Non-Abelian T-duality and the AdS/CFT correspondence

64 Pages·2013·0.58 MB·English

by  Georgios Itsios

Availableonlineatwww.sciencedirect.com
NuclearPhysicsB873(2013)1–64
www.elsevier.com/locate/nuclphysb
Non-Abelian T-duality and the AdS/CFT
N =
correspondence: New 1 backgrounds
Georgios Itsiosa,c, Carlos Núñezb,∗, Konstadinos Sfetsosc,a,
Daniel C. Thompsond,e
aDepartmentofEngineeringSciences,UniversityofPatras,26110Patras,Greece
bSwanseaUniversity,SchoolofPhysicalSciences,SingletonPark,SwanseaSA28PP,UK
cDepartmentofMathematics,UniversityofSurrey,GuildfordGU27XH,UK
dTheoretischeNatuurkunde,VrijeUniversiteitBrussel,Belgium
eInternationalSolvayInstitutes,Pleinlaan2,B-1050Brussels,Belgium
Received25February2013;accepted2April2013
Availableonline6April2013
Abstract
Weconsidernon-AbelianT-dualityonN =1supergravitybackgroundspossessingwellunderstoodfield
theoryduals.ForthecaseofD3-branesatthetipoftheconifold,wedualisealonganSU(2)isometry.The
resultisatype-IIAgeometrywhoselifttoM-theoryisofthetyperecentlyproposedbyBahetal.asthedual
tocertainN =1SCFTquiversproducedbyM5-braneswrappingaRiemannsurface.Inthenon-conformal
caseswefindsmoothdualsinmassiveIIAsupergravitywithaRomansmassnaturallyquantised.Weinitiate
theinterpretationofthesegeometriesinthecontextofAdS/CFTcorrespondence.Weshowthatthecentral
chargeandtheentanglemententropyareleftinvariantbythisdualisation.Thebackgroundssuggestaform
ofSeibergdualityinthedualfieldtheorieswhichalsoexhibitdomainwallsandconfinementintheinfrared.
©2013ElsevierB.V.Allrightsreserved.
1. Introduction
Within the context of gauge/string duality, solution generating techniques in supergrav-
ity are an extremely powerful tool. Prominent examples include the use of bosonic and
fermionicT-dualitiestoshowdualsuperconformalsymmetryatstrongcoupling[1,2];theT–s–T
* Correspondingauthor.
E-mailaddresses:[email protected](G.Itsios),[email protected](C.Núñez),[email protected]
(K.Sfetsos),[email protected](D.C.Thompson).
0550-3213/$–seefrontmatter ©2013ElsevierB.V.Allrightsreserved.
http://dx.doi.org/10.1016/j.nuclphysb.2013.04.004
2 G.Itsiosetal./NuclearPhysicsB873(2013)1–64
transformationsthatarethestringanalogueofβ-deformationsingaugetheory[3]andwhichcan
alsobeusedtoconstructgravitydualsforsomenon-relativisticfieldtheories[4–6];theuseofG
structurerotationstoobtainsolutionswith/withoutback-reactedsourcebranesinconifoldrelated
geometries[7–9].Evidentlysomeofthesetechniques,namelyU-dualities,areunderstoodtobe
symmetriesoftheunderlyingstringtheory.FermionicT-duality,however,providesanexample
wherethesymmetryisonlyvalidattree-levelinstringperturbationtheorybutnonethelesshas
applicationsinAdS/CFTwhenconsideringjusttheplanarlimit.
PerformingaT-dualitywithrespecttoanon-Abelianisometrygroupisalsoasolutiongen-
erating technique of supergravity. Rather like the case of fermionic T-duality it is not expected
tobeafullsymmetryofstringperturbationtheory.Butitisnonethelessnaturaltoaskwhatrôle
itmighthavewithinthegauge/stringcorrespondence.Thisstudywasinitiatedin[10]inwhich
thedualisationofAdS ×S5withrespecttoanSU(2)isometrygroupwascarriedout.Theresult
5
wassomewhatsurprising;thedualwasfoundtobeasolutionoftype-IIAsupergravitywhoselift
toM-theorycapturessomeuniversalpropertiesofthesolutionsfoundbyGaiottoandMaldacena
in[11],asdualgeometriestothegeneralisedN =2quiverSCFTsproposedbyGaiottoin[12].
Further progress and works in studying non-Abelian T-duality in this context can be found in
[13–17]andabriefreviewofelementaryaspectsofnon-AbelianT-dualityin[18].
Motivatedbythis,inthispaperweshallinvestigatenon-AbelianT-dualityappliedtosolutions
withminimalsupersymmetrywhosefieldtheorydualiswellunderstood.
TheKlebanov–Witten(KW)solution[19]providesthefirstsuchexample;thissolutionrepre-
sentstheSU(N)×SU(N)conformalfieldtheoryonD3-braneslocatedatthetipoftheconifold.
Wearealsointerestedingaugetheorieswhicharenotconformalbutrather,havenon-trivialRG
flows.TheprototypicalexampleistheKlebanov–Tseytlin(KT)solution[20]whichincorporates
fractionalbranes(D5-braneswrappedontheshrinkingtwo-cycleoftheconifold)andisagood
modelfortheUVdynamicsofanSU(N)×SU(N+M)theory.AsoneflowstowardstheIRthe
theoryundergoesasequenceofSeibergdualitiestoeverdecreasinggaugegroupranks.IntheIR
thesolutionofKTissingular,afactwhichisremediedwhenM isamultipleofN (N =kM),
as it occurs in the Klebanov–Strassler (KS) solution [21], wherein strong coupling effects take
holdandremovethesingularitybyreplacingtheconifoldwithitsdeformation.IntheIRthethe-
ory exhibits R-symmetry breaking (or rather Z →Z ), confinement,domain walls and other
2n 2
interesting phenomena. One level up in complexity is the construction of the gravity dual for
the case in which the KS field theory is exploring its baryonic branch; in this case there exists
aone-parameterfamilyofregulardeformations[22]interpolatingbetweentheKSsolutionand
thewrappedD5-branesolutionsin[23,7,24,25].
AlloftheseexamplespossessrichisometrygroupscontainingatleastanSU(2)factoralong
which we will dualise. There is also a U(1) isometry of the metric (at least in the KW and
KT solutions) that may be understood in the dual field theory as the R-symmetry. The Killing
spinorsofthebackgroundareinvariantundertheSU(2)action,inasensewhichweshallexplain.
Thiscorrespondstothefactthatthesupersymmetriesareunchargedundertheglobal(flavour-
like) symmetries in the field theory and because of this performing the dualisation preserves
supersymmetry.ThisshouldbecontrastedwithperforminganAbelianT-dualityintheinternal
space which would either destroy supersymmetry or result in a singular background.Since the
SU(2) isometrygrouphasthreegeneratorsonewillarriveatsolutionsin(massive)IIA.Letus
nowsummarisewhathappensineachcaseinascendingorderofcomplexity.
WefindthatinthecaseofKWthedualgeometrycanbeliftedtoM-theoryandcanbedirectly
matched to some solutions recently proposed by [26,27], generalising the eleven-dimensional
solutionsof[28],asdualtotheN =1SCFTsobtainedfromM5-braneswrappedonaRiemann
G.Itsiosetal./NuclearPhysicsB873(2013)1–64 3
surface.IncludedinthisclassofSCFTaretheso-calledSicilianquiversof[29].Thisisadirect
N =1 analogue of the dualisation of AdS ×S5 to Gaiotto–Maldacena-like geometries that
5
wasperformedin[10].Indeed,onecanobtaintheKWtheorybyconsideringtheN =2gauge
theorydualtotheorbifold AdS ×S5/Z addingarelevantdeformationandflowingtotheIR.
5 2
WeessentiallyfindaT-dualcomplementofthisrelation.
For the dualisation of the KT solution one finds that the resultant geometry is a solution of
massiveIIAsupergravityandtheRomansmassisnaturallyquantisedbythenumberoffractional
branes. The reason for this can be understood intuitively by the fact that there is a component
of the RR three-form with legs along all the SU(2) directions. Upon dualisation,this then gets
converted to a zero-form. Since this is a solution of massive IIA it has no lift to M-theory; the
fractionalbranesofthetype-IIBsolutionrepresentsomeobstructiontothis.
To get a better handle on this novel background we perform a number of checks. The first
is to look at the central charge before and after the dualisation, following the method of [30]
and [31]. We find that, up to a subtlety that depends on the global properties of the geometry,
thecentralchargebeforeandafterthenon-Abelianduality,matches.Aswewillexplain,thiscan
√
be understood by the fact that the measure, ge−2φ, is an invariant of the duality (just as it is
for Abelian T-duality). The same invariance is present for the entanglement entropy. By using
probe branes one can define a gauge coupling. A strange feature is that this suitably defined
gaugecouplingdoesnotbehavelikethoseofarenormalisable4dQFTwhereg−2∼lnr (asin
theKTcase),insteadgoinglikeg−2∼(lnr)3/2 whichhintsataratherunusualdualfieldtheory
(eitherthat,orthecouplingsodefineddoesnotrepresenttheusualgaugeinteractions).Finally
one can consider the Page and Maxwell charges after duality. Essentially what was D3-brane
charge becomes D6-brane charge. The Maxwell charge of D6-branes changes logarithmically.
As we will discuss, this is one among other similarities with the KS-cascade. We will discuss
below,aformofSeibergdualitythatappearsaftertheduality.
Toprobethelowenergyphysics,oneneedstolookatthedualoftheKSgeometry.Inthiscase
thingsarerathermoreinvolved,butweverifythattheIRsignaturesofconfinementanddomain
wallsarepreservedafterthedualisation.Thesamepatternshows-upifwestartwiththesolution
describingD5-braneswrappingSUSYtwo-cycles[33]anddualiseit.
Interestingly,forthetype-IIBsolutiondescribingthebaryonicbranchoftheKSfieldtheory,
something qualitatively different happens. After the dualisation, we find that the large radius
asymptoticsofthemetricisnolonger(logarithmically)approachingAdS .Weprovidetwosug-
5
gestionsastothefieldtheoreticinterpretationofthis;eitherthisisduetopresenceofanirrelevant
operatorinthedualQFTormoreconservativelythatthistheoryceasestohaveabaryonicbranch.
Letuspresentnowa“roadmap”thatsummarisesthepointsaboveandlayoutthegeneralidea
behindthislongandtechnicalpaper.SeeFig.1.
1.1. Generalideaandroadmap
Westartwith AdS ×S5 orbetteryet,with AdS ×S5/Z ,with N unitsoffluxofthefive-
5 5 2
form. The field theories associated are N =4 SYM or theN =2 version of the two groups
quiver SU(N)×SU(N) and adjoint matter. Following the paper [10] we can perform a non-
AbelianT-dualityonthegeometrytoobtainatype-IIA/M-theorygeometryoftheformproposed
byGaiottoandMaldacena[11]withthefollowingcharacteristics(see[10]fordetails):
• ItcontainsafactorofS2 insteadofahyperbolicplaneH2.
• Theresultinggeometryissingular.Thedilatonfielddivergesatagivenangularposition.
4 G.Itsiosetal./NuclearPhysicsB873(2013)1–64
Fig.1.Aroadmapforthepaper.
• Incorrespondencewiththepreviouspoint,the‘chargedistribution’inthelanguageof[11]
isλ(η)=η,whichimpliesaquiveroftheform
SU(2)×SU(3)×SU(4)×···×SU(N)×SU(N+1)×···. (1.1)
A natural first step taken in this paper, is to apply the non-Abelian T-duality to examples pre-
serving N =1 SUSY.We choosethe Klebanov–Wittengeometry[19]whose dualfieldtheory
isthemassdeformationofthe N =2 fieldtheorydescribedabove.Thenon-Abeliandualityis
performedinSection3.Someinterestingthingsare:thatthesolutionisnon-singular,preserves
N =1andfallswithintheclassofgeometriesproposedin[29,26,27](originallyfoundin[32]).
These geometries have been proposed to be dual to the mass deformation of the Maldacena–
Gaiotto theories. Our geometry presented in Section 3 falls within this class, for the case in
whichwehaveanS2 factorinsteadofanH .Thedualfieldtheoryseemstobelessunderstood
2
inthatcase.
ThefollowingstepistocontinuewiththeknowndeformationsoftheKlebanov–Wittenthe-
ory/geometry. We then study the case of the Klebanov–Tseytlin geometry, Klebanov–Strassler
geometry, baryonic branch and the background of D5-branes wrapping a two-cycle inside the
resolvedconifold.WeobtaininthiscasenewbackgroundsinmassiveIIAwithaquantizedmass
parameter,proportionaltothenumberoffivebranesN ,the‘deformation’fromtheconformal
c
point. We present arguments for the non-singular behaviour of these new solutions (the trans-
formed of the KT-solution is obviously singular as the seed solution is) and ‘define’ their field
theorydualcalculatingobservableswiththebackground.
Inmoredetail,thestructureofthepaperisthefollowing:InSection2wedevelopthetechnol-
ogyrequiredtoimplementthesenon-Abeliandualitytransformations.InSection3weapplythis
totheKWbackground.InSections4,5and6,weturnourattentiontothedualisationsandfield
theory analysis of the non-conformalbackgroundsdescribedabove.We concludein Section7,
presentingsomeopenquestionsandfuturetopicsforresearch.Weprovidegenerousappendices
describingourconventionsandgenericBuscher-likerulesfordualisation.
G.Itsiosetal./NuclearPhysicsB873(2013)1–64 5
2. Non-AbelianT-dualitytechnology
In this section we give details of the dualisation procedure used. The hurried reader who
simplywantstogetthephysicalresultsshouldfeelfreetoreadthefollowing“Non-AbelianT-
duality101”andskippasttherestofthesectionreturningwhenhewishestoknowmoreofthe
technicalities.
2.1. Non-AbelianT-duality101
T-dualitystatesequivalencebetweenstringtheoriespropagatingontwodifferenttargetspace–
times containing some Abelian isometries. In its simplest form, it is the equivalence between
strings on circle of radius R with those on a circle radius 1/R. More generally T-duality pro-
vides a map, known as the Buscher rules, between one solution of supergravity and a second
solution.Apowerfulapproachtoderivingtheserulesisthepathintegralapproach(orBuscher
procedure)[34].Thisprocedureisathreesteprecipe:onebeginswiththestringsigmamodelfor
the first space–time and gauges a U(1) isometry of this space–time; second, one invokes a flat
connectionforthisgaugefieldbymeansofaLagrangemultiplier;finally,oneintegratesbyparts
toyieldanactionwithanon-propagatinggaugefieldthatcanbeeliminatedbyitsequationsof
motiontoproducetheT-dualsigmamodel.
The Buscher procedure can be naturally generalised to the case of a target space equipped
with a non-Abelian isometry group G. One follows exactly the same steps but in this case the
gaugefields are valuedin the algebraof G. Doingso producesa mapbetweenone solutionof
supergravity and another. It is in this spirit of solution generating that we employ non-Abelian
T-dualityinthispaper.
DespitethefactthatthedualisationprocedureisrathersimilarbetweentheAbelianandnon-
Abelian cases there are some important differences. Generically the isometry (and potentially
supersymmetry)enjoyedbythestartinggeometryis,atleastpartially,destroyed.Howeverthis
lostisometrymayberecoveredasanon-localsymmetryinthesigmamodelandthecorrespond-
ingsigmamodelsarecanonicallyequivalent.Asecondpointistherathersubtleeffectofglobal
issues that arise when performing the Buscher procedure on worldsheets of arbitrary genera.
These global concerns mean one should not view non-Abelian duality as a full symmetry of
string (genus) perturbationtheorybutjust atree-levelsymmetry.Nonetheless,if one’sfocus is
onsupergravity(asitwillbeinthispaper)ortheplanarlimitthenonemaystillharnessitspower
asasolutiongeneratingsymmetry.
Early work on non-Abelian duality can be found in [35–41] in context of purely Neveu–
Schwarzbackgrounds.Thissubjecthashadsomethingofarevivalfollowingtheworkof[10],in
whichthisprocedurewasextendedtogeometriescontainRRfluxes.Aparticularlycuriousresult
camefromperforminganon-AbeliandualisationofanSU(2)isometrygroupthatactswithinthe
sphereofAdS ×S5.AfterdualisationtheresultantgeometrywasasolutionofIIAwhoseliftto
5
M-theory bore a very close resemblance to the Giaotto–Maldacena geometries that come from
consideringM5-braneswrappedonRiemannsurfaces.
Weclosethissectionbygivinganexampletogetthereaderinthespirit.Considertheround
metriconthe S3 whichpossesses SO(4)=SU(2) ×SU(2) isometryandmaybewritten(in
L R
Eulerangles)as
ds2=dθ2+dφ2+2cosθdφdψ+dψ2. (2.1)
6 G.Itsiosetal./NuclearPhysicsB873(2013)1–64
AfterperformingthedualisationwithrespecttosaytheSU(2) isometryonefindsageometry
L
thatinterpolatesbetweenR×S2 andR3 withmetricgivenby
d(cid:2)s2=dr2+ r2 (cid:3)dθ2+sin2θdφ2(cid:4). (2.2)
1+r2
Inaddition,inthisexamplethedualgeometryissupportedbyanNStwo-formanddilatongiven
by
B(cid:2)= r3 vol(cid:3)S2(cid:4), Φ(cid:2)=−1ln(cid:3)1+r2(cid:4). (2.3)
1+r2 2
Thisexampleservestoillustratesatwokeyfeaturesthatwewillencounter.FirstlytheSO(4)=
SU(2) ×SU(2) isometrygetsreducedtojustSU(2)thatisreflectedbythepresenceoftheS2
L R
inthedual.Secondlytherehasbeenaserioustopologychange,indeedthedualgeometrycontains
a non-compact direction. Whilst this example does not represent, evidently, a full solution of
supergravity on its own, it may be embedded into true supergravity solutions and indeed it is
prototypicalofthedualisationsthatwewillperform.
AnimportantingredientinthispaperwillbetheincorporationofRRfields.Letusillustrate
howthisworksbysupposingthatintheexampleabovetheinitialgeometryissupportedbyan
RRthree-form
(cid:3) (cid:4)
F =vol S3 . (2.4)
3
Toextractthedualfluxesonemayusethefollowingformula,
eΦ(cid:2)F(cid:2)/ =F/.Ω−1, (2.5)
wheretheslashesindicatetheRRpolyform(sumofRRforms)contractedwithgammamatrices
to form a bispinor, i.e.F/ =Γ123, and Ω is a matrix, the construction of which we describe in
detailinthefollowingsection,givenbyinthiscase
(cid:3) (cid:4)
1
Ω−1= √ −Γ123+rΓr . (2.6)
1+r2
Fromthisoneascertainsthatthedualgeometrywillcontainazero-formandtwo-form:
r3 (cid:3) (cid:4)
F =1, F = vol S2 . (2.7)
0 2 1+r2
Notice that this would, when embedded into a true type-II supergravity background, lead to
a solution in massive type-IIA (the F is the Romans mass and comes when the Γ123 in Ω
0
annihilate the same factor inF/ ). We shall see the same phenomenon happens in a number of
3
the examples in this paper. The fact that the type of the supergravity changed from IIB to IIA
is due to the fact that the isometry group dualised had an odd dimension (if it were to be even
dimensionalthetypewouldhaveremainedthesame).
We now present details of how to technically compute the dualisation rules for the non-
Abelianduality.
2.2. Non-AbelianT-duality;somenutsandbolts
WewishtoconsiderbackgroundsthatsupportanSU(2)isometrysuchthatthemetriccanbe
castas
G.Itsiosetal./NuclearPhysicsB873(2013)1–64 7
ds2=G (x)dxμdxν+2G (x)dxμLi+g (x)LiLj, (2.8)
μν μi ij
where μ=1,2,...,7 and Li are the Maurer–Cartan forms. Our group theory conventionscan
befoundinAppendixA.TheNSsectorcomprisesalsothe2-form
1
B=B (x)dxμ∧dxν+B (x)dxμ∧Li+ b (x)Li∧Lj (2.9)
μν μi ij
2
andadilaton
Φ=Φ(x). (2.10)
Hence,allcoordinatedependenceontheSU(2)Euleranglesθ,φ,ψ iscontainedintheMaurer–
Cartan one-forms whilst the remaining data can all be dependent on the spectator fields xμ.
Notice that we could have taken b =0,1 however, we will not do that since in the specific
ij
exampleswewillencounteritisnecessaryforaclearpresentationofthevariousresults.
Inwhatfollowsitwillbeconvenienttouseaparametrisationoftheframefieldsgivenby
eA=eAdxμ, ea=κa Lj +λa dxμ, (2.12)
μ j μ
whereA=1,2,...,7anda=1,2,3.Bydemandingthat
ds2=η eAeB+eaea, (2.13)
AB
weobtainthat
λaλa=K , η eAeB =G −K ,
μ ν μν AB μ ν μν μν
κa κa =g , κa λa =G , (2.14)
i j ij i μ μi
whereη istheseven-dimensionalMinkowskimetric.Notethatκa andλa dependingeneral
AB i μ
onthexμ’s.
2.2.1. Thenon-AbelianT-dualoftheNS-sector
The corresponding Lagrangian density for the NS sector metric and antisymmetric fields is
givenby
L0=Qμν∂+Xμ∂−Xν+Qμi∂+XμLi−+QiμLi+∂−Xμ+EijLi+Lj−, (2.15)
where,inaccordancewith(A.4),Li±=−iTr(tig−1∂±g)andwehavealsodefined
Q =G +B , Q =G +B ,
μν μν μν μi μi μi
Q =G +B , E =g +b . (2.16)
iμ iμ iμ ij ij ij
Toperformthenon-AbelianT-dualitywereplacederivativeswithcovariantderivativesaccording
to
1 Toseethatnotethatin(2.9)therelevanttermbecomes
√
BμidXμ∧Li+(cid:14)ijkbkLi∧Lj=BμidXμ∧Li+ 2bidLi
√ √ (cid:3) (cid:4)
=(Bμi− 2∂μbi)dXμ∧Li+ 2d biLi , (2.11)
whereinthesecondlinewehaveperformedapartialintegration.Hence,thelasttermhasnocontributiontothefield
strengthdBandwemayaswelldenotethefirsttermbyBμidXμ∧Li.
8 G.Itsiosetal./NuclearPhysicsB873(2013)1–64
∂±g→D±g=∂±g−A±g, (2.17)
andaddtheLagrangemultiplierterm
−iTr(vF±), F±=∂+A−−∂−A+−[A+,A−]. (2.18)
Thenthetotalactionisinvariantunder
g→h−1g, v→h−1vh, A±→h−1A±h−h−1∂±h, (2.19)
foragroupelementh(σ+,σ−)∈SU(2).Underthistransformationthefieldsxμ stayinertand
thusarecalledspectators.AftersomepartialintegrationstheLagrangemultipliertermtakesthe
form
Tr(i∂+vA−−i∂−vA+−A+fA−), fij =fijkvk. (2.20)
We now can integrate out the gauge fields to produce a dual theory that still depends on
θ,φ,ψ,v and the spectators. One must now gauge fix the SU(2) isometry to remove three of
i
thesevariables.Theobviouswaytoproceedistoset
g=I, (2.21)
i.e.θ =φ=ψ =0inthenotationofAppendixA,whichleavesanactionintermsofjustthev
i
andthespectators.Thereareothergaugefixingchoicesthatmaybemorerevealingby,forin-
stance,makingmanifestsomeresidualisometries.Differentgaugefixingchoicesmayberelated,
atleastlocally,throughcoordinatetransformationsas willdemonstratebelowinSection2.2.3.
For the time being we proceed with the gauge fixing choice g =I. Integrating out the gauge
fieldsgives
(cid:3) (cid:4) (cid:3) (cid:4)
Ai+=iMj−i1 ∂+vj +Qμj∂+Xμ , Ai−=−iMi−j1 ∂−vj −Qjμ∂−Xμ , (2.22)
wherewehavedefinedthematrix
M=E+f. (2.23)
SubstitutingbackintotheactiongivesthedualLagrangian
(cid:3) (cid:4) (cid:3) (cid:4)
L(cid:2)=Qμν∂+Xμ∂−Xν+ ∂+vi+∂+XμQμi Mi−j1 ∂−vj −Qjμ∂−Xμ . (2.24)
FromthiswereadoffthebackgroundfieldsoftheNS-sectorfortheT-dualtheoryas
Q(cid:2) =Q −Q M−1Q , E(cid:2) =M−1,
μν μν μi ij jν ij ij
Q(cid:2) =Q M−1, Q(cid:2) =−M−1Q . (2.25)
μi μj ji iμ ij jμ
Additionally one finds that the dilaton receives a contribution at the quantum level just as in
Abelianduality
Φ(cid:2)(x,v)=Φ(x)− 1ln(detM). (2.26)
2
It is clear from the above that the inverse of the matrix M determines the dual geometry.
Since we are working with SU(2) isometries it is simple enough to evaluate this explicitly. In
three-dimensionsanantisymmetricmatrixisdualtoavector,hencewemaywrite
b =(cid:14) b . (2.27)
ij ijk k
G.Itsiosetal./NuclearPhysicsB873(2013)1–64 9
√
Rescalingalsov →v / 2wehavetoinvertthematrixwithelements
i i
M =g +(cid:14) y , y =b +v . (2.28)
ij ij ijk k i i i
Tocomputetheinversedefineanantisymmetricdensityandavectoras
(cid:5) yi yi
e¯ = detg(cid:14) , zi = √ = . (2.29)
ijk ijk
detg detκ
Inthiswaywemayusethematrix g tolowerandraiseindicesin zi sincenowtheindexhas
ij
beentransformedintoacurvedindex.Then
M =g +(cid:14)¯ zk. (2.30)
ij ij ijk
Thentheinverseisfoundtobe
(cid:3) (cid:4) (cid:3) (cid:4)
1
M−1 ij = gij +zizj −(cid:14)¯ij zk , z2=zizjg =ziz . (2.31)
1+z2 k ij i
Returningtoouroriginalvariables
(cid:3) (cid:4) (cid:3) (cid:4)
1
M−1 ij = detggij +yiyj −(cid:14) y¯ , (2.32)
detg+y2 ijk k
wherey¯ =g yj andy2=yiyjg =y¯ y¯ gij.
i ij ij i j
2.2.2. ComputingtheLorentztransformation
Bymakinguseof(2.22)onecanestablishthattheworldsheetderivativestransformunderthe
non-AbelianT-dualityas
(cid:3) (cid:4) (cid:3) (cid:4)
Li+=− M−1 ji ∂+vj +Qμj∂+Xμ ,
(cid:3) (cid:4)
Li−=Mi−j1 ∂−vj −Qjμ∂−Xμ ,
∂±Xμ=invariant. (2.33)
These relations, in fact, provide a canonical transformation in phase space between pairs of T-
dualsigmamodels[37,40].
Crucialtouswillbethatbyvirtueof(2.33),leftandrightmovershavedifferenttransforma-
tionrulesandwilldefinetwodifferentsetsofframefields.However,sincetheseframefieldswill
describethesamegeometrytheymustberelatedbyaLorentztransformation.Explicitlywefind
thattheframesin(2.12)transforms,usingthe“plus”andthe“minus”transformations(2.33),to
theframes
(cid:3) (cid:4)
e→eˆ+=−κM−T dv+QT dX +λdX,
e→eˆ−=κM−1(dv−QdX)+λdX, (2.34)
where(Q) =Q .Writing
iμ iμ
eˆ+=Λeˆ−, (2.35)
whereΛistheLorentztransformationmatrixtobecomputed.Wefindfromequatingtheterms
proportionaltodvin(2.34)that
Λ=−κM−TMκ−1=−κ−TMM−TκT. (2.36)
10 G.Itsiosetal./NuclearPhysicsB873(2013)1–64
ThetermsproportionaltodX equateidenticallywithnoextracondition.Toexplicitlycompute
Λnotefirstthat
(cid:3) (cid:4)
κ−TMκ−1 =δ +(cid:14) ζc, (2.37)
ab ab abc
where
ζa=κa zi, (2.38)
i
istheflatindexcoordinate.Then
(cid:3) (cid:4) (cid:3) (cid:4)
1
κM−1κT ab= δab+ζaζb−(cid:14) ζc . (2.39)
1+ζ2 abc
Thenwecomputethat
ζ2−1 2 (cid:3) (cid:4)
Λab= δ − ζaζb+(cid:14) ζc , (2.40)
ζ2+1 ab ζ2+1 abc
where ζ2=ζ ζa. Notethatthishasexactlythesameformas in[10]forthecaseofthePCM.
a
Moreover,itisanO(3)rotationasithasdetΛ=−1.Theeffectofthenon-trivialextracouplings
g and b is to “dress up” the original Lagrange multipliers and is hidden into the definition
ij ij
ofζa.
ThisLorentztransformationalsoinducesanactiononspinorsgivenbyamatrixΩ obtained
byrequiringthat
Ω−1ΓaΩ=Λa Γb. (2.41)
b
Onefindsthat2
−Γ +ζ Γa
Ω=Γ (cid:5)123 a , (2.42)
11
1+ζ2
where Γ istheproductofalltenGammamatrices,thatanticommuteswitheachoneofthem
11
andforMinkowskispace–time,itsquarestounity.NotealsothatΩ leavesinvarianttheGamma
matricesΓA correspondingtotheseven-dimensionalspectatorspace–timeanditisofthesame
formasthecorrespondingmatrixin[10].
2.2.3. Generalgaugefixingandcoordinatetransformations
As noted above, gauge choices different than (2.21) might be more convenient in certain
applications. To expound this point let us consider a more general situation where the target
admitsanisometrygroupGandwedualisewithrespecttoasubgroupH (forthecaseathand
wedualisethefullSU(2)isometryandsodimG=dimH =3).Specificgaugechoicesamong
the original dimG+dimH variables are of the form f (g,v)=0, i =1,2,...,dim(H). This
i
leavesdimGvariablesfortheT-dualmodel.Nevertheless,onemayshowthatthedifferentgauge
choices are related by coordinate transformations. This can be done by defining the “dressed”
Lagrangemultipliersas
vˆ =D vj, (2.43)
i ji
2 ThegeneralexpressionforΩforafreelyactinggroupGcanbefoundin[14].