Self-dual Tensors in Six-Dimensional Supergravity

arXiv:hep-th/9812042v4 25 Jan 1999

ROM2F-98/42

Self-dual Tensors in Six-Dimensional Supergravity

Fabio Riccioni and Augusto Sagnotti

Dipartimento di Fisica Universit`a di Roma “Tor Vergata” I.N.F.N. - Sezione di Roma “Tor Vergata” Via della Ricerca Scientifica, 1 00133 Roma ITALY

Abstract We review some properties of the field equations of six-dimensional (1,0) supergravity coupled to tensor and vector multiplets, and in particular their relation to covariant and consistent anomalies and a peculiar Noether identity for the energy-momentum tensor. We also describe a lagrangian formulation for this system, obtained applying the Pasti-Sorokin-Tonin prescription.

Based on talks presented at the 1998 Trieste Conference on “Superfivebranes and Physics in 5+1 Dimensions”, at the 1998 DESY Workshop on “Conformal Field Theory of D branes” and at the 1998 Como-Centro Volta RFBF-INTAS School on “Advances in Quantum Field Theory, Statistical Mechanics and Dynamical Systems”

Self-dual Tensors in Six-Dimensional Supergravity Fabio Riccioni and Augusto Sagnotti Dipartimento di Fisica, Universit` a di Roma “Tor Vergata” I.N.F.N., Sezione di Roma “Tor Vergata” Via della Ricerca Scientifica, 1 00133 Roma ITALY We review some properties of the field equations of six-dimensional (1,0) supergravity coupled to tensor and vector multiplets, and in particular their relation to covariant and consistent anomalies and a peculiar Noether identity for the energymomentum tensor. We also describe a lagrangian formulation for this system, obtained applying the Pasti-Sorokin-Tonin prescription.

1

Introduction

Six-dimensional (1, 0) supergravity is built out of four types of multiplets 1,2 . + Aside from the gravitational multiplet (eaµ , ψL,µ , Bµν ), one has the option to − add tensor multiplets (Bµν , χR , ϕ), vector multiplets (Aµ , λL ) and hypermultiplets. This talk is devoted to some notable properties of the low-energy couplings between the first three types of multiplets, and is meant to complement our previous short review 3 . Couplings between several tensor multiplets have entered the stage relatively recently, since they emerge rather naturally only in perturbative type-I vacua 4 . On the other hand, perturbative heterotic vacua always involve a single tensor multiplet, that is responsible for the corresponding Green-Schwarz mechanism 5 . The couplings between vector and tensor multiplets are rather unconventional, since they are “classically anomalous” and conspire to implement a generalized Green-Schwarz mechanism 6 . Limitedly to the gauge anomalies, in six dimensions the Green-Schwarz mechanism is visible in the low-energy field equations, and brings about a number of oddities, most notably some singularities in the moduli space of tensor multiplets 6 . These signal an important new phenomenon, a phase transition 7 related to strings of vanishing tension, likely to be of quite some interest in the coming years. In formulating the low-energy couplings between tensor and vector multiplets, one has two natural options. The first is related to covariant field equations and to the corresponding covariant anomalies 6,8 . It has the virtue of respecting gauge covariance and supersymmetry, but the resulting field equations are not integrable. The second is related to consistent, and thus integrable, field equations 9,10 . These may be derived from an action principle that satisfies Wess-Zumino consistency conditions, and as a result embody a supersymmetry anomaly. The complete field equations, first obtained in this framework 10 , are 2

¯ µ λ)2 , not unique a . A quartic coupling for the gauginos, proportional to (λγ is undetermined, while the gauge algebra contains an extension that makes the construction consistent for any choice of it. We would like to stress that all these features are determined by local couplings in these “classical” field equations, that are thus a remarkable laboratory for current algebra. Indeed, a closer scrutiny of their properties 8 brought about one further surprise: in a theory with gauge and supersymmetry anomalies, gravitational anomalies are not directly related to the divergence of the energy-momentum tensor. Thus, in a theory without gravitational anomalies the energy-momentum tensor need not be divergenceless. We shall return to the energy-momentum tensor in the next Section, where we shall also review our recent work on the completion of the covariant equations 8 . Aside from this general lesson, there are also a number of subtleties related to the rigid limit of the coupled equations. These are not considered here, but have been discussed at length in ref. 12 . Actually, in mentioning an action principle, we have been somewhat cavalier about the presence of (anti)self-dual antisymmetric tensors, that bring about a number of traditional difficulties. The long-standing problem of giving a covariant action principle to this type of fields has recently been given a compact solution, in terms of a single auxiliary scalar mode, by Pasti, Sorokin and Tonin (PST) 13 . The last Section is devoted to the completion of the consistent action principle of ref. 10 using their method. This new result, not contained in the original presentation, is included here for completeness. 2

Consistent and covariant field equations and their anomalies

In this Section we would like to review some basic properties of six-dimensional (1, 0) supergravity coupled to vector and tensor multiplets. This model is rather peculiar, since it embodies gauge anomalies induced by tensor couplings 6,9,10,8 , to be disposed of by fermion loops that one is actually not accounting for. Consequently, in formulating the field equations, one is faced with the familiar choice between covariant and consistent anomalies. The former originate from a set of covariant non-integrable equations invariant under local supersymmetry, while the latter, generally regarded as the most interesting ones, originate from a set of integrable equations, with a consistent supersymmetry anomaly related to the consistent gauge anomaly by Wess-Zumino consistency conditions. To lowest order in the fermi fields, in the conventions of ref. 10 , the aA

number of fermionic couplings, also to hypermultiplets, with additional singularities related to those of ref. 6 , were anticipated in ref. 11 .

3

fermionic equations are 6,9 i γ µνρ Dν Ψρ + vr H rµνρ γν Ψρ − xm H rµνρ γνρ χm 2 r i 1 + xm ∂ν v r γ ν γ µ χm − √ vr crz trz (Fστ γ στ γ µ λ) = 0 2 r 2

,

i 1 vr H rµνρ γµνρ χm − xm H rµνρ γµν Ψρ 12 2 r i i rz µν − xm ∂ν v r γ µ γ ν Ψµ − √ xm r c trz (Fµν γ λ) = 0 2 r 2

(1)

γ µ D µ χm −

(2)

and 1 1 vr crz γ µ Dµ λ + (∂µ vr )crz γ µ λ + √ vr crz Fαβ γ µ γ αβ Ψµ 2 2 2 i m rz 1 rz µν m + √ xr c Fµν γ χ − c Hrµνρ γ µνρ λ = 0 , 12 2 2

(3)

while the consistent bosonic equations are 9 1 1 r H s ν αβ Rµν − gµν R + ∂µ v r ∂ν vr − gµν ∂α v r ∂ α vr − Grs Hµαβ 2 2 1 +4vr crz trz (Fαµ F α ν − gµν Fαβ F αβ ) = 0 , 4 2 m µ r r sαβγ rz αβ xm − xm )=0 r Dµ (∂ v ) + xr vs Hαβγ H r c trz (Fαβ F 3

(4) (5)

and Dµ (vr crz F µν ) − crz Grs H sνρσ Fρσ − −

′ 1 νραβγδ z z′ ǫ cr Fρα crz ωβγδ =0 . 12e

′ 1 νραβγδ z ǫ cr Aρ crz trz′ (Fαβ Fγδ ) 8e

(6)

Moreover, the tensor fields satisfy (anti)self-duality conditions, conveniently summarized as 9 1 (7) Grs H sµνρ = ǫµνραβγ Hrαβγ , 6e m where Grs = vr vs + xm r xs . In this talk, as in refs. 6,9,10,8 , we are confining our attention to residual anomalies, that correspond to reducible traces. In type-I vacua, these are the contributions left over by tadpole conditions. In String Theory, all residual 4

anomalies are absent a priori, since they would draw their origin from the non-planar one-loop amplitude, that is regulated by the momentum flow. On the other hand, in Field Theory the residual anomalies draw their origin from fermion loops, and are disposed of by the Green-Schwarz couplings that we are actually discussing. In particular, eqs. (1)-(6) embody a reducible gauge anomaly 9 ′ 1 AΛ = − ǫµναβγδ czr crz trz (Λ∂µ Aν )trz′ (Fαβ Fγδ ) , 4

(8)

and a corresponding supersymmetry anomaly ′

′

Aǫ = −crz crz [trz (δǫ A A)trz′ (F 2 ) − 2trz (δǫ A F )ω3z ] ,

(9)

determined by Wess-Zumino consistency conditions 14 . The complete field equations were obtained 10 from the commutator of two supersymmetry transformations on the fermi fields, in the spirit of refs. 15,2 . However, in this case one is actually solving Wess-Zumino consistency conditions, and these do not fix a cubic contribution to the gaugino equation and a related quartic contribution to the vielbein equation. This lack of uniqueness, a rather surprising phenomenon in supergravity, reflects a familiar property of anomalies, that are defined up to the variation of a local functional. It would be interesting to investigate further this issue directly in String Theory. The gauge anomaly AΛ = δΛ L naturally satisfies the condition AΛ = −tr(ΛDµ J µ )

,

(10)

where J µ = 0 is the complete field equation of the vector field. One can similarly show that the supersymmetry anomaly is related to the field equation of the gravitino, that we write succinctly J˜µ = 0, according to Aǫ = −(¯ ǫDµ J˜µ )

.

(11)

Moreover, these equations embody an amusing Noether identity for the energy-momentum tensor, a general result in Field Theory, that came as a little surprise to us. In a theory with gauge and supersymmetry anomalies, the gravitational anomaly is not simply related to the divergence of the energymomentum tensor 8 , since ¯ ν Dµ J˜µ ) . Aξ = δξ L = 2ξν Dµ T µν + ξν tr(Aν Dµ J µ ) + ξν (Ψ

(12)

In particular, in our case we are not accounting for gravitational anomalies, that would result in higher-derivative couplings, and indeed one can verify that 5

the divergence of the energy-momentum tensor does not vanish, but satisfies the relation 1 ¯ν 1 (13) Dµ J˜µ ) . Dµ T µν = − tr(Aν Dµ J µ ) − (Ψ 2 2 Thus, the lack of conservation of matter currents feeds an algebraic inconsistency in the Einstein equations. We are not aware of a previous discussion of this simple but neat result in the literature. Consistent and covariant gauge anomalies are related by the divergence of a local functional 16 . In six dimensions, to lowest order in the fermi fields, the residual covariant gauge anomaly is 6 Acov Λ =

1 µναβγδ rz z′ ǫ c cr trz (ΛFµν )trz′ (Fαβ Fγδ ) 2

,

(14)

and is related to the consistent anomaly according to Acons + tr[ΛDµ f µ ] = Acov Λ Λ

(15)

,

where

′ 1 1 ′ ′ ′ Fγδ ) + Fνα ωβγδ } . f µ = −czr crz ǫµναβγδ { Aν trz′ (Fαβ 4 6 Comparing eq. (16) and eq. (9), one can see that

Aǫ = tr(δǫ Aµ f µ ) ,

(16)

(17)

and this implies that the transition from consistent to covariant anomalies turns a model with a supersymmetry anomaly into one without any 6,9 . Indeed, sixdimensional supergravity coupled to vector and tensor multiplets was originally formulated in this fashion in ref. 6 to lowest order in the fermi fields, extending the work of Romans 2 . This observation can be generalized to all orders in the fermi fields. The complete supersymmetry anomaly has the form Aǫ = tr(δǫ Aµ f µ ) + δǫ eµ a g µ a

,

where fµ

′ 1 1 ′ ′ ′ = czr crz trz′ {− ǫµναβγδ Aν Fαβ Fγδ − ǫµναβγδ Fνα ωβγδ 4 6 ie ¯ ′ γ µνρ λ′ ) + ie (λγ ¯ µνρ λ′ )F ′ + ie(λγ ¯ ν λ′ )F ′µν + Fνρ (λ νρ 2 2 ′ exm csz 3i ¯ µ ′ ¯ ′ m e ¯ µ ν ρ ′ ¯′ √ (λγ γ γ λ )(λ γν Ψρ ) + s tz′ [− √ (λγ λ )(λ χ ) − vt c 2 2 2 2

6

(18)

i ¯ i ¯ µνρ ′ ¯ ′ ′ ¯ ′ µν m √ (λγ λ )(λ γνρ χm ) − √ (λγ ν λ )(λ γ χ )] 4 2 2 2 ¯ ′ γ µνρ λ′ ) + iα(λγ ¯ µνρ λ′ )Fˆ ′ − 6iα(λγ ¯ ν λ′ )Fˆ ′ ] + e[−iαFˆνρ (λ νρ µν −

+ + +

sz ′ √ exm s c ¯ νρ χm )(λ ¯ ′ γ µνρ λ′ )] ¯ µ λ′ )(λ ¯ ′ χm ) + iα √ (λγ 2(λγ ′ [−iα tz vt c 2 2 sz iα ¯ µνρ ′ ¯ ′ iα ¯ µ ′ ¯ ′ m exm s c λ )(λ χ ) − √ (λγ λ )(λ γνρ χm ) [ √ (λγ vt ctz 2 2 2 iα ¯ ′ ¯ ′ µν m √ (λγ (19) ν λ )(λ γ χ )]} 2

differs from the expression in eq. (16) by the addition of higher-order fermionic terms, while e ¯ µνρ ¯ ′ αeeµ a ¯ ν ′ ¯ (λγ λ)(λ γaνρ λ′ ) + (λγ λ )(λγν λ′ )} . (20) 32 2 It should be appreciated that both expressions depend on the arbitrary parameter α. Modifying the vector equation so that ′

g µ a = czr crz trz,z′ {

µ (eq. Aµ )(cov) ≡ J(cov) =

δL − fµ δAµ

,

(21)

and similarly for the Einstein equation, the resulting theory is supersymmetric but no longer integrable. The divergence of the complete covariant vector equation satisfies µ tr(ΛDµ J(cov) ) = −Acov , (22) Λ where

Acov Λ

= + + + − − −

′ 1 ′ ′ ¯ ′ γ µνρ Dµ λ′ ) crz czr trz,z′ { ǫµναβγδ (ΛFµν )(Fαβ Fγδ ) + ieΛFνρ (λ 2 ie ¯ µνρ λ′ )F ′ + eΛDµ {i(λγ ¯ ν λ′ )F ′µν − iαFˆνρ (λ ¯ ′ γ µνρ λ′ ) ΛDµ (λγ νρ 2 1 ¯ µ ν ρ ′ ¯′ ¯ µνρ λ′ )Fˆ ′ − 6iα(λγ ¯ ν λ′ )Fˆ ′µν − √ (λγ γ γ λ )(λ γν Ψρ ) iα(λγ νρ 2 2 sz ′ xm i ¯ µνρ ′ ¯ ′ 3i ¯ µ ′ ¯ ′ m s c λ )(λ χ ) − √ (λγ λ )(λ γνρ χm ) [− √ (λγ vt ctz′ 2 2 4 2 i ¯ iα ¯ ′ ¯ ′ µν m m ¯ ′ µνρ ′ √ (λγ λ) ν λ )(λ γ χ ) + √ (λγνρ χ )(λ γ 2 2 2 2 m sz √ iα ¯ µ ′ ¯ ′ m ¯ µ λ′ )(λ ¯ ′ χm )] + xs c [ √ iα 2(λγ (λγ λ )(λ χ ) vt ctz 2 iα ¯ iα ¯ µνρ ′ ¯ ′ ′ ¯ ′ µν m √ (λγ λ )(λ γνρ χm ) + √ (λγ (23) ν λ )(λ γ χ )]}} 2 2 2

7

now contains higher-order fermi terms 8 . Finally, one can study the divergence of the Rarita-Schwinger and Einstein equations in the covariant model. With the covariant equations obtained from the consistent ones by the redefinition of eq. (21) and by (eq. eµ a )(cov) =

δL − gµa δeµ a

,

(24)

where g µ a is defined in eq. (20), the usual procedure shows that the divergence of the Rarita-Schwinger equation vanishes for any value of the parameter α. On the other hand, the divergence of the energy-momentum tensor presents further subtleties that we would like to address. In particular, it vanishes to lowest order, while it gives a covariant non-vanishing result if all fermion couplings are taken into account. The subtleties have to do with the transformation of the vector under general coordinate transformations, δξ Aµ = −ξ α ∂α Aµ − ∂µ ξ α Aα

(25)

and with the corresponding (off-shell) form of the identity of eq. (12). Starting again from the consistent equations, one finds ¯ ν Dµ J˜µ ) . Aξ = 2ξν Dµ T µν + ξν tr(Aν Dµ J µ ) + ξν tr(F µν Jµ ) + ξν (Ψ

(26)

Reverting to the covariant form then eliminates the divergence of the RaritaSchwinger equation and alters the vector equation, so that the third, “off-shell” term, has to be retained. The final result, 1 1 µν µ Dµ T(cov) = − tr(Aν Dµ J(cov) + fµ F µν + Aν Dµ f µ ) − eνa Dµ g µ a , 2 2

(27)

is nicely verified by our equations. As anticipated, this implies that the diverµν gence of T(cov) vanishes to lowest order in the fermi couplings. 3

PST construction and covariant Lagrangian formulation

In the previous Section we have reviewed a number of properties of 6d (1, 0) supergravity coupled to vector and tensor multiplets 6,9,10,8 . We have always confined our attention to the field equations, thus evading the traditional difficulties met with the action principles for (anti)self-dual tensor fields. In this Section we would like to complete our discussion and present an action principle for the consistent field equations. What follows is an application of a general method introduced by Pasti, Sorokin and Tonin (PST), that have shown how 8

to obtain Lorentz-covariant Lagrangians for (anti)self-dual tensors with a single auxiliary field 13 . Alternative constructions 17 , some of which preceded the work of PST, need an infinite number of auxiliary fields, and bear a closer relationship to the BRST formulation of closed-string spectra 18 . This method has already been applied to a number of systems, including (1, 0) six-dimensional supergravity coupled to tensor multiplets 19 and type IIB ten-dimensional supergravity 20 , whose (local) gravitational anomaly has been shown to reproduce 21 the well-known results 22 of Alvarez-Gaum´e and Witten. Still, an action principle for the consistent equations reviewed in the previous Section is of some interest since, as we have seen, these six-dimensional models have a number of unfamiliar properties. Let us begin by considering a single 2-form with a self-dual field strength in six-dimensional Minkowski space. The PST lagrangian 13 L=

1 1 − ∂ µ φHµνρ H −σνρ ∂σ φ , Hµνρ H µνρ − 12 4(∂φ)2

(28)

where H = dB and H − = H −∗H, is invariant under the gauge transformation δB = dΛ, as well as under the additional gauge transformations δB = (dφ)Λ′

(29)

and δφ = Λ , δBµν =

Λ H − ∂ ρφ (∂φ)2 µνρ

.

(30)

The last two types of gauge transformations can be used to recover the usual field equation of a self-dual 2-form 13 . Indeed, the scalar equation results from the tensor equation contracted with − Hµνρ ∂ ρφ (∂φ)2

,

(31)

and consequently does not introduce any additional degrees of freedom. The invariance of eq. (30) can then be used to eliminate the scalar field. This field, however, can not be set to zero, since this choice would clearly make the Lagrangian of eq. (29) inconsistent. With this proviso, using eq. (29) one can see that the only solution of the tensor-field equation is precisely the self-duality condition for its field strength. 9

We now want to apply this construction to six-dimensional supergravity coupled to vector and tensor multiplets. In the conventions of ref. 10 , the theory describes a single self-dual 2-form ˆ r − i (χ ˆ µνρ = vr H H ¯m γµνρ χm ) µνρ 8

(32)

and n antiself-dual 2-forms i m rz m ¯ ˆ µνρ ˆr H = xm r Hµνρ + xr c trz (λγµνρ λ) . 4

(33)

The complete Lagrangian is obtained adding the term −

1 ∂ µ φ∂ σ φ ˆ − ˆ − νρ m+ ˆ m+ νρ ˆ µνρ [Hµνρ Hσ + H H σ ] 4 (∂φ)2

(34)

to eq. (2.6) of ref. 8 . It can be shown 19 that the 3-form σ

ˆ µνρ = H ˆ µνρ − 3 ∂[µ φ∂ φ H ˆ− K νρ]σ (∂φ)2

(35)

is identically self-dual, while the 3-forms σ

∂[µ φ∂ φ ˆ m+ m m ˆ µνρ ˆ µνρ K =H −3 Hνρ]σ (∂φ)2

(36)

are identically antiself-dual. With these definitions, we can display rather simply the complete supersymmetry transformations of the fields. Actually, only the transformations of the gravitino and of the tensorinos are affected, and become 3i ˆ µνρ γ νρ ǫ + i (χ ˆ µǫ + 1 K ¯m γµνρ χm )γ νρ ǫ − (¯ ǫχm )γµ χm δΨµ = D 4 32 8 i i 9i − (¯ ǫγµν χm )γ ν χm + (¯ ǫγ νρ χm )γµνρ χm − vr crz trz [(¯ ǫγµ λ)λ] 8 16 8 i i ǫγ ν λ)γµν λ] − vr crz trz [(¯ ǫγµνρ λ)γ νρ λ] , + vr crz trz [(¯ 8 16 i 1 i ˆ m αβγ ˆr α δχm = xm Kαβγ γ ǫ + xm crz trz [(¯ ǫγα λ)γ α λ] , (37) r ∂α v γ ǫ + 2 12 2 r while the scalar field φ is invariant under supersymmetry 19,20 . It can be shown that the complete lagrangian transforms under supersymmetry as dictated by the Wess-Zumino consistency conditions. 10

We now turn to describe the corresponding modifications of the supersymmetry algebra. In addition to general coordinate, gauge and supersymmetry transformations, the commutator of two supersymmetry transformations on r Bµν now generates two local PST transformations with parameters Λ′rµ =

∂σφ − ˆ σµα ˆ m+ α − xm (vr H r Hσµα )ξ (∂φ)2

,

Λ = ξ α ∂α φ

.

(38)

The transformation of eq. (38) on the scalar field φ is opposite to its coordinate transformation, and this gives an interpretation of the corresponding commutator 19,20 [δ1 , δ2 ]φ = δgct φ + δP ST φ = 0 , (39) that vanishes consistently with the invariance of φ under supersymmetry. Finally, the commutator on the vielbein determines the parameter of the local Lorentz transformation, that is now ˆ νab − i (χ Ωab = −ξ ν (ωνab − K ¯m γνab χm )) 8 1 m 1 m + (χ ¯ ǫ1 )(χ ¯m γab ǫ2 ) − (χ ¯ ǫ2 )(χ ¯m γab ǫ1 ) 2 2 +vr crz trz [(¯ ǫ1 γa λ)(¯ ǫ2 γb λ) − (¯ ǫ2 γa λ)(¯ ǫ1 γb λ)]

.

(40)

All other parameters remain unchanged while, aside from the extension 10 , the algebra closes on-shell on the modified field equations of the fermi fields. Of course, the resulting field equations reduce to those of previous Section once one fixes the PST gauge invariances in order to recover the conventional equations for (anti)self-dual tensor fields. For completeness, we conclude by displaying the lagrangian of six-dimensional supergravity coupled to vector and tensor multiplets, 1 1 1 1 s − ∂µ v r ∂ µ vr − vr crz trz (Fµν F µν ) e−1 L = − R + Grs H rµνρ Hµνρ 4 12 4 2 1 µναβγδ z r i ¯ µνρ 1 − ǫ cr Bµν trz (Fαβ Fγδ ) − (Ψµ γ Dν [ (ω + ω ˆ )]Ψρ ) 8e 2 2 i ˆ rµνρ (Ψ ¯ µ γν Ψρ ) + i vr [H + H] ˆ rαβγ (Ψ ¯ µ γ µναβγ Ψν ) − vr [H + H] 8 48 i i m µ r ˆ µνρ (χ ¯m γ µνρ χm ) (χ ¯ γ Dµ (ˆ ω)χm ) − vr H + 2 24 1 m ¯ µ γ ν γ µ χm ) − 1 xm [H + H] ˆ rµνρ (Ψ ¯ µ γνρ χm ) + x [∂ν v r + ∂νˆv r ](Ψ 4 r 8 r 1 m ¯ µD ˆ rµνρ (Ψ ¯ α γαµνρ χm ) + ivr crz trz (λγ ˆ µ λ) + x [H + H] 24 r 11

+ + + − − − + + + −

i ¯ µ γ νρ γ µ λ)] √ vr crz trz [(F + Fˆ )νρ (Ψ 2 2 i m m ˆ r sz ¯ µνρ λ) − i xm crz trz [(χ ¯ µ γν λ)] xr xs Hµνρ c trz (λγ ¯m γ µ γ ν λ)(Ψ 12 2 r 1 1 rz ¯ µνρ λ)(χ ¯m γ µν λ)Fˆµν ] vr crz trz (λγ ¯m γ µνρ χm ) + √ xm r c trz [(χ 16 2 i m rz ¯ µνρ λ) (χ ¯ γµν Ψρ )xm r c trz (λγ 8 3 1 vr crz trz [(χ ¯m γµν λ)(χ ¯m γ µν λ)] − vr crz trz [(χ ¯m λ)(χ ¯m λ)] 16 8 rz n sz 3 xm 1 m µνρ m ¯ r c xs c trz [(χ ¯m λ)(χ ¯n λ)] + (χ ¯ γ χ )(Ψµ γν Ψρ ) 4 vt ctz 8 m rz n sz 1 m µ n m 1 xr c xs c trz [(χ ¯m γµν λ)(χ ¯n γ µν λ)] − (χ ¯ γ χ )(χ ¯ γµ χn ) tz 8 vt c 8 1 ¯ ¯ µ λ′ )(λγ ¯ µ λ′ )] ¯ µνρ λ) − 1 vr vs crz csz′ trz,z′ [(λγ (Ψµ γν Ψρ )vr crz trz (λγ 4 2 α rz z′ ¯ µ λ′ )(λγ ¯ µ λ′ )] c cr trz,z′ [(λγ 2 ∂ µ φ∂ σ φ ˆ − ˆ − νρ m+ ˆ m+ νρ ˆ µνρ [Hµνρ Hσ + H H σ ] , (41) 4(∂φ)2

where α is the (undetermined) coefficient of the quartic coupling for the gauginos, and the corresponding supersymmetry transformations δeµ a = −i(¯ ǫγ a Ψµ ) ,

r ¯ [µ γν] ǫ) + 1 xmr (χ δBµν = iv r (Ψ ¯m γµν ǫ) − 2crz trz (A[µ δAν] ) , 2 δvr = xm ¯m ǫ) , r (χ i δAµ = − √ (¯ ǫγµ λ) , 2 3i ˆ µǫ + 1 K ˆ µνρ γ νρ ǫ + i (χ δΨµ = D ¯m γµνρ χm )γ νρ ǫ − (¯ ǫχm )γµ χm 4 32 8 i 9i i ǫγµν χm )γ ν χm + (¯ ǫγ νρ χm )γµνρ χm − vr crz trz [(¯ ǫγµ λ)λ] − (¯ 8 16 8 i i + vr crz trz [(¯ ǫγ ν λ)γµν λ] − vr crz trz [(¯ ǫγµνρ λ)γ νρ λ] , 8 16 i i ˆ m µνρ 1 ˆr µ δχm = xm Kµνρ γ ǫ + xm crz trz [(¯ ǫγµ λ)γ µ λ] , r ∂µ v γ ǫ + 2 12 2 r rz rz 1 1 xm 1 xm r c r c δλ = − √ Fˆµν γ µν ǫ − (χ ¯m λ)ǫ − (χ ¯m ǫ)λ sz sz 2 vs c 4 vs c 2 2

12

+

rz 1 xm r c (χ ¯m γµν ǫ)γ µν λ sz 8 vs c

.

(42)

One further comment is in order. Kavalov and Mkrtchyan 17 obtained long ago a complete action for pure d=6 (1,0) supergravity in terms of a single tensor auxiliary field. Their work may be connected to this special case of our result via an ansatz relating their tensor to the PST scalar. Still, the PST formulation has the virtue of simplicity and makes it manifest that the extra degrees of freedom may be locally eliminated via additional gauge transformationsb . Acknowledgments A.S. would like to thank the Organizers of the 1998 Trieste Conference on “Superfivebranes and Physics in 5+1 Dimensions”, the Organizers of the 1998 DESY Workshop on “Conformal Field Theory of D branes” and the Organizers of the 1998 Como-Centro Volta RFBF-INTAS School on “Advances in Quantum Field Theory, Statistical Mechanics and Dynamical Systems” for their kind invitation to present these results. References 1. 2. 3. 4.

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