Geometric coupling of scalar multiplets to D=4, N=1 pure supergravity

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1 Interntonl eserh Journl of Engneerng nd Tehnology (IJET) e-issn: olue: 0 Issue: 09 De-05 p-issn: eoetr ouplng of slr ultplets to D=, N= pure supergrvty Polo D S,,3 Adunt Professor, Deprtent of Phlosophy, Eduton nd Psyhology, Unversty of eron, eron, Itly Professor, gher Insttute of elgous Sene, Bolzno, Itly 3 Meer, ISEM, Insttute for Sentf Methodology, Plero, Itly Astrt - In ths pper we onsder the ouplng of the slr ultplets (ultplets of Wess-uno) to ( z, z ); D=, N= pure supergrvty. We use the geoetr. () pproh, tht s ll felds re onsdered s superfors n superspe nd we use the onepts of We ntrodue lso for lter use the hrl proetons of spnors nd : supersyetry, superspe nd rheono prnple. The Bnh denttes re nlyzed nd resolved, 5 5 ; ; endng wth the Bnh dentty of grvtno. ; (5) Key Words: Supergrvty, Slr (Wess-uno) Multplets, Supersyetry, Superspe, heono Prnple, eoetr Couplng, Dfferentl eoetry.. INTODUCTION Wth N= supergrvty (N s the nuer of supersyetry genertors) n the geoetr pproh we work wth the veren, the grvtno nd the spn onneton. Fro the prtle pont of vew, nd desre the N= grvttonl ultplet [, 3/] [,]. The n gol of ths pper s the ouplng of suh ultplet to n ultplets of Wess-uno: [/, 0 +, 0 - ], () desred y the set of 0-fors: [, A, B ], ( =,.., n). () A nd B re rel slr nd rel pseudo-slr respetvely, s Morn spnor. Fro phenoenologl pont of vew, the slr ultplets ontn qurks, leptons nd ggs prtle together wth ther superprtners,.e. squrks, sleptons, ggsno. It s possle to ntrodue set of oplex felds z : def z A B ; z ( z ) A B, (3) nd we n onsder the s oordntes of n- densonl oplex nfold M wth Kählern struture. On M the Kähler potentl s ntrodued: T 5 ; 5 ; C 0 ( ) ; (6) 5 5 ; ; ; (7) T 5 ; 5 ; 0 ( ) ( ) 0 ( 5 5 ) 0 ( C ; () ) ; (9) 5 5 ( ) 0 ( ) 0 ( ) ; (0) ( ) 0 ( 5 0 ( 5 ( ) ) ; () ). (). COUPLIN OF SCALA MULTIPLETS TO PUE SUPEAITY The ouplng of slr ultplets to pure supergrvty orresponds to the onstruton of ross-seton of the fer undle B( /, M ), whh hs the N= superspe / s support spe nd the Kähler nfold M s fer [3]. The oordnte z s superfeld []: z z ( x, ), (3) therefore t every pont ( x, ) / of the support t s ssoted pont z M of the fer. Developng the ss (, ), we fnd: dz n 05, IJET.NET- All ghts eserved Pge 7

2 Interntonl eserh Journl of Engneerng nd Tehnology (IJET) e-issn: olue: 0 Issue: 09 De-05 p-issn: dz ; () (D d ), dz. (5) d, () In dong tht we posed the rheonoy, wrtng the out oponent of dz s, tht s the feld of spn /. Therefore s vetor feld nd left-hnded spnor feld. The ton of the Kähler trnsforton on feron felds n e defned s hrl rotton [5]. The Kähler onneton s defned s: Q ( dz dz ). (6) The urvture of the Kähler onneton s the -for K defned s: K dq g dz dz. (7) Usng reltons (,5) nd defnng the qunttes: K g [ ], () T g, (9) g, (0) C ( ) g we n wrte:, () K K T. () The exteror dervtves of felds of tter z e re the orrespondent of urvtures, e of the supergrvty felds. More presely, t s helpful to defne s urvture of the ovrnt dervtve, whh s ovrnt oth wth respet to Lorentz trnsfortons, nd wth respet to Kähler trnsfortons nd to oordntes trnsfortons on the Kähler nfold. In generl, ll ovrnt dervtves of ferons ontn the Kähler onneton, for eng ovrnt under Kähler trnsfortons. Therefore the set of supergrvty urvtures of Wess-uno ultplets re gven y: D, (3), (5) def ( z) dz, (6) def ( ), (7) wth:, () d Q k d dz Q. (9) k 3. BIANCI IDENTITIES OF SUPEAITY COUPLED TO SCALA MULTIPLETS The Bnh denttes of pure supergrvty oupled to ultplets of Wess-uno re gven y [6]: D 0, (30) D 0, (3) K 0, (3) d d z 0, (33) k k K d z d z 0. (3). SOLUTIONS OF BIANCI IDENTITIES AND AUXILIAY FIELDS In Bnh denttes (30-3) we nsert the followng nfortons: ) the equton of the torson: 0 ; (35) ) the rheono ondton: 05, IJET.NET- All ghts eserved Pge 79

3 Interntonl eserh Journl of Engneerng nd Tehnology (IJET) e-issn: olue: 0 Issue: 09 De-05 p-issn: dz. (36) The frst ondton s knet onstrnt tht n e posed redefnng the spn onneton, whle the rheono ondton defnes who re the supersyetr prtners of z. To gve these two ondtons s equvlent to fx the supersyetry trnsfortons [7]. They set the supersyetry trnsforton lws of e z : z, (37). (3) The felds e whh pper n reltons (,5) re the oponents long the verens of undelne tht s not equl to dz e d z. We z, s would hppen f the theory ws forulted only on spe-te. Indeed, proetng relton () long the dfferentls of spe-te oordntes, we get: z z. (39) In the oponent pproh s lso lled superovrnt dervtve of z. For solvng the Bnh denttes (30-3) we ust nsert the ost generl rheono pretrztons of urvtures e, whh re gven y: d d K, (0). () For pure supergrvty, vlues of hve een fxed s [7]: K pq d K, K,, 0, () pqrs rs 5 d δ [p d q]st st 5. (3) Wth the presene of tter felds, vlues of prevous tensors y lso depend fro the tter felds, euse we n even use for ther onstruton the nner felds nd [6]. It s therefore pproprte to solve the Bnh denttes of the supergrvttonl off-shell ultplet,.e. wthout gvng the explt for of these tensors. Then, ntrodung the new generl preterztons n the oplete Bnh denttes of pure supergrvty + Wess-uno ultplets (30-3), we otn the explt for of, K,, n ters of the tter felds nd supergrvty felds. Prelnrly we solve the grvttonl off-shell Bnh denttes. Insertng equton (35) n (30) nd onsderng the nellton of ters, we otn the se soluton of pure supergrvty [7]. The setor gves: K 0. () Ths equton reks n two prts: { } 0, (5) [ ] K. (6) Eq. (5) s solvle deoposng n oplete Dr ss: ( ) ( ) / ( ) ( ) 5 / 5 d The generl soluton s gven y: A d. (7) 5 5 u S 5 u, () wth A nd A two xl vetors, S nd slr nd pseudo-slr respetvely. eplng Eq. () n (6), we get: K d 5 S d. (9) The setor of (30) rngs to: 0 0, (50) wth:. (5) Wrtng, n oplete generlty:, (5) nd usng the deoposton n rredule representtons []: () (), (53) 05, IJET.NET- All ghts eserved Pge 0

4 Interntonl eserh Journl of Engneerng nd Tehnology (IJET) e-issn: olue: 0 Issue: 09 De-05 p-issn: we get: () [ () ] wth Morn spnor. Introdung the oplex feld: (), (5), (55) S S, (56) nd usng the hrl notton, we otn: 0 ; (57) A S ; (5) S d d ] ( [ [ ] ) S d d. (59) The se of pure supergrvty s reoverle fro the prevous f: A S 0 0. (60) It follows tht n the theory oupled to tter the uxlry felds A,, S nd ust e dentfed wth onvenent funtons of the tter felds z nd. In pure supergrvty the Bnh denttes re nvrnt wth respet to the sle trnsfortons [,7]:, (6) w, (6) w. (63) These trnsfortons re extended n onsstent wy to the tter felds y plng: wth rel onstnt preter. Nng w ( ) the slng power of every feld, we n rewrte reltons (6-63) s: w ( ), w ( ), w( ) 0. (66) Fro the rheono ondtons (,5) we get: w ( z ) w( ) w( ) w( ), (67) w( z ) w( ) w( ) w( ). (6) On the other hnd, sne z re oordntes of the ennn nfold of slrs, they re sle nvrnt. Ths leds to: w ( z ) 0 w( ), (69) w( ). (70) The renng vlues re: w ( A ) w( ), (7) w ( S), (7) w ( ). (73) 5. BIANCI IDENTITY OF AITINO The soluton of Bnh denttes s opleted wth the nlyss of grvtno, the supersyetr prtner of grvton. The preterzton of the grvtno urvture s gven y relton (5). We prelnrly oserve tht n (5) we n put 0, sne the grvtno urvture ontns the Kähler onneton Q, whh ws not n the defnton of n sene of tter. The ddtonl ter Q s -for whose oponent long n e dentfed wth. Therefore there s no restrton puttng 0. We n so rewrte relton (5) s: A S. (7) The orrespondng Bnh dentty s: z z, (6), (65) D g dz dz 0. (75) 05, IJET.NET- All ghts eserved Pge

5 Interntonl eserh Journl of Engneerng nd Tehnology (IJET) e-issn: olue: 0 Issue: 09 De-05 p-issn: Introdung n (75) the pretrztons (7) nd (59) for, n setor we fnd: A S S d ( ) ( ) 0 ( S d ) g. (76) Consderng the vldty of Ferz denttes [,7]: 0, (77) 0, (7) nd tht t holds: 0, (79) for self-dulty, we note tht the feld S does not gve ontruton. Consderng tht: ( ), (0) ( ), () ( ), () ( ), (3) fro relton (76) we get: ( A ) g - ( ) - () Therefore: ( A ) ( ) 0. () () fro whh t s: A g () ( g ) 0, (5). (6) Consderng tht the expresson on the rght sde of relton (6) s the only oet wth orret weght sle, to whh y e dentfed the felds A nd, they re on ft not ndependent. In prtulr we hoose A 0. We onsder now the setor of the se equton: ( (0,) (0,) ) ( S S ) ( ) g ( ) 0, (7) where wth the syol (0,) we en the oeffent long of the exteror dervtve. The nellton of the oeffent of the urrent l rngs to: S 0, () wth: p' ; ( ). (9) In ths se t s: S S S 0, ( p ' ) (90) nd ths s equvlent to wrte: ( S e e S) 0, (9) whh ples: ( ) 0 e S e S f ( z), (9) wth f (z) n rtrry nlytl funton. Therefore: S f ( z) e. (93) On the other hnd, the S feld ust e pure gnry for the Morn ondtons, then the nlyt funton f (z) ust e equl to tes onstnt, denoted y e. The fnl soluton s therefore: S ee. (9) In the l setor we hve: 05, IJET.NET- All ghts eserved Pge

6 Interntonl eserh Journl of Engneerng nd Tehnology (IJET) e-issn: olue: 0 Issue: 09 De-05 p-issn: l l ( ( 0,) 6 ) l g l 3 - l l = 0. (95) l Multplyng oth sdes of (95) for nd onsderng the reltons: l d l 5 δ [ [l ] ] δ l, (96) d, (97) we get: ( ( 0,) ) 3 g 0. (9) In the setor of urrent wth one ndex of Eq. (75) we fnlly hve: l e ( ( 0,) l ) ( ) e l l g 0. (99) k = d + dz Q. (03) k The setor of the Bnh dentty (3) onnets, whh s ssued s funon of the slr felds ( z, z ), to the uxlry feld of supergrvty S: ( + ) = ( S S k d T d ) k p T 0. (0) In the setor, Eq. (03) eoes: = S, (05) tht s: = S, (06) Multplyng oth sdes of (99) for, we rrve to the equton of oton of grvtno. The Bnh dentty (33) ssoted to the slr feld z n e used for the deternton of the pretrzton of the ovrnt dervtve. We wrte, n full generlty: +, (00) where the oeffents.. depend y the feld. Consderng the exteror dervtve of the rheono pretrzton () nd nsertng relton (00), fro setor we hve: - = 0, (0) whh rngs to: ; = 0; = free. (0) The free funton s the uxlry feld of the slr ultplet. Its Kähler weght s p = /, therefore the explt defnton of ts ovrnt dervtve s: wth: = +. (07) Ths dfferentl equton s resolvle through the nstz: = wth: g g g e, (0). (09) Insertng (0) n (07) nd reeerng (9), the equton s stsfed for: e e; p. (0) p It hs een therefore deterned, y the nlyss of Bnh denttes, the dependene of the grvtno feld nd of the uxlry feld of spn / fro the Kähler potentl ( z, z) : S eexp( ) ; () 05, IJET.NET- All ghts eserved Pge 3

7 Interntonl eserh Journl of Engneerng nd Tehnology (IJET) e-issn: olue: 0 Issue: 09 De-05 p-issn: = e( g )exp( ). () 6. CONCLUSIONS In ths pper we hve onsdered n geoetr pproh the ouplng of the slr ultplets of Wess- uno to D=, N= pure supergrvty. Supergrvty s the effetve theory of superstrng theory [-]. The geoetr pproh onsders ll felds s superfors n superspe. We hve used the onepts of supersyetry, superspe, rheono prnple, Bnh denttes. The found results llow the onstruton of the oplete ton of pure supergrvty oupled to slr ultplets n thetlly very elegnt nd generl wy. EFEENCES [] P. D S, Supergrvtà nel superspzo: pnor generle e nls ten, o, Itly: Arne Edtre, 0. [] F. Mndl,. Show, Quntu feld theory, UK: John Wley & Sons, 9. [3] P. D S, elevnt tools n uldng supergrvty theores, Integrted Journl of Brtsh, vol., ssue 3, pp. -5, My-June 05. [] P. D S, Aout the portne of supersyetry nd superspe for supergrvty, Interntonl Journl of Innovtve Sene, Engneerng nd Tehnology (IJISET), vol., ssue, pp , Aprl 05. [5] P. D S, Kähler nfolds s trget spe of supergrvty theores: hrtersts for D=, N= theory, Interntonl Journl of Current eserh, vol. 7, ssue, pp , Jn 05. [6] P. D S, heono prnple, Bnh denttes nd supergrvty, Interntonl Journl of Innovtve Sene, Engneerng nd Tehnology (IJISET), vol., ssue 5, pp , My 05. [7] P. D S, eoetr onstruton of D=, N= pure supergrvty, Interntonl eserh Journl of Engneerng nd Tehnology (IJET), vol., ssue, pp. 6-0, July 05. [] D.. Freedn, A. n Proeyen, Supergrvty, Crdge: Crdge Unversty Press, 0. [9] M. B. reen, J.. Shwrz, E. Wtten, Superstrng Theory: 5 th Annversry Edton, Crdge: Crdge Monogrphs on Mthetl Physs, Crdge Unversty Press, 0. [0] P. D S, Extree Physs nd Infortonl / Coputtonl Lts, Journl of Physs, Conferene Seres, vol. 306, p. 0067, 0. [] P. D S, Extng Peulrtes of the Extree Physs, Journl of Physs, Conferene Seres, vol., ssue, p. 006, 03. BIOAPY Polo D S s urrently Adunt Professor y the Unversty of eron (Itly), Professor t ISS, Bolzno (Itly) nd eer of ISEM, Plero (Itly). e otned st level Lure n Metphyss, nd level Lure n Theoretl Physs, PhD n Mthetl Modellng ppled to Nno-Bo-Tehnology. e nterested n Clssl Quntu eltvst Nnophyss, Plnk Sle Physs, Supergrvty, Quntu eltvst Inforton, Mnd Phlosophy, Quntu eltvst Eonophyss, Phlosophy of Sene, Sene Eduton. e wrote 7 pultons, s revewer of thets ooks, revewer of nterntonl ournls, Awrds otned, nluded n Who's Who n the World 05 nd 06. e s eer of 6 nterntonl sentf soetes nd eer of Interntonl Advsory/ Edtorl_Bords. p olo.ds@gl.o 05, IJET.NET- All ghts eserved Pge