All N= 4 Conformal Supergravities

All N=4 Conformal Supergravities Daniel Butter1 , Franz Ciceri1 , Bernard de Wit1,2 and Bindusar Sahoo3 1

3

Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands 2 Institute for Theoretical Physics, Utrecht University, Princetonlaan 5, 3584 CC Utrecht, The Netherlands Indian Institute of Science Education and Research, Thiruvananthapuram, Kerala 695016, India

arXiv:1609.09083v1 [hep-th] 28 Sep 2016

All N=4 conformal supergravities in four space-time dimensions are constructed. These are the only N=4 supergravity theories whose actions are invariant under off-shell supersymmetry. They are encoded in terms of a holomorphic function that is homogeneous of zeroth degree in scalar fields that parametrize an SU(1, 1)/U(1) coset space. When this function equals a constant the Lagrangian is invariant under continuous SU(1, 1) transformations.

Conformal supergravities in four dimensions are invariant under the local symmetries associated with the superconformal algebra su(2, 2|N ). They are supersymmetric generalizations of the conformally invariant action of general relativity quadratic in the Weyl tensor. The transformation rules and the corresponding invariant Lagrangians are known for N = 1 and 2 [1, 2]. For N = 4 the full non-linear transformation rules of the fields, which constitute the so-called Weyl supermultiplet, have been determined [3]. This is the largest possible conformal supergravity that can exist in four space-time dimensions [4], and so far a complete action was not known. A unique feature is the presence of dimensionless scalar fields that parametrize an SU(1, 1)/U(1) coset space. The U(1) factor is realized as a local symmetry with a composite connection, which acts chirally on the fermions. Hence the R-symmetry group is extended to SU(4) × U(1). As explained below there are good reasons to expect that a large variety of these theories will exist. This Letter reports important progress on this question as we derive the most general invariant action, which turns out to depend on a single arbitrary holomorphic and homogeneous function of the coset fields. Here we will present its purely bosonic terms; full results will be reported elsewhere. When this function is constant these bosonic terms turn out to agree with a recent result derived by imposing supersymmetry on terms that are at most quadratic in the fermions [5]. Before continuing, let us briefly discuss the issue of ‘non-minimal’ couplings, where the terms quadratic in the Weyl tensor are multiplied by a function of the scalar fields. The possible existence of such Lagrangians was suggested long ago in [6, 7]. Meanwhile indirect evidence came from string theory and from dyonic N = 4 black holes, where the threshold corrections in the effective action of IIA string compactifications on K3 × T 2 reveal the presence of terms proportional to the square of the Weyl tensor multiplied by a modular function [8]. The same terms emerge in the semiclassical approximation of microscopic degeneracy formulae for dyonic BPS black holes [9–11]. Finally higher-derivative couplings derived for N = 4 Poincar´e supergravity [12, 13] do also exhibit

non-trivial scalar interactions. We briefly summarize the field content of N = 4 conformal supergravity, which comprises 128 + 128 bosonic and fermionic off-shell degrees of freedom. Space-time indices are denoted by µ, ν, . . ., tangent space indices by a, b, . . . and SU(4) indices by i, j, . . .. Among the bosonic fields are the vierbein eµa , SU(4) gauge fields Vµi j and a gauge field bµ associated with dilatations. Furthermore there are three composite bosonic gauge fields, namely the spin connection ωµab , the gauge field fµa associated with conformal boosts and the gauge field aµ associated with the U(1) symmetry. In addition to the gauge fields, the bosonic fields comprise complex antiselfdual tensor fields Tabij transforming in the 6 representation of SU(4), whose complex conjugates are the selfdual fields Tabij , complex scalars Eij , and pseudo-real scalars Dijkl transforming in the 10 and the 20′ representation of SU(4), respectively. Finally there exists a doublet of complex scalars φα , which are invariant under dilatations and transform under rigid SU(1, 1) transformations (α = 1, 2). They are subject to the SU(1, 1) invariant constraint, φα φα = 1 ,

φ1 ≡ (φ1 )∗ ,

φ2 ≡ −(φ2 )∗ ,

(1)

and as a result the fields φα and φα parametrize SU(1, 1) matrices. Because these fields are subject to the local U(1) symmetry, they describe two physical degrees of freedom associated with an SU(1, 1)/U(1) coset space. The positive chirality fermions consist of the gravitini ψµi (the gauge fields of Q-supersymmetry), a composite gauge field φµi associated with S-supersymmetry, and two spinor fields, Λi and χijk , transforming in the 4, the 4, the 4 and the 20 representation, respectively. The negative chirality fermions transform under the corresponding conjugate representations. We will refrain from discussing the superconformal transformations of the various fields in any detail. These results have already appeared in [3, 5]. The Weyl multiplet defines a doublet of chiral superfields with lowest components φα , but no other independent chiral supermultiplets can exist coupled to conformal supergravity. In view of their relevance for the present paper we first present some further details regarding the fields φα

2 and φα , which we refer to as the holomorphic and the anti-holomorphic fields, respectively. The holomorphic fields carry U(1) charge equal to −1 and transform under Q-supersymmetry into the positive chirality spinors Λi , which themselves carry U(1) charge −3/2, δφα = −¯ ǫi Λi εαβ φβ .

φα → e−iλ(x) φα ,

structure of the three left-invariant vector fields associated with the group SU(1, 1), ∂ ∂ − φα , ∂φα ∂φα ∂ ∂ D† = φα εαβ , D = −φα εαβ , β ∂φ ∂φβ

D0 = φα

(2)

The supercovariant constraint that determines the U(1) gauge field aµ and the generalized supercovariant derivatives of the coset fields, Pa and P¯a , are defined by ¯ i γa Λ i , φα Da φα = − 14 Λ α

β

Pa = φ εαβ Da φ ,

  which satisfy the commutation relations D0 , D = 2 D  and D, D† = D0 . Using these definitions the supersymmetry variation and the supercovariant derivative of arbitrary functions H(φα , φβ ) can be written as

(3) P¯a = −φα ε

αβ

Da φβ ,

  δH = − ǫ¯i Λi D + ¯ǫi Λi D† H ,   ¯ i γa Λ i D 0 H . Da H = P¯a D + P¯a D† + 1 Λ

where Da denotes the fully superconformal covariant derivative. Note that Pa and P¯a carry Weyl weight +1 and U(1) weights +2 and −2, respectively. From these definitions one may derive the supercovariant extension of the Maurer-Cartan equations associated with the SU(1, 1)/U(1) coset space,
¯ i γ[a Db] Λi − h.c. , F (a)ab = − 2i P[a P¯b] − 21 i Λ ¯ i γ[a Λi Pb] + 1 Λ ¯ i R(Q)ab i , D[a Pb] = − 1 Λ (4) 2

4

h

1 2

4

abcd

R(M )

R(M )− abcd + R(V )

ab i

j

j R(V )− ab i +

− P¯ a Da Db P b + P 2 P¯ 2 + 31 (P a P¯a )2 − +

1 48

+

1 2

1 6

After these definitions we present the bosonic terms of the full class of superconformal Lagrangians, which are given by the real part of the following expression,

1 8

Dij kl Dkl ij +

1 4

Eij D2 E ij − 4 Tabij Da Dc T cbij

P a P¯a Eij E ij − 8 Pa P¯ c T abij Tbcij −

[Eij E ij ]2 + T abij Tab kl T cd ij Tcdkl − T abij Tcdjk Tab kl T cd li −

Eij T ab kl R(V )abi m εjklm −

1 16

(6)

The class of Lagrangians presented below involves a function H(φα ) that is homogeneous of zeroth degree in the holomorphic variables, so that D† H(φα ) = 0 and D0 H(φα ) = 0. Using the above commutation relations, it then follows that D† Dn H(φα ) ∝ Dn−1 H(φα ) for n > 1, and vanishes for n = 1 so that DH(φα ) is holomorphic while D2 H is not.

where F (a)ab and R(Q)ab i denote the supercovariant U(1) and Q-supersymmetry curvatures, respectively. Note that the expressions (2) and (3), when combined with those for the anti-holomorphic fields, reflect the

e−1 L = H

(5)

1 2

Eij Ekl T abmn Tab pq εikmn εjlpq −

1 16

Eij E jk Ekl E li

E ij T ab kl R(V )abm i εjklm 1 16

E ij E kl T ab mn Tab pq εikmn εjlpq

i

− 2 T ab ij P[a Dc] Tb c kl + 61 P c Dc Tab kl + 31 Tab kl Dc P c εijkl − 2 T abij P¯[a Dc] Tb c kl − 12 P¯ c Dc Tab kl εijkl h
abcd + DH 41 Tab ij Tcd kl R(M ) εijkl + Eij T ab ik R(V )abj k − 81 Dij kl T ab mn Tab kl εijmn − 12 Eim Ejn εklmn i 1 Eij E ij T ab kl Tab mn εklmn − 16 E ij Tab kl T ac mn T b c pq εiklm εjpqn + T ab ij Ta c kl R(V )bcm k εijlm − 24 h 1 + D2 H 61 Eij Tab ik T ac jl T b c mn εklmn − 18 Eij Ekl Tab ik T ab jl + 384 Eij Ekl Emn Epq εikmp εjlnq i 1 1 + 32 T ab ij T cd pq Tab mn Tcd kl εijkl εmnpq − 64 T ab ij T cd pq Tab kl Tcd mn εijkl εmnpq h i + 2 H ea µ fµ c ηcb P a P¯ b − P d P¯d η ab , (7)

where R(M )−abcd and R(V )−ab i j denote anti-selfdual supercovariant curvatures. When suppressing the fermionic

terms and imposing the gauge bµ = 0, they become equal to the Weyl tensor and the SU(4) field strengths.

3 Let us now turn to the derivation of this result. It makes use of the fact that any supersymmetric component Lagrangian can be written as the Hodge dual of a four-form built in terms of the vierbein, gravitini, and possibly other connections, multiplied by supercovariant coefficient functions that we will treat as composite fields. This approach is known as the superform method [14]. Since for N = 4 supergravity chiral superspace does not exist, we aim to construct such a density formula directly, assuming that only the vierbein and gravitini may appear explicitly within the four-form. Schematically we will thus consider a four-form decomposed into five types of forms, namely ψ 4 , e ψ 3 , e2 ψ 2 , e3 ψ and finally e4 . The Weyl weight of these forms ranges from w = −2 for the first one to w = −4 for the last one. This last form will be multiplied by a composite coefficient function with w = 4 that contains all the purely bosonic terms of the Lagrangian specified in (7) (as well as fermionic terms). The structure of the Lagrangian is dictated by the transformation of the lowest-dimensional supercovariant composites. We thus start by considering the quartic gravitino forms, which we postulate to be of the following type,

(8)

where the supercovariant composites Aijkl , C ijkl , and C¯ ijkl are assumed to be S-supersymmetric, and are therefore also invariant under conformal boosts. All three composites have w = 2 and belong to the 20′ representation of SU(4). The correctness of these assumptions will be confirmed at the end of the calculation. To elucidate the calculation we start by evaluating the Q-supersymmetry variations that remain proportional to four gravitini. For this we note that gravitino fields transform under Q-supersymmetry as (9)

where the derivative Dµ is covariant under the bosonic gauge transformations with the exception of the conformal boosts. At the quartic gravitino level we may ignore the second term, but neither the third nor the first. The reason the first term is relevant is that, in writing the variation of the action in a manifestly supercovariant form, we must integrate the derivative by parts and reconstruct supercovariant quantities. This then leads to further gravitino terms in two ways. The first is when the derivative hits another gravitino, which must be converted into the supercovariant Q-supersymmetry curvature by adding appropriate terms, D[µ ψν] i = 12 R(Q)µν i − 41 εijkl ψ¯[µj ψν]k Λl + · · · .

Here we write the Q-supersymmetry transformations of the scalar composites as δC ijkl = ǫ¯m Ξijkl,m + ǫ¯m Ξij,mkl , δAijkl = ǫ¯m Ωijkl,m + ǫ¯m Ωij,mkl ,

(10)

(12)

with fermionic composites Ξ and Ω. Naturally the transformations (12) also induce variations of (8) proportional to ψ 4 times Ξ and Ω. Finally there is yet another way to generate variations quartic in gravitini originating from a four-form of the type e ψ 3 , induced by the transformation δeµa = ¯ǫk γ a ψµk + ǫ¯k γ a ψµk . Note that connections other than the gravitini will also be generated, for example from (10), but those turn out to cancel at the end. Collecting all the resulting variations proportional to the various possible quartic gravitini four-forms and requiring them to vanish imposes the following constraints on the traceless parts of the fermionic composites in (12),

60

4

δψµi = 2 Dµ ǫi − 12 Tabij γ ab γµ ǫj − εijkl ¯ ǫj ψµk Λl ,

2

[Ξijkl,m ]60 = [2Λm Aij kl ]60 , [Ωij kl,m ] = [Λm C¯ ij kl ] ,

L = −i ε ψ¯µi ψνj ψ¯ρ k ψσl Aijkl − 1 i εµνρσ ψ¯µi ψνj ψ¯ρk ψσl εklrs C ijrs µνρσ

− 41 i εµνρσ ψ¯µi ψνj ψ¯ρk ψσl εklrs C¯ rsij + · · · ,

The second way is when the derivative hits a supercovariant composite, such as C ijkl , which we rewrite as   Dµ C ijkl = Dµ C ijkl + 1 ψ¯µm Ξijkl,m + ψ¯µm Ξij,mkl . (11)

60

[Ξij,mkl ]60 = 0 , (13)

along with their complex conjugates, where [•]r denotes projection onto the SU(4) representation r. The remaining terms in the fermionic composites Ξ and Ω lie in the 20 and 20 representations and must be proportional to the fermionic composites multiplying the e ψ 3 four-forms. We refrain from giving explicit formulae as the general pattern should be clear. Note that so far we have only made use of the transformations of the vierbein and the gravitini. This procedure must be continued by considering the remaining Q-supersymmetry variations proportional to the four-forms e ψ 3 , e2 ψ 2 , etc., to determine the relations between all the supercovariant composites in the Lagrangian and their transformation rules. This calculation will make use of the Q-supersymmetry transformations of almost all the Weyl multiplet fields specified in [3, 5]. The fact that no inconsistencies arise at this level is a first indication that our original assumptions regarding the ψ 4 four-forms are correct. Finally, we must check that the derived transformation rules of the composites satisfy the same off-shell superconformal algebra as the Weyl multiplet. This is a straightforward but technically involved calculation for which we made extensive use of the computer algebra package Cadabra [15, 16]. In this process one also identifies the missing S-supersymmetry variations of the composites, which so far were only specified for Aijkl and C ijkl . This then provides a complete density formula built upon (8) and (13), that is invariant under all local superconformal symmetries. We emphasize that this

4 density formula makes no assumptions about the specific dependence of the supercovariant composites on the Weyl multiplet fields. At this point we have fully confirmed all the assumptions that are at the basis for the above calculations. To obtain the final results we express the composites Aijkl and C ijkl in terms of the supercovariant fields of the N = 4 Weyl multiplet; they then yield corresponding expressions for all the composites through their transformation rules. As it turns out there exist only four scalar S-supersymmetric expressions in the 20′ representation, X(1)ijkl = Dijkl , X(2)ijkl = 21 Tabij T ab mn εklmn − 41 εijmn Emk Enl ¯ [k χijl] − traces , + 2Λ ¯ k γ ab Λl + 1 εijmn Em[k Λ ¯ l] Λn − traces , X(3)ijkl = − 41 Tabij Λ 4 ¯ n Λl , ¯ k Λm Λ X(4)ijkl = − 1 εijmn Λ (14) 24

each of which is homogeneous in the Weyl multiplet fields. The first one is pseudo-real and the others are complex. The composites Aijkl and C ijkl must then be ¯ (n)ijkl written as linear combinations of X(n)ijkl and X multiplied by apriori arbitrary functions of the coset scalars with the appropriate U(1) weights. From these expressions one determines the corresponding fermionic composites via (12), and subsequently imposes the constraints (13). This then leads to two linearly independent solutions for Aijkl and C ijkl . One solution turns out to correspond to a Lagrangian that is a total derivative. The other one depends on an arbitrary holomorphic function H(φα ) that is homogeneous of zeroth degree and an associated potential K(φα , φβ ), which obeys DD† K = H. The function H is uniquely determined as it appears in C ijkl as a distinctive term equal to 12 i X(2)ijkl H. The holomorphicity of H can then be seen as a direct consequence of the constraints (13), which also determine how derivatives of H and K appear within C ijkl and the other composites. At the end the potential K is removed by splitting off a total derivative. In deriving the result (7) we have introduced additional total derivative terms in order to bring the formula into a concise form at the cost of generating terms that explicitly depend on the conformal boost gauge field fµa . In this Letter we have described the construction of a class of Lagrangians of N = 4 conformal supergravity encoded in an arbitrary holomorphic function that is homogeneous of zeroth degree. There are strong indications that no other invariant Lagrangians can exist. We will return to this question in a forthcoming paper where we also present the details of this calculation, including the full density formula and all relevant supersymmetry transformations. In this construction many consistency checks were carried out. Perhaps the most stringent one is the comparison of the bosonic Lagrangian (7) for the case that the holomorphic function equals a constant, to

the result of [5]. Indeed both expressions agree up to a total derivative. The results derived here can be directly incorporated into Poincar´e supergravity as a four-derivative coupling following the same construction carried out originally in [17], by including the superconformal Lagrangian of this Letter before proceeding to the standard gauge choices. The SU(1, 1) symmetry will then become entangled with an electric-magnetic duality transformation in the vectormultiplet sector. This set-up should offer a good starting point for discussing subleading corrections to BPS black hole entropy in a manifestly N = 4 supersymmetric formulation. The above Lagrangian can alternatively be obtained in one step by applying the superform method directly based on an extended field configuration consisting of at least six (on-shell) vector supermultiplets and the Weyl supermultiplet, where one must bear in mind that the supersymmetry algebra for this field representation will no longer close off shell. It is then possible to compare the corresponding expressions with the R2 -couplings for N = 4 Poincar´e supergravity of [12, 13], which may also depend non-trivially on the coset fields. In the context of the Poincar´e theory the higherderivative couplings are primarily studied as potential counterterms that could render the theory finite. At this moment there is agreement that this theory is not finite at the four-loop level [18]. The presence of a U(1) anomaly and the non-trivial dependence on the coset fields plays an important role in this discussion, as was also extensively discussed in [19]. We thank Kasper Peeters for frequent correspondence regarding the use of Cadabra, and Zvi Bern, Renata Kallosh and Kellogg Stelle for informative discussions on anomalies and divergences of N = 4 supergravity. B.S. thanks Nikhef Amsterdam for hospitality extended to him during the course of this work. We thank the 2015 Simons Summer Workshop where this work was initiated, and the 2016 “Supergravity: what next?” workshop at the Galileo Galilei Institute for Theoretical Physics where it was completed, and the INFN for partial support. This work is also partially supported by the ERC Advanced Grant no. 246974, “Supersymmetry: a window to nonperturbative physics” and by the Marie Curie fellowship PIIF-GA-2012-627976.

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