Complexified Gravity in Noncommutative Spaces

Transcription

1 CAMS/00-04 Complexified Grvity in Noncommuttive Spces Ali H. Chmseddine Center for Advnced Mthemticl Sciences (CAMS), nd Physics Deprtment, Americn University of Beirut, Lebnon ABSTRACT The presence of constnt bckground ntisymmetric tensor for open strings or D-brnes forces the spce-time coordintes to be noncommuttive. This effect is equivlent to replcing ordinry products in the effective theory by the deformed str product. An immedite consequence of this is tht ll fields get complexified. The only possible noncommuttive Yng-Mills theory is the one with U(N) guge symmetry. By pplying this ide to grvity one discovers tht the metric becomes complex. We show in this rticle tht this procedure is completely consistent nd one cn obtin complexified grvity by guging the symmetry U(1,D 1) insted of the usul SO(1,D 1). The finl theory depends on Hermitin tensor contining both the symmetric metric nd ntisymmetric tensor. In contrst to other theories of nonsymmetric grvity the ction is both unique nd guge invrint. The results re then generlized to noncommuttive spces. 1

2 1 Introduction The developments in the lst two yers hve shown tht the presence of constnt bckground B-field for open strings or D-brnes led to the noncommuttivity of spce-time coordintes ([1],[2],[3], [4],[5],[6],[7]). This cn be equivlently relized by deforming the lgebr of functions on the clssicl world volume. The opertor product expnsion for vertex opertors is identified with the str (Moyl) product of functions on noncommuttive spces ([8],[9]). In this respect it ws shown tht noncommuttive U(N) Yng-Mills theory does rise in string theory. The effective ction in presence of constnt B-field bckground is 1 Tr(F F ) 4 where nd the str product is defined by F = A A + ia A ia A f (x) g (x) =e i 2 θ ζ η f (x + ζ) g (x + η) ζ=η=0 This definition forces the guge fields to become complex. Indeed the noncommuttive Yng-Mills ction is invrint under the guge trnsformtions A g = g A g 1 g g 1 where g 1 is the inverse of g with respect to the str product: g g 1 = g 1 g =1 The contributions of the terms iθ in the str product forces the guge fields to be complex. Only conditions such s A = A could be preserved under guge trnsformtions provided tht g is unitry: g g = g g =1. It is not possible to restrict A to be rel or imginry to get the orthogonl or symplectic guge groups s these properties re not preserved by the str product ([7],[10]). I will ddress the question of how is grvity modified in the low-energy effective theory of open strings in the presence of bckground fields. It hs been shown tht the metric of the trget spce gets modified by contributions of the B-field nd tht it becomes nonsymmetric ([11],[7]). If we think of grvity s resulting from locl 2

3 guge invrince under Lorentz trnsformtions in the tngent mnifold, then the previous resoning would suggest tht the vielbein nd spin connection both get complexified with the str product. This seems inevitble s the str product ppers in the opertor product expnsion of the string vertex opertors. We re therefore led to investigte whether grvity in D dimensions cn be constructed by guging the unitry group U(1,D 1). In this rticle we shll show tht this is indeed possible nd tht one cn construct Hermitin ction which governs the dynmics of nonsymmetric complex metric. Once this is chieved, it is strightforwrd to give the necessry modifictions to mke the ction noncommuttive. The pln of this pper is s follows. In section two the ction for nonsymmetric grvity bsed on guging the group U(1,D 1) is given nd the structure of the theory studied. In section three the equtions of motion re solved to mke connection with the second order formlism. In section four we give the generliztion to noncommuttive spces. Section five is the conclusion. 2 Nonsymmetric grvity by guging U(1,D-1) Assume tht we strt with the U(1,D 1) guge fields ωb.theu(1,d 1) group of trnsformtions is defined s the set of mtrix trnsformtions leving the qudrtic form (Z ) ηb Z b invrint, where Z re D complex fields nd η b = dig ( 1, 1,, 1) with D 1 positive entries. The guge fields ωb must then stisfy the condition ( ω b ) = η b c ω c d ηd The curvture ssocited with this guge field is R b = ωb ωb+ ωcω b c ωcω b c Under guge trnsformtions we hve ω b = M c ω c d M 1d b M c M 1c b 3

4 where the mtrices M re subject to the condition: (Mc ) ηb Md b = ηd c The curvture then trnsforms s R b = M c R c d M 1d b Next we introduce the complex vielbein e nd its inverse e defined by which trnsform s e e = δ e e b = δ b ẽ = M b eb ẽ = ẽ b M 1b It is lso useful to define the complex conjugtes e ( e ) With this, it is not difficult to see tht trnsforms to e d M 1d M e R e e (e ) e R bηce b c f M 1f b ηc b ( ) M 1l c e l nd is thus U(1,D 1) invrint. It is lso Hermitin ( e R bηce b c) = e c ηbη c er b e f ηe f = e R bηce b c The metric is defined by stisfy the property g = ( e ) η b e b g = g When the metric is decomposed into its rel nd imginry prts: g = G + ib 4

5 the hermiticity property then implies the symmetries G = G B = B The guge invrint Hermitin ction is given by I = d D x Ge R b ηb c ec This ction is nlogous to the first order formultion of grvity obtined by guging the group SO(1,D 1) One goes to the second order formlism by integrting out the spin connection nd substituting for it its vlue in terms of the vielbein. The sme structure is lso present here nd one cn solve for ωb in terms of the complex fields e resulting in n ction tht depends only on the fields g. It is worthwhile to stress tht the bove ction, unlike others proposed to describe nonsymmetric grvity [12] is unique nd unmbiguous. The ordering of the terms is lso done in wy tht generlizes to the noncommuttive cse. The infinitesiml guge trnsformtions for e is δe =Λ b eb which cn be decomposed into rel nd imginry prts by writing e = e 0 + ie 1, nd Λ b =Λ 0b + iλ 1b to give δe 0 = Λ 0b eb 0 Λ 1b eb 1 δe 1 = Λ 1b eb 0 +Λ 0b eb 1 The guge prmeters stisfy the constrints (Λ b ) = ηc bλc d ηd the two constrints which implies (Λ 0b )T = ηc b Λc 0d ηd (Λ 1b) T = ηcλ b c 1dη d From the guge trnsformtions of e 0 nd e 1 one cn esily show tht the guge prmeters Λ 0b nd Λ 1b cn be chosen to mke e 0 symmetric in nd nd e 1 ntisymmetric in nd. This is equivlent to the sttement tht the Lgrngin should be completely expressible in terms of G nd B only, fter eliminting ωb through its equtions of motion. In relity we hve G = e 0 eb 0 η b + e 1 eb 1 η b B = e 0 eb 1 η b e 1 eb 0 η b 5

6 In this specil guge, where we define g 0 = e 0 eb 0 η b, g 0 g0 λ = δ λ, nd use e 0 to rise nd lower indices we get B = 2e 1 G = g B κb λ g κλ 0 The lst formul ppers in the metric of the effective ction in open string theory [11]. 3 Second Order Formultion We cn express the Lgrngin in terms of e only by solving the ωb equtions of motion e eb ωb c + e b ec ω b eb e ω b c e b ec ω b = 1 ( ) G (e e c e e c ) G X c where X c stisfy (Xc ) = X c. One hs to be very creful in working with nonsymmetric metric g = e e g = e e g g ρ = δ ρ but g g ρ δ. ρ Cre lso should be tken when rising nd lowering indices with the metric. Before solving the ω equtions, we point out tht the trce prt of ωb (corresponding to the U(1) prt in U(D)) must decouple from the other guge fields. It is thus undetermined nd decouples from the Lgrngin fter substituting its eqution of motion. It imposes condition on the e 1 ( ) G (e e e e ) X =0 G We cn therefore ssume, without ny loss in generlity, tht ωb ( ω =0 ). is trceless Multiplying the ω eqution with e κ e ρ c we get δ κ ω ρ + δ ρ ω κ ω κρ ω ρ κ = X ρκ 6

7 where ω ρ = e e ρb ω b X ρκ = e ρc e κx c Contrcting by first setting = κ then = ρ we get the two equtions 3ω ω ρ ρ + ω ρ = X ρ +3ω ρ = X ρ These could be solved to give ωρ = 1 ( 3X 8 ρ X ρ) ωρ = 1 ( X 8 ρ +3X ρ) Substituting these bck into the ω eqution we get ω κρ + ω ρκ = 1 ( 8 δ κ 3X ρ X 1 ( ρ) + 8 δ ρ X κ +3X ) κ X ρκ Y ρκ We cn rewrite this eqution fter contrcting with e c e c σ to get ω κρσ + e e ce c σ ω ρκ = g σy ρκ Y σρκ By writing ωρκ = ω ρκe we finlly get ( δ α κ δρ β δσ γ + g β g σ δρ α δκ γ ) ωαβγ = Y σρκ To solve this eqution we hve to invert the tensor M αβγ κρσ = δ α κ δ β ρ δ γ σ + g β g σ δ α ρ δ γ κ In the conventionl cse when ll fields re rel, the metric g is symmetric nd g β g σ = δσ β αβγ so tht the inverse of Mκρσ is simple. In the present cse, becuse of the nonsymmetry of g this is firly complicted nd could only be solved by perturbtive expnsion. Writing g = G + ib nd from the definition g g ρ = δ ρ we get where g = + ib = ( G + B κ G κλ B λ ) 1 = G G κ B κλ G λσ B ση G η + O(B 4 ) 7

8 b = G κ B κλ G λ + G κ B κλ G λσ B στ G τρ B ρη G η + O(B 5 ) We hve defined G G ρ = δ ρ. This implies tht The inverse of M αβγ κρσ g α g α δ + L L = ig ρ B ρ 2G ρ B ρσ G σα B α + O(B 3 ) defined by N σρκ αβγ M α β γ σρκ = δα α δβ β δγ γ is evluted to give N σρκ αβγ = 1 ( ) δγ σ δ ρ β 2 δκ α + δβδ σ αδ ρ γ κ δαδ σ γδ ρ β κ 1 ( ) δ κ 4 βδαl σ ρ γ + δαδ κ γ σ L ρ β δκ γ δβl σ ρ α + 1 ( ) L κ γ 4 δσ β δρ α + Lκ β δσ α δρ γ Lκ α δσ γ δρ β 1 ( ) δα κ 4 Lσ γ δρ β + δκ γ Lσ β δρ α δκ β Lσ α δρ γ + O(L 2 ) This enbles us to write nd finlly ω αβγ = N σρκ αβγ Y ρσκ ωb= e β e γ b ω βγ It is cler tht the leding term reproduces the Einstein-Hilbert ction plus contributions proportionl to B nd higher order terms. The most difficult tsk is to show tht the Lgrngin is completely expressible in terms of G nd B only. The other components of e 0 nd e 1 should dispper. We hve rgued from the view point of guge invrince tht this must hppen, but it will be nice to verify this explicitly, to leding orders. We cn check tht in the flt pproximtion for grvity with G tken to be δ,theb field gets the correct kinetic terms. First we write e = δ i 2 B nd the inverses e = δ + i 2 B e = δ i 2 B e = δ + i 2 B 8

9 The ω eqution implies the constrint X = (e e e e )=0 This gives the guge fixing condition B =0 We then evlute X ρκ = i 2 ( ρb κ + κ B ρ ) This together with the guge condition on B gives Y ρκ = i 2 ( ρb κ + κ B ρ ) nd finlly ω ρ = i 2 ( B ρ + B ρ ) When the ω ρ is substituted bck into the Lgrngin, nd fter integrtion by prts one gets L = ω ρ ω ρ ω ρ ω ρ = 1 4 B 2 B This is identicl to the usul expression where 1 12 H ρh ρ H ρ = B ρ + B ρ + ρ B We hve therefore shown tht in D dimensions one must strt with 2D 2 rel components e, subject to guge trnsformtions with D 2 rel prmeters. The resulting Lgrngin depends on D 2 fields, with D(D+1) 2 symmetric components G nd D(D 1) 2 ntisymmetric components B. 4 Noncommuttive Grvity At this stge, nd hving shown tht it is perfectly legitimte to formulte theory of grvity with nonsymmetric complex metric, bsed on the ide of guge 9

10 invrince of the group U(1,D 1). It is not difficult to generlize the steps tht led us to the ction for complex grvity to spces where coordintes do not commute, or equivlently, where the usul products re replced with str products. First the guge fields re subject to the guge trnsformtions where M 1b is now ω b = M c ω c d M 1d b is the inverse of M b M c M 1c b with respect to the str product. The curvture R b = ωb ωb + ω c ω b c ω c ω b c which trnsforms ccording to R b = M c R c d M 1d Next we introduce the vielbeins e nd their inverse defined by b which trnsform to e e = δ e e b = δ b ẽ = M b e b ẽ = ẽ b M 1b The complex conjugtes for the vielbeins re defined by e ( e ) Finlly we define the metric e (e ) g = ( e ) η b e b The U(1,D 1) guge invrint Hermitin ction is I = d D x G ( e R bηc b e c) This ction differs from the one considered in the commuttive cse by higher derivtives terms proportionl to θ. It would be very interesting to see 10

11 whether these terms could be rebsorbed by redefining the field B,orwhether the Lgrngin reduces to function of G nd B nd their derivtives only. The connection of this ction to the grvity ction derived for noncommuttive spces bsed on spectrl triples ([13],[14],[15]) remins to be mde. In order to do this one must understnd the structure of Dirc opertors for spces with deformed str products. 5 Conclusions We hve shown tht it is possible to combine the tensors G nd B into complexified theory of grvity in D dimensions by guging the group U(1,D 1). The Hermitin guge invrint ction is direct generliztion of the first order formultion of grvity obtined by guging the Lorentz group SO(1,D 1). The Lgrngin obtined is function of the complex fields e nd reduces to function of G nd B only. This ction is generlizble to noncommuttive spces where coordintes do not commute, or equivlently, where the usul products re deformed to str products. It is remrkble tht the presence of constnt bckground field in open string theory implies tht the metric of the trget spce becomes nonsymmetric nd tht the tngent mnifold for spce-time does not hve only the Lorentz symmetry but the lrger U(1,D 1) symmetry. The results shown here, cn be improved by computing the second order ction to include higher order terms in the B expnsion nd to see if this cn be put in compct form. Similrly the computtion hs to be repeted in the noncommuttive cse to see whether the θ contributions could be simplified. It is lso importnt to determine link between this formultion of noncommuttive grvity nd the Connes formultion bsed on the noncommuttive geometry of spectrl triples. To mke such connection mny points hve to be clrified, especilly the structure of the Dirc opertor for such spce. This nd other points will be explored in future publiction References [1] A. Connes, M. R. Dougls nd A. Schwrz, JHEP 9802:003 (1998). [2] M. R. Dougls nd C. Hull, JHEP 9802:008 (1998). 11

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