Journl of Modern Physcs 88-95 http://dxdoorg/6/jmp9 Pulshed Onlne Septemer (http://wwwscrporg/journl/jmp) Noneln Dulzton of Plne Wve Bckgrounds Ldslv Hlvtý Mroslv Turek Fculty of Nucler Scences nd Physcl Engneerng Czech Techncl Unversty n Prgue Prgue Czech Repulc Eml: hlvty@fjfcvutcz turekm@kmfjfcvutcz Receved June ; revsed July ; ccepted August ABSTRACT We nvestgte plne-prllel wve metrcs from the pont of vew of ther (Posson-Le) T-dulzlty For tht purpose we reconstruct the metrcs s ckgrounds of nonlner sgm models on Le groups For constructon of dul ckgrounds we use Drnfel d doules otned from the sometry groups of the metrcs We fnd dlton felds tht enle to stsfy the vnshng et equtons for the duls of the homogenous plne-prllel wve metrc Torson potentls or B-felds nvrnt wrt the sometry group of Lochevsk plne wves re otned y the Drnfel d doule constructon We show tht certn knd of plurlty dfferent from the (tomc) Posson-Le T-plurlty my exst n cse tht metrcs dmt severl sometry sugroups hvng the dmenson of the Remnnn mnfold An exmple of tht re two dfferent ckgrounds dul to the homogenous plne-prllel wve metrc eywords: Sgm Model; Strng Dulty; pp-wve Bckground Introducton Sgm models cn serve s models of strng theory n curved nd tme-dependent ckgrounds Soluton of sgm-models n such ckgrounds s often very complcted not to sy mpossle On the other hnd there re mny ckgrounds whose propertes were thoroughly nvestgted nd t s therefore nterestng to fnd f they cn e trnsformed to some others Importnt exmple of such trnsformton s so clled Posson Le T-dulty In ther semnl work [] lmčík nd Šever set condtons for dulzlty of ckgrounds nd gve formuls for ther trnsformton Snce then severl exmples of dulzle sgm models were constructed see eg [- ] Unfortuntely most of the exmples re not physclly nterestng The purpose of ths pper s to show tht physcl ckgrounds tht dmt suffcently lrge group of sometres re nturlly dulzle nd therefore equvlent n sense to some others In ths pper we re gong to nvestgte four-dmensonl plne-prllel wve metrcs [5-8] from ths pont of vew The sc concept used for constructon of dulzle sgm models s Drnfel d doule-le group wth ddtonl structure The Drnfel d doule for sgm model lvng n curved ckground cn sometmes e found from the knowledge of symmetry group of the metrc More precsely n the Drnfel d doule there re two eqully dmensonl sugroups whose Le lgers re sotropc suspces of the Le lger of the Drnfel d doule In cse tht the metrc hs suffcent numer of ndependent llng vectors the sometry group of the metrc (or ts sugroup) cn e tken s one of the sugroups of the Drnfel d doule The other one then must e chosen eln n order to stsfy the condtons of dulzlty Short summry of the dulzton procedure descred eg n [9] s gven n the next secton Elements of Posson-Le T-Dul Sgm-Models Let G e Le group nd ts Le lger Sgm model on the group G s gven y the clsscl cton SF d F where F s second order tensor feld on the Le group G The functons : R R dm G re de- termned y the composton x g where g : R g G nd x : Ug R re components of coordnte mp of neghorhood U of element g g G Equvlently the cton cn e expressed s where R : SF g d xr g E g R g re rght-nvrnt felds R g gg T The reltonshp etween E nd F s gven y the formul F x e g x E gxe gx () where () () e g x re the components of rght nvrnt Copyrght ScRes
L HLAVATÝ M TURE 89 forms dgg The equtons of moton de- e rved from the cton () hve the followng form () where re components of the Lev-Cvt connecton ssocted wth the second order tensor feld F Ths tensor feld s composton of the metrc ( symmetrc prt) nd the torson potentl (n ntsymmetrc prt) The condton of dulzlty of sgm-models on the level of the Lgrngn s gven y the formul [] jk F v c v F (5) v F j k jk where c re structure coeffcents of the dul lger nd v re left-nvrnt felds on the Le group G The lgers nd then defne the Drnfel d doule tht enles to construct tensor F stsfyng (5) The Drnfel d Doule nd Posson-Le T-Dulty As mentoned n the Introducton the Drnfel d doule D s defned s connected Le group whose Le lger cn e decomposed nto pr of sulgers mxmlly sotropc wth respect to symmetrc d-nvrnt nondegenrte lner form on Under the condton (5) the feld Equtons () for the -model cn e rewrtten s equton for the mppng l from the world-sheet R nto the Drnfel d doule D where suspces l l (6) spn T E e Tj j spn T E e T j re orthogonl wrt nd spn the whole Le lger T T re the j ses of nd Due to Drnfel d there exsts unque decomposton (t lest n the vcnty of the unt element of D) of n rtrry element l of D s product of elements from nd The solutons of Equton (6) nd soluton x g of the Equton () re relted y l g h D (7) where g G h G fulfl the equtons hh gg E (8) c c c gd g c gd g hh gg E (9) The mtrx E g of the dulzle -model s of the form E g E g () where E s constnt mtrx formul g g g t g g s gven y the () g nd mtrces g d g re gven y the djont representton of the Le sugroup G on the Le lger of the Drnfel d doule n the ss T T Ad g t g g dg Let us note tht E s the vlue of E g e G e j () n the unt of the group ecuse e The dul model cn e otn y the exchnge G G g g E E () Solutons of the equtons of moton of dul models re mutully ssocted y the relton l g h () g h Posson-Le T-Plurlty Generlly more thn two decompostons (Mnn trples) of Le lger of the Drnfel d doule cn exst Ths posslty leds to Posson-Le T-plurlty Let ˆ s nother decomposton of the Drnfel d lger nto pr of mxml sotropc sulgers Then the Posson-Le T-plurl sgm model s gven y the followng formuls [] ˆ g g g Eˆ gˆ Eˆ ˆ gˆ ˆ ˆ ˆ ˆ ˆ E R Q E S Eˆ (5) (6) where the mtrces QR S determne the reltonshp etween the ses of the pproprte decompostons nd ˆ T QTˆ T (7) R S T The reltonshp etween the clsscl solutons of the two Posson-Le T-plurl sgm-models s gven y posslty of two decompostons of the element l D s The superscrpt t mens trnsposton of the mtrx ˆ Two decompostons lwys exst Copyrght ScRes
9 L HLAVATÝ M TURE l h g gˆ h The Posson-Le T-dulty s then specl cse of Posson-Le T-plurlty for S Q R j x x du d x (8) Homogenous Plne Wve Metrcs Homogenous plne wve s generlly defned y the metrc of the followng form [56] ds dud A u u (9) where dx s the stndrd metrcs on Euclden spce d d E nd x E The form of ths metrc seems to e smple ut explct constructon of sgm models cn e very complcted Therefore we hve focused on the specl cse of sotropc homogenous plne wve metrc A u ds dud u x du d x Metrc () hs numer of symmetres mportnt for the constructon of the dulzle sgm models It dmts the followng llng vectors T X ux () R x x where u stsfes u j j () () The llng vectors R re genertors of orthogonl d rottons n E For specl choce of k u k const () u there re further sometres relted to the sclng of the lght-cone coordntes u u () The specfc form of enles us to clculte the functon u explctly The llng vectors of the k metrc () for u re u T X u u x X u u x (5) D uu R x x j j where D s the genertor ssocted wth the sclng symmetry nd k In the followng we shll nvestgte the cse d It mens tht the metrc tensor n coordntes u x y reds k x y u G u x y (6) Ths metrc s not flt ut ts Gussn curvture vnshes Note tht t hs sngulrty n u It does not stsfy the Ensten equtons ut the conforml nvrnce condtons equtons for vnshng of the -functon mn R j HmnH j k k H H k k Rk k HkmnH k k kmn (7) (8) (9) where the covrnt dervtves k Rcc tensor R nd Guss curvture R re clculted from the metrc G tht s lso used for lowerng nd rsng ndces Torson H n ths cse vnshes nd dlton feld s [5] cu lnu () The metrc (6) dmts the followng llng vectors u x u x u y u y u x u x () 5 u y u y 6 uu x y 7 y x One cn esly check tht the Le lger spnned y these vectors s the semdrect sum where Spn6 7 nd del Spn 5 The lger s eln nd ts genertors cn e nterpreted s dlton n u nd rotton n x y Genertors of the lger commute s two-dmensonl Hesenerg lger wth the center Constructon of Dul Metrcs As explned n Secton dulzle metrc cn e con- If e k then 5 Copyrght ScRes
L HLAVATÝ M TURE 9 structed y vrtue of Drnfel d doule For ths gol the Le lger of the Drnfel d doule cn e composed from the four-dmensonl Le sulger somorphc to the four-dmensonl sulger of llng vectors nd four-dmensonl Aeln lger Moreover the four-dmensonl sugroup of sometres must ct freely nd trnstvely [] on the Remnnn mnfold M where the metrc ( ) s defned so tht M G Usng the method descred n [] for semsmple lgers we fnd tht up to the trnsformton e k k there re sx clsses of four-dmensonl sulgers of the sometry lger of the homogeneous plne wve metrc somorphc to 5 Spn 7 Spn 6 7 Spn 6 Spn 5 6 Spn 5 5 7 6 where re rtrry prmeters Infntesml form of trnstvty condton cn e formulted s requrement tht four ndependent llng vectors cn e tken s ss vectors of four-dmensonl vector dstruton n M In other words these llng vectors must form ss of tngent spce n every pont of M It mens tht n every pont of M there s n nvertle mtrx A u x y tht solves the equton A u x y X 6 7 () where u x y nd X form ss of the sulger Infntesml form of requrement tht the cton of the sometry sugroup s free sys tht f n ny pont of M there s vector of the correspondng Le sulger such tht ts cton on the pont vnshes then t must e null vector By nspecton we cn fnd tht the only four-dmensonl sulgers tht generte trnstve ctons on M re somorphc to Spn 6 7 or Spn 5 6 Ther non-vnshng commutton reltons re 6 7 6 7 () nd It s esy to see tht the Equton (5) s then fulflled 6 6 6 5 5 () respectvely where nd re rel prmeters One cn lso check tht the cton of oth correspondng groups of sometres s free In the followng we shll fnd metrc dul to (6) tht follows from ts Drnfel d doule descrpton where s somorphc ether to lger spnned y 6 7 or y 5 6 Let us strt wth constructon of the Drnfel d doule followng from the lger somorphc to () nd dul Aeln lger Assume tht the Le lger s spnned y elements X X X X wth commutton reltons X X X X XX X (5) X X X X where nd re rtrry rel prmeters The ss of left-nvrnt vector felds of the group generted y s x e x x x e cos x e sn x x x (6) x x sn e x e cos x x x x where x x x x rmetrzton g e e e e re group coordntes used n p- xx xx xx xx (7) To e le to otn the metrc (6) y the Drnfel d doule constructon frst we hve to trnsform t nto the group coordntes Trnsformton etween group coordntes x x x x nd geometrcl coordntes u x y s x u e x x xx e x xcos x x x y x cos x x sn x sn (8) It converts the llng vectors 6 7 nto the left-nvrnt vector felds (6) nd the metrc (6) nto the form Copyrght ScRes
9 L HLAVATÝ M TURE F x x x x x x xx x x x x x x x (9) tht s otnle y () nd () To get the mtrx E necessry for constructon of the dul model we note tht t s gven y the vlue of E g n the unt of the group e y vlue of F for x x x x E () The dul tensor on the Aeln group G constructed y the procedure explned n the Secton nmely y usng () () nd () s F x x x x x x x x x x x x x x x x x x () One cn B see tht the dul tensor hs lso ntsymmetrc prt ( -feld or torson potentl) B F F j () nd ts torson H db s H dxdx dx x () The Guss curvture of ts symmetrc prt vnshes ut the Rcc tensor s nontrvl Dul metrc tht s symmetrc prt of () does not solve the Ensten equtons ether ut gn we cn stsfy conforml nvrnce condtons (7)-(9) y the dlton feld x Cln ln x () x If we use the sulger of sometres spnned y 5 6 nsted of tht spnned y 6 7 then the trnsformton etween group coordntes x x x x nd geometrcl coordntes u x y s x u e x x x x e x x y x x (5) the mtrx E gets gn the form () nd we get nother tensor dul to (6) F x x x x x x x x x x x x x x (6) Even though t s not symmetrc ts torson s zero It stsfes the conforml nvrnce condtons (7)-(9) wth the dlton feld x Cln ln x x (7) Lochevsky Plne Wves Another type of metrcs tht hve rther lrge group of sometres re so clled Lochevsky plne wves [78] They re of generl form G u x y Hu x y x x x x x (8) They stsfy Ensten equton wth cosmologcl con- stnt ff Copyrght ScRes
L HLAVATÝ M TURE 9 y x x x Hu x y Hu x y H u x y (9) The Guss curvture of ths metrc s For specl forms of functon H the metrc (8) dmts vrous sets of llng vectors All of them re sulgers of vector spce spnned y I II u u III u IV (5) y V y u y VI u x y u x y VIII u x y ux uy u x y A t surprsngly ll these seven ndependent vector felds found n [7] form Le lger even though they re not llng vectors of the sme metrcs (t depends on the form of H uxy ) We re nterested n metrcs tht dmt t lest four ndependent llng vectors ecuse they cn e nterpreted s dulzle ckgrounds for sgm models n four dmensons As mentoned n the Secton for constructon of dulzle metrcs we need four-dmensonl sulger of llng vectors tht genertes group of sometres tht cts freely nd trnstvely on the four-dmensonl Remnnn mnfolds Here we shll nvestgte metrcs of the form (8) where tht H x e x x G u x y x (5) x x It solves the Ensten equton wth the cosmologcl constnt for [] Constructon of the Dul Metrc The metrc (5) hs fve-dmensonl Le group of sometres generted y the llng vectors I II III IV V VI Ther nonzero commuttors red III V IV IV V I IV VI IV I VI I III VI III V VI V (5) Four-dmensonl sulgers of the Le lger (5) for generc re somorphc to one of the followng lgers: Spn I III IV VI V Spn I III IV V Spn I IV V I II VI It s esy to check tht the only sulger of these tht stsfy the condton of trnstvty () n every pont of M s the frst one Its cton s free on M s well so tht we cn use t for dulzton of the metrc (5) In the followng we shll consder the cse ecuse do not rng nythng qulttvely dfferent It mens tht for dulzton we shll use the lger spnned y I III IV VI wth nonzero commutton reltons I VI I III VI III IV VI IV (5) The correspondng Drnfel d doule s generted y the lger defned y the commutton reltons (5) nd four-dmensonl Aeln lger The ss of leftnvrnt vector felds of the group generted y x x x e e e x x x x s (5) where x x x x re group coordntes used n prmetrzton xx xx xx xx g x x x x e e e e nd X X X X re genertors of stsfyng X X X X X X X X X (55) Trnsformton etween group coordntes nd coordntes u x y of the Lochevsky mnfold s y x x x ux x x lnx x (56) Ths trnsformton converts the llng vectors I III IV VI nto the left-nvrnt vector felds (5) nd the metrc (5) nto Copyrght ScRes
9 L HLAVATÝ M TURE F x x x x x x x x x x x x x x xx The vlue of ths metrc for x x x x e n the unt of the group gves the mtrx E (57) Hvng ths mtrx we cn construct the dul tensor It s gn otned usng () () nd () nd hs the form 6 x x x x x x x x x x x x( ) F x x x x x x x 6 6 x ( x x ) x x x x x x x( ) x x x x x x x x x x x Ths tensor hs nonzero nd nonconstnt Guss curv re nd torson tu B-Feld The Drnfel d doule constructon enles to dd the B- feld (torson potentl) to the metrc so tht the resultng tensor G G B s nvrnt wth respect to the sme sometry group s the metrc tself Nmely chngng E to 5 5 Other ntsymmetrc elements do not chnge torson E (58) nd pplyng the formul () () we get covrnt tensor tht fter the trnsformton (56) cqures the form Copyrght ScRes
L HLAVATÝ M TURE 95 G u x y x x x x x x x x x (59) Its symmetrc prt s the metrc (5) Ths tensor s gn nvrnt wth respect to the sometry group generted y I III IV VI For the nvrnt group cn e extended y the genertor V Torson H db otned from the ntsymmetrc prt of G s H duddydudxdy (6) ddxdy As the tensor (59) ws otned y the Drnfel d doule constructon t s possle to dulze t ut the result s too extensve to dsply 5 Conclusons Isometry groups of metrcs cn e used for constructon of ther (no neln) T-dul ckgrounds Suffcent condton for tht s tht the metrc hve n sometry sugroup whose dmenson s equl to the dmenson of the Remnnn mnfold nd ts cton on the mnfold s trnstve nd free We hve shown tht for the plne wve metrcs (6) nd (5) such sometry sugroups exst nd the metrcs cn e dulzed y the Posson-Le T-dulty trnsformton We hve determned th e metrcs nd B-felds dul to the plne wves For homogeneous plne wves (6) we hve lso found the dlton feld tht gurntees conforml nvrnce of the dul metrc Metrcs tht possess sometry group whose dmenson s greter thn the dmenson of the Remnnn mnfold my hve severl duls More precsely f the metrc dmts vrous sometry sugroups wth ove gven propertes then we cn construct severl ckgrounds dul to the metrc Ths phenomenon s nother knd of plurlty of sgm models dfferent from the Posson-Le T-plurlty descred n the Secton An exmple of ths type of plurlty s provded y the plne wve metrc (6) wth sometry sugroups generted y llng vectors 6 7 or y 5 6 (see () producng two dul ckgrounds () nd (6)) To decde f ths plurlty s dfferent from the Posson-Le T-plurlty one hs to check whether the eght-dmensonl Drnfel d doule s gener- ted y the four-dmensonl eln lger nd lgers spnned y 6 7 or 5 6 re somorphc y trnsformton tht leve the constnt mtrx () nvrnt Ths s however very dffcult tsk tht mght e nvestgted n the future 6 Acknowledgements Ths work ws supported y the reserch pln LC57 of the Mnstry of Educton of the Czech Repulc Consultton wth P Wnterntz nd L Šnol on clssfcton of sulgers re grtefully cknowledged REFERENCES [] C lmčík nd P Šever Dul Non-Aeln Dulty nd the Drnfeld Doule Physcs Letters B 995 pp 55-6 [] M A Lledo nd V S Vrdrjn SU() Posson-Le T-Dulty Letters n Mthemtcl Physcs Vol 5 No 998 pp 7-57 do:/a:798898 [] Sfetsos Posson-Le T-Dulty eyond the Clsscl Level nd the Renormlzton Group Physcs Letters B Vol No - 998 pp 65-75 do:6/s7-69(98)666- [] L Hlvtý nd L Šnol Posson-Le T-Plurlty of Three- Dmensonl Conformlly Invrnt Sgm Models II: Nondgonl Metrcs nd Dltonpuzzle Journl of Hgh Energy Physcs No [5] G Ppdopoulos J G Russo nd A A Tseytln Solvle Model of Strngs n Tme-Dependent Plne-Wve Bckground Clsscl nd Quntum Grvty pp 969-6 [hep-th/89] [6] M Blu nd M O Loughln Homogeneous Plne Wves Nucler Physcs B Vol 65 No - pp 5-76 do:6/s55-()55-5 [7] S T C Sklos Lotchewsk Plne Grvttonl Wves n Glxes Axsymmetrc Systems nd Reltvty M A H McCllum Ed Cmrdge Unversty Press Cmrdge 985 p 7 [8] J Podolský Interpretton of the Sklos Solutons s Exct Grvttonl Wves n the Ant-De Stter Unverse Clsscl nd Quntum Grvty Vol 5 No 998 pp 79-7 do:88/6-98/5//9 [9] C lmčk Posson-Le T-Dulty Nucler Physcs A 996 pp 6- [hepth95995] [] R von Unge Posson-Le T-Plurlty Journl of Hgh Energy Physcs [hepth55] [] J Pter P Wnterntz nd H Zssenhus Contnuous Sugroups of the Fundmentl Groups of Physcs I Generl Method nd the Poncré Group Journl of Mthemtcl Physcs Vol 6 No 8 975 pp 597-6 [] V R gorodov Ensten Spces of Mxmum Molty Sovet Physcs Dokldy Vol 7 96 p 89 Copyrght ScRes