstandthedimensionalreductionofthesecond termin(4).Thisisano-derivativeterm,soitonlychangesthe “po-tential” forthe

φ,

B

,

c fields. Inthispaper we will set

φ

=

const

and c

=

0. The complete dimensional reduction is carried out in

[14].Settingthesize oftheextradimensionstoaconstant corre-sponds to

φ

=

const. At the same time, it is clear from (8) that the 3-formfield c is“conjugate” to

φ

andso settingthisfield to constantjustifiessettingc tozero.

Tocomputethevolumeformcorrespondingto(6)(withc

=

0) we needtowritethis3-forminSO

(

3

)

notations.Thisisachieved bydecomposingallmatrix-valuedfieldsintermsoftheSU

(

2

)

gen-erators

τ

i

_{= −}

₍

_i

_/

₂

_)σ

i_,where

_σ

i _{arethe usual Paulimatrices. So,}

wewrite W

=

Wi

τ

i_{etc.Thisgives}

C

=

φ

3

6

i jk_Wi_Wj_Wk

₊

_φ

_Wi_Bi

_.

₍₉₎

The metric gC and thus the volume form volC are then easiest

computedbyputtingthisC intoitscanonicalform.Asthe canon-icalformwetake

C

=

1

6

i jk_ei_ej_ek

₊

_ei

i

_.

₍₁₀₎

1

=

e45

−

e67

,

2

=

e46

−

e75

,

3

=

e47

−

e56

.

Thenotationhereiseab

=

eaeb,withwedgeproductimplied.Itis theneasytocheckthatforC initscanonicalform(10),themetric definedbyC via(2)isds2

C

=

7

a=1

(

ea

)

2.

To compute the metric for (9) we need to rewrite it in the canonical form (10). This is done by choosing a convenient parametrisationof Bi _{fields.Toestablishthisparametrisation, we}

note that the triple of 2-forms Bi defines a metric on the base inwhich theseformsare anti-self-dual (ASD). This isthe Urban-tkemetric[16].Infact,a simplecalculationwiththeformula(2)

showsthattheUrbantkeformula

g

(ξ,

η

)

vol

= −

1 6

i jk_i

ξ

i

∧

iη

j

∧

k (11)

arisesasthemetriconthe basefrom(2),withC inits canonical form(10).Thisclearlypointstowardsa7-dimensionaloriginofthe Urbantkeformula.The2-formsBicanthenalwaysbeparametrised as

Bi

=

√

Xi j

j

,

(12)

whereone usesan arbitrarybranchofthe squarerootof a sym-metricmatrix Xi j, andtwo-forms

i are orthonormal

i

∧

j

∼

δ

i j.Here

iisanorthonormalbasisofASD2-formsforthemetric defined(viaUrbantkeformula)by Bi.ThematrixXi j isdefinedas thatof thewedge productsof Bi. Wehave Bi

∧

Bj

= −

2Xi jvol,

wherevol isthevolumeformofthemetricwhoseASD2-forms

are

i.Substitutingtheparametrisation(12)into(9)we seethat the3-formcanbewritteninthefollowingway

C

=

ρ

1 6

i jk_ei_ej_ek

₊

_ei

i

,

(13)

with

ρ

=

(

det

(

X

))

1/4

,

ei

=

φ

ρ

√

Xi jWj

.

(14)

Thisputs C into a form that is a multiple of the canonical. The metric gC isthen

ρ

2/3 time themetricforwhichtheaboveea is

theframe.Thisgives

ds2_C

=

φ

2Wi X

i j

(

det

(

X

))

1/3W

j

₊

₍

_det

₍

_X

₎₎

1/6_ds2

.

(15)

Thevolumeformforthismetricis

volC

=

φ

3

6

i jk_Wi_Wj_Wk

₍

_det

₍

_X

₎₎

1/3_vol

.

(16)

We now put all pieces together and write the dimensionally reduced4DLagrangian,whichis(4)ontheansatz(6)(withc

=

0), dividedbythevolumeofthefibre.Wehave

L4D

=

φ

4BiFi

+

φ

2

2 B

i_Bi

₊

₃

_φ

3

₍

_det

₍

_X

₎₎

1/3_vol

.

(17)

This is a Lagrangian of the type “BF theory plus a potential for the B field”. From general considerations in [17] we know that thisisa 4D gravitytheory, inthesense that the onlydegreesof freedom it describes are, as in GR, the two polarisations of the graviton. Wealso know[18] that forsufficiently smallWeyl cur-vatures (i.e. sufficiently low energy) any such theory of gravity reducesto GR.

Wewouldnowliketoquantifytheregimeinwhichtheabove theoryreducesto GR.Tothisend,let usrewritethe lasttermin

(17)byintroducingtwoauxiliaryfields.Wehave

6

(

det

(

X

))

1/3vol

ˆ= −

Hi jBiBj

−

2

μ

(

det

(

H

)

−

1

)

vol

,

wherethemeaningofthehatsymbolovertheequalitysignis “on-shell”.Indeed,varyingtheright-hand-sidewithrespectto Hi j we get Xi j

₌

_μ

_det

₍

_H

₎₍

_H−1

₎

i j_{. The condition det}

₍

_H

₎

₌

_{1 imposed by}

theLagrange multiplier

μ

then sets

μ

=

(

det

(

X

))

1/3_{. Substituting}

theresulting solution Hi j

=

(

det

(

X

))

1/3

(

X−1

)

i j intothe firstterm wereproducetheleft-hand-side.

Using the above wayof writingthe last term in (17)we can rewritetheeffective4DLagrangianasfollows

L4D

/φ

4

=

BiFi

−

1 2M

i j_Bi_Bj ₍₁₈₎

−

2

μ

(

det

(

I

+

φ

2M

)

−

φ

3

)

vol

.

Here we defined a new matrix Mi j _{so that}

_φ

_H

₌

_I

₊

_φ

2_{M, and}

redefinedtheLagrangemultiplier

μ

.Thekeypointnowisthatthe constraintdet

(

I

+

φ

2M

)

=

const,whenexpandedinpowers ofM, approximates the constraintTr

(

M

)

=

const, andthisis knownto giveGeneralRelativityinitsPlebanskiformulation[4,5].

For readers lost in the sequence of field redefinitions per-formed to achieve (18) we describe in words the origin of all the fields. The field

φ

is the scalar field parametrising the size of the extra dimensional S3 _{in the 3-form (6). The 2-form field} Bi is also explicitly present in (6) as B

=

g−1Bi

τ

i_{g. The object} Fi

:=

d Ai

+

(

1

/

2

)

i jk_Aj_Ak_{isthecurvatureoftheSO}

₍

₃

₎

_connection Ai _{definedby C,with W from(6)givenby W}

₌

_m

₊

_g−1_Ai

_τ

i_g.

The auxiliary fields Mi j and

μ

where introduced in such a way that their elimination by their algebraic field equationsproduces thedimensionalreductionofthe“potentialterm”,i.e.thelastterm in (4). The volume formvol isdefined by Bi via formulas (12)

and(11).

We now claim that (18) describes Riemannian signature GR, plushigherordercorrectionsimmaterialintheregime ofnot too highWeylcurvatures.Toseethis,letusremindthereaderthe Ple-banskiLagrangian[4]

L_Pleb

=

M2_p

BiFi

−

1

2

i j

+

3

δ

i j

BiBj

.

(19)

Here M2p

=

1

/

8

π

G is thePlanckmass, G istheNewton’s, and

isthecosmologicalconstant.Here Bi _{isadimensionlessfieldthat}

describesthe metric (viaUrbantke formula).We now absorb the Planckmasssoastomake Bi(andthusthemetric)dimensionful. ThedimensionfulmetricmeasuresdistancesinunitsofthePlanck length.Thus,weredefineBi

→

Bi

/

M2_p

,

→

M2_p

,

→

M2_p

.

L_Pleb

=

BiFi

−

1

2

i j

+

3

δ

i j

BiBj

.

(20)

The new

,

are dimensionless. The object

is the (anti-self-dualpartof) theWeylcurvature,measuredinPlanck units.So,it satisfies

1 inall situationsin whichGR hasbeen tested, or canbetrusted.

Toexhibitclosesimilarlyto(18)wefurtherrewritethe Pleban-skiLagrangian by introducing an auxiliary field

μ

to impose the constraintthat

i j istracefree.Thus,wewrite

LPleb

=

BiFi

−

1 2M

i j_Bi_Bj

₋

₂

_μ

₍

_Tr

₍

_M

₎

₋

₎

_vol

.

(21)

The onlydifferencebetweenthisLagrangianand(18),apartfrom theoverallfactor,isthattheconstraintsimposedontheauxiliary matrixMi j aredifferent.

Toquantifythedifference,weparametrisethematrixMi j _via

Mi j

=

i j

+

3

δ

i j

_.

₍₂₂₎

Lagrangianwithrespectto

μ

.Thus,inthecaseofGRinPlebanski formalismtheconstraintsimplystatesthat

isaconstant —the cosmologicalconstant.Inthecaseofthetheory(18)oneobtains

asanon-trivialfunctionof

i j instead.Thisistheparametrisation inwhichthisclass of4D gravitytheories wasdiscovered in[18]. For(18)theconstraintgives

1

+

φ

2

3

3

−

3 2

1

+

φ

2

3

φ

4Tr

(

2

)

(23)

+

φ

6det

()

=

φ

3

.

We can use this as an equation to solve for

in terms of the tracefreepart.Underassumptionthat

1 andtoorder

2the solutionis

()

3

=

φ

−

1

φ

2

+

φ

2Tr

(

2

₎

₊

_O

₍

3

_).

₍₂₄₎

Itisnowclearthat(18)isthetheoryofthesametypeas(20), the only difference being that

is not a constant, but a func-tion

()

givenby(24).However, for

1 the dependenceof

()

on

in (24)can be neglected and

()

becomes a con-stant.Thisshowsthatthetheory(18)isindistinguishablefromGR initsform(21)inallsituationswhereGRhasbeentestedand/or canbe trusted. Notethatthe O

(

2

)

termin(24)isneglected as compared to

i j termin (22), not as compared tothe constant, which can be small. Detailed study of effects of modification of GRsuchas(24)one.g.thesphericallysymmetricsolutioncanbe foundin[19].

Theaboveargumentcanberephrasedasfollows.Thetheory(4)

or(18)isaclassicaltheorywithnoscaleinit.Themetricthatthe field Bi determinesmeasuresdistancesthatare dimensionless.As

(24)shows, thistheory behaves asGRwithnon-zero dimension-lesscosmologicalconstant

(φ

−

1

)/φ

2 forsmall

1 values of thedimensionlessWeylcurvature.Tocomparethiswiththeusual GR inwhich the distances are dimensionful,one needs to intro-duceascaleintotheory(18).AlternativelyonecanredefineallGR fields using some scale to make GR quantities dimensionless, as wehavedone inthepassagefrom(19)to(21).Thisscale canbe chosen to be the Planck scale, and then the theory (18) will be indistinguishablefromGRforcurvaturessmallerthanPlanckian.

Thus,choosingthescaleforidentifying(18)withGRtobethe Planckscale,thedimensionless

in(24)becomes the cosmolog-ical constant measured in Planck units, and is the extraordinary small number

∼

10−120 that embodies the cosmological con-stant problem. Ourtheory(18) givessmallcosmologicalconstant forvaluesofradiusofcompactification

φ

closetounity.Thismust holdtoextraordinaryhighaccuracy

φ

−

1

=

3M2p

,

(25)

wherewenowreinstatedthePlanckmasssothatthisistheusual dimensionful

,andomittedhigherorderterms.In(24)onecan alsogetsmall

forlarge

φ

,butthisgiveslargeGRmodifications asinthiscasetheTr

(

2

)

termin(24)cannolongerbeignored.

Thereareseveralopen questionsthat needtobe addressedto convert the model studied here into a realistic theory.First and foremost, one must find a dynamical mechanismfor driving the compactificationradius

φ

tounitytoproduceasmallcosmological constant. Similar issue is present in the usual Kaluza–Klein sce-narioswhereone needstoprovidea mechanismforspontaneous compactification.

We note, however, that the situation in theory (4) is some-what better than inthe usual KK setup. In the lattercase, apart from the case of compactification on S1, the pure gravity the-ory in 4

+

D dimensions usually doesnot have solutions of the

formoftheproductofMinkowskispacetimeand(compact) inter-nalmanifolds.Forthisreasononeusuallyextendsthepuregravity theory in 4

+

D dimensionswithextra fields,e.g. by considering theEinstein–Yang–Millssystem.Thestress–energytensorofthese extrafieldsthenallowsforsolutionsoftherequiredproductform, see e.g. [20], Section 3. Probably the mostfamous compactifica-tion mechanismisthat duetoFreund andRubin[21], wherethe 3-formfield ofthe 11Dsupergravityisdoing thejob.Incontrast, thetheory(4)admitsthesolutionthatisthe S3 fibrationover S4, see[14]foranexplicitdescription.Thus,atleastthereisasolution of(4)ofthedesiredtypewithouthavingtointroduceextrafields. However, the cosmological constant for the S3 fibration over S4

solutionistoolarge,see[14].Thisissimilartothesituationwith theFreund–Rubinsolution.

Thus, a compactification mechanism that would result in an appropriately smallcosmological constant is a very serious open issueforoursetup. Itispossible thatthe onlywayforwardisto add other fields.Wethenremarkthat thereisa verynatural ex-tensionofthetheory(4)thataddsformsofallodddegrees.Thisis thetheory thatappeared in[12],formula (29). Itwouldbe inter-esting to study4D compactificationsof thismoregeneraltheory. Wehopetoanalysethisinthefuture.

Anotheropen problemofthe presentapproach isthatof cou-pling to matter. Again, a naturalway toproceed is suggestedby supergravity.Onedoesnotcouplesupergravitytoextrafields,one simply studies what the modes already present become when viewedfromthe4Dperspective.Inparticular,whencompactifying on a coset manifold all modesrelatedto isometriesof the inter-nalspaceareknowntobeimportant.Indeed,recallthatthegauge groupthatarisesintheKKcompactificationisthegroupof isome-tries of the internal manifold, and its dimension may be larger than the dimension ofthe internal spaceitself. In thispaper we haveconsidereda compactificationonagroupmanifold,butonly retainedhalfofthe relevantisometriesby consideringthe invari-ant dimensionalreductionansatz. Itisclearthat additionalfields will arise by enlarging the ansatz by takinginto account all the isometries.Inthiscase,however,onemustbecarefulaboutthe is-sueofconsistenttruncation,seee.g.[22]foracleardescriptionof all theissuesarising.Weleave astudyofthedimensional reduc-tiononS3 _{viewedasacoset S}3

₌

_SO

₍

₄

_)/

_SO

₍

₃

₎

_{tofutureresearch.}

Third, there is a question of how to describe Lorentzian sig-nature metrics using this formalism. To do this one must make the3-formC complex-valued,andthen imposesomeappropriate reality conditions.Similarissues existinall Plebanski-related for-mulations.Wepostponetheirresolutiontofuturework.

Finally, to avoid confusion, we would like to say that our presentuseofG2structures(3-formsin7D)isdifferentfromwhat

onecanfindintheliteratureonKaluza–Kleincompactificationsof supergravity. In our approach a 3-form is not an object that ex-istinadditiontothemetric—itistheonlyobjectthatexist.The metric, andinparticularthe 4Dmetric, isdefinedby the3-form via(2).Also,inthesupergravitycontexta 7DmanifoldwithaG2

structure isusedforcompactifying the11Dsupergravitydown to 4D.Incontrast,wecompactifyfrom7Dto4D.

Acknowledgements

References

[1]L. Freidel, K. Krasnov, R. Puzio, BF description of higher dimensional gravity theories, Adv. Theor. Math. Phys. 3 (1999) 1289, arXiv:hep-th/ 9901069.

[2] Y. Herfray, K. Krasnov, C. Scarinci, 6D interpretation of 3D gravity, Class. QuantumGravity34 (4)(2017)045007,http://dx.doi.org/10.1088/1361-6382/ aa5727,arXiv:1605.07510[hep-th].

[3]Y.Herfray,K.Krasnov,Topologicalfieldtheoriesof2- and3-formsinsix di-mensions,arXiv:1705.04477[hep-th].

[4]J.F.Plebanski,OntheseparationofEinsteiniansubstructures,J. Math.Phys.18 (1977)2511.

[5] K.Krasnov,Plebanskiformulationofgeneralrelativity:apracticalintroduction, Gen.Relativ.Gravit.43(2011)1,http://dx.doi.org/10.1007/s10714-010-1061-x, arXiv:0904.0423[gr-qc].

[6]N.J.Hitchin,Stableformsandspecialmetrics,arXiv:math/0107101[math-dg]. [7]I.Agricola,OldandnewontheexceptionalgroupG2,Not.Am.Math.Soc.55

(2008)922.

[8]T.Friedrich, I. Kath,A.Moroianu, U.Semmelmann, OnnearlyparallelG(2) structures,J. Geom.Phys.23(1997)259–286.

[9]D.A.Salamon,T.Walpuski,Notesontheoctonions,arXiv:1005.2820[math.RA]. [10]R.L. Bryant, Metrics with exceptional holonomy, Ann. Math. 126 (1987)

525–576.

[11] A.A.Gerasimov, S.L. Shatashvili, Towardsintegrabilityoftopologicalstrings. I. Three-forms on Calabi–Yau manifolds, J. High Energy Phys. 0411 (2004) 074, http://dx.doi.org/10.1088/1126-6708/2004/11/074, arXiv:hep-th/ 0409238.

[12]N.Nekrasov,AlarecherchedelaM-theorieperdueZtheory:chasingM/f theory,arXiv:hep-th/0412021.

[13] C.Beasley,E.Witten,AnoteonfluxesandsuperpotentialsinMtheory com-pactificationsonmanifoldsofG(2)holonomy,J. HighEnergyPhys.0207(2002) 046,http://dx.doi.org/10.1088/1126-6708/2002/07/046,arXiv:hep-th/0203061. [14]K.Krasnov,Dynamicsof3-formsinsevendimensions,arXiv:1705.01741

[hep-th].

[15] M.J.Duff,B.E.W.Nilsson,C.N.Pope,Kaluza–Kleinsupergravity,Phys.Rep.130 (1986)1,http://dx.doi.org/10.1016/0370-1573(86)90163-8.

[16]H.Urbantke,OnintegrabilitypropertiesofSU(2)Yang–Millsfields.I. Infinitesi-malpart,J. Math.Phys.25(1984)2321.

[17] K.Krasnov,Plebanskigravitywithoutthesimplicityconstraints,Class. Quan-tum Gravity 26 (2009) 055002, http://dx.doi.org/10.1088/0264-9381/26/5/ 055002,arXiv:0811.3147[gr-qc].

[18]K. Krasnov, Renormalizable non-metric quantum gravity?, arXiv:hep-th/ 0611182.

[19] K. Krasnov, Y. Shtanov, Non-metric gravity. II. Sphericallysymmetric solu-tion,missingmassandredshiftsofquasars,Class.QuantumGravity25(2008) 025002, http://dx.doi.org/10.1088/0264-9381/25/2/025002, arXiv:0705.2047 [gr-qc].

[20] D. Bailin, A.Love, Kaluza–Klein theories, Rep.Prog. Phys. 50 (1987) 1087, http://dx.doi.org/10.1088/0034-4885/50/9/001.

[21] P.G.O.Freund,M.A.Rubin,Dynamicsofdimensionalreduction,Phys.Lett.B97 (1980)233,http://dx.doi.org/10.1016/0370-2693(80)90590-0.