de Sitter supersymmetry revisited

The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.

Citation Anous, Tarek, Daniel Z. Freedman, and Alexander Maloney. “De Sitter Supersymmetry Revisited.” J. High Energ. Phys. 2014, no. 7 (July 2014).

As Published http://dx.doi.org/10.1007/JHEP07(2014)119

Publisher Springer-Verlag

Version Final published version

Citable link http://hdl.handle.net/1721.1/94598

Terms of Use Creative Commons Attribution

Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP07(2014)119 Springer July 24, 2014 May 16, 2014 June 30, 2014 : : : Received Published Accepted d,e 10.1007/JHEP07(2014)119 doi: 1) superconformal group, but | 2 , Published for SISSA by 2) group of conformal symmetries, which maloney@physics.mcgill.ca , , 1) as a subgroup. The theories are ghost-free , and Alexander Maloney is positive. SUSY Ward identities uniquely select } † = 1 superconformal field theories in four-dimensional α a,b,c N ,Q α Q { [email protected] α , P . 3 1403.5038 The Authors. Daniel Z. Freedman a c Supersymmetric gauge theory, Conformal and W Symmetry We present the basic

, [email protected] = 4 theory, which is presented in appendix B. = 1 theories are classically invariant under the SU(2 N N E-mail: Stanford Institute for Theoretical Physics,382 Department Via of Pueblo Physics, Mall, StanfordDepartments Stanford, University, of USA Physics, McGill3600 University, rue University, Montreal, Canada Center for the Fundamental17 Laws of Oxford St., Nature, Harvard Cambridge, University, USA Center for Theoretical Physics, Massachusetts77 Institute Massachusetts of Ave., Technology, Cambridge, USA Department of Mathematics, Massachusetts77 Institute Massachusetts of Ave., Technology, Cambridge, USA b c e d a Open Access Article funded by SCOAP ArXiv ePrint: The this symmetry is brokenin by the radiative corrections. However, no such difficulty isKeywords: expected tiplet theory with gauge interactionspercharges or and cubic superpotential. are invariant These under theoriesincludes the have full the eight SO(4 de su- Sitterand isometry the group anti-commutator SO(4 the Bunch-Davies vacuum state.transformations, despite the This fact vacuum that state de is Sitter space invariant has under non-zero superconformal Hawking temperature. Abstract: de Sitter space-time, namely the non-abelian super Yang-Mills theory and the chiral mul- Tarek Anous, de Sitter supersymmetry revisited JHEP07(2014)119 5 = 1 N ]. 2 . , 4 1 2 anti-commutator closes = 1 unitary, supersym- = 0, so any non-trivial } } N ¯ Q † α Q, 7 { ,Q α 16 Q 11 { α P 7 8 15 13 5 symmetry – 1 – 18 R 5 ) there are no known 4 4 1) de Sitter isometry group — has no unitary rep- , 17 ]): 2 , 1 = 1 SYM theory 6 13 N = 1 SCFT’s in dS 1 N = 4 SYM with manifest SU(4) Lorentzian signature SUSY theories in curved space, doThe not usual exist de in dS Sitteron super-algebra the generators — of inresentations. the which SO(4 the Indeed,representation one of can the show de Sitter that superalgebra has negative norm states [ Majorana Killing spinors, which are generally necessary for the construction of • • N C.1 Maximally symmetric spacetimes 6.1 Ward identities 3.4 Chiral multiplet with gauge interactions 3.1 Non-abelian 3.2 The superconformal3.3 chiral multiplet Superpotentials In four dimensional demetric Sitter theories, space while (dS suchtwo theories main obstacles abound are in (see Minkowski [ and anti-de Sitter space. The 1 Introduction B C Conformal Killing spinors 6 Propagator Ward identities and the Vacuum 7 Hawking temperature and dSA SUSY Spinor and Dirac matrix conventions 4 The SUSY algebra for5 conformal Killing spinors The supercharge algebra and unitarity 3 The basic Contents 1 Introduction 2 De Sitter supersymmetry and conformal Killing spinors JHEP07(2014)119 , . 1 4 9 2) , (2.1) (1.1) . We 4 . It is 4 1) Killing , in flat space. 4 ] have studied 7 ]; appendix A.1 – 5 ]. Several authors 13 – 8 = 1 superconformal is locally conformal , 7 11 4 N 1), and , )): x ], and its first realization in 18 , which we expect enjoys exact – , ..., 0 B 16 ≥ ] Φ + ]) exploits, as we also do, the fact that µ ]. However, the known examples have 4 K [ D = 1 theories are conformal invariant only ) = 1 superconformal theories in dS Q 21  , µ N = N 20 γ [ 0 in appendix } 2), rather than SO(4  α , 4 (¯ – 2 – 4 † ], some progress has been made in writing down − ] (see also [ ,Q 3 19 α [ Q 4 ]Φ = {  α ). We also explore the SUSY Ward identities that relate , δ = 1 superconformal theories cannot be constructed in dS X 1), we suspect this theory is non-unitarity. 0 ,  1.1 δ N has the structure (on any field Φ( [ = 4 SYM on dS . These theories are consistent because dS ,  0 1) contains a conformal energy (i.e. a generator of conformal time 4  , N satisfy ( algebra closes on SO(4 ] also develops de Sitter symmetry from superfields by embedding dS 4 SO(4 } / 15 , ¯ Q 2) 14 , ] is a formally conserved conformal Killing charge with positive integrand Q, { K [ Q 4) supersymmetry. | 2 ], in particular, proposes an approach to de Sitter supersymmetry similar to our own. , In the following sections we will construct the non-abelian super Yang-Mills and the Recently, in light of the dS/CFT correspondence [ There is interesting previous work. First, the positive energy theorem for de Sitter Our primary focus is the basic classical Recent work [ = 2 superconformal supersymmetry in conformally flat spacetimes such as dS 13 = 2 supersymmetry. 1 ] and superconformal symmetry in Lorentzian curved spacetimes [ As the super-algebra closes on SO(4 In a general supersymmetricwith field spinor parameters theory, the commutator of two SUSY transformations propagators of boson and fermionat fields. the classical Our level, sinceflow. perturbative We radiative also corrections present introduceSU(2 a scale and an RG 2 De Sitter supersymmetry and conformal Killing spinors supersymmetric higher spin theoriesN in dS chiral multiplet theories in defield Sitter space. theories on We will dS discuss why the have also considered10 theories with rigidof supersymmetry [ in Euclidean curved spaces [ terms of a higher spin theory in dS the coset SO(4 translations) with positive spectrum. Second,N de Medeiros and Hollands [ puzzling that they state that The geometries that support conformal Killing spinors are studied in [ in which when expressed as an integral over a time slice. space proven by Kastor and Traschen [ spinors, for which thereone obtains is 8 no conserved supercharges difficultycharges whose in anti-commutators along give imposing with the a 10conformal 5 SO(4 Majorana algebra conformal condition. of Killing debecause charges From the Sitter these necessarily space. to Thus fill the out second the obstacle full noted SO(4 above is avoided conformal theories in dS to Minkowski space. Indeed,appropriate Weyl the transformation actions of and the flat transformation space rules theory. areavoid determined the by first an obstacle noted above by constructing our theories using conformal Killing In this note we consider a modest exception to the above no-go results: global super- JHEP07(2014)119 The (2.4) (2.5) (2.6) (2.7) (2.2) (2.3) . where satisfy 2 a  µ = γ 0  + + + +) B Y − = ¯ A in a constant neg- µ Y  satisfies K can be viewed as the ab  AB 4 γ µ η = diag( γ . 0 , one finds the equivalent )  µ µνab  µ AB 0 γ µ  = ¯ R η , (¯ ¯ γ 1 4 µ ¯ γ a µν 1 a 2 i = K g = 0 2 Then AdS i  a −  . ] . . The usual Killing spinor condition ) is given by . = ρ ν 4 are Cartesian coordinates in a 5-  = = . , − µ 4 / K 3 D ρ 2 µ = 0 ,D ρ . , = 0 a D ←− µ  2 K D µν ←− ¯  D A − µ , ρ ¯  g D 1 4  1 1 2 K D , / = D + + + ν µ µν − D − γ g B µ = 0 – 3 – ) are 1 2 1 4 on the right side differs by a chiral rotation of the spinors Y + D µν µ A ν ν = −  g γ γ  5 Y µ K ,A µ µ γ µ γ µ A + γ D = diag( K a AB , which has constant positive curvature. There is a i D → ν ν Y a 4 2  η 1 3 2 µ D D AB µ = iγ η 2 = + γ .   4 ν µ  µ K D D µ ) cannot be used, because their bilinears are complex, and their D and dS 2.4 ]. 4 Suppose that 3 is usually a Killing vector. With this perspective in view let us compare The embedded hyperboloid of dS ) is not consistent with reality properties of the fields. .  4 µ 2.1 ? γ 4 0  = ¯ µ K 2). This leads to the well known and understood theory of AdS SUSY. , This problem can be bypassed by working with conformal Killing spinors (CKS) whose This is one clue that superconformal SUSY is needed in de Sitter space. However are now coordinates in a 5-dimensional space with metric We present our Dirac matrixThe alternate conventions in condition in appendix which ; the “weights” (the coefficients of 2 2 3 A µν condition and is thus equivalent [ definition, namely is compatible with the Majorana property. Contracting with These bilinears are conformalg Killing vectors, since the right hand side isspinors proportional that to satisfy ( appearance in ( is required bymaximal integrability set in of dS solutions.there are However, the no Majorana Killing spinor spinor solutions. equation Further, is the essentially spinor complex, bilinears so Y would-be Killing spinor condition It is a KillingSO(3 vector, and the set ofSUSY such in bilinears spans dS the set of 10 Killing vectors of are compatible with the integrabilityative condition curvature spacetime. [ They aremaximal also set real of (in Majorana a solutions. Majorana Further, representation) the and bilinear admit a the situation in AdS SUSY in AdS dimensional space with metric embedded hyperboloid satisfying and its Dirac adjoint where JHEP07(2014)119 ] ],  22 , δ (2.8) (2.9) 0  (2.10) (2.11) δ . 4 ) and learn is associated that satisfies )): 2.9 4 1 1). The CKV’s η , . 0 t/a is associated with x − 2 η SO(4 . /  exp( 2 2) a , , x ~  . d ν 0 ). We will obtain the full set  + T ≡ − ˆ 0 . ). Thus we have a basis con- ) 2 γ C 0 ) ) i x ˆ 2 0 x ( γ 2.7 η ν 0 dx ˆ µ 1 x ( γ 2 gK µ − −  x + of the hyperboloid. It is worthwhile to i 2 . √ + ) ∂ A 2 x 0 5 1 3 a ) satisfy (  Y x η d are CKS’s, the SUSY commutator [ ( ( which is not future-directed. ) of all bilinears in the basis ( =  2.9 i Z ρ =  ∂ /a – 4 – i γ ,  0 2 0 0 2). Its Lie algebra is realized by the ten KV’s of − 0  x  , x x . This is clearly future-directed and time-like. ~ 1 (¯ 0 = d ρ − + ∂ } D 0 t/a p α 1 4 = 2 ∂ † j e 0 x is SO(4  D ,Q + ) = K ij 0 4 α δ x 2 ∂ are constant Majorana spinors. The spinor ( Q  dt = { 2 ≡ i − , albeit with admixtures of KV’s. As we will argue below, this  , the spinor covariant derivatives are: α 4 0 1) plus the five CKV’s of the coset SO(4 µ , η K X = , 1 D η 2 dx -supersymmetry transformations. The simplest way to obtain the 0 a ds S x − -supersymmetry transformations in flat space, while = Q is to use the Weyl transformation from flat spacetime to define , where ˆ µ 2 is a future directed time-like CKV. Such conformal Killing charges are con- e 4 η ν ˆ µ 0] is chosen so that spatial volumes increase with increasing γ , K ˆ µ x −∞ [ + The Lorentz boost Translation of patch time The conformal group of dS One can calculate the weights In flat spacetime the most general CKS, given, for example, by Wess and Zumino [ Many coordinate systems are useful to discuss physics in de Sitter space. Since con- ) are Majorana conformal Killing spinors (see appendix 1 i. ∈ ii. η 0 2.4 are translations of thedisplay embedding them coordinates as vector fields on the surface. In patch coordinates the five CKV’s are: served formally, i.e. if boundary conditionstraceless. permit, and Positivity if holds the if stresscondition. the tensor We is stress elaborate conserved tensor on and this of point the in theory section satisfiesthe the isometry group dominant SO(4 energy contains the CKV’s ofleads dS to a supercharge anti-commutator of the schematic form in which It is straightforward totaining check 8 that real spinors supercharges. ( that they do not vanish. Thus, when In the frame is with the usual conformally related CKS’s for dS Its structure is similarx to the Poincar´epatch of anti-de Sitter space. The time coordinate ( of Majorana spinors in a simpler way, but thisformal requires symmetry a choice is of central(obtained metric from for on planar dS us coordinates we by the use transformation the conformally flat “Poincar´epatch metric” One can show that the real and imaginary parts of any complex spinor in dS JHEP07(2014)119 ). (3.7) (3.3) (3.4) (3.5) (3.6) (3.1) (3.2) 4 2.11 ) to be a 2. x √ (  and is given by metric and spin . 2  4  a ¯ FF D . Readers are invited zz/a a µ 2¯ + . ] and show for a general D a K − . is a CKS and thus satis- 23 , also future-directed and ¯ 1 2 µ zz λ a  x ~ 2 λ D a 2 ∂ ρσ a + 1 4 6 = a ·  F ρσ a ) − F  x ~ µ a µ ¯ 0 zz/ / γ Dλ γ x . The extension of the flat space D R / a = 1 de Sitter SUSY theory is thus Dχ ρσ 5 ¯ λ ρσ − ¯ χ γ γ 1 2 iγ + 2 N µ 2 1 µ γ 0 are the main contribution to ( + + ( ∂ − D ←− 2 ) z, χ, F / ¯  a D z 2 ←− ρσ . g K ) ¯  ν aµν F a a 0 g − λ F x ¯ z∂ ρσ − µ √ / µ a γ Dλ and µν 4 – 5 – x √ ∂ 5 1 2 ¯ γ + ( 4 F 0 x 2  d µν 1 4 4 ¯ γ x ~ 1 K g d  2 2 · √ = 4. The first Z 1 1 g − √ √ x ], except that SUSY parameters are multiplied by ~ − ] are linear combinations of the 10 KV’s, namely 3 Z 2 d 0 − is the index of the gauge group. The action is  23 2 1 √ g = = = √ a = ( 1 2 √ x . − a a a µ 4 8 2 µ − 0 for d √ ∂ δλ − K δA δD x µ SYM theory = 4 Z → x d = , where − = a ρσ δS Z = 1 CKV,CKV s γ = δS includes the conformal coupling K ,D = 4) N = 1 SCFT’s in dS a 4 − ) to write gauge kin , λ N d S is contracted with the conserved supercurrent. Now choose ¯ a µ S 2.6 A µ = ( D ←− µ  γ ρσ = 1 chiral multiplet contains the fields ). We simply “covariantize” the flat space calculation of [ they are genuine CKV’s with non-vanishing weights γ µ time-like. 4 N 2.6 γ The special conformal It is now very easy to show that the action is invariant if The Lie brackets [ We use the conventions of Ch. 6 of [ ) that 4 iii. x ( The kinetic action to dS But established! 3.2 The superconformal chiral multiplet Note that ¯ CKS and use ( fies (  The fields are This is just the flatconnection. space The action same extended holds by minimal for coupling the to SUSY the variations: dS scale transformation 3 The basic 3.1 Non-abelian In dS to compute these weights. The CKV’s space transformations, 3 rotations, 3 spatial special conformal transformations and the JHEP07(2014)119 , W → F S σ 2 (3.8) (3.9) µν (3.10) (3.11) (3.12) − g e +∆ , W →  S ) zχ respectively. ( z ν z, F σ D − 2 and its conjugate  , µν e and ¯ / ) is invariant because γ  ), the sum 2 z ) is 2 3 z / = D ( 3.7 0 ¯ z mz + z 3.8 . 2 1 W . ] χ / 00 = D . µ → + ) are each conformal Killing 3) γ L / x W ¯ z 0 ) F ( cubic in ) P W  ¯ , F χ 2 + 2) ¯ L W zW / − ¯ ) is fixed by the requirement that z σ/ 00 e / − D χP (¯ 3.8 / cancels in the sum of the three terms 3 is superconformally invariant, so it  Dz ). The de Sitter covariant extension of and = 1 2 / zW ( W z R 0 3 σ (  (  z µ P − in ( g − W χ . R ∂ 0 0 W − + R P = → χ , /  W √ D . The kinetic action (  R ( +  / x  FW DP ¯ with i χ W 4 – 6 – [  a  / 2 P g D ) d g ¯ z W − = ¯ − = = ¯ 1 2 zχ S Z (¯ √ ¯ ¯ √ z i F does not vanish for general ν x + a / + x 3 is cubic. However, as an example of the problems with D 4 4 D d W F d − W µν W S , consider the mass term δS + Z = γ Z + 3 2 W χ , δ ) are generated by the improved supercurrent = = W / Dz L + S χ , δ kin  W 3.8 W L S ∆ χ / L -traceless. S DP µ δS  P P γ = γ ) S = ¯ = ¯ = F ) and thus also is chosen as the scalar term One may check to see that − δz vanishes when δχ 2.4 δF W Under this transformation χ. ¯ z S 2 . W / D σ/ S ( µν 3  g − ) L e x ( P σ = . 2 = 0 e χ L χ µ Let us look briefly at the possibility of generating supersymmetry using Killing spinors The transformations ( = ) that satisfy ( J δP → x 0 µν ( Notice that ∆ this construction for general specified by the sum  it is manifestly realspinors. and Although the the variation realis and invariant imaginary if parts ∆ of It is easy to verify that its variation under the transformations ( This vanishes only for a cubic superpotential. The complete interacting theory is then consider a general holomorphic superpotential its flat space interaction term is 3.3 Superpotentials The chiral multiplet consideredZumino above model is with a cubicshould theory superpotential have of an free extension fields. to de In Sitter flat space. space We the begin Wess- the discussion more broadly and which is conserved and The coefficient of thethe term sum proportional of to allg terms transforms withχ weight 3/2 underin the Weyl transformation It is invariant under the transformation rules: JHEP07(2014)119 χ . and R (4.2) (4.3) (4.1) χ (3.16) (3.17) (3.13) (3.14) (3.15) L χ P a µ L P ∗ A involving P ∗ a t κγ ). We define 1 2 = ¯ δF + i z − 2.7 χ δz a R χ ¯ 2 zt P L a µ B P ) D iD µν γ + → ) ) are now taken to trans-
im/a µ χ a K R − . The kinetic term is exactly ν zλ P a 2 a t µ  D z t m a  R z, χ, F t − / a + D ν is a complex Killing spinor. λ ¯ 2 z χP = 1 chiral and gauge multiplets in R K  1 2 − /a µ coupling χ, D N P L S + χ D (2 2¯ ( P L 1 2 + F a µ √ 1 8 P a A + + − kin a + 2 ¯ t zt S χ a χ A / Dz L – 7 – + ¯ ) λ L + χ  L P κF χ 2 / L ) 2 L DP ) . For the chiral multiplet, the commutator of the µ  P P √ P im/a − gauge / µ κz K D − = ¯ = = ¯ S 5 µ h + F D + γ ) 0 g 2 χ D = µ ¯  δz z 1 4 L δF 2) → − ) m ) are conformal Killing spinors that satisfy ( ( S K µ − √ x √ χ 3 2 µ δP 0 + ( / x L 0 K ) D 2 4 + µ P / 1 4 D d µ ,  5 iB /a D χ ) 1 4 L Z x (2 γ + ( P (¯ + 2( 1 2  = 1 4 µ A + ( = 1 chiral multiplet to include gauge interactions is a straightforward F = z D z, D = µ µ µ = ( a µ N ) gives V D κ D A K z µ µ a 3.8 coupling t = K K S and 2. After elimination of the auxiliary fields one finds a non-hermitean scalar + are not related by complex conjugation if χ = 0 / = z L  2 χ ), except that the derivatives are replaced by the usual gauge covariant derivatives µ µ ¯ z P F z ] ] ] R . In addition, the following coupling term ∂ z γ m 3.7 , δ , δ , δ P 0 0 0 → δ δ δ = ¯ . We assume that = [ [ [ = ¯ z 4 µ and µ ¯ z ¯ ∂ In this section wedS study the SUSYK algebra of the variations of ( then proceeds much as in the flat space case. 4 The SUSY algebra for conformal Killing spinors must be added to thethe action kinetic to action. cancel The the proof SUSY of variation of invariance the of gauge the fields full appearing action in include the usual gauge covariantλ derivatives as well as an additional term in The transformation rules 3.4 Chiral multipletGeneralizing with the gauge interactions exercise that follows from theform flat in a space representation case. ofas The the in fields gauge ( group ( with generators potential (note with complex masses.δ A further pathology is that the SUSY variations W JHEP07(2014)119 , . ) R 4 4 1.1 (5.4) (5.1) (5.2) (5.3) (4.4) (4.5) (4.6) (which ) of dS 0 0). Then x , 0 2.8 λ , ∗ is covariantly 0 κγ µ 3 2 /a, J 0 x + satisfy the expected − λ κ µν is covariantly conserved = ) leads to γ . 0 ) µν 3.2 n 0 . It is accompanied by the µ . The last term is the U(1) Θ . µ K χ ν gJ ν α K = ( K  − .... D α µ − √ ¯ n Q − ν x = ] + 0 3 ν  = d µ Θ ν K 0 ν γ J α µ 0 Z  J [¯ D gK α ( = ¯  − g 1 8 i Q is the determinant of the full metric of dS µ √ J − − . For the Poincar´epatch metric ( g + x µ µ D √ – 8 – 3 λ ) n Thus we deal with the conformal Killing charges x d ) ] = µ 3 . γ n µ β d Z is conserved if the supercurrent  K , the current √ K is conserved and traceless. For a conformal Killing where β µ ν x = 0 µ Z − µ ¯ α 3 Q g D K µ ]. The first term is the Lie derivative that effects the D d , J µν 1 4 = − 4 1 α α J 22 ( ] =  Z √ α µ Q 2 3 γ K α is the volume element of the spatial 3-volume orthogonal to − Q − + 2( 0 [ = ¯  α + [¯ γ Q ¯  = = D µ λ µρ √ µ 0 J µ F Q x D 3 µ D γn µ d µ K √ K K = = = ρ and . There is a local Lorentz transformation on -traceless, i.e. D A λ ] ] ] 0 γ , δ , δ , δ ) is given by 0 0 0 x δ δ δ a/x [ [ [ ( ), the current z, χ, F µ x = provided that the current falls off sufficiently fast at spatial infinity. ( J  0 0 n x For a conformal Killing vector The structure and the coefficients agree with the commutators of flat space super- For the gauge multiplet the analogous computation from ( = µ Then the operator algebra statementthe that previous corresponds section to is the commutators of variations in spinor conserved and and supercharges n Thus we obtain thetime second expression for the charge. Thisprovided charge is that independent the of stress the tensor Θ In the first expression the future-pointing timelike unitthis normal volume may beis taken negative to in our be conventions). the spatial The slice normal vector of is constant conformal time In this section we expressat the the SUSY algebra classical in level. termscurrent of The conserved Noether charges and charge derive associated ( with a covariantly conserved vector transformation of the superconformalrelations algebra. between the (The R-charges coefficients offrom of the the various conventional fields, values.) but Similar they comments are apply scaled to by the the5 factor gauge 3/2 multiplet. The supercharge algebra and unitarity conformal algebra compiled indiffeomorphism [ generated by theconformal conformal weight Killing term whose vector coefficientsthe are fields the scale dimensions 1, 3/2, 2 respectively for JHEP07(2014)119 η, η (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) (5.11) (5.12) that must and trace, ˆ 0 µν appear in the iγ Θ ˆ 0 ) is positive in ν iγ K † . ) µ . 5.12 i ν 2 n ν . S , 0 ( K ν ) ˆ 0 0 4 ρ )Θ x Θ − iγ ˆ 0 . ˆ ρ β γ ν γ ) α = 0 ( ν x 0 i ) = )) Θ ( ¯ 0 i S x ν /a K ν ( 0 S 4 j ν x 1 xη − 0 0 gK S γ β β · − ) ν − J J γ x x β β ) ( p √ i α α x 0) = x ¯ − S ( 0 (5.13) , ≡ 3 i )) )) ν 0 ¯ d x x ) S x ≥ , ( ( ( Tr( x 0 ), it is the integral of ) and define the two spinor supercharges: 0 1 2 g ( g Z , x ¯ ¯ 2 S S µν − − ( ( 5.1 − S 2.9 ) becomes Θ ), we multiply from the right by g g √ 4(1 4(2 √ ν – 9 – = x x − − 5.9 − − B 3 3 1.1 } µ d √ √ d α x x † i A 3 3 /a Z ) = ) = Z 0 d d ˆ 0 ˆ 0 i ,Q x 1 γ γ = − ) ) Z Z − iα } x x α α 2 1 = ( ( α Q p ¯ ¯ η η † i 1 2 } { S S β j ] for the electromagnetic field and the proof is immediately ν ν ≡ ≡ ≡ α 4 matrices whose Dirac adjoints ¯ ,Q Q γ γ X ) 24 α α ) ) , × iα 1 4 2 1 x x x ) can be replaced by the supercharge anti-commutator ( ( ( Q iα ], but it does not hold for conformally coupled scalar of the chiral Q Q 1 = 1 theories. Given ( 1 2 { α α 2 1 S ¯ ¯ Q 5.4 S S 24 with no sum. We find the traced anti-commutator ¯ ¯ η η { α N j X ) are 4 Tr( Tr( x = ]. ( 2 i is any pair of future pointing timelike or null vectors. The dominant energy 27 is the conformal Killing vector for time translation or time-like special con- ν – ,S ν ) 25 2, the traces are x ,B K , ( µ 1 S A ). We refer to the specific spinors of ( = 1 We now argue that the the operator on the right hand side of ( Since we are interested in deriving ( We now need to be more specific and strip off the constant Grassmann parameters ¯ i 5.4 where condition is well knownapplicable to [ Yang-Mills. Dominantthe energy free is scalar also field validmultiplet [ for [ the canonical stress tensor of tensor of our basic be non-negative. Let’s firstWe look note into that the the stronger standard condition dominant of energy positivity condition of requires the that integrand. in which formal transformation in the Poincar´epatch. the classical approximation in which we include only the bosonic contribution to the stress In each case we findend one of section of 2. the future So pointing the conformal general Killing form vectors of described ( at the For and we choose where supercharges. Then ( in ( JHEP07(2014)119 = 0 0 0 (5.20) (5.21) (5.14) (5.18) (5.22) (5.19) (5.15) (5.16) (5.17) )
K and try φ i i  φ φ i 2  D . We work φ 2 ). For the 2 φD 2 2 φ 0 φD φ a 1 2 0 /a 6 D a 1  5.11 ( D 2 i R +  i )–( − K φ µν = 12  0 K g 2 φ + i
1 2 R D 5.10 . + φ 0 D  0 i + 2  .  ) D 2 φ 2 . and φD  0 . µν 2 φ φ φ φD 1 3 )
x 0 2 1 3 R 0 R 1 µν φ a 1   x − 0 − g   2 − ( µν + φ φD 2 ) , obtaining φ φ ( g 0 a 0 1 φ φ φ − i  1 4 + φD i i 6 φ ρ D D i φ K µν + φ i D i i i =  i + g i 2 2 φD φD ) i ) K D 2 φD φ 0 ρ i D φD − . The CKV equation implies a φD µν φ 1 3 x φD i ν 1 3 D i 0 ( 1 3 i D x ν ~ D R ∂ 1 6 ( D D ) and find D 1 3 D − , D D µ 1 6 − = 1 2 ( ) + µ = φ − − φ 2 + + ) φ φ i φ ) − 0 i D i 2 i 2 φ 5.21 − φD
0 2 φ 0 a φ  K 1 3 0 φ x D φ /a φD 0 i ( – 10 – 0 φD  φD 1 6 0 ( 0  − φD φD 0 D 0 = ) 2 φD φD − 0 D φD and 0 2 3 ) for a real conformally coupled scalar field = (2 a φ 1 D 0 ) 2 3 K φ D ρ 2 i 0 0 φ φ 1 2  D φ D  0   ρ i D 2 3 ν K D i  1 2 g  5.12 i 0 φD 0 D 3 1 K 0  2 ρ K ) as K −  2 φD D ) K D 0 ν 2 0 ρ + K 0 + 0 D 1 3 √ a  + x (  K x ( D 2 Θ K g 5.16 3  ( ) −  µν 2 − 0 d g g − g g a µν x − − 1 6 ) to the two positive timelike CKV’s in ( g ( √ − = Z 1 2 x g √ √ √ φ − 3 x x 0 − x d 3 3 5.21 − . We regroup terms in ( φ ] = 3 0 d d √ D ν 0 d φ Z x ν K 3 /x Z Z [ Z − φD 1 d Q µ − − φD − − Z µ D = ] = = 2 3 D ] = i 0 ] = K [ ] = = = K [ K K Q i [ K [ Q µν D Q Q 1 3 Θ We now apply ( Let us look at the integral in ( generator of translations of conformal time, and which is non-negative. Next we substitute We assume that thepartial fields integration on without the picking time upin slice boundary the decay contributions. first fast We line integrate enough and the at the 3rd spatial 2nd term term infinity in to the allow second line above to obtain to rewrite the integrand in ( The first step is to use the equation of motion in the form where in the secondin line the we Poincar´epatch and used consider the conformal Killing charge to show it is positive. The improved stress tensor, which is both conserved and traceless, is: JHEP07(2014)119 is ]), 1), . } = 0 23 ± (6.1) β is re- , 0  † 2 (5.23) (5.24) (5.25) 0 ν 2 , K ! i ,Q 2 0 K α , D  2 Q φ {  0 φ D 0 2 ) D 0 0 2 . ) for any CKV that K ) x ( summed over spinor 0 , satisfies 2 i x ) x ~ } x ~ 0 5.21 ∂ K α 0 i † · K + ( x D 2 x ~ x ~ ,Q 1 3 , · 0 = ( ) is replaced by (1 α x 2 x ~ + Q  of a massive scalar field which {  φ i . + 2 5.10 + 2 5  )  φ K 0 . In this case one can verify that g i x φ ~ ∂ + in ( x ~ 0 ) − D x, y · D 2 2 ν 0 ( 1 3 ] for more information. D ) √   x ~ δ 0 2 K  28 2 φ ) x ) 0 0 + 2 0 ) x x 2 D 0 ) = x 1) invariant vacua. ) · + ( i , although their integrated charges are not ( 0 , x 0 j i x x + ( ~  x δ K K 0 x, y ( · 2 0 x ~ ( + ( x ) – 11 – x − · x ~ + 0 2 G ) ) x x ~ i φ = = 2 i 2 = ( a K x ) − m D + ( i 2 2 · j ( φ  x ~ − x K 2 ·  ) ) actually have the common form  0 ,K x ~ ( + j x ) which describes a special conformal transformation in the  1). In both cases these are future-pointing null vectors, so ( x g 5.23 µν ij ±

− δ , 2 0 0 xη √ a ) , where , = 0 and ( · = x 0 K 2 0 3 0 x , x  ) d ( i ) and ( g 0 ( K − i − x/ Z K ν D · = (1 √ x 5.22 x x µ µ 3 ] = x v − 2 d ). They are also satisfied by the three Lorentz boost CKV’s discussed in (2 K = µ [ Z i v 2 Q 5.12 K i ] = D K [ 3 1 We begin with a brief discussion of the propagators So far we have considered the SUSY anti-commutator The results ( The special conformal CKV, We do not specify the time ordering prescription. See [ Q 5 satisfy 6 Propagator Ward identities andIn a the supersymmetric Vacuum fieldboson theory and there fermion fields. are Ward Weselect will identities the argue that Bunch-Davies in vacuum relate this state section the whichthe that is propagators superconformal well the of Ward known unique identities one-parameter conformal invariant family state of among SO(4 placed by null direction the dominant energy conditionmultiplet implies we positivity were not for able the to gauge extend multiplet. the analytic For proof the of chiral positivity above. components. We wouldclassically like positive. to show We that summarizeresentation our each of progress individual the on Dirac diagonal this algebrathe term (and question. diagonal of in terms In the take the a Majoranathe usual simple representation sum Weyl form. of of rep- Sec The the 3.3.1 CKV conformal of [ time translation and a space translation. Similarly, These conditions are satisfied byalgebra the ( two CKV’s thatsection appear in 2, the namely traced denecessarily Sitter positive. SUSY This form of the integrated conformalsatisfies Killing charge matches with ( again non-negative. and JHEP07(2014)119 ) y c. x ( φ → (6.6) (6.7) (6.2) (6.3) (6.4) (6.5) ) ) and and ]). Of 2 x c x ( /a 0 35 A , x = 0. In de , ( Y ] defined in 2 31 −
 – 28 )) 2 u ) = arccos y 28 , ( )[ ) = 2 A , x, y x ( ( Y − x, y . In Poincar´epatch 0 ( 2 . D φ , h ) then corresponds to P )+ + x ( h P/a 6.5 (see e.g. [ A ) = 0 − 1 4 u Y F 1 ( ), and simply replace G . ≡ 2 2 2 . 6.1 c ) is given by . u )  . 4 + y ν u ( ma 2 )
( a B c y 2 u − 2 − Y ) is related by to the flat space propaga- 2 − ) − m ) u x x ) in dS ( 1 u + 0 ( ( − ( , 0 G y µ A 2 1 u 0 ) 9 4 , x, y G Y y -function in ( x ( )  − )). Alternatively, this can be seen by noting δ 2 r u 2 D − = 0, AB – 12 – , h a ± 2.8 η − x c 2 1 + ( π 3 2 h 8 ) simplifies significantly at the conformally coupled µν ) = = η 1 F 6.5 ± are antipodally separated points is not preserved by a ) = = ) + 4(1 x, y h -vacua of a field theory on dS ( u 1 2 u u y ( α c ( = 0. The second term is singular when one point is null P  00 G 2 ) G ) is given by: ) )) − and , determined by the Weyl transformation y u 2 h 2 ( a 6.4 x 2 A − /a )Γ( where we find: ) π Y ) has the expected singularity for null separated points 0 + (2 2 ) into the hypergeometric equation y 16 u h − 0 6.5 /a x ) = 0 gives the Bunch-Davies vacuum. 6.1 Γ( x c ( = 2 takes the simple form A ) = 2 u Y u m ( = 1 in order to normalize to the . We use instead the chordal distance variable G 2 1 c /a ) ), i.e. ( The hypergeometric solution ( y ( x, y B ( tor by the factor ( of the field tothat the the Poincar´epatch condition (see that conformal ( transformation on de Sitterare space coincident (though, is of preserved). course the condition that In this case the Bunch-Davies choiceple is way to the see unique this conformally is invariant to vacuum. notice One that, sim- when The one-parameter family ofthe de family Sitter of invariant propagators Mottola-Allenthese in or the ( choice mass point The first term inY ( separated from the antipodal reflectionSitter of space the this other, unphysical i.e. term when cannotarated ( be by discarded a because horizon, two antipodal andwe points the take are singularity sep- is undetectable to geodesic observers. Henceforth where The general solution to ( coordinates, One can convert ( terms of the embedding coordinates of the hyperboloid by: The geodesic distance between twoP points A de Sitter invariant solution of this equation will be a function of JHEP07(2014)119 ). 2.9 (7.2) (7.1) define (6.10) . ) ) of ( x ) (6.8) ) (6.9) ( x ( y y ψ  ( ( 2 1 1 / 3 − − 1 2 ) S S ˆ 0 ˆ 0 /a γ γ 0 ) ) x x x ( ( − 1 2 S S i i . This allows us to derive ) ) ˆ 0 ) = ( 2 y y = 0 it is easy to derive a γ . ( ( x ). A calculation (Mathemat- Ω 2 z z ( i . In fact we use this to ˆ µ 0 d ) ) S ) i 2 γ ) 6.7 2 ψ a x x y r ( ( ( y µ 0 ( ) ¯ ¯ z z z has unbroken supersymmetry. We y ) for the general CKS + h h )¯ y χ 0 )¯ i x 1 1 a a = + 2 x ( 0 − x 3.8 r | 2 2 ( ) are mutually consistent only for the χ − x p h χ . − ( dr ) + ) δ h / 6.9 y y DS 2 1 ( ( u 1 πa 1 1 = + 3 )–( 2 − − 1 2 1 a i 2 and S S 2 ) , namely: = ) ) 6.8 dt y i π ˆ 0 – 13 – x x ) ( ) 8 γ ( ( H 2 z y 1 1 2 r ) ( T S = S S x χ 2 a i i ( − )¯ i ) ) ) χ x − y y h y ( (1 ( ( ( δ χ z z = − χ h ) ) )¯ = 0. This is natural since only this vacuum is conformal 1 x x = x ( ( c ( ¯ ¯ z z 2 / h h χ 2 DS h x x a ds / / D D . As we saw in section 6, the Bunch-Davies state is consistent with − − i 0 | = = ), it is easy to show that i ) y 6.9 is the scalar two-point function given in ( ( i χ ) )–( ) and noting that )¯ , corresponding to the Weyl transformation y 2 x ( / ( ) and note that z 6.8 3.8 3 ) . We consider the SUSY transformations ( χ ) i h 2 x ) 5.7 ( y ¯ z /a ( h ) χ 0 )¯ We begin by recalling the derivation of Hawking radiation in de Sitter space. The It would be interesting to consider the propagator Ward identities for the gauge multi- Using ( y x 0 ( x χ the SUSY Ward identities. Thatconsider is an to inertial say, the observerstate. state in To this de observer, we Sitter can space associate who a static makes patch observations of de in Sitter this space, with vacuum coordinates One might worry that thesymmetry, non-zero because Hawking in temperature flat of spaceargue de a that Sitter finite this space temperature is breaks not breaks super- the supersymmetry. case. We willHilbert now space of quantum field theoryDavies in vacuum state a fixed de Sitter background is built on the Bunch- metry. An observer in dethe Sitter de space Sitter will observe horizon a at bath the of Hawking thermal temperature particles emitted from plet. However, the analysis becomesinclude more a complicated new because term due the consistency to conditions the SUSY variation7 of the gauge-fixing term Hawking in temperature the and action. We dS now comment SUSY on the relationship between Hawking radiation and de Sitter supersym- Notice again that this(( propagator is related to the flat space propagator by the factor of where ica!) then shows thatBunch-Davies the vacuum where two conditionsinvariant, ( but it is satisfying to see how conformal SUSY forces this choice. two independent expressions for We are now readyequations to ( discuss propagatorconsistency Ward condition identities satisfied for by the theh chiral propagator multiplet. Using We use ( 6.1 Ward identities JHEP07(2014)119 . . t 0 H x ) is (7.3) (7.4) out . Note t , where H ( O in ρ H = 0. The dash-dotted , where r out Poincar´ecoordinates cover ) would break SUSY if ⊗ H = 1. The coordinate patch 7.4 in r H Right: = ∂ ∂t H = | 0 H ih 0 | out H – 14 – πaH 2 − ] then gives e = Tr ) is slightly different from that of a finite temperature 1 ρ Z 35 , 7.4 = 34 ρ and the arrows represent the flow of the time coordinate r which is inside the static patch can be written as Tr = 0 and the de Sitter horizon is at O r static patch coordinates cover one quarter of de Sitter space and are good for ) is the standard form for a finite temperature density matrix. However, 7.4 slices become spacelike in the northern diamond. Left: r is SUSY invariant. is the Hamiltonian which generates time translation in the static time coordinate 1 covers that part of de Sitter space which is causally accessible to the observer. . Here we present the de Sitter Penrose diagram and how various coordinate systems cover i H 0 | r < Equation ( is the reduced density matrix ≤ density matrix in flat space.were In the standard flat Hamiltonian space, which the generates matrixfunctions time-translations. density in For ( example, this correlation vacuumto would see not this obey is standardso to SUSY its Ward note identities. expectation that this value A Hamiltonian simple would is way vanish the in square an of SUSY a invariant supersymmetry state. generator, In de Sitter space, where We emphasize that this Hawking radiationstate computation is correct, even though the vacuum we note that the interpretation of ( obtained by tracing overHawking the radiation degrees computation [ of freedom outside the static patch. The standard Because the static patchQFT does Hilbert not space can cover be anthe written Hilbert entire as space Cauchy a of surface degrees tensor of productvalue in freedom of inside de an (outside) Sitter of observable the space,ρ static the patch. The expectation that constant half of de Sitter space. The dashed lines represent spacelike surfacesThe of constant observer conformal sits time at 0 Figure 1 the manifold. describing static observers whose worldlinelines coincides with represent the lines south of pole constant at JHEP07(2014)119 i 2) 0 | , (A.4) (A.1) (A.2) (A.3) denote the the square of conformal 0 not ˆ ν . . . K µ, is the Hamiltonian which . H 5 1) de Sitter isometries can be γ , 2 − 1 ]. Dirac matrices satisfy is given by or hatted indices ˆ 23 = ), where χ . . R ] 7.4 ν . µν g ˆ 0 , γ )–( γ µ a, b, . . . † γ = 2 [ 7.3 iχ } 1 2 ν – 15 – ≡ ≡ , γ ¯ χ µ ,P µν γ 5 γ { γ 2 1 + . Latin indices a to denote curved space Dirac matrices, namely those which = γ µ a L e which generates static-patch time translation is P ≡ as well as the projection operators H µ µ, ν . . . ˆ 3 γ γ ˆ 2 γ ˆ 1 γ ˆ 0 iγ ≡ 5 γ An analogous situation would arise in flat space if we considered an observer undergoing Finally the Dirac adjoint of a fermionic field such as contain a frame field local Lorentz frame. We define We use Throughout the main text we use the conventions of [ We use greek indices supported in part by the U.S.DE-FG02-05ER41360. Department of Energy under cooperative research agreement A Spinor and Dirac matrix conventions We thank Dionysios Anninos,Jaume Bernard Gomis, de Daniel Jafferis, Wit,discussions. Matthew Lorenzo Headrick di The and Pietro, Andrew researchand Strominger Thomas of Cosmology, for Dumitrescu, AM the very is useful Foundationalresearch supported Questions of by Institute DZF NSERC, is and New supported the in Frontiers Simons in part Foundation. by Astronomy NSF The grant PHY-0967299. Both DZF and TA are generates translations in the Rindlerpresent time even coordinate. though the This vacuum finite state temperature exactly radiation preserves is supersymmetry. Acknowledgments time translation vanishes in the ground state. constant acceleration. In thiswould case, appear the to usual emitin SUSY a invariant the finite flat temperature causal space bath vacuumcomputed region precisely state of associated with Hawking the with quanta. density matrix the Indeed, ( observables accelerating observer (the Rindler wedge) are of a supersymmetry generator.vanish in a Thus supersymmetric it state.written is as Indeed, the none not square of subject of the(and a to SO(4 hence fermionic the generator. vanish in The usual thewhich only requirement operators ground are that which state) not can it are be regular generators so de of written Sitter conformal isometries. isometries in For SO(4 example, the generator however, the operator JHEP07(2014)119 . . i 4 C (B.8) (B.4) (B.5) (B.1) (B.2) (B.3) invariant R .  , a i when applied to to denote SU(4) ! X (B.7) 4 anti-symmetric 2 ) a i 0 , β αβ × i . iσ  X ] C 2 α, β ! 2 β 1 2 a  iαβ 0 0 iσ L C iσ + α a − P R 2 a β i λ

P 0 α ] is an exact superconformal iσ i 6. The 4 X ) , + j µ − +∆ 5 33 β C ]. . D , # , i  = = ) ] a i C 33 β 32 6 3 αβ cβ i X (B.6) = 4 ] . We will use λ µ / C i i D + ( ,C R aβ L D ,C β X P λ P : 1 2  iαβ ! , four Majorana gauginos whose chiral α ! bα )( R a µ C symmetry  3 + iαβ a ¯ aα i λ P L σ R A 1 0 0 λ C a α β R 0 1 P X − − α λ R R matrices are written as iαβ µ ) = 4 theory [ ,P L + P P 3 i .

j C D 0 ] is hermitian. α µ 0 α P σ i β – 16 – C when applied to right-handed spinors. Note that C µ  c  j γ µ N i + cβ − γ L ∗ α / X ( λ C D c β ) + ¯ ¯ D  P 0 b µ = = R λ i [( a β − c αβ j − γ αβ i L i P 2 5 λ X α P C a α C X c aα j  bα ( b b α i αβ λ i R ¯ ,C λ ¯ λ a X ¯ λ ρσ L P C X b ≡ i ( F + P = 1 theories discussed in the main text break conformal and ! L ,C αβ 3 and imaginary when iαβ a i i X µ 2 , and six real scalars abc , P 0 ρσ 0 C γ C c iαβ 2 X f ! α N [ 0 iσ γ µνa aα α ,  1 a 1 2 1 2 C i ¯ [¯  + 0 1 2 ab λ 2 σ F X f 0 − " − − − c β R 1 a iσ = 1 µν λ σ 0 abc abc i = = = = L F

,P − f f i 4 1 P a a µ a α i a α a α 1 2 1 4 b α  −

λ λ ¯ λ δλ − − − L δA δX ∆ = = P αβ i 4 1 = = = C C C 2 3 4 are L L L i C = 4 SYM with manifest SU(4) are real when are the usual Pauli matrices. The i i N C σ The action is invariant under the transformation rules that involve an SU(4) quartet The cubic Lagrangian contains the (modified) set of six ’t Hooft instanton matrices The theory contains the gauge potential of Majorana conformal Killing spinors and left-handed spinors and as the operator [ The matrices potential: projections are indices, which we will placeconfusion down with or up the according mainconsists to text, chirality of (we where the hope greek sum this letters of will label not a cause spinor quadratic indices). kinetic term, The a Lagrangian cubic Yukawa term and quartic scalar Radiative corrections in the superconformal symmetry. However, the theory in Minkowski spacetimeTherefore we and present the this de property Sitterform extension is closely of related this expected theory to in to the a notation be manifestly and SU(4) maintained formulation in in [ dS B JHEP07(2014)119 = . λ  d ] ν = (C.6) (C.3) (C.4) (C.5) (C.1) (C.2) ,D ). Note / µ D D 2.4 . γβ ) and ( j , one can construct  C 2 compared to those 2.2 √ iαγ = 1 description. To be  . C ν N γ  . =  β 1) = 0, and therefore R α )  − j µ µν  , d R d g C P i  µ contain both chiralities and do not 4( / 1) C D R D λ − 1 4 1) − 1 1) / = D. d must also satisfy additional constraints − − µ −  and (  d R 2( d γ 2 d R d  and ∆ 1 d / 4( aux. fields in the D +  4( jγβ − − 1 d δλ − D C µν µ s R – 17 – = r αγ i D ±  − C  , and ∓ 2 =  =  / 1) = D F µ = 0 also satisfies ± 1 d 2) = β P − λ  R α µ ± − d ) = d  j ) can be derived using the integrability condition[ P d ) we determine  4( C i  2( − C.2 C C.3 = =  in these formulas are scaled by a factor of /  D  µ satisfying D  . . From ( λ ± λ . We therefore determine that a CKS  ab γ Using the integrability conditions one can further derive Conformal Killing spinors are defined to live in the kernel of the Penrose operator µνab R 1 4 with eigenvalues spacetimes of constant positivethat curvature. in a This constant curvature matches spacetime, givena equations a pair conformal ( Killing of spinor Killing spinors which are real for any spacetime of constant negative curvature, and pure imaginary for given by A conformal Killing spinor is alsofor a some Killing constant spinor if it satisfies the added condition where the second equality in ( Note that any CKS C Conformal Killing spinors We present here awe compendium wish of to formulae be satisfied as by general conformal as Killing possible, we spinors. given Since our presentation in arbitrary dimension clear on notation, we writeThe ( SUSY parameters in the main text. Notefully that conform the to formulas the for down/up conventionclose stated only above. on We note shell. that these transformations The last term contains the effects of JHEP07(2014)119 , , , ), 30 , C.5 (C.7) (C.8) (C.9) (C.10) JHEP , ) we may also , ]. ) simplifies con- . In lieu of ( C.7 / C.6 D SPIRE Class. Quant. Grav. IN ]. , ][ . From (  SPIRE  . µν 2 IN , equation ( γ /  ]. D ][ µν 1) µ µ g − γ γ  , d R d R ν 1 d ( µ  . γ d γ SPIRE de Sitter Superalgebras and Supergravity µ 4 = ]. = γ 1) hep-th/0206105 IN 1) 2)  d [ = µν ][ R − 2 µ ]. − ] − R R γ d ν − d ( d SPIRE ( ]. ]. ( D d 1) d = IN 4 µ R [ 4 −  – 18 – SPIRE ][ R ] − D arXiv:1305.0499 ]. d IN / = [ D [ = SPIRE SPIRE (2002) 5901 , 4(   µ µ IN IN Gravitational energy, dS/CFT correspondence and cosmic − and ν 19 D ][ ][ SPIRE Conformal symmetry superalgebras Superconformal quantum field theory in curved spacetime [  D = / DD µ IN hep-th/0109057 A Positive energy theorem for asymptotically de Sitter ), which permits any use, distribution and reproduction in µν [ [ Rigid Supersymmetric Theories in Curved Superspace All Possible de Sitter Superalgebras and the Presence of Ghosts D / g Supersymmetry on curved spaces and superconformal anomalies D ]. (1985) 105 µ 1) − D (2013) 175015 R d 98 ( arXiv:1307.6567 d 4 SPIRE [ 30 IN (1985) 382 − (2001) 010 [ CC-BY 4.0 Rigid supersymmetry, conformal coupling and twistor spinors = 11 arXiv:1302.7269 arXiv:1105.0689 This article is distributed under the terms of the Creative Commons ) [ [ ν Class. Quant. Grav. B 151 , D (2013) 025 µ ( JHEP , D 10 is a CKS for some maximally symmetric spacetime, then so is  (2011) 114 Class. Quant. Grav. JHEP 06 arXiv:1209.4043 (2013) 175016 Phys. Lett. space-times no hair Commun. Math. Phys. We use this to derive D. Cassani and D. Martelli, G. Festuccia and N. Seiberg, P. de Medeiros, P. de Medeiros and S. Hollands, P. de Medeiros and S. Hollands, D. Kastor and J.H. Traschen, T. Shiromizu, D. Ida and T. Torii, K. Pilch, P. van Nieuwenhuizen and M.F. Sohnius, J. Lukierski and A. Nowicki, [8] [9] [5] [6] [7] [3] [4] [1] [2] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. and hence derive and finally this makes it clear that thegive real two and independent imaginary conformal parts Killing of spinors. any Killing spinor in de Sitter space In any maximally symmetric spacetimesiderably where Thus, if C.1 Maximally symmetric spacetimes JHEP07(2014)119 ] B 70 ]. Class. 08 ] , , ] Phys. (2013) 218 ]. , SPIRE (2013) , 10 IN JHEP [ , , SPIRE Nucl. Phys. A 46 ]. IN , arXiv:1205.3855 . [ JHEP ][ (2014) 577 , hep-th/0106113 SPIRE (1985) 754 Supersymmetry in 327 [ IN arXiv:1302.5228 J. Phys. [ [ , D 31 ]. ]. (2012) 1230013 ]. gr-qc/9602052 (2001) 034 [ (2013) 057 SPIRE 10 SPIRE IN A 27 ]. 05 IN SPIRE , Cambridge Univ. Pr., Cambridge, U.K., ][ Phys. Rev. IN , hep-th/0106109 Exploring Curved Superspace , ][ JHEP Les Houches lectures on de Sitter space Higher Spin Realization of the dS/CFT ]. N=1 de Sitter supersymmetry algebra , SPIRE Supersymmetry on Three-dimensional Lorentzian JHEP Commun. Math. Phys. (1996) 6233 IN , , – 19 – ][ ]. (1988) 33 [ SPIRE Supergravity D 54 ]. IN 36 [ SPIRE http://www.aei.mpg.de/ rendall/gr.pdf , Does back reaction enforce the averaged null energy condition IN Int. J. Mod. Phys. gr-qc/0003025 astro-ph/0210603 , SPIRE ][ [ [ ]. ]. IN ]. Extended Supersymmetry on Curved Spaces Scalar fields, energy conditions and traversable wormholes Energy conditions and their cosmological implications ]. ][ Supersymmetric Higher Spin Theories Phys. Rev. ]. Supergauge Transformations in Four-Dimensions gr-qc/0506099 , [ SPIRE SPIRE SPIRE Fortsch. Phys. IN IN N=1 Super-symmetry Lagrangian in the de Sitter space SPIRE , IN ]. SPIRE [ ][ ][ (2003) 013 IN [ Non-Gaussian features of primordial fluctuations in single field inflationary (2000) 3843 IN arXiv:1108.5735 [ , The dS/CFT correspondence 05 Extended Higher Spin Superalgebras and Their Realizations in Terms of General Relativity 17 (2005) 217 de Sitter Musings SPIRE Particle Creation in de Sitter Space arXiv:1205.1115 Quantum gravity in de Sitter space IN [ [ ]. ]. ]. arXiv:1208.6019 JHEP [ , B 627 SPIRE SPIRE SPIRE IN IN IN arXiv:1207.2181 arXiv:1308.1102 hep-th/0110007 [ in semiclassical gravity? gr-qc/0001099 Quant. Grav. (1974) 39 pg. 607, 2012. Quantum Operators 214022 [ models Correspondence arXiv:1311.7218 Curved Spaces and Black Hole[ Holography Lett. [ (2012) 141 Lorentzian Curved Spaces and Holography [ M. Spradlin, A. Strominger and A. Volovich, D. Anninos, E. Mottola, M. Visser and C. Barcelo, C. Barcelo and M. Visser, D.Z. Freedman and A. Van Proeyen, A.D. Rendall, E.E. Flanagan and R.M. Wald, M.A. Vasiliev, E. Sezgin and P. Sundell, J. Wess and B. Zumino, J.M. Maldacena, D. Anninos, T. Hartman and A. Strominger, M.R. Masoumi Nia, E. Witten, A. Strominger, K. Hristov, A. Tomasiello and A. Zaffaroni, A. Pahlavan, S. Rouhani and M.V. Takook, D. Cassani, C. Klare, D. Martelli, A. Tomasiello and A. Zaffaroni, C. Klare and A. Zaffaroni, T.T. Dumitrescu, G. Festuccia and N. Seiberg, [28] [29] [30] [26] [27] [23] [24] [25] [20] [21] [22] [18] [19] [15] [16] [17] [13] [14] [11] [12] [10] JHEP07(2014)119 , B ]. SPIRE Nucl. Phys. , IN [ (1985) 3136 ]. ]. D 32 ]. SPIRE IN SPIRE ][ IN SPIRE [ IN [ Phys. Rev. Conformal vacua and entropy in de Sitter space , – 20 – Supersymmetric Yang-Mills Theories Supersymmetry, Supergravity Theories and the Dual (1977) 2738 Cosmological Event Horizons, Thermodynamics and (1977) 253 hep-th/0112218 [ D 15 B 122 ]. Phys. Rev. (2002) 104039 SPIRE , IN Nucl. Phys. [ , D 65 Vacuum States in de Sitter Space (1977) 77 Particle Creation Phys. Rev. 121 Spinor Model R. Bousso, A. Maloney and A. Strominger, L. Brink, J.H. Schwarz and J. Scherk, F. Gliozzi, J. Scherk and D.I. Olive, G.W. Gibbons and S.W. Hawking, B. Allen, [35] [32] [33] [34] [31]