Contents

Off-Shell Poincaré Supergravity

Daniel Z. Freedman, Diederik Roest, and Antoine Van Proeyen
SITP and Department of Physics, Stanford University, Stanford, California 94305 USA

Center for Theoretical Physics and Department of Mathematics,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Van Swinderen Institute for Particle Physics and Gravity,

University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands

KU Leuven, Institute for Theoretical Physics,

Celestijnenlaan 200D, B-3001 Leuven, Belgium

Abstract

We present the action and transformation rules of Poincaré supergravity coupled to chiral multiplets with off-shell auxiliary fields. Starting from the geometric formulation of the superconformal theory with auxiliary fields, we derive the Poincaré counterpart by gauge-fixing the Weyl and chiral symmetry and -supersymmetry. We show how this transition is facilitated by retaining explicit target-space covariance. Our results form a convenient starting point to study models with constrained superfields, including general matter-coupled de Sitter supergravity.  E-mails: dzf@math.mit.edu, d.roest@rug.nl, Antoine.VanProeyen@fys.kuleuven.be

## 1 Introduction

In this paper we reformulate supergravity theory coupled to chiral multiplets using ingredients that are manifestly covariant under complex diffeomorphisms of the Kähler target space. This simplifies the action and transformation rules at the superconformal level and streamlines the passage to the physical theory, which is invariant under local Poincaré supersymmetry. We explicitly retain the auxiliary fields of the chiral multiplets, keeping these off-shell. This formulation permits the construction of supergravity theories in which supersymmetry is realized nonlinearly due to constraints on superfields.

Nonlinear supersymmetry breaks the usual degeneracy between bosonic and fermionic degrees of freedom. It can be seen to follow from imposing supersymmetric constraints on the superfield, leading to constrained superfields. The present constructions include a nilpotent field coupled to the supergravity multiplet [1, 2] as well as physical chiral and vector multiplets [3, 4, 5]. Moreover, extensions have been proposed involving additional constrained superfields subject to orthogonal constraints [6, 7]. An overview of such possibilities can be found in [8, 9]. Finally, one can construct de Sitter supergravity by imposing constraints involving the supergravity multiplet as well [10, 11].

Given these prolific developments in constrained superfields, it is clearly advantageous to solve for any auxiliary fields at the latest possible stage, in order to allow for as many constraints. This is exactly what our current formulation provides, with the final Poincaré result containing the auxiliary fields and of the supergravity and chiral multiplets. The latter correspond to the order parameters of supersymmetry breaking. In the case of linearly realized supersymmetry, these are taken as auxiliary fields of a non-constrained superfield and take their Gaussian values in terms of the physical components. In the case of a non-linear realization, however, the auxiliary field is a free parameter and instead some of the otherwise physical components are solved in terms of this parameter.

The proposed framework allows for arbitrary numbers of constrained and independent superfields. For instance, it can be used to analyze the situation with a number of nilpotent superfields in addition to independent chiral multiplets. Similarly, it allows for arbitrary couplings between the different types of fields.

An important role in our derivation is played by the covariance of supersymmetry transformations and auxiliary fields. We will employ a formulation in which these transform covariantly under reparametrizations of the target space spanned by the scalars of the theory. This formulation was emphasized very recently in [12], where further details and motivation can be found. The key point is that symmetry transformations (which can include Killing isometries, supersymmetry, ….) generically do not transform covariantly under reparametrizations. To remedy this situation, one can introduce covariant transformations and derivatives defined by

 ^δχi=δχi+Γijkχjδϕk,∇μχi=∂μχi+Γijkχj∂μϕk. (1.1)

These are the unique quantities that transform covariantly both under target space reparametrizations as well as other symmetries.

Previous formulations of the supergravity theories, some of which also retain aspects of the auxiliary fields [13, 14], were not manifestly covariant under the reparametrizations of these two manifolds. With the methods of [12] we can rewrite all essential formulas in a more geometric fashion, such that reparametrization invariance is manifest. This formulation is very useful in the context of the transition from the superconformal formulation of matter-coupled supergravity to the super-Poincaré theory. A key point is that the covariant transformations of the superconformal theory maintain covariance after gauge-fixing to the super-Poincaré theory. This allows us to obtain the supersymmetry transformations of the fields in the physical Poincaré supergravity theory in a straightforward way from those in the superconformal theory.

Though many ingredients of our work apply to both and theories, we will restrict ourselves in this paper to supergravity coupled to chiral multiplets. The Weyl multiplet is the gauge multiplet that allows local supersymmetry. Its vector gauge field is an auxiliary field, and we will use its field equations. On the other hand, the auxiliary fields of the chiral multiplets will remain off-shell. We will derive the covariant supersymmetry transformations and the action for the full field content of chiral multiplets. Gauge multiplets can be included in this new formulation, but they will rather appear as spectators in the theory of the chiral multiplets, and we omit them here.

The Kähler manifolds in and supergravity are projective manifolds embedded in larger Kähler manifolds that have conformal properties. The formulation of Poincaré supergravity as a broken superconformal theory makes use of ‘compensating fields’, whose presence allows the super-Poincaré group to be promoted to a superconformal group. We start with chiral multiplets, with complex scalar fields , . This includes the compensating scalar field and others scalars that will be physical in the super-Poincaré theory. However, we do not want to specify which of the fields is the compensator. That is part of the reparametrization invariance that we do not want to break. We will use the name ‘embedding manifold’ for the scalar manifold with complex fields, and ‘projective manifold’ for the complex -dimensional manifold that describes the super-Poincaré theory.

We start in section 2 by reviewing the main general results needed for a covariant formulation, as found in [12]. These will be exploited in full in this paper. In section 3 we discuss the superconformal theory in the covariant formulation, emphasizing the role of the various symmetries. This extends the results of section 3 in [12] to include superconformal transformations. We present the covariant transformation rules and the covariant form of the action. In section 4 we introduce the convenient variables to discuss the super-Poincaré theory. We discuss the gauge fixing that leads to the Poincaré group and the resulting projective space. The relation between superconformal and Poincaré supersymmetry is presented in section 5. With these preliminaries in place, one can derive the transformation laws of the Poincaré fields smoothly, given the superconformal transformations. We obtain the full transformation rules, and the part of the action relevant for auxiliary fields, in section 6. We conclude with a synopsis, and a brief discussion of applications to constrained multiplets in section 7.

## 2 Geometrisation of transformations

We summarize the main ideas and results of [12] for a theory that contains scalar fields that are maps from spacetime to coordinate charts on a Riemannian target space with metric and Christoffel connection . We require covariance under reparametrizations

 ϕi→ϕ′i(ϕ). (2.1)

The theory may as well contain composite vectors , such as Killing vectors, and other fields such as the fermions of supersymmetric theories, which transform as sections of the tangent bundle of . Their transformation laws111We use the notation , to indicate an equation that is only valid when is a function of the scalars and not of other fields of the theory.

 Vi(ϕ)→V′i(ϕ′(ϕ))=∂ϕ′i∂ϕjVj(ϕ),χi→χ′i=∂ϕ′i∂ϕjχj, (2.2)

are similar, but there is an important difference that we discuss shortly. As in most treatments of supersymmetry, the fields and are considered as independent, so that form a basis of the field space.

Readers are probably familiar with the following definition of covariant spacetime derivatives222We use torsionless connections. of :

 ∇μϕi = ∂μϕi, ∇μVi(ϕ) = (∂μϕj)∇jVi(ϕ)=(∂μϕj)[∂jVi(ϕ)+ΓijkVk(ϕ)], ∇μχi = ∂μχi+Γijkχk(∂μϕj). (2.3)

Using (2.1-2.2) and the transformation property of , one can show that these covariant derivatives transform as vectors.

Consider an infinitesimal symmetry operation

 δϕi(ϕ,χ),δχi(ϕ,χ), (2.4)

on our system such as spacetime translations or supersymmetry. Then (2.1-2.2) show that is a vector, but and the induced transformation are not. Again we need a connection term to define covariant transformations in the last two cases: [15], [16, App.14B]

 ^δVi(ϕ) ≡ δVi(ϕ)+ΓijkδϕjVk(ϕ)=δϕj(∂jVi+ΓijkVk(ϕ), ^δχi ≡ δχi+Γijkδϕjχk (2.5)

We can now observe the difference between the covariant rules for composite vectors and vector-valued fields such as . Only the former can be expressed in terms of covariant derivatives on . To make this clear we repeat

 ∇μVi(ϕ)=(∂μϕj)∇jVi(ϕ),^δVi(ϕ)=(δϕj)∇jVi(ϕ). (2.6)

Note that the covariant rules defined above for vectors can be easily extended to covectors and tensors. For the metric of which are the Christoffel symbols:

 ∇μgij(ϕ)=(∂μϕk)∇kgij=0,^δgij=(δϕk)∇kgij=0. (2.7)

We now state a principle that is both obvious when thought about and powerful in operation: If an action is built as a scalar from vectors and tensors, then invariance under a symmetry operation is equivalent to invariance under the covariant transformation .

Ordinary derivatives and transformations commute by definition:

 δ∂μ=∂μδ, (2.8)

but the commutator of and gives rise to curvature terms.

 ^δ∇μVi=∇μ^δVi+RkℓijVj(δϕk)(∂μϕℓ). (2.9)

Furthermore, curvature terms appear also in the commutator of covariant derivatives and the commutator of covariant transformations. This relation as well as those below, which were derived in [12], are valid both for composite and vector-valued fields such as ;

 [∇μ,∇ν]Vk=Rijkℓ(∂μϕi)(∂νϕj)Vℓ, [δ1,δ2]Vi=δ3Vi ⟹ [^δ1,^δ2]Vi=^δ3Vi+RkℓijVj(δ1ϕk)(δ2ϕℓ), (2.10)

where is a shortcut for , …and is the function of and determined by the structure of the symmetry algebra.

We close this section with an exercise for interested readers. Let be a Killing vector on that acts on fermions fields as . By the rules stated above the covariant form of this symmetry operation is . Without peeking at [12], show that .

## 3 Covariant superconformal theory

In the first stage of the superconformal approach to supergravity, a set of chiral multiplets, denoted by , with Weyl weight 1, is coupled to the Weyl multiplet . The complex scalar fields are coordinates of a Kähler manifold with conformal symmetry. A conformal Kähler manifold obeys certain homogeneity conditions, which we explain in the next subsection. These conditions constrain the allowed reparametrizations and also induce a chiral symmetry. We apply the geometric methods of section 2 to define covariant derivatives and covariant transformations, which transform properly under homogeneous reparametrizations and chiral transformations. We then focus on covariant superconformal transformations of and also write the superconformal action that determines their dynamics. We refer to this setting as the geometric superconformal theory. This prepares the way for a covariant treatment of the physical supergravity theory in section 4.

### 3.1 Superconformal Kähler manifolds

We first discuss the embedding space spanned by the scalars of the superconformal theory, together with a covariant formulation of its symmetries. In outline, our discussion follows [12], but we emphasize two new ingredients, namely homogeneity and chiral symmetry, called -symmetry.

The scalars are coordinates of a Kähler manifold. Its metric is determined by a Kähler potential as usual:

 GI¯J=NI¯J≡∂I∂¯JN(X,¯X). (3.1)

In order to apply it to the physical supergravity theory, as we will discuss it in section 4, the metric should have signature , which corresponds to the index values ). The negative direction corresponds to the conformal compensator, but in this fully covariant approach we need not identify it more specifically.

#### Homogeneity.

An important condition imposed by superconformal symmetry is that the Kähler potential is homogeneous of weight one333This means that the manifold possesses a ‘closed homothetic Killing vector’ [17] (summarized in[16, section 15.7]). The presence of such a vector implies conformal symmetry, and in a Kähler manifold, it further implies a Killing vector for the -symmetry: . In this paper we choose coordinates where this closed homothetic Killing vector is aligned in the direction . A generalization to more general coordinates is possible. in both the holomorphic and the anti-holomorphic coordinates. The homogeneity condition requires the equations (in a notation where subscripts on indicate derivatives)

 N(X,¯X) = XINI=¯X¯IN¯I=NI¯JXI¯X¯JNI=NI¯J¯X¯J, XINIJ = 0NIJ¯K¯X¯K=NIJXKNKI¯J=0. (3.2)

An important consequence of homogeneity is that geometrical quantities, such as the connection and curvature tensor444In general, closed homothetic Killing vectors are zero-modes of the curvature. for the Kähler manifold, have zero vectors, viz

 ΓIJK = GI¯LN¯LJK,XJΓIJK=0, (3.3) RI¯JK¯L = NI¯JK¯L−NIK¯MG¯MNNN¯J¯L,XIRI¯JK¯L=0. (3.4)

Kähler transformations of are not permitted since holomorphic additional terms do not satisfy the homogeneity requirement.

#### Chiral Symmetry.

Homogeneity of the superconformal Kähler manifold, together with its complex structure, imply that there are separate dilatation and chiral symmetries under which the scalars transform as

 δXI=(λD+iλT)XI,δ¯X¯I=(λD−iλT)¯X¯I. (3.5)

These are the Weyl scaling and the chiral -symmetry, which is the R-symmetry of the conformal supersymmetry algebra. The Lagrangian contains auxiliary connections for both symmetries. The -connection is the gauge field of the Weyl multiplet.555It is called in this section in which we include only its action on the scalars. We focus on the -connection, since the dilatation gauge field will be set to zero when gauge fixing the special conformal transformations in the passage to the Poincaré theory. The scalar Lagrangian is therefore

 L=−NI¯Jgμν∇μXI∇ν¯X¯J, (3.6)

where

 ∇μXI=∂μXI−iAμXI. (3.7)

and the T-connection transforms as . The connection is an auxiliary field whose field equation is solved by

 iAμ=12N(NI∂μXI−N¯J∂μ¯X¯J). (3.8)

After substitution of this result in (3.6), we find the equivalent Lagrangian

 L=−14N∂μN∂μN−N(∂μXI)(∂μ¯X¯J)∂I∂¯JlnN. (3.9)

Upon redefining this can be interpreted as a cone over a projective manifold. Note that the radial direction has a kinetic term of the wrong sign. However, this corresponds to the conformal compensator and not to a physical field and does not pose a problem.

The composite connection in (3.8) must be included in our covariant definitions as follows. A general vector of chiral weight satisfies

 δTVI=icλTVI, (3.10)

and we extend the definition of covariant derivatives , (1.1), with the -connection:

 ∇μVI≡(∂μ−icAμ)VI+ΓIJKVJ∂μXK. (3.11)

This transforms as a tangent vector under coordinate reparametrizations, and has chiral weight . Similarly, we define covariant transformation rules as

 ^δVI≡δVI+ΓIJKVJδXK, (3.12)

which also transform as a tangent vector.

In the special case of composite vectors on the target space, i.e. , the -transformation is implemented as the Killing symmetry

 δTVI(X,¯X)=iλT(XJ∂J−¯X¯J∂¯J)VI(X,¯X). (3.13)

Covariant spacetime derivatives and transformation rules satisfy relations to covariant derivatives on the Kähler manifold, i.e.

 ∇JVI=∂JVI+ΓIJKVK,∇¯JVI=∂¯JVI, (3.14)

which generalize (2.6),

 ∇μVI(X,¯X)=∇μXJ∇JVI+∇μ¯X¯J∇¯JVI,^δVI(X,¯X)=δXJ∇JVI+δX¯J¯¯¯¯¯∇¯JVI, (3.15)

since the -connections follow this pattern on account of (3.13). The last equation is valid separately for all transformations, e.g. both for supersymmetry and for -transformations. Note also that (3.7) is consistent with (3.11) because of (3.3), and that .

In the rest of this paper we develop covariant formulas under homogeneous reparametrizations666If we use results of actions and transformations in a more general frame than frames where , the restriction to homogeneity can be removed. of the target space. This means that coordinates transform as vectors under this class of reparametrizations,

 XI → X′I(X)=XI∂IX′I(X), (3.16)

which is not true for the more general coordinate transformations of section 2.

### 3.2 Covariant superconformal transformations

In this section we obtain the covariant form of the supersymmetry variations given in [16, (17.3)]. The rules for target space covariance are essentially as given in section 3 of [12], but we extend them to incorporate two features of superconformal supergravity. First we have local -supersymmetry with parameter . Second, the superconformal covariant derivatives include connections of the Weyl multiplet. The covariant transformations are777We use the notation that is left-handed, i.e. and .

 ^δXI =δXI=1√2¯ϵΩI, ^δΩI =1√2PL(⧸DXI+^FI)ϵ+√2XIPLη, ^δ^FI =1√2¯ϵ[γμ^DμΩI+12RJ¯LIKΩ¯L¯¯¯¯ΩJΩK], (3.17)

where we introduced the covariant auxiliary field

 ^FI=FI−12ΓIJK¯ΩJΩK. (3.18)

The superconformal covariant derivatives depend on the fields of the Weyl multiplet: the frame field , the gravitino , the dilatation gauge field and the -gauge field . These are independent fields. After the action is constructed, the field equation of sets it equal to as in (3.8) plus a fermionic part given in (6.6) below. These derivatives are

 DμXI= ∂μXI−bμXI−iAμXI−1√2¯ψμΩI, (3.19) DμΩI= (∂μ+14ωμabγab−32bμ+12iAμ)ΩI−1√2PL(⧸DXI+FI)ψμ−√2PLXIϕμ,

where is the gauge field of the -supersymmetry. This is dependent on the other fields

 ϕμ=−12γνR′μν(Q)+112γμγρνR′ρν(Q),R′μν(Q)=2(∂[μ+14ω[μabγab+12b[μ−32iA[μγ∗)ψν]. (3.20)

The spin connection used here is the conformal connection with gravitino and torsion

 ωμab=2eν[a∂[μeν]b]−eν[aeb]σeμc∂νeσc+2eμ[aeb]νbν+12¯ψμγ[aψb]+14¯ψaγμψb. (3.21)

The geometric covariant derivative is888Since of [12] was a coordinate and not a vector, we did not define a covariant derivative . Here is also a vector, and thus in principle we can define the geometric covariant . However, due to (3.3), this is equal to .

 ˆDμΩI=DμΩI+ΓIJKΩJDμXK. (3.22)

The calculations to arrive at (3.17) are the same as in section 3 of [12]. We only have the extra -supersymmetry. The covariance of the transformation part999In a frame with an arbitrary closed homothetic Killing vector, the -supersymmetry transformations of would be of the form , and transforms to other frames as a vector. of , is the statement that behaves as a vector. This is only true for transformations of the form (3.16).

There is no -transformation of in (3.17), because does not transform and the -transformations of the fermions in (3.18) do not contribute due to (3.3).101010If we would formulate the transformations with an arbitrary closed homothetic Killing vector, would transform under , and the term in this extra part of , which is then , would cancel the -transformations of , such that would still be -invariant.

The commutator relation applied on is similar to (2.10)

 [^δ1,^δ2]ΩI = ^δ3ΩI+RK¯LIJΩJ[12¯ϵ1ΩK¯ϵ2Ω¯L−(1↔2))] (3.23) = ^δ3ΩI−14RK¯LIJ[ϵ1(¯ΩJΩK)¯ϵ2Ω¯L−(1↔2))]

Since can be viewed as a vector, we might expect that should include an analogous curvature term. But it vanishes due to (3.4).

### 3.3 Superconformal action

There are two independent parts of the superconformal action for chiral multiplets. The first is the covariant kinetic action, called because it is a -term which requires the Kähler potential as input data. The second part is the superpotential action, an -term called , which is determined by the holomorphic superpotential . Each part is invariant under the transformation rules (3.17).

The kinetic action of [16, (17.19)] simplifies considerably when one uses covariant derivatives and . In this geometric formulation, it can be written as (it actually includes kinetic terms for graviton and gravitino in the last line)

 [N]De−1=NI¯J(−DμXIDμ¯X¯J−12¯ΩI⧸ˆDΩ¯J−12¯Ω¯J⧸ˆDΩI+^FI¯^F¯J) (3.24) +14RI¯KJ¯L¯ΩIΩJ¯Ω¯KΩ¯L+[12√2¯ψ⋅γ(NI¯J^FIΩ¯J−NI¯J⧸D¯X¯JΩI) +18iεμνρσ¯ψμγνψρ(NIDσXI+12NI¯J¯ΩIγσΩ¯J+1√2NI¯ψσΩI)+h.c.] +16N(−R(ω)+12¯ψμγμνρR′νρ(Q))−16√2(NI¯ΩI+N¯I¯Ω¯I)γμνR′μν(Q).

The superpotential is a homogeneous, holomorphic function of Weyl weight 3:

 XIWI=3W,WI≡∂∂XIW. (3.25)

The -term action is given by

 [W]Fe−1=WI^FI−12WI;J¯ΩIΩJ+1√2WI¯ψ⋅γΩI+12W¯ψμPRγμνψν+h.c., (3.26)

where the semicolon sign is used for covariant derivatives, e.g. . The Ricci scalar is calculated with the spin connection in (3.21), and is given in (3.20).

### 3.4 On shell transformation of auxiliary fields

The auxiliary field can be eliminated using the algebraic field equation111111Note that on shell is valid both in the rigid limit and in supergravity when . The reason is that the term on the 2nd line of (3.24) cancels with a similar term in the supercovariant derivative of , see (3.19), in the fermion kinetic term.

 ^FIon−shell=−¯¯¯¯¯¯W¯KGI¯K=−¯¯¯¯¯¯WI. (3.27)

The covariant transformation of the right side gives

 ^δ(^FIon−shell)=−1√2¯¯¯¯¯¯WI;¯J¯ϵ¯Ω¯J. (3.28)

We now show that this result is consistent with (3.17) when the fermion equation of motion is used. To check this, it is useful to rewrite the transformation of the auxiliary field as

 ^δ^FI= 1√2¯ϵ[γμ^DμΩI+12RJ¯LIKΩ¯L¯¯¯¯ΩJΩK+¯¯¯¯¯¯WI;¯JΩ¯J+γ⋅ψ(¯¯¯¯¯¯WI+^FI)] −1√2¯ϵ[γ⋅ψ(¯¯¯¯¯¯WI+^FI)+¯¯¯¯¯¯WI;¯JΩ¯J]≈−1√2¯¯¯¯¯¯WI;¯J¯ϵΩ¯J. (3.29)

The first line of this equation is equal to the field equation for the fermionic fields. Further, the round brackets in the second line enclose the field equation for the auxiliary fields. The net result is indeed the on-shell transformation (3.28)!

## 4 Projective space and gauge fixing

### 4.1 Introducing Poincaré coordinates

The next important step is to move toward physical variables by the substitution

 XI=yZI(zα). (4.1)

This relates the complex fields to new variables with and . The are the physical scalars, which are coordinates on an -dimensional projective Kähler manifold. Because we require invariance under the reparametrization , the are arbitrary functions, but subject to the requirement that

 (ZI∂αZI) (4.2)

is a non-singular matrix.

When the Poincaré coordinate Ansatz is substituted in (3.9), we find that cancels in the second term (see [16, Sec 17.3.4]), and the Lagrangian takes the form

 L=−14N∂μN∂μN−N(∂μzα)(∂μ¯z¯β)∂α∂¯βln[ZIGI¯J¯Z¯J],N=y¯yZIGI¯J¯Z¯J. (4.3)

Interpreting this as a cone, the projective manifold is a non-linear -model whose metric is given by the Kähler potential

 K(z,¯z)=−aln[−a−1ZI(z)GI¯J¯Z¯J(¯z)]=aln[−ay¯yN], (4.4)

where we have included an arbitrary constant for later convenience. Moreover, in these coordinates, the chiral connection (3.8) takes the form (see [16, Ex.17.10])

 iAμ=12a−1(∂μ¯z¯α∂¯αK−∂μzα∂αK)+12∂μln(y/¯y). (4.5)

Since this Lagrangian is simply a reparametrization of (3.6), it inherits its chiral and dilatation symmetries.

However, there is now a new symmetry since the split of coordinates (4.1) is not unique: it is invariant under

 ZI→e−f(z)/aZI,y→ef(z)/ay. (4.6)

This induces Kähler transformations on the potential:

 K(z,¯z)→K(z,¯z)+f(z)+¯f(¯z). (4.7)

We consider the transformations with and as independent. Since we have these transformations we can choose the dilatations and -transformation to act only on and not on ;

 δDy=λDy,δTy=iλTy,δDzα=δTzα=0. (4.8)

Hence and have chiral weights and , respectively. More generally, functions of the scalars and can have weights , and and transform as

 (δT[λT]+δK[f])V=(icλT−a−1w+f(z)−a−1w−¯f(¯z))V, (4.9)

under and Kähler transformations.

In order to define covariant derivatives, we introduce auxiliary connections for Kähler transformations121212Note that in [16, (17.162)] we used another normalization for and , because we introduced them as gauge fields of the symmetry . Here we consider and as independent transformations, where the first one has only holomorphic gauge connection as in the equation below, and the second one has only anti-holomorphic gauge connection.

 ωμ=ωα∂μzα,ωα=∂αK,¯ωμ=¯ω¯α∂μ¯z¯α,¯ω¯α=∂¯αK. (4.10)

The covariant derivatives are then defined as

 ∇μV≡(∂μ−icAμ+a−1w+ωμ+a−1w−¯ωμ)V. (4.11)

For tensors there are the extra terms such as the last term in (3.11). The weights for various fields and quantities are given in the left part of Table 1.

In particular, the covariant derivative is thus

 ∇μy=(∂μ−iAμ−a−1ωμ)y=12y∂μ(lny¯y)−12ay∂μK, (4.12)

while

 ∇μZI=(∂μ+a−1ωμ)ZI. (4.13)

The latter two equations show that these covariant derivatives are an extension of (3.7), consistent with (4.1).

### 4.2 Gauge fixing

Superconformal symmetries that are not part of the Poincaré superalgebra must be gauge fixed so as to maintain covariance under target space reparametrization. In this section we discuss the fixing of the bosonic symmetries, namely dilatation, chiral, and special conformal131313Special conformal transformations are fixed by eliminating one field from the Weyl multiplet: the gauge field of dilations. symmetries.

Dilatations are fixed by requiring that is constant, i.e.

 N(X,¯X)=−a. (4.14)

For supergravity, the value canonically normalizes the Einstein–Hilbert term. This translates into the following condition on the magnitude of :

 y¯y=−a[ZIGI¯J¯Z¯J]−1oraln(y¯y)=K(z,¯z). (4.15)

With this gauge choice, (4.12) implies

 ∇μy=0. (4.16)

To define all variables in terms of and , we must determine both the modulus of as in (4.15), and its phase. This is done by the chiral symmetry gauge fixing condition

 y=¯y=eK/2a. (4.17)

As required, this expresses the field in terms of the Poincaré fields .

An important consequence of this gauge choice is that the auxiliary chiral connection (4.5) can now be expressed as the pullback of covariant derivatives on the projective manifold. For this purpose we write

 Aμ=Aα∂μzα+A¯α∂μ¯z¯α,Aα=i∂αlny=12ai∂αK. (4.18)

The last equation follows from (4.15).

### 4.3 Covariant derivatives in projective space

The coordinates of the projective manifold are , and their complex conjugates . This manifold has a Kähler potential , and corresponding metric

 gα¯β=∂α∂¯βK(z,¯z). (4.19)

Functions and tensors on the projective space depend on the spacetime points via their dependence on and . Therefore we define the split covariant derivatives as

 ∇μV(z,¯z)≡∇αV∂μzα+¯¯¯¯¯∇¯αV∂μ¯z¯α, (4.20)

for any scalar quantity .

After the gauge fixing, and are both related to