DDedicated to I. V. Tyutin anniversary
Matter-coupled de Sitter Supergravity
Renata Kallosh
Stanford Institute of Theoretical Physics and Department of Physics,Stanford University, Stanford, CA 94305 USA
Abstract
De Sitter supergravity describes interaction of supergravity with general chiral and vector mul-tiplets as well as one nilpotent chiral multiplet. The extra universal positive term in the potentialdue to the nilpotent multiplet, corresponding to the anti-D3 brane in string theory, supports deSitter vacua in these supergravity models. In the flat space limit these supergravity models includethe Volkov-Akulov model with a non-linearly realized supersymmetry. The rules for constructingpure de Sitter supergravity action are generalized here in presence of other matter multiplets. Wepresent a strategy to derive the complete closed form general supergravity action with a givenK¨ahler potential K , superpotential W and vector matrix f AB interacting with a nilpotent chiralmultiplet. It has the potential V = e K ( | F | + | DW | − | W | ), where F is a necessarily non-vanishing value of the auxiliary field of the nilpotent multiplet. De Sitter vacua are present undersimple condition that | F |− | W | >
0. A complete explicit action in the unitary gauge is presented. e-mail: [email protected] a r X i v : . [ h e p - t h ] S e p Historical note
I was fortunate to start my career in physics under the great influence of Igor Tyutin’s workin early 70’s. Igor in collaboration with E. Fradkin and I. Batalin at the Lebedev PhysicalInstitute has made series of groundbreaking discoveries about the nature of quantum fieldtheories with local gauge symmetries: Yang-Mills theories and gravity. This field received atremendous boost in 1971 with the publication of the paper by ’t Hooft suggesting that gaugetheories with spontaneous symmetry can be renormalizable [1]. However, for a completeproof of renormalizability and unitarity of these theories it was necessary to prove equivalencebetween the results obtained in renormalizable gauges, where unitarity was hard to establish,and the unitary gauge, where renormalizability was not apparent. The equivalence theoremwas proven in 1972 in our paper with Igor Tyutin, using path integral methods [2]. Anindependent combinatorial proof of this result was given a month later by ’t Hooft andVeltman [3], with a reference to our work [2].One of the most significant works by Tyutin was a discovery of what is now known asBRST symmetry. T in BRST reflects his preprint which was published in 1975 as a LebedevInstitute preprint
The complete action of the supergravity multiplet interacting with a nilpotent goldstinomultiplet was recently constructed in [6, 7]. The action was named ‘Pure de Sitter Super-gravity’ in [6] since it has maximally symmetric classical de Sitter solutions, even in theabsence of fundamental scalars in the theory. The action has a non-linearly realized localsupersymmetry. In the flat space limit the action becomes that of the Volkov-Akulov (VA)theory [8] with a global non-linearly realized supersymmetry.The recent interest to nilpotent supergravity in application to cosmology was raised in[9] for VA-Starobinsky supergravity and developed in [10] in the context of the chiral scalarcurvature superfield R subject to a certain superfield constraint. Important implications ofthe nilpotent multiplet on the bosonic action and on cosmology were explained there. How-ever, a complete and general locally supersymmetric component action including fermionsin this approach was not yet constructed.A general approach to supergravity with a nilpotent multiplet was proposed in [11] usingthe superconformal approach in the form developed in [12]. It is described in detail in the2extbook [13], where also earlier references to superconformal derivation of supergravity weregiven. Using this approach it was possible to construct a complete locally supersymmetricsupergravity action interacting with a nilpotent multiplet [6, 7]. In particular, the differencebetween the pure dS supergravity actions in [6] and in [7] is due to a different choice ofthe superconformal gauges for the local Weyl symmetry, local R symmetry and local specialsupersymmetry. One can also derive de Sitter supergravity starting with a complex lineargoldstino superfield [14].The interest to a nilpotent goldstino in an effective N = 1 supergravity is enhancedby the better understanding of the relation to superstring theory, specifically to the KKLTconstruction of dS vacua [15, 16, 17]. The role of an anti-D3 brane placed on the topof the O3 (or O7) plane at the tip of the warped throat is now better understood [17].The corresponding string theory constructions of the nilpotent goldstino and an associatedspontaneous breaking of supersymmetry by the anti-D3 brane action may lead to a UVcompletion of the effective 4-dimensional de Sitter supergravity interacting with a nilpotentmultiplet.An additional reason for using a nilpotent multiplet in supergravity is to describe inflation[9, 11]. The nilpotent multiplet is particularly useful when constructing cosmological modelswith dark energy and susy breaking which are in agreement with the Planck/BICEP datasince they have a flexible level of gravitational waves and a controllable level of susy breakingparameter [18, 19].The newly discovered dS supergravity action [6] in the limit to a flat space leads to agoldstino model in the form of a nilpotent chiral multiplet as known from [20, 21, 22], whichis equivalent to the original globally supersymmetric VA goldstino model [8]. An interestingfeature of the pure de Sitter supergravity [6, 7] is that in the unitary gauge where the localsupersymmetry is gauge-fixed by the choice of the vanishing fermionic goldstino, the actionis e − L| goldstino=0 = 12 κ (cid:2) R ( e, ω ( e )) − ¯ ψ µ γ µνρ D (0) ν ψ ρ + L SG , torsion (cid:3) + 3 m κ − f + m κ ¯ ψ µ γ µν ψ ν (2.1)in notation of [6], where f and m are constants. The action (2.1) for Λ = f − m /κ > f (cid:54) = 0and Λ = f − m /κ ≤ f = 0 the action (2.1) has a restored linearly realized localsupersymmetry and becomes a well known AdS supergravity action [23].The non-trivial nilpotent multiplet is consistent only when its auxiliary field is not van-ishing. The nilpotency condition for the field S ( x, θ ) = 0 for S ( x, θ ) = s ( x ) + √ θψ s ( x ) + θ F s ( x ) includes three equations in terms of the component fields, a scalar, sgoldstino s ( x ),a fermion, goldstino ψ s ( x ) and an auxiliary field F s ( x ) s F s − ψ s = 0 , s ψ s = 0 , s ( x ) = 0 (2.2)There are 3 distinct possibilities to resolve the nilpotency condition : the first one, witha non-vanishing auxiliary field, leaves us with a fermion goldstino without a fundamental3calar. F s (cid:54) = 0 , ⇒ ψ s (cid:54) = 0 , s (cid:54) = 0 , s = ψ s F s solves all three equations (2.3)This solution is used in the locally supersymmetric action including a nilpotent multipletwithout a fundamental scalar. The second solution still with a non-vanishing auxiliary field,has a solution where only the auxiliary field does not vanish, but both the scalar and thefermion vanish. F s (cid:54) = 0 , ⇒ ψ s = 0 , s = 0 solves all three equations (2.4)This solution is the one where the local supersymmetry is gauge-fixed in the unitary gaugewith ψ s = 0 and s = 0.The third possibility, with the vanishing auxiliary field, is a trivial solution, goldstino aswell as sgoldstino both have to vanish, to solve all three equations (2.2) F s = 0 ⇒ s = ψ s = 0 solves all three equations (2.5)The purpose of this paper is to outline the strategy for the derivation of a completematter-coupled supergravity action with one of the multiplets constrained to be a nilpotentone. The price for a relatively easy procedure of finding dS vacua with spontaneously brokensupersymmetry is the non-linear fermion terms in the action. We will find, however, thatall these terms can be presented in a closed form. The upshot of the result obtained in thispaper is that we will describe a complete de Sitter supergravity with a non-linearly realizedlocal supersymmetry, which has the following potential V = e K ( F s K s ¯ s ¯ F ¯ s + D i W K i ¯ k ¯ D ¯ k W − W W ) ≡ e K ( | F | + | DW | ) − m / , (2.6)where F s (cid:54) = 0 is the non-vanishing auxiliary field of the nilpotent multiplet. For the vacuumto have a positive cosmological constant it is sufficient to require that at the minimum e K | F | − m / > . (2.7)In the past de Sitter vacua in models without a nilpotent multiplet were constructed byrequiring that | DW | − m / >
0, see for example [24] and more references for such modelsin string theory inspired supergravity in [17]. The common feature of all such models isthat to provide the positive value of | DW | − m / in the vacuum requires a significantengineering and practically always some computer computations. Several chiral multipletsare required and a choice of the K¨ahler potential and a superpotential has to be madecarefully. The advantage of our de Sitter models with a nilpotent multiplet is that all weneed is that one constant parameter, the value of | F | at the vacuum is larger that anotherconstant parameter, the value of 3 m / at the vacuum.Thus we proceed with the strategy to derive de Sitter supergravity: general chiral andvector multiplets and a nilpotent multiplet, coupled to supergravity multiplet.4 Superconformal theory with a nilpotent multiplet
We start with the underlying superconformal action in [11]: the SU (2 , | N = 1 supergravity coupled to chiral multiplets X I and to Yang–Mills vectormultiplets λ A superconformally. It consists of 4 parts, each of which is conformally invariantseparately. The superconformal action with all supersymmetries linearly realized is given bythe following expression: L sc = [ N ( X I , ¯ X ¯ I )] D + [ W ( X I )] F + (cid:2) f AB ( X )¯ λ AL λ BL (cid:3) F + (cid:2) Λ ( X ) (cid:3) F . (3.1)The first 3 terms are standard and in this form are described in [12] and in detail in thetextbook [13]. We will use here the notations of [13]. The non-standard last term dependson the Lagrange multiplier chiral superfield Λ. As different from other standard chiralsuperfields X I , it is not present in the K¨ahler manifold of the embedding space N ( X I , ¯ X ¯ I ).Therefore it can be eliminated on its algebraic equations of motion. The three functions N ( X I , ¯ X ¯ I ), W ( X I ), f AB ( X ) as well as all chiral superfields transform in a homogeneousway under local Weyl and R transformations: Here X I include the chiral compensatingmultiplet { X , Ω , F } , a chiral goldstino multiplet { X , Ω , F } , generic matter multiplets { X i , Ω i , F i } i = 2 , ..., n and a Lagrange multiplier multiplet { Λ , Ω Λ , F Λ } . Equations ofmotion for the Lagrange multiplier multiplet Λ consist of 3 component equations, for eachof its components. One of these equations for Λ( x ) is [6]2 X F − ¯Ω P L Ω + √ ψ µ γ µ X P L Ω + ¯ ψ µ P R γ µν ψ ν ( X ) = 0 . (3.2)It is solved by X ( x ) = Ω ( x ) P L Ω ( x )2 F ( x ) ≡ (Ω ) F , ( X ( x )) = 0 , (3.3)which also solves the remaining equations for Ω Λ ( x ) and F Λ ( x ).The next step in producing supergravity action starting from the superconformal action(3.1) is to impose the constraint (3.3), to eliminate the auxiliary fields F I and to fix localsymmetries of the superconformal action by gauge-fixing local Weyl and R symmetry and alocal special supersymmetry. The procedure is well known in the absence of the constraint(3.3). The non-vanishing value of the auxiliary fields is according to eq. (17.21) in [13]¯ F ¯ IG = N ¯ II ( −W I + 12 N I ¯ K ¯ L ¯Ω ¯ K ¯Ω ¯ L + 14 f ABI ¯ λ A P L λ B ) . (3.4)This is due to a Gaussian dependence of the action (17.19) on F of the kind L G ( F ) = N I ¯ J F I ¯ F ¯ J + [ F I ( W I − N I ¯ K ¯ L ¯Ω ¯ K ¯Ω ¯ L − f ABI ¯ λ A P L λ B ) + h.c. ] , (3.5)or equivalently L G ( F ) = N I ¯ J F I ¯ F ¯ J + F I ¯ F I G + ¯ F ¯ I F I G . (3.6)The new situation when one of the chiral multiplets is nilpotent may be studied first usingthe superconformal framework in eq. (3.1). Note that N ( X I , ¯ X ¯ I ), W ( X I ) and f AB ( X ) are5lgebraic functions of X I . The presence of the last term in (3.1), [Λ ( X ) ] F suggests that N ( X I , ¯ X ¯ I ), W ( X I ) and f AB ( X ) can only depend on X ¯ X ¯1 on X and on ¯ X ¯1 , since accordingto (3.3) ( X ( x )) = 0. We design the dependence on X , ¯ X ¯1 in N ( X, ¯ X ) such that it dependsonly on X ¯ X ¯1 .First, we will notice that in the expression F ¯ F G a dependence in ¯ F G on X will comeas proportional to F X = (Ω ) which makes such terms F independent. Or if there areterms in ¯ F G of the form X ¯ X ¯1 , they also become F -independent. Thus terms in L G ( F )which depend on F X or F X ¯ X ¯1 can be moved out from the F -dependent part of theaction into an F -independent part.However, term in ¯ F G depending on ¯ X ¯1 cannot be removed that way. Thus we have tokeep in mind that in F ¯ F G we keep only terms linear in ¯ X ¯ F G ( ¯ X ) = ¯ F G ( ¯ X ¯1 = 0) + ¯ F G, ¯1 ¯ X ¯1 (3.7)and a conjugate. Using this fact one can look at the complete superconformal action in eq.(17.22) in [13] and check that it can be given in the form e − L = ( F I − F IG ) N I ¯ I ( ¯ F ¯ I − ¯ F ¯ IG ) − F IG N I ¯ I ¯ F ¯ IG + ¯ X A c X + X ¯ B c + B c ¯ X ¯1 + C c . (3.8)We define e − L X ≡ − F IG N I ¯ I ¯ F ¯ IG + ¯ X A c X + X ¯ B c + B c ¯ X ¯1 + C c . (3.9)The part depending on auxiliary fields ( F I − F IG ) N I ¯ I ( ¯ F ¯ I − ¯ F ¯ IG ) we present as( F − F G ) N ( ¯ F ¯1 − ¯ F ¯1 G ) + [( F − F G ) N i ( ¯ F ¯ i − ¯ F ¯ iG ) + h.c. ] + ( F i − F iG ) N i ¯ k ( ¯ F ¯ k − ¯ F ¯ kG ) . (3.10)Dependence on F i and ¯ F ¯ i is Gaussian since e − L X does not depend on these fields. We canintegrate over them and find that ¯ F i − ¯ F iG = 0 (3.11)and get the action in the form e − L = ( F − F G )( N ) − ( ¯ F ¯1 − ¯ F ¯1 G ) + e − L X . (3.12)where ( N ) − = N − N k N − k ¯ k N k ¯1 . (3.13)The second term in eq. (3.13) is proportional to X ¯ X . Therefore, all terms in (3.12) of theform ( F − F G )( − N k N − k ¯ k N k ¯1 )( ¯ F ¯1 − ¯ F ¯1 G ) add to L X since F X = Ω /
2. This appears tomodify L X into ˜ L X . However, we may take into account that F − F G is non-vanishing onlydue to a non-Gaussian nature of the dependence of the action on F and this difference isproportional to goldstino. But the term N k N − k ¯ k N k ¯1 already has all possible dependnece ongoldstino via X ¯ X . Therefore these terms do not change the action.Thus, the remaining expression depending on auxiliary field is e − L = ( F − F G ) N ( ¯ F ¯1 − ¯ F ¯1 G ) + e − L X . (3.14)6ne more step is required to understand the possible role of the terms in (3.7). We rewritethe first term in (3.14) as follows( F − F G )( ¯ F − ¯ F G ) = ( F − F G )( ¯ F − ( N ¯ F ¯1 G + N k ¯ F ¯ kG ) , (3.15)using (3.11) and taking into account that ¯ F G = N ¯ F ¯1 G + N k ¯ F ¯ kG and according to discussionaround eq. (3.7) it is X -independent but may have some ¯ X -dependence. Here we take intoaccount that N is X and ¯ X independent and that N k is proportional to ¯ X . This meansthat linear dependence of ¯ F G on ¯ X translates into a linear dependence on ¯ X in ¯ F ¯1 G .Now something interesting takes place: let us define ¯ F ¯1 + ¯ F ¯1 G, ¯1 ¯ X ¯1 ≡ ( ¯ F ¯1 ) (cid:48) . It appearsthat the relevant dependence on ¯ X ¯1 = ¯Ω F ¯1 = ¯Ω
2( ¯ F ¯1 ) (cid:48) does not see the difference between F and F (cid:48) since ¯ X ¯1 ¯Ω = 0. An analogous shift can be made with regard to F which will absorbthe term in F G depending on X . This brings us to the following action e − L = ( F − F G ) N ( ¯ F ¯1 − ¯ F ¯1 G ) | X =0 + e − L X . (3.16)One more step is useful to reduce the problem to the one which was already solved in [6]. N can depend on moduli and we would like to redefine ( F − F G ) by absorbing the factor √ N . This brings us finally to the action in the form e − L gen = ( F − F G )( ¯ F − ¯ F G ) − F G ¯ F G + ¯ X A X + X ¯ B + B ¯ X + C . (3.17)with X = Ω F . Here F G = F G | X = ¯ X =0 (3.18)The explicit identification of all entries in eq. (3.17) requires a significant effort. This will bedone in a separate work [25]. Here we have pointed out the important steps and explainedwhy we can make the same construction as in the pure de Sitter case. The final action wherethe auxiliary field F is eliminated in the symbolic form using all entries in (3.17) is verysimple. It is given by e − L gen | δ L δF =0 = − F G ¯ F G + C + (cid:104) ¯ X A X + X ¯ B + B ¯ X − F G ¯ F G | ¯ X ( AX + B ) | (cid:105) X = Ω22 FG , (3.19)or, equivalently, − F G ¯ F G + ¯Ω F G A Ω F G − Ω F G ¯ B − B ¯Ω F G + C − F G ¯ F G (cid:12)(cid:12)(cid:12) ¯Ω F G (cid:16) A Ω F G + B (cid:17)(cid:12)(cid:12)(cid:12) . (3.20)This action consist of the original action with X replaced by ¯Ω F G , and higher order inspinors term given by the last term in eq. (3.20).The relation between supergravity moduli z α , α = 1 , ..., n and the superconformal vari-ables X I is explained in the superconformal gauge in κ = 1 units N ( X, ¯ X ) = − , y = ¯ y , N I Ω I = 0 , (3.21)introduced in [12] is explained in detail in [13]. One can start with the superconformal actionderived here, gauge-fix it and derive the corresponding supergravity model where one of themultiplets in a nilpotent one.Alternatively, one can start with the known supergravity action in case of chiral andvector multiplets, and find the relevant modifications in case of one of the multiplets beinga nilpotent one. This strategy will be described in the next section.7 Supergravity action
A general supergravity action with chiral and vector multiplets derived from the supercon-formal one is given in sec. 18.1 in [13]. But the result we would like to present here is thedeformation of the standard action in sec. 18.1 in [13] for a given K ( z, ¯ z ), W ( z ) and f ABα due to the fact that the superconformal chiral superfield X satisfies the constraint (3.2).We will assume that the moduli z α with α = 1 , ..., n are split into z = X X = S and the rest z i = X I X , i = 2 , ..., n . We will study the class of models such that K ( z i , ¯ z ¯ i ; S, ¯ S ) = K ( z i , ¯ z ¯ i ) + g S ¯ S ( z, ¯ z ) S ¯ S , (4.1) W ( z, S ) = g ( z ) + Sf ( z ) , f AB ( z, S ) = p AB ( z ) + S q AB ( z ) . (4.2)In view of the fact that S = 0 both holomorphic functions W and f AB depend only onlinear functions of S . These are the most general form of such holomorphic functions, aswas recognized in [18], since all powers of S n for n > S and ¯ S . We will not consider suchmodels, our K¨ahler potential will depend on S ¯ S .The conceptual simplicity in the superconformal theory of the single nilpotent chiralmultiplet is due to a clear off-shell local supersymmetry transformation of the supermultipletsin the action (3.1). For example, for the off-shell chiral goldstino multiplet { X , Ω , F } ofthe Weyl weight ω = 1 with the off-shell Lagrange multiplier { Λ , Ω Λ , F Λ } the local Q-andS-supersymmetry rules is δ Ω = 1 √ P L ( /D + F ) (cid:15) + √ X P L η . (4.3)This allows an unambiguous identification of the solution of the nilpotency constraint as X = (Ω ) √ F which was used above. Meanwhile at the supergravity level, the fermion of thecorresponding multiplet transform as δχ = 1 √ P L (ˆ /∂z − e K/ g β ∇ ¯ β W ) (cid:15) + cubic in fermions terms , (4.4)as shown in eq. (18.22) in [13]. We may, therefore, identify the on-shell value of F as F G with − e K/ g β ∇ ¯ β W + fermions and proceed with all steps described in the superconformalcase. These include the analysis of i) the dependence in e K/ g β ∇ ¯ β W + fermions on z and ¯ z ¯1 and ii) the fact that the moduli space metric is field-dependent and non-diagonal in direction1 and the rest of chiral matter.In particular, we may exclude the terms linear in z in F G as they originate from theS-supersymmetry preserving gauge-fixing. The same was done in the superconformal modelwhere we have also noticed that the shift of F by aX does not affect the solution X = Ω F = Ω F + aX ) . Thus, if we would like to associate e K/ g β ∇ ¯ β W + fermions with − F G , wemay ignore terms proportional to z but not proportional to ¯ z ¯1 . The terms in e K/ g β ∇ ¯ β W which depend on ¯ z are of a different nature. But such terms will be contracted with ¯ F ¯1 .Therefore the relevant terms proportional to ¯ z ¯1 will be multiplied by ¯ F ¯1 which will make8hem ¯ F -independent, as we have observed in the superconformal case above. Thus we mayidentify e K/ g β ∇ ¯ β W + fermions taken at z = ¯ z ¯1 = 0 with the relevant part of the auxiliaryfield of the nilpotent multiplet. In this process one should carefully identify all other termsin the supergravity action where the chiral nilpotent multiplet is taken off shell, so that thetotal action is not the one in sec. 18.1 in [13], but its partially off shell version with regardto a nilpotent multiplet. In such case we would arrive to the action of the form shown ineq. (3.17) with X replaced by z . If we would ignore that z = χ F and integrate out F for F -independent z we will get the standard supergravity in sec. 18.1 in [13]. But in case wetake into account the nilpotency condition, starting with eq. (3.17) we will find an actionwith non-linear terms in χ given in eq. (3.19).In practical terms to find a complete action for matter coupled supergravity means to findthe explicit expressions for the entries into action (3.17) for F G , A, B, C for specific K ( z, ¯ z ), W ( z ) and f AB . Once these are known, the complete non-linear in fermions action is givenin eq. (3.20). At this point no further steps are required: all local symmetries like the Weyland U (1) and special superconformal symmetry are already fixed.Therefore it appears advantageous to use the strategy developed above directly in thesupergravity setting. In supergravity interacting with matter multiplets, including a nilpotent one, the action hasthe following terms mixing gravitino ψ µ with goldstino v , the combination of other fermions¯ ψ µ γ µ v + h.c. = ¯ ψ µ γ µ (cid:20) √ e K ( χ i D i W + ψ s D s W ) + 12 iP L λ A P A (cid:21) + h.c. . (5.1)Here ψ s is a VA fermion, χ i are fermions from chiral matter multiplets and λ A are gaugini,using notation in [13]. All matter fermions transform under local supersymmetry without aderivatives on (cid:15) ( x ), including the VA fermion [6]. Therefore any gauge algebraic in fermions,which does not involve the gravitino, is a unitary gauge since it requires no ghosts. Thepreferable gauge from the point of view of the absence of mixing of gravitino with otherfermions at the minimum of the potential is the gauge where the goldstino v is vanishing v = 1 √ e K ( χ i D i W + ψ s D s W ) + 12 iP L λ A P A = 0 . (5.2)However in models with matter multiplets this gauge, in general, for P A (cid:54) = 0 and D α W (cid:54) = 0at the minimum of the potential, leaves us with an extremely complicated action, with highlevel of non-linearity on fermions originating from the VA theory. The advantage of the gauge v = 0 is that gravitino is not mixed with other fermions of the theory. The disadvantage isthat, in case that D i W (cid:54) = 0 and P A (cid:54) = 0 at the vacuum, the action is very complicated sincein such case ψ s = − D s W (cid:16) χ i D i W + 12 iP L λ A P A (cid:17) , (5.3)and the action as a function of independent spinors χ i and λ A has many higher order fermioncouplings. 9eanwhile, the possible choice of the unitary gauge ψ s = 0 (5.4)in general simplifies the action significantly due to absence of fermion terms beyond quartic,even though there is a gravitino-fermion mixing for models with some D i W (cid:54) = 0 and P A (cid:54) = 0at the minimum. This problem can be taken care by the change of the basis to decouple χ and λ from ¯ ψ µ γ µ .In this gauge the complete matter-coupled supergravity with chiral and vector multipletsand one nilpotent chiral multiplet, has the following action. First, we note that the originalaction for all unconstrained multiplets is defined for example in sec. 18.1 in [13], let us callit e − L book . It depends on e aµ , ψ µ , z α , ¯ z ¯ α , χ α , χ ¯ α , A Aµ , λ A . The complete action with a localsupersymmetry is complicated due to a non-Gaussian nature of the auxiliary field F of thenilpotent multiplet ( z = s, χ = ψ s , F = F s ). However, in the unitary gauge (5.4) thisfeature of the theory disappears: all terms with a non-Gaussian dependence on F enter via ψ s F s and vanish in this gauge. The complete action, defined by K given in (4.1) and W, f AB given in (4.2) has to be constructed according to the standard rules. It is given for examplein the book [13] in equations (18.6)-(18.19) and takes the form e − L book (cid:16) K ( z i , ¯ z ¯ i ; S, ¯ S ) , W ( z, S ) f AB ( z, S ) (cid:17) ⇒ e − L book ( e aµ , ψ µ , z i , ¯ z ¯ i , s, ¯ s, χ i , χ ¯ i , ψ s , ¯ ψ s , A Aµ , λ A ) . (5.5)The locally supersymmetric action with the S multiplet nilpotent has ψ s -dependent termswhich are not present in the standard action. However, all these terms are absent in theunitary gauge (5.4) where the physical fields of the S multiplet are absent. Therefore thecomplete matter-coupled supergravity action with a nilpotent multiplet in the gauge (5.4)is given by e − L unitary = e − L book ( e aµ , ψ µ , z i , ¯ z ¯ i , s, ¯ s, χ i , χ ¯ i , ψ s , ¯ ψ s , A Aµ , λ A ) | s = ψ s =0 . (5.6)Despite the physical fields of the nilpotent multiplet, s and ψ s , are both absent in the unitarygauge, the auxiliary field F s = − e K/ g β ∇ ¯ β W + fermions is still present as shown in eq. (2.6)and does affect the action in the unitary gauge. The complete action is different from theone in which the nilpotent multiplet is absent.For example, there are extra positive terms in the potential due to | F s | = | D S W | . Thereare additional terms in the mass formula of physical fermions due to terms with D S W and incase that f AB depends on S . However, all of these terms are included in the standard action(5.5), which is computed in a way that a summation over all chiral multiplets, including z = s , is performed in the derivation of the action.The unitary gauge (5.4) is useful and convenient in case that at the vacuum D i W = 0and only D s W (cid:54) = 0 and the Killing potentials are absent, P A = 0. In the case of a singleinflaton chiral multiplet Φ, present in the theory in addition to a nilpotent one, the condition D Φ W = 0 is fulfilled in inflationary models constructed in most models in [18, 19], where Φ isthe inflaton multiplet. When more general matter multiplets z i are present, it is possible to10rovide this condition taking canonical K¨ahler potential z i ¯ z ¯ i and quadratic superpotential( z i ) and providing the minimum at z i = 0, as suggested in [19]. Thus for such models thegauge (5.4) is very useful.In general, some other classes of gauges fixing local supersymmetry may be also useful.We postpone a discussion of other gauges for future studies. Such studies can be performedafter the explicit form of the complete locally supersymmetric action will be derived in [25],following the strategy proposed in this paper. We have found here that it is possible to include the nilpotent multiplet in the generalsupergravity models and to derive the complete locally supersymmetric action, includingfermions. Our main result is shown in eq. (3.19) and it is explained in the paper why thisaction is valid for matter coupled supergravity. The elimination of auxiliary fields from theaction despite their non-Gaussian dependence can be done using the same procedure as inthe case of pure de Sitter supergravity [6]. An explicit realization of this procedure andconstruction of such general de Sitter type supergravities will be presented in [25]. It willrequire us to find the actual values of the symbolic expressions in (3.19) which would bevalid for general matter-coupled supergravities. But when they are known, the result followseither in the form (3.19), or in the form (3.20). A particular feature of this answer is thatall higher order fermion terms in this locally supersymmetric action with the non-linearlyrealized supersymmetry are given in the closed form.We have also observed that in the unitary gauge, where the fermion from the nilpotentmultiplet vanishes, the full action is very simple and given in eq. (5.6). Fermion couplingsare not higher than quartic in this gauge. However, other interesting gauges are possiblewith some complementary properties. It would be interesting to derive the action in othergauges, when the detailed locally supersymmetric action will be available, and to study thephysics of these models with non-linearly realized local supersymmetry.The class of models with general matter-coupled supergravity and de Sitter vacua asso-ciated with a non-linearly realized supersymmetry lead to interesting phenomenology bothin cosmological setting as well as in particle physics [18, 19]. These models have a potentialfor explaining dark energy, inflation and susy breaking. It is therefore interesting to find outthat such theories have a rather universal form for a general matter coupling.
Acknowledgments
We are grateful to E. Bergshoeff, J. J. M. Carrasco, K. Dasgupta, S. Ferrara, D. Freedman,A. Linde, F. Quevedo, A. Uranga, and especially to A. Van Proeyen and T. Wrase, forstimulating discussions and for the collaboration of related projects. This work is supportedby the SITP, by the NSF Grant PHY-1316699 and by the Templeton foundation grant‘Quantum Gravity Frontiers’. 11 eferences [1] G. ’t Hooft, “Renormalizable Lagrangians for Massive Yang-Mills Fields,” Nucl. Phys.B , 167 (1971).[2] R. E. Kallosh and I. V. Tyutin, “The Equivalence theorem and gauge invariance inrenormalizable theories,” Yad. Fiz. , 190 (1973) [Sov. J. Nucl. Phys. , 98 (1973)],see http://web.stanford.edu/~rkallosh/KalloshTyutin1972 [3] G. ’t Hooft and M. J. G. Veltman, “Combinatorics of gauge fields,” Nucl. Phys. B ,318 (1972).[4] I. V. Tyutin, 1975: “Gauge Invariance in Field Theory and Statistical Physics in Oper-ator Formalism,” arXiv:0812.0580 [hep-th].[5] C. Becchi, A. Rouet and R. 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