Chern Simons D = 3, N = 6 superfield theory

Transcription

1 Physics Letters B 66 28) Chern Simons D = 3 N = 6 superfield theory B.M. Zupni Bogoliuov Lortory of Theoreticl Physics JINR Dun Moscow Region 498 Russi Received 29 Novemer 27; received in revised form 3 Decemer 27; ccepted 3 Decemer 27 Aville online 2 Jnury 28 Editor: M. Cvetič Astrct We construct the D = 3 N = 5 hrmonic superspce using the SO5)/U) U) hrmonics. Three guge hrmonic superfields stisfy the offshell constrints of the Grssmnn nd hrmonic nlyticities. The corresponding component supermultiplet contins the guge field A m nd n infinite numer of osonic nd fermionic fields with the SO5) vector indices rising from decompositions of guge superfields in hrmonics nd Grssmnn coordintes. The non-aelin superfield Chern Simons ction is invrint with respect to the N = 6 superconforml supersymmetry relized on the N = 5 superfields. The component Lgrngin contins the Chern Simons interction of A m nd n infinite numer of iliner nd triliner interctions of uxiliry fields. The fermionic nd osonic uxiliry fields from the infinite N = 5 multiplet vnish on-shell. 28 Elsevier B.V. All rights reserved. PACS:.3.P;.5.T Keywords: Hrmonic superspce; Grssmnn nd hrmonic nlyticities; Chern Simons theory. Introduction Supersymmetric extensions of the three-dimensionl Chern Simons CS) theory were discussed in Refs. ].TheN = CS theory of the spinor guge superfield 2] ws constructed in the D = 3 N = superspce with rel coordintes x m where m = 2 is the 3D vector index nd = 2is the SL2R) spinor index. The N = CS ction cn e interpreted s the superspce integrl of the Chern Simons superform da A3 in the frmewor of our theory of superfield integrl forms 3 5]. The Aelin N = 2 CS ction ws first constructed in the D = 3 N = superspce ]. The non-aelin N = 2 CS ction ws considered in the D = 3 N = 2 superspce in terms of the Hermitin superfield Vx m ) prepotentil) 39] where nd re the complex conjugted N = 2 spinor coordintes. The corresponding component-field Lgrngin includes the osonic CS term nd the iliner terms with fermionic nd sclr fields without derivtives. The unusul dulized E-mil ddress: zupni@theor.jinr.ru. form of the N = 2 CS Lgrngin contins the second vector field insted of the sclr field ]. The D = 3 N = 3 CS theory ws first nlyzed y the hrmonic-superspce method 67]. Note tht the off-shell N = 3 nd N = 4 vector supermultiplets re identicl 3]; however the superfield CS ction is invrint with respect to the N = 3 supersymmetry only. Nevertheless the N = 3CS equtions of motion re covrint under the 4th supersymmetry. The field-component form of the N = 3 CS Lgrngin ws studied in 8]. The off-shell D = 3 N = 6 SYM theory rises y dimensionl reduction of the D = 4 N = 3 SYM theory in the SU3)/U) U) hrmonic superspce 2]. Three sic prepotentils of the D = 3 N = 6 guge theory contin n infinite numer of uxiliry fields with the SU3) indices nd coupling constnt of this model hs dimension /2. We do not now how to construct the D = 3 N = 6 CS theory from these guge hrmonic superfields. Note tht the SU3)/U) U) nlytic hrmonic superspce hs the integrtion mesure of dimension in the cse D = 3. In this Letter we consider the simple D = 3 N = 5 superspce which cnnot e otined y dimensionl reduction /$ see front mtter 28 Elsevier B.V. All rights reserved. doi:.6/j.physlet

2 B.M. Zupni / Physics Letters B 66 28) of the even coordinte from ny 4D superspce. The corresponding hrmonic superspce using the SO5)/U) U) hrmonics is discussed in Section 2. The Grssmnn-nlytic D = 3 N = 5 superfields depend on 6 spinor coordintes so the nlytic-superspce integrl mesure hs zero dimension. It is shown tht five hrmonic derivtives preserve the Grssmnn nlyticity. In Section 3 we consider five sic guge superfields in the D = 3 N = 5 nlytic superspce nd the corresponding guge group with nlytic superfield prmeters. To simplify the superfield formlism of the theory one cn introduce dditionl off-shell hrmonic-nlyticity constrints for the guge-group prmeters nd guge superfields. These hrmonic constrints yield dditionl relity conditions for components of superfields. In this convenient representtion two guge superfields vnish nd we use only three sic guge superfields prepotentils). The Chern Simons superfield ction cn e constructed from these D = 3 N = 5 guge superfields. We show tht this CS superfield ction is invrint with respect to the D = 3 N = 6 superconforml supersymmetry trnsformtions relized on the N = 5 superfields. The superfield guge equtions of motion hve only pure guge solutions y nlogy with the N = 2 3 superfield CS theories. The field-component structure of our D = 3 N = 6 Chern Simons model is nlyzed in Section 4. In the Aelin cse the sic guge superfield includes the guge field A m nd the fermion field ψ in the SO5) invrint sector nd n infinite numer of fermionic nd osonic fields with the SO5) vector or tensor indices. The component Lgrngin contins the Chern Simons term for A m nd the simple iliner nd triliner interctions of other fermionic nd osonic fields. The field strength of the guge field nd ll other fields vnish on-shell. The preliminry version of the D = 3 N = 5 hrmonicsuperspce guge theory without the hrmonic-nlyticity conditions ws presented in our tl 4]. This model descries the interction of the N = 5 Chern Simons multiplet with some unusul mtter fields. 2. D = 3 N = 5 hrmonic superspce The CB-representtion of the D = 3 N = 5 superspce uses three rel even coordintes x m m = 2) nd five twocomponent odd coordintes where = 2 is the spinor index of the group SL2 R) nd = is the vector index of the utomorphism group SO5). We use the rel trceless or symmetric representtions of the 3D g m mtrices γ m ) = ε ρ γ m ) ρ = γ m) γ m ) γ m) ργ = δ ρ δγ + δρ δγ γ m γ n ) = γ m) ρ γ n) ρ = γ m) ρ γ n ) ρ = η mn δ + ε mnpγ p ). 2.) One cn consider the ispinor representtion of the 3D coordintes nd derivtives x = γ m ) x m = γ m) m. The N = 5 CB spinor derivtives re D = + i = δ δ. The N = 5 supersymmetry trnsformtions re δ ɛ x m = iɛ γ m ) δ ɛ = ɛ. 2.2) 2.3) 2.4) We shll use the SO5)/U) U) vector hrmonics defined vi the components of the rel orthogonl 5 5mtrix U K = U ) U ) U ) U ) U ) ) 2.5) where is the SO5) vector index nd the index K = corresponds to given comintions of the U) U) chrges. These hrmonics stisfy the following conditions: U K U L = gkl = g LK g KL U K U L = δ g 5 = g 24 = g 33 = g = g 2 = =g 45 = g 55 = 2.6) where g LK is the ntidigonl symmetric constnt metric in the spce of chrged indices. Let us introduce the following hrmonic derivtives: KL = U K glm 2.7) U M U L gkm U M = LK IJ KL] = g JK IL + g IL JK g IK JL g JL IK 2.8) which stisfy the commuttion reltions of the Lie lger SO5). We will minly use the five hrmonic derivtives nd the corresponding U) U) nottion 2 = 2) = U ) / U ) 3 = ) = U ) / U ) U ) 23 = ) = U ) / U ) 4 = 2) = U ) 25 = 2) = U ) / U ) / U ) U ) / U ) U ) / U ) / U ) U ) / U ) U ) / U ). The Crtn chrges of two U) groups re descried y the neutrl hrmonic derivtives = U pq) = pu pq) 2 = U pq) = qu pq). 2.9) The hrmonic integrl hs the following simple properties: du = du U pq) U r s) = 5 δ δ pr δ qs. 2.) Let us define the hrmonic projections of the N = 5 Grssmnn coordintes where η mn = dig ) is the 3D Minowsi metric nd ε mnp is the ntisymmetric symol. K = U K = ) ) ) ) ) ). 2.)

3 256 B.M. Zupni / Physics Letters B 66 28) Using the hrmonic-superspce method one cn define the coordintes of the N = 5 nlytic superspce with only three spinor coordintes ζ = xa m ) ) ) x m A = xm + i ) γ m ) + i ) γ m ) δ ɛ x m A = iɛ) γ m ) ) 2iɛ ) γ m ) 2iɛ ) γ m ) 2.2) 2.3) where ɛ K = ɛ U K re the hrmonic projections of the supersymmetry prmeters. Generl superfields in the nlytic coordintes depend lso on dditionl spinor coordintes ) nd ). The hrmonized prtil spinor derivtives re ) = / ) ) = / ) ) = / ) ) = / ) ) = / ). 2.4) Ordinry complex conjugtion connects hrmonics of the opposite chrges U ) = U ) U ) = U ) U ) = U ). 2.5) We use the comined conjugtion in the hrmonic superspce U pq) = U p q) pq) = p q) xm A = xa m pq) sr) ) s r) = p q) fx A ) = fx A ) 2.6) where f is the ordinry complex conjugtion. The nlytic superspce is rel with respect to the comined conjugtion. One cn define the comined conjugtion for the hrmonic derivtives of superfields ±) A ) = ± ) Ã ± ) A ) = ±) Ã ±2) A ) = ±2) Ã ±2) A ) = 2) Ã. 2.7) The nlytic-superspce integrl mesure contins prtil spinor derivtives 2.4) dμ 4) = 64 du d3 x A ) ) 2 ) ) 2 ) ) 2 = du d 3 x A d 6 4) d 6 4) )) 2 ) ) 2 ) ) 2 =. It is pure imginry dμ 4) ) = dμ 4) d 6 4)) = d 6 4). 2.8) 2.9) The hrmonic derivtives of the nlytic sis commute with the genertors of the N = 5 supersymmetry D ) = ) i ) ) ) ) + ) ) D ) = ) i ) + ) ) = D )) D 2) = D ) D )] = 2) 2i ) ) ) ) ) ) ) + ) ) D 2) = 2) + ) ) ) ) D 2) = D 2)) = 2) + ) ) ) ). Note tht hrmonic derivtives D ±2) chnge the second U) chrge; these opertors do not ct on xa m in distinction with other hrmonic derivtives. It is useful to define the ABrepresenttion of the U) chrge opertors D Apq) = pa pq) D 2 Apq) = qa pq) where A pq) is n ritrry hrmonic superfield in AB. The spinor derivtives in the nlytic sis re D ) D ) = ) + 2i ) = ) + 2i ) 2.2) D ) = ) + i ) D ) = ) D ) = ). 2.2) The nlytic superfields Λζ U) depend on hrmonics nd the nlytic coordintes nd stisfy the Grssmnn nlyticity conditions G: D ±) Λ =. The ction of the five hrmonic derivtives D ±) D 2) D ±2) preserves this G-nlyticity. 3. Chern Simons model in N = 5 nlytic superspce 2.22) 2.23) Using the hrmonic-superspce method 2] we introduce the D = 3 N = 5 nlytic mtrix guge prepotentils corresponding to the five hrmonic derivtives 2.23) V pq) ζ U) = V ) V ) V 2) V ±2)] V ) ) = V ) V 2) ) = V 2) V 2) = V 2)] 3.) where the Hermitin conjugtion includes conjugtion of mtrix elements nd trnsposition. The infinitesiml guge trnsformtions of these prepotentils depends on the nlytic nti-hermitin mtrix guge prmeter Λ δ Λ V ±) = D ±) Λ + V ±) Λ ] δ Λ V 2) = D 2) Λ + V 2) Λ ] δ Λ V ±2) = D ±2) Λ + V ±2) Λ ]. 3.2)

4 B.M. Zupni / Physics Letters B 66 28) We shll consider the restricted guge supergroup using the supersymmetry-preserving hrmonic H ) nlyticity constrints on the guge superfield prmeters H : D ±2) Λ =. 3.3) These constrins yield dditionl relity conditions for the component guge prmeters. We use in this Letter the hrmonic-nlyticity constrints on the guge prepotentils H 2: V ±2) = D 2) V ) = V ) D 2) V ) = 3.4) nd the conjugted constrints comined with reltions 3.).It is evident tht the G- nd H -nlyticities of the prepotentils re preserved y the restricted guge trnsformtions 3.3). Now we hve only three guge prepotentils in complete nlogy with the lgeric structure of the guge theory in the N = 3 D = 4 hrmonic superspce 2]. The superfield CS ction cn e constructed in terms of these H -constrined guge superfields S = 2i 3g 2 { dμ 4) Tr V 2 D ) V ) + V D 2) V ) + V D ) V 2) + V 2 V ) V )] 2 V 2) V 2) } 3.5) where g is the dimensionless CS coupling constnt. The corresponding superfield guge equtions of motion hve the following form: F 3 = D ) V 2) D 2) V ) + V ) V 2)] = F 3 = D ) V 2) D 2) V ) + V ) V 2)] = 3.6) V 2) = D ) V ) D ) V ) + V ) V )] ˆV 2). 3.7) The lst prepotentil cn e composed lgericlly in terms of two other sic superfields. Using the sustitution V 2) ˆV 2) in 3.5) we cn otin the lterntive form of the ction with only two independent prepotentils V nd V S 2 = 2i { 3g 2 dμ 4) Tr V D 2) V ) + D ) V ) D ) V ) 2 + V ) V )]) } ) It is evident tht the superfield ction 3.5) is invrint with respect to the sixth supersymmetry trnsformtion defined on our guge prepotentils δ 6 V ±) V 2)] = ɛ 6 D) V ±) V 2)] 3.9) where ɛ6 re the corresponding spinor prmeters. Thus our superfield guge model possesses the D = 3 N = 6 supersymmetry. The D = 3 N = 5 superconforml trnsformtions cn e defined on the nlytic coordintes. For instnce the specil conforml K-trnsformtions re δ x A = 2 xγ A γρx ρ A + 2lx A δ ) = 2 x A )γ γ δ ) = 2 x A )γ γ + i 4 ) ) 2 ) 3.) 3.) where = m γ m ) re the corresponding prmeters. The K-trnsformtions of the hrmonics hve the form δ U ) = λ ) U ) δ U ) = λ ) U ) + λ 2) λ 2) λ ) U ) U ) δ U ) = λ ) U ) U ) δ U ±) = λ ) = i ) ) λ ) = i ) ) λ 2) = i ) ). 3.2) The specil supersymmetry trnsformtions of ll coordintes cn e otined vi the Lie rcet δ η =δ ɛ δ ]. It is esy to chec tht the nlytic integrl mesure μ 4) 2.8) is invrint with respect to these superconforml trnsformtions. The specil conforml trnsformtions of the hrmonic derivtives hve the following form: δ D ) = 2 λ) D + D2) ) λ D 2) δ D ) = 2 λ ) D D2) ) λ D 2) δ D 2) = λ 2) D2 + λ ) D ) λ ) D ) δ D 2) = δ D 2) = δ D = δ D2 = 3.3) nd the SO5) nd specil supersymmetry trnsformtions cn e defined nlogously. The K-trnsformtions of the guge prepotentils re δ V ) = δ V ) = δ V 2) = λ ) V ) λ ) V ) = δ ˆV 2) 3.4) where ˆV 2) is the composite prepotentil 3.7). It is esy to chec directly the superconforml invrince of the guge ctions S 3.5) nd S 2 3.8). The clssicl superfield equtions 3.6) nd 3.7) hve only pure guge solution V ±) = e Λ D ±) e Λ V 2) = e Λ D 2) e Λ 3.5) where Λ is n ritrry nti-hermitin mtrix superfield stisfying the conditions 3.3).

5 258 B.M. Zupni / Physics Letters B 66 28) Hrmonic component fields in the N = 6 Chern Simons model Let us consider the U) guge group. The pure guge degrees of freedom in the Aelin prepotentil V ) cn e eliminted y the trnsformtion δv ) = D ) Λ.Intheguge we hve Λ = ix A ). The hrmonic decomposition of the HA-constrined prepotentil V ) in the -guge hs the following form: V ) = V ) + V ) + O U 2) D 2) V ) = V ) ] V ) = ) γ m )) A m + i )) 2 ) ψ 4.) V ) = )) 2 U ) B + ) γ m )) U ) Cm + i ) ) )γ ) U ) Ψγ + i ) )) ) U ) ξ i ) )) ) U ) ξ + i )) 2 ) ) 2 U ) R + i ) )) )) 2 U ) R + i ) γ m )) )) 2 U ) G m 4.2) where ll terms re prmetrized y the rel off-shell osonic fields A m B Cm R nd G m or the rel Grssmnn fields ψ Ψγ nd ξ ). The higher hrmonic terms in V contin n infinite numer of the SO5) tensor fields. In the guge group SUn) ll component fields re the Hermitin trceless mtrices. Two other U) prepotentils contin the sme component fields in the -guge V ) = D 2) V ) = V ) + V ) + O U 2) ˆV 2) = D ) V ) D ) V ) = V 2) + V 2) + O U 2). 4.3) The superfield terms Tr V ) D 2) V ) ] V ) V ) ]) 2 D ) V ) D ) V ) in the ction S 2 3.8) yield the following contriution to the component Lgrngin: L = ε mnr Tr A m n A r + i ) 4.4) 3 A na r ] i 3 Tr ψ ψ. The superfield terms { Tr V ) D 2) V ) + D ) V ) 2 D ) V ) + } V ) V ) ] ) + V V ) ]) 2 + Tr { D ) V ) D ) V ) ) V ) V ) ]} 4.5) give us the Lgrngin for the SO5) vector fields L = 2 5 Tr C m m B + ia m B ] ) 8 5 Tr C m Gm 4 4.6) 5 Tr B R + i 6 Tr ξ ξ i 2 Tr Ψ γ Ψγ. It is not difficult to construct the component Lgrngin for the SO5) tensor fields. The N = 6 CS equtions of motion for the lowest SO5) component fields re ε mnr n A r r A n + ia n A r ] ) = ψ = Cm = B = R = G m = ξ = Ψ γ =. 4.7) All SO5) tensor uxiliry fields lso vnish on-shell. The superfield representtion of this pure guge solution in the guge is V ±) = e i D ±) e i V 2) = e i D 2) e i. 5. Conclusion 4.8) We considered the superfield model with the D = 3 N = 6 superconforml supersymmetry. The ction of this model is constructed in the N = 5 hrmonic superspce using the Grssmnn nd hrmonic nlyticity conditions. The clssicl superfield equtions of motions for the nlytic Chern Simons guge prepotentils hve the pure guge solution only. In the fieldcomponent representtion the ction of this model contins the Chern Simons term for the vector guge field nd n infinite numer of the interction terms for the uxiliry osonic nd fermionic fields. All uxiliry fields vnish on-shell. The superfield representtion is useful for the quntiztion nd perturtive clcultions. Note dded P.S. Howe informed me tht the hrmonic-superspce description of the N = 5 6 Chern Simons theories ws considered in the pper 5]. It should e stressed tht our hrmonic constrints 3.3) reduce the SO5)/U) U) spce to the SO5)/U2) spce proposed in 5]. Acnowledgements I m grteful to E.A. Ivnov for interesting discussions nd to P.S. Howe for the importnt comments. This wor ws prtilly supported y DFG grnt 436 RUS 3/669-3 y RFBR grnts nd y INTAS grnt nd y grnt of the Heisenerg Lndu progrmme. References ] W. Siegel Nucl. Phys. B ) 35. 2] J. Schonfeld Nucl. Phys. B 85 98) 57. 3] B.M. Zupni D.G. P Teor. Mt. Fiz ) 97 Theor. Mth. Phys ) 7. 4] B.M. Zupni D.G. P Clss. Quntum Grv ) ] B.M. Zupni Phys. Lett. B ) 27; B.M. Zupni Teor. Mt. Fiz ) 253 Theor. Mth. Phys ) 9.

6 B.M. Zupni / Physics Letters B 66 28) ] B.M. Zupni D.V. Khetselius Yd. Fiz ) 47 Sov. J. Nucl. Phys ) 73. 7] B.M. Zupni Hrmonic superspces for three-dimensionl theories in: J. Wess E. Ivnov Eds.) Supersymmetries nd Quntum Symmetries in: Lecture Notes in Physics vol. 524 Springer 998 p. 6 hep-th/ ] H.-C. Ko K. Lee Phys. Rev. D ) 469; H.-C. Ko Phys. Rev. D 5 994) ] E.A. Ivnov Phys. Lett. B ) 23. ] H. Nishino S.J. Gtes Int. J. Mod. Phys ) 337. ] J.H. Schwrz JHEP 4 24) 78 hep-th/477. 2] A. Glperin E. Ivnov S. Klitzin V. Ogievetsy E. Sotchev Clss. Quntum Grv. 984) 469; A. Glperin E. Ivnov S. Klitzin V. Ogievetsy E. Sotchev Clss. Quntum Grv ) 55; A. Glperin E. Ivnov V. Ogievetsy E. Sotchev Hrmonic Superspce Cmridge Univ. Press Cmridge 2. 3] B.M. Zupni Nucl. Phys. B ) 365 hep-th/99238; B.M. Zupni Nucl. Phys. B ) 45 Errtum. 4] B.M. Zupni rxiv: hep-th]. 5] P.S. Howe M.I. Leeming Clss. Quntum Grv. 994) 2843 hep-th/