In gnral, w do not xpct th xistnc of th lctric-agntic duality in a (+)- dinsional thory. This is basd on an obsrvation that, unlik th (3+)-dinsional c

Transcription

1 Magntically chargd solutions via an analog of th lctric-agntic duality in (+)-dinsional gravity thoris Youngjai Ki (a) and Dahl Park (b) (a) Dpartnt of Physics Sjong Univrity Soul , KOREA E-ail: (b) Dpartnt of Physics KAIST Tajon 35-7, KOREA E-ail: Abstract W nd an analog of th lctric-agntic duality, which is a Z transforation btwn agntic and lctric sctors of th static and rotationally sytric solutions in a class of (+)-dinsional Einstin-Maxwll-Dilaton gravity thoris. Th thoris in our considration includ, in particular, on paratr class of thoris continuously conncting th Banados-Titlboi- Zanlli (BTZ) gravity and th low nrgy string ctiv thory. Whn thr is no U() charg, w havo() or O(; ) sytry, dpnding on a paratr that spcis ach thory. Via th Z transforation, w obtain xact agntically chargd solutions fro th known lctrically chargd solutions. W xplain th rlationship btwn th Z transforation and O(;Z) sytry, and cont on th T -duality of th string thory. 4..Jb, 4.6.Kz,.3.Er Typst using REVTEX

2 In gnral, w do not xpct th xistnc of th lctric-agntic duality in a (+)- dinsional thory. This is basd on an obsrvation that, unlik th (3+)-dinsional cas [], th nubr of indpndnt coponnts for th lctric ld in (+)-dinsion is dirnt fro that of th agntic ld. If w rstrict our attntion only to static and rotationally sytric ld congurations in (+)-dinsional thoris, howvr, w hav th sa nubr of th lctric-agntic ld coponnt and it is concivabl that so analog of th lctric-agntic duality ay xist []. Thr ar so rasons why w ar intrstd in this issu. Th T -duality of th string thory has bn incrasingly playing a signicant rol in rcnt dvlopnts of th string thory [3]. Sinc th targt spac ctiv action of th string thory contains U() gaug lds fro th opn string sctor, th T - duality ay iply so analog of th lctric-agntic duality in a targt spac gotry. Convrsly, th study of th lctric-agntic duality ay lad to a bttr undrstanding of th T -duality in th low nrgy string thory. In particular, Cadoni [4] rcntly found an O() sytry in a class of (+)-dinsional Kaluza-Klin typ thoris that includs th low nrgy string ctiv thory without U() gaug lds and th (unchargd) Banados- Titlboi-Zanlli (BTZ) thory [5] [6]. It was furthr suggstd thr that th discrt vrsion of th sytry, i.., O(;Z), ay b rlatd to th O(; ;Z) duality fro th string thory dscription of th (+)-dinsional gravity thoris [7]. Thr is also a or practical rason why w ar intrstd in an analog of th lctric-agntic duality. Whil w hav anubr of xact lctrically chargd solutions in (+)-dinsional gravity thoris, for xapl, as can b found in Rf. [8], agntically chargd solutions ar rlativly lss undrstood. If w nd an analog of th lctric-agntic duality, w can us it to nd agntically chargd solutions fro th known lctrically chargd solutions. In cas of th BTZ thory, th agntically chargd solutions wr obtaind in [9]. As thir rsults show, th proprtis of th agntic solutions ar too dirnt fro th lctric solutions to idiatly uncovr any rlations btwn th. Howvr, th possibility of an analog of th lctric-agntic duality was suggstd in furthr studis, for xapl, in [] for th BTZ thory and in [] for th cas of th Einstin-Maxwll-Dilaton thory without th

3 cosological constant tr. In this not, w nd an analog of th lctric-agntic duality for th static and rotationally sytric solutions of th thoris givn by th (+)-dinsional action I = Z d 3 x qg (3) (R (3) f@ f + bf + 4 f F ) () whr R (3), f, F dnot th (+)-dinsional scalar curvatur, th dilaton ld and th curvatur two for for a U() gaug ld, rspctivly. W us (+ ) signatur for th (+)-dinsional tric g (3). W also hav th cosological constant and two paratrs b and, th spcication of which givs us a particular gravity thory. For xapl, th choic b = = producs th BTZ thory, whil th choic b = = p yilds th (+)- dinsional low nrgy string ctiv action aftr a rscaling of th tric g (3). Undr th assuption of th rotational sytry, w can writ th (+)-dinsional tric as ds = g dx dx 4 d ; () whr th two-dinsional longitudinal tric g and th conforal factor of th angular part of th tric is indpndnt of th aziuthal angl. W choos to dscrib th rsulting -dinsional longitudinal gotry of th spac-ti in trs of a conforal gaug, thrby stting g dx dx = xp()dx + dx. W can now rduc th action Eq. () to a class of -dinsional dilaton gravity thoris [] by intgrating out th coordinat []. In this procss, (;) coponnts of th Einstin quations, that can not b capturd in th rsulting -dinsional action, rducs to a condition F + F =: (3) To gt static solutions, w assu all th physical variabls in our considration dpnds only on a spac-lik variabl x = x + + x. This in particular iplis F + = F or, in othr words, th two for curvatur F of th gaug ld consists of th purly lctric ld F + and th purly agntic ld F. Eq. (3) thrfor shows th solutions of our probl hav ithr lctric charg or agntic charg, but not both at th sa ti. For our furthr 3

4 considration, w introduc a ld A that, in lctrically chargd cas, is dnd to satisfy F + = da=dx, and F = da=dx in agntically chargd cas. Explicitly, th action for th lctrically chargd sctor can b writtn as I = Z dx + b f + 4 f 4 f A + 4 ; (4) whil w hav th following action for th agntically chargd sctor. Z I = dx + b f + 4 f + 4 f +4 A + : (5) 4 Hr w introduc a nw spatial coordinat x via dx = xp( + bf=)dx and th pri dnots th dirntiation with rspct x. W should also ipos th static vrsion of th gaug constraints for ach cas that rsults fro our choic of a conforal gaug. Th undrlying sytry of th thoris in our considration is ost apparnt whn w introduc a st of ld rdnitions to nw lds X, Y and u givn by whr th atrix T is T = X Y u b p p b j4 b j = T p f p p j4 W dn (4 b )=j4 b j, which bcos + for jbj < and for jbj >. Th dtrinant of th atrix T is jb 4j =. Thrfor as long as th paratr b is not 4, th ld rdnitions, Eq. (6), ar wll-dnd. In what follows, w nd it convnint to us a vctor notation ~ X =(X; Y ) on th -dinsional spac of lds X and Y with an innr product ~ X ~ X = X + Y. Th action for both th lctric sctor and th agntic sctor, thn, bcos I = Z dx u ( ~ X ~ X u b j p 4 b + p 4 xp(~ d ~ X 4 b j4 b b j : (6) b +4 4 b u)a + ); (7) 4

5 whil th gaug constraints can b writtn as ~X ~ X u 4 b + p 4 xp(~ d ~ X b +4 4 b u)a =: (8) 4 upon using th static quations of otion. Th only dirnc btwn th action for th lctrically chargd sctor and th agntically chargd sctor is th choic of a - dinsional vctor ~ d and a paratr p =. For th lctrically chargd sctor, w hav th vctor ~ d = ~ d whr ~d = p (; q +b j4 b j ) (9) along with p =. For th agntically chargd sctor, w hav ~ d = ~ d whr ~d = p (b + ; b b qj4 b j ) () along with p = +. W notic that th action (7) has a sytry undr th transforation x! x + for an arbitrary constant. Th rol of th gaug constraint (8) is to st th Nothr charg of this sytry to zro []. Eq. (7) anifstly shows th O() sytry for jbj < ( X ~ X=X ~ +Y ) and O(; ) sytry for jbj > ( ~ X ~ X = X Y ) whn w st A =, i.., whn thr is no U() charg. (Whn =, w hav an nhancd sytry. Thn, by a rdnition of x and a rscaling of u, O(; ) sytry bcos anifst. W also not th O() sytry was rst obsrvd in [4].) Ths sytris corrspond to th rotation in (X; Y ) spac. Furthror, togthr with th two translational sytris of X and Y, which ar also prsnt whn A =,thy constitut th -dinsional Euclidan Poincar group or th - dinsional Minkowskian Poincar group, dpnding on th valu of b. Whn w introduc an lctric charg, w nd up choosing a particular vctor d ~ in th -dinsional (X; Y ) spac. This braks th Poincar invarianc to th Z sytry, that corrsponds to th rction about th lctric axis ~ d, and a singl translational sytry of th lds (X; Y ) along th dirction prpndicular to th vctor ~ d. A siilar story holds whn w introduc 5

6 a agntic charg; w brak th unchargd Poincar invarianc to th Z sytry about th agntic axis ~ d plus on translational sytry. W concntrat on th cas b<, which contains two pri xapls of our concrn, naly, th BTZ odl (b = = ) and th targt spac ctiv action of th string thory (b = = p ). For unchargd cass, w thn hav th O() sytry. A ky obsrvation that shows th xistnc of an analog of th lctric-agntic duality is that th quality ~d d ~ = d ~ d ~ +b + =8 () 4 b holds. Thrfor, by th rction about th axis that biscts th lctric axis and th agntic axis, w can transfor th lctric action xactly into th agntic action and vic vrsa. Of cours, just as in th cas of th lctric-agntic duality of th 4-dinsional Minkowskian spac-ti, w nd to ip th sign of th A tr along with th rction. This Z transforation, which is also a subgroup of th unchargd O() group, is an analog of th lctric-agntic duality in cas of th (+)-dinsional gravity thoris. An iportant application of our rsults is to obtain agntically chargd solutions fro th known lctrically chargd solutions. For this purpos, it is convnint to work out th rprsntation of th abov transforations that acts on th spac of (; ; f) lds. Using th atrix T and th abov considrations, w nd f f = = whr th atrics ar givn by = f f +b + ; ; f f = f = f (b + ) b + (b + ) 4 +b + () 6

7 = +b + = = +b + +b + (b + ) b+ 4(b + ) +b + (b + ) b + 4(b + ) +b + (b+) b+ (b + ) 4 +b + Th subscripts and for ach ld rprsnt th solutions in th lctrically chargd sctor and in th agntically chargd sctor, rspctivly. W can straightforwardly vrify that transfors Eq. (5) into Eq. (4), transfors Eq. (4) into Eq. (5), lavs Eq. (4) invariant, and lavs Eq. (5) invariant. Sinc ( ) dnots th slfduality within th lctrically chargd sctor (agntically chargd sctor), thy satisfy = = and dt = dt =. W can also straightforwardly vrify that = = and dt = dt =. Anothr intrsting obsrvation is th siultanous transforation b +! and! (b+) xchangs and subscripts. Using Eq.(), it is straightforward to obtain agntically chargd solutions fro th known lctrically chargd solutions. Fro [8], so xact lctrically chargd solutions for th b = cas ar availabl. W rcast th in th conforal gaug, apply, and go back to th original gaug, to obtain th following xact agntically chargd solutions. Th tric is coputd to b ds =( r M)dt r ( r M) b = "r + N( r M) b =4 l l l "r + N( r M) b =4 N l N dr d (3) 7

8 and th dilaton ld f turns out to b bf = k b = bf ( r l M) b = (4) whr l, M, k, N and f ar constants. In cas of th BTZ thory, b = =,wtak th liit b! of Eqs. (3) and (4) stting M! M, l! l and N! 4Q M =b. Th dilaton ld in this cas bcos a constant f = f and th tric bcos ds =( r l M )dt r ( r l M )(r + Q M ln j r l M j) dr (r + Q M ln j r l M j)d ; which is xactly th sa as th solutions of [9]. For othr cass including th ost iportant string ctiv thory, b = = p, w obtain nw non-trivial agntically chargd solutions. It is intrsting that th drastically dirnt solutions in ach sctor ar rlatd by an analog of th usual lctric-agntic duality unlik th (3+)-dinsional cas. In fact, whn =, it was found in [] that th agntically chargd solutions in [3] is rlatd to th lctrically chargd solutions in [4] via an analog of th lctric-agntic duality. Actually, w can also gnrat, for xapl, th dual of th known lctrically chargd solutions via. W plan to addrss th dtaild study of th nw solutions and th rlationships btwn solutions rlatd by th dual transforations in a futur publication. W can undrstand or of th structur of th transforations w nd so far by noting th rlation cos = ~ d ~ d ~d ~ d = + b + b = +b + (5) that shows th angl btwn th lctric and th agntic vctor for a givn st of paratrs (b; ). By varying th valu of fro zro to, w for a disjoint collction of paths in (b; ) spac. On distinctiv cas is whn a path is rprsntd by b +=. W thn hav cos =, which ans th ~ d points to th opposit dirction to th vctor ~d. Th BTZ thory with b = = blongs to this cas. Clarly, th atrix satisfy =, thrby =, sinc th two rotations, ach of th by 8 dgrs, cobin to bco th idntity opration. W also hav =, sinc thy rprsnt th rction 8

9 about th lctric and agntic axis, ach of which points to th opposit dirction to ach othr. Th atrics and, that cout with ach othr in this cas, gnrat th dirct su of two Z groups, th Klin four group. In gnral, whn ==n for an intgr n, th sallst positiv intgr that satiss = is n. Thus, togthr with and, gnrats th Dn, th sytry group of n-gon. Th ost intrsting cas is whn n = 4 that givs a situation whr th vctor d ~ is prpndicular to th vctor d ~. In this cas, = =, thus + b + b = = (6) and th 8-lnt nonablian group D 4 is isoorphic to O(;Z). Th low nrgy string ctiv thory with b = = p satiss Eq.(6). Sinc O(;Z) is both a natural subgroup of O(; ; Z) and O(), th transforations w nd ay b considrd as a natural subgroup of th T -duality fro th string thory. For othr valus of n, including n = for th BTZ thory,, and gnrat a group dirnt fro O(;Z). Whn cobind with b =, which is clar fro th fact that th cosological constant tr rsults fro th closd string sctor and th U() ld, fro th tr lvl opn string sctor, Eq.(6) uniquly dtrins b = = p. Both sts of th paratrs corrctly giv th kintic tr of th dilaton ld f for th low nrgy ctiv string thory aftr a transforation f! f in Eq.(). This shows an intrsting possibility; th paratrs (for xapl, b and in our cas) apparing in th low nrgy string ctiv action ay b dtrind to so xtnt by th rquirnt of prsrvation of th T -duality. A siilar analysis in highr dinsional cass to th on givn hr ay furthr vrify this ida and th work in this rgard is in progrss. 9

10 REFERENCES [] G. Horowitz, in String Thory and Quantu Gravity '9, Procdings of th Trist Spring School and Workshop, J. Harvy t. al., d., (World Scintic, 993) and rfrncs citd thrin. [] D. Park and J.K. Ki, KAIST prprint. [3] R. Dijkgraaf, E. Vrlind and H. Vrlind, Nucl. Phys. B37, 69 (99); G. Vnziano, Phys. Ltt. B65, 87 (99); A.A. Tsytlin and C. Vafa, Nucl. Phys. B37, 443 (99). [4] M. Cadoni, INFN prprint, INFN-TH96, gr-qc/96648, to appar in Phys. Rv. D. [5] M. Banados, C. Titlboi, J. Zanlli, Phys. Rv. Ltt. 69, 849 (99); M. Banados, M. Hnnaux, C. Titlboi, J. Zanlli, Phys. Rv. D48, 56 (993). [6] S. Carlip, Class. Quant. Grav., 853 (995). [7] A. Givon, M. Porrati and E. Rabinovici, Phys. Rp. 44, 77 (994). [8] K.C.K. Chan and R.B. Mann, Phys. Rv. D5, 6385 (994); rratu, D5, 6 (995). [9] E.W. Hirschann and D. L. Wlch, Phys. Rv. D53, 5579 (996). [] M. Cataldo and P. Salgado, Phys. Rv. D54, 97 (996), gr-qc/ [] T. Banks and M. O'loughlin, Nucl. Phys. B36, 649 (99); R.B. Mann, Gn. Rl. Grav. 4, 433 (99). [] D. Park and Y. Ki, Phys. Rv. D53, 553 (996). [3] J. D. Barrow, A. B. Burd and D. Lancastr, Class. Quantu Grav. 3, 55 (986). [4] J. R. Gott, III, J. Z. Sion and M. Alprt, Gn. Rl. and Grav. 8, 9 (986).