Preprint DFPD 98/TH/29

DFTT 29/98

hep-th/9806140

June 1998

, Supergravity: Lorentz–invariant actions and duality

Gianguido Dall’Agata^{1}^{1}1,
Kurt
Lechner^{2}^{2}2 and
Mario Tonin^{3}^{3}3

Dipartimento di Fisica Teorica, Università degli studi di Torino,

via P. Giuria 1,I-10125 Torino

Dipartimento di Fisica, Università degli Studi di Padova,

and

Istituto Nazionale di Fisica Nucleare, Sezione di Padova,

Via F. Marzolo, 8, 35131 Padova, Italia

We present a manifestly Lorentz invariant and supersymmetric component field action for , type supergravity, using a newly developed method for the construction of actions with chiral bosons, which implies only a single scalar non propagating auxiliary field. With the same method we construct also an action in which the complex two–form gauge potential and its Hodge–dual, a complex six–form gauge potential, appear in a symmetric way in compatibility with supersymmetry and Lorentz invariance. The duals of the two physical scalars of the theory turn out to be described by a triplet of eight–forms whose curvatures are constrained by a single linear relation. We present also a supersymmetric action in which the basic fields and their duals, six–form and eight–form potentials, appear in a symmetric way. All these actions are manifestly invariant under the global –duality group of , supergravity and are equivalent to each other in that their dynamics corresponds to the well known equations of motion of , supergravity.

PACS: 04.65.+e; Keywords: Supergravity, ten dimensions, duality

## 1 Introduction and Summary

The obstacle which prevented for a long time a Lagrangian formulation of , supergravity is the appearance of a chiral boson in the spectrum of the theory, i.e. a four–form gauge potential with a self–dual field strength. A possible way to overcome this obstacle was presented in [1], by extending Siegel’s action for two–dimensional chiral bosons [2] to higher dimensions. In this approach one gets, through lagrangian multipliers, not the self–duality condition as equation of motion for the chiral bosons, but rather its square. However, in dimensions greater than two, the elimination of the lagrangian multipliers seems problematic [2] and, moreover, at the quantum level, for example in the derivation of the Lorentz anomaly, it seems to present untreatable technical and conceptual problems. On the other hand, a group manifold action for supergravity has been obtained in [3]. In general, when a group manifold action is restricted to ordinary space–time one gets a consistent supersymmetric action for the component fields; but if a self–dual (or anti self–dual) tensor is present, the restricted action loses supersymmetry [4].

A new method for writing manifestly Lorentz–invariant and supersymmetric actions for chiral bosons (–forms) in has been presented in [5]: it uses a single non propagating auxiliary scalar field and involves two new bosonic symmetries; one of them allows to eliminate the auxiliary field and the other kills half of the degrees of freedom of the –form, reducing it to a chiral boson. This method turned out to be compatible with all relevant symmetries, including supersymmetry and –symmetry [6, 7] and admits also a canonical coupling to gravity, being manifestly Lorentz invariant. As all lagrangian formulations of theories with chiral bosons, the method is expected to be insufficient for what concerns the quantization of these actions on manifolds with non–trivial topology [8], but it can be successfully applied, even at the quantum level, on trivial manifolds. As an example of the efficiency of this method also at the quantum level, we mention that the effective action for chiral bosons in two dimensions, coupled to a background metric, can easily be computed in a covariant way [9], and that it gives the expected result, namely the effective action of a two–dimensional complex Weyl–fermion. This implies, in turn, that also the Lorentz– and Weyl–anomalies due to a chiral boson, as derived with this new method, coincide with the ones predicted by the index theorem. Work regarding the Lorentz–anomaly in higher dimensions is in progress.

In this paper we present a manifestly
Lorentz–invariant and supersymmetric action for ,
supergravity, based on this method^{4}^{4}4The covariant action for
the bosonic sector of type supergravity has already been
presented in [13]..
Apart from the above mentioned new features, the basic ingredients
are
the equations of motion and SUSY–transformations of the basic
fields, which are well known [10, 11, 12], and can be most
conveniently derived in a superspace approach [10, 3].
In addition to the metric, the bosonic fields in this theory are two
complex scalars (–forms), which parametrize the coset
, a complex two–form
gauge potential
and
the real chiral four–form gauge potential.

Since the discovery of –branes and their coupling to RR gauge potentials, the Hodge duals to the zero– and two–forms (the four–form is self–dual) i.e. the eight– and six–forms acquired a deeper physical meaning. It is therefore of some interest to look at a Lagrangian formulation with manifest duality i.e. in which the zero and eight–forms and the two and six–forms appear in a symmetric way. The method presented in [5] appears particularly suitable to cope also with this problem. Indeed, a variant [14, 15] of this method allowed in the past to construct a manifestly duality invariant Lagrangian for Maxwell’s equations in four dimensions [14] as well as for supergravity [15].

In this paper we shall also construct an action with manifest duality between the gauge potentials and their Hodge duals. Upon gauge fixing the new bosonic symmetries, mentioned above, one can remove the six– and eight–forms and recover the (standard) formulation with only zero– and two–forms. On the other hand, a Lagrangian formulation in which only the six and eight–forms appear, instead of the zero and two–forms, is not accessible for intrinsic reasons i.e. the presence of Chern–Simons forms in the definition of the curvatures.

The case of the eight–forms requires a second variant of the method. As we will see, manifest invariance under the global –duality group of the theory requires the introduction of three real eight–form potentials, with three nine–form curvatures, which belong to the adjoint representation of . Two –invariant combinations of the three nine–form curvatures are related by Hodge–duality to the two real (or one complex) one–form curvatures of the scalars. The third one is determined by an –invariant linear constraint between the three curvature nine–forms. While the first variant of our method allows to treat Hodge–duality relations between forms at a Lagrangian level, the second variant allows to deal, still at a Lagrangian level, with linear relations between curvatures. This is then precisely what is needed to describe the dynamics of the eight–forms through an action principle.

The general validity of the method is underlined also by the fact that all these lagrangians are supersymmetric. To achieve supersymmetry one has to modify the SUSY transformation laws of the fermions in a very simple and canonical way, the modifications being proportional to the equations of motion, derived e.g. in a superspace approach. The on–shell SUSY algebra can then be seen to close on the two new bosonic symmetries mentioned at the beginning.

In section two we present the superspace language and results for , supergravity, following mainly [10]. There, we construct also the dual supercurvatures and potentials in a covariant way. These results are used at the component level, in section four, to write a manifestly Lorentz–invariant action for the theory, using only the scalars, two and four–forms. In section three we give a concise account of the new method itself (for more details see [5, 6]). In section five we write an action in which the two and six–form potentials appear in a symmetric way and prove its invariance under supersymmetry. Section six is devoted to the construction of an action in which all gauge potentials appear in a symmetric way, paying special attention to the new features exhibited by the eight–forms. Section seven collects some concluding remarks and observations.

## 2 Superspace results

The superspace conventions and results of this paper follow mainly [10]. The , superspace is parametrized by the supercoordinates where the are sixteen complex anticommuting coordinates. Here and in what follows the ”bar” indicates simply complex conjugation and in case transposition. The cotangent superspace basis is indicated by , where and , and indicates the complex gravitino one–superform. All superforms can be decomposed along this basis. The Lorentz superconnection one–form is given by with curvature .

The two physical real scalars of the theory parametrize the coset , where is the global –duality symmetry group of , supergravity, and the is realized locally. The coset is described by two complex scalars which are constrained by such that the matrix

(2.1) |

belongs to and the fields form an doublet. The Maurer–Cartan form decomposes then as

(2.2) |

where and are invariant one–forms:

(2.3) | |||||

(2.4) |

Since the weights of are , i.e. has weight , is a –connection and , which has to be considered as the curvature of the scalars, has weight 4. We can then introduce a and covariant derivative which acts on a –form with weight as

(2.5) |

For a list of the weights and representations of the fields see the table at the end of the section.

For a –form with purely bosonic components

we introduce its Hodge–dual, a –form, as

(2.6) |

where

In particular, on a –form we have

The other bosonic degrees of freedom are carried by the following superforms. We introduce a complex two–form , where constitute an doublet, and its dual which is a complex six–form , where constitutes also a doublet. The real four–form in the theory, the ”chiral boson”, is denoted by . As anticipated in the introduction the duals of the scalars are parametrized by three real eight–forms which are described by a complex eight–form and a purely imaginary one . The three forms form an triplet, i.e. they belong to the adjoint representation of . All these forms are singlets.

The curvatures associated to these forms maintain their and representations and are given by

(2.7) | |||||

(2.8) | |||||

(2.9) | |||||

(2.10) | |||||

(2.11) |

Again form an triplet.

The superspace parametrizations of these curvatures are more conveniently given in terms of the invariant combinations which one can form using the scalars and . Including also the curvatures for the scalars these invariant curvatures are given by

(2.12) | |||||

(2.13) | |||||

(2.14) | |||||

(2.15) | |||||

(2.16) | |||||

(2.17) |

and carry charge 4, and carry charge 2 and and carry charge 0 and are respectively real and purely imaginary, while all other are complex. The associated Bianchi identities are

(2.18) | |||||

(2.19) | |||||

(2.20) | |||||

(2.21) | |||||

(2.22) | |||||

(2.23) |

For the connection we have

(2.24) |

it is also useful to notice that

(2.25) |

Defining the torsion as usual by

(2.26) |

it satisfies the Bianchi identities

(2.27) | |||||

(2.28) |

The superspace parametrizations of the curvatures in (2.12)–(2.17) can now be written, for , as

(2.29) |

where indicates the purely bosonic part

(2.30) |

and the –forms involve the gravitino one–form and the complex spinor , which completes the fermionic degrees of freedom of , supergravity (contraction of spinorial indices is understood):

(2.31) | |||||

(2.32) | |||||

(2.33) | |||||

(2.34) | |||||

(2.35) | |||||

(2.36) |

Actually, and contain also a contribution with only bosonic vielbeins. These amount, however, only to a redefinition of and . These redefinitions are convenient for what follows, see eqs. (2.43)–(2.46) below. It is also convenient to decompose the forms as

(2.37) |

where indicates the parts which depend on and the parts which are independent of , in particular .

The parametrizations of the torsions and of become

(2.38) | |||||

(2.39) | |||||

(2.40) |

Here parametrizes the part of with only bosonic vielbeins and

(2.41) |

indicates the self–dual part of :

(2.42) |

For the parametrization of , see [10].

All these parametrizations and the form of the Bianchi identities are dictated by the consistency of the Bianchi identities for the torsion and for the curvatures themselves.

Since the closure of the SUSY–algebra sets the theory on shell one gets also the following (self)–duality relations between the curvatures, which become extremely simple when expressed through the defined in (2.29)-(2.36):

(2.43) | |||||

(2.44) | |||||

(2.45) | |||||

(2.46) |

The first relation is the equation of motion for the four–form . Equation (2.44), which relates the curvature of to the curvature of , promotes the Bianchi identities (2.19) and (2.21) to equations of motion for and ,respectively.

Equation (2.46) constitutes the linear constraint between and mentioned in the introduction, and allows to express — through (2.17) — the curvature , and hence , as a function of . Substituting this expression for in (2.16) one can compute and as a function of . At this point the duality relation (2.45) promotes (2.18) and (2.22) to equations of motion for and respectively. The complex eight–form is thus dual to the two real scalars contained in and . These fields are,in fact, constrained by and are subjected to the local invariance. Once this invariance is fixed only two real physical scalars survive. It is clear that the elimination of breaks manifest invariance and that a manifestly invariant action principle for the dual scalars has to be based on three eight–forms, i.e. (2.10)–(2.11), (2.16)–(2.17) and (2.45)–(2.46).

The occurrence of three eight–forms can also be understood from the following point of view. Since the theory possesses a global invariance there must exist three conserved currents which belong to the adjoint representation of . The Hodge duals of these currents, which have to be closed and hence locally exact, are just given by and and their explicit expressions can be derived from (2.10) and (2.11) using (2.45) and (2.46).

An expression for the dual curvatures and their Bianchi identities was given also in [16], in a non–manifestly and covariant formulation. In this formulation it is sufficient to introduce only two eight–form potentials because the invariance has been gauge fixed.

The equations of motion for the gravitino, for and for the metric can be found in [10]; their explicit expressions are not needed here since the action is completely determined by SUSY invariance, by the knowledge of the Bianchi identities and by the superspace parametrizations given above.

The and representations of the basic fields and their charges are:

, | , | ||||||||

0 | 1 | 3 | -2 | 0 | 0 | 0 | 2 | 4 | |

1 | 1 | 1 | 2 | 2 | 3 | 1 | 1 | 1 |

## 3 The covariant method

From now on we will work in ordinary space–time but still continue to use the language of forms to avoid the explicit appearance of Lorentz indices. In particular, our actions will be written as integrals over ten–forms. In the next section we will perform the reduction of the superspace results of the preceding section to ordinary bosonic space–time.

In this section we will present the basic ingredients which allow to write covariant actions for equations (2.43)-(2.46) concentrating in particular on the self–duality equation of motion (2.43).

This equation is of the type

(3.1) |

where

(3.2) |

and is independent of ^{5}^{5}5In the case of ,
supergravity we have , now in ordinary space–time..

The covariant method [5] requires the introduction of a scalar auxiliary field and the related vector

(3.3) |

satisfying . We introduce also the one–form

(3.4) |

and indicate with the interior product of a –form with the vector field .

Defining

(3.5) |

the action which reproduces (3.1) can be written as

(3.6) | |||||

The form of this action is selected, and fixed, by the following symmetries:

(3.7) | |||||

(3.8) |

where and are transformation parameters, respectively a scalar and a three–form. The action is, actually, invariant also under finite transformations of the type i.e. under . This fact becomes relevant in what follows.

The equation of motion for and are respectively given by

(3.9) | |||||

(3.10) |

The symmetry promotes the auxiliary field to a ”pure gauge”
field and allows to gauge--fix^{6}^{6}6Typical non–covariant gauges
are
where is a constant vector. it to an arbitrary function provided that . Correspondingly, the equation of
motion (3.9) can easily be seen to be a consequence of (3.10).

The general solution of (3.10) is . Since under a finite transformation we have

choosing we get

(3.11) |

Due to the identity decomposition on a –form

(3.12) |

one gets the identity

(3.13) |

and hence (3.11) is equivalent to the self–duality equation of motion for (3.1).

This concludes the proof that the action describes indeed interacting () chiral bosons in ten dimensions. If the fields composing are themselves dynamical, one has to complete the action by adding terms which involve the kinetic and interaction terms for those fields, but not itself because otherwise the symmetries and are destroyed.

Another five–form, which will acquire an important role in establishing supersymmetry invariance, is given by

(3.14) |

## 4 The complete action for , supergravity

In this section we write a covariant and supersymmetric component level action for supergravity in its canonical formulation, i.e. when the bosonic degrees of freedom are described by , and , incorporating the dynamics of according to the method presented in the preceding section.

The component results are obtained from the superspace results of section two in a standard fashion setting . Whenever we use the same symbols as in section two we mean those objects evaluated at . In particular the differential becomes the ordinary differential. Every form can now be decomposed along the vielbeins and the gravitino reduces to . The supercovariant connection one–form is naturally introduced, via equation (2.38) now evaluated at , as

(4.1) |

This determines as the metric connection, augmented by the standard gravitino bilinears. The supercovariant curvature two–form is now with given in (4.1). It is also convenient to introduce the , , supercovariant curvatures as ()

(4.2) |

where the are given in (2.7)-(2.9) and (2.12)-(2.14) and the are defined in (2.31)-(2.33). More precisely

(4.3) | |||||

(4.4) | |||||

(4.5) |

Since we write the Lagrangian as a ten–form it is also convenient to define the –forms

(4.6) |

In particular .

The action for type supergravity with the canonical fields can now be written as follows:

(4.7) | |||||

In the first line we have the kinetic terms for the metric, the gravitino and the field . The second line contains the action of the preceding section, augmented by a term proportional to which compensates the gauge transformation of , but is –independent, as required. Since all the other fields are required to be invariant under the transformations , and appears in (4.7) only in the combination , this action gives as (gauge fixed) equation of motion for just (3.1), i.e. (2.43).

The third and fourth lines in (4.7) contain, between square brackets, the kinetic and interaction terms for and respectively. These particular combinations are just the ones which respect the dualities , as we will see in the next section. Variation of (4.7) with respect to and produces, as equations of motion, just the Bianchi identities (2.21)-(2.23) of section two. The remaining terms in the action above are quartic in the fermions and are fixed by supersymmetry, which also fixes the relative coefficients of all the other terms.

The supersymmetry transformations of the fields can again be read from the superspace results. Introducing the transformation parameter , the on–shell SUSY transformations of the component fields are given by covariantized superspace Lie–derivatives of the corresponding superfields, evaluated at :

(4.8) |

For the graviton, gravitino, and , we get from (2.38),(2.39),(2.40),(2.25) and the parametrization of

(4.9) | |||||

(4.10) | |||||

(4.11) | |||||

(4.12) | |||||

(4.13) |

The term can be easily evaluated by substituting, in the r.h.s. of (2.39), and respectively with and .

For what concerns the –forms, due to gauge invariance and Lorentz invariance, (4.8) would reduce simply to . However, the presence of the Chern–Simons forms in (2.7)–(2.11) requires compensating SUSY transformations for the potentials . It is convenient to parametrize generic transformations for these potentials in such a way that the curvatures (and ) transform covariantly. For later use we give here a complete list of all the combined transformations:

(4.14) | |||||

(4.15) | |||||

(4.16) | |||||

(4.17) | |||||

(4.18) |

where parametrize generic transformations.

The corresponding (invariant) transformations for the curvatures are:

(4.19) | |||||

(4.20) | |||||

(4.21) | |||||

(4.22) | |||||

(4.23) |

The transformations for the are easily obtained from their definitions (2.13)–(2.17).

For supersymmetry transformations we have to choose, here for ,

(4.24) |

For the curvatures, this leads to

(4.25) |

and

(4.26) |

i.e., again to the covariant Lie derivative. Expressing the in terms of the , whose super–space parametrizations are known, in particular , the transformations (4.26) can be easily evaluated.

It remains to choose the SUSY transformation law for the auxiliary field . Since this field, being non propagating, has no supersymmetric partner, the simplest choice turns out to be actually the right one. We choose

(4.27) |

This concludes the determination of the on–shell SUSY transformation laws for the fields. Due to the chirality condition (2.43), which is an equation of motion of the on–shell superspace approach, some of these transformation laws could change by terms proportional to , or equivalently, to . As we will now see, SUSY invariance of the action, and therefore the closure of the SUSY algebra on the transformations and , requires, indeed, such modifications, but, in the present case, only for the gravitino supersymmetry transformation.

In practice the SUSY variation of the action (4.7), which is written as , can be performed by lifting formally the ten–form to superspace, applying then the operator to and using the superspace parametrizations and Bianchi identities of section two to show that vanishes. The unique term for which this procedure does not work is , because cannot be lifted to a superfield; therefore the supersymmetry variation of this term has to be performed ”by hand”. In particular, one has to vary explicitly the vielbeins contained in .

We give the explicit expression for the SUSY variation of the terms in which depend on and (the second line in (4.7)). This will be sufficient to guess the correct off–shell SUSY transformation law for the gravitino (the term proportional to is included to get a gauge–invariant expression)