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Pure De Sitter Supergravity Pure de Sitter supergravity The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Bergshoeff, Eric A., Daniel Z. Freedman, Renata Kallosh, and Antonie Van Proeyen. "Pure de Sitter supergravity." Phys. Rev. D 92, 085040 (October 2015). © 2015 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevD.92.085040 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/99486 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 92, 085040 (2015) Pure de Sitter supergravity † ‡ Eric A. Bergshoeff,1,* Daniel Z. Freedman,2,3, Renata Kallosh,2, and Antoine Van Proeyen4,§ 1Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh 4, 9747 AG Groningen, Netherlands 2SITP and Department of Physics, Stanford University, Stanford, California 94305, USA 3Center for Theoretical Physics and Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 4KU Leuven, Institute for Theoretical Physics, Celestijnenlaan 200D, B-3001 Leuven, Belgium (Received 30 July 2015; published 27 October 2015) Using superconformal methods we derive an explicit de Sitter supergravity action invariant under spontaneously broken local N ¼ 1 supersymmetry. The supergravity multiplet interacts with a nilpotent Goldstino multiplet. We present a complete locally supersymmetric action including the graviton and the fermionic fields, gravitino and Goldstino, no scalars. In the global limit when the supergravity multiplet decouples, our action reproduces the Volkov-Akulov theory. In the unitary gauge where the Goldstino vanishes we recover pure supergravity with the positive cosmological constant. The classical equations of motion, with all fermions vanishing, have a maximally symmetric solution: de Sitter space. DOI: 10.1103/PhysRevD.92.085040 PACS numbers: 04.65.+e, 11.30.Pb, 95.36.+x I. INTRODUCTION Volkov-Akulov (VA) Goldstino theory [6] coupled to a supergravity background. The global supersymmetry is The cosmological constant is known to be negative or realized nonlinearly. This recent development indicates that zero in pure supergravity, if there are no scalar fields [1]. a scalar independent de Sitter supergravity might exist. Pure supergravity with a positive cosmological constant Another indication of the existence of such a supergravity without scalars was not previously known. In this paper we was presented in [7], where the proposal to couple the VA present the locally N ¼ 1 supersymmetric action and Goldstino theory [6] to supergravity was made. However, a transformation rules of such a theory. De Sitter space is complete action and transformation rules that describe this a homogeneous solution of the bosonic equations of coupling have never been presented. The supersymmetric motion. Supersymmetry is spontaneously broken, so there coupling of the gravitino and Goldstino in D ¼ 10 at the is no conflict with no-go theorems that prohibit linearly quadratic level in fermions was studied in [8,9]. The curved realized supersymmetry [2].1 superspace formulation of the VA Goldstino theory was The main motivation for this work is an increasing studied soon after the discovery of this theory; see for amount of observational evidence for an accelerating exampleareviewpaper[10] or an application of the Universe where a positive cosmological constant is a good constrained superfield formalism in superspace in [11]. fit to data. The next step toward a better understanding of The relation between the superspace approach and non- dark energy is not expected before the ESA space mission linearly realized supersymmetries was investigated in [12]. Euclid launches in 2020. It is therefore desirable to find a All earlier theories were not yet developed to the level simple version of de Sitter supergravity as a natural source of a component supergravity action with spontaneously for the positive cosmological constant. broken local supersymmetry, generalizing the globally The Kachru, Kallosh, Linde and Trivedi (KKLT) uplifting supersymmetric VA model. To construct such an action procedure for constructing de Sitter (dS) vacua in string is the purpose of our paper. We will do this by decoding theory was proposed in [3]. It was recently updated to the the superconformal action underlying dS supergravity, status of a manifestly supersymmetric uplifting using the proposed in [13]. Such a decoding procedure, in addition 3 D -brane on top of an O3-plane at the bottom of a warped to a standard gauge-fixing of local Weyl, R-symmetry throat [4,5]. It corresponds to a globally supersymmetric and special supersymmetry requires an elimination of the auxiliary field F of the Goldstino multiplet from the action *[email protected] which has a non-Gaussian dependence on F. † [email protected] ‡ The important step for our ability to derive the complete [email protected] action of a pure dS supergravity is the observation made in §[email protected] 1 N [14,15] that VA theory canffiffiffi be described using a chiral Note that there exist -extended de Sitter superalgebras for p 2 even N but they have a noncompact R-symmetry group and superfield Sðx; θÞ¼X þ 2θχ þ θ F of global N ¼ 1 therefore do not allow unitary representations. supersymmetry that satisfies the nilpotent constraint 1550-7998=2015=92(8)=085040(13) 085040-1 © 2015 American Physical Society BERGSHOEFF et al. PHYSICAL REVIEW D 92, 085040 (2015) 2 θ 0 χ¯ χ 2 One passes to the physical form of the theory by fixing S ðx; Þ¼ . The constraint sets X ¼ PL = F and thus pffiffiffi eliminates the would-be fundamental scalar partner of the the conformal gauge using X0 ¼ 3=κ, thus introducing Goldstino χ. The Komargodski–Seiberg (KS) model con- ’ κ2 8π −2 4 Newton s constant ¼ G ¼ MPl . It is then convenient structed in this way [15] is equivalent to the original VA to redefine our parameters as follows: a ¼ κm and b ¼ κ2f. geometric model. That model with the action det E, where The new parameters m and f have mass dimension 1 and 2, E is a supersymmetric 1-form, is related to the model of respectively, and we take them to be real.5 The cosmo- [15] by the nonlinear change of variables presented in [16]. logical constant Λ and the Lagrangian mass term of the χ 0 The fact that X is Grassmann valued so that XPL ¼ gravitino are greatly simplifies the construction of [15] and of our locally 2 2 2 m μν supersymmetric extension. Λ f − 3m M ;L ψ¯ μγ ψ ν; : ¼ Pl m ¼ 2κ2 ð1 3Þ The superconformal approach to pure de Sitter super- gravity suggested in [13] is the following: The model at the where ψ μ has dimension 1=2. superconformal level contains the chiral compensating multi- The physics of the model depends on the relation 0 0 0 1 1 1 plet fX ;χ ;F g; a chiral Goldstino multiplet fX ;χ ;F g; between these quantities. When m ¼ 0, f ≠ 0,wehave and a Lagrange multiplier multiplet fΛ; χΛ;FΛg, interacting the pure de Sitter model with nonlinearly realized super- with the Weyl gravitational multiplet. The action is symmetry discussed above. When m ≠ 0, f ¼ 0, which ¯ 1 2 requires that the fermion of the nilpotent multiplet vanishes, L N X; X W X Λ X ; : 1 ¼½ ð ÞD þ½ ð ÞF þ½ ð Þ F ð1 1Þ χ ¼ 0; for consistency, we have the basic anti–de Sitter where the notation of Chapter 16 of [17] is used2 and all three supergravity theory with linearly realized supersymmetry chiral supermultiplets are unconstrained. All supersymme- [1,7]. In all other cases there is nonlinearly realized tries in (1.1) are linearly realized and manifest. Models of this supersymmetry, and the sign of Λ determines whether the type differ from the generic models in [17], and in other homogeneous bosonic geometry is de Sitter, Minkowski, or textbooks, in that the Kähler manifold of the embedding space anti–de Sitter spacetime. Nonlinearly realized supersym- NðX; X¯Þ does not depend on the superfield Λ but does depend metry (essentially the same as spontaneous breaking) means on XI, I ¼ 0, 1. Therefore the equation of motion for Λ is that the vacuum expectation value of the SUSY transform χ δχ ≠ 0 algebraic and can be solved producing the superfield con- of the Goldstino field does not vanish, h i . straint ðX1Þ2 ¼ 0.Thisinturnleadstoanongenericsuper- In Sec. II of the paper we present the main result, the gravity: the elimination of the auxiliary field F1 requires a novel pure dS supergravity action and its local supersym- metry. In Sec. III we explain the main logical steps in the more complicated procedure since its algebraic equation of derivation of the supergravity theory from the supercon- motion contains both positive and negative powers of F1,the formal model in [13], with the details given in the Appendix. latter due to the relation X1 χ¯1P χ1=2F1 which arises as ¼ L In Sec. IV we study features of dS supergravity. We perform the solution of the constraint. Therefore, the knowledge of the the limit of our new supergravity theory to flat spacetime, Kähler potential K and the superpotential W at the super- where fields of the gravity multiplet are decoupled and gravity level is not sufficient in the presence of the nilpotent → 0 3 m . We show how the VA theory is recovered via its Goldstino multiplet to produce the full fermionic action. KS version. In the same section we look at the possible At the superconformal level, the dynamics of our pure gauge-fixing of the local supersymmetry.
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