Scale invariance of the Î· deformed AdS<inf>5</inf>Ã S5 superstring, T duality and modified type II equations

(1)

ScienceDirect

Nuclear Physics B 903 (2016) 262–303

www.elsevier.com/locate/nuclphysb

Scale

invariance

of

the

η

-deformed

AdS

₅

×

S

5

superstring,

T-duality

and

modified

type II

equations

G. Arutyunov

a,b,1

_{S. Frolov}

c,1

_,

_{B. Hoare}

d,∗

_,

_{R. Roiban}

e

_,

_{A.A. Tseytlin}

f,2 a_{II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany}

b_{Zentrum für Mathematische Physik, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany} c_{Hamilton Mathematics Institute and School of Mathematics, Trinity College, Dublin 2, Ireland} d_{Institut für Theoretische Physik, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland}

e_{Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA} f_{The Blackett Laboratory, Imperial College, London SW7 2AZ, UK}

Received 30November2015;accepted 21December2015

Availableonline 23December2015

Editor: StephanStieberger

Abstract

Weconsider the ABF background underlying the η-deformedAdS5×S5 sigma model. This back-ground fails to satisfy the standard IIBsupergravity equationswhich indicatesthat the corresponding sigmamodelisnotWeylinvariant,i.e.doesnotdefineacriticalstringtheoryintheusualsense.Weargue thattheABFbackgroundshouldstilldefineaUVfinitetheoryonaflat2dworld-sheetimplyingthatthe η-deformedmodelisscaleinvariant.ThispropertyfollowsfromtheformalrelationviaT-dualitybetween theη-deformedmodelandtheonedefinedbyanexacttype IIBsupergravitysolutionthathas6isometries albeitbrokenbyalineardilaton.WefindthattheABFbackgroundsatisfiescandidatetype IIBscale in-varianceconditionswhichfortheR–Rfieldstrengthsareofthesecondorderinderivatives.Surprisingly, wealsofindthattheABFbackgroundobeysaninterestingmodificationofthestandardIIBsupergravity equationsthatarefirstorderinderivativesofR–Rfields.Thesemodifiedequationsexplicitlydependon KillingvectorsoftheABFbackgroundand,althoughnotuniversal,theyimplytheuniversalscale invari-anceconditions.Moreover,weshowthatitispreciselythenon-isometricdilatonoftheT-dualsolutionthat

* _{Corresponding}_author.

E-mail addresses:[email protected](G. Arutyunov),[email protected](S. Frolov),[email protected]

(B. Hoare),[email protected](R. Roiban),[email protected](A.A. Tseytlin).

1 _{Correspondent}_fellow_at_Steklov_Mathematical_Institute,_Moscow. 2 _Also_at_Lebedev_Institute,_Moscow.

http://dx.doi.org/10.1016/j.nuclphysb.2015.12.012

(2)

leads,afterT-duality,tomodificationoftype IIequationsfromtheirstandardform.Weconjecturethatthe modifiedequationsshouldfollowfromκ-symmetryoftheη-deformedmodel.Allourobservationsapply alsotoη-deformationsofAdS3×S3×T4andAdS2×S2×T6models.

1. Introduction

Thestudy of integrabledeformationsof theAdS5×S5 superstring sigmamodelis an im-portantdirection inthe searchfor newsolvable examplesof AdS/CFTduality. Aninteresting one-parameter integrablegeneralisationof theclassical AdS5×S5 Green–Schwarz action re-latedtothequantumdeformationoftheunderlyingsupergroupsymmetrywasfoundin [1].Just fromtheconstructionofthis“η-model”(basedonaparticular current–currentdeformationof thesupercosetaction [2]generalisingthebosonicmodelof [3])thereisnoapriorireasonwhyit shoulddefineascaleinvariant(UVfinite)2dtheoryand,moreover,whyitshouldpreservethe conformal(Weyl)invarianceandhencestillcorrespondtoaconsistentsuperstringtheoryasthe undeformedAdS5×S5modeldoes.3

Theonlyindicationinthisdirectionisthattheη-modelaction,liketheoriginalAdS5×S5

action,isinvariantunderaversionoffermionicκ-symmetry [1],whichreducesthenumberof fermionsbyhalf.However,the usualclaimthat κ-symmetryimpliesthecorrespondingaction canbeinterpretedasthatofaGSsuperstringpropagatinginabackgroundthatisaconsistent type IIsupergravitysolution(andhencedefinesaconsistentcriticalsuperstringtheory)assumes thattheκ-symmetryisofthestandardGS“projector”form [5].Thisismostprobablynotthe casefor theη-modelathigherordersinfermions.Indeed,itwasfoundin [6,7]thatthetarget spacebackgroundcorrespondingtotheη-modelaction [1],interpretedasaGSaction,doesnot representatype IIBsupergravitysolution.

StartingwiththeGSLagrangianwritteninsuperspaceform(ZM=(xm,θα))

L=(√hhabE_Mr E_Nsηrs−abBMN)∂aZM∂bZN , (1.1)

onecansolvethestandardtypeIIsuperspaceconstraintsandBianchiidentitiesforE(Z),B(Z)

(whichimplythesupergravityequations)inordertoexpresstheGSactionintermsofcomponent fields.Onethenobservesthatthe dilatonφ (whichispartof thedilatonsuperfield (Z)that isintroducedintheprocessofsolving theconstraints)entersthe world-sheetaction(i)inthe combinationF=eφF withtheR–Rfieldstrengthsstartingatorderθ2 and(ii)viaderivatives

∂mφ startingat order θ4 (see [8] and the references therein). Thisactionhas classical Weyl invarianceandκ-symmetry,whichwillbebroken,ingeneral,byquantum corrections.Asfor thebosonicstring [9],tocancelthe2dstresstensortraceanomalyrequiresaddingthefamiliar 1-loopdilatoncounterterm∼d2z√hR(2) (Z)(see [10,11]andthereferences therein).4

Thecaserelevanttoourdiscussionbelowisaspecialisometrictype IIsolutionforwhichthe metricGmn,B-fieldBmnandR–Rfields Fm1...mn areinvariantwhileφislinearintheisometric

3 _This_is_in_contrast,_e.g.,_to_the_integrable_deformation_[4]_based_on_TsT_duality_{transformations,}_which_preserve

conformality.Inparticular,theTsTdeformedbackgroundisasolutionoftype IIBsupergravity.

4 _This_additional_term_is_certainly_required_to_reproduce_the_standard_1-loop_{Weyl-invariance}_conditions_for_the_G_and

(3)

directions.InthiscasetheGSactionwilldependontheisometriccoordinatesonlythroughtheir 2dderivativesandcanthusbeT-dualised.Asweshallsee,inthiscasetheT-dualmodelwillbe scaleinvariantbutmaynotbeWeylinvariant(onemaynotbeabletocanceltheWeylanomaly byalocalcounterterm),i.e.maynotcorrespondtoatype IIsupergravitysolution.

The ABF background[6,7]includes the10d metric G,the B-fieldandthe R–R fields Fn (n=1,3,5)thatareextractedfromthequadraticfermionicpartoftheactionof [1]putintothe usualGSform,

A= −T

d2z

₁

2(η ab_G

mn−abBmn)∂axm∂bxn

+iθ ¯ mDθ ∂xm+ ¯θ mF· nθ ∂xm∂xn+. . .

. (1.2)

For the standard GS actionin atype IIB supergravity background Fn are interpreted as the productsofthedilatonandtheR–Rfieldstrengths Fn=eφFnbutintheη-modelcasethereis noindependentinformationaboutthe dilaton,andthereexistsnodilatonfieldthatcompletes

G,B,FnoftheABFbackgroundtoatype IIBsolution [7].

Whilenotsolvingthestandardtype IIBequationsdirectlythisABF backgroundstill turns outtobeveryspecial:itisrelatedbyT-dualitytoanexacttype IIBsupergravitysolution [12, 13].ThelatterHTbackgroundinvolvesanon-diagonalmetricGˆ,animaginary5-formFˆ5and

the dilatonφˆ,andtheT-dualityappliedinall6 isometricdirections actsonlyonthefields Gˆ

andFˆ5=eφˆFˆ5 enteringthe correspondingGSaction (1.2)onaflat 2dbackground. TheGS

actionforanytype IIsolution(andthusfortheHTbackground)shouldbeWeylinvariantand,in particular,scaleinvariant.AstheT-dualityappliedtotheGSaction [14]isasimplepathintegral transformation,theT-dualityrelationbetweenthe ABFandHTbackgroundsimplies [12]that theη-modelactionshoulddefineascaleinvariant2dtheoryatleastto1-looporder.

However,theremaybeaproblemwithWeylinvariancefortheη-modelactiononacurved 2dbackground.TheHTdilatonφˆ hasatermlinearlydependingontheisometricdirectionsof

ˆ

GandFˆ5andthusonecannotdirectlyapplythestandardT-dualitytransformationrules [15]to

thefullHTbackgroundtogetafullT-dualsupergravitysolution,andthustheWeylinvariance oftheT-dualsigmamodelrequiresfurtherinvestigation.5Thisisofcourseconsistentwiththe observation [7]thatthe ABFbackgrounddoesnotsatisfythefullsetof type IIBsupergravity equations.

The aim of the present paper istofurther clarifyandextend theseobservations.Weshall demonstratethattherelationbyformalT-dualitybetweentheABFandHTbackgroundsimplies thattheformer,whilenotasupergravitysolution,shouldsatisfythefollowingtwogeneralisations or“modifications”ofthetype IIsupergravityequations:

(i)thescaleinvarianceconditionsforthetype IIsuperstringsigmamodel(withequationson theR–RfieldsFbeingof2ndorderinderivatives)

5 _The_1-loop_Weyl_invariance_conditions_of_the_NSR_or_GS_{type II}_superstring_sigma_model_are_believed_to_be_equivalent

tothefieldequationsoftype IIsupergravity.Whilethisisawell-establishedfactintheNS–NSsector[9,16]thiswas neverdemonstrateddirectlywiththeR–Rcouplingsincluded(forsomerelatedwork,mostlyfortheheteroticstring,see

(4)

(ii)asetofequations thatarestructurallysimilartothoseoftype IIsupergravity(with 1st-orderequationsfortheR–Rfields F)butinvolving,insteadofderivativesofthedilaton,acertain co-vectorZmplayingnowtheroleofthedilatonone-formandaKillingvectorImresponsible forthe“modification”oftheequationsfromtheirstandardform.6

Whilethe scaleinvarianceconditions areuniversal,the secondsetof equations(whichwe shallrefertoas“I-modified”type IIequations)onlyapplytoparticularbackgroundswith iso-metricG,Band F-fields,whicharerelatedbyformalT-dualitytoatype IIsolution(G,ˆ B,ˆ Fˆ,φˆ) withthe dilatonφˆ containing aterm linearintheisometriccoordinates.Suchadilaton back-ground,breakingisometriesbyalineartermonly,isspecial.Asthetype IIsupergravityequations writtenintermsoftheF-fieldsonlydependonthedilatonthroughitsderivatives,theyremain independentoftheisometricdirections.Asaresult,thestandardtype IIsupergravityequations fortheT-dualsolution(G,ˆ B,ˆ Fˆ, ˆφ)canbere-interpretedascertainmodifiedtype IIequations fortheoriginalfields(G,B,F),alsodependingonthevectorsZ andI.TheKillingvectorIm

dependenceisfixedbythetermlinearinisometriccoordinatesinthedilaton,whilethevector

ZmisdeterminedbyapplyingthestandardT-dualityrulestothepartofthedilatonindependent oftheisometriccoordinates.7

ItispossibletoexpressthemodifiedequationsfortheNS–NSfieldsintermsofjustonesingle vectorXm=Zm+Im,whichisthevectorthatappearsinthescaleinvarianceconditions.The su-perstringscaleinvarianceconditionsgeneralisethefamiliarone-loopscaleinvarianceconditions forthebosonicsigmamodelwithcouplingsGmnandBmn(cf. (1.2))

β_mnG ≡Rmn−1₄HmklHnkl= −DmXn−DnXm, (1.3)

β_mnB ≡1₂DkHkmn=XkHkmn+∂mYn−∂mYn. (1.4)

Here the terms involving Xm [17] andYm do not contribute toon-shell UV divergencesor, equivalently, reflect the freedom of renormalisation by reparametrisations andB-fieldgauge transformations.The Xm terms drop outof the actionif thesigmamodel field xm issubject to the classical equations, or, equivalently, they can be absorbed in a field renormalisation,

xm→xm+Xmlog.TheoriginoftheXkHkmntermcanbeunderstoodeitherbystartingwith acountertermproportionalto(DmXn+DnXm)∂axn∂bxn+. . .,integratingbypartsandusing theequations ofmotion for xm,orbyobservingthat Bmn transformsunder acombinationof reparametrisationsandgaugetransformationsasXk∂kBmn+∂mXkBkn−∂nXkBkm+∂mYn−

∂nYm =XkHkmn+∂mYn−∂nYm whereYm or Ym drop out of the sigmamodel actionupon integrationbyparts.

TheWeylinvarianceconditionsareequivalenttothevanishingof thetraceofthe2dstress tensoroperatoronacurved2dbackground.FortheNSRtype IIsuperstringsigmamodeltheycan besatisfiedprovidedoneaddsthedilatonterm ∼R(2)φ(x)[9,19–21,16]:theyareastrongerform ofthescaleinvarianceconditions (1.3), (1.4)withXmandYmnolongerarbitrary,butgivenby

Xm=∂mφ , Ym=0. (1.5)

TheWeylinvarianceequations (1.3), (1.4), (1.5)implythe“centralcharge”identity [9,22]

∂mβ¯φ=0, β¯φ≡R−₁₂1H_mnk2 +4D2φ−4∂mφ∂mφ , (1.6)

6 _In_what_follows_we_shall_not_distinguish_between_co-vectors_and_vectors,_referring_to_both_Xm_and_Xm_as_vectors. 7 _Let_us_stress_that_Fˆ_and_φˆ_explicitly_depend_on_the_isometric_coordinates._It_is_G,ˆ _B,ˆ _Fˆ_,_d_φˆ_,_and_G,_B,_F_,_Z_that_are

(5)

i.e.thattheeffectivedilaton“β-function”isaconstant(whichshouldbezeroincriticalstring theory). The fullsetof Weylinvariance equations for G,B andφ follows fromthe effective actionwiththesameformastheNS–NSsectorofthetype IIsupergravityaction(F≡eφF)

S=

ddx√Ge−2φβ¯φ+F F+. . .

=

ddx√G e−2φβ¯φ+FF+. . ., (1.7)

wherewehaveindicatedthepresenceoftheR–Rfieldstrengthtermsforfuturereference. The generalisationof thescaleinvariance conditionstothepresenceof R–Rfieldsisgiven by (1.3), (1.4)withextraFF terms,togetherwithasetofsecond-derivativeequationsforthe R–R fields F that directlyenter the GSaction (1.2), 1₂D2F+. . .=X∂F +F∂X. Herethe r.h.s.standsforreparametrisation(Liederivative)termswiththesameX-vectorasin (1.3), (1.4) anddotsindicatenon-linearterms.InthespecialcasewhenXm=∂mφtheseequationsarethe consequenceof the type IIB equations or Weylinvariance conditions, whichare 1st orderin

F =e−φF,i.e.d F+. . .=0 anddF+. . .=0.8Theseuniversalscaleinvarianceconditions willbesatisfiedbytheABFbackgroundforaparticularchoiceofthevectorsXmandYm.

Toexplaintheoriginofthesecond“I-modified”setofequationsletusfirstignoretheR–R fieldsandassumethatthereexiststhefollowingmetric-dilatonbackgroundthatsolvestheWeyl invarianceequations(i.e.Rmn+2DmDnφ=0,β¯φ=const)

ˆ

ds2=e2a(x)ˆ [dyˆ+ ˆAμ(x)dxμ]2+gμν(x)dxμdxν , φˆ= −cyˆ+f (x) . (1.8)

Herethemetrichasanisometrywhichisbroken bythelinearterminthedilaton(c=const). Examplesofsuchnon-trivialsolutions9canbefoundbytakingspeciallimitsofgaugedWZW backgrounds [13].T-dualisingthismetric,wefindadiagonalmetricGandB-field,i.e.

ds2=e2a(x)dy2+gμν(x)dxμdxν, B= ˆAμ(x) dy∧dxμ, a= −ˆa . (1.9)

Forc=0 (i.e.whenφˆ isisometric)thesefieldstogetherwiththeT-dualitytransformeddilaton

φ= ˆφ− ˆa would solve the standardWeylinvariance equations (1.3), (1.4)with Xm=∂mφ,

Ym=0.Fornon-zero ctheequationRˆmn+2DˆmDˆnφˆ=0 (fortheoriginalbackground (1.8)) expressedintermsofthedualfieldsG,Bwillcontainadditionalc-dependenttermsobstructing (fornon-constanta(x))theintroductionofanewdilatonscalar.Still,theycanbeputinamore generalformRmn+DmXn+DnXm=0 withaspecialvectorXgivenby10

Xmdxm≡Imdxm+Zmdxm=c e−2ady+ ∂μ(φˆ− ˆa)+cAˆμ

dxμ. (1.10)

Thedilatonequationβ¯φ=0 fortheoriginalbackground (1.8)alsocanberewrittenasthe fol-lowinggeneralisedequation(cf. (1.6))11

8 _The_relation_between_the_1st-order_and_2nd-order_equations_on_F_has_the_same_spirit_as_the_relation_between_the_Dirac

andtheKlein–Gordon(squaredDirac)equationsforspinorfields.

9 _It_is_important_that_dilaton_has_a_linear_term_in_a_“warped”_isometric_direction_of_the_metric,_i.e._a(x)_,_Aμ(x)_are

non-constant,otherwisetheeffectofaddingthelineardilatonwouldbetrivial.

10 _The_need_to_introduce_the_vector_Xm_,_which_is_not_simply_a_gradient_of_a_scalar,_is_therefore_directly_related_to_the

feature∂y_ˆφˆ= −c =0.

11 _Note_that_this_equation_is_not_present_in_the_list_of_scale_invariance_conditions,_and_Weyl_invariance_conditions_require

(6)

¯

βX≡R−₁₂1H_mnk2 +4DmXm−4XmXm=0, (1.11)

thatissatisfiedfortheT-dualbackground.

TheT-dualbackground(G,B)definesasigmamodelthatisscaleinvariantonaflat2d back-ground(satisfyingequations (1.3), (1.4)withYm=Xm)butwhichisnot Weylinvariant.The traceof stress tensorT =β_mnG ∂axm∂axn+βmnB ab∂axm∂bxn is atotalderivative T = ∇aNa,

Na=2(Xm∂axm+abYm∂bxm)(uptotermsproportionaltothexmequationsofmotion).This cannotbecancelledbyalocalcounterterm(theclassicaldilatonterm)unlessXm=∂mφ,Ym=0

[19,20],whichisnotthecasefortheABFbackground.Thesigmamodelsbasedon (1.9)(with explicitbackgroundsgivenbelow)thusrepresentparticularexamplesof2dscaleinvariant theo-riesthatavoidtheZamolodchikov–Polchinskitheorem [23]duetotheirnon-compactness(and/or non-unitarity related tothe presence of time-like directions).It thus remains unclear if such backgroundsrelatedbyformalT-dualitytoWeylinvariantmodels (1.8)canalsobeassociated somehowwithaconsistentcriticalstringtheory.

Asweshallseebelow,asimilargeneralisationofthefullsetofthebosonictype IIsupergravity equationsalsoexistsinthepresenceofR–Rfields Fnthathavethesameisometriesasthemetric (i.e.when (1.8)isextendedtoananalogoftheHTsolution [12]).Thusingeneral,givenatype II solution withnon-isometric lineardilatonthere will bean associated (“T-dual” or ABF-like) backgroundsolvingsuchamodifiedsetoftype IIequations.

Therestofthispaperisorganisedasfollows.Insection2weshallpresentthegeneralscale invarianceconditionsforthecouplingsG,Bofthesigmamodel (1.2)thatgeneralise (1.3), (1.4) tothepresenceoftheR–Rfields FandshowthatthereexistsuchvectorsX=Z+I andY that theseequationsaresatisfiedbytheABFbackground.Insection3weshallderiveamodification ofthe standard1st-orderIIBsupergravityequations ofthe R–Rfluxesthatis “driven”bythe specialisometryvectorI andwhicharesatisfiedbytheABFbackground.Insection4weshall showthatcombiningthese1st-orderequationsonecanfind2nd-orderequationsforFthathave therightstructure(whengeneralisedtoarbitraryvectorX)tobeinterpretedasscaleinvariance conditionsontheR–Rcouplings.Insection5weexplainhowthestandardtype IIsupergravity equationsforasolutionwiththedilatonlinearalongtheisometricdirectionsismappedtothe modifiedequationsforT-dualsolution.

Ournotationandsomeusefulrelationsaresummarisedin Appendix A.In Appendix Bwe present the explicit form of the ABF backgroundand the T-dual type IIB HT solution. Ap-pendix Ccontainsthederivationoftheidentity∂mβ¯X=0 fromthemodifiedtype IIequations whichiscloselyrelatedtotheon-shellconservationofR–Rstresstensor.In Appendix D,starting withthemodifiedtype IIequations,wederivethe2nd-orderequationsfortheR–Rfieldsthatare candidatesforthecorrespondingscaleinvariance conditions.In Appendix Ewe remarkonan alternativederivationoftherelation (2.13)forthevectorZ,whichplaystheroleofthedilaton one-forminthemodifiedequations.In Appendix FwesummarisetheanalogsoftheABFand HTbackgrounds intheAdS2×S2×T6andAdS3×S3×T4casesandgivethecorresponding expressionsforthevectorsX,Y andI thatsolvethescaleinvarianceandmodifiedtype II equa-tions.In Appendix Gweexplainhowthe2nd-orderequationsfortheR–RcouplingsFemerge astheone-loopconditionsofscaleinvariancefortheGSsigmamodel (1.2).

2. Scale invariance conditions and modified type II equations: NS–NS sector

(7)

Appendix G).Theθ¯Fθ ∂x∂xtermsintheGSaction (1.2)shouldleadtoone-loopdiagrams(with onebosonicandonefermionicline)contributinglogarithmicUVdivergences ∼FF∂x∂x.These termswillproduceextra FFtermsintheβ-functionsin (1.3)and (1.4).Inparticular,theanalog of theEinsteinequation (1.3)shouldpickuptheR–RstresstensortermandtheB-field equa-tion (1.4),theFFtermasintheIIsupergravityequations.12ThisisexpectedasforXm=∂mφ,

Ym=0 theresultingequationsaretheWeylinvarianceequationsthatshouldbeequivalenttothe type IIsupergravityequations.

ThescaleinvarianceequationsfortheF-fields(tobediscussedinsection4)willnot,however, havethefamiliarsupergravityformof1st-orderequationsforF (theseshouldfollowfromthe Weylinvarianceconditions).Insteadtheywillbeof2ndorder,D2F+. . .=X-dependentterms, andforXm=∂mφwillbeaconsequenceofthe1st-ordersupergravityequations.

Explicitly,thescaleinvarianceconditions (1.3)and (1.4)generaliseto

β_mnG ≡Rmn−1₄HmklHnkl−Tmn= −DmXn−DnXm, (2.1)

β_mnB ≡1₂DkHkmn+Kmn=XkHkmn+∂mYn−∂nYm, (2.2)

Tmn≡1₂FmFn+1₄FmpqFnpq+₄_×1₄_!FmpqrsFnpqrs−₂1Gmn(1₂FkFk

+ 1 12FkpqF

kpq_{) ,}

(2.3)

Kmn≡1₂FkFkmn+₁₂1FmnklpFklp. (2.4)

HereFm,Fmnk,FmnklpareR–Rfieldsoftype IIBsupergravity(fornotationsee Appendix A). ForXm=∂mφ,Ym=0 theseequationsfollowfromtype IIBsupergravityaction (1.7).Tmnis thefamiliarstresstensorthatfollowsfromthetype IIBaction (1.7)uponvariationoverGmn.13

Aswas notedin [12],the existence ofthe HTsolution relatedtotheABF backgroundby T-duality,suggeststhattheGSsigmamodelforthelatterdefinedonaflat2dbackgroundshould be scaleinvariant(atleasttoleading,1-loop,order).Our keyobservationisthat indeedthere existvectorsXm andYm such thateqs.(2.1) and(2.2)aresatisfied for theABF background

(B.1).ThevectorXmrequiredtosatisfy (2.1)turnsouttobe(see Appendix Bfornotation)

X≡Xmdxm=c0

1+ρ2

1−2_ρ2dt+c1ρ 2

sin2ζ dψ2+c2

ρ2cos2ζ

1+2_ρ4_sin2_ζdψ1

+c3

1−r2

1+2_r2dϕ+c4r

2_sin2_{ξ dφ} 2+c5

r2cos2ξ

1+2_r4_sin2_ξdφ1

+ 2ρ4sin 2ζ

2(1+2_ρ4_sin2_{ζ )}dζ+

1

ρ

1− 3

1−2_ρ2+

2 1+2_ρ4_sin2_ζ

dρ

+ 2r4sin 2ξ

2(1+2_r4_sin2_{ξ )}dξ+

1

r

1− 3

1+2_r2+

2 1+2_r4_sin2_ξ

dr . (2.5)

Xmcanbesplitinthefollowingway

Xm=Im+Zm, DmIn+DnIm=0, DmIm=0, (2.6)

12 _For_an_argument_supporting_this_in_the_NSR_formalism_see_[18]_.

13 _Note_that_in_the_first_(NS–NS)_term_of_(1.7)_one_does_not_need_to_vary_the√_G_factor_as_its_contribution_vanishes_after

(8)

whereIm=6_i₌₁ci(I(i))m. Theindex i labelsthe 6 isometricdirections yi =(t,ψ2,ψ1,ϕ, φ2,φ1)ofthe 10dABF metricandci arearbitrary constantcoefficients.(I(i))m arethe6 in-dependent commuting Killing vectors of the ABF background: the Lie derivatives of the G,

B and F-fieldsin [6]along Imallvanish.Ifwe splitthecoordinatesas xm=(yi,xμ)where

μ =1,2,3,4 labelsthenon-isometricdirectionsxμ=(ζ,ρ,ξ,r),then

Im=

6

i=1

δ_mi ciGii(xμ) , Im=δimci=const, Zm=δmμZμ(xν) . (2.7)

ThevectorYmrequiredtosatisfy (2.2)ontheABFbackgroundisfoundtobe14

Y≡Ymdxm=4

1+ρ2

1−2_ρ2dt+2

ρ2cos2ζ

1+2_ρ4_sin2_ζdψ1

+4 1−r 2

1+2_r2dϕ−2

r2cos2ξ

1+2_r4_sin2_ξdφ1

+ 2ρ4sin 2ζ

2(1+2_ρ4_sin2_{ζ )}dζ+

1

ρ

1− 3

1−2_ρ2+

2(−1_c 2−1)

1+2_ρ4_sin2_ζ

dρ

+ 2r4sin 2ξ

2(1+2_r4_sin2_{ξ )}dξ+

1

r

1− 3

1+2_r2−

2(−1c5+1)

1+2_r4_sin2_ξ

dr . (2.8)

Weobservethatifwefixciin (2.5)tothefollowingspecificvalues

c0=c3=4 , c1=c4=0, c2= −c5=2 , (2.9)

thenYmandXmcoincide

Ym=Xm. (2.10)

Thenextsurprisingobservationisthatforthesespeciallychosenvaluesofci in (2.9)thevector

Xmsatisfiesalsoadirectgeneralisation (1.11)ofthedilatonequation (1.6)(∂mφ→Xm)15:

¯

βX_≡_R₋ 1 12H

2

mnk+4DkXk−4XkXk=0. (2.11)

Asweshallshowin Appendix Cthisβ¯Xsatisfiesthegeneralisationofthedilatonidentity (1.6)

∂mβ¯X=0. (2.12)

Thereasonforthisparticularchoiceofciin (2.9)canbetracedtotheformofthelinearterms inthedilatonφˆ of theT-dual HTsolution (B.3).That isthepresenceof theI-terminXm in

(2.6)reflectsthe presenceofthenon-isometriclineartermsinφˆ.Therefore,theseterms drive themodificationof theequationssatisfiedbytheABFbackgroundfromtheirstandardtype II form.InthissensetheZmpartofXmmaybeinterpretedastheanalogof∂mφinthemodified equations.Indeed,onecancheckthat forIm in (2.7)withci chosenasin (2.9)thefollowing relationissatisfied

∂mZn−∂nZm+IkHkmn=0. (2.13)

14 _Y_is_of_course_defined_modulo_a_total_derivative. 15 _Since_DnXn₌_DμZμ_, _Xm_Xm₌_Gij_c

(9)

Thismaybeinterpretedasamodified“dilatonBianchiidentity”:ifImisformallysettozerothen

Zmbecomesaderivativeofascalar,∂mφ.Ingeneral,assumingthatImrepresentsanisometryof theB-field,i.e.theLiederivative(LIB)mn=Ik∂kBmn+Bkn∂mIk−Bkm∂nIkvanishes(modulo agaugetransformationterm∂mUn−∂nUm),wecansolve (2.13)as16

Zm=∂mφ+BkmIk, (2.14)

where∂mφ termrepresentsthetrivial “zero-mode”solution.In theparticularcaseof theABF backgroundwithZmandImgivenby (2.5), (2.6), (2.7)andci fixedasin (2.9)wefind

Xm=Ym=Im+Zm=∂mφ+(Gkm+Bkm)Ik, (2.15)

φ=1₂log (1−κ

2_ρ2₎3₍₁₊_κ2_r2₎3

(1+κ2_ρ4_sin2_{ζ )(}₁₊_κ2_r4_sin2_{ξ )}. (2.16)

Thescalarφin (2.16)ispreciselytheonethatisfound [12]byapplyingthestandardT-duality transformationruletotheisometricpartofthedilatonφˆoftheHTsolutionin (B.3)(cf. (1.10)).

3. Modified type II equations: first-order equations for R–R couplings

Let us now explorewhat modificationof the type IIBequations for the R–R couplings is satisfiedbytheABFbackground.

Thestandardequationsoftype IIBsupergravity [28]intheR–Rsectorwrittenintermsofthe rescaled F=eφF fieldstrengthsarepairsof dynamicalequations andBianchiidentities(see Appendix Afornotation)17

DmFm−ZmFm−1₆HmnpFmnp=0, dF1−Z∧F1=0, (3.1)

DpFpmn−ZpFpmn−1₆HpqrFmnpqr=0, dF3−Z∧F3+H3∧F1=0,

(3.2)

DrFrmnpq−ZrFmnpq+₃₆1εmnpqrstuvwHrstFuvw=0, dF5−Z∧F5+H3∧F3=0.

(3.3)

HereZ=Zmdxm=dφisthedilatonone-form.Thefive-form F5isalsorequiredtosatisfythe

self-dualityequationF5=F5 whichimpliestheequivalenceofthefirstandsecondequation

in(3.3).

An apriori surprisingobservation isthat thereexist directgeneralisationsof the 1st-order equations (3.1)–(3.3)involvingZ=ZmdxmandI=Imdxmin (2.5), (2.6),withfixedvaluesof thecoefficientsci asgivenin (2.9),whicharesolvedbytheABFbackground (B.1).Explicitly, theequationsfortheone-formF1in (B.1)are

DmFm−ZmFm−1₆HmnpFmnp=0, ImFm=0, (3.4)

(dF1−Z∧F1)mn−IpFmnp=0. (3.5)

16 _In_general,_we_find_Zm₌_∂mφ₊_BkmIk₋_Um_._Under_gauge_{transformations}_of_B_the_vector_Um_transforms_so_that φmaybeassumedtobeinvariant.IntheparticularcaseoftheABFbackground(B.1)withtheB-fieldchoseninthe manifestlysymmetricformwehaveUm=0.

17 _Note_that_all_equations_including_(2.2)_are_invariant_under_the_simultaneous_change_of_sign_of_H

(10)

WehaveaddedtheconditionImFm=0 asanindependentequationon F1.18

Similarly,theequationsthatgeneralise (3.2)andaresatisfiedforthethree-form F3in (B.1)

arefoundtobe

DpFpmn−ZpFpmn−1₆HpqrFmnpqr−(I∧F1)mn=0, (3.6)

(dF3−Z∧F3+H3∧F1)mnpq−IrFmnpqr=0. (3.7)

TheequationssatisfiedbyF5oftheABFbackgroundarefoundtobe

DrFrmnpq−ZrFrmnpq+₃₆1εmnpqrstuvwHrstFuvw−(I∧F3)mnpq=0, (3.8)

(dF5−Z∧F5+H3∧F3)mnpqrs+1₆εmnpqrstuvwItFuvw=0. (3.9)

Thesetwoareequivalentinviewoftheself-dualityofF5.

Thesemodified equations (3.4)–(3.9)reduceback to (3.1), (3.2), (3.3)ifwedrop allterms withImandassumethatdZ=0,i.e.ifweset

Zm→∂mφ , Im→0. (3.10)

Thestructureof (3.4)–(3.9)supportstheinterpretationofZasageneralised“dilatonone-form”, whiletheisometryvectorI effectivelydrivesthedeformationofthestandardtype IIBequations. Aninterestingobservationisthatthereexistcertaincombinationsoftheequations (3.4)–(3.9) thatdependonZ andI onlythroughthecombinationX=Z+I,whichenteredtheNS–NS equationsoftheprevioussection.Thesearefoundbyaddingtogetherequationsofequalform degree,forexample,theequationofmotionfortheR–Rthree-formandtheBianchiidentityfor theR–Rone-form.TheresultingX-dependentequationsaregivenby

DmFm−XmFm−1₆HmnpFmnp=0, (3.11)

DpFpmn−XpFpmn−1₆HpqrFmnpqr+(dF1−X∧F1)mn=0, (3.12)

DrFrmnpq−XrFrmnpq+₃₆1εmnpqrstuvwHrstFuvw+(dF3−X∧F3+H3∧F1)mnpq

=0. (3.13)

Usingtheself-dualityofF5thelastequationcanbealsowrittenas

(dF5−X∧F5+H3∧F3)pqrlmn−1₆εpqrlmnvst u(DvFstu−XvFstu−FvHst u)=0.

(3.14)

Aswillbediscussedbelow,thesethreeequationsarealreadysufficientforderivingcandidates forthescaleinvarianceequationsfortheF-fields,whichare2ndorderinderivatives.

Itisusefultorewrite (3.1)–(3.3)inthenotationofforms(see Appendix Aforconventions). Todosoweintroducethedualformsdefinedby

F1=F9, F3= −F7, F5=F5, F7= −F3, F9=F1. (3.15)

Thenthecompletesetofthetype IIsupergravityequationsforR–RstrengthsandBianchi iden-tities (3.1)–(3.3)isgivenby19

18 _{Alternatively,}_one_can_derive_this_equation_from_the_Bianchi_equation_(3.5)_,_the_invariance_of_F

1undertheisometry,

theorthogonalityofI andZ,andtheconditionthatZisnotanexactone-form.Indeed,multiplying(3.5)byImone finds∂n(ImFm)−ZnImFm=0.Thus,ifImFm =0 thenZ=dln(ImFm).Wefind,however,itmoreconvenientto

addImFm=0 asanindependentequation,andinferfromittheorthogonalityofIandZ. 19 _We_assume_that_F

(11)

dF2n+1−Z∧F2n+1+H3∧F2n−1=0, n=0,1, . . . ,

d F2n+1−Z∧F2n+1−H3∧F2n+3=0, n=0,1, . . . , (3.16)

whereZ=dφ.

The“I-modified”equations (3.4)–(3.9)aregivenby20

dF2n+1−Z∧F2n+1+H3∧F2n−1−(I∧F2n+3)=0, n= −1,0, . . . ,

d F2n+1−Z∧F2n+1−H3∧F2n+3+(I∧F2n−1)=0, n=0,1, . . . .

(3.17)

Dueto (3.15)thetwoequationsin (3.16)areequivalentandthesameistruefor (3.17).

LetusnotethatthedeformedR–Requations (3.17)togetherwiththerelation (2.13)ordZ+ ιIH3=0 implythefollowingrelation

LIF2n+1=(I·Z)F2n+1. (3.18)

ThustheconditionthattheF-fieldsareinvariantundertheisometryI isequivalenttothe con-ditionI ·Z=0, whichis clearlysatisfiedfor the ABF backgroundas is evidentfrom (2.5), (2.7).

4. Second-order equations for R–R couplings as scale invariance conditions

LetusreturntothediscussionofthescaleinvarianceconditionsforthecouplingsoftheGS sigmamodel (1.2)insection2andconsidertheequationsfortheR–RcouplingsF thatshould followfromtherequirementof(1-loop)UVfinitenessofthe2dmodel.Onecanarguethatthe conditionsanalogoustoeqs.(2.1), (2.2)fortheGandB-fieldcouplingsshouldhavetheform

β_kF

1...ks ≡ 1 2D

2_F

k1...ks +. . .=X m_∂

mFk1...ks +

i

Fk1...m...ks∂kiX

m_, _(4.1)

wherewehaveomittedpossiblenon-lineartermssuchas RF+DHF+. . . onthel.h.s.The

X-dependentLiederivativetermonther.h.s.reflects,asin (2.1), (2.2),thereparametrisation(or off-shell xm-renormalisation) freedom.Forexample, startingwiththe linearised RGequation

dFn(x)

dt =βnF=12∂ 2_F

n(x),t=loganddoingthecoordinateredefinitionxm→xm+t Xm,one endsupwith dFn_dt(x)=1₂∂2Fn(x) −Xm∂mFn−Fm∂nXm.

Weshall discussthe computationof1-loop logarithmicUVdivergencesforthe GSaction (1.2)in Appendix GclarifyingthestructureofβF.

ForXm=∂mφtheequations (4.1)shouldbeaconsequenceofstrongerWeylinvariance con-ditions,21whichshouldbeequivalenttothetype IIsupergravityequations (3.1)–(3.3)or (3.16) whereZ=X=dφ.Indeed,combining(“squaring”)thefamiliardF+. . .=0,d F+. . .=0

20 _Note_that_here_we_include_n_{= −}_{1 as}_in_the_deformed_theory_it_is_no_longer_trivial:_it_gives_the_second_equation_in (3.4),i.e.(I∧F1)=ImFm=0.

21 _For_example,_using_the_NSR_approach_on_a_flat_background_we_may_consider_the_R–R_vertex_operators_built_out_of

(12)

equationsleadstod d F+d dF+. . .=0 orD2F+. . .=0,wheretheleadingtermisthe Hodge–deRhamoperator.

Moreover, the same equations should follow also from the modified type II equations (3.4)–(3.9)or (3.17)(as,e.g., theABF backgroundthat solvesthe modifiedequations should alsobeasolutionofthescaleinvarianceconditions).Thisshouldprovideanon-trivial consis-tencycheck:afterproperly“squaring” (3.4)–(3.9)thedependenceontheZandI vectorsinany candidatescaleinvarianceequationsshouldappearonlythroughtheirsumX=Z+I asin (2.1), (2.2).

StartingfromthemodifiedtypeIIequations (3.4)–(3.9)(whichincludethestandardtype IIB supergravityequationsasaspecialcase (3.10),Im=0),letusoutlinethederivationofthe2nd orderequationsfortheR–Rcouplingsthatshouldbeequivalenttothescaleinvarianceconditions for FnoftheGSsigmamodel (1.2).Tobeacandidateforthescaleinvarianceconditionsthese equationsshouldhavethefollowingproperties:

(i)vanishonthesupergravityequations (2.1), (2.2), (2.11), (3.1)–(3.3)withX=dφ,Y=0 (ii)dependonZandI throughX=Z+I

(iii)dependonXthroughLiederivatives.22

Startingwiththemodifiedequations (3.17)andactingwithdonthefirstequationandd

onthesecondandthenusingthemodifiedequations(asdescribedin Appendix D)wearriveat thefollowingequation,whichsatisfiestheaboveproperties

d d F2n+1+d dF2n+1+1₄R∧F2n+1−1₈ (H3∧H3)∧F2n+1

−H3∧(H3∧F2n+1)−(H3∧(H3∧F2n+1))

−d (H3∧F2n+3)−(H3∧dF2n+3)+d (H3∧F2n−1)+H3∧d F2n−1

=LXF2n+1+LXF2n+1−(d X)∧F2n+1+βB∧F2n−1−(βB∧F2n+3) .

(4.2)

HereβBisthe2-formanalogof (2.2),i.e.

βB≡1₂ d H3+K=(X∧H3)+dY . (4.3)

ThisisthenacandidateforthescaleinvarianceequationfortheR–Rform F2n+1.

Usingtheidentity

LXF2n+1=LXF2n+1+(d X)∧F2n+1+βG·F2n+1,

βG·F2n+1≡

i

β_mG

inFm1...mi−1

n

mi+1...m2n+1, (4.4)

whereβ_mnG isdefinedin (2.1),wefindthat (4.2)becomes

d d F2n+1+d dF2n+1+1₄R∧F2n+1−1₈ (H3∧H3)∧F2n+1

−H3∧(H3∧F2n₊1)−(H3∧(H3∧F2n₊1))

−d (H3∧F2n+3)−(H3∧dF2n+3)+d (H3∧F2n−1)+H3∧d F2n−1

=2LXF2n+1+βG·F2n+1+βB∧F2n−1−(βB∧F2n+3) . (4.5)

22 _Moreover,_since_the_R–R_fields_F_are_invariant_under_the_isometries_generated_by_I_,_their_Lie_derivatives_along_I

(13)

ThedependenceoftheseequationsonXratherthanseparatelyonZandIcanberelatedtotheir closeconnectiontotheparticularX-dependentcombinationsofthemodifiedequationsin (3.11), (3.12), (3.13),i.e.to(heren ∈Zasin (3.17))

2n≡dF2n−1−X∧F2n−1+H3∧F2n−3

+(−1)n (dF9−2n−X∧F9−2n+H3∧F7−2n)=0. (4.6) Wealsodefineasin (1.11), (2.2)

¯

βB≡1₂ d H3+K−(X∧H3)−dX=0,

¯

βX≡R−1₂ (H3∧H3)+4 d X−4 (X∧X)=0. (4.7)

Deconstructingthederivationin Appendix D,wefindthatthe2nd-orderequationfortheR–R fluxes (4.2)canalsobewrittenas

d2n−X∧2n+H3∧2n₋2−F2n₋1∧ ¯βB

+(−1)n (d8−2n−X∧8−2n+H3∧6−2n−F7−2n∧ ¯βB

+1

4F2n+1∧ ¯β

X

=0. (4.8)

Finally,letuspresenttheexplicitformofeq.(4.5)incomponents.For F1wefind

D2Fm−RmnFn+1₄(R−3₄H2)Fm

+1 2H

pnk_H

mpnFk−1₆DmHpnkFpnk−1₂HpnkDpFnkm

=2(XpDpFm+DmXpFp)+βmnGF

n₋1

2β

B nkF

nk

m. (4.9)

Using the identity D_[mHnpk] = 0 the term ₆1DmHpnkFpnk in (4.9) can be replaced by

1

2DpHmnkF

pnk_._The_equation_for_F

3maybewrittenas

D2Fnkm−Ra[nFakm]+Rab[nkFabm]+

1 4(R−

3

4H

2₎_F

nkm

+1 2H

abc_H

ab[nFkm]c−1₂HabcHa[nkFm]bc

+DaHa[nkFm]+Ha[nkDaFm]−FaDaHnkm

−1 6D[nH

abc_F

km]abc−1₂HabcDaFbcnkm

=2(XaDaFnkm+D[nXaFkm]a)+βaG[nFakm]+β_[BnkFm]−1₂βabBFabnkm, (4.10) whiletheequationfor F5canbeputintotheform

D2Fij klm−Ra[iFaj klm]+Rab[ijFabklm]+

1 4(R−

3 4H

2₎_F

ij klm

+1 2H

abc_H

ab[iFj klm]c−1₂HabcHa[ijFklm]bc

+DaHa_[ijFklm_]+Ha_[ijDaFklm_]−Fa_[ijDaHklm_]

+ 1

12εij klmbdef(DaH

abc_Fdef₊_Habc_D

aFdef−FabcDaHdef)=

=2(XaDaFij klm+D[iXaFj klm]a)+βaG[iF a

j klm]+β_[BijFklm]

+ 1

12εij klmabcde(β

B₎ab_Fcde_.

(4.11)

Thisexpressionisconsistentwiththeself-dualityof F5(inparticular,thethirdandforthlines

(14)

These2nd-orderequationsfor F1, F3and F5exhibitobviousstructuralsimilarities.In

par-ticular,theycontaintheexpectedHodge–deRhamoperatortermsandthevectorXonlyenters throughthereparametrisationterms as in(4.1).The βG andβB terms intheseequationsare definedasin (2.1), (2.2)butcanalsobereplacedbyexpressionsonther.h.s.of (2.1), (2.2).

Asweshall discussin Appendix G,similarequations comeout ofthe computationof the one-loopbeta-functionsfortheR–RcouplingsintheGSsigmamodel (1.2).

5. Origin of modified equations: T-duality relation to type II equations for backgrounds with non-isometric linear dilaton

Givenascaleinvariantsigmamodelinflat2dspaceT-dualityinanisometricdirectionshould alsoproduceascaleinvariantsigmamodel.Similarly,givenaWeylinvariantsigmamodelon curved2dspacewithallcouplingsincludingthedilatonbeingisometrictheT-dualbackground shouldalso beWeyl invariant(provided thedilatontransforms intheusualway [32,33]).As discussedintheintroduction,ingeneralthisisnotsoifthe dilatonisnotisometric:T-duality will still preserve scale invariance but not Weyl invariance. Thusgiven asolution of type II supergravityequations whichhas linearnon-isometric termin thedilatonits T-duality image will no longer solve the standardtype II equations but will satisfy instead amodified setof type IIequationsasdiscussedabove.

5.1. Simpleexamples

Hereweshallmaketheoriginofthemodifiedequationsexplicitbyshowingthatthey repre-senttheoriginaltype IIequationsforasolutionwithnon-isometriclineardilaton,rewrittenin termsofthefieldsoftheT-dualbackground.Toexplainhowthishappensinsimpletermsletus firststartwithabosonicbackground (1.8)withAμ=0,i.e.

ˆ

ds2=e−2a(x)dyˆ2+gμν(x)dxμdxν, φˆ= −cyˆ+ϕ(x)−1₂a(x) . (5.1)

ThenthecorrespondingWeylanomalycoefficients

¯

β_mnG = ˆRmn+2DˆmDˆnφ ,ˆ β¯φ= ˆR+4Dˆ2φˆ−4Gˆmn∂mφ∂ˆ nφ ,ˆ (5.2)

havethefollowingcomponentsunderthexˆm=(y,ˆ xμ)splittingofcoordinates23 ¯

β_μνG =Rμν−∂μa∂νa+2DμDνϕ , β¯_yG_ˆ_y_ˆ=e−2a(DμDμa−2∂μa∂μϕ) , (5.3)

¯

β_yμG_ˆ = −2c ∂μa , β¯φ=R−∂μa∂μa+4D2ϕ−4∂μϕ∂μϕ−4c2e2a. (5.4)

Weseethatifc=0,i.e.thedilatonisisometric,thentheWeylinvarianceconditionsβ¯_μνG =0, ¯

βφ=0 areinvariantunderT-dualityiny,i.e.underaˆ= −a→a,φˆ→ ˆφ+aor ϕ→ϕ.The

c= −∂y_ˆφˆ dependenttermsin (5.4)thusrepresentobstructionstomappingoneWeylinvariant modeltoanother.TheT-dualmetricthensolvesweaker,modified,equations

Rmn+DmXn+DnXm=0, β¯X=R+4DmXm−4XmXm=0, (5.5)

withXmbeing(cf. (1.10))

23 _Here_R

(15)

Xμ=Zμ=∂μφ=∂μ(ϕ+1₂a) , Xy=Iy= −Gyy∂y_ˆφˆ=ce2a. (5.6)

Similar conclusions are reached in the casewe havea non-diagonalmetric in (1.8)(see the generaldiscussionbelow).Thepresenceofnon-zeroAˆμisinfactnecessarytohaveasolution oftheWeylinvarianceconditionswhenc =0 (cf. (5.4))andthetargetspaceshouldthusbeat least3-dimensional.Anexample ofsuchasolutionwasfoundin [13].Itrepresentsalimitof thebackgroundassociatedwiththeSO(4)/SO(3)gWZWmodel,whichhasthefollowingmetric anddilaton [34]

ˆ

ds2=dt2+tan

2_{t dp}2₊_cot2_{t dq}2

1−p2₋_q2

=dt2+cot2t (dθ+tanψcotθ dψ )2+tan 2_t

sin2θdψ 2_,

(5.7)

ˆ

φ= −lnb2₋_p2₋_q2_{sin 2}_t_{= −}_ln_sin_θ _cos_ψ _{sin 2}_t_, _(5.8)

wherep=sinψ,q=cosθcosψandt,ψ,θareanglesofthecosetparametrisationoftheSO(4)

groupelement.Thisbackground(whichsolvestheWeylinvarianceconditionβ¯G=0 withβ¯φ=

const)has noisometries.Oneoption togeneratean isometryistosett =iz andthenshiftz

byaninfiniteconstant.Doingsowegetlineardilatoninz,butthezdirectiondecouplesinthe metric.Anon-trivialalternativeistosetψ=iyˆ andtoshiftyˆ byan infiniteconstant(which correspondstoinfiniterescalingofp,q generatingascalingisometryinthemetric (5.7)).The resultingbackground(wedropaninfiniteconstantinthedilaton)

ˆ

ds2=dt2−tan

2_{t dp}2₊_cot2_{t dq}2

p2₊_q2 =dt

2₊_cot2_{t (dθ}₊_cot_{θ d}_y)_ˆ 2₋tan2t

sin2θdyˆ

2_, _(5.9)

ˆ

φ= −lnp2₊_q2_{sin 2}_t_{= − ˆ}_y₋_ln_sin_θ _{sin 2}_t_, _(5.10)

isthereforeofthesametypeasin (1.8)anddefinesaconformalsigmamodel.24Similarhigher dimensionalbackgrounds canbe constructedstarting fromSO(n)/SO(n−1)gWZWmodels withn>4[13].

T-dualisingthemetric (5.9)alongyˆwegeta(G,B)backgroundthatwillsolvethemodified

(G,B)equations (2.1), (2.2)withnon-trivialXm=Im+Zm,whereIy= −∂y_ˆφˆandZmisgiven by (2.14),withφ= ˆφ−1₂logG_y_ˆ_y_ˆ.ThesemodifiedequationswillbetheoriginalWeylinvariance conditionsrewrittenintermsofthedualGandB-fields.

Given the2d CFT in(5.9)with2d stress-tensordefinedtaking into accountthe dilatonin (5.10),onemayformallycompactifyyˆ andaskifthisCFT hasT-dualityasasymmetryofits spectrum.Theanswerappearstobenoasthe2dstress-tensorwillnotbeinvariantunderT-duality (i.e.mapping momentumintowindingmodes).25 Thisiscompatiblewithourexpectationthat formallyT-dualisingthemetric (5.9)willnotleadtoaconsistentCFT.

24 _The_central_charge_for_this_d₌_{3 conformal}_model_is_given_by_c₌_d₋3 2α(R−

1

12H2+4D2φ−4DmφDmφ)+. . .=

3−3₂α×12+. . ..Herethescaleofthespacewassettoone,sothatαisthentheinverseoftheWZWlevelk.Thisisin agreementwiththeusualcountofthecentralchargeforthe SO(4)/SO(3)gWZWmodelc=6k/(k+4)−3k/(k+2)=

3−18/k+. . .,whichshouldbeunchangedinthecoordinatelimitleadingfrom(5.7)to(5.9).

25 _Given_a_free_compact_scalar_CFT_L₌_r2_(∂φ)2_with_φ_≡_φ₊₂_π_the_spectrum_of_dimensions_of_primary_operators

(16)

ThesameconclusionsarereachedfortypeIIsolutionswithalineardilatonandnon-zeroR–R fluxes(withisometricG,Band Fn=eφFn),suchastheHTsolutiondualtoABFbackgroundin theAdS5×S5caseanditscounterpartsintheAdS2×S2×T6andAdS3×S3×T4casesdiscussed in Appendix F.Explicitly,inthecaseofasolution(G,ˆ B,ˆ Fˆn,φ)ˆ withseveralisometriesbroken onlybythelineardilatonterm

ˆ

φ=φ0−ciyˆi+f (xμ) , (5.11)

wewill getageneralisationof thetypeIIsupergravityequations,dependingonthefollowing twovectorsZandI (cf. (2.6), (2.7), (2.14))

I=ciGyiyidy

i_, _Z₌_dφ₊_ι

IB , φ=f−

1 2

i

logGˆ_y_ˆ_i_y_ˆ_i . (5.12)

HereGandBarethebackgroundfieldsoftheT-dualbackground.

Belowweshallillustratetheserelationsinthegeneralcasewithoneisometry.

5.2. NS–NSsector

Weconsiderthefollowingtwod-dimensionalbackgrounds(hereweuseKμinsteadofAˆμin

(1.8))

ds2=e2a(dy+Aμdxμ)2+gμνdxμdxν,

B=Kν(dy+1₂Aμdxμ)∧dxν+1₂bμνdxμ∧dxν,

φ= −c y+ϕ+1₂a , Iy= ˆc , Iμ=0, (5.13)

ˆ

ds2=e−2a(dyˆ+Kμdxμ)2+gμνdxμdxν,

ˆ

B=Aν(dyˆ+1₂Kμdxμ)∧dxν+1₂bμνdxμ∧dxν ,

ˆ

φ= −ˆcyˆ+ϕ−1₂a , Iˆyˆ=c , Iˆμ=0. (5.14)

Here y and yˆ are the directions that are assumed tobe (shift) isometries of their respective metricsandB-fields.Weusetheindicesμ,ν,. . .=1,. . . ,d−1 andm,n,. . .=1,. . . ,d.For

c= ˆc=0 (5.13)and (5.14)arerelatedbystandardT-duality, suchthat ϕ istheanalogofthe duality-invariantdilatonfield.Letusalsodefine

Z=dφ+ιIB= −c dy+dϕ+1₂da+ ˆc Kμdxμ,

X=Z+I=(−c+ ˆc e2a)dy+dϕ+1₂da+(c Kˆ μ+ ˆc e2aAμ)dxμ,

ˆ

Z=dφˆ+ι_I_ˆBˆ= −ˆc dyˆ+dϕ−1₂da+c Aμdxμ,

ˆ

X= ˆZ+ ˆI=(−ˆc+c e−2a)dyˆ+dϕ−1₂da+(c Aμ+c e−2aKμ)dxμ, (5.15)

where

Z·I= ˆZ· ˆI = −cc .ˆ (5.16)

restoredinthe“doubled”formulationifthelineardilatontermweregivenbyqφ+ ˜qφ˜whereφ˜isthedualfield(with

1 √

rq↔ _r

√

(17)

Thetwo(G,B)backgroundsin (5.13)and (5.14)areT-dualtoeachotherandthusforc= ˆc=0 solve theequivalentWeylinvariance equations(see,e.g., [37]andthereferences therein).We willnowshowhowthisrelationalsoextendstothemoregeneralcasewithlineardilatons.

Letusfirstconsiderthegeneraliseddilatonequation

R−₁₂1H2+4DmXm−4XmXm=0. (5.17)

The questionwe wanttoaddress is:if thebackground (5.13) satisfies (5.17) does that imply that (5.14)satisfies (5.17).Asthetwobackgrounds (5.13)and (5.14)arerelatedbytheobvious symmetry

a→ −a , Aμ↔Kμ, c↔ ˆc , y↔ ˆy , (5.18)

itissufficienttocomputetheleft-handsideof (5.17)for (5.13)andcheckthatitisinvariant(or atleastcovariant)under (5.18).

For (5.13)wehave

Xy= −c+ ˆc e2a, Xμ=∂μϕ+1₂∂μa+ ˆc Kμ+ ˆc e2aAμ. (5.19)

Itwillalsobeusefultodefinethefollowingobjects

Fμν≡∂μAν−∂νAμ, Hμν≡∂μKν−∂νKμ,

hμνρ≡(db+1₂A∧dK+1₂K∧dA)μνρ , (5.20)

whereweobservethathisinvariantunder (5.18).Nowusingthedimensionalreductionformulae in Appendix Awefind

R−₁₂1H2+4DmXm−4XmXm

=R−∂μa∂μa−₁₂1h2−1₄e2aFμνFμν−1₄e−2aHμνHμν+4∇μ∂μϕ−4∂μϕ∂μϕ

+4c∇μAμ+4cˆ∇μKμ−8(c Aμ+ ˆc Kμ)∂μϕ

−4c2(AμAμ+e−2a)−4cˆ2(KμKμ+e2a)+8cc (ˆ 1−AμKμ) , (5.21) whichisindeedinvariantunder (5.18).Therefore,if (5.17)issatisfiedforbackground (5.13)itis satisfiedforbackground (5.14)andviceversa.26

LetusnowturntothemodifiedmetricandB-fieldequationstoshowthatthetwo combina-tionsappearingin (1.3)and (1.4)

Rmn−1₄HmpqHnpq+DmXn+DnXm, (5.22)

1

2D

p_H

mnp−XpHmnp−DmXn+DnXm, (5.23)

are covariantunderthesymmetry (5.18).27 Theniftheyvanishforthebackground (5.13)this impliesthattheyvanishfor (5.14)andviceversa.As (5.22)issymmetricand (5.23)is antisym-metric,wemayjustconsidertheirdifference

26 _This_generalises_the_usual₍_c_{= ˆ}_c₌₀₎_discussion_of_the_T-duality_invariance_of_the_string_effective_action_(1.7)_with

√

G e−2φ= √g e−2ϕ.

27 _Here_we_assume_Ym_in_(1.4)_,_(2.2)_is_equal_to_Xm_in_(1.3)_,_(2.1)_as_is_the_case_for_the_I_-modified_equations

sat-isfiedbytheABFbackground.Moregenerally,givenascaleinvariantsigmamodelwithanisometryandtheGand

B-fieldcouplingssatisfying(1.3),(1.4),itsT-dualcounterpartwillalsosatisfy(1.3),(1.4)withtherolesofXmandYm

(18)

Cmn≡Rmn−1₄HmpqHnpq−1₂DpHmnp+2DmXn+XpHmnp, (5.24)

whichwecandecomposeintoapartindependentofcandcˆ(C(mn0))andapartlinearincandcˆ (Cmn(1))as

Cmn=Cmn(0)+2Cmn(1), (5.25)

C_mn(0)=Rmn−₄1HmpqHnpq−1₂DpHmnp+2DmXn(0)+X(0)pHmnp, (5.26)

C_mn(1)=DmX(n1)+12X

(1)p_H

mnp. (5.27)

Herewehaveusedthefactthatallthecand ˆcdependenceiscontainedinX=X(0)+X(1),where

X(0)isthec- and ˆc-independentandX(1)isthec- andcˆ-dependentpart.Usingthespecificform ofXforthebackground (5.13),asgivenin (5.19),wehave

X_y(0)=0, X(_μ0)=∂μϕ+1₂∂μa , Xy(1)= −c+ ˆc e2a, Xμ(1)= ˆc Kμ+ ˆc e2aAμ. (5.28)

Usingtheformulaein Appendix AwefindthefollowingrelationsforCmn(0) andCmn(1)evaluated onthebackground (5.13)

C_yy(0)=2e2a∂μϕ∂μa−e2a∇μ∂μa+1₄e4aFμνFμν−1₄HμνHμν ,

C_yμ(0)−Gyμ Gyy

C_yy(0)=e2a(1₂∇ν+∂νa−∂νϕ)Fμν+1₄hμνρHνρ

+(1₂∇ν−∂νa−∂νϕ)Hμν+1₄e2ahμνρFνρ,

C_μy(0)−Gμy Gyy

C_yy(0)=e2a(1₂∇ν+∂νa−∂νϕ)Fμν+1₄hμνρHνρ

−(1₂∇ν−∂νa−∂νϕ)Hμν−1₄e2ahμνρFνρ ,

C_μν(0)−Gμy Gyy

C_yν(0)−Gyν Gyy

C_μy(0)+Gμy Gyy

Gyν

Gyy

C_yy(0)

=Rμν−∂μa∂νa+2∇μ∂νϕ−1₂e2aFμνFνρ−₂1e−2aHμρHνρ

−1 2∇

ρ_h

μνρ−1₄hμρσhνρσ+hμνρ∂ρϕ , (5.29)

C_yy(1)=e2a(cAμ+ ˆc Kμ)∂μa ,

C_yμ(1)−Gyμ Gyy

C_yy(1)= −1₂(e2aFμν+Hμν)(cAν+ ˆc Kν)+(c− ˆc e2a)∂μa ,

C_μy(1)−Gμy Gyy

C_yy(1)= −1₂(e2aFμν−Hμν)(cAνc Kˆ ν)+(c+ ˆc e2a)∂μa ,

C_μν(1)−Gμy Gyy

C_yν(1)−Gyν Gyy

C_μy(1)+Gμy Gyy

Gyν

Gyy

C_yy(1)

=1

2c(∇μAν+ ∇νAμ)+ 1 2c eˆ

2a₍_∇

μAν− ∇νAμ)

+1

2c(ˆ ∇μKν+ ∇νKμ)+ 1 2ce−

2a₍_∇

(19)

Thenusing themap (5.18)betweenthe backgrounds (5.13)and (5.14),we findthe following relationsforCmn

Cyy Gyy = −

ˆ

Cy_ˆyˆ

ˆ

Gy_ˆyˆ

, 1

Gyy Cyμ−

Gyμ GyyCyy

=1

ˆ

Gy_ˆyˆ ˆ

C_yμ_ˆ −Gˆyμˆ

ˆ

Gy_ˆyˆ ˆ C_y_ˆ_y_ˆ,

1

Gyy

Cμy−G_Gμy yyCyy

= −1

ˆ

Gy_ˆyˆ ˆ

C_μ_y_ˆ−Gˆμyˆ

ˆ

Gy_ˆyˆ ˆ Cy_ˆyˆ

,

ˆ

Cμν−G_Gμy yyCˆyν−

Gyν GyyCˆμy+

Gμy Gyy

Gyν

GyyCˆyy= ˆCμν−

ˆ

Gμyˆ

ˆ

G_y_ˆ_y_ˆCˆyνˆ −

ˆ

Gyν_ˆ

ˆ

G_y_ˆ_y_ˆCˆμyˆ+

ˆ

Gμyˆ

ˆ

G_y_ˆ_y_ˆ

ˆ

Gyν_ˆ

ˆ

G_y_ˆ_y_ˆCˆyˆyˆ, (5.31)

wherethe left-handside isevaluatedon (5.13)andthe right-handside on (5.14).From these equalities it follows that the vanishingof the tensors (5.22), (5.23)on thebackground (5.13) implies their vanishingalso on the background(5.14), andin thissense are covariantunder T-duality.

LetusbrieflycommentonthegeneralisationwhenIμ= ˆIμ =0 inthebackgrounds (5.13) and (5.14) (i.e.,when thereareextraisometries inxμ directions).Runningthroughthe same analysiswefindthattheresultstillholdsonlyifIμsatisfiescertainproperties.Inparticular,the T-dualityrelationbetweentheequationsfor (5.13)and (5.14)stillholdsif

IμAμ=IμKμ=0. (5.32)

ThisrequirementisalsosufficientfortheT-dualityofthemodifiedequationsofmotionforthe R–RfieldsdiscussedinthefollowingsectiontocontinuewhenIμ= ˆIμ =0.

OnecancheckthattheserelationsarevalidateachstageinthesequenceofT-dualities re-quiredtotransformfromthesupergravityHTsolutionsof [12]totheABFbackground (B.1)and itsAdS3×S3andAdS2×S2counterparts (F.4), (F.13).

5.3. R–Rsector

Letusnow considerthecaseofnon-zeroisometricR–Rfields Fn.Thecontributionofthe R–RfieldstothemetricandB-fieldequationsappearsintheusualunmodifiedformandhence wecanfocusourattentiononthemodifiedequationsofmotionfortheR–Rfields (3.17).Written in terms of the forms fk≡e−a/2Fk these take the followingform (dropping the distinction betweenyandyˆ)

Ek≡dfk−Z∧fk+H3∧fk−2− ˆc ιyfk+2=0, (5.33)

ˆ

Ek≡dfˆk− ˆZ∧ ˆfk+ ˆH3∧ ˆfk₋2−c ιyfˆk₊2=0, (5.34)

wherewehaveintroduced(K=Kμdxμ,A=Aμdxμ)

Z=Z−1

2da= −c dy+dϕ+ ˆc K , Zˆ

_{≡ ˆ}_Z₋1

2daˆ= −ˆc dy+dϕ+c A , (5.35) whicharerelatedtoeachotherunderT-dualityas

ˆ

Z=Z+c(dy+A)− ˆc(dy+K)≡Z+δZ . (5.36)

RecallthattheinvarianceoftheR–Rformsundertheisometryalongyrequires