Tensorial superspace approach to higher spin theories Igor A. Bandos - PowerPoint PPT Presentation

OSp ( 1 | n ) as AdS-TSSP κ and OSp ( 1 | 2 n ) Intro Tensorial SSP Hspin eqs in TSSP SUGRA in TSSP HSpin eqs in extended TSSPs Concl. Tensorial superspace approach to higher spin theories Igor A. Bandos Department of Theoretical Physics, University of the Basque Country, UPV/EHU, Bilbao, Spain, & IKERBASQUE, the Basque Foundation for Science, Bilbao, Spain Based on the papers with J. Lukierski, D. Sorokin, M. Tonin, P . Pasti, X. Bakaert, J. de Azcárraga, M. Tsulaia, C. Meliveo (time ordering - from 1998 till present- is used) Conference ”Higher Spins, Strings and Duality”, GGI, Firenze, 6-9/05/2013 May 9, 2013

OSp ( 1 | n ) as AdS-TSSP κ and OSp ( 1 | 2 n ) Intro Tensorial SSP Hspin eqs in TSSP SUGRA in TSSP HSpin eqs in extended TSSPs Concl. Introduction 1 Flat tensorial superspace Σ ( n ( n + 1 ) / 2 | n ) 2 4D Tensorial superspace Σ ( 10 | 4 ) Higher D tensorial superspace Σ ( n ( n + 1 ) | n ) 2 Preonic superparticle in tensorial superspace Σ ( n ( n + 1 ) | n ) 2 Higher spin equations in tensorial superspace Σ ( n ( n + 1 ) / 2 | n ) 3 Higher spin equations in 4D tensorial superspace Higher spin equations in 10D tensorial superspace OSp ( 1 , n ) as AdS generalization of Σ ( n ( n + 1 ) / 2 | n ) 4 AdS HSpin equations on OSp ( 1 | n ) supergroup manifold AdS HSpin equations on Sp ( n ) group manifold Preonic superparticle on OSp ( 1 , n ) supergroup manifold Preonic properties and OSp ( 1 | 2 n ) superconformal symmetry of tensorial 5 superparticle κ symmetry and SUSY preserved by preonic BPS state OSp ( 1 | 2 n ) symmetry of Σ ( n ( n + 1 ) / 2 | n ) and OSp ( 1 | n ) superparticles Searching for the interacting theory: Supergravity in tensorial superspace 6 Preonic superparticle and SUGRA constraints in M ( n ( n + 1 ) | n ) 2 Supergravity in tensorial superspace Higher spin equations in extended tensorial superspaces 7 Conclusions 8

OSp ( 1 | n ) as AdS-TSSP κ and OSp ( 1 | 2 n ) Intro Tensorial SSP Hspin eqs in TSSP SUGRA in TSSP HSpin eqs in extended TSSPs Concl. Outline Introduction 1 Flat tensorial superspace Σ ( n ( n + 1 ) / 2 | n ) 2 4D Tensorial superspace Σ ( 10 | 4 ) Higher D tensorial superspace Σ ( n ( n + 1 ) | n ) 2 Preonic superparticle in tensorial superspace Σ ( n ( n + 1 ) | n ) 2 Higher spin equations in tensorial superspace Σ ( n ( n + 1 ) / 2 | n ) 3 Higher spin equations in 4D tensorial superspace Higher spin equations in 10D tensorial superspace OSp ( 1 , n ) as AdS generalization of Σ ( n ( n + 1 ) / 2 | n ) 4 AdS HSpin equations on OSp ( 1 | n ) supergroup manifold AdS HSpin equations on Sp ( n ) group manifold Preonic superparticle on OSp ( 1 , n ) supergroup manifold Preonic properties and OSp ( 1 | 2 n ) superconformal symmetry of tensorial 5 superparticle κ symmetry and SUSY preserved by preonic BPS state OSp ( 1 | 2 n ) symmetry of Σ ( n ( n + 1 ) / 2 | n ) and OSp ( 1 | n ) superparticles Searching for the interacting theory: Supergravity in tensorial superspace 6 Preonic superparticle and SUGRA constraints in M ( n ( n + 1 ) | n ) 2 Supergravity in tensorial superspace Higher spin equations in extended tensorial superspaces 7 Conclusions 8

OSp ( 1 | n ) as AdS-TSSP κ and OSp ( 1 | 2 n ) Intro Tensorial SSP Hspin eqs in TSSP SUGRA in TSSP HSpin eqs in extended TSSPs Concl. Introduction The interacting theory of higher spin fields was constructed by Misha Vasiliev in late 80th. [Fradkin & Vasiliev 87, Vasiliev 88-89] Misha’s interacting massless h-spin theory is formulated with the use of noncommutative star product and has quite a complicated structure. Not so many exact solutions of this theory are known. The known action principle [P . Sundell, N. Boulanger, N. Colombo] is quite unusual. Some properties are to be clarified. This stimulates not only its extensive study, but also a search for alternative frameworks to reformulate it/to construct interacting higher spin theories. One of such frameworks is provided by ’tensorial superspace’.

OSp ( 1 | n ) as AdS-TSSP κ and OSp ( 1 | 2 n ) Intro Tensorial SSP Hspin eqs in TSSP SUGRA in TSSP HSpin eqs in extended TSSPs Concl. Introduction The interacting theory of higher spin fields was constructed by Misha Vasiliev in late 80th. [Fradkin & Vasiliev 87, Vasiliev 88-89] Misha’s interacting massless h-spin theory is formulated with the use of noncommutative star product and has quite a complicated structure. Not so many exact solutions of this theory are known. The known action principle [P . Sundell, N. Boulanger, N. Colombo] is quite unusual. Some properties are to be clarified. This stimulates not only its extensive study, but also a search for alternative frameworks to reformulate it/to construct interacting higher spin theories. One of such frameworks is provided by ’tensorial superspace’. Its brief review will be the subject of the present talk.

OSp ( 1 | n ) as AdS-TSSP κ and OSp ( 1 | 2 n ) Intro Tensorial SSP Hspin eqs in TSSP SUGRA in TSSP HSpin eqs in extended TSSPs Concl. Outline Introduction 1 Flat tensorial superspace Σ ( n ( n + 1 ) / 2 | n ) 2 4D Tensorial superspace Σ ( 10 | 4 ) Higher D tensorial superspace Σ ( n ( n + 1 ) | n ) 2 Preonic superparticle in tensorial superspace Σ ( n ( n + 1 ) | n ) 2 Higher spin equations in tensorial superspace Σ ( n ( n + 1 ) / 2 | n ) 3 Higher spin equations in 4D tensorial superspace Higher spin equations in 10D tensorial superspace OSp ( 1 , n ) as AdS generalization of Σ ( n ( n + 1 ) / 2 | n ) 4 AdS HSpin equations on OSp ( 1 | n ) supergroup manifold AdS HSpin equations on Sp ( n ) group manifold Preonic superparticle on OSp ( 1 , n ) supergroup manifold Preonic properties and OSp ( 1 | 2 n ) superconformal symmetry of tensorial 5 superparticle κ symmetry and SUSY preserved by preonic BPS state OSp ( 1 | 2 n ) symmetry of Σ ( n ( n + 1 ) / 2 | n ) and OSp ( 1 | n ) superparticles Searching for the interacting theory: Supergravity in tensorial superspace 6 Preonic superparticle and SUGRA constraints in M ( n ( n + 1 ) | n ) 2 Supergravity in tensorial superspace Higher spin equations in extended tensorial superspaces 7 Conclusions 8

OSp ( 1 | n ) as AdS-TSSP κ and OSp ( 1 | 2 n ) Intro Tensorial SSP Hspin eqs in TSSP SUGRA in TSSP HSpin eqs in extended TSSPs Concl. 4D Tensorial superspace Σ ( 10 | 4 ) Fronsdal [1985]: tensorial space Σ ( 10 | 0 ) = { x m , y mn } , y mn = − y nm m , n = 0 , 1 , 2 , 3 is the natural space for the 4D massless (=)conformal higher spin theories. The reason is clearer if we notice that Σ ( 10 | 0 ) = { X αβ } , X αβ = X βα , α, β = 1 , .., 4 + 1 X αβ = X βα ⇒ X αβ = x m γ αβ 2 y mn γ αβ mn . m The first dynamical model in the superspace generalization of Σ ( 10 | 0 ) , Σ ( 10 | 4 ) = { x m , y mn , θ α } = { X αβ , θ α } , α, β = 1 , .., 4 was constructed in 1998 [I.B. + J. Lukierski MPLA 1999]. Its quantization [I.B. + J. Lukierski + D. Sorokin 1999] gave the tower of conformal massless higher spin fields in D=4.

OSp ( 1 | n ) as AdS-TSSP κ and OSp ( 1 | 2 n ) Intro Tensorial SSP Hspin eqs in TSSP SUGRA in TSSP HSpin eqs in extended TSSPs Concl. Higher D tensorial superspace Actually this ’generalized superparticle model’ [I.B.+ J. Lukierski 1999] was formulated in Σ ( n ( n + 1 ) | n ) = { X αβ , θ α } , α, β = 1 , .., n 2 where n is dim. of a min. spinor representation in D dimensions. It is D dimensional as far as x m = ∝ Γ m αβ X αβ , m = 0 , 1 , ..., ( D − 1 ) . The additional tensorial coordinates y m 1 ... m p = ∝ Γ m 1 ... m p X αβ αβ correspond to tensorial central charges of most general D-dim SUSY m 1 ... m p algebra, { Q α , Q β } = P αβ = Γ m αβ P m + Γ Z m 1 ... m p . ( αβ ) m 1 ... m p m 1 ... m p Only Z m 1 ... m p with p , D obeying Γ = Γ are present. Hence αβ ( αβ ) n ( n + 1 ) = # of central y m 1 ... m p D n Z m 1 ... m p 2 charges y mn 4 4 10 = 4 + 6 Z mn Z I (= 1 , 2 , 3 ) y mnp 6 8 36 = 6 + 30 , mnp I y m 1 ... m 5 10 16 136 = 10 + 126 Z m 1 ... m 5 y mn , y m 1 ... m 5 11 32 528 = 11 + 517 Z mn , Z m 1 ... m 5 � d τλ α λ β ( ˙ X αβ ( τ ) − i ˙ θ ( α θ β ) ) The action of [I.B.+ J.L. 1999]: S = contains a huge amount of additional coordinate functions in X αβ ( τ ) .

OSp ( 1 | n ) as AdS-TSSP κ and OSp ( 1 | 2 n ) Intro Tensorial SSP Hspin eqs in TSSP SUGRA in TSSP HSpin eqs in extended TSSPs Concl. Preonic superparticle Preonic superparticle action In addition to coordinate functions X αβ = X αβ ( τ ) , θ α = θ α ( τ ) , � � � X αβ − i ˙ d τλ α λ β ( ˙ θ ( α θ β ) ) = λ α λ β Π αβ S = d τ L = Π αβ = dX αβ − id θ ( α θ β ) , Π αβ ( τ ) := d τ Π αβ τ contains auxiliary bosonic spinor λ α = λ α ( τ ) . ∂ L The canonical momentum P αβ := X αβ is expressed through λ α , ∂ ˙ P αβ = λ α λ β ⇐ ’twistorial dimensional reduction’: momentum d.o.f.s: n ( n + 1 ) �→ n . 2 4 D : 10 �→ 4, 6 D : 36 �→ 8, 10 D : 136 �→ 16, 11 D : 528 �→ 32, In D=4,6,10 (but not in D=11) we have also another two effects = λ Γ m λ is light–like, p m p m = 0. ⇐ famous Γ α ( β Γ γδ ) a = 0. p m ∝ P αβ Γ αβ m m p m p m = 0 suggests that the spectrum of the quantum states of the model consists of masseless particles. But to this end one has to prove the spectrum is discreet.