rxiv:0909.0294v1 hep-th 1 Sep 2009 A Detiled Disussion of Superfield Supergrvity Prepotentil Perturbtions in the Superspe of the AdS 5 /CFT 4 Correspondene J. Ovlle Deprtmento de Físi, Universidd Simón Bolívr, Crs, Venezuel. Abstrt This pper presents detiled disussion of the issue of supergrvity perturbtions round the flt five dimensionl superspe required for mnifest superspe formultions of the supergrvity side of the AdS 5 /CFT 4 Correspondene 1 Introdution The importne of deeply understnding the superspe geometry of five dimensions, hs reeived ttention during the lst few yers 1, 2, 3, motivted minly by the postulte of the AdS/CFT orrespondene 4. In this respet, the study of supergrvity theories represents n unvoidble issue 5, 6, 7, even more keeping in mind the existene of the supergrvity jovlle@usb.ve 1
side of the AdS 5 /CFT 4 orrespondene. Indeed the works of 6, 7 present omplete nonliner desriptions of suh superspes bsed on prtiulr hoie of ompenstors. It hs long been known 8, tht the superspe geometry hnges when different ompenstors re introdued. So one of the gols of the urrent study is to begin the proess of looking t wht fetures of the work of 6, 7 re universl (i.e. independent of ompenstor hoie). When the superspe pproh is used, the onventionl representtion for Grssmnn vribles for SUSY D = 5, N = 1 (often denominted N = 2) onsiders these vribles obeying pseudo-mjorn relity ondition, then the spinor oordintes re dotted with n SU(2) index. Thus the onventionl pproh first doubles the number of fermioni oordintes, by the introdution of the SU(2) index, then hlves this number by the imposition of the pseudo-mjorn relity ondition. However, s noted previously 1, there is no fundmentl priniple tht demnds the use of Mjornsympleti spinors for desribing the fermioni oordinted. Indeed, it ws demonstrted in 1 tht omplex 4-omponents spinors provide n dequte bsis for desribing suh fermioni oordinte. Building on this previous work, in this pper it will be shown tht it is possible to develop suessfully geometril pproh to five dimensionl N = 1 supergrvity theory, using this unonventionl representtion for Grssmnn vribles. The geometril pproh to supergrvity involves lulting fields strengths to determine the form of the torsions nd urvtures of the theory. With this informtion in hnd, we n set onstrints suh tht the super spin onnetions nd the vetor-supervetor omponent of the inverse supervierbein beome dependent vribles of the theory. One this is omplished, the route to deduing the prepotentils of the supergrvity theory re opened. On the other hnd, there is n lterntive pproh whih is bsed in the torsion superfield CAB C ssoited to the superspe derivtive E A, nmely, the super-nholonomy E A, E B } = CAB CE C. Using the superspe derivtive A to lulte the super (nti)-ommuttor A, B }, we will be ble to write ll super-torsion omponents in terms of the nholonomy nd the spinoril onnetion. Through hoie of suitble onstrints on superspe through some super-torsion omponents, we will be ble to write the spin onnetion superfield in terms of the nholonomy, eliminting by this wy the spin onnetion s independent fields. One this is omplished, 2
the linerized theory is onsidered through perturbtions round the flt superspe. In this linerized regime, ll super holonomy omponents n be obtined in terms of semi-prepotentils. Hene the torsion nd urvture of the theory re determinted in terms of these semi-prepotentils. 2 Superspe geometry: the unonventionl representtion Let us strt onsidering the superoordinte Z A = (x m, θ µ, θ µ ), where the bosoni nd fermioni oordintes re given respetively by x nd θ, where m = 0,..., 4 nd µ = 1,..., 4. As lredy mentioned, unlike the onventionl representtion for Grssmnn vribles for SUSY D = 5, N = 1, the unonventionl representtion for the Grssmnn vribles (θ µ, θ µ ) given in 1 where there is not n SU(2) index ppended to the spinor oordintes of the superspe, will be used. Under this unonventionl spinoril representtion for SUSY D = 5, N = 1, the spinoril superovrint derivtives is given by (for ll detils onerning this lgebr, see 1) D µ = µ + 1 2 (γm ) µν C ν σ θ σ m, Dµ = µ 1 2 (γm ) µσ θ σ m, (1) whih stisfies the lgebr {D µ, D ν } = (γ m ) µν m, {D µ, D ν } = { D µ, D ν } = 0. (2) In order to onstrut the SUGRA D = 5, N = 1 version ssoited with this representtion, it is neessry to onsider the supervetor derivtive A, superspe supergrvity ovrint derivtive, whih is ovrint under generl superoordinte nd superlol Lorentz groups, given by A = E A + Υ A ; Υ A 1 2 ω d A M d + Γ AZ, A = (,, ᾱ) (3) This supergrvity superderivtive is written in terms of superderivtive E A, spin super onnetion ωa d, the Lorentz genertor Md, entrl hrge 3
super onnetion Γ A nd the entrl hrge genertor Z. It n be seen there is n bsene of ny SU(2) onnetion, whih is hrteristi of the unonventionl fermioni representtion onsidered here. We should lso mention one other possibility (though we will not study it in this work). Sine the bosoni dimension is five, it follows tht the field strength of 5-vetor guge field is thus two-form. Hene in superspe there must be super two-form field strength (s ppers in (2.5) below). However, by Hodge dulity, there should be expeted to be formultion of supergrvity here where the entrl hrge onnetion Γ A n be set to zero nd insted there is introdued super two-form Γ AB. The superspe derivtive E A is given through the super vielbein E M A by E A = E M A D M, (4) with D M being the supervetor (this is the flt sitution) whih omponents re ( m, D µ, D µ ) stisfying the lgebr (2). The supertorsion TAB C, urvture superfield RAB d nd entrl hrge superfield strength re given through the lgebr A, B } = TAB C C + 1 2 R AB d M d + F AB Z. (5) Agin, there is no urvture ssoited with SU(2) genertors. The nholonomy superfield C E A is given by AB C ssoited to the superspe derivtive E A, E B } = C C AB E C. (6) This superfield struture will ply fundmentl role in our nlysis. The first step will be to lulte the super (nti)-ommuttor (5) using the superspe derivtive (3). By this wy we will be ble to write ll supertorsion omponents in terms of the nholonomy nd the spinoril onnetion. Then hoosing suitble onstrints on superspe through some super-torsion omponents, we will be ble to write the spin onnetion superfield in terms of the nholonomy, eliminting thus the spin onnetion s independent fields. Finlly eliminted the spin onnetion, the next step will be to obtin speifi form for ll omponents of the nholonomy superfield. To rry out this, it will be neessry to provide n speifi struture to the super vielbein 4
EA M. This struture will be bsed in ll fundmentl geometri objets whih pper in D = 5 N = 1 SUSY. These fundmentl objets re the following spinor metri nd gmm mtries η β ; (γ ) β ; This will be explined in detil lter. (σ b ) β. (7) Let s strt now with the first step, whih is to lulte the super (nti)- ommuttor (5) using the superspe derivtive (3) nd then identify ll super-torsion omponents. The super (nti)-ommuttor n be written s A, B } = E A, E B } + E A, Υ B } + Υ A, E B } + Υ A, Υ B }, (8) using the nholonomy definition we hve A, B } = C C AB E C + E A, Υ B } + Υ A, E B } + Υ A, Υ B }. (9) Hene finlly we hve the expliit form of the lgebr, given by (see ppendix to detils), b = Cb C C ωb + 1 C C b 2 ω ef C + E ω ef b E b ω ef ω e ω f b Mfe + Cb C Γ C + E Γ b E b Γ Z. (10) {, β } = Cβ C C + ı 4 + 1 2 C C β ωc d ω d (σ d ) γ β + ω β d (σ d ) γ + 1 2 E ω d β + 1 2 E βω d γ + ω b ω βb d Md + C C β Γ C + E Γ β + E β Γ Z, (11) {, β } = C C β C + ı 4 + 1 2 C C β d ω β (σ d) γ ω d C γ + ı 4 ω d + 1 2 E ω d β (σ d) γ β γ + 1 2Ēβω d + 1 2 ω b ω d βb M d + C C β Γ C + E Γβ + ĒβΓ Z, (12) 5
, b = Cb C C ωb ı 4 ω d (σ d ) γ b γ C C b ωc d + E ωb d E b ω d ω e ω de b Md + 1 2 + Cb C Γ C + E Γ b E b Γ Z. (13) Compering the lgebr (10)-(13) from (5), the super torsion omponents n be identified in terms of the super nholonomy nd super spin onetion omponents, s it is shown below T b = C b + ω b ω b ; T γ b = C γ b ; T γ b = C γ b. T β = C β ; T γ β = C γ β + ı 4 ω βd(σ d ) γ + ı 4 ω d(σ d ) γ β ; T γ β = C γ β. T β = C β ; T γ β = C γ β + ı 4 ω βd(σ d ) γ ; T γ β = C γ β + ı 4 ω d(σ d ) γ β. T b = C b ω b ; T γ b = C γ b + ı 4 ω bd(σ d ) γ ; T γ b = C γ b. (14) In order to eliminte the spin onnetions s independent fields, it is neessry to impose some restritions on the torsion superfield. To omplish this, the following suitble onstrints re onsidered, through whih we re ble to write the spin onnetion in terms of the nholonomy: T b = 0 ω b = 1 2 (C b C b C b ); (15) T b = 0 ω b = C b, (16) leving thus the spin onnetions s dependent fields. It is worth notiing tht keeping in mind generl reltivity (torsion free theory) s low energy limit of SUGRA, the onstrint (15) seems nturl hoie. 6
3 Perturbtion round the flt superspe The supergrvity theory we re building up is represented by the lgebr (10)-(13), whih expliitly gives the field strengths nd urvture of the theory. After the spin onnetion is eliminted s independent field, this lgebr essentilly depends of the nholonomy. Then the following logil step is to find n speifi form for ll omponents of the nholonomy superfield in terms of simpler funtions. When this is omplished, the onstruted SUGRA theory will be desribed by these funtions, whih will ontin ll the bsi physil informtion. To rry out this, we need to provide n speifi struture to the super vielbein EA M using ll fundmentl geometri objets whih pper in D = 5 N = 1 SUSY, nmely, those given by (7). First of ll let us onsider E A = E M A D M, (17) nd let us strt onsidering its vetoril omponent, whih is writing s E = E M D M = E m m + E µ D µ + Ē µ D µ, (18) now expnding E m round the flt solution δ m we hve E m = δ m hene finlly we obtin the perturbtive version of (18) + H m, (19) E = + H m m + H µ D µ + H µ D µ. (20) It is not omplited to relize tht the fields H m, H µ nd its onjugted H µ nnot be expressed in terms of the fundmentl geometri objets given by (7), thus in some sense they re onsidered s fundmentl objets of the theory. Indeed these field re identified s the grviton H m nd the grvitino H µ with its onjugted H µ. On the other hnd, the spinoril omponent E of the superfield E A is written s E = E M D M = E m m + E µ D µ + Ē µ D µ, (21) 7
gin expnding E µ µ round the flt solution δ we hve hene we obtin the perturbtive version of (21) E µ = δ µ + H µ, (22) E = D + H µ D µ + H µ D µ + H m m. (23) The fields H µ nd H µ n be expressed s liner ombintion of the fundmentl objets (7) by nd H µ = δ µ ψ 1 + ı(γ ) µ ψ 1 + 1 4 (σb ) µ ψ 1 b (24) H µ = δ µ ψ 2 + ı(γ ) µ ψ 2 + 1 4 (σb ) µ ψ 2 b. (25) The oeffiients ψ s re the so lled semi-prepotentils of the theory. We will see lter tht it is possible to obtin n expliit form to some semiprepotentils in terms of H s fields by imposing dditionl suitble onstrints on supertorsion omponents. Using (24) nd (25) in (23) we obtin the spinoril omponents of the superspe derivtive in terms of the semi-prepotentils ψ s nd the fields H m hene E = D + δ µ ψ1 + ı(γ ) µ ψ1 + 1 4 (σb ) µ + δ µ ψ2 + ı(γ ) µ ψ2 + 1 4 (σb ) µ ψ2 b ψ1 b Dµ Dµ + H m m, (26) Ē = D + δ µ (ψ2 ) ı(γ ) µ (ψ2 ) 1 4 (σb ) µ (ψ2 b ) D µ + δ µ (ψ1 ) ı(γ ) µ (ψ1 ) 1 4 (σb ) µ (ψ1 b ) Dµ + H m m; (27) Now using (20), (26) nd (27) in (6) nd keeping liner terms, we re ble to express ll the nholonomy omponents in terms of the grviton, grvitino, the semi-prepotentil fields ψ s, nd the fields H m. Hene we hve C b = H b 8 b H ; (28)
C γ b = H γ b b H γ ; (29) C γ b = H γ b b H γ. (30) Cβ = δ γ ψ2 + ı(γ ) γ ψ2 + 1 4 (σb ) γ ψ2 b (γ ) βγ + D Hβ + ( β); (31) C γ β C γ β = D = D δ γ β ψ1 + ı(γ ) γ β ψ1 + 1 4 (σb ) γ β ψ1 b + ( β); (32) δ γ β ψ2 + ı(γ ) γ β ψ2 + 1 4 (σb ) γ β ψ2 b + ( β); (33) C β = η β ıη m ((ψm 1 ) ψm 1 ) + 1 4 ηµν (D µ H ν + D ν Hµ ) +(γ ) β δ (1 + ψ1 + (ψ 1 ) ) H + ı 4 ηm δ n (ψ1 mn (ψ1 mn ) ) + X 1 +(σ b ) β 2 ηm η b (ψm 1 + (ψ1 m ) ) 1 8 ǫmnb (ψmn 1 + (ψ1 mn ) ) + X b ; (34) C γ β = D X 1 4 (γ ) β (D H β + D β H ); (35) X b 1 8 (σb ) β (D H β + D β H ); (36) δ γ β (ψ2 ) ı(γ ) γ β (ψ2 ) 1 4 (σb ) γ β (ψ2 b ) + D β δ γ ψ1 + ı(γ ) γ ψ1 + 1 4 (σb ) γ ψ1 b (γ ) β H γ ; (37) C γ β = D δ γ β (ψ1 ) ı(γ ) γ β (ψ1 ) 1 4 (σb ) γ β (ψ1 b ) + D β δ γ ψ2 + ı(γ ) γ ψ2 + 1 4 (σb ) γ ψ2 b (γ ) β H γ ; (38) C γ b C b = H γ b (γ ) γ + D Hb b H = D H γ b b δ γ ψ1 + ı(γ ) γ ψ1 + 1 4 (σ ) γ ψ1 ; (39) ; (40) 9
C γ γ b = D H b b δ γ ψ2 + ı(γ ) γ ψ2 + 1 4 (σ ) γ ψ2 ; (41) With ll the omponents of the nholonomy written in terms of the semiprepotentil fields ψ s nd fields H s, the next step will be to impose some suitble onstrint to write some ψ s fields in terms of H s fields. First of ll, in order to keep rigid supersymmetry, we impose whih by (14) mens T β = (γ ) β, (42) C β = (γ ) β. (43) Using (34) in the expression (43), we obtin (γ ) β = η β ıη m ((ψm 1 ) ψm 1 ) + 1 4 ηµν (D µ H ν + D ν H +(γ ) β δ (1 + ψ1 + (ψ 1 ) ) H + ı 4 ηm δ n µ ) (ψ1 mn (ψ1 mn ) ) + X +(σ b ) β 1 2 ηm η b (ψ 1 m + (ψ1 m ) ) 1 8 ǫmnb (ψ 1 mn + (ψ1 mn ) ) + X b, (44) showing thus tht the rigid onstrint (42) leds to the following three independent equtions ıη m ((ψ 1 m ) ψ 1 m ) + 1 4 ηµν (D µ H ν + D ν Hµ ) = 0, (45) δ (ψ 1 + (ψ 1 ) ) H + ı 4 ηm δ n (ψmn 1 (ψmn) 1 ) + X = 0, (46) 1 2 ηm η b (ψ 1 m + (ψ1 m ) ) 1 8 ǫmnb (ψ 1 mn + (ψ1 mn ) ) + X b = 0. (47) From the equtions (45) nd (46) we hve respetively nd ψ 1 (ψ 1 ) = ı 1 4 η η β (D H β + D β H ) (48) ψb 1 (ψ1 b ) = ıη Hb ı 1 4 η (γ b ) β (D H β + D β H ). (49) From (47) it is found the following two expressions ψ 1 + (ψ 1 ) = 1 16 (σ ) β (D H β + D β H ); (50) 10
ψb 1 + (ψ1 b ) = 1 12 ǫ bkl(σ kl ) β (D H β + D β H ). (51) Thus from (48) nd (50) we obtin ψ 1 = 1 ıη η β + 1 8 4 (σ ) β (D H β + D β H ), (52) nd from (49) nd (51) we hve ψb 1 = ı 1 2 η Hb + 1 ıη (γ b ) β + 1 8 3 ǫ bkl(σ kl ) β (D H β + D β H )(53) In order to obtin ψ 2 nd ψ2 b, the following onstrint is imposed whih leds to T β = 0, (54) C β = 0. (55) Thus using the expression (31) in the ondition (55), we finlly obtin ψ 2 = 1 16 (σ ) β D H β ; (56) ψ 2 b = 1 12 ǫ bde(σ de ) β D H β. (57) Anlyzing the supertorsion omponents (14) nd the super nholonomy omponents, we n relize tht there is no wy to impose ny dditionl onstrint on superspe whih leds to determinte the semi-prepotentils ψ 1, ψ 2 s funtions of other bsi fields. Hene they remin s indeterminte fields. Indeed, it n be seen through the symmetriztion of (31) tht the slr semi-prepotentil ψ 2 never pper in the nholonomy in single wy. 4 The Binhi identities So fr we hve impose some restritions on superspe through some onstrints on the supertorsion omponents. It ws neessry to impose the onstrints (15) nd (16) to leve the spin onnetions s dependent fields of 11
the nholonomy, the onstrint (42) to keep rigid supersymmetry nd (54) to write the semi-prepotentils in terms of the smller set of superfields of the theory. When ll these onstrint re imposed, the geometry of the superspe is restrited, in onsequene the Binhi identities, whih n be written by A, B, C }}+( 1) A(B+C) B, C, A }}+( 1) C(A+B) C, A, B }} = 0, (58) now ontin non trivil informtion. This informtion n be red by the following three equtions A TBC F + ( 1) A(B+C+D) T D BC T F AD + ( 1) A(B+C)1 2 R d +( 1) A(B+C) B TCA F + ( 1)C(B+A)+BD T D CA T F BC ΦdA F BD + ( 1)C(A+B)1 2 R d CA Φ db F +( 1) C(A+B) C TAB F + ( 1) CD T D AB T F CD + 1 2 R d AB Φ F dc = 0; (59) BC D R d BC d ( 1) A(B+C+D) T AD + AR +( 1) C(A+B)+BD TCA D R BD d + BRCA d +( 1) CD TAB D RCD d + C RAB d = 0; (60) where Φ bc D = ( 1) A(B+C+D) T BC D CA D D +( 1) C(A+B)+BD T F AD + A F BC F BD + ( 1) A(B+C) B F CA +( 1) CD TAB F CD + ( 1) C(A+B) C F AB = 0; (61) ( Φ d b 0 0 Φ δ bγ ) = ( η δ d b 0 0 ı 1 2 (σ b) δ γ ). It is well known 9 tht it is suffiient to nlyze the Binhi identities (59) nd (61), sine ll equtions ontined in (60) re identilly stisfied when (59) nd (61) hold. Hene using the onstrints (15), (16), (42) nd (54) in the Binhi identities (59) nd (61) we will be ble to obtin the urvture nd field strength superfield omponents in terms of the smller set of superfields of the theory. 12
5 Symmetries nd semi-prepotentils In order to obtin some informtion on ψ s, let s see the behviour of them under the sle, U(1) nd Lorentz Symmetry, whih re represented respetively by E E = e f 0 E ; E E = e 1 2 f 0 E ; Ē Ē = e 1 2 f 0 Ē (62) E E = eıf 0 E ; E E = eı1 2 f 0 E ; Ē Ē = e ı1 2 f 0 Ē (63) E E = Λ b E b; E E = e1 8 Λb (σ b ) β E β ; Ē Ē = e1 8 Λb (σ b ) β Ē β (64) Let us begin onsidering the sle trnsformtion E E = e f 0 E, E E = e1 2 f 0 E, Ē Ē = e1 2 f 0 Ē. (65) Considering the infinitesiml version of (65) nd the perturbtive expression of (E, E, Ē) round the flt solution, we hve E E = (1 + f 0)( + H m m + H µ D µ + H µ D µ ), (66) E E = (1 + 1 2 f 0)(D + H m m + H µ D µ + H µ D µ ), (67) Ē Ē = (1 + 1 2 f 0)( D + H m m + H µ D µ + H µ D µ ), (68) whih n be written s Hene we hve E E = + (f 0 δ m + H m ) m + H µ D µ + H µ D µ, (69) E E = D + H m m + ( 1 2 f 0δ µ + H µ )D µ + H µ D µ, (70) Ē Ē = D + H m m + H µ D µ + ( 1 2 f 0δ µ + H µ ) D µ. (71) E E = ef 0 E H m H m + f 0 δ m, (72) 13
E E = e1 2 f 0 E H µ H µ + 1 2 f 0δ µ, (73) Ē Ē = e 1 2 f 0 Ē H µ H µ + 1 2 f 0δ µ. (74) Now using the expliit form of H µ nd H µ given in (24) nd (25), we hve E E = e1 2 f 0 E ψ 1 ψ 1 + 1 2 f 0, (75) Ē Ē = e 1 2 f 0 Ē ψ 2 ψ 2 + 1 2 f 0. (76) Thus the sle trnsformtion produes shift on ψ 1 nd ψ 2. However if the two unknown funtions ψ 1 nd ψ 2 re written in terms of the two unknown funtions ˆψ nd ψ s shown below we hve tht ψ 1 = 1 2 ( ˆψ 1 + ı ψ 1 ) ψ 2 = 1 2 ( ˆψ 2 ı ψ 2 ), (77) E E = e1 2 f 0 E 1 2 ( ˆψ 1 + ı ψ 1 ) 1 2 ( ˆψ 1 + ı ψ 1 ) + 1 2 f 0, (78) Ē Ē = e 1 2 f 0 Ē 1 2 ( ˆψ 2 ı ψ 2 ) 1 2 ( ˆψ 2 ı ψ 2 ) + 1 2 f 0, (79) hene it s esy to see tht E E = e1 2 f 0 E ˆψ 1 ˆψ 1 + f 0 ; ı ψ 1 ı ψ 1. (80) Ē Ē = e 1 2 f 0 Ē ˆψ 2 ˆψ 2 + f 0 ; ı ψ 2 ı ψ 2. (81) Therefore the sle trnsformtion produes shift on ˆψ 1 nd ˆψ 2 leving invrint ψ 1 nd ψ 2. Let s onsider now the U(1) trnsformtion E E = eıf E, E E = eı1 2 f E, Ē Ē = e ı1 2 f Ē. (82) 14
Considering the infinitesiml version of (65) nd the perturbtive expression of (E, E, Ē) round the flt solution, we hve E E = (1 + ıf)( + H m m + H µ D µ + H µ D µ ), (83) E E = (1 + ı 1 2 f)(d + H m m + H µ D µ + H µ D µ ), (84) Ē Ē = (1 ı1 2 f)( D + H m m + H µ D µ + H µ D µ ), (85) whih n be written s Hene we hve E E = + (ıfδ m + H m ) m + H µ D µ + H µ D µ, (86) E E = D + H m m + (ı 1 2 fδ µ + H µ )D µ + H µ D µ, (87) Ē Ē = D + H m m + H µ D µ + ( ı 1 2 fδ µ + H µ ) D µ. (88) E E = eıf E H m H m + ıfδ m, (89) E E = eı1 2 f E H µ H µ + ı1 2 fδ µ, (90) Ē Ē = e ı1 2 f Ē H µ H µ ı 1 2 fδ µ. (91) Now using the expliit form of H µ nd H µ given in (24) nd (25), we hve E E = e ı1 2 f E ψ 1 ψ 1 + ı 1 f, (92) 2 Ē Ē = e ı1 2 f Ē ψ 2 ψ 2 ı 1 f. (93) 2 As in the previous se, the U(1) symmetry produes shift on ψ 1 nd ψ 2. Considering gin the redefinition (77), we hve tht E E = eı1 2 f E 1 2 ( ˆψ 1 + ı ψ 1 ) 1 2 ( ˆψ 1 + ı ψ 1 ) + ı 1 f, (94) 2 Ē Ē = e ı1 2 f Ē 1 2 ( ˆψ 2 ı ψ 2 ) 1 2 ( ˆψ 2 ı ψ 2 ) ı 1 f, (95) 2 15
hene it s esy to see tht E E = e ı1 2 f E ˆψ 1 ˆψ 1, ı ψ 1 ı ψ 1 + ıf. (96) Ē Ē = e ı1 2 f Ē ˆψ 2 ˆψ 2, ı ψ 2 ı ψ 2 + ıf. (97) Therefore the U(1) trnsformtion produes shift on ψ s leving invrint ˆψ s. Now let s onsider the Lorentz trnsformtion on the vetor omponent E E E = Λ b E b. (98) Now we onsider the infinitesiml Lorentz trnsformtion Λ b = δ b + ǫ b ; ǫ b = ǫ b (99) ting on the perturbtive expression of E round the flt solution Hene E E = (δ b + ǫ b )( b + Hb m m + H µ b D µ + H µ b D µ ) (100) E E = + (ǫ m + H m ) m + H µ D µ + H µ D µ (101) E E = Λ b E b H m H m + ǫ m, (102) showing thus tht the Lorentz trnsformtion produes shift on the grviton. Now onsidering the Lorentz trnsformtion on the spinoril omponents E E = eı1 8 Λb (σ b ) β E β, (103) Ē Ē = e ı1 8 Λb (σ b ) β Ē β. (104) Considering the infinitesiml trnsformtion on the perturbtive expression of E β nd Ēβ round the flt solution, we hve E E = δ β Ē Ē = δ β + ı 1 8 Λb (σ b ) β D β + H m β m + H µ β D µ + H µ β D µ, (105) ı 1 8 Λb (σ b ) β D β + H m β m + H µ β D µ + H µ β D µ. (106) 16
Thus E E = D + H m m + H µ + ı1 8 Λb (σ b ) µ D µ + H µ D µ (107) Hene Ē Ē = D + H m m + H µ D µ + H µ ı1 8 Λb (σ b ) µ D µ. (108) E E = eı1 8 Λb (σ b ) β E β H µ H µ + ı1 8 Λb (σ b ) µ (109) Ē Ē = e ı1 8 Λb (σ b ) β Ē β H µ H µ ı 1 8 Λb (σ b ) µ. (110) Now using the expliit form of H µ nd H µ given in (24) nd (25), we hve E E = eı1 8 Λb (σ b ) β E β ψ 1 b ψ1 b + ı1 2 Λ b, (111) Ē Ē = eı1 8 Λb (σ b ) β Ē β ψ 2 b ψ2 b ı1 2 Λ b (112) Thus the Lorentz trnsformtion produes shift on ψ 1 b nd ψ2 b. However if them re written in terms of the funtions ˆψ b nd ψ b s shown below s before we obtin ψ 1 b = 1 2 ( ˆψ 1 b + ı ψ 1 b ) ψ 2 b = 1 2 ( ˆψ 2 b ı ψ 2 b), (113) E E = eı1 8 Λb (σ b ) β E β ˆψ 1 b ˆψ 1 b ; ψ1 b ψ 1 b + Λ b.(114) Ē Ē = e ı1 8 Λb (σ b ) β Ē β ˆψ 2 b ˆψ 2 b ; ψ2 b ψ 2 b + Λ b (115) Therefore the Lorentz trnsformtion produes shift on ψ b leving invrint ˆψ b. 17
6 Conlusions Using the notion of superspe, geometril pproh to five dimensionl N = 1 supergrvity theory ws disussed in detil. There ws not used the onventionl representtion for Grssmnn vribles, bsed in spinors obeying pseudo-mjorn relity ondition. Insted, the unonventionl representtion for the Grssmnn vribles (θ µ, θ µ ) given in 1 for SUSY D = 5, N = 1 representtion, ws suessfully extended for supergrvity theory, dispensing with the use of SU(2) index to the spinor oordintes of the superspe. The omponents of the torsion nd urvture superfield were found through the super (nti)-ommuttor of the superspe supergrvity ovrint derivtive, finding these superfields s funtion of both the nholonomy nd the spin onnetion. Imposing suitble onstrints on superspe through some super-torsion omponents, the spin onnetion ws written in terms of the nholonomy, eliminting thus the spin onnetion s un independent field. Tking perturbtion round the flt superspe, the omponents of the superspe derivtive were found s the sum of the rigid (SUSY) prt nd perturbtive terms. These perturbtive terms, rising from the super vielbein omponents, were written in terms of funtions when the vetoril omponent of the superspe derivtive ws onsidered. These funtions were two vetoril omponent superfield nd its supersymmetri prtner, nmely, the grviton nd grvitino t quntum level. On the other hnd, when the spinoril omponent of the superspe derivtive ws onsidered, it ws possible to write the perturbtive terms s liner ombintion of fundmentl geometri objets of SUSY N = 1, introduing thus the semiprepotentil of the theory. Using the perturbtive version (linerized theory) of the superspe derivtive, it ws possible to find ll omponents of the super nholonomy in terms of the simpler set of superfields, nmely, the grviton, the grvitino nd semi-prepotentils. Demnding onsistene with rigid SUSY nd one dditionl suitble onstrin on superspe through super-torsion omponent, ll semi-prepotentils were written in terms of the smller set of superfields of the theory, leving the two slr semi-prepotentils ψ 1 nd ψ 2 super- 18
fields. Using the Binhi identities, three set of equtions written in terms of superfields were found. Two of these set ontining enough informtion to determinte the urvture nd field strength superfield omponents in terms of the smller set of superfields of the theory. It ws explined in detil the behviour of the semi-prepotentils under the tion of the sle, U(1) nd Lorentz Symmetry. It ws found tht the sle trnsformtion produes shift on the rel prt of the slr semiprepotentils, leving invrint their imginry prt. The opposite behviour ws found when the U(1) trnsformtion ws onsidered. On the other hnd, when the Lorentz trnsformtion ws onsidered, it ws found tht its tion produes shift on the imginry prt of the two vetoril indies semiprepotentils ψb 1 nd ψ2 b, leving invrint the rel prt. Appendix hene The tion of the genertor M b on spinors Ψ nd vetors X is given by M b, Ψ = ı 2 (σ b) γ Ψ γ ; M b, X = η X b η b X, (116) M b, Φ d = ı 2 (σ b) γ Φ γd + η Φ bd η b Φ d + η d Φ b η db Φ b. (117) For instne let us onsider {, β } = C C β E C + {E, Υ β } + {Υ, E β } + {Υ, Υ β }, (118) thus omputing eh ntiommuttor {E, Υ β } = 1 2 (E ωβ d )Md + ı 4 ω β d (σ d ) γ E γ + E Γ β Z, (119) {Υ, Υ β } = 1 4 {ω b M b, ω e βd M d e } + ı 4 ω b (σ b ) γ β Γ γz + ı 4 ω e βd (σ d e ) γ Γ γ Z. (120) 19
Using (119) nd (120) in (118) we hve with, β = C Σ β = = ω b β C E C + 1 2 (E ωβ d )Md {ω b M b, ω e βd M d ω d β + 1 2 (E βω d )M d + ı 4 ω β d (σ d ) γ E γ + E Γ β Z + ı 4 ω d (σ d) γ β E γ + E β Γ Z + 1 4 {ω b M b, ω e βd M d e } + ı 4 ω b (σ b ) γ β Γ γz e } = 4ω b ω βbdm d + ı 2 + ı 4 ω e βd (σ d e ) γ Γ γz, (121) M b, M d + ω b M b, ω βd M d + ωβ b M b, ω d M d (σb ) γ ω β b + (σ b ) γ β ω b ωγd M d (122) nd E C = C 1 2 ω d C M d Γ C Z (123) in (121), we finlly obtin {, β } = Cβ C C + ı 4 + 1 2 C β C ωc d + 1 2 E ωβ d ω d (σ d) γ + 1 2 E βω d β + ω β d (σ d) γ γ Md + ω b ωβb d + C C β Γ C + E Γ β + E β Γ Z. (124) Aknowledgments The uthor thnks to Dr. S. J. Gtes, Jr. for vluble disussions nd suggestions, nd the Center for String & Prtile Theory t the University of Mrylnd for the hospitlity nd finnil support. This work ws prtilly supported by Desrrollo Profesorl de l Universidd Simón Bolívr. 20
Referenes 1 S. Jmes Gtes, Jr., Lubn Rn, Russ.Phys.J. 45 (2002) 682-689; Izv.Vuz.Fiz. 2002N7 (2002) 35; rxiv:hep-th/0208105v1 2 Sergei M. Kuzenko, Nul.Phys.B745:176-207,2006; rxiv:hep-th/0601177v3 3 Sergei M. Kuzenko, Gbriele Trtglino-Mzzuhelli, Nul.Phys.B785:34-73,2007; rxiv:0704.1185v3 hep-th. 4 Jun M Mlden, Adv.Theor.Mth.Phys.2:231-252,1998. 5 Sergei M. Kuzenko, Gbriele Trtglino- Mzzuhelli,Phys.Lett.B661:42-51,2008; rxiv:0710.3440v4 hep-th 6 Sergei M. Kuzenko, Gbriele Trtglino- Mzzuhelli,JHEP0802:004,2008; rxiv:0712.3102v2 hep-th. 7 Sergei M. Kuzenko, Gbriele Trtglino-Mzzuhelli, JHEP0804:032,2008; rxiv:0802.3953v3 hep-th. 8 W. Siegel nd S. J. Gtes, Jr., Nul.Phys.B147 (1979) 77. 9 N. Drgon, Z. Phys. C 2, 29 (1979). 21