A Chern–Simons gravity action in d=4

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Physics

Letters

B

www.elsevier.com/locate/physletb

A

Chern–Simons

gravity

action

in

d

=

4

F. Izaurieta,

I. Muñoz,

P. Salgado

∗

DepartamentoofFísica,UniversidaddeConcepción,Casilla160-C,Concepción,Chile

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received2July2015

Receivedinrevisedform10August2015 Accepted12August2015

Availableonline14August2015 Editor: M.Cvetiˇc

Recently,Antoniadis,KonitopoulosandSavvidyhaveintroducedinRefs.[1–4]aproceduretoconstruct background-free gauge invariants, using non-abelian gauge potentials described by forms of higher degree. Their construction is particularly useful because it can be used in both, odd- and even-dimensional spacetimes.Usingtheirtechnique,wegeneralizetheChern–Weiltheoremand constructa gauge-invariant,(2n+2)-dimensionaltransgressionform,andstudyitsrelationshipwiththegeneralized Chern–SimonsformsintroducedinRefs.[1,2].

UsingthemethodsforFDAmanipulationanddecompositionin1-formsdevelopedinRef.[5]andapplied inRefs.[6]and[7],weconstructafour-dimensionalChern–Simonsgravityaction,whichisoff-shellgauge invariantundertheMaxwellalgebra.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Manymodels instringtheorypredictan infinitetowerof par-ticlesofarbitraryhighspinintheirspectrum(e.g.,seeRefs.[8,9]). Forinstance, inthe low energylimit of open string theory with Chan–Patoncharges[10],themasslessstatescanbeidentifiedwith Yang–Millsquanta.

A text book example of the situation is provided by the so-called Kalb–Ramond field (also known as NS–NSB field). It is a 2-form quantum field, i.e. an antisymmetric tensor field Bμν

withtwo indices.The gaugetransformation

δ

Bμν

(

x

)

=

∂

μ

ε

ν

(

x

)

−

∂

ν

ε

μ

(

x

)

leaves invariant the 3-form field strength Hμνρ

(

x

)

=

∂

μBνρ

+

∂

νBρμ

+

∂

ρBμν.Therefore whena backgroundmetricis

provided,it ispossibleto constructthe invariantaction principle

[8]SKR

= −

₁₂1

dDx

√

|

g

|

HμνρHμνρ

= −

12

H

∧ ∗

Hforthefields.1 Thesefieldsof ranktwo andhigher are interesting aspartof the spectrum of a QFT. This motivates a generalization of Yang– Millssymmetry toincludeformsofhigher degreeasnon-abelian gaugefields.Inrecenttimes,importantresearchhasbeencarried outinthisdirection,seeforinstanceRefs.[11–15].

In particular, in Ref. [1] this idea of using forms of higher degree as non-abelian gauge fields was used to construct gauge

*

Correspondingauthor.

E-mailaddresses:fi[email protected](F. Izaurieta),[email protected](I. Muñoz), [email protected](P. Salgado).

1 _{Fromnowonandtomakethenotationlesscumbersome,thewedge‘∧’product} betweendifferentialformswillbeimplicitlyassumed.

invariant Lagrangian forms which are independent ofthe metric. Theseforms areanalogoustothe Pontryagin-formsinYang–Mills gaugetheory.

TheseresultsweregeneralizedinRefs.[2–4].Therewerefound closed invariant forms, similar to the Pontryagin–Chern forms in non-abelian tensor gauge field theory. These forms are based on non-abelian tensorgaugefieldsandare polynomialon the corre-spondingcurvatureforms.

It is the purpose of this paper to extend these results to the case of the Chern–Weil theorem and transgression forms in

(2

n

+

2)dimensions.Usingthisgeneralizedcase,wewillconstruct a four-dimensional Chern–Simons gravity action invariant under theMaxwellalgebra.Thisisaccomplishedusingtheformalism de-velopedinRefs.[1–4]andafterwardsusingthemethodsdeveloped inRef.[5],andwhichwerelaterappliedinRefs.[6]and[7].

This paper is organized as follows. In Section 2 we briefly review the usual Chern–Simons theory and the non-abelian ten-sor gauge theory. In Section 3 we study a generalization of the Chern–Weiltheorem.Takingthistheoremasthestartingpoint,it ispossibletoconstructgeneralized

(2

n

+

2)-dimensional trangres-sion forms,whichallowsustoreproducethe

(2

n

+

2)-dimensional Chern–Simons forms obtainedin Refs. [1,2]. These mathematical resultsare usedin Section 4to studytheconstruction ofan off-shellgauge-invariantChern–Simons–Antoniadis–Savvidyactionfor gravity ind

=

4. We finishinSection 5withsome final remarks andsomeconsiderationsonfuturepossibledevelopments.

http://dx.doi.org/10.1016/j.physletb.2015.08.030

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

2. Gravitationasagaugetheory

Quantum chromodynamics and the Weinberg–Salam electro-weakunificationaredescribedbygaugetheories.Thesameistrue for GUTs. However, the standard theory of General Relativity for gravitydoesnotcorrespond toagenuinely off-shellgauge invari-anttheory,eventhoughGeneralRelativityandYang–Millstheories havesimilargeometricfoundations.

Yang–Mills theories require a background metric in order to construct the action principle. On the contrary, an authentic gauge invariant theory of gravity requires a background-free ac-tion principle. An action for gravity fulfilling this condition is providedby Chern–Simons andTransgressionformgravities.This kind of theories have been extensively studied (see for instance Refs. [16–25]), but the construction is possible only in the case ofodd-dimensionalspacetimes.Inthisworkweconsiderthe con-structionofa transgressionformin evendimensions anda four-dimensional Chern–Simonsgravity action.The resulting theoryis off-shellgaugeinvariantundertheMaxwellalgebra.

2.1. Chern–Weiltheorem:Chern–Simonsandtransgressionsforms

Chern–Simons

(2

n

+

1)-forms canbe obtainedaslocal poten-tialsforthe

(2

n

+

2)-Pontryagin–Chernforms

P

2n+2

=

Fn+1

,

(1)

where

· · ·

stands for a multilineal symmetricinvariant polyno-mialfortheLiealgebra

· · · :

g

n+1

→

R

,

asforinstancetheoneprovidedbythesymmetrizedtraceinsome matrix representationof the Liealgebra (see [26]), i.e.,

Fn+1

=

Str

(

F

∧ · · · ∧

F

)

n+1

.

The Pontryagin–Chern forms (1) which satisfy the condition d

P

2n+2

=

0, where F

=

dA

+

A2 is the 2-formYang–Mills field-strength ofthe 1-formvector field A. From thePoincaré lemma, we know that locally there exists a

(2

n

+

1)-form

C

2n+1 such that

P

2n+2

=

d

C

2n+1.This

(2

n

+

1)-form

C

2n+1 iscalleda Chern– Simonsform.

ItiseasilycheckedthattheChern–Simonsformislocally quasi-invariantundergaugetransformations(i.e.invariantmoduloclosed forms)[32].InordertofindanexplicitexpressionfortheChern– Simons form, we have to make use ofthe Chern–Weil theorem, whichwewillsketchbriefly.

2.1.1. Chern–Weiltheorem

Let A0 and A1 be two one-formgauge connections ona fiber bundle over a

(2

n

+

1)-dimensionalbasemanifold M,and let F0 and F1 be the corresponding curvatures. Then, the difference of Pontryagin–Chernformsisexact,

Fn₁+1

−

F₀n+1

=

d

T

(2n+1)

(

A1

,

A0

) ,

(2) where

T

(2n+1)

(

_A1,A0)

=

(

_n

+

₁

)

1 0 dt

F_tn

(3)

is called a transgression

(2

n

+

1)-form, where

=

A1

−

A0 and

At

=

A0

+

t

.

The 2-form Ft stands forthe field-strength ofthe 1-formconnection At, Ft

=

dAt

+

AtAt.

Setting A0

=

0 and A1

=

A in(3),we obtainthe well known Chern–Simons

(2

n

+

1)-form

C

2n+1

(

A

)

=

T

(2n+1)

(

A

,

0

)

=

(

n

+

1

)

1 0 dt

A

tdA

+

t2A2

n

.

(4)

From the Chern–Weil theorem it is straightforward to show thatundergaugetransformationsd

δ

C

(2n+1)

(

A

)

=

0,i.e.theChern– Simons formisquasi-invariant. However,it isimportantto stress that since a connectioncannot be globally setto zerounless the bundle (topology) is trivial, Chern–Simons forms turn out to be onlylocallydefined.

ThesepropertiesimplythatChern–Simonsformshavenice fea-turesasLagrangians:

(

i

)

they leadtogaugetheorieswitha fiber-bundle structure,whoseonlydynamicalfield isaone-formgauge connection A, and

(

ii

)

they do change by only a total deriva-tiveundergaugetransformations.Whenwechoose

g

=

so

(2

n

+

2) we can write A

=

1_lea_P

a

+

1₂

ω

abJab, and therefore the Chern– Simons form provides with a background-free gravity theory in

d

=

2n

+

1.The main drawbackof theconstruction isthat trans-gression and Chern–Simons Lagrangians seem to be intrinsically odd-dimensional.Inthefollowingsectionswewillshowhowthis issuecanbecircumvented.

2.2. Non-abeliantensorgaugefields

TheideaofextendingtheYang–Millsfieldstohigherrank ten-sorgaugefieldswasusedinRef.[1]inordertoconstructgauge in-variantandmetricindependentformsinhigherdimensions.These formsareanalogoustothePontryagin–ChernformsinYang–Mills gaugetheory.

These results were generalized in Refs. [2–4], where the au-thorsfoundclosedinvariantformssimilartothePontryagin–Chern forms in non-abelian tensor gauge field theory. These forms are based on non-abelian tensor gauge fields andare polynomial on thecorrespondingcurvatureforms.

ALiealgebravalued,1-formconnectionA canbewritten mak-ing moreorlessexplicitdependence ontheLiealgebragenerator basis Taorthebasisof1-formsdxμ,

A

=

Aμ

⊗

dxμ

=

AaμTa

⊗

dxμ

.

The sameis trueforthe gauge potential 2-form B

=

₂1_!Bμν

⊗

dxμdxν

=

₂1_!Ba

μνTa

⊗

dxμdxν. The corresponding 2-form and 3-form “curvatures” are given by F

=

₂1_!Fμν

⊗

dxμdxν and H

=

1

3!Hμνλ

⊗

dxμdxνdxλrespectively,where

F

=

dA

+

A2

,

H

=

DB

=

dB

+ [

A

,

B

]

.

(5)

Thecurvatures F andHsatisfytheBianchiidentities,

DF

=

0, (6)

DH

+ [

B

,

F

] =

0. (7)

The infinitesimal, non-abelian gauge transformations of the generalizedgaugefieldsaregivenby

δ

A

=

Dξ0

,

(8)

δ

B

=

Dξ1

+ [

B

, ξ

0

]

,

(9)

where

ξ

0isa0-formgaugeparameter

ξ

0

=

ξ

aTaand

ξ

1isa1-form gaugeparameter

ξ

1

=

ξ

aμTa

⊗

dxμ[1].Underthesegauge transfor-mations,thecurvaturestransformas[2]

δ

F

=

D(δA

)

=

[F

, ξ

0] (10)

2.3.Chern–Simonsformsin

(2

n

+

2)dimensions

InRefs.[1,2]therewerefoundclosedinvariantformssimilarto thePontryagin–Chernformsinnon-abeliantensorgaugefield the-ory.Inparticular, itwas foundthatthere existsagauge invariant metric-independentinvariant

(

A

)

in

(2

n

+

3)-dimensional space-time

2n

₊3

=

FnH

(12)

whereH

=

dB

+

[A

,

B] isthe3-formfield-strengthtensorforthe rank-2 gaugefield B.Bydirectcomputation ofthederivativeitis possibletoprovethat

2n+3isaclosedform, d2n+3

=

0 (seethe proof in Ref. [2]). According to the Poincaré lemma, this implies that

2n+3 can be locally written asan exterior differential of a certain

(2

n

+

2)-form.Inordertofindthispotential

(2

n

+

2)-form, thevariationof

2n+3 inducedbya variationof A and B is com-puted.Since

δ

F

=

D

(δ

A

),

δ

H

=

D

(δ

B

)

+ [

δ

A

,

B

]

,

(13)

thevariation

δ

2n+3isgivenby

δ

2n+3

=

δ

F Fn−1H

+ · · · +

Fn−1

δ

F H

+

Fn

δ

H

=

d

δ

A Fn−1H

+ · · · +

Fn−1

δ

A H

+

Fn

δ

B

(14) FollowingRef.[26],weintroduceaone-parameterfamilyof po-tentialsandstrengthsthroughtheparametert,0

≤

t

≤

1:

At

=

t A

,

Ft

=

t F

+

(

t2

−

t

)

A2

,

Bt

=

t B

,

Ht

=

t H

+

(

t2

−

t

)

[

A

,

B

]

.

Whena variationofthe form

δ

=

δ

t

(∂/∂

t

)

ischosen, wehave

δ

At

=

δ

t A andBt

=

δ

t B.Fromeq.(14),wehave

2n

₊3

=

FnH

=

d

C

(2n_ChSAS+2)

,

(15) wherethe

(2

n

+

2)-form

C

_ChSAS(2n+2), iswhat we will call a “Chern– Simons–Antoniadis–Savvidy”form,anditisgivenexplicitly by

C

(2n_ChSAS+2)

(

A

,

B

)

=

1

0

dt

A Fn_t−1Ht

+

. . .

+

Ftn−1A Ht

+

FtnB

.

(16) This resultis analogous to the usual Chern–Simons form(4), butin even dimensions[2]. From eq. (16), we havefor the case

n

=

1[2],

C

(4)_ChSAS

=

1

0

dt

A Ht

+

FtB

=

F B

.

(17) Thismeans that the four-dimensional Chern–Simons–Antonia-dis–Savvidyactionisgivenby

S

(

A

,

B

)

=

M4

F B

.

(18)

Using eqs. (10) and (9), it is direct to prove that the action eq. (18) is gauge invariant (modulo boundary terms) under the transformationseqs.(8)and(9).

3. Transgressionsformsin

(

2n

+

2

)

dimensions

In this section we prove that it is possible to generalize the transgression form and the Chern–Weil theorem to the

(2

n

+

2)-dimensional case. The theorem ingredients are:

(

i

)

Two Lie-algebra valued, connection 1-forms A0 and A1. Their cur-vatures are given by F0

=

dA0

+

A02 and F1

=

dA1

+

A21, re-spectively.

(

ii

)

Two Lie-algebra valued, generalized connection 2-forms B0 and B1. Their generalized curvatures are given by

H0

=

dB0

+ [

A0

,

B0

]

and H1

=

dB1

+ [

A1

,

B1

]

, respectively

(

iii

).

Intermsofthesefundamental ingredients,itispossibletodefine thedifferences

θ

=

A1

−

A0 and

=

B1

−

B0,andthe interpolat-ing connections At

=

A0

+

t

θ

and Bt

=

B0

+

t

.

Theircurvatures aregivenby

Ft

=

dAt

+

A2t

,

(19)

Ht

=

DtBt

=

dBt

+ [

At

,

Bt

]

.

(20) Theysatisfytheconditions

d

dtFt

=

Dt

θ

(21)

d

dtHt

=

Dt

+

[θ,Bt] (22) 3.1. GeneralizedChern–Weiltheorem

Let A0 and A1 be two gauge connection 1-forms, andlet F0 and F1 be their corresponding curvature 2-forms.Let B0 and B1 betwogaugeconnection2-formsandlet H0 andH1 betheir cor-respondingcurvature3-forms.Then,thedifference

_2n(1)₊₃

−

_2n(0)₊₃ isanexactform,

_2n(1)₊₃

−

_2n(0)₊₃

=

F₁nH1

−

F0nH0

=

d

T

(2n+2)

(

A0,B0

;

A1,B1), (23) where

T

(2n+2)

(

A0,B0

;

A1,B1)

=

1 0 dt

n

Fn−1

θ

Ht

+

Fnt

(24)

iswhatwecalla“Antoniadis–Savvidytransgressionform”.

Proof. LetusstartwritingtheLHSofeq.(23)as

F₁nH1

−

Fn0H0

=

1 0 dtd dt

F_tnHt

.

Usingeqs.(21)and(22),

_2n(1)₊₃

−

_2n(0)₊₃

=

1 0 dt

nFn_t−1dFt dt Ht

+

F_tndHt dτ

,

=

1 0 dt

n

Fn_t−1Dτ

θ

Ht

+

d

F_tn

−

(

−

1

)

p

[Bt

, θ

]Ftn

.

Since n

F_tn−1Dτ

θ

Ht

=

nd

Fn_t−1

θ

Ht

−

θ

Bt,F_tn

,

wehave

F₁nH1

−

F0nH0

=

1 0 dt

nd

F_tn−1

θ

Ht

+

d

F_tn

−

(

−

1

)

p

Bt

, θ

Ftn

,

=

d 1 0 dt

n

F_tn−1

θ

Ht

+

F_tn

.

Therefore, defining the

(2

n

+

2)-Antoniadis–Savvidy-transgres-sionformas

T

(2n+2)

(

A0

,

B0

;

A1

,

B1

)

=

1 0 dt

n

Fn−1

θ

Ht

+

Ftn

,

wehave

F₁nH1

−

F0nH0

=

d

T

(2n+2)

(

A0,B0

;

A1,B1).

2

Following the procedure followed in the case of the Chern– Simons forms,we define the

(2

n

+

2)-Chern–Simons–Antoniadis– Savvidyformas

C

(2n_ChSAS+2)

=

T

(2n+2)

(

A

,

B

;

0,0)

=

1 0 dt

n A Fnt−1Ht

+

B Ftn

.

Thisresultagrees withtheexpressionfoundbyAntoniadisand Savvidy in Refs. [1,2]. It is interesting to notice that transgres-sionforms(both,standardonesandtheabovegeneralization)are defined globally on the spacetime basis manifold of the princi-palbundle,andareoff-shellgaugeinvariant.Chern–Simonsforms (both, standard ones and the Antoniadis–Savvidy generalization) are locally defined and are off-shell gauge invariant only up to boundary terms (i.e., quasi-invariants). Physical consequences of this subtle difference between Chern–Simons and transgression formshas beenstudied inthe literature forthecase ofstandard odd-dimensional Chern–Simons gravity in Refs. [33,34]. What it couldimplyinthecurrentapproachisworkinprogress,asitwill require a deeper exploration of the phenomenology of this kind of theories for specific symmetries. For this reason, in the next sectionwewillstudytheconstructionoffour-dimensionalgravity usingtheAntoniadisandSavvidy[1,2]expressionfor

C

(2n_ChSAS+2)with

n

=

1 andtheMaxwellalgebraasgaugesymmetry.

4. Chern–Simons–Antoniadis–SavvidyformfortheMaxwell algebra

We have seen that the four-dimensional Chern–Simons–Anto-niadis–Savvidyactioncorrespondsto

SChSAS

(

A

,

B

)

=

M4

F B

,

(25)

and it is invariant (modulo boundary terms) under the gauge transformationseqs. (8) and(9) [1,2].Now we willuse this con-structionforthe particularcaseof theMaxwell algebra, in order toshowtheconnectionbetweeneq.(25)andgravityind

=

4.

4.1. Maxwellalgebra

Theso-called Maxwellalgebrawasintroducedintheearly sev-enties(seeRefs.[28,29])asanalgebraencodingthesymmetriesof

aparticlemovinginaconstantelectromagneticfield.Thisalgebra is generateby

{

Pa

,

Jab

,

Zab

}

where Pa are not common Poincaré translations.Infact,thecommutationrelationsoftheMaxwell al-gebraread

[

Pa

,

Pb

] =

Zab

,

[

Jab

,

Pc

] =

ηbc

Pa

−

ηac

Pb

,

[

Jab

,

Jcd

] =

ηbc

Jad

+

ηad

Jbc

−

ηac

Jbd

−

ηbd

Jac

,

[

Jab

,

Zcd

] =

ηbc

Zad

+

ηad

Zbc

−

ηac

Zbd

−

ηbd

Zac

.

Thisalgebraanditsinvariantpolynomialscanbestudiedinthe context ofS-expansions (whereitcorresponds tothe

B

4 algebra, seeRefs.[30,31]).

In order to write down a four-dimensional Chern–Simons– Antoniadis–Savvidy action forMaxwell algebra we start fromthe gaugeconnections AandB.Theconnection1-formA isexpressed intheMaxwellbasisas

A

=

1 le a_P a

+

1 2

ω

ab_J ab

+

1 2k ab_{Zab, (26)}

whereeaisidentifiedasthevierbein1-form,

ω

ab_isthespin con-nection 1-form,andkab _{isan extraantisymmetricbosonic1-form} field.Thecorresponding2-formcurvature F

=

dA

+

A A isgivenby

F

=

1 lT a_P a

+

1 2R ab_J ab

+

1 2F ab_Z ab

,

(27)

where Ta _andRab _{arethestandardtorsionandLorentzcurvature} 2-forms, Ta

=

dea

+

ω

abeb

,

Rab

=

d

ω

ab

+

ω

a_cωcb

,

Fab

=

Dωkab

+

1 l2e a_eb

.

(28)

From eq. (28)in thecase Ta

=

Rab

=

Fab

=

0, we recoverthe Maurer–CartanequationsfortheMaxwellalgebra,

dea

+

ω

a beb

=

0, (29) d

ω

ab

+

ω

a_cωcb

=

_{0, (30)} Dωkab

+

1 l2e a_eb

=

_{0. (31)}

Forthetwo-formB,wecanwrite

B

=

BaPa

+

1 2B ab_J ab

+

1 2

β

ab_{Zab, (32)}

where Ba

,

Bab

,

β

ab are2-forms thatwemustdetermine.The cor-responding3-formcurvatureH

=

DB

=

dB

+ [

A

,

B

]

isgivenby

H

=

HaPa

+

1 2H ab_J ab

+

1 2

ab_Z ab where, Ha

=

DωBa

−

1 lB a beb Hab

=

DωBab

ab

=

Dω

β

ab

+

kacBcb

+

kbcBac

+

1 l

[

e a_Bb

−

_eb_Ba

]

Theseequationsareanalogoustoequation (2.13)ofRef.[5],or to equation (III.6.47) of Ref. [27], and therefore it is not a free differential algebra (FDA). But when the condition Ha

=

Hab

=

DωBa

−

1 lB a beb

=

0, (33) DωBab

=

0, (34) Dω

β

ab

+

kacBcb

+

kbcBac

+

1 l

[

e a_Bb

−

_eb_Ba

] =

_{0. (35)} Thesetofequations(29),(30),(31),(33),(34),(35)correspondto anFDAforthefields

{

ea

,

ω

ab

,

_kab

,

_Ba

,

_Bab

,

β

ab

}

_.

The problem now is to express the form B defined by the FDA relations (33), (34) and (35) in terms of the one-forms

{

ea

,

ω

ab

,

_kab

}

_{oftheMaxwellalgebra.}

To express the 2-forms

Ba

,

Bab

, β

ab

as the wedge product ofthe1-forms

ea

,

ω

ab

,

_kab

_{we followaprocedure developedin} Refs.[5,6].Weimposetheansatz

Ba

=

a1 2l

ω

a beb

+

a2 2lk a beb

,

(36) Bab

=

b1 2l2e a_eb

+

b2 2

ω

a ckcb

+

b3 2k a ckcb

+

b4 2

ω

a_cωcb

,

(37)

β

ab

=

c1 2l2e a eb

+

c2 2

ω

a ckcb

+

c3 2k a ckcb

+

c4 2

ω

a_cωcb

,

(38)

wherea1

,

a2

,

b1

,

. . . ,

b4

,

c1

,

. . . ,

c4 are arbitraryconstants.Inorder tofixthem,weimposethatthefieldsmustsatisfytheFDA condi-tionsgivenbyeqs.(29),(30),(31),(33),(34),(35).

Introducingeqs.(36)and(37)ineqs.(33)and(34)wefind

a1

=

b4, a2

= −

b1, b2

=

b3

=

0

,

(39) andusingnoweqs.(36),(37)and(38),weobtain

c2

=

2a1, c3

=

2a2. (40) ItmeansthattheFDAfieldsaregivenby

Ba

=

a1 2l

ω

a beb

+

a2 2lk a beb

,

(41) Bab

=

a1 2

ω

a c

ω

cb

−

a2 2l2e a_eb

,

₍₄₂₎

β

ab

=

c1 2l2e a_eb

+

a1 2

ω

a ckcb

+

a1 2

ω

b ckac

+

a2 2k a ckcb

+

a2 2k b ckac

+

c4 2

ω

a c

ω

cb

.

(43)

There are four arbitrary constants in the FDA expansion in termsof 1-forms; the fields given by eqs. (41), (42),(43) repre-sent the most general solution that can be built fromthe fields

{

ea

,

ω

ab

,

_kab

}

_{.Any choice ofthe constants representasolution to} theFDA.

Itisinterestingtonotethatifc1isaconstantthenitispossible towrite,c1

=

a1

+

γ

,where

γ

isanotherconstant.Choosinga2

=

c4

=

γ

=

0 leadstothesolutiongivenby

B

=

a1

2

[

A

,

A

]

.

(44)

4.2.Chern–Simons–Antoniadis–SavvidyLagrangian

UsingtheinvarianttensorfoundinRef.[31],

JabJcd

=

α

0l2

ε

abcd

,

JabZcd

=

α

2l2

ε

abcd

,

(45) being

α0

and

α2

arbitrary constants,the Chern–Simons–Antonia-dis–Savvidy Lagrangian 4-form

L

(4)_ChSAS

≡

C

(4)_ChSAS in 4D eq. (17) is explicitlygivenby

L

(4) ChSAS

=

1 4

α

0l 2

ε

abcdRabBcd

+

1 4

α

2l 2

ε

abcdRab

β

cd

+

1 4

α

2l 2

ε

abcdDωkabBcd

+

1 4

α

2

ε

abcdB ab_ec_ed

.

₍₄₆₎

IntroducingtheFDAexpansiongivenbyeqs.(41),(42)and(43)

in(46),the Chern–Simons–Antoniadis–Savvidy Lagrangianforthe Maxwellalgebratakestheform

L

(4) ChSAS

=

μ

8

ε

abcdR ab_ec_ed

+

ν

l2 8

ε

abcdR ab

ω

c f

ω

f d

−

σ

8l2

ε

abcd(e a_eb_ec_ed

+

_2l2_kab_Tc_ed

−

_2l4_Rab_kc fkf d

)

+

τ

8

ε

abcd(

ω

a f

ω

f beced

+

l2Dωkab

ω

cf

ω

f b

)

−

σ

8d

(

ε

abcdk ab_ec_ed

),

₍₄₇₎ where

μ

=

α2

c1

−

a2α0,

ν

=

(

α0

+

2

α2

)

a1

+

c4α2,

σ

=

a2α2 and

τ

=

a1α2.

From eq. (47), we can seethat when

μ

=

0 i.e.,

α2

c1

=

α0

a2, theChern–Simons–Antoniadis–SavvidyLagrangian fortheMaxwell algebracontainstheEinstein–Hilbertterm.

An interesting solution can be obtainedchoosinga1

=

a2

=

0. Inthiscasethefieldsofeqs.(41),(42),and(43)taketheform

Ba

=

0, (48) Bab

=

0, (49)

β

ab

=

c1 2l2e a_eb

+

c4 2

ω

a c

ω

cb

.

(50)

Under this choice, the Chern–Simons–Antoniadis–Savvidy La-grangian forMaxwellalgebratakesthecompactform

L

(4) ChSAS

=

μ

8

ε

abcdR ab_ec_ed

+

ν

8l 2

ε

abcdRab

ω

cf

ω

f d

,

wherewecanseethatinthelimitl

→

0,weobtaintheEinstein– HilbertLagrangian,

L

(4) ChSAS

=

μ

8

ε

abcdR ab_ec_ed

.

₍₅₁₎

Another case particularly interesting choice is given by a1

=

c1

=

c4

=

0. Inthiscasethe fieldsofeqs. (41),(42)and(43)are givenby Ba

=

a2 2lk a be b

,

(52) Bab

= −

a2 2l2e a eb

,

(53)

β

ab

=

a2 2k a ckcb

+

a2 2k b ckac

,

(54)

and the Chern–Simons–Antoniadis–Savvidy Lagrangian for the Maxwellalgebraisgivenby

L

(4) ChSS

=

μ

8

ε

abcdR ab_ec_ed

−

σ

8l2

ε

abcd

(

e a_eb_ec_ed

+

_2l2_kab_Tc_ed

−

_2l4_Rab_kc fkf d

)

−

σ

8d

(

ε

abcdk ab_ec_ed

),

₍₅₅₎

wherethecasekab

=

0 leadsto thestandardEinstein–Hilbert La-grangianwithcosmologicalconstant.

5. Concludingremarks

In Refs. [1,2] there were found invariants similar to the Pontryagin–Chern forms in non-abelian tensor gauge field the-ory [11,12]. The firstseries ofexact

(2

n

+

3)-forms are givenby

2n+3

=

FnH3

=

d

C

_ChSAS(2n+2) where H3

=

dB

+ [

A

,

B

]

isthe3-form field-strengthtensorfortherank-2gaugefieldB.Thesecondseries ofinvariantformsaredefinedin2n

+

4 dimensionsandaregiven by

2n+4

=

FnH4

=

d

C

(2nChSAS+3)wherethecorrespondingsecondary

(2

n

+

3)-form

C

(2n_ChSAS+3) is defined in terms of the 4-form H4

=

dC

+[

A

,

C

]

asthefield-strengthtensorfortherank-3gaugefieldC. The third series of forms is defined in

(2

n

+

6) dimensions [3]

2n+6

=

FnH6

+

n

Fn−1H42

=

d

C

(2n+5)

ChSS .The fourth seriesof in-variantclosedforms

2n+8 in

(2

n

+

8) dimensionsisgivenby[4]

2n+8

=

FnH8

+

3n

Fn−1H4H6

+

n

(

n

−

1)

Fn−2H34

=

d

C

(2n+7) ChSS . Allforms

2n+3,

2n+4,

2n+6 and

2n+8 are analogoustothe Pontryagin–Chern invariants

P

2n in the Yang–Mills gauge theory inthesensethattheyaregaugeinvariant,closedandmetric inde-pendent.

In Refs. [2–4] there were found explicit expressions for these invariants in terms of higher order polynomials of the curva-ture forms on a vector bundle. As with standard Chern–Simons forms,thesecondaryforms

C

(2n_ChSAS+m)arebackground-freebut quasi-invariant and only locally defined (and therefore defined only up to boundary terms,

C

(2n_ChSAS+m)

∼

C

_ChSAS(2n+m)

+

d

σ

(2n+m−1)_{). In the} presentarticlewehaveconstructedthe

(2

n

+

2)-dimensional ana-logue oftransgressionforms andthe Chern–Weiltheorem. These transgression forms are defined globally and are off-shell gauge invariant, but the price to pay is the doubling in the num-ber of fields. From this theorem is straightforward to recover the generalized

(2

n

+

2)-dimensional Chern–Simons–Antoniadis– SavvidyformsfromRefs.[1,2]settingtozerohalfofthefields.The 2-formfield Bcanbediscomposed intermsofcomponentsofthe 1-form A.It isperformedinaself-consistent wayby considering thegeneralization ofMaurer–Cartan approach to formsof higher order,i.e.freedifferentialalgebras,andbyfollowingtheprocedure usedinRefs.[5,6]and[7].

The final result is a four-dimensional gravity action principle equation (47), which is gauge quasi-invariant under the general-izedgaugetransformationseqs.(8),(9)fortheMaxwellalgebra.

The dynamicsof thesystem willbe presented elsewhere, but it is clear that the non-linear couplings with the kab field does generateingeneralnon-vanishingtorsion,inawaysimilartothe one presented in Ref. [35]. A non-vanishing torsion may lead to highly non-trivial consequences incosmology (see Refs. [35–37]), whereatthe very enditplays the role ofan extra stress-energy tensorinEinsteinfieldequations.

Acknowledgements

ThisworkwassupportedinpartbyFONDECYTgrants1130653 and 1150719 from the Government ofChile. One of the authors (I.M.) was supported by grants from Universidad de Concepción, Chile.

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