Chiral Supergravity Supergravity Andrew J

Journal of High Energy Physics

Related content

- Exact wave solutions to 6D gauged chiral Chiral supergravity supergravity Andrew J. Tolley, C.P. Burgess, Claudia de Rham et al. To cite this article: Melanie Becker et al JHEP10(2009)004 - UV caps and modulus stabilization for 6D gauged chiral supergravity Cliff P. Burgess, Doug Hoover and Gianmassimo Tasinato

View the article online for updates and enhancements. - Convoluted solutions in supergravity A M Ghezelbash

Recent citations

- Critical N $$ \mathcal{N} $$ = (1, 1) general massive supergravity Nihat Sadik Deger et al

- An extremal superconformal field theory Nathan Benjamin et al

- Critical solutions of topologically gauged $ \mathcal{N} $ = 8 CFTs in three dimensions Bengt E. W. Nilsson

This content was downloaded from IP address 128.194.86.35 on 02/10/2018 at 21:14 JHEP10(2009)004 or appropri- October 2, 2009 August 13, 2009 : : September 15, 2009 September 17, 2009 : : , Received Published Revised Accepted 10.1088/1126-6708/2009/10/004 Physics, Texas A& M University, e chiral behavior, so that chiral y. = 1 topologically massive super- doi: N Published by IOP Publishing for SISSA [email protected] , . Linearized gravitino ﬁelds are explicitly constructed. F 3 0906.4822 AdS Supergravity Models, Models of Quantum Gravity We study the linearized approximation of [email protected] SISSA 2009 George P. and Cynthia W.College Mitchell Station, TX Institute 77843, for U.S.A. Fundamental E-mail: [email protected] c

Abstract: Melanie Becker, Paul Bruillard and Sean Downes Chiral supergravity gravity around ArXiv ePrint: ate boundary conditions, the conservedgravity charges can demonstrat be consistently extended to Chiral Supergravit Keywords: JHEP10(2009)004 6 8 1 4 6 9 24 24 10 11 12 15 18 20 21 24 15 ] for the latest pected to have 2 ], and New Massive 7 – 4 ymmetries surrounded by an even ting that some theories might ndependent of String Theory. , we are interested in the latter. antum theories. One might wonder lass of theories: renormalizable and ]. That is, the set of effective theories ], Log Gravity [ 1 3 – 1 – = 8 Supergravity in four dimensions (see [ N ] may be examples of such theories, although Log Gravity is ex 8 = 1 topologically massive supergravity In three dimensions, gravity is highly constrained, sugges B.1 Clifford algebraB.2 and spinor representation Discrete symmetries 5.1 Bosonic charges:5.2 energy Fermionic charges: supercharge N 4.1 Linearized supergravity 4.2 Linearized gravitons 4.3 Linearized gravitini 3.1 Isometries 3.2 Supersymmetry 1 Introduction and motivation The immense number oflarger vacua number in of the vacua contained String in Theory the Swampland Landscape [ is A Notation and conventions B Clifford algebras, spinor representations, and discrete s 5 Energy and supercharge in TMSG 6 Conclusion 4 The perturbative expansion 2 3 The classical background Contents 1 Introduction and motivation if, in the gravitational scenery, there couldfully exist consistent a third quantum c theoriesCandidate of theories gravity that include might be i be consistent at the quantum level.Gravity Chiral [ Gravity [ that appear valid semiclassically, but are inconsistent qu status) and Supergravity in three dimensions. In this paper JHEP10(2009)004 3 3 n- s than sh and sence of is work in 2 Topologically . We would like to 3 1 ] showed positivity of ension of Chiral Grav- two dimensional super- ] and cosmologically ex- 19 e supersymmetric exten- ]. Though retaining all 3 9 13 ity. ]. Techniques from Super- ies emerged from a critical ce. However, in supersym- e theories are more compli- of Log Gravity im [ ssive) gravitons. For generic otic boundary conditions for ], Chiral Supergravity might rgy. Consequently, the AdS 18 ty (TMG). 970s. At the time, the positive . this Na¨ıvely, would suggest 15 2 ion fields to zero, and culminated otal energy does not have to be ], TMG allows for new solutions. Q . 2 10 Q ndary conditions in the susy theory in AdS = 1 extremal = Σ N =Σ H H Supergravity [ ]. ] had still not been proven. Supersymmetry ] will be disregarded in favor of that of ordinary Einstein 3 – 2 – 17 9 18 , 17 The Hamiltonian of supersymmetric theories is ex- ], have no primaries other than the identity of dimension les ] of using properties of supersymmetric theories to under- 16 ] which has logarithmic fall-off to asymptotic AdS 21 . 11 ∗ k ] extended this proof to Supergravity with a cosmological co 20 ] that at the critical or “chiral point,” these energies vani = 12 ]. As in simple AdS 3 c case, the total energy is not positive definite, due to the pre 14 2 R + R ]. We focus on the simplest supersymmetric extension of TMG, ]. 12 22 It was later realized that supersymmetric higher derivativ It was argued in [ TMG was constructed by Deser, Jackiw and Templeton [ There are several reasons which motivated us to construct th The goal of this paper is to construct the supersymmetric ext Abbott and Deser [ However, the sign conventions of Deser et. al. [ Extremal SCFT, as defined in [ Evidence supporting the existence of logarithmic-type bou 2, with central charge 1 3 2 / ∗ positive energy for TMSGmetric for theories generic with points higher inpositive, derivative parameter as interactions, became spa the evident from t energy the theorem literature of of the general late relativity 1 [ point in the parameter space of Topologically Massive Gravi is the exact pp-wave solution of [ a non-unitary dual conformal field theory. The first two theor extend our thanks to the referee for pointing this out. k stant. This lead to the idea [ seemed an interesting path to follow, and Deser and Teitelbo stand bosonic supersymmetrizable theories by setting ferm gravity. solutions of Einstein gravity, likeAround the an BTZ Anti-de black Sitter background, holevalues it [ in has its propagating parameter (ma space,background is these unstable. modes have negative ene progress [ possess a holographic dual. Specifically, a chiral, the total energy in simple Supergravity using in Schoen and Yau’sgravity proof then inspired of Witten’s the new positive proof energy [ theorem [ cated. In the pressed as a sum of squares of supercharges, that the resulting theorythe is metric stable. two Depending theories on emerged: the Chiral asympt Gravity and Logity, Grav while a detailed study of theMassive Supergravity supersymmetric (TMSG), extension constructed bytended Deser by and Deser Kay in [ [ conformal field theory. sion of Chiral Gravity: (1) Positive energy theorem. ghosts [ JHEP10(2009)004 3 AdS ravity . This 2 −Q tion 6 with 1 stable at the Q = = 1 Topologically Q t-moving sector, or m for TMSG can be N n calculations for the and und was derived. The . 2 g Theory. We will make ee chiral supermultiplets. 2 t. The Hamiltonian and an outlook. H ated to String Theory for ifferent possibilities of having the Abbott-Deser-Tekin etry properties of the multiplet vanishes, and the concrete model at hand. − 6∂λ . In section 4.2, we review escribe the linearized theory. 1 2 med into a new question, the lation matches with the recent ¯ λ total energy is not guaranteed. H n our conclusions and will leave ntain ghosts. These can lead to etric theory mimics the bosonic i c extension of Chiral Gravity— ]. tions, and conformal weights. In ] for the non-linear theory where = − 16 2 23 , H φ ] and in section 4.3, we compute the , 3 15 11 ∗ 2 φ − Chiral Supergravity may have an interesting 1 6∂λ – 3 – 1 ¯ λ Having found a supersymmetric generalization of i + 1 φ is not positive even though the theory is supersymmetric. ], where it was shown that TMG has ghosts for generic 2 ∗ 1 24 φ H − 1 2 1 H = ]: For generic points in parameter space, there is a massive g = L 3 2 Q ]. Positivity of the total energy indicates that TMSG is only = 1, even though the theory is supersymmetric. We finish in sec = tr 25 , µℓ H 20 = 1 supersymmetry either in the left-moving sector, the righ This paper is organized as follows: In section 2, we describe To illustrate this idea, consider a theory containing two fr It is an important question to ask if a positive energy theore N In the abovesupercharge equation, are a the difference of second two positive multiplet quantities, describes a ghos section 5, we calculate theapproach energy and [ supercharge of TMSGchiral us point, explicit form of the gravitini excitations, their wave func the positivity of thepuzzle of energy energy could positivity nottype was be not of solved shown, solutions buttheory but admitted only indicate a transfor by that lower at the the bo theory linearized equations considered level, in of the [ supersymm motion.supermultiplet with negative Our energy, ow andHowever, positivity at of the the chiraltotal point energy the is positive. energyChiral contribution Thus, Supergravity— a of exists. consistent this The supersymmetri resultwork of of our Andrade energy calcu andvalues Marlof of the [ couplings (except at the(2) chiral point). Uniqueness of string theory. Chiral Gravity one may wonder if it could be embedded in Strin The total Lagrangian of this theory is some observations regarding the relationthe to String detailed Theory i check offuture work. whether Chiral Supergravity can(3) be Extremal rel conformal fielddual theories. extremal SCFT description along the lines of [ Massive Supergravity (TMSG). We discussing in some detail the d implies that both. In section 3, we describe the isometries and supersymm classical background we areIn interested in. section 4.1, In sectionthe we 4, graviton derive we excitations the d that linearized solve these equations equations of [ motion In short, higher derivative supersymmetric theories may co negative enregy. The positivity of the energy depends onderived. the This issue has been recently explored in [ some conclusions, some comments about Log Supergravity and JHEP10(2009)004 ) g µ 0) 0) f ψ , , (2.1) (2.2) (2.4) (2.3) (2.5) (2.6) e, ( = (1 ω = (1 ) defined . e N ( N . ν ω λ f is equal to the µ ψ γ is then enhanced ν λ αβ ermined using the γ R Γ µ ) at the torsional spin ¯ f , . − R µ , ] by the addition of a i β a 2 , b c ρ ψ 13 = 0 describing the ω ψ define the functional form + c ab first present the µ SL(2 , b νb ν ρ γ pin connection = 0 er to the connection γ e ω a ngth given by ) ψ × ab α b ) el connection. νb ¯ a µ ψ ω ψ L e ψ ) 4 ω 1 ν ) ng the theory) the spin connection e, + γ 2 3 g, R e ( ( a ( , ℓ + 1 2 + β ψ ab µ ab µ µab b ω ψ ω − κ γ ρba SL(2 α µ ν + ∂ ω ¯ ψ ≈ D )+ a ρ )+ ab ν e = g e − ( ω ( ) 2) β µ b µ , ψ ∂ ¯ ρ µν ψ – 4 – ψ ψ µab ] in the mid-1980s, Deser and Kay constructed a α Γ g, ω 9 a ρ γ ( D µνρ e µ is the three-dimensional Newton’s constant. The µνρ iǫ ρ µν ¯ ] provided the cosmological extension of the theory. ψ − ǫ )= Γ − G a ν µ 14 ψ 1 i 4 e 2 − µ e, a ν ∂ ( − e )= µ , and = ℓ ∂ µab . It is also possible to include gravitinos in the right movin ψ 2Λ ; where a ν ω R ]) giving e = e, β ) − µ ( a ν 26 ψ R R D e , α µ L µab D κ D xe xe 3 3 SL(2 µαβ d d ǫ × is the cosmological constant, the gravitino mass parameter Z Z = L 2 ) 1 ℓ 0) extension, Topologically Massive Supergravity [ µ , R f πG πG 1 1 − ) representing the standard Christoffel connection. Note th ) to be precisely that of simple supergravity. This can be det g 2; | ψ 16 16 ( = 1 topologically massive supergravity (1 = (1 e, ρ µν = = ( N ab µ S N Osp ω theory is to 2 Shortly after the appearance of TMG [ sector or in both the right and the left moving sector. We will its single Majorana gravitino; DeserThe [ underlying AdS symmetry group SO(2 Here, Λ = In this expression (as well asinvolves in all torsion. other expressions defini We workof in the second-order formalism and theory and discuss the other two cases after that. The action Palatini formalism (see [ We observe that torsion comes from gravitini, and we will ref where appearing in the action is the dual of the gravitino field stre as the torsional spin connection as opposed to the standard s by the vielbein postulate reciprocal of the AdS radius with Γ connection also satisfies the torsional constraint which also serves as the definition of the torsional Christoff JHEP10(2009)004 R 1) , ψ mal (2.8) (2.7) , (2.10) (2.13) (2.11) (2.12) (2.14) L = (0 ψ N ons ). In this case we . 2.1 a 1) supersymmetry by . The action of parity , , ψ ρ ) B.2 γ R to have been discussed in µb = (1 o following from the action ψ er of the Cotton tensor, the e ( ll give its explicit expression . spin connection to be e action ( N ab µ studying the linearized theory, + s. A detailed discussion of the itten in terms of the torsionful κ . a der supersymmetry. and (2.9) ψ the fermionic (up to 6th order in R . µν b , )+ g γ σν L µν ǫ , µ ρ g µ ψ δ F = 0 = 0 4 1 ( γ + Λ 2 µ ℓ 1 ab µ − µν R , C κ F − , so that all of the previous formulas apply 1 µ µν µ σν ℓ ǫ g λνab + ψ R µ )+ 2 1 + a R e D µν ( – 5 – ν and ρ − ǫγ C ab λνρ µ ψ ǫ ∇ µ 1 µ ω = 2 = ¯ µν µν 8 1 ρσ µ a µ R γ + µ ℓ − 0) theory, it is an easy matter to describe the ǫ ψ )= 1 δe 2 = , δ ρ µν L f = G 1) model incorporates two Majorana gravitinos, − ψ ν µν , , 0) theory is presented in appendix G µ µν = (1 D , R f C is the standard covariant derivative of a spinor. The confor µν ψ N = (1 ǫ γ e, ρ = (1 ab ( γ is the cosmologically modified Einstein tensor, N γ ab µ 2 1 N ab µ ω µν ω = G 4 1 µ 0) is invariant under the local supersymmetry transformati + C , ǫ 1) theory after the corresponding sign changes. µ , ∂ = (1 = = (0 ǫ N µ N D is the Cotton tensor, The The action for the The non-linear field equations for the graviton and gravitin ) are µν 2.1 theory taking into accountaction that of parity parity relates on the both theorie effectively reduces to a sign reversal in can apply the Palatini formalism to determine the torsional where In these equations, Having the formulas for the fermions) contribution to theto graviton the field relevant order equation when andCottino, needed. is wi The the supersymmetric vector-spinor partn for the (topological) part of the action is separately invariant un ( C with mass terms of oppositethe literature, sign. as Such far as an we actionwe know. does can However, easily for not the extend seem purpose the of including previous an formulas extra to gravitino a with theory opposite with mass term into th The covariant derivative appearing inChristoffel this connection. expression Furthermore, is we wr denote by JHEP10(2009)004 ors (3.1) (3.4) (3.6) (3.2) (3.3) (3.5) (2.15) (2.16) (2.17) reviously = 1 models we , , vacuum for which 2 ) ) N 3 ρ ρ , which generate the dρ i∂ i∂ µ + urbative expansion, but K ∓ ± , 2 φ φ 0 ght-moving algebras respec- L ρ∂ ρ∂ ρdφ ted in. The supersymmetry ch we elaborate on in the next 2 mmetric case, we are interested tino fields due to this torsional he form of the (super)symmetry form of the bosonic and fermionic ] = 2 coth , 1 R µ + coth − − + sinh τ τ ψ ǫ , ǫ . ,L 2 a µ µ 1 ρ∂ ρ∂ γ γ ¯ ǫγ L ℓ ℓ 1 1 ρdτ − 2 − + , , L µ (tanh (tanh = 1 supersymmetric AdS ǫ ǫ ) ) ψ ) ) µ µ ; [ – 6 – φ φ cosh a φ φ N 1 D ∂ ∂ D + − − ǫγ ± τ τ ( ( + − i i L 2 = ¯ = 2 = 2 τ τ ± ± ℓ ± e e ∂ ∂ a µ L µ R µ ( ( ]. i i 2 2 = 3 ψ ψ δe i i 2 2 ]= δ ν δ = 1) theory are = 1 , = = ± µ µ dx ∂ ∂ µ µ µ ) group of isometries. When acting on scalars, their generat ,L ∂ ∂ 1) 1) R 0 = (1 dx ± ± , µ µ µ µ L ( (0) ( (0) [ µν ¯ ¯ N K K K K g SL(2 = = = = = ¯ × 1 0 1 0 2 are the torsion-free spin connection and the contorsion as p ) ¯ ± ± L L ds ]: κ R ¯ L L , 3 , we calculate the energy and supercharge of the three SL(2 5 . ) and e ≈ ( 3.2 ω is maximally symmetric and thus has six Killing vectors, 2) , 3 take the forms [ SO(2 while the gravitino vanishes. Ingenerators that this will section, be we used describe laterwave t to functions calculate along the the explicit lines of [ the metric in global coordinates takes the form defined. Eventual interactions betweencoupling left and would right show gravi upthey at are fourth irrelevant and fortransformations higher for the orders the linearized in the theory pert we are interes and similarly for the right movingin operators. this algebra In the as supersy extendedsection to a super-Virasoro algebra, whi where where unbarred and barredtively. operators These refer generators to satisfy the the left- conformal and algebra ri 3.1 Isometries AdS just discussed. 3 The classical background Topological massive supergravity has an In section JHEP10(2009)004 . ¯ h ]. tion − (3.7) (3.8) (3.9) , it is 35 h (3.10) (3.11) – = 0 and = 31 ℓ/G L S , c 1. For the 5 , 4 | ≥ ] that . Representations | µℓ 3 | mensional Virasoro y behave as in pure S , 0 ≥ | n, . + and their spin , are described as primary E i m ]. Unitary representations ¯ mands h ms these central charges so δ h | 1 ular boundary conditions are µℓ 29 + orresponds to large -moving gravitational physics , , h y that is trivial [ m . i ) 28 1+ = − ¯ h 0 ¯ h | garithmic mode is a counterexample. m of all states) imposes constraints 3 ¯ h E s when the standard Brown and Henneaux h, m ditor for pointing out this technicality. ℓ G = . 3 n> 2 c Gravitons, i ℓ ¯ h G and describe non-propagating degrees of = 1 4 | 3 12 2 = 1, it was noticed in [ 0 5 R ¯ L = 0 0, or equivalently + = µℓ ] that for ordinary Einstein theory, the central i n R ≥ ¯ h , 27 + , c | c – 7 – are called “massive” and describe propagating i n h m ) algebra can be used to classify the states that | = h ¯ L L R | S 1 | , ) h L µℓ = c n i = − 1 and − h i | 1 h E > n ] and references therein) | ≥ m 0 L c 30 ℓ L G 3 2 ]: ] = ( 27 m = L ) with ,L ] that the SL(2 c n . As will be checked later, primary states in both representa 3 S L c, h [ . This suggests that right-moving gravitational physics ma ℓ/G It was shown by Brown and Henneaux [ It was shown in [ The conformal algebra in the bulk is enhanced to an infinite di Strictly speaking these modes are only refered to as massles Although not all solutions can be obtained in this way, the lo = 3 5 4 R gravity (albeit with twicemight the be trivial. central In charge) otherimposed, and work, it that it is was left indeed shown that possible once to partic obtain a left-moving theor interesting to note that unitarity of the boundary theory de boundary conditions are applied. We would like to thank the e The gravitational Chern-Simons term appearingthey in are TMG no defor longer equal (see [ appear in TMSG. c satisfy the three-dimensional equations of motion. states with helicity and similarly for the barred algebra. specific value of the Chern-Simons coupling, freedom. Representations with charges of the left- and right-moving algebra are equal: algebra on the boundary [ exist for all values ( states of this algebra and are labeled by the weights ( Taking into account that the semiclassical approximation c and satisfy that saturate this bound are called “massless” Equivalently, we can label these states by their energy Unitarity of the representation (e.g. positivityon of the the nor central charge and weight of the primary fields [ JHEP10(2009)004 (3.14) (3.20) (3.18) (3.15) (3.16) (3.12) (3.13) (3.17) (3.19) , s + r L sector algebra. = 2 , , } s 0 ! n, ρ 1 , G ary fields of the super- + ie r m

G δ can be expressed in terms , s the central charge of the { 2 0 / not be needed to obtain the ) m ebra on the boundary left sector, the right sector or m n, ρ ciated with the fermionic gen- , L + + − r m ic ﬁelds depending on a compact iu , d in the NS sector of the algebra. 3 , these Killing spinors are simply + ( δ . is simply the directional derivative 1 1 − m ) and m . µ − e φ ± G = 0 1 4 0 ¯ L K 3.9 L = ǫ , can be expressed in terms of Killing , c r − 2 0 µ φ . ∗ L / γ ¯ 2 1 µ ), ( ξ L 1 12 − . ℓ , and ∂ 1 ± m 2 µ 1 n 3.8 / 2 + µ 1 G m , r> − 1 + K n ǫ K ± L m and + – 8 – c µ = G m 3 1 = 0 , G D φ 1 ]= i L 1 r + ± ) h ! n on a scalar ﬁeld | L ρ = 2 n n r 1 , G − ,L + ie 1 µ L G 0 − m m − 2 ψ L m 2. Similarly, as in the bosonic case, we expect this algebra L L δ m [

coordinate) can be either periodic or anti-periodic under / 1 2 / φ ) = 2 , ± ρ ]= ] = ( } n 0) global subalgebra is generated by − n n = , n + iu ( m ,L , G e , G ]). 0) theory the Killing spinor equation takes the form r, s L m m , = (1 m ) = 36 L L [ n [ G L N { ξ − = (1 . Like the Killing vectors . There are two Killing spinors that solve this equation: 1 and φ a m ± N γ , + a µ e , corresponding either the Ramond (R) or Neveu-Schwarz (NS) τ ] = ( π = 0 = n = µ + 2 u γ ,L m,n m φ In the supersymmetric case graviton and gravitinos are prim L [ → of Killing vectors, the fermionic generators, Since the above spinors areIn anti-periodic, this we sector, are intereste the The left-moving superconformal algebra is φ complex conjugates of oneerators another. of Killing the spinors super-Virasoro are algebra.coordinate asso In (in general, this fermion case the Although the explicit formgraviton of and the gravitino fermionic wave functions, generators notice will that just as The central charge ofbosonic the theory supersymmetric (see theory [ is the same a spinors. For example, the action of both. For the Virasoro algebra that satisfy the constraints ( 3.2 Supersymmetry We consider again all three cases, i.e. supersymmetry in the with where where to be enhanced to an inﬁnite dimensional super-Virasoro alg JHEP10(2009)004 o 1) , (4.1) (4.2) es of (3.21) (3.22) (3.23) = (1 order in th N n the iﬀord algebra, see . se perturbations. First, ! when performing such an ct is central to all variants i ρ . e − avity and the corresponding that take the form

also be expanded in powers of 3 rbation theory. In this section, . ations about some background 2 sponding work on Topologically λ / ive regime as to make headway. ) damental ﬁeld is the metric, the ρ 3 ity, and then proceed to determine + λ O iv . ( + 1) yields the resulting Killing spinor O φ . − , µ e µν + ∂ = 0 j background by considering perturbations = L 2 ǫ ξ = (0 λ 3 (2) µ µ ∗ R µ γ ψ γ ) in the relevant expansions. In supergravity + ℓ 2 N 1 n 2 λ – 9 – AdS = λ , ξ µν ( + + φ ! 2 λh ǫ O µ ρ / i µ e 1 + D λψ

G is 2 µν = / g ) φ ρ µ = ¯ − is a viable supersymmetric background for an ]: Compute the explicit form of linear graviton and gravitin ψ on iv 3 3 ( µν 2 e g / 1 = AdS G R ξ , and we observe again that both spinors are complex conjugat φ for details. A study of the − 1) theory involves an inequivalent representation of the Cl , τ , which is a known solution of the theory, as follows B.1 = µν is a small parameter to be used as a book-keeping device. Thus g = (0 v λ N Expansion in small fluctuations about some fundamental obje Thus, we see that Next, we analyze the stability of the . It is important to note that the gravitino is a fermion and so appendix Similarly, the action of of perturbation theory. Instarting gravitational point theories, of the perturbative gravity fun metric, is ¯ to expand in fluctu fields, then computehowever, conserved we must charges discuss to some generalities. second order in4 the The perturbative expansion Due to the nonlinearitysupergravity present theories, in one Topologically is MassiveIndeed, often the Gr forced analysis into leading to a ChiralMassive perturbat Gravity Supergravity and relies our heavily corre onwe the will methods briefly of recapitulate the pertu basicsthe of perturbative perturbative grav spectrum of the supersymmetric theory. equation: The Here, perturbation theory is tantamounttheories to there is a secondλ field, the gravitino, which should There are again two Killing spinors that solve this equation expansion the background term is identically zero: Here, one another. extension of TMG. around it. We proceed as in [ JHEP10(2009)004 (4.9) (4.4) (4.8) (4.3) (4.7) (4.6) (4.5) ns of ], and (4.10) (4.12) (4.11) 3 , , , σν h µ 3 βν ¯ λ ely from the gravitinos. ∇ h σ ¯ 2Λ ckground metric. Here we O ∇ one to work order by order , s completely decouple. This d, we will review the metric − + + . , isely the one found by [ which is a second order effect ) ds of the theory, one constructs tion. In general, given a multi- h σµ µν =0 ( ) = 0 (2) , h µν... λ h ν

ψ . ( (1) M ) ¯ ∇ µ 2Λ 2 R ψ σ λ ψ (1) µ − ) = 0 ¯ 2 ∇ βν g, h , C + ) h ¯ ( g 1 ( n h µℓ 1 µ + 4 1 ( 2Λ 4 h (1) ∂λ µν (1) + µν... − µν... ν − (1) − C ν ) ¯ ∇ ) M R 1 h µ ψ µν µ λM n ( ψ – 10 – h ( µν ¯ ∂ µν ∇ ν ¯ + g ¯ γ (1) βν ! )+ ¯ 2 1 ∇ (1) − ℓ 1 ρ n h R 1 , µ 2 f ( (0) − is given by ν µν... ¯ β µν ∇ = ) − ¯ h α ψ D (1) µν ) M 2 h ) + α ) ¯ n G ( ∇ ( µν ¯ n µν... ψ ∇ ¯ h µ ( D µν... ( ¯ γ 2 (1) µν )= − ρ M αβ ¯ M ∇ ¯ γ (1) ǫ R ψ µ µαβ 1 2 2 1 f − ǫ 0) theory. In the linearized approximation the field equatio g, , ( )= )= ), its formal perturbative expansion can be written as h h )= )= )= )= ( ( h h ψ ψ = (1 ψ µν... ( ( ( ( (1) (1) µν µν g, µ M N (1) ( (1) µν G (1) C µ (1) R f R C µν... M and generate the perturbative spectrum of the theory. . This allows us to study the graviton wave functions separat λ λ where barred quantities are expressed with respect to the ba this theory take the form Applying such an expansion toin the equations of motion allows with 4.1 Linearized supergravity Consider first the where the functional form of Given such perturbative expansions for the fundamental fiel perturbative expansions for alllinear objects map appearing in the ac The form of the linearized bosonic equation of motion is prec so before proceeding tofluctuations. the case of the Rarita-Schwinger fiel Note that at linearis order due the to bosonic gravitons andin and fermionic gravitini equation coupling through torsion, have introduced the notation and JHEP10(2009)004 ) n n- ( K s. In (4.16) (4.20) (4.21) (4.15) (4.13) (4.19) (4.14) (4.17) (4.18) , which 2 defined in ∇ 2 2 ) = 0, so that ℓ n µ ( h − ¯ K . = , ) 2 µλ n and is found to be µ ( ¯ h L e. At the linear level K + µν ν ) 2 whose explicit form will h n ( L ors . ρ lar gauge is chosen. It was K λ t form ee and traceless: , ¯ celess condition = 0 ative along a vector field ∇ can be determined to be ,    2 + , which take the form ) σν a µν . 3 2 . . ia ρ h . iSa λν ( ) Casimir, − ρ M . h R ¯ ρ ∇ µν , AdS 2 2 = 0 1 S = 0 = 0 iSa ) = 0 µ ρσ M ) h µ ) µν µν n 2 µν ǫ 1 S ( cosh ia h h 1 µν h Sφ 1 µ 1 ρ µ K g    + ¯ λ ¯ L 1 ∇ ) + + ¯ ¯ ρ ¯ Eτ – 11 – ∇ L ( = ( sinh i µν , ), the tensor µν ± f − h µν h e + = 1 ( h L = 0 a 1 µ = 4.18 )= L ¯ µν h ρ ∇ 2 ( 2 h µν ℓ ) algebra to find the solutions to the equation of motion. λ h µν R ¯ + ∇ ) can be determined through the application of the primary , 2 ρ M ) ( ¯ ∇ n ] that implementation of this gauge condition reduces the li λ ( 3 µν are taken to be Lie derivatives along the Killing vector field K M n = ¯ L , µν n ]: h L n 37 L ] to use the SL(2 is a symmetric two-index tensor depending only on 3 ] that a convenient gauge choice is the divergence free-gaug 3 ). With the lower sign of ( µν M 2. Furthermore, 3.5 ± )-( = 3.2 Gravitons are described by harmonic functions on where with 4.2 Linearized gravitons The graviton field equation becomes much simpler if a particu shown in [ it reads earized bosonic equation of motion to the linearized bosonic field can be chosen to be divergence fr This equation can be written in terms of the SL(2 motivated [ Combining this with the bosonic field equation yields the tra The f fields in these equations are taken to be the Killing vect be calculated. The spin is determined by the gauge choice for It was further shown in [ S ( field constraints These constraints can be rearranged into the more convenien takes the form [ The generators, particular, when acting on a two-index tensor, the Lie deriv JHEP10(2009)004 ]: 3 ). In (4.24) (4.23) (4.25) (4.28) (4.26) (4.27) (4.22) ρ is some ( unction 2) is the f κ − , ized equation . As with the ρ , where ] that for suitable ) = (2 κ 3 ) µ γ E,S ℓ 1 . 2 , ±    . 2 µ own gauge freedom. Specif- gued in [ oceeds in a similar fashion. a 2 uged away so that the theory D ia hts or energy and spin can be 2) iSa p to overall normalization [ − of motion. ) = 0 ± race-free gauge condition yields ess graviton. In fact, applying a + ( ρ ( 2 2 1 S µ , iSa µℓ, f ψ ) , ρ 2 1 S ρ ± . ia 2 1 ( → ρ    ψ µ ρ µ 0 . ρ ζ 2 ρ E ) ¯ ρ γ ψ 2 E Sφ 2 cosh 2) is the massive graviton. The final case, − = 0 = 1, the wave function of the massive mode + sinh cosh 0 cosh − µ ρ ψ sinh µℓ, Eτ cosh ρ 1 ψ µℓ ( – 12 – ) i γ 2 2) or (1 µ )= − ¯ γ Sφ ± ρ e sinh = ¯ ( , + 2 sinh = f ). Expanding this condition yields the relationship Eτ ψ ( µ E i ) = (1+ 0) theory. The left-moving gravitino wave functions are ψ − , )=(2 4.27 e ) allows one to determine the matrix prefactor, b )+ ρ N E,S ( E,S = (1 4.18 f ( = ρ N ∂ 2 is a two-component spinor depending only on µν , h 1 3 , 2) is referred to as the left-moving graviton, ( can be determined by inserting this solution into the linear 2), is not considered a solution to the theory since the wave f , AdS − = 0 E µ µℓ, ) = (2 − with E,S µ ζ ) = (1 ], ( It is understood that the Rarita-Schwinger field carries its 3 E,S In [ of motion. Upon restrictingfixed to to normalizable modes, the weig right-moving graviton, and ( ( is non-normalizable. At thecoincides chiral with point, the one of the left-moving graviton. It was ar The energy value vector-spinors on Taking the upper sign of ( Combining these results yields the graviton wave function u boundary conditions this left movingbecomes wave function chiral, can with be only ga a right-moving degree of4.3 freedom. Linearized gravitini Deriving the gravitinoConsider wave again functions first at the the linear level pr spinor field. This gauge freedom allows one to fix particular, one arrives at the differential equation where gravitons, a suitably chosen gauge simplifies the equations ically, equivalent physical states are obtained by This is the natural choicesupersymmetry for the transformation superpartner to of the the linearizedthe tracel graviton gamma-traceless t condition ( which admits the solution JHEP10(2009)004 it 1 ψ (4.31) (4.34) (4.33) (4.36) (4.30) (4.39) (4.32) (4.38) (4.37) (4.29) ]: and 37 0 ψ . Inserting the µ . algebra to reduce D λ 3 µ ψ , γ , . ) ρ λ = ¯ λ n λ ( ψ = 0 ψ , = 0 (4.35) ) K 6D µ ρ 1 ) µ cosh ) ( ψ n ψ λ n ( ¯ ρ λ . ∇ = 0 ( ρ ! ionic equation reduces to iF K ¯ µ tions K 2 1 µ ! ℓ µ 1 ψ . ¯ + γ and ρ ¯ i 4 sinh ∇ i ¯ e ∇ µ . they satisfy the Dirac equation 3 2 , 2 2 cosh

ψ − 3 ρ ρ ℓ = 0 1 ) + = )= 2 + 4 are known. To determine 2 ρ αβ + µ / ρ µ ( µ γ 1 S 1) ( 1 = 0 ψ ψ − )¯ AdS µ 2 ψ +1 ψ 2 sinh ψ β µ 2 ℓ − F ) ) E 2 λ ψ ρ n λ 6D ( ρ/ ¯ E γ 1) ¯ γ iSφ ( and ℓ − 1 , F K ℓ 1 operators are Lie derivatives along the Killing − ℓ e 2 − cosh ) can now be ﬁxed by choosing the positive sign 1 ¯ ) L 0

α 2 1 µ – 13 – ρ coth ρ 1 ψ ¯ ( E 1 ( ∇ − µ iEτ ± ¯ + L ( − ( − − λ 1 )= F F 1 2 , λ µ − e ρ ρ ¯ 1 − L D ¯ f ( ψ D µ once + L N 6D ψ F )= ) 2 µ ) ℓ n ρ ), one finds that ℓ 1 λ tanh n ( ψ ψ 1 = 2 ( λ 2 ( λ 1 K ¯ µ E K ¯ − F − D 4.29 ψ ) = µ = are Killing vectors, one can apply the AdS n + ( λ 6D ( ψ µ ) µ 1 )= 1 ψ K n λ ψ ρ ψ ( n ( ρ n ℓ − ¯ = K L are harmonic functions on 0 ∂ ¯ acting on a vector-spinor. Specifically, they are given by [ L , µ F E µ ) µ 1 ψ n ψ λ ¯ ( n K L K , µ 1 K is some overall normalization. ), which leads to the differential equation f N 4.29 Notice that since and and which can be used to determine Here, However, when vector fields Choosing the minus sign in ( with solution where the Feynman slash notation has been adopted so that is sufficient to apply the lowest-weight/primary-field condi After fixing the gamma-traceless gauge, the linearized ferm in ( these expressions to where, as in the bosonic case, the linearized modes into this equation yields JHEP10(2009)004 2 3 µ ψ hts − = (4.45) (4.41) (4.43) (4.44) (4.42) (4.40) . The 1 S F ) and the 4.26 ge for , ! , 2 θ 1 ρ ) θ . . 1) theory, by taking the rstood that fermions are ) µ , . ) left moving gravitino, the ence satisfies the requisite . Sφ ψ ρ cosh e functions are the real (or, ) ( Sφ , 2 3 rgravity preserves the chiral tric version of Chiral Gravity, s chiral behavior will extend ond mode corresponds to the + ρ 3 2 ssive gravitino that propagates 3 2 al” temporal component of tric version of Log Gravity. As + = (0 iF ! − µℓ, ρ Eτ om. As in the bosonic theory, we µℓ, N sinh ie Eτ 1 + µℓ, − + 1 2 cos ( +

1 2 )= sin ( ρ ) 2 1 ρ e ρ ( ( 2

µ ρ F and or , F iSφ ) ρ sinh ) or ( − – 14 – ρ 2 into account. Given the ansatz ( ( 2 3 2 / 2 3 2 3 F ρ/ ℓ iEτ − and there are implicit Grassmann-valued numbers , +1 , − ± − 3 2 E e , = 1 the wave functions of the massless and massive e µ 2 3 ) f )= 2 3 ψ and f ρ N µℓ ( N cosh 1 µ ( ) = ( = )= F )= µ − ), it is straightforward to show that the spin is fixed to Re ψ E,S 1) theory will contain three gravitinos with conformal weig E,S . The energy and spin of the left-moving gravitini are ( , ( )= E,S 4.29 E )= ( ρ 0 ( 0 ψ = (1 ( ]. F 3 , and similarly for the other components of i N Re θ j θ − = j ) is given by the same expression as before. Note the sign chan θ i ρ θ ( F Note that in the classical supergravity analysis, it is unde The previous analysis can be carried over for the to the conserved charges,structure hence found in Topologically [ Massive Supe gravitino coincide. As will be shown in the next section, thi Even though in this paper we arethere interested are in the first supersymme indications for the existence of a supersymme Grassmann-valued Majorana spinors. Thus,alternatively, the imaginary) physical parts wav of associated with all fermion spinor components. The “physic corresponding sign changes in which fixes the value of where primary field constraint ( and that right-moving gravitino fields are given by and These states correspond to a right-moving gravitinoin and a the ma bulk. The may be written as equations of motion then fix the energy and spin to be where The former is clearlyDirac a equation solution with to appropriatesupersymmetric simple mass. partner of supergravity the It and leftso-called corresponds h moving fermionic to graviton. “massive” propagating the The degreeobserve sec of that freed at the chiral point given by JHEP10(2009)004 3 of ill µν gh g (5.2) (5.1) AdS ng cos- ] through = 1. One 3 l [ µℓ MSG on an ine the supercharge. , the energies of some µ 6= 1. sitive energy theorem. As ges as defined by Abbott ation was that for a non- s, one can speak of Chiral s: a background value ¯ uation of motion into three chiral behavior allowing us µℓ ned in the introduction, this ues of . r derivative theories. Indeed, gically Massive Supergravity o rectify the situation, these heory. However, at the chiral the chiral point t satisfies Einstein’s equations mptotically flat. This leads to tional energy fails and they sub- = 0 ows for the definition of the energy ve semi-definite, and so the theory µν . g , µ µν ψ + Λ h ) R + τ, ρ µν ( µν g y g 1 2 – 15 – = = ¯ − µ µν ψ µν g R ≡ µν G ] shows that TMG has ghosts for ], there is an additional solution to the gravitino equation 24 . A detailed study of this supersymmetric logarithmic mode w 34 ρ , 4 , which does not have to be small, but needs to vanish fast enou background is stable against metric perturbations. 3 µν ] Abbott and Deser noted that in the presence of a non-vanishi h ln cosh 20 AdS − ]. The energy was previously calculated for the bosonic mode 20 iτ − = y Once the energy is computed, we extend the analysis to determ We first compute the energy of individual modes and show that T mological constant, the conventional definitionsequently of determined gravita a modifiedvanishing cosmological definition. constant, the Theirthe space-time key failure is observ of not asy conservationauthors constructed for conserved the charges for conventionalwith a charges. arbitrary background cosmological tha constant T might have hoped to make usemay of be supersymmetry to inferred, arrive one atfailure is a po can not be able to tracedthe show back positivity. to recent calculation the As of existence explai [ of ghosts in highe Insertion of this expansion into themomentum Einstein pseudo-tensor. equations all This is done by partitioning the eq All conserved charges ofto the show supersymmetric that the theorySupergravity. theory exhibit is stable against perturbations.5.1 Thu Bosonic charges:In energy their work [ at infinity appear in a future publication. 5 Energy and superchargeIn in this TMSG sectionis the analyzed. stability under Thisand perturbations is Deser of done [ through Topolo the study of conserved char in the bosonic case [ with background is unstable in general. Stability is restored at of the modes arepoint, negative, the indicating energies an for all instabilitydefined linear in on perturbations the an are t positi Hamiltonian methods, where it was shown that for generic val motion at the chiral point To construct the charges, the metric is divided into two part and a deviation JHEP10(2009)004 ) of (5.8) (5.7) (5.4) (5.9) (5.3) (5.6) (5.5) (5.11) (5.10) 2.9 , so that µν T ]). They later , 38 , (2) µν 25 T lution of the vacuum ) due to the absence of . Since the background )= he gravitational energy- λ ψ ( µν momentum pseudo-tensor ( h e modified by the presence iton field equation ( , O m pseudo-tensor (2) µν ear in metric fluctuation, and . pseudo-tensor is the collection 3 . F tely. One then takes the terms . λ µν − ν . gravities (see [ 3 h ξ ) λ O ν Λ µν 0 is the Cotton tensor with only terms + T − . O h, ψ xT ( . µν = 3 µν (1) + j (2) µν µν 2 (1) ed R µν T C λ ) = 0 (2) µ = 0 C µν ν 1 . The expressions for the linearized Einstein Z µ ψ (1) . To obtain the explicit form of the energy ¯ ξ g + = 2 µν 4 C 2 1 µν λ − µν 0) theory, we apply the perturbation expansion – 16 – T µν 1 1 h µ (1) πG µν ) , µ T − + 8 ( G ¯ λh + ∇ µ µ (1) µν ≡ h, ψ ∂ + = (1 ( µν R λψ ) ν µν (1) (2) µν ξ = N = g ( G G µ ) to denote the expansion of the relevant object containing E = ¯ (1) µν − , one finds ψ n . It is straightforward to show that the energy momentum G 2 ]: µν µν λ g )= h 20 j ( , [ O in ν (1) µν ξ n C 1 µ )+ j is the linearized Einstein tensor, ( (1) µν 4 µν G (1) G and Cotton tensors were given in section is Einstein the zerothnonlinear order in contribution metric vanishes fluctuations immedia to define the energy momentu of all terms quadratic or higher order in linear in metric fluctuations retained and the stress energy a final term containing all terms quadratic or higher order in pieces: a piece dependent only on the background, a piece lin the equation of motion reads momentum pseudo-tensor of the of section applied their analysis toof TMG the . Cotton In tensor, thisequations case, of and TMG. the the Insertion equations background of ar is the expansion now yields taken to be a so TMSG and working to Here, A similar analysis can be carried out for higher-derivative Here, taking the superscript ( terms only of order and where background fermions. Applying these expansions to the grav When the Killing vectormomentum density, is thus the taken gravitational to energy be time like, this defines t We remind the reader that the gravitino expansion starts at pseudo-tensor satisfies the background Bianchi identity One can obtain conservedwith currents a Killing by vector contracting the energy JHEP10(2009)004 ) 4.25 (5.18) (5.17) (5.14) (5.19) (5.20) (5.15) (5.16) (5.12) (5.13) , , er, the fermionic , ) . . ψ ( 0 Lµν Rµν ˙ ˙ h h ) Mµν ) Mµν (2) νµ ˙ ˙ h h S M M µν L µν R h h ψ βν M ψ βν M µν 0 ( 0 ( h 0 h 0 0 ¯ ¯ µ µ ∇ ∇ 0 β 0 β xǫ ashion exactly agree with the x x (2) (2) νµ νµ , , 3 3 3 ght, and massive modes ( ) xǫ ) xǫ S S 3 3 ed L R ed ed µν µν ψ Rµν ψ Lµν 0 0 ed ed in the fashion discussed above. To ( ( ˙ Z ˙ h h Z Z 0 0 . xǫ xǫ 1) theories are related by parity, it is Z Z 2 µν R 3 µν L 3 , ν (2) (2) νµ νµ λ h h 2 1 1 1 S . 1 S ed 0 ed 0 ψ µℓ µℓ ) ρ ¯ ¯ ∇ ∇ µν µν O − − E γ = (0 Z Z 0    0 x x 2 µ 2 i 0 0 3 3 1+ ℓ 1+ can be determined by the general procedure ℓ − 1) 1) ¯ xǫ xǫ N ψ . Given these explicit forms, one can obtain 2 2    3 ) 3 − − ed ed 4 1 µ µ A n − − ( µν , given by the antisymmetric part of the torsional – 17 – ed ed = Z Z ρ = C µℓ µℓ µ Z Z ρ µν ξ πG πG πG πG 1 1 1 1 0) and 1+4(1 S ) denotes expansion to second order in perturbation 1 1 µν ) (2 ) (2 , µℓ µℓ 64 32 32 64 and S 1 1 µℓ µℓ µℓ ) µℓ − − + + + + n 5.13 1 µℓ ( µν 1 = (1 1 − − 2 − − 1) theory the energies take the form G − − , 1 1 (1 (1 − N 2 2 1 = (0 ] and explicitly by 1 1 πG πG 1 1 1 1 πGℓ πGℓ πGℓ πGℓ 26 N 32 8 8 8 8 32 1 πGℓ 8 ======L L R R = M M E E E E E E F E ], the Hamiltonian approach was used. Next, to quadratic ord 3 ) and are given explicitly in appendix 4.3 Taking into account that the ( As usual, the (2) superscript in ( where the functional forms for do so, one identifies the time-like Killing vector as theory. Recalling the energywe and find spin the values total for energies the are left, given ri by the Abbott-Deser-Tekin gravitational energy to contribution to the energy is given by The bosonic contributions to theresults energy of determined [ in this f where we have introduced the torsion, easy to see that in the Christoffel connection [ JHEP10(2009)004 (5.23) (5.25) (5.21) (5.22) (5.26) (5.24) are the 1) theory , B,i ]. Similarly, E 24 = (1 N , , ) 3 λ ( F,M ons do not contribute to ext order in perturbation O ains ghosts [ E e energy will be negative, , , roduction, it is known that e begin by considering the + ]. Here, of course, it is im- 1) , , which is the gravitino field n of motion, we can calculate µ lutions to TMSG with Brown- ergies of the F,R 0 ) 24 F,L ], we conclude that for generic − to second order. Schematically, J semi-definite as one may naively 3 ψ E E n concurrent work, Andrade and > ( Q µℓ . B (2) µ 1 1 0 E µℓ ) (2 µℓ F 1) model become positive semi-definite xJ , µℓ − − 3 = 2 )+ 1 1 − ed R dependence factored out, and (2) = (1 ψ − Z + µ ( + (1 N , E – 18 – (1) µ 1 B,L B,R πG F 8 E E B,M = 0 E 2 = M λ 1 1 1 Q µℓ µℓ E ) with the − )+ = − 2 ψ 1 1+ ]. L ℓ ( 3 5.13 2 so that E µ (1) µ 2 − − λ λF = = = L = 1 such that R = M E E µ E µℓ F is as defined in ( is the temporal component of the supercurrent F,i 0) theory. As with the energy, we can calculate , 0 E J The evaluation of the fermionic energy indicates that fermi = (1 are given by At the linear level the corresponding expressions for the en where Here values of the coupling constants, theexpect energy for is not a positive supersymmetricwhen theory. there As are elaboratedthereby ghosts in spoiling in the the the int usual positivityMarolf theory, arguments. showed their that Indeed, contribution TMG i at toas general in th values the of bosonic the model, coupling the cont energies of the plicit that to getHenneaux the boundary signs conditions for have the been energies chosen. stated above, so 5.2 Fermionic charges: supercharge Using the a perturbativethe expansion supercharge of to the order gravitino equatio at the chiral point quadratic order, though wetheory. expect By them the to same contribute reasoning to as the in n the bosonic model [ equation contracted withN the appropriate Killing spinor. W bosonic energies as given in [ the gravitino field equation takes the form in agreement with the disappearance of ghosts found in [ JHEP10(2009)004 ) , 2 λ ν (5.27) (5.35) (5.33) (5.31) (5.38) (5.36) (5.37) (5.30) (5.34) (5.28) (5.39) (5.29) (5.32) ψ ) µν a h ψ 1) theory: 2 ρ , ℓ 1 ¯ γ 4 µ b e − = (0 + µ a , N (2) ψ b R ρ γ ℓ µ ρ 1 ψ 2 . . δ , of the mn (2 γ i µ h,L , ab h,M h,M )¯ h,R , 0) supercharges (to order h Q ρ Q Q Q , − ρ ( (1) σν ψ ψ R . . . λ mn ) 1 1 1 mn ¯ 1 µℓ γ λ µℓ µℓ = (1 µℓ γ (1) ν (1) )¯ ψ νλ − ) ω = 0 − − = 0 ρ h σνρ − N h β b ( ¯ ǫ γ 1 1 1 νρ 1 R e M νρ 0 µ i α a σλ 0 ǫ Q nm 8 e Q h + ¯ ξǫ + − − µ ν , (1) ν ¯ − , ξ x δ ω ) x 3 ) ν i i µναβ ω,L 3 + ω,R ψ ω,M ω,M ed R Q – 19 – µνρ , h , h σ ed Q i Q Q ǫ i µν ψ Z 1 4 h λ Z = ( , ψ , ψ ℓ 1 i ¯ i 1 γ µℓ 1 2 1 1 =0 and ξ ξ µℓ µℓ µℓ =0 and ( ( ρλ 1 )= πG πGℓ 1 − + L (1) µνab M h ω h − requires expanding the gravitino field equation to order − − ψ 1 ρ µ ν ( 32 32 R Q Q Q Q 1 1 1 δ µ µ ψ = = − (2) (2) = mn )= )= = = = σ F R γ h,i L ω,i )¯ ψ R ρ h M M Q dependent terms. The result is Q Q ). ( ¯ Q γ Q Q µ ν = 1, we have = 1, we have ψ ξ,ψ,h ξ,ψ,h h mn ( ( 2.10 ( µℓ µℓ h ω 2 (1) ν 1 ℓ Q Q ω 1 4 + µνρ ǫ given in ( µ )= (1) ψ ( F µ Applying a parity transformation leads to the supercharges (2) , and extracting the 2 F where with λ At the chiral point, At the chiral point, where the subindex labels individual modes. and and and After plugging in the first-order modes, we find the In the above formulas, we defined JHEP10(2009)004 (5.41) (5.40) (5.43) (5.42) imilar to its . he conserved charges ], as well as the recent t would be of value to rk in this direction was = 0 5 . , , uld not be proven, at the rbative regime around the a classical supersymmetric M on for Chiral Supergravity. tinguished point in param- = 1 supersymmetric exten- i) de Sitter radius. In Chiral h,M more work on the supersym- Q vanishing ine if the quantum theory is h,R h,L ves a scale hierachy, and the N tring Theory might be to map Q ], one may anticipate this par- Q Q cosmological Supergravity with 5 ’ chiral behavior. nction [ = 1 partition function calculated Given that Chiral Supergravity is 1 and 1 µℓ 1 µℓ µℓ N − , − 1 1+ 1 h,R + Q − + 2 − ], it seems likely that Chiral Supergravity is a ω,L ω,R ω,M – 20 – Q Q Q 39 ω,R Q is stable against perturbations. 1 1 1 1), charges are found to be µℓ µℓ µℓ , 3 = 2 − − R 1 1 1+ = (1 N ]. They found a consistent truncation of higher dimensional = = = , Q L 40 R M Q Q Q = 0 ]. 3 L Q = 1, at which the theory acquires a chiral nature in a fashion s µℓ background. The wave functions have been constructed, and t ]. 3 Several important questions remain: Given the existence of Although the positivity of energy of chiral Supergravity co If Chiral Supergravity is consistent at the quantum level, i It would also be interesting to compute the partition functi 41 sion of Chiral Gravity.AdS The theory has been studied in a pertu were computed to second order.eter These space, charges picked out a dis Similarly, at the linear level Therefore, the fermionic charges share the bosonic charges 6 Conclusion In this paper we have constructed Chiral Supergravity, the extension of Chiral Gravity,consistent. it would Also, be given interesting the to calculation exam of the partition fu tition function to correspond to the chiral part of the chiral point, it was shown that AdS claim about renormalizability of TMG [ consistent theory of gravity even at the quantum level. Some Supergravity with matter fields to pure three-dimensional, metric theory needs to be done to confirm this. see whether or notdone it by is Gupta derivable and from Sen String [ Theory.a Some gravitational wo Chern-Simons term.Chern-Simons coupling is Their assumed truncation toSupergravity, be invol unfortunately, these smaller paramteres than are the equal. (ant three-dimensional, an alternative way toChiral find Supergravity a to relation a to string S sigma model. Due to the recent work of Maloney, Song and Strominger [ At the chiral point only the right moving supercharge is non- bosonic counterpart [ in [ JHEP10(2009)004 ] ], ify 43 42 +). The , + . , ], there is an − β σ δ 34 2 , , − 4 2 avity, there are first = e University of Texas nancial support under dρ ex structure such that er for useful discussions. . Moreover, we chose a µασ + work was presented. We ǫ 2 and curved coordinates 2 ∞ r, M. Henneaux, C. Keller, ic case [ kind hospitality during the a possible dual logarithmic tions in the future. of signature ( , this project was carried out. f this project. Many thanks nizers of the “Workshop on ge were reported, it would be µαβ 1 = η s have been discussed in [ ǫ ρdφ f Log Gravity (a detailed dis- ecify the boundary conditions. he chiral point , 2 ρ is cyclic. In such a geometry, one = 0 ], where difficulties in constructing and φ 16 + sinh , , 2 µ ψ ) αβ 2. Moreover, the manifold parameterized λσ ρdτ δ a, b, . . . in global coordinates: , 2 − τ, ρ 1 3 ( , y = – 21 – cosh = AdS − = 0 µλσ µ ǫ ψ 2 ℓ µαβ = ν µ, ν,... , ǫ = 2 version of TMG. Kaura and Sahoo had one attempt [ dx µ N dx = +1 . The fermionic boundary terms of the action are needed to ver µν ρ g 012 = ¯ ǫ 2 − ds ln cosh = − 012 ǫ iτ − = y Though we focused on the supersymmetric version of Chiral Gr In light of the recent work of Gaberdiel et. al. [ = 2 extremal conformal field theories with large central char are labeled by Greek indices, background curved space is taken to be by the flat coordinates is endowed with a Minkowski metric interesting to construct an with N but clearly more work needs to be done. indications for the existencecussion of is a provided supersymmetric in version a o future publication). As in the boson and references therein. We hope to address some ofAcknowledgments these ques It is a pleasure toW. Li, thank D. T. Marolf, Andrade, A. G. Maloney,We D. M. Compère, thank Grumille Rocek, E. W. Song Sezgin and forto A. his Stroming the collaboration Erwin at Institute Schrödinger the3D (Vienna) early Gravity” and stages (April to o 09) theespecially for would orga like a to very thankprogram exciting the “Fundamental Aspects meeting KITP of Santa where Superstring Barbara Theory” this This for where uts work was supportedA&M. by M. NSF B. under would grant likeNational PHY-0505757 to Science and thank Foundation th under KITP Grant Santa No. Barbara for PHY05-51164. partial fi A Notation and conventions Throughout the course offlat this coordinates paper are we labeled have by worked Latin with indices, an ind additional solution to the gravitino equation of motion at t and the Lorentzian theory is considered so that only that this mode obeys theThis variational would principle allow and for better asuperconformal sp detailed field study of theory. Log Fermionic Supergravity and boundary condition can speak of aspace-time orientation conformal such boundary that understood to lie at JHEP10(2009)004 χ , ), we . β 4.2 b ] µc α , ω h rac adjoint ) ac ν µ [ and ω µν ¯ ∇ g − ρ , g ν λ b νc ¯ ∇ ψ − ω ac + µ λ αβ ing useful identities: µρ ω χ , g Γ ν µ ν ] + γ ∂ − µ ν h β ab µ + β [ ψ ¯ φγ ω ν ¯ ρν ab ∇ g , and tensors are ∂ = γ are taken to lie in the Majorana µ µ α smann-valued spinors. Taking ab φ α − ∂ ψ ν ( ¯ ω ∇ γ ab ν 1 4 µ λρ ω g µ + αβ ¯ χγ 2 1 ∂ h β ) and the Rarita-Schwinger field ( ψ = = + α ) , ν ∂ . ab χ , 4.1 α µν µ λ ( h µν µν = νσ β Γ h ¯ φγ R β – 22 – ¯ X Λ ∇ , vector-spinors ψ − µ ǫ , h , ] σ µρ α , , − α , = Γ ) ) ) h 2Λ µν h h ) φ β − ( ( [ h µα µ ψ − ¯ g ¯ ( ∇ µν σρ , (1) (2) ǫ , D ¯ ρ χγ νβ h µν and 0) theory: X + , R R ab h 2 h , 3 (2) µν γ ℓ a ν ν σ µν µν µν κ − e αβ ν µσ ¯ ab µ ¯ ¯ ρ g g Γ ∇ b µ − ) h . Γ ω µ 2 2 1 1 ¯ e ν = (1 ¯ ∇ ) φχ , T − 4 1 νβ σb ¯ ) ∇ ¯ h ∇ − ¯ e g − − , ∗ ( = N νρ a ν ) + ) ) , b ν + )+ χ αβ e µα e h h X ǫ ω h (2) µν h h ( ( h a µ ( ¯ ( µ µ χφ µν 2 + ( µ e R = ( ∂ ∂ , g 2 ¯ ¯ ab µν ∇ (2) (1) µν µν are three-dimensional gamma matrices and bar denotes the Di ∇ † (2) ℓ µν νb 1 µν = = 4 2 1 e χ R R g G − R a µ ν b ǫ γ e νρ ∂ µ a µ = = 2Λ¯ = 6Λ = = Λ µ a a ν e )= ) = ¯ )= )= )= X D e e ¯ R ab µ µν h h h h (1) = ( ( ( ( ¯ , with = = R µν ∇ R (1) µναβ 0 µ h, ψ ¯ γ ( R to be Grassmann-valued Majorana spinors, we have the follow γ (2) (2) (2) (1) µν µν µν R R ab µ † G G ω R R χ φ The curvatures and connections are given by (2) µν G = ¯ where representation. Moreover all fermionsand are implicitly Gras The theories under consideration all involve spinors which χ find the tensors of the Covariant derivatives acting on spinors Upon a perturbative expansion of the metric ( JHEP10(2009)004 . ) b , , , ρ ↔ ) a ρ ν ) . 02 ( , h 12 (1) βc ↔ − 22 h e 2 cosh ) h µ ν α 2sinh 2 1 − βc ∂ ψ , ( µν ¯ e τ ¯ g c µ = = = α ¯ γ β ¯ e ) + ( h ρ ∂ ψ β b ( h ¯ Λ ψ ν ¯ e ( c µ (1)0 ρ (1)1 ρ (1)2 ρ ¯ , γ . ¯ α a e e − α ¯ e β σρτ σ 0) theory satisfies (1) a βλ ¯ ǫ ψ 1 2 , ein along with our linearized ψ G ψ µν (1) b ν λ e h µσ αβ bc − ¯ γ g α a . µ γ ρ ) (1) να ¯ ǫ 2¯ e = (1 , ¯ ψ Γ 2 1 Λ (1) µb 2 . . − i = 3Λ µνbc 8 N e µτ αβ λ ρ ¯ µ ν g , e µ µν R σ λ ρ ∂ ψ )+ ǫ (2) − ψ βν and h h 4 1 , e ρ τ , O ψ ρ ) G ρστ βc ¯ γ − − µ 1) ǫ ρ ¯ e ¯ ψ + σ ∇ + ¯ , 01 γ ) νσµ ¯ ( e α ¯ b ν ψ − h λ λαβ 2 h (1) bν a ρ ∂ 11 µν − ¯ ǫ ( 12 i ψ e ¯ ψ ( R h h 2 cosh E µ µℓ (1) µ h c µ (2) βν ρστ ( λ µλ ρµ ¯ 8 ∂ + e (1) βν 2sinh 2 ǫ 1 − κ − ( h – 23 – h ¯ β b ℓ λe 1 µνab ∇ ρ ν ν a λ ν ν ¯ e µν − ¯ = = = e ¯ ψ R ¯ ∇ h ¯ + g α ∇G ¯ − = ρ ∇ ρ 2 1 )+ α a µ ¯ γ λ µ ψ = (1) a h αβ 2 ¯ e + µ ν ∇ λσµ h ( e (1)0 φ (1)1 φ (1)2 φ ) (1) µ µ a ¯ ¯ γ ν σ ǫ ψ 1 2 f )+ E = ¯ ψ ρν αβ σ ¯ ¯ 2 (2) h R βν ∇ ] ¯ µ h µb − ψ a µ ν − G ǫ ¯ µ e e ρ + = − )+ α ν λµ ¯ ∇ ψ (1 ¯ ∂ λ ν g ¯ , D σ µ βc ∇ ν )+ µ h ¯ ¯ e h e ¯ γ − h µ 2 α αβ λρ D σλρ , e µ ( i [ ∂ µ ¯ µ ǫ ¯ h ∇ bν ψ ( µℓ ǫ , e ρ ¯ e ¯ ¯ e 2 1 c ∇ (2) 2 µν i ℓ µ , ρ i 4 2 4 2 1 1 − C (1) µ ∂ , e 00 σ − ( e h ¯ 01 ∇ ν β b 02 ¯ h e 2 cosh )= )= )= )= )= )= h (1) a α a h h h 1 ψ 2sinh 2 ψ e ¯ − e ( ( ( ( ( 2 1 2 1 λ λ ρ h, ψ = = = ( (2) − µν (2) µν − (2) µν (1) µν (2) µν C F Γ Γ = κ (2) µν (1)1 τ (1)2 τ (1)0 τ e e e C (1) µab ω At the linear level, the fermionic field strength of the The vielbein decomposes as Employing the relationship between themetric metric solution and we find the vielb The linearized spin connection in terms of the vielbeins is It is useful to note and JHEP10(2009)004 (B.1) ymmetries by . is a hypersurface. is a curved space. , 3 3 µνρ ! ǫ 2+2 1 ” index represents the R AdS − AdS = − onents. ] 1 0 0 , ρ onent Weyl spinors. These γ

ρ ν ˆ γ pinors on the global group of mmetries. We use the block γ = . When dealing with fermions µ 2 [ µ µνρ γ ds in three dimensions. Fermions 2. The “ γ ǫ of which , − 1 = , . . We have used the notation → − 0 2+2 , ρ , , ]= and R µ µνρ , ν . AB − ! γ ˆ γ µν acting on fermions. η A , g 0 m , ! = cosh γ µ γ L P R γ ρ = 2 A [ˆ ! m µ A ψ ψ 0 = 2 e ˆ γ and }

2. The hypersurface in } B

0 1 1 0 , , – 24 – ν = sinh 1 Γ on this curved space: γ , ρ

ℓ µ , , = µ γ ) γ µ A Ψ= γ 1) and = µ = A γ , { Γ γ 1 = 0 Γ 1 µνρ { ν ǫ , 012 γ 1 m ). If we deﬁne a Dirac spinor on ǫ − − , γ ]= , ν ν B.1 1 ! γ , for − , γ to deﬁne , so that the gamma matrices explicitly read: µ m µ γ γ 1 0 γ m µ ( [ 0 1 2+2 e − − 1 2! R

= = = diag( = m η 0 γ µν 2). This is the isometry group of γ γ , , and ˆ I = − γ = ˆ − B Clifford algebras, spinor representations, and discrete s In TMG, a parity transformation effectively takes in TMSG, we must also consider the operator B.1 Clifford algebraThe and structure spinor of representation the gravitinoisometries, fields SO(2 descends from defining s Four component “Dirac” spinors areWeyl decomposed spinors into have two the comp correct dimensionality forare spinor taken fiel to be Majorana spinors andB.2 have anticommuting comp Discrete symmetries The Dirac notation isdiagonal form convenient from for equation discussing ( discrete sy The Clifford algebra of gamma matrices is Three-dimensional gamma matrices satisfy Here the metric is We employ the dreibein and in addition γ where again, with curved indices with additional direction in JHEP10(2009)004 , , ] , y at , (2008) 082 , (2000) 409] 04 ]. , 201 JHEP arXiv:0905.2326 , [ SPIRES [ ]. ]. The ultraviolet behavior of using Dirac notation. In this ]. Erratum ibid. iban, ]. C SPIRES SPIRES ] [ ] [ ]. (2009) 081301 . L µ ψ R µ SPIRES SPIRES , . hep-th/0509212 ψ 103 . ] [ = ] [ . R µ , . ! = Massive gravity in three dimensions ψ Ψ R µ L (1988) 406] [ µ SPIRES 0 C 0 1 0 ψ L γ ψ µ γ T ] [ Γ 1 0 ψ i 0 − γ 0 1 The black hole in three-dimensional space-time 185 iγ ˆ γ Chiral gravity, log gravity and extremal CFT iγ = = −

– 25 – Topologically massive gauge theories Ψ= arXiv:0805.2610 L µ R ¯ µ = Ψ=Ψ = [ = arXiv:0901.1766 ψ P [ ψ Chiral gravity in three dimensions L µ C R µ C C ψ ψ Phys. Rev. Lett. pp waves of conformal gravity with self-interacting source Exploring AdS waves via nonminimal coupling hep-th/9204099 hep-th/0512074 Instability in cosmological topologically massive gravit [ P [ P , ]. Erratum ibid. hep-th/0409150 [ [ ]. (2008) 134 07 (2009) 201301 SPIRES (1992) 1849 SPIRES [ ] [ (2006) 104001 102 69 (2005) 175 (1982) 372 JHEP , 317 140 D 73 The string landscape and the swampland ]. ]. supergravity at four loops =8 Phys. Rev. Lett. SPIRES SPIRES arXiv:0801.4566 the chiral point Ann. Phys. Phys. Rev. Lett. Ann. Phys. Phys. Rev. arXiv:0903.4573 [ N [ [ [3] W. Li, W. Song and A. Strominger, [1] C. Vafa, [4] D. Grumiller and N. Johansson, [2] Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson and R. Ro [5] A. Maloney, W. Song and A. Strominger, [6] E. Ayon-Beato and M. Hassaine, [7] E. Ayon-Beato and M. Hassaine, [8] E.A. Bergshoeff, O. Hohm and P.K. Townsend, [9] S. Deser, R. Jackiw and S. Templeton, [10] M. C. Bañados, Teitelboim and J. Zanelli, and then the parity operator acts on this spinor as It is also usefulcase: to define the charge conjugation operator In Weyl language, we have In Weyl language, we have References The Majorana condition is then JHEP10(2009)004 , ] D 2) 2 , 20 ]. (1977) 249 = (2 ]. N 39 (1983) 97 , , (1978) 207 ]. , SPIRES arXiv:0807.2613 arXiv:0909.0727 [ [ SPIRES ]. ]. , (2006) 022 ] [ B 120 Extremal B 73 01 SPIRES [ SPIRES SPIRES [ Class. Quant. Grav. [ , JHEP Phys. Rev. Lett. Ooguri, , , Phys. Lett. (2008) 205005 Phys. Lett. , , , , 25 arXiv:0805.4216 ]. , arXiv:0803.3998 (1981) 189 (1986) 105 [ 68 Cosmological topologically massive gravitons ]. 103 arXiv:0706.3359 ]. , , Brandeis preprint 82-0692. SPIRES Conformal invariance, supersymmetry and ]. The general supersymmetric solution of Energy and supercharge in higher derivative gravity – 26 – Boundary analysis of topologically massive log SPIRES [ ]. ]. ]. SPIRES [ (2009) 075008 Unitary representations of the Virasoro and Phys. Rept. , Class. Quant. Grav. (1986) 93 [ 26 SPIRES [ , Central charges in the canonical realization of asymptotic SPIRES SPIRES SPIRES [ [ ] [ Supergravity has positive energy (1986) 207 No chiral truncation of quantum log gravity? ]. ]. B 271 Stability of gravity with a cosmological constant (1981) 381 Positivity of the total mass of a general space-time Holographic gravitational anomalies Commun. Math. Phys. Witten-Nester energy in topologically massive gravity Supergravity Topologically massive supergravity 104 80 Energy in topologically massive gravity , (1979) 1457 (1982) 76 SPIRES SPIRES (1986) 336 [ 43 ] [ Nucl. Phys. Positivity of the energy in einstein theory Class. Quant. Grav. gr-qc/0307073 B 195 , , , work in progress. B 168 Three-dimensional gravity revisited A simple proof of the positive energy theorem Cosmological topological supergravity ]. ]. ]. ]. ]. Nucl. Phys. Phys. Rev. Lett. hep-th/0508218 SPIRES SPIRES SPIRES SPIRES SPIRES (2003) L259 [ and photons [ Commun. Math. Phys. [ super-Virasoro algebras Phys. Lett. arXiv:0903.3779 [ [ [ symmetries: an example from three-dimensional gravity Commun. Math. Phys. string theory [ supergravity conformal field theories and constraints of modularity topologically massive supergravity [23] E. Sezgin and Y. Tanii, [26] P. Van Nieuwenhuizen, [30] P. Kraus and F. Larsen, [31] S. Carlip, S. Deser, A. Waldron and D.K. Wise, [24] T. Andrade and D. Marolf, [25] S. Deser and B. Tekin, [29] P. Goddard, A. Kent and D.I. Olive, [15] E. Witten, [20] L.F. Abbott and S. Deser, [22] D.G. Boulware, S. Deser and K.S. Stelle, [27] J.D. Brown and M. Henneaux, [28] D. Friedan, E.J. Martinec and S.H. Shenker, [14] S. Deser, [18] R. Schoen and S.-T. Yau, [19] S. Deser and C. Teitelboim, [21] M.T. Grisaru, [12] M. Becker, P. Bruillard and S. Downes, [16] M.R. Gaberdiel, S. Gukov, C.A. Keller, G.W. Moore[17] and H. E. Witten, [13] S. Deser and J.H. Kay, [11] G.W. Gibbons, C.N. Pope and E. Sezgin, JHEP10(2009)004 ]. ]. , , , ] , SPIRES [ ty (2008) 084 , SPIRES [ 09 , , supergravity (2002) L143 3 Anti-de Sitter/CFT JHEP (1998) 085020 , 19 AdS arXiv:0901.2874 [ black holes ]. arXiv:0808.0506 ]. D 58 , . Ortiz, ]. ]. ]. arXiv:0905.1536 ]. , (2 + 1) theory of SPIRES [ SPIRES 0) SPIRES ] [ SPIRES , SPIRES ]. ]. Phys. Rev. ] [ ] [ (2 SPIRES ] [ Topologically massive AdS gravity , ] [ (2009) 081502R Class. Quant. Grav. Asymptotically Anti-de Sitter spacetimes in Canonical analysis of cosmological topologically , SPIRES SPIRES D 79 – 27 – ] [ ] [ Topologically massive gravity at the chiral point is not arXiv:0806.4185 , arXiv:0901.3609 [ hep-th/0212292 Supersymmetry of the [ hep-th/9310194 arXiv:0807.0486 [ [ Supergravity at the boundary of AdS supergravity Phys. Rev. arXiv:0807.4703 ]. [ Quantum gravity partition functions in three dimensions , ]. ]. ]. Consistent truncation to three dimensional (super-)gravi Boundary S-matrix in a Energy in generic higher curvature gravity theories arXiv:0710.4177 arXiv:0809.4603 [ [ SPIRES (1994) 183 (2008) 272 SPIRES SPIRES SPIRES [ ] [ (2009) 085006 (2003) 084009 ] [ ] [ (2008) 045 72 A simple proof of the chiral gravity conjecture 10 B 666 D 79 Constraint dynamics and gravitons in three dimensions D 67 (2008) 002 (2008) 015 A note on Lie-Lorentz derivatives ]. Renormalizability of topologically massive gravity 12 03 JHEP , Phys. Rev. Phys. Lett. arXiv:0805.4328 hep-th/0206159 hep-th/9805165 SPIRES [ massive gravity at the chiral point [ JHEP Phys. Rev. Lett. topologically massive gravity [ JHEP arXiv:0712.0155 Phys. Rev. [ chiral correspondence in three-dimensional supergravity [43] A.J. Amsel and G. Compere, [45] D. Grumiller, R. Jackiw and N. Johansson, [46] M.-I. Park, [34] M. Henneaux, C. Martinez and R. Troncoso, [41] A. Maloney and E. Witten, [42] P. Kaura and B. Sahoo, [44] O. Coussaert and M. Henneaux, [35] A. Strominger, [37] T. Ort´ın, [38] S. Deser and B. Tekin, [33] G. Giribet, M. Kleban and M. Porrati, [36] M. K. Bañados, Bautier, O. Coussaert, M. Henneaux and M [39] I. Oda, [40] R.K. Gupta and A. Sen, [32] S. Carlip, S. Deser, A. Waldron and D.K. Wise,