SIT-LP-13/01 January, 2013 Composite representations of new SUSY algebra 3 1 under nonlinear SUSY transformations in curved spacetime 0 2 n a J 1 3 Kazunari Shima ∗ and Motomu Tsuda † ] h t Laboratory of Physics, Saitama Institute of Technology - p Fukaya, Saitama 369-0293, Japan e h [ 1 v 4 Abstract 5 5 7 We discuss composite representations of new supersymmetry (SUSY) al- . gebra under nonlinear (NL) SUSY transformations in curved space-time from 1 0 the viewpoint of nonlinear supersymmetricgeneral relativity (NLSUSY GR)/ 3 SGM scenario. We show in the linearization of NLSUSY how various ba- 1 sic fields in linear SUSY theories/supergravity are expressed as functionals : v in terms of a vierbein and Nambu-Goldstone fermion fundamental fields in i X NLSUSY GR. r a PACS:04.50.+h, 12.60.Jv, 12.60.Rc, 12.10.-g Keywords: supersymmetry, Nambu-Goldstone fermion, graviton, composite unified theory ∗e-mail: [email protected] †e-mail: [email protected] Supersymmetry(SUSY)[1]isafundamentalandpromissingnotiontowardsaunified field theories of space-time and matter beyond the standard model (SM). Various basic fields in linear (L) SUSY theories are representations of SUSY algebra where LSUSY transformations satisfy the closed commutator SUSY algebra. The SUSY algebra is localized (gauged) in supergravity (SUGRA) [2, 3] by means of the spin- connection formalism used in general relativity (GR) coupled with fermions. Onthe other hand, the localization of SUSY algebra is realized [4, 5] in nonlinear supersymmetric general relativity (NLSUSYGR) [4] by introducing new nonlinear (NL) SUSY transformations for a vierbein and Nambu-Goldstone (NG) fermion fields in curved space-time. NLSUSYGR was constructed based on the GR prin- ciple and NLSUSY [6] as a representation of SO(10) super-Poincar´e (SP) algebra which minimally accomodates all observed particles in SM and the graviton [4, 7]. The fundamental action of NLSUSYGR is given as an Einstein-Hilbert (EH) form with a cosmological term by defining a unified vierbein in a (unstable) space-time whose tangent space-time are denoted by SL(2,C) coset (superGL(4,R)/GL(4,R)) Grassmann coordinates in addition to SO(3,1) Minkowski ones. Corresponding to the spontaneous breakdown of the space-time inNLSUSYGR to the Riemann space- time with (massless) NG fermions, NLSUSYGR is described by means of the EH action of GR and the highly nonlinear interactions of the ordinary vierbein (gravi- ton) and the NG fermions. Based on the (localized) SUSY algebra, NLSUSYGR and LSUSY theories / SUGRA would relate with each other through the linearization of NLSUSY, where various basic fields in LSUSY theories/SUGRA are expressed as functionals (com- posites) of an ordinary vierbein and the NG fermions in NLSUSYGR. In flat space- time, the relation between Volkov-Akulov (VA) NLSUSY model [6] and various LSUSY theories (NL/LSUSY relation) are systematically obtained by means of the linearizationofNLSUSY[8]-[10]andwehave showntheNL/LSUSYrelationinmore realistic N = 2 SUSY (QED and Yang-Mills) theories in flat space-time [11]-[14]. Since the action in NLSUSYGR reduces to the VA NLSUSY one in flat space-time, the NLSUSYGR/SGM scenario [4, 5, 7, 15] gives new insights into the origin of mass and (bare) gauge coupling constant from the NL/LSUSY relation [16, 17]. We have also discussed the linearization of NLSUSY in curved space-time for a relation between the (most) general supermultiplet of N = 1 SUGRA and the fundamental fields in NLSUSYGR [18, 19]. In order to develop the argument of NL/LSUSY relation in curved space-time, 2 we discuss in this letter the new SUSY algebra under the NLSUSY transformations in NLSUSYGR and the composite representations for various basic fields of LSUSY theories/SUGRA in terms of the vierbein and NG fermion fields. We show that the basic fields in LSUSY theories/SUGRA have to be represented as functionals of the vierbein, the NG fermions and their (anticommutative) first-order derivatives in the linearization of NLSUSY. For self-contained arguments, let us briefly review a fundamental action of NL- SUSYGR and NLSUSY transformations in curved space-time [4]. The space-time structure of NLSUSYGR is a curved one whose tangent space-time is denoted by means of SO(3,1) Minkowski and SL(2,C) Grassmann (NG fermion) coordinates (xa,ψi). A unified vierbein wa is defined as µ wa = ea +ta , (1) µ µ µ where ea is an ordinary vierbein in Riemann space-time and ta = −iκ2ψ¯iγa∂ ψi µ µ µ with a constant κ whose dimension is (length)2 = (mass)−2. The action (Lagrangian density) in NLSUSYGR is expressed as an EH form with a cosmological term, c4 L = − |w|(Ω+Λ), (2) 16πG where |w| = detwa , Ω is a scalar curvature in terms of (wa , w µ) and Λ means a µ µ a cosmological constant. In the (Riemann-)flat space-time case (ea → δa), the action µ µ (2) reduces to a VA NLSUSY action, 1 1 1 L = − |w| = − 1+ta + (ta tb −ta tb )+··· , (3) a a b b a 2κ2 2κ2 (cid:26) 2 (cid:27) c4Λ where the constant κ is fixed as κ−2 = . 8πG The NLSUSYGR action (2) is invariant under the following (new) NLSUSY transformations of the NG fermions ψi and the vierbein ea, µ 1 δ ψi = ζi −iκζ¯jγµψj∂ ψi, ζ µ κ δ ea = 2iκζ¯iγνψi∂ ea , (4) ζ µ [µ ν] with global spinor parameters ζi, which induces GL(4,R) transformations of wa , µ δ wa = ξν∂ wa +wa ∂ ξν, (5) ζ µ ν µ ν µ 3 where ξµ = −iκζ¯iγµψi. The NLSUSY transformations (4) satisfy a closed (and localized) commutator algebra to GL(4,R) [5], [δζ1,δζ2] = δGL(4,R)(Ξµ) (6) with a GL(4,R) transformation parameters Ξµ = 2(iζ¯iγµζi −ξνξρeµ∂ ea). (7) 1 2 1 2 a [ν ρ] The NLSUSY GR action (2) possesses (promissing) large symmetries accomodating SO(N) (SO(10)) SP group, namely, they are invariant under [4, 5, 15] [new NLSUSY]⊗[local GL(4,R)]⊗[local Lorentz] ⊗[local spinor translation]⊗[global SO(N)]⊗[local U(1)N]⊗[chiral]. (8) Let us discuss below possible composite representations of the new SUSY algebra (6) under the NLSUSY transformations (4). In flat space-time, we assume that various basic fields in LSUSY theories are expressed as functionals of ψi and its first- and higher-order derivatives (∂ψi, ∂2ψi, ···) in the linearization of NLSUSY as, f = f (ψi,∂ψi,∂2ψi,···), g = g (ψi,∂ψi,∂2ψi,···) (A,B = a,ab,···,etc.) A A B B (9) which satsify the familiar closed commutator algebras under the NLSUSY transfor- mations of ψi, [δ ,δ ]f = Ξa∂ f , [δ ,δ ]g = Ξa∂ g , (10) ζ1 ζ2 A a A ζ1 ζ2 B a B with translational parameters Ξa = 2iζ¯iγaζi. Note that f and g include the case 1 2 A B of spinor functionals. Then, commutator algebras for the products f g are also A B closed as [δ ,δ ](f g ) = Ξa∂ (f g ). (11) ζ1 ζ2 A B a A B This means that allfunctionals intermsof (ψi,∂ψi,∂2ψi,···)aretherepresentations of SUSY algebra, [δ ,δ ] = δ (Ξa), since NLSUSY transformations of (ψi, ∂ψi, ζ1 ζ2 P ∂2ψi, ···) satisfy the SUSY algebra, respectively. 4 In the curved space-time case, let us first consider functionals of ψi and ea , µ fA = fA (ψi,ea ), gB = gB (ψi,ea ), (M,N = µ,µν,···) (12) M M µ N N µ with satisfying the NLSUSY algebra (6) as [δ ,δ ]fA = Ξρ∂ fA + fA ∂ Ξρ, ζ1 ζ2 M ρ M Nk(ρ) µk Xk [δ ,δ ]gA = Ξρ∂ gA + gA ∂ Ξρ. (13) ζ1 ζ2 M ρ M Nk(ρ) µk Xk In Eq.(13), N (ρ) = µ µ ···ρ···µ and the summation means k 1 2 n Xk fA ∂ Ξρ = fA ∂ Ξρ +fA ∂ Ξρ +··· Nk(ρ) µk N1(ρ) µ1 N2(ρ) µ2 Xk = fA ∂ Ξρ +fA ∂ Ξρ +···. (14) ρµ2··· µ1 µ1ρ··· µ2 Then, thecommutatoralgebrasfortheproductsfA gB arealsoclosedtoGL(4,R) M N as [δ ,δ ](fA gB ) = Ξρ∂ (fA gB )+ fA ∂ ΞρgB +fA gB ∂ Ξρ ζ1 ζ2 M N ρ M N Lk(ρ) µk N M Lk(ρ) µk Xk Xk = δGL(4,R)(fAMgBN)(Ξρ). (15) Since the NLSUSY transformations (4) for ψi and ea satisfy the NLSUSY algebra µ (6), allfunctionalsintermsofψi andea aretherepresentations oftheSUSYalgebra µ (6) from Eqs.(13) and (15). As for commutator algebras for the derivatives of ψi and ea under the NLSUSY µ transformations (4), only for ∂ ψi and ∂ ea are closed as µ [µ ν] [δζ1,δζ2]∂µψi = Ξν∂ν(∂µψi)+(∂νψi)∂µΞν = δGL(4,R)(∂µψi)(Ξν), [δ ,δ ]∂ ea = Ξρ∂ (∂ ea )+(∂ ea )∂ Ξρ +(∂ ea )∂ Ξρ ζ1 ζ2 [µ ν] ρ [µ ν] [ρ ν] µ [µ ρ] ν = δGL(4,R)(∂[µeaν])(Ξν). (16) The commutator algebra for the higher-order derivative terms of ψi and ea , i.e. for µ (∂2ψi, ∂2ea , ···), are not closed to GL(4,R). Therefore, we conclude that in the µ linearization of NLSUSY the composite representations of the SUSY algebra (6) in curved space-time are given by means of the functionals (including the spinor ones), fA = fA (ψi,ea ;∂ ψi,∂ ea ) (17) M M µ µ [µ ν] 5 from the same arguments as Eqs.(13) and (15). Note that the spin connection and the scalar curvature, etc. in GR are also the case, for they are the functionals of ea µ and ∂ ea . [µ ν] Wesummarize ourresultsasfollows. Wehavediscussed inthisletter thepossible composite representations of the new SUSY algebra (6) under the NLSUSY trans- formations(4)inthelinearizationofNLSUSY. Basedonthecommutator algebrafor the vierbein, the NG fermions and their (anticommutative) first-order derivatives in Eqs.(6) and (16), we conclude the functionals (17) is the representations by means of the arguments (13) and (15). This is in contrast with the flat space-time case (9) depending for all higher-order derivatives of ψi. 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