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Hypergravity in Ads3 Physics Letters B 739 (2014) 106–109 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Hypergravity in AdS3 Yu.M. Zinoviev Institute for High Energy Physics, Protvino, Moscow Region, 142280, Russia a r t i c l e i n f o a b s t r a c t Article history: Thirty years ago Aragone and Deser showed that in three dimensions there exists a consistent model Received 19 August 2014 describing interaction for massless spin-2 and spin-5/2fields. It was crucial that these fields lived in Received in revised form 18 September a flat Minkowski space and as a result it was not possible to deform such model into anti-de Sitter 2014 space. In this short note we show that such deformation becomes possible provided one compliment Accepted 17 October 2014 to the model with massless spin-4 field. Resulting theory can be considered as a Chern–Simons one Available online 24 October 2014 with a well-known supergroup OSp(1, 4). Moreover, there exists a straightforward generalization to the Editor: L. Alvarez-Gaumé OSp(1, 2n) case containing a number of bosonic fields with even spins 2, 4, ..., 2n and one fermionic field with spin n + 1/2. © 2014 Published by Elsevier B.V. This is an open access article under the CC BY license 3 (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP . 1. Introduction and show that in AdS background it describes interacting system 5 of massless spin-2, spin-4 and spin- 2 fields. Thirty years ago Aragone and Deser showed [1] that in three di- mensions there exists a consistent model describing interaction for Notations and conventions. We will use a multispinor frame-like 5 massless spin-2 and spin- 2 fields. It was crucial that these fields formalism where all gauge fields are one-forms but with all lo- lived in a flat Minkowski space and as a result it was not possible cal indices replaced with the completely symmetric spinor ones. to deform such model into anti-de Sitter space. Taking into ac- αβ Spinor indices α, β = 1, 2will be raised and lowered with ε = count crucial role that AdS background plays in all massless higher βα αβ α −ε such that ε εβγ =−δ γ . For the AdS3 background we will spin theories it is natural to look for generalization of such model αβ βα use background frame e = e and AdS3 covariant derivative D admitting deformation into AdS space. In this note we show that normalized so that such deformation becomes possible provided one compliment to 2 amodel with massless spin-4 field. Resulting theory can be con- α λ α βγ D ∧ Dξ = e γ ∧ e ξβ sidered as a Chern–Simons one with a well-known supergroup 4 OSp(1, 4). Moreover, there exists a straightforward generalization In what follows we will not write the symbol ∧ explicitly. to the OSp(1, 2n) case containing a number of bosonic fields with even spins 2, 4, ..., 2n and one fermionic field with spin n + 1/2. The paper is organized as follows. In Section 2 in our current 2. Gravity in AdS3 formalism (see notations and conventions below) we reproduce the well-known fact (see e.g. [2]) that the three-dimensional gravity in In this section we consider frame formulation of gravity in AdS3 αβ αβ AdS background can be considered as aChern–Simons theory with background. We need two one-forms h and ω which are sym- group SO(2, 1) ⊗ SO(2, 1). Then in Section 3 we start directly with metric bi-spinors. The Lagrangian (being three-form) looks like: the Chern–Simons theory with OSp(1, 2) supergroup and show that 2 in AdS background it corresponds to minimal (1, 0) supergravity α βγ αβ λ α βγ L = ωαβ e γ ω + ωαβ Dh + hαβ e γ h (for the general case of extended (M, N) supergravities, see [3]). 4 At last, in Section 4 we consider straightforward generalization of 2 β γ α a0λ β γ α such supergravity model to the well-known OSp(1, 4) supergroup + a0hα ωβ ωγ + hα hβ hγ (1) 12 where a0 is a coupling constant. This Lagrangian is invariant under E-mail address: [email protected]. the following local gauge transformations: http://dx.doi.org/10.1016/j.physletb.2014.10.041 0370-2693/© 2014 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. Yu.M. Zinoviev / Physics Letters B 739 (2014) 106–109 107 2 1 i αβ αβ λ (α β)γ L = αβ + α δω = Dη + e γ ξ ΩαβdΩ ΨαdΨ 4 2 2 a λ2 a0 β γ α αβ (α β)γ 0 (α β)γ + Ωα Ωβ Ωγ + b0ΨαΩ Ψβ (12) + a0ω γ η + h γ ξ 4 3 αβ αβ (α β)γ δh = Dξ + e γ η and corresponding gauge transformations: (α β)γ (α β)γ + a0ω γ ξ + a0h γ η (2) αβ αβ (α β)γ (α β) δΩ = dη + α1Ω γ η + α2Ψ ξ Now let us introduce new variables: α α αβ αβ δΨ = dξ + α3η Ψβ + α4Ω ξβ (13) αβ αβ λ αβ ˆαβ αβ λ αβ ωˆ = ω + h , h = ω − h (3) αβ 2 2 Variations of the Lagrangian under the η transformations have the form: and similarly for the parameters of gauge transformations: α βγ λ λ δηL = 2(α1 − a0)dΩαβ Ω γ η ηˆαβ = ηαβ + ξ αβ , ξˆαβ = ηαβ − ξ αβ (4) 2 2 αβ + (iα3 + 2b0)dΨαη ψβ In terms of these variables the Lagrangian becomes a sum of two ˆ + − αβ γ independent Lagrangians for ωˆ and h fields, each one having its 2b0(α3 α1)ΨαΩ ηβ Ψγ own gauge symmetry. In what follows we restrict ourselves with so we have to put one field ωˆ only. Corresponding Lagrangian has the form: 1 λ a ia0 αβ α βγ 0 β γ α α1 = α3 = a0, b0 =− L = ωˆ αβ Dωˆ + ωˆ αβ e γ ωˆ + ωˆ α ωˆ β ωˆ γ (5) 2 2 3 2 α and is invariant under the following gauge transformation: At the same time variations under the ξ transformations look like: λ δωˆ αβ = Dηˆαβ + e(α ηˆβ)γ + a ωˆ (α ηˆβ)γ (6) γ 0 γ α β 2 δξ L = 2(α2 + b0)dΩαβ Ψ ξ Now let us introduce convenient combination αβ + (iα4 − 2b0)dΨαΩ ξβ λ ˆ αβ αβ αβ αβ γ ω0 = ω0 + e (7) + 2(b α − a α )Ψ Ω Ω ξ 2 0 4 0 2 α β γ αβ where ω0 is an AdS3 background Lorentz connection, so that so we obtain ˆ αβ + ˆ α ˆ βγ = ia0 dω0 ω0 γ ω0 0(8)α2 = , α4 =−a0 2 2 Here and in what follows d is a usual external derivative d = 0. Thus all coefficients in the Lagrangian and gauge transformations Then introducing new variable are expressed in terms of just one coupling constant a0. Now let αβ 1 αβ αβ us consider equations that follow from this Lagrangian: Ω = ωˆ 0 + ωˆ a0 αβ α βγ ia0 α β dΩ + a0Ω γ Ω + Ψ Ψ = 0 the Lagrangian, gauge transformations and equations can be 2 rewritten in a very simple form: α α β dΨ + a0Ω β Ψ = 0 (14) 1 αβ a0 β γ α L = ΩαβdΩ + Ωα Ωβ Ωγ (9) 2 3 It is easy to see that AdS3 background is a solution of these equa- αβ αβ (α β)γ tions, namely: δΩ = dηˆ + a0Ω γ ηˆ (10) dΩαβ + a Ωα Ωβγ = 0 (11) αβ 1 αβ α 0 γ Ω0 = ωˆ 0 ,Ψ0 = 0 a0 Thus we have a Chern–Simons gauge theory with the Sp(2) ∼ αβ 1 αβ αβ SO(2, 1) group. Nice feature of such formulation beyond its sim- Again using Ω = ωˆ 0 + ω we obtain the Lagrangian for a0 plicity is its background independence, while AdS3 background ap- (1, 0) supergravity in AdS3 background: pears just as a particular solution of equations. In the next two sections we will use the reverse procedure, namely we will start L = L0 + L1 directly with such background independent formulation and then 1 αβ λ α βγ to see the field content we will go back to AdS3 background. L0 = ωαβ Dω + ωαβ e γ ω 2 2 3. OSp(1, 2) supergravity i α iλ α β + Ψα DΨ + Ψαe β Ψ (15) 2 4 As is well known [3] all (N, M) extended supergravities in AdS3 a0 β γ α ia0 αβ can be considered as Chern–Simons theories with the supergroups L1 = ωα ωβ ω − Ψαω Ψβ (16) 3 γ 2 OSp(N, 2) ⊗ OSp(M, 2). The simplest example is the (1, 0) super- gravity that contains just two massless fields: spin-2 and spin 3/2 where L0 is just the sum of the free Lagrangians for massless 3 L ones. Here we reproduce this model to illustrate our formalism. spin-2 and spin- 2 fields, while 1 describes self-interaction of αβ 3 Model contains two one-form fields: spin-2 Ω and spin- 2 graviton and its interaction with gravitino. Corresponding gauge Ψ α ones. Let us consider the following ansatz for the Lagrangian transformations take the form: 108 Yu.M. Zinoviev / Physics Letters B 739 (2014) 106–109 αβ αβ λ (α β)γ δω = Dη + e γ η over all spinor indices denoted by the same letter is assumed.
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