Exotic dual of type II double field theory

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University of Groningen

Exotic dual of type II double field theory

Bergshoeff, Eric A.; Hohm, Olaf; Riccioni, Fabio

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Physics Letters B

DOI:

10.1016/j.physletb.2017.01.081

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Citation for published version (APA):

Bergshoeff, E. A., Hohm, O., & Riccioni, F. (2017). Exotic dual of type II double field theory. Physics Letters

B, 767, 374-379. https://doi.org/10.1016/j.physletb.2017.01.081

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Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Exotic

dual

of

type

II

double

ﬁeld

theory

Eric

A. Bergshoeff

a

,

∗

,

Olaf Hohm

b

,

Fabio Riccioni

c

a_Centre_for_Theoretical_Physics,_University_of_Groningen,_Nijenborgh_4,₉₇₄₇_AG_Groningen,_The_Netherlands b_Simons_Center_for_Geometry_and_Physics,_Stony_Brook_University,_Stony_Brook,_NY_11794-3636,_USA

c_INFN_Sezione_di_Roma,_Dipartimento_di_Fisica,_Università_di_Roma_“La_Sapienza”,_Piazzale_Aldo_Moro_2,₀₀₁₈₅_Roma,_Italy

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received18December2016 Accepted3January2017 Availableonline16February2017 Editor:M.Cvetiˇc

WeperformanexoticdualizationoftheRamond–RamondfieldsintypeIIdoublefieldtheory,inwhich theyareencodedinaMajorana–WeylspinorofO(D,D).Startingfromafirst-ordermasteraction,the dualtheoryintermsofatensor–spinorof O(D,D) isdetermined.Thistensor–spinorissubjecttoan exotic version ofthe (self-)duality constraint neededfor ademocraticformulation. We show that in components,reducing O(D,D) toG L(D),one obtainsthe expectedexoticallydualtheoryintermsof mixedYoungtableauxfields.Tothisend,wegeneralizeexoticdualizationstoself-dualfields,suchasthe 4-formintypeIIBstringtheory.

1. Introduction

Stringtheorycomprisesarichspectrumofstatesorfields.The masslessfieldsincludethemetric,Kalb–Ramond2-formandscalar (dilaton), together withvarious p-forms, depending onthestring theory considered, butthere is also an infinite tower ofmassive ‘higher-spin’ fields, often taking values in mixed Young tableaux representations. Even when restricting to the massless sector, it is sometimes necessary to go beyond the minimal field content in order to couple the various branes present in the full (non-perturbative)stringtheory.Forinstance,inD

=

10 a6-formneeds tobeintroducedastheon-shelldualoftheKalb–Ramond2-form in order to describe the NS5 brane. In recent years it has been argued from different angles that the various dualities of string theory imply also the existence of ‘exotic branes’ [1], which in turncouple toﬁeldsof a moreexoticnature, typically belonging tomixedYoungtableauxrepresentations[2].

Recently, we showed how to describe, at the linearized level, such exoticdual ﬁeldsin doubleﬁeld theory (DFT) [3–5] in a T-duality or O

(

D

,

D

)

covariant way [6]. In DFT the Kalb–Ramond ﬁeld is uniﬁed with the metric into a generalized metric

H

M N, with O

(

D

,

D

)

indices M

,

N

=

1

,

. . . ,

2D. Therefore, dualizing the 2-form requires also dualizing the graviton, which in turn leads toamixedYoungtableauxﬁeld [7,8].Moreover,additionalmixed

*

Correspondingauthor.

E-mailaddresses:e.a.bergshoeff@rug.nl(E.A. Bergshoeff),

ohohm@scgp.stonybrook.edu(O. Hohm),Fabio.Riccioni@roma1.infn.it(F. Riccioni).

Young tableauxﬁeldsemergethat canbeinterpreted asso-called ‘exotic duals’ of the 2-form, implementing the dualization pro-cedure of [9,10]. Remarkably, in DFT the various mixed Young tableaux representations under G L

(

D

)

organize into completely antisymmetric O

(

D

,

D

)

tensors,includinga4-indextensor DM N K L fortheNSsector.

In this letter, we extend the results of [6] by including the Ramond–Ramond (RR)sector oftype II string theory. The differ-encetotheNSsectoristhatinordertomakeO

(

D

,

D

)

manifestas a locallyrealizedsymmetry itisnecessarytoincludeforeachRR p-formitsdual

(

D

−

p

−

2

)

-form,requiringademocratic formula-tion[11].TheRRﬁeldsthenorganizeintoaMajorana–Weylspinor of O

(

D

,

D

)

, forwhich a completeDFT formulation exists [12,13]

(see [14] formassivedeformations and[15,16] forearlierrelated results). Thus, the RR ﬁeldsand their conventional duals already enter in an O

(

D

,

D

)

complete form, without theneed to invoke exoticdualizations.However,itisneverthelesspossibletoperform anexoticdualizationfortheRRﬁelds,asindeedisnecessaryin or-dertodescribecertainexoticbranes[17]andisalsosuggestedby theKac–Moodyapproachtosupergravity[8].TheexpectedG L

(

D

)

representationsfortheexoticallydualﬁeldscanbeorganizedinto a simple O

(

D

,

D

)

representation,a tensorspinor EM Nα [17]. We willshowherethatDFTprovidespreciselysuchaformulation.

This letterisorganized as follows.Insec. 2 webriefly review the exoticdualization procedure, following [10], and discuss the generalization to self-dual fields.Fordefiniteness andinorder to simplifythediscussion,weanalyzeindetailthesimplercaseofa self-dual vector in D

=

4,assuming euclideansignature. Insec. 3

wereviewtypeIIDFT,andinsec.4wepasstoanunconventional http://dx.doi.org/10.1016/j.physletb.2017.01.081

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first-ordermasteractioninordertoperformtheexoticdualization. Webriefly discusshow theresulting dualtheory interms ofthe fieldEM Nα reproducesincomponents,breakingO

(

D

,

D

)

toG L

(

D

)

, theexpectedresult. Wecloseinsec. 5withabriefsummary and outlookoffurtherexoticﬁeldsneededinstringtheory.

2. Exoticdualizationofself-dualﬁelds

We consider here the exotic dualization of ﬁeldsthat are al-ready subject to a self-duality condition, as is the case for the 4-formin type IIB string theory or the 2-formin

(

2

,

0

)

theories inD

=

6.Forsimplicity,weanalyzethecaseofa self-dualvector inD

=

4,whichexistsforeuclideansignature.

Westartbyreviewingtheexoticdualizationoftheconventional Maxwell theory [10]. The action in terms of the ﬁeld strength Fmn

= ∂

mAn

− ∂

nAmisrewritten,uptoboundaryterms,as

S

= −

1₄

d4x FmnFmn

=

d4x

−

1₂

∂

mAn

∂

mAn

+

1₂

(∂

mAm

)

2

,

(2.1) andthen promotedtoa ﬁrst-orderaction,intermsofﬁelds Pm,n andEmn,k

≡

E[mn],k,asfollows: S

=

d4x

−

1₂Pm,nPm,n

+

1₂

(

Pm,m

)

2

−

Emn,k

∂

mPn,k

.

(2.2) Theﬁeldequationsfor Pm,n andEmn,k imply,respectively,

∂

kEkm,n

=

Pm,n

−

η

mnPk,k

,

∂

_[mPn],k

=

0

.

(2.3) Solving the second equation by setting Pm,n

= ∂

mAn and re-inserting into the action, we recover Maxwell’s theory. Equiva-lently,actingontheﬁrstequation with

∂

m andusingthe‘Bianchi identity’

∂

m

∂

kEkm,n

=

0 weget

∂

mPm,n

− ∂

nPm,m

=

0

,

(2.4)

whichforPm,n

= ∂

mAn isequivalenttotheMaxwellequations.On theotherhand,solvingtheﬁrstequationforP ,

Pm,n

= ∂

kEkm,n

−

13

η

mn

∂

kEkl,l

,

(2.5)

andback-substitutinginto(2.2) oneobtainsasecond-orderaction forE,whoseﬁeldequationsareobtainedbyinserting(2.5)intothe secondequation of(2.3).Notethatthe Maxwellgauge invariance

δ

λAm

= ∂

m

λ

elevatestoagaugeinvarianceoftheﬁrstorderaction givenby

δ

λPm,n

= ∂

m

∂

n

λ ,

δ

λEmn,k

=

2

η

k[m

∂

n]

λ .

(2.6) Thereisalsoanextragaugeinvarianceassociatedto E,

δ

Emn,k

= ∂

l

lmn,k

,

(2.7)

withparameter

mnk,l

≡

[mnk],l.

We now investigate the dual theory in terms of E in more detail.Letusﬁrstdecomposethisﬁeldintoirreducible representa-tionsas

Emn,k

=

12

mnpq_C

pq,k

+

2

δ

k[mBn]

,

C[mn,k]

≡

0

,

(2.8) where the Maxwell gauge invariance (2.6) acts on the new vec-tor Bm,

δ

λBm

= ∂

m

λ

. Inserting thisdecomposition into (2.5), one obtains

Pm,n

= ∂

nBm

−

₃1_!

mpqkFpqk,n

,

Fmnk,p

≡

3

∂

[mCnk],p

.

(2.9) Thesecond-orderﬁeldequationfollowingfromthedualactionfor E isequivalentto

∂

[mPn],k

=

0,i.e. to

0

=

mnkl

∂

kPl,p

= ∂

p

Fmn

(

B

)

+ ∂

kFmnk,p

,

Fmn

(

B

)

≡

1₂

mnklFkl

(

B

) ,

(2.10) whereFmn

(

B

)

≡

2

∂

_[mBn].UsingtheBianchiidentity

∂

m

Fmn

(

B

)

=

0, weconcludebytakingthetracethat

∂

kFmnk,m

=

0

,

(2.11)

which is the correct ﬁeld equation for a

(

2

,

1

)

ﬁeld describing spin-1 in D

=

4[10].

WenextinvestigatethisexoticdualizationforMaxwell’stheory subjectto aself-duality constraint,assuming euclidean signature. Thus,theﬁeldstrengthsatisﬁes

Fmn

=

1₂

mnklFkl

.

(2.12)

In the ﬁrst-order formulation, we then haveto impose the con-straint

P_[m,n]

=

21

mnklPk,l

,

(2.13)

whichreducesto(2.12)whensolvingtheBianchiidentityforPm,n.

Letusshowthattheintegrabilityconditionsofthisﬁrst-order re-lationarecompatiblewiththesecond-orderequations.Tothisend weactwith

∂

p on(2.13)andusetheBianchiidentity

∂

[mPn],k

=

0,

∂

pPm,n

− ∂

pPn,m

= ∂

mPp,n

− ∂

nPp,m

=

mnkl

∂

pPk,l

.

(2.14) Contractingthisnowwith

η

mp_,_we_get

∂

mPm,n

− ∂

nPm,m

=

mnkl

∂

mPk,l

=

0

,

(2.15) usingagaintheBianchiidentityinthe laststep.Thisagrees with the second-order equations (2.4). It is instructive to write the (self-)duality constraint explicitly in terms of the decomposition

(2.8).Wecomputefrom(2.9)

2P_[m,n]

= −

Fmn

(

B

)

−

12

mnpqFpqk,k

,

(2.16) wherewe used the Schoutenidentity 0

=

_[mpqkFpqk,n].The con-straint(2.13)thenimplies

Fmn

(

B

)

−

Fmn

(

B

)

=

Fmnp,p

−

1₂

mnklFklp,p

.

(2.17)

Thus,theanti-self-dualpartofthefieldstrengthofthevector Bm isequaltotheanti-self-dualpartofthetraceofthe‘fieldstrength’ of theexotically dualfield Cmn,k. In particular, we donot obtain

a first-orderconstraint forthisfield alone. Therefore,there is no formulationforonlya(irreducible)mixed-Young-tableauxfield in D

=

4 that describesthedegreesoffreedomofaself-dualvector, notevenon-shell.Extraﬁeldslikethenewvector Bmareneeded. This canbe understood by notingthat forthe gauge symmetries

(2.7)thereisnoinvariant first-orderfield strengthforthe mixed-Young-tableauxfieldCmn,k,andhencetherecannotbeafirst-order

self-dualitycondition.

Letus finally note that this discussion generalizes straightfor-wardly to self-dual fields in other dimensions. For instance, for theself-dual 4-formCmnkl in typeIIBstringtheory onepromotes its derivative to a field Pm,klpq and imposes a Bianchi identity

∂

_[mPn],klpq

=

0 withaLagrangemultiplierﬁeld Emn,klpq,which en-codesthemixedYoungtableauxﬁeldinthedualformulation.

3. Ramond–RamondﬁeldsintypeIIdoubleﬁeldtheory

In this section we briefly review the Ramond–Ramond (RR) fields of type II double field theory, which are encoded in a Majorana–Weylspinorof O

(

D

,

D

)

. Ourspinorconventionsfollow

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{

M

,

N

} =

2

η

M N

,

η

M N

=

0 1 1 0

,

(3.1)

isrealizedintermsoffermionicoscillators

ψ

i,

ψ

i,with

(ψ

i

)

†

= ψ

i, as

i

=

√

2

ψ

i,

i

=

√

2

ψ

i,satisfying

{ψ

i

, ψ

j

} = {ψ

i

, ψ

j

} =

0

,

{ψ

i

, ψ

j

} = δ

ij

.

(3.2) WedeﬁnetheDiracoperatorwitharelativefactorforlater conve-nience,

/

∂

≡

√1 2

M

_∂

M

= ψ

i

∂

i

+ ψ

i

˜∂

i

,

(3.3)

where

˜∂

i _denotes _the _derivative _with_respect _to_the _dual coordi-nate.We recall the strong constraint

η

M N

_∂

M

∂

N

=

0, whichholds actingon arbitrary objects, andwhich impliestogether withthe Cliffordalgebrathat

∂

/

2

=

0.

Wealsoneedthe chargeconjugation matrixC ,whose explicit expression can be found in [11,13]. For our purposes here it is suﬃcienttorecallthat C†

=

C−1 and

C

ψ

iC−1

= ψ

i

,

C

ψ

iC−1

= ψ

i

,

(3.4)

whichimpliesfortheGammamatrices

C

MC−1

= (

M

)

†

,

C−1

MC

= (

M

)

†

.

(3.5) ThespinorrepresentationisconstructedfromtheCliffordvacuum

|

0

satisfying

ψ

i

|

0

=

0

∀

i

.

(3.6)

By taking the conjugate of this equation we also conclude that

0

|ψ

i

=

0 foralli.Ageneralstateisthengivenby

χ

=

D

p=0 1 p

!

Ci1...ip

ψ

i1

· · · ψ

ip

|

₀

,

_(3.7)

which encodes the RR p-forms C(p). States including only even formsareofpositivechiralityandstatesincludingonlyoddforms areofnegativechirality.Wealsousethecommonnotation

¯

χ

≡

χ

†C

=

D

p=0 1 p

!

Ci1...ip

0

|ψ

ip

· · · ψ

i1C

.

(3.8) ThegroupsPin

(

D

,

D

)

andSpin

(

D

,

D

)

arethetwo-foldcovering groupsofO

(

D

,

D

)

andS O

(

D

,

D

)

,respectively.Foragivenelement ofthe covering group S

∈

(

D

,

D

)

, thereis a corresponding el-ement h

≡

ρ

(

S

)

∈

O

(

D

,

D

)

, where

ρ

:

(

D

,

D

)

→

O

(

D

,

D

)

is a grouphomomorphism,deﬁnedimplicitlyby

S

MS−1

= (

h−1

)

MN

N

.

(3.9)

Note that

+

S and

−

S project to the same O

(

D

,

D

)

element h. A particular Spin

(

D

,

D

)

element that will be useful belowis

K

, whichisthespinorrepresentativeofthegeneralizedmetric

H

M

N withoneindexraised:

ρ

(

K

)

=

H

•_•

=

bg−1 _g

₋

_bg−1_b g−1

−

g−1b

∈

O

(

D

,

D

) ,

(3.10) where g and b arethe metric andKalb–Ramond 2-form. Denot-ingthespin representativeoftheoriginalgeneralizedmetric

H

_•• by

S

andusingthatthechargeconjugationmatrix C under

ρ

ac-tuallyprojectstothe O

(

D

,

D

)

metric

η

M N (viewedasamatrixin O

(

D

,

D

)

),wehave

K

=

C−1

S .

(3.11)

Theconstraintson

H

,whichread

(H

•_•

)

2

=

1 and

H

_••

=

H

t

••,

cor-respondtothefollowingconstraintson

S

orequivalently

K

,1

S

†

_{= S ,}

_K

2

₌

₁

_⇒

_K

−1

₌

_K

_.

_(3.12)

Wecanthinkof

S

asbeingconstructedfrom

H

,inwhichcasewe write

S

=

S_H,butitwasarguedin[12,13]thatamoreuseful per-spectiveistotreat

S

asthefundamentalﬁeld,satisfyingtheabove constraints.Ausefulrelationfollowsbyspecializing(3.9) to

K

,

K

M

=

H

MN

N

K

.

(3.13)

We arenow readytodeﬁne theRR action,forwhichwe take theNSsectortobe ﬁxed,givenby aconstantbutotherwise arbi-trarybackground

H

.Theactionreads

SRR

=

14

d2DX

(/

∂

χ

)

†

S /∂

χ

=

1 8

d2DX

∂

M

χ

¯

M

K

N

∂

N

χ

,

(3.14) wherethesecondformfollowswitheqs.(3.5) and(3.8).Wehave to subjectthe action to (self-)dualityrelations, since we are us-ingademocraticformulation.Thesecanbewritteninan O

(

D

,

D

)

covariantformas[15]

(

1

+

K

)/

∂

χ

=

0

.

(3.15)

Theactionanddualityrelationsaremanifestlyinvariantunderthe gaugetransformations

δ

λ

χ

= /∂λ ,

(3.16)

due to

∂

/

2

=

0. The gauge parameter here is a Majorana–Weyl spinorwiththechiralitytheoppositetothatof

χ

.

It was shown in [13] how to evaluate the above action in components,after solvingthe strongconstraintby setting

˜∂

i

=

0, which we brieﬂy reviewin thefollowing.To thisendone hasto use an explicit parametrizationof the generalizedmetric and its spinrepresentative,

S =

S_H

=

S†_bS−_g1Sb

,

(3.17) where Sb

=

e− 1 2bi jψiψj

_,

S−_g1

ψ

i1

· · · ψ

ip

|

₀

=

_σ

√

_{g g}i1j1

· · ·

_gipjp

_ψ

j1

· · · ψ

jp

|

₀

,

(3.18) where

σ

= −

1 forLorentzian signatureand

σ

= +

1 foreuclidean signature.Herewe havegivenonlytheactionof Sg onoscillators actingonthevacuum,whichissufficientforourpurposesbelow. Wefirstobservethatthenaiveabelianfieldstrengthsareencoded asfollows, F

≡

√1 2

M

_∂

M

χ

_˜∂= 0

= ψ

i

_∂

i

χ

⇒

F

=

dC

,

(3.19)

using the familiar notation in which forms of different rank are combined into a single object C . It is now easy to see, using eq. (3.18), that in the RRLagrangian the action of Sb inside SH changesthistotheeffectiveﬁeldstrength

F

=

e−b2

∧

_F

_,

_(3.20)

which is the gauge invariant field strength, given that the RR fields transform under the b-field gauge symmetry. Using again

1 _In_general_dimension_K2_{= ±}_1,_but_consistency_of_the_self-duality_constraint_to

beintroducedbelowrequiresK2₌_1._In_the_following_we_assume_that_we_are_in

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eq.(3.18),itistheneasytocheckthattheRRLagrangianreduces to

L

RR

_˜∂=₀

= −

14

√

g D

p=1 1 p

!

g i1j1

· · ·

_gipjp

_F i1...ip

Fj1...jp

,

(3.21) whichisthe standardaction fortheRRpotentials.Similarly, itis straightforwardtoverifythateq.(3.15)reducestotheconventional dualityrelationsfor

˜∂

i

₌

_0.

4. First-orderactionandexoticdual

Wenow turnto a ﬁrst-orderformof theRRaction discussed intheprevioussectioninordertodeﬁnetheexoticdual.Westart fromtheexpression(3.14) andintegrateby partstwice,toobtain theequivalentLagrangian

L

RR

=

18

∂

N

χ

¯

M

K

N

∂

M

χ

,

(4.1)

usingthat

K

isconstant.Notethatinthisformtheactionisonly gaugeinvariant up to boundaryterms. Nextwe promote

∂

M

χ

to anindependent‘vector–spinor’ﬁeld PM ofthesamechiralityas

χ

andaddaLagrangemultiplierterm,

L

1st

=

1₈P

¯

N

M

K

NPM

+

1₂

∂

MP

¯

NEM N

,

(4.2) whereEM N

₌

_E[M N] _is _a_{tensor–spinor}_of_the_same_chirality_as _P

forevenD andtheoppositechiralityforoddD.Asforthe second-orderformulation, we have to subject the ﬁeld equations to the (self-)dualityconstraint,nowwrittenintermsof P :

(

1

+

K

) /

P

=

0

,

(4.3)

where

/

P

=

MPM.Varyingtheﬁrst-orderactionw.r.t. EM Nwe ob-taintheconstraint

∂

_[MPN]

=

0

.

(4.4)

ThisimpliesPM

= ∂

M

χ

,anduponre-insertioninto(4.2)and(4.3) we recoverthe RR action inthe form(4.1) andthe duality rela-tions,respectively.Ontheotherhand,varyingw.r.t. P oneobtains

1 2

M

_K

N_P

M

= ∂

MEM N

,

(4.5)

whicharethe ‘exotic’ dualityrelations.Actingwith

∂

N andusing theBianchiidentity

∂

M

∂

NEM N

=

0 weobtaintheintegrability con-dition

M

K

∂

/

PM

=

0

,

(4.6)

whichbyuseof(4.4),writingPM

= ∂

M

χ

,isequivalenttothe orig-inalﬁeldequationfor

χ

.Inthefollowingwewillbeinterestedin thetheory fortheexoticdualﬁeld EM N,obtainedby eliminating P usingeq.(4.5).

Letusinvestigate the gauge symmetriesof the ﬁrst-order ac-tioncorrespondingto(4.2).First,theactionisinvariant,uptototal derivatives,underthenewgaugesymmetry

δ

EM N

= ∂

K

M N K

,

(4.7)

with

M N K

=

[M N K]. Second, theaction is alsoinvariant under the original RR gauge symmetry (3.16), which acts in the ﬁrst-orderformulationas

δ

λPM

= ∂

M

∂λ ,

/

δ

λEM N

=

[M

K

N]

∂λ .

/

(4.8) Inordertoprovethisgaugeinvariance,weﬁrstconsiderthe vari-ationoftheﬁrst-orderform(4.2)oftheRRterm,2

2 _Here_we_used_that_the_variation_of_both_{P factors}_gives_the_same_{contribution,}

uptototalderivatives,whichcanbeveriﬁedincomponentform.

δ

λ

L

RR

=

1₄P

¯

N

M

K

N

∂

M

∂λ

/

=

1 2P

¯

N

[ M

_K

N]

_∂

M

∂λ

/

+

1₄P

¯

N

K

M

∂

M

∂λ

/

= −

1 2

∂

MP

¯

N

[ M

_K

N]

_{∂λ .}

_/

(4.9)

Here weused

∂

/

2

=

0 andintegratedby partswith

∂

M inthelast step. We then observe that the termin the last line isprecisely canceled bythevariationofEM Ninthesecondtermof(4.2),while the

λ

gauge variation of P in that term drops out by the anti-symmetry ofEM N.Thisproves thegauge invarianceoftheaction correspondingto(4.2).

Letusnowreturntotheﬁeldequations(4.5)inordertosolve for P in terms of E. We ﬁrst rewrite the left-hand side, using eq.(3.13),andbringtheresulting

H

totheothersideofthe equa-tion:

1 2

M

K

_K

_P

M

=

H

KN

∂

MEM N

.

(4.10)

Next, we contract this equation with

K and use

K

M

K

=

−

2

(

D

−

1

)

M,toobtain

M

K

PM

= −

1 D

−

1

H

K N

K

∂

MEM N

.

(4.11)

Returning to (4.10) we use the Clifford algebra andcompute for theleft-handside

1 2

M

K

PM

=

12

{

M

,

K

}

K

PM

−

12

K

M

K

PM

=

K

PK

+

1 2

(

D

−

1

)

H

P Q

K

P

∂

LEL Q

,

(4.12)

wherewe insertedeq.(4.11) inthesecond line.Sincethisequals theright-handsideof(4.10),wecansolvefor

K

PM _in_terms_of_E,

K

PM

=

H

M N

∂

KEK N

−

1 2

(

D

−

1

)

H

K L

M

L

_∂

NEN K

.

(4.13) Using

K

2

=

1 wecanﬁnallysolveforPM,obtainingtheresult

PM

=

QM

(

H

,

E

) ,

(4.14) wherewedeﬁned QM

≡

H

MN

K

∂

KEK N

−

1 2

(

D

−

1

)

H

K L

K

M

K

_∂

NEN L

.

(4.15) Amorecompactformofthisexpressionisobtainedbyintroducing the

gaugeinvariant‘ﬁeldstrength’

G

M

_≡

_K

_∂

NEN M

,

(4.16)

satisfying theBianchi identity

∂

M

G

M

=

0. Using eq. (3.13) in the secondtermof(4.15)twice,weobtain

QM

=

H

M N

G

N

₋

1 2

(

D

−

1

)

N

K

G

K

.

(4.17)

Back-substitution of (4.14) into the Lagrangian (4.2) gives the second-orderactionforthedualﬁeld EM N.Itsﬁeld equationsare equivalentto

∂

[MQN]

=

0 andthusfollowfromthedualityrelation

(4.14) andthe Bianchi identity (4.4). Conversely, we can use the dualityrelation(4.14)toderivethesecond-orderequationsforthe originalﬁelds.Tothisend,weneedtheBianchiidentityofthe QM deﬁnedin(4.15)whichreads

M

K

∂

/

QM

≡

0

.

(4.18)

Thiscanbe veriﬁedbya directcomputation,usingeq.(3.13) and the Clifford algebra together with the Bianchi identity

∂

M

∂

NEM N

=

0. The duality relation (4.14) then immediately im-plies the original second order equation (4.6) in terms of P . As

(6)

usual, the duality transformations thereforeswap ﬁeld equations andBianchiidentities.

WerecallthattheequationsforthedualﬁeldsE arestill sub-jecttotheﬁrst-orderconstraint(4.3),upon eliminating P accord-ingto (4.14),i.e.

(

1

+

K) /

Q

=

0.It isinstructivetoverifythatthe integrabilityconditionsofthis(self-)dualityconstraintare compat-ible with the second-order equations obtainedfrom the pseudo-action,eitherintermsoftheoriginalﬁeldsorthedualﬁeldsEM N_. Tothisend,weactwith

∂

M on(4.3)toobtain

(

1

+

K

)

N

∂

MPN

=

0

⇒ (

1

+

K

)/

∂

PM

=

0

,

(4.19) usingtheBianchiidentity(4.4) inthelaststep.Actingwith

M

K

on the second equation, using

K

2

=

1 and the Bianchi identity again,weobtain

0

=

M

K

∂

/

PM

+

√1₂

M

N

∂

NPM

=

M

K

∂

/

PM

+

√1₂

∂

MPM

.

(4.20) Duetothe Bianchiidentity PM

= ∂

M

χ

,thelast termvanishes by thestrongconstraint,andindeedwerecovertheexpectedeq.(4.6). We close this section by verifying that in components, upon solving the strong constraint and thereby breaking O

(

D

,

D

)

to G L

(

D

)

, we recover the expected exotic dualizations. In order to simplifythe presentation we will focus on a vector, subjectto a self-dualityconstraintinfoureuclideandimensions,andmatchthe resultswiththoseinsec.2.WethusassumethattheﬁeldsPM and EM N_have_only_the_{non-vanishing}_components

Pm

=

Pm,n

ψ

n

|

0

,

Emn

=

Emn,k

ψ

kC−1

|

0

,

(4.21) where thefactor of C is necessary inorder for E to lead tothe sametensorstructureasusedinsec. 2.3 _Let_us_verify_that _{E has} therightchirality.Toseethisnotethatwiththe‘numberoperator’ NF

≡

k

ψ

k

ψ

kaquickcomputationyieldsfortheaboveansatz

NFPm

=

Pm

⇒ (−

1

)

NFPm

= −

Pm

,

(4.23)

showing that Pm has negative chirality, as it should be since it correspondstoan oddform(1-form).Thus,in D

=

4, Emn should alsohavenegativechiralityand, indeed,astraightforward compu-tationgives for the above ansatz NFEmn

= (

D

−

1

)

Emn andthus

(

−

1

)

NF_Emn

= −

_Emn_,_as_required._The_ﬁrst-order_form_(4.2) _of_the RRkinetictermsthenreducesto

L

RR

=

1₄

(

Pn

)

†C

ψ

m

K

ψ

nPm

=

1 4Pn,kPm,l

0

|ψ

kC

ψ

m_C−1_S H

ψ

n

ψ

l

|

0

=

1 4

√

g Pn,kPm,lgnpglq

0

|ψ

k

ψ

m

ψ

p

ψ

q

|

0

=

1 4

√

g

Pm,nPm,n

− (

Pn,n

)

2

,

(4.24)

whereweused(3.18)andthattheCliffordrelations(3.2)and(3.6)

imply

0

|ψ

k

ψ

m

ψ

p

ψ

q

|

0

= δ

mp

δ

kq

− δ

mq

δ

kp

.

(4.25) WeinferthatthisreducespreciselytotheP2 _terms_in_the_master

action(2.2),up toan irrelevantpre-factor.Similarly, theLagrange multipliertermin(4.2)reducesas

3 _{Equivalently,}_we_could_write _Emn₌_Emn k1...kD−1ψ

k1_{. . . ψ}kD−1|₀_,_in_which_case

thetermintheLagrangianwouldbeproportionalto

L ∝k1...kD_Emn

,k1...kD−1∂mPn,kD, (4.22)

cf. thediscussioninsec. 5.1.3in[13].Thedeﬁnitionin(4.21)avoidstheexplicit epsilontensor. 1 2

∂

MP

¯

NE M N

₌

1 2

∂

mPn,kEmn,l

0

| ψ

kC

ψ

lC−1

|

0

=

1 2

∂

mPn,kEmn,k

,

(4.26)

whereweused(3.4),givingthesametermasintheMaxwell mas-ter action(2.2). Wethus recover the master actionthat was the starting point for the exotic dualization in sec. 2. Moreover, the dualityconstraint(4.3)yieldsincomponentsthesameself-duality constraint (2.13) as forthe self-dual vector (cf. the discussion in sec. 5.1.3in[13]).Wethereforehaveshownthattheresultsofthis section provide the proper O

(

D

,

D

)

covariant exotic dualizations oftheRRﬁeldsinDFT.

5. Conclusionsandoutlook

Inthisletterwehaveappliedtheexoticdualizationprocedure of[10]totheRRfieldsindoublefieldtheory.Thisgeneralizesthe analysisof[6],whereitwasshownthatthedualizationofthe gen-eralized metricnaturally yields, together withthe standardduals ofthe2-formandthegraviton,alsotheexoticdualofthe2-form. The differencebetweentheresultsof[6]andtheanalysiscarried out in thisletteristhat in thecaseof theRRfields the dualiza-tionprocedureisalreadyexoticinthedoubledspace,whileinthe caseofthegeneralizedmetriconeperformsastandarddualization inthedoubledspace,whichincludestheexoticdualizationofthe 2-formwhenwrittenincomponents.

Anaturalcontinuationofthisworkwouldbetoapply the du-alization procedure discussed in this letter to the ﬁeld DM N P Q, whichitselfisthedualofthegeneralizedmetric

H

M N.The dual-ization carriedout in [6]givesan action for DM N P Q in termsof its gaugeinvariant ﬁeld strength.Proceedingasinthisletter,one can writedownaDFT actionforthisﬁeld intermsofthe gauge-dependentquantity

G

M,N1...N4

= ∂

MDN1...N4

,

(5.1)

satisfyingtheBianchiidentity

∂

[M1

G

M2],N1...N4

=

0

.

(5.2)

In a ﬁrstorderformulation, the Lagrange multiplier forthis con-straintwouldbethedualpotentialFM1M2,N1...N4.Thisﬁeld decom-posesunderG L

(

10

)

preciselyintothemixed-symmetrypotentials givenintab. 10of[18].Such potentialscan bewrittenina com-pactformas F8+n,6+m,m,n,whereeach entrydenotesa setof

an-tisymmetricindicesinthemixed-symmetryrepresentation,andm andn takeallthepossiblevaluesthatareallowedbythefactthat thenumberofindicesineachsetcanbeatmost10,withthe fur-therrestrictionthateachsethastobegreaterorequaltothenext. Asexpected,oneofthecomponentsistheﬁeld F8,6,whichisthe

exoticdualofD6,thatinturniscontainedinDM N K L.

Onecanalsoapplythedualizationproceduretotheﬁeld EM Nα discussedinthisletter,therebywritingtheDFTactionforthisﬁeld intermsof

˜

Q

M,N Pα

= ∂

MEN Pα

,

(5.3)

satisfyingtheBianchiidentity

∂

_[M

Q

˜

N],P Qα

=

0

.

(5.4)

The Lagrange multiplier in this case is a ﬁeld GM N,P Qα . In terms of mixed-symmetry potentials, this ﬁeld decomposes as G8+m,8+m,2n,m,m inthe IIBcaseandG8+m,8+m,2n+1,m,m inthe IIA case. Inparticular,form

=

n

=

0 thisgivesapotential G8,8 inthe

IIBcasewhichistheexoticdual ofthepotential E8 containedin

(7)

Acknowledgements

Wewouldliketoacknowledgethehospitalityofthe GGI (Flo-rence) and thank the organizers of the workshop “Supergravity: whatnext?”,wherepartofthisworkhasbeencarriedout,for cre-ating a stimulating atmosphere. OH andFR would like to thank theUniversityof Groningenforhospitalityduringthe completion ofthiswork.

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