Projective multiplets and hyperkahler cones in conformal supergravity

Projective superspace provides a natural framework for the construction of actions coupling hypermultiplets to conformal supergravity. We review how the off-shell actions are formulated in superspace and then discuss how to eliminate the infinite number of auxiliary fields to produce an on-shell N=2 supersymmetric sigma model, with the target space corresponding to a generic 4n-dimensional hyperkahler cone. We show how the component action coupling the hypermultiplets to conformal supergravity may be constructed starting from curved superspace. The superspace origin of the geometric data -- the hyperkahler potential, complex structures, and any gauged isometries -- is also addressed.


Introduction
The hypermultiplet is the most general on-shell matter multiplet with rigid N = 2 supersymmetry in four dimensions. It consists of a complex doublet of scalar fields and a pair of Weyl fermions (equivalently, a single Dirac fermion), whose supersymmetry transformations close only on the equations of motion. For this reason, specifying the hypermultiplet is equivalent to specifying the action, or equivalently, specifying the target space of the sigma model parametrized by the hypermultiplet scalars. It has long been known that such target spaces must be hyperkähler manifolds [1,2]. For locally supersymmetric theoriesby which one usually means those coupled to supergravity with a canonically normalized Einstein-Hilbert Lagrangian -the target space must instead be quaternion-Kähler [3]. This distinction recalls that between rigid N = 1 theories, where the target space is Kähler [4], and their locally supersymmetric counterparts, which must be Hodge-Kähler [5]. 1 It is known that off-shell formulations of the hypermultiplet, employing N = 2 superspace and involving (an infinite number of) auxiliary fields, provide a means to generate the required target space geometry from some unconstrained generating function. To understand this better, let us elaborate first on the rigid N = 1 case. Recall that the chiral multiplet φ, the natural N = 1 cousin of the hypermultiplet, admits a simple off-shell representation {φ, ζ α , F } with a single complex auxiliary field. (We employ a convention where a single symbol φ may stand for both a superfield and its lowest component.) Zumino showed that the most general two-derivative function of off-shell chiral multiplets φ a is described by a superspace integral [4] d 4 x d 2 θ d 2θ K(φ,φ) (1.1) for an arbitrary real function K. 2 When the auxiliary fields F a are eliminated, the component Lagrangian takes a sigma model form 3 L = −g ab ∂ m φ a ∂ mφb − i 4 g ab ζ a σ m D mζb +ζbσ m D m ζ a + 1 16 R aābb (ζ a ζ b )(ζāζb) , ζ a α := D α φ a , D m ζ a := ∂ m ζ a + Γ bc a ∂ m φ b ζ c . (1. 2) The metric is determined in terms of the function K, g ab := ∂ a ∂bK; of course, this is the well-known Kähler metric -in using chiral multiplets we have automatically diagonalized the complex structure -and this construction gives the explicit proof that N = 1 sigma models in four dimensions must possess a Kähler target space geometry [4]. The key point is that the general off-shell supersymmetric action (1.1) provides both the means to construct the on-shell action and the generating function for the target space geometry. This narrative can be repeated for N = 2 theories. There one has two complementary formulations of a general off-shell hypermultiplet, depending on whether one uses harmonic 1 See also the work of Komargodski and Seiberg for a new perspective on these results [6,7]. 2 For now we ignore the possibility of a superpotential or gauged isometries. 3 The fermion ζ a α is normalized here in an unconventional way. We use the same normalization for N = 2 models, following [8].
superspace [8,9] or projective superspace [10][11][12]. In harmonic superspace, which augments N = 2 superspace with an auxiliary S 2 manifold with harmonic coordinates u i± , one has n pseudoreal superfields q a+ , q a+ = q b+ Ω ba where Ω is the canonical symplectic form of Sp(n), obeying the analyticity condition This q + hypermultiplet is the natural off-shell matter multiplet of harmonic superspace and is defined globally on the S 2 . The most general two-derivative action is du d 4 x d 4 θ + 1 2 Ω ab q a+ D ++ q b+ + H +4 (q + , u ± ) (1.4) with H +4 (q + , u ± ) an arbitrary real function [13,14]. The striking resemblance to a Hamiltonian system was explained in [14] (see also [15]), where it was shown that any hyperkähler manifold locally possesses a Hamiltonian structure.
For projective superspace, one uses the same auxiliary manifold CP 1 ∼ = S 2 as in harmonic superspace, but superfields are taken to be holomorphic on an open domain of CP 1 rather than globally defined on S 2 , which is the main distinction between the two approaches. 4 We denote the auxiliary coordinates by v i± to distinguish them from harmonic superspace. Sigma models are described by n complex arctic superfields Υ I+ and their antarctic conjugatesΥĪ + = Υ I+ . These are Grassmann analytic, D + α Υ I+ = 0, and holomorphic on an open domain of CP 1 , D ++ Υ I+ = 0. Arctic multiplets are taken to be holomorphic near the north pole and antarctics are holomorphic near the south pole. The combination of Υ I+ andΥĪ + , collectively known as a polar multiplet, serves as the general off-shell matter multiplet in projective superspace, and the most general two-derivative action is given by [11] (see also [21] for a recent discussion) involving an arbitrary real function F ++ that is homogeneous of weight two in its parameters, F ++ (λΥ + , λΥ + , λv i+ ) = λ 2 F ++ (Υ + ,Υ + , v i+ ). This function can be interpreted as the generating function for symplectic transformations on the hyperkähler manifold [22] (see also the pioneering work of [23] and the recent approach of [24]) and turns out to possess a simple relationship to the harmonic Hamiltonian H +4 [25]. In both the projective and harmonic cases, the data necessary to define the hyperkähler geometry are encoded in their respective superspace Lagrangians. In evaluating the component actions, one must find a way to eliminate the infinite number of auxiliary fields. In doing so, one generates a hyperkähler metric and associated complex structures that describe the sigma model parametrized by the physical scalars. In principle this can be done explicitly for a specific generating function, although only certain classes of hyperkähler metrics have been explicitly constructed in this way. Once the auxiliaries are eliminated and the geometric data constructed, the component action takes its final form. These are distinct steps because the component reduction is just as easily done for a general geometry as a specific one, since one can treat the conversion from generating function to hyperkähler geometry completely formally. Our concern in this paper will be with the general action and with the formal elimination of the auxiliaries; we will return to the subject of specific solvable classes in the conclusion.
What about locally supersymmetric theories? In this case, there are two options, depending on whether one couples minimally to conformal supergravity or to Poincaré supergravity with a canonical Einstein-Hilbert term. In the N = 1 setting, a sigma model coupled to conformal supergravity is given by except now the superconformal algebra includes both dilatations (D) and chiral U(1) R rotations (A) under which φ a and K must transform as δφ a = Λ D χ a + 2i 3 Λ A χ a , δK = δφ a K a + h.c. = 2Λ D K . (1.7) The chiral function χ a (φ) describes a homothetic conformal Killing vector, where ∇ a and ∇ā are the target space covariant derivatives. The Kähler potential is K ≡ χ a χbg ab and describes a Kähler cone [26]. At the component level, the Lagrangian includes a contribution from the Ricci scalar of the form 1 6 RK. Usually one prefers instead a canonically-normalized Einstein-Hilbert term, which can be achieved by imposing the dilatation gauge K = −3 and simultaneously fixing the U(1) R symmetry. This eliminates two scalar fields from the sigma model and converts the Kähler cone of dimension 2n into a Hodge-Kähler manifold of dimension 2(n − 1).
In the N = 2 setting, a similar picture emerges. A sigma model coupled to N = 2 conformal supergravity must be a hyperkähler cone [27] (see [26,28] for the rigid superconformal case and [29] for a discussion in general dimensions), which is a hyperkähler manifold possessing a homothetic conformal Killing vector and SU(2) R isometries that rotate the complex structures; such spaces are also known as Swann bundles [30]. There exists a one-to-one correspondence between 4n-dimensional hyperkähler cones and 4(n−1)dimensional quaternion-Kähler manifolds [30] (see also [31] as well as [32,33] for recent discussions and references). The component action for a hyperkähler cone coupled to conformal supergravity was given in [27], where its relation to the Poincaré supergravitycoupled quaternion-Kähler action of [3] was also discussed: the elimination of four scalars in the target space comes from fixing the dilatation and SU(2) R gauges.
Our goal in this paper is to reproduce in a systematic way the component action of [27] for a hyperkähler cone coupled to conformal supergravity directly from (curved) projective superspace. 5 For this reason, aside from a brief discussion in the conclusion, we will always be referring to conformal supergravity when we mention locally supersymmetric sigma models. We will employ the formalism of [16], which is a slight generalization of the approach of [17][18][19][20]. The superspace action is given by We explicitly parametrize the SU(2) contour C by the coordinate τ , and the measure E −− is a superdeterminant of the relevant superspace vielbein. 6 The generating function F ++ possesses no explicit dependence on v i+ : this generalizes to curved superspace the superconformal version of (1.5), constructed originally in flat space [36]. The condition that F ++ is independent of any explicit v i+ dependence ensures that the target space describes a hyperkähler cone [37]. The major barrier to this calculation is that the usual method of eliminating the infinite number of auxiliary fields in the flat superspace action (1.5) depends on introducing an intermediate N = 1 superspace. This is easy to do in flat superspace and has been accomplished recently in AdS superspace [38][39][40][41], but it is quite daunting in a general curved geometry. 7 Instead, we will take inspiration from rigid harmonic superspace and proceed directly from N = 2 superspace to the component action. The elimination of the auxiliary fields will seem rather different at first glance from the N = 1 approach, but will actually involve solving the same set of equations; this implies that the coupling to conformal supergravity will in no way affect the elimination of the hypermultiplet auxiliaries. 8 Along the way, we will derive explicit formulae in projective superspace for all of the geometric quantities necessary for describing the hyperkähler cone and its sigma model. This paper is laid out as follows. Section 2 provides a review of how hyperkähler geometry can be derived from flat projective superspace via the N = 1 superspace method. Although this method seems to be useful mainly for rigid supersymmetric spaces such as Minkowski or AdS, many of the same formulae and notation will reoccur in later sections, so some familiarity will be necessary. In section 3, we describe how the full N = 2 superfield equations of motion lead to on-shell N = 2 hypermultiplets whose target space is a hyperkähler cone, and we derive all the geometric data we will need from projective superspace. We then describe in section 4 how to restrict the N = 2 equations of motion to only the auxiliary sector, eliminating the infinite tail of auxiliary fields while keeping the physical fields off-shell. In section 5 we test this approach by deriving the component action in the rigid supersymmetric case, before addressing the curved case in section 6. We comment on our results and speculate about some open questions in the conclusion.
Two appendices are included. The first gives our conventions for vector multiplets and their component fields (these conventions were absent in [16]). The second provides some 6 In the flat space limit, E −− = v + i dv i+ /dτ so that dτ E −− reduces to the flat measure v + i dv i+ . Further details can be found in [16]. 7 The approach of reducing curved N = 2 superfields to N = 1 superfields has been discussed in [42,43], but its application in this case would seem to be very difficult. 8 In the quaternion-Kähler case, this is more subtle. As is evident from the corresponding harmonic [35] and projective [33] descriptions, a hypermultiplet compensator plays the role of the effective auxiliary variable for the on-shell hypermultiplet superfields.
technical details necessary to calculate the final component action.

A review of hyperkähler geometry from flat projective superspace
Let us begin by reviewing how projective superspace permits the construction of hyperkähler sigma models in flat space. This material is well-known and we refer the reader to the lecture notes [44] as well as [22] and [21,45] for further details and the relevant references.
We begin with the flat projective superspace Lagrangian F ++ depending on arctic multiplets Υ I+ , antarctic multipletsΥĪ + and the coordinate v i+ . Such a Lagrangian is analytic and holomorphic, D + α F ++ = D ++ F ++ = 0, by construction. The action is This can be evaluated as an integral in N = 1 superspace. To do this, recast all superfields so that they depend solely on the complex coordinate ζ = v 2+ /v 1+ rather than v 1+ and v 2+ separately. For example, one introduces a new arctic superfield Υ I (ζ), If we interpret the components in this expansion as N = 1 superfields, then Φ I is chiral and Σ I is complex linear, while the infinite tail of superfields Υ I n are unconstrained N = 1 superfields. The antarctic superfield possesses a similar expansion Rewriting the projective superspace Lagrangian as F ++ = iv 1+ v 2+ F(ζ), the action becomes in terms of the N = 1 superspace Lagrangian L . Note that while the original function F ++ had to be homogeneous of weight two in its parameters, no such constraint is imposed on F. Because the superfields Υ I n with n ≥ 2 are unconstrained N = 1 superfields, their equations of motion are purely algebraic 9 C dζ 2πiζ Imposing these, the Lagrangian becomes solely a function of Φ I , Σ I and their complex conjugates. Now perform a duality transformation, exchanging the complex linear superfield Σ I for a chiral superfield Ψ I , The equation of motion for Ψ I enforces the complex linearity of Σ I , recovering (2.5). Alternatively, we can eliminate Σ I using its own equation of motion, effecting a Legendre transformation The resulting function K is a Kähler potential with complex coordinates φ a = (Φ I , Ψ I ). This Kähler potential describes a hyperkähler manifold. In addition to the manifest N = 1 supersymmetry, there is a hidden second supersymmetry on-shell, which manifests in N = 1 language as [49] δφ a = ω abραDα K b = ω abραDαφb (2.9) for constantρα ≡ξα 2 , the second supersymmetry parameter. The tensor ω ab is antisymmetric, covariantly constant, and obeys ω ab ωb c = ω ab ω bc = −δ a c . The special coordinates φ a = (Φ I , Ψ I ) are Darboux coordinates for which [22,45] The presence of such an antisymmetric covariantly constant tensor ensures that the Kähler manifold is actually hyperkähler, with a triplet of closed hyperkähler two-forms Ω ij , The three hyperkähler two-forms are related to three covariantly constant complex structures (J ij ) µ ν = g µρ (Ω ij ) ρν , which obey the multiplication rule Introducing J A := −i(τ A ) i j J i j , A = 1, 2, 3 with the Pauli matrices (τ A ) i j , the multiplication rule becomes that of the quaternions, 14) The complex structures in this form are given by Note that J 3 = −2iJ 12 is the complex structure associated with the manifest N = 1 supersymmetry. We will eventually be interested in the case where the model is superconformal [37]. This amounts to the condition which is equivalent to requiring the projective Lagrangian F ++ to be independent of v i+ . In contrast to the N = 1 situation, there is no requirement that F ++ be separately homogeneous in Υ I+ . This is because arctic and antarctic multiplets are both inert under U(1) R , so there is no superconformal symmetry that distinguishes between them. 10 Now the Kähler potential K turns out to possess a chiral homothetic conformal Killing vector (CKV) χ a obeying (1.8), implying that the Kähler potential can be chosen (up to a Kähler transformation) as K = χ a χ a . For the Darboux coordinate system, the homothetic conformal Killing vector takes the simple form χ a = (Φ I , Ψ I ). The presence of χ a ensures that the hyperkähler manifold is actually a hyperkähler cone. In addition to the two supersymmetry transformations, it admits a full set of N = 2 superconformal transformations, including dilatation and SU(2) R transformations. These manifest as [45] δφ a = Λ D χ a + λ 12 χ a − λ 22 ω ab χ b , (2.17) where Λ D is the scale parameter and λ i j is the SU(2) R transformation parameter. The fields φ a are inert under U(1) R .
3 Hyperkähler geometry and on-shell N = 2 superfields Our goal in this section is to establish the geometric properties of the target space geometry of (1.9) (including the results of the previous section) without explicitly reducing to N = 1 superspace. This is necessary in order to derive the component action in the presence of supergravity, where an N = 1 superspace is not readily available. Our goal will be to reduce the arctic superfields to on-shell N = 2 superfields: we will define these as φ a = (Φ I , Ψ I ) in analogy to the N = 1 superfields of the previous section. Our starting point is equivalent to that discussed in [50] and [22]: we will analyze the full N = 2 superfield equations of motion. If the action (1.9) is stationary under arbitrary variations of Υ I+ , then ∂F ++ /∂Υ I+ must itself be an arctic superfield. (The integral vanishes in this case since the contour C can be retracted to the origin without encountering singularities.) This result holds both in the rigid and locally supersymmetric situations. This leads one to introduce superfields Γ + I andΓ + I , defined by The equations of motion require Γ + I andΓ + I to be, respectively, arctic and antarctic. 11 The superfields Φ I and Ψ I correspond to the leading terms in the ζ expansions of Υ I+ and Γ + I , and can be defined equivalently via contour integration, 12 We will assume that the contour C winds around the north pole (and thus the south pole as well) exactly once and that the arctic (antarctic) multiplets possess no singularities in the northern (southern) chart bounded by C. Consistency with the flat space N = 1 analysis implies that the on-shell N = 2 superfields Υ I+ andΥĪ + must be given by power series in the N = 2 superfields φ a = (Φ I , Ψ I ) and their complex conjugates. Because no fields of the conformal supergravity multiplet appear in the solution of the power series, the sigma model for local supersymmetry will be identical as for the rigid superconformal case. The N = 2 superconformal transformations of Φ I and Ψ I can be derived from their definitions (3.4), where δ consists of any local (super)symmetry transformation. Consistency dictates that δΥ I+ can equivalently be calculated by Using only these results, let us briefly discuss how the geometry of the target space follows.

Hyperkähler geometry
Here we follow closely the approach of [22]. The two-form 13 11 These can be interpreted as dual superfields; see [51] for a discussion of polar-polar duality. 12 In defining the N = 2 superfields Φ I and ΨI , we implicitly choose to work in the central gauge of projective superspace as discussed in [16]. 13 For target space quantities such as Ω ++ , we follow the standard conventions for differential forms rather than the superspace conventions of e.g. [52].
is both arctic and antarctic when the full N = 2 equations of motion are imposed, so it must be globally-defined on the auxiliary manifold. This means it is given by Ω ++ = Ω ij v i+ v j+ for a triplet of two-forms Ω ij . It is obvious from its definition that Ω 11 = dΦ I ∧ dΨ I and Ω 22 = dΦĪ ∧ dΨĪ, while Ω 12 can only be a (1, 1) form, Ω 12 = 1 2 dφ a ∧ dφbg ab , for some tensor g ab . The expression for Ω ++ can then be written as identifying These relations also hold upon replacing Υ I+ →ΥĪ + and Γ + I →Γ + I . Because Ω ++ is closed, it follows that both ω ab and g ab must be closed when viewed respectively as (2, 0) and (1, 1) forms. (The closure of ω ab is obvious in the Darboux coordinates.) The closure of g ab implies that it must be the second derivative of a function K. This function can be chosen as in the explicit N = 1 reduction as (3.10) Σ I should be identified as the second term in the expansion of Υ I+ . If one similarly identifies Σ I as the second term in the expansion of Γ + I , one can check that from which one can show that Let us next establish that ω ab = g ac ωcdgd b where ωāb is the inverse of ωāb, given by A proof of this follows by using the explicit antarctic expression for g ab and writing (3.14) This expression must be independent of v i+ , so we can discard all terms that go as negative powers of ζ. Using we see that the only terms that contribute are which is what we wished to establish. This result actually guarantees that g ab is invertible because both sides of (3.16) must have non-vanishing determinant. In other words, the non-degeneracy of the metric g ab is implied if we can solve the equations (3.1) completely in terms of the coordinates Φ I and Ψ I . This equality also guarantees that ω ab is covariantly constant. It follows that the manifold is hyperkähler.

Gauged isometries from projective superspace
Suppose that the projective superspace Lagrangian possesses some gauge invariance -that is, the arctic multiplets possess gauged holomorphic isometries of the form for an arctic function J I+ r , and similarly for the antarctic multiplets, with r labelling the adjoint representation of the gauge group. Let us show how these descend to triholomorphic isometries in the hyperkähler manifold, rederiving the results of [49]. 14 Because the projective Lagrangian is gauge-invariant, 15 we may introduce a real quantity This is the N = 2 moment map (or Killing potential) in projective superspace. 16 By construction, the gauge transformation of Γ + I is 14 A treatment based on N = 1 superspace methods can be found in [53] and [54] for flat 4D and 5D cases, and [40,41] for AdS geometries. 15 It is actually not necessary for the Lagrangian to be fully gauge invariant, provided that one can consistently introduce a naked prepotential to counter its gauge transformation property. As discussed in [49], the prepotential can be absorbed in a covariant framework by introducing a fictitious multiplet that drops out of the action (its metric vanishes), except for its modification of the moment map. In this way, we retain the description above with only covariant hypermultiplets and no prepotentials. We thank Martin Roček for pointing out this important subtlety. 16 The N = 1 formulation of the N = 2 Killing potential appeared in [49]. Harmonic and projective superspace formulations appeared explicitly in [34] and [54]. and so it follows that D ++ r transforms in the co-adjoint of the gauge group, When the equations of motion are imposed, the gauge transformations of the arctic fields must be manifested on the target space of complex fields. As a consequence of the transformation properties of Υ I+ and Γ + I , the two-form Ω ++ is gauge-invariant. Interpreting the gauge transformation as a target space transformation, it follows that L J Ω ++ vanishes, ensuring that gauged isometries in projective superspace descend to triholomorphic isometries in the target space. To derive this explicitly, observe that for Now D ++ r must be both arctic and antarctic and so must be given by The quantity D r is the N = 1 Killing potential and is related to the holomorphic quantity Λ r and the Killing vector J a r via [49] The relations (3.24) can equivalently be written From this equation, one can prove that the Killing vector J µ r is triholomorphic. It is worth mentioning that if F ++ is independent of v i+ , so that the target space is a hyperkähler cone, J I+ r must also be independent of v i+ and homogeneous in Υ I+ of degree one, Υ J+ ∂ J+ J I+ r = J I+ r . The N = 2 moment map can then equivalently be defined as This can be derived from Ω ++ , replacing one d with D 0 and the other with the gauge generator X r , leading to an explicit expression for its components, (3.27) in terms of the homothetic conformal Killing vector χ µ .

Superconformal isometries
Now let us analyze the superconformal properties of the target space, following roughly the same approach as [37,45], to which we refer for further details. The arctic multiplet Υ I+ transforms locally under dilatations and SU(2) R transformations as and similarly forΥĪ + . Both are inert under U(1) R , and so are their dual fields Γ + I andΓ + I . The above transformation should map to the target space as for some choice of vectors k a D := Dφ a and k ij a := I ij φ a . (Obviously, φ a must be inert under U(1) R because Υ I+ is.) Our goal in this section is to review how to recover the properties of k a D and k ij a using projective superspace. Superconformal invariance dictates that the projective Lagrangian F ++ must transform as which in turn implies that F ++ is homogeneous of degree two in the projective multiplets, and possesses no explicit dependence on v i+ . It follows that the fields Γ + I andΓ + I transform in the same way as Υ I+ andΥĪ + under superconformal transformations.
From the definitions (3.4) of Φ I and Ψ I , it is clear that they possess unit dilatation weight, so we establish A similar calculation with Σ I , the second component in the ζ-expansion of Υ I+ , establishes that it also has unit dilatation weight. This leads immediately to using the definition (3.10) of the Kähler potential. Next, we establish the SU(2) R transformation properties of the fields. Consider first the diagonal U(1) subgroup of SU(2) R generated by I 1 1 = −I 2 2 ; this transformation was called the shadow chiral rotation in [37,45]. From its action on the arctic and antarctic multiplets, one can show that while Σ I transforms as I 1 1 Σ I = 1 2 Σ I . The Kähler potential (3.10) is inert; the proof amounts to showing that L is inert using its contour integral definition, while Σ I Ψ I is obviously inert. Now interpreting K as a function of φ a , one finds k a D ∂ a K − kā D ∂āK = 0. Together with (3.33), this implies that k a D ≡ χ a is a homothetic conformal Killing vector. Now consider the off-diagonal SU(2) R component I 1 2 = I 22 . From its action on an arctic multiplet, one can show that From its action on an antarctic multiplet, one can show thatφā is inert.
Putting these results together, we deduce that where J i j is defined in (2.12). An immediate consequence of this is that the Kähler potential is invariant under all of the SU(2) generators.
At this stage, we should point out how these isometries act on the two-form Ω ++ . Noting that Ω ++ possesses weight two under dilatations, it is easy to see that Similarly, an SU(2) transformation acts on Ω ++ as demonstrating that the SU(2) transformation rotates the complex structures. This is consistent with the target space transformations These comprise the isometries required of a hyperkähler cone.

Supersymmetry and fermion transformations
In addition to the 2n complex bosons φ a parametrizing the target space, there must be 2n Weyl fermions. It will be convenient for us to define the fermions to be consistent with the N = 1 reduction -that is, we will associate one left-handed Weyl fermion with each of the φ a and one right-handed Weyl fermion with each of theφā. Using the on-shell superfields φ a andφā, we define (using the curved superspace spinor derivatives ∇ i α and∇α i ) The set of fields {φ a ,φā, ζ a α ,ζāα} constitute the on-shell field content of the supersymmetric sigma model. Our goal in this section is to derive their supersymmetry transformations.
We first establish the action of the spinor derivatives on the scalars φ µ : 17 17 Such superfields φ µ were called deformed Fayet-Sohnius multiplets in [55].
We will focus on deriving the first line of (3.41), the second following by complex conjugation. The first equality is just a definition. To get the others, we take the partial pullback of Ω ++ to the supermanifold, replacing one of the exterior target space derivatives with a spinor derivative, The left-hand side vanishes from the analyticity of the arctic superfields. Dividing by a factor of (v 1+ ) 2 , and isolating the terms involving dφ a leads to This must vanish for all values of ζ, leading to the first line of (3.41). The terms involving dφā vanish similarly. The spinor derivatives of ζ a α andζā α can be derived directly. Noting, for example, that αφb one can evaluate any spinor derivative of ζ a α by exploiting the fact that each spinor derivative annihilates either φ a orφb. This leads to as well as their complex conjugates The superfield W r appearing above is the chiral field strength of the N = 2 vector multiplet, involving components {W r , A r m , λ r αi , Y r ij }. We collect their definitions and supersymmetry transformations in appendix A.
We will also need the dilatation, U(1), and SU(2) R transformations of the fermions: Their S-supersymmetry transformations are given by All of the matter fields are invariant under the special conformal generator K a .

Sp(n)-covariant formulation and summary
We have not commented yet on one important feature of hyperkähler manifolds: the tangent space group is actually Sp(n) × Sp(1) (due to the existence of the covariantly constant holomorphic tensor ω ab ) and the Sp(1) part of the target space connection vanishes. Following [56] (see also [3]), we can introduce a tensor f µi a and its inverse f a i µ , with an Sp(n) index a = 1, · · · , 2n. These obey the conditions and allow one to convert any vector They are related to the metric, the hyperkähler two-forms and the complex structures via Requiring f µi a to be covariantly constant, defines the Sp(n) connection Γ µb a . We are actually interested in the situation where the indices a, b, · · · are not quite flat tangent space Sp(n) indices, but rather complex world indices in the coordinate system that diagonalizes the complex structure J 3 . We impose the pseudoreality condition where g ab is the Kähler metric associated with J 3 . In our conventions, the tensors f µi a , f µ iā and their inverses are given by 52) and the Sp(n) connection is identical to the Kähler connection. The advantage of introducing the tensors f µi a is that they simplify the equations given in the preceding sections. For example, the spinor derivatives of φ µ in (3.41) become equivalent to the supersymmetry transformations then the supersymmetry and S-supersymmetry transformations of the fermions can be written compactly as where η iα andη iα are the S-supersymmetry parameters, J ri a := J r µ f µi a is the Killing vector associated with the gauged isometries, and ∇ a includes the Sp(n) connection. For reference, we also give the transformations of the fermions under gauged isometries, Note that the scalar fields φ µ and the fermions ζ b α transform into each other (and into the components of the vector multiplet) under supersymmetry. The conditions (3.41) (equivalently (3.53)) have eliminated the hypermultiplet auxiliary fields and placed the entire multiplet on-shell: in particular, one can check that the supersymmetry algebra closes only up to the equations of motion. These results match those of [27], up to differences in conventions discussed at the end of section 6.

Building blocks of a component reduction
In the previous section, we addressed the on-shell structure of the target space multiplets φ µ , where the auxiliary fields have been completely eliminated. In a manifestly supersymmetric setting -which we have implicitly been using -this actually is a much stronger condition than what we want. It corresponds, in the N = 1 situation, to specifying not only the algebraic equations of motion for the auxiliaries F a but also the dynamical equations of motion for the physical fields φ a and ζ a α . Our goal in this section is to describe how to analyze the superfield equations of motion for the arctic multiplet Υ I+ so that only the auxiliary fields are placed on-shell.

Component expansions of the projective multiplets
We begin by analyzing the component structure of the projective multiplets. The conventional approach is to reduce to N = 1 superfields and superspace, but as we have already discussed in the introduction, this is far more difficult in the curved case. Instead, it will be more convenient to reduce directly to component fields. Much insight can be gleaned from the harmonic superspace approach to sigma models (see section 11.4 of [8]), where no N = 1 subspace is readily available. The first step is to choose an appropriate system of coordinates. In projective superspace, the following coordinates are well-defined in the north chart and suitable for arctic superfields [39,40]: These lead to the simple expressions implying that holomorphic analytic multiplets are independent of z −− 1 and θ α− . An arctic multiplet Υ I+ , well-defined in the north chart, is then simply specified as a function Υ I+ (x N , v 1+ , ζ, θ + ) possessing an arctic expansion in ζ. Schematically, such a superfield admits a decomposition (following closely the approach of [8]) where the component fields Ψ I α , · · · , P I(−3) depend on x N , v 1+ and ζ = v 2+ /v 1+ . Their dependence on v 1+ is indicated by their charge, that is, and all expressions are arctic in ζ.
In a curved background, it is more convenient to define the components of Υ I+ in a covariant way. The closest analogue to the expansion given above is to define Each of these components is manifestly holomorphic and well-defined in the north chart. The corresponding formulae for the components of an antarctic multipletΥĪ + are found by α . An inconvenient feature of these component fields is that the arctic and antarctic fields are naturally defined in terms of different sets of spinor derivatives. Moreover, when we actually analyze the component action, we will encounter the spinor derivatives it is straightforward to determine the relation between expressions like ∇ − α Υ I+ and ∇ 1 α Υ I+ . The expressions for the lowest few components are rather simple: Recall that the arctic function J I+ r arises from the action of the gauge generator on Υ I+ , so some of the terms in the second line above are present only when isometries are gauged. The expressions with three spinor derivatives are and involve the covariant conformal supergravity fields W αβ and χ αi as well as the gaugino λ r αi . The term with four spinor derivatives is the most complicated: It is the only term where the conformal supergravity auxiliary field D and the vector multiplet auxiliary Y rij appear. Among the numerous complicated terms involving gauged isometries is the first expression in the second-to-last line, written explicitly using the product of two generators. It can equivalently be rewritten in terms of J I+ r using Similar equations to (4.6) arise if we replace the arctic Υ I+ with the antarcticΥĪ + -the only change is the replacement of

Auxiliary field equations of motion
It should be emphasized that the arctic nature of the fields Ψ I α , · · · , P I(−3) is a consequence of the arctic nature of Υ I+ . Now we need to understand what conditions arise from placing only the auxiliary fields on-shell. It helps to recall again the N = 1 situation where (in the absence of a superpotential) the on-shell equation of motion is given by (taking a flat superspace for simplicity) The chiral superfieldD 2 K a is constrained to vanish. At the component level, this corresponds to three distinct equations, corresponding to the equations of motion of the three components of φ a . Setting to zero the lowest component ofD 2 K a amounts to constraining the auxiliary field, while constraining the higher two components leads to dynamical equations of motion. The situation can be rendered schematically as in table 1. These statements apply equally well in the N = 2 setting. There we have the equations (3.1), equivalently written as which place the arctic multiplet on-shell and imply that the composite Γ + I is an arctic multiplet. If we introduce the component fields Ψ αI , · · · , P The key issue here is that the auxiliary field equations consist of all but the final two lines of table 2, which for dimensional reasons must contain the field equations for the physical fermions and scalars. Keeping in mind that Γ + I is a composite quantity, it follows that setting the components Γ + I through N − I to be arctic must correspond to fixing them and the original quantities Υ I+ through N I− to be given by their on-shell expressions. For the on-shell superfields discussed in the previous section, these expressions are easy to work out. We already know that  Table 2. N = 2 component equations of motion defines the lowest components φ a = (Φ I , Ψ I ). The components at next order in θ are We have given two equivalent expressions for each of Ψ I α and their conjugates; analogous expressions follow for Ψ Iα by replacing Υ I+ with Γ + I . The θ 2 components are and 13a) These conditions, as well as the corresponding ones for the components of Γ + I andΓ + I , constitute the set of auxiliary equations of motion for the arctic multiplet.
We cannot specify anything about the components Ξ I−− α and P I(−3) , aside from their arctic nature, without placing the physical fields on-shell.

Some relevant quantities
In performing the component reduction, there are a number of geometric quantities that we will encounter that arise from simple expressions in terms of the arctic superfields. Rather than discuss them piecemeal, one at a time as we come across them, we present them here together to emphasize their common features. They can be grouped loosely into three classes: quantities corresponding to pullbacks of the hyperkähler two-forms, which exist even in the flat limit; quantities associated with gauged isometries; and quantities that arise from the cone structure.

Hyperkähler two-forms and their pullbacks
Recall that the hyperkähler two-forms can be represented equivalently as (4.14) We can interpret the second line as a superfield expression with on-shell superfields φ a provided we only work at most to order θ 2 . Taking the pullback to the supermanifold and using two spinor derivatives of opposite chirality, we find where the second line follows from (4.11) and (3.9b). Similar expressions for other quantities can be derived: Note that each of the quantities Ω αβ , Ω αβ and Ωαβ are harmonic-independent at lowest order in their θ expansion. Similar pullbacks can be defined using the vector derivatives, or with the gauge generator, It is also possible to define pullbacks using only the bosonic quantities, such as Ω ++ rb or Ω ++ ab , but these will not play a major role in our discussion so we do not give their explicit forms here.
It can be useful to employ a condensed notation to simplify the right-hand sides of (4.17) and (4.18). Using the Sp(n) vielbeins f µi a introduced in section 3.5, we define Then one can alternatively write (using also the fields A i a and J ri a ), It is also useful to note some relations between these various quantities. For example, ∇ j β Ω αα and∇ jβ Ω αα are found by taking Relations such as these will be extremely useful in our analysis.

Components of the N = 2 moment map
Recall that the N = 2 moment map D ++ r is given in projective superspace by (3.19). With the auxiliary field equations imposed, D ++ r is a globally defined superfield to order θ 2 . Its lowest component was given already in (3.23). Its higher components can be analyzed by applying spinor derivatives to (3.19). A single spinor derivative gives For two spinor derivatives, one finds which along with its complex conjugate implies

Hyperkähler cone potential and hyperkähler one-forms
Recall that the hyperkähler cone possesses a globally defined function, the hyperkähler potential K. In terms of superfield quantities, K is given by the pullback of Ω ++ onto the auxiliary SU(2) manifold, Let us prove this result. Because F ++ is independent of v i+ in the superconformal case, this expression is equivalent to its conjugate so K is real. It is easy to check that D ++ K = 0, so K is independent of v − i . Because it is both arctic and antarctic, it must actually be independent of the harmonics. To prove that K is the hyperkähler potential, we note that an SU(2) R transformation acts on projective multiplets and on the target space respectively as For projective multiplets Υ I+ and Γ + I , this implies It follows that (4.25) can be written From (2.13), one finds (J ++ J −+ ) µ ν = 1 2 (J ++ ) µ ν and (J −− J ++ ) µ ν = − 1 2 δ µ ν − (J +− ) µ ν . This implies that K = 1 2 χ µ χ µ , so K is indeed the hyperkähler potential. It should be emphasized that the expression (4.25) defines the hyperkähler potential to order θ 2 if we impose only the auxiliary equations of motion. This is because at higher order, Γ + I ceases to be arctic (without imposing all the equations of motion) and so K ceases to be harmonicindependent.
It will be useful to have explicit expressions for various derivatives of K. One can check that where ∇ b carries the target space connection. Although we will not explicitly make use of them, it is interesting to note that the hyperkähler one-forms k ij can be written in superspace as These are given by a partial pullback of Ω ++ , replacing one d with the derivative D 0 of the auxiliary manifold. Because they are globally defined, Ω ++ is exact. Because Γ + I and Υ I+ are both weight one under dilatations, we can rewrite this on the target space in the familiar way,

The component action in rigid projective superspace
Now that we have established a great deal of preliminary material, we now can turn to deriving the component action from superspace. This is fairly involved, so we have chosen to separate the task into two distinct stages. In this section, we will derive the component action from rigid projective superspace. This calculation will yield a subset of the terms we actually need. Of course, the result can already be derived via a reduction to N = 1 superspace as discussed in section 2, and we will be able to compare our result to the component version of that action. The point of this exercise is to introduce the techniques we will need in the curved case. In fact, it will turn out that reconstructing the rigid terms is actually more involved than finding the additional supergravity contributions! For this reason, we will be rather explicit in the calculation. To emphasize the applicability to the rigid case, we will avoid assuming in this section that the target space is a cone. The action we seek to evaluate is with D A the gauge covariant derivative associated with rigid projective superspace. It will be convenient for later reference in the curved case to refer to L −− above as T 0 := In the curved case, there will be additional terms.
The function F ++ depends on the collective set of projective multiplets Q + = (Υ I+ ,ΥĪ + ). We will suppress any index on Q + to keep the notation as compact as possible, denoting derivatives of the function F ++ as F + Q , F QQ , etc. The Lagrangian T 0 can now be equivalently written There is some ambiguity in how to write this. In the first three lines, we have chosen to write explicit spinor derivatives of F + Q rather than expanding them out. The reason is that F + Q = (iΓ + I , −iΓ + I ) is arctic or antarctic to order θ 2 , and applying the auxiliary field equations tells us quite a bit about these quantities. In contrast, we cannot say anything about D + α (D + ) 2 F + Q without applying dynamical equations of motion, so we have written T 0 in a particular way to avoid such terms.
Even in the rigid case, the expressions for the various terms we will encounter can be involved. To simplify the analysis, we will first consider only those terms that contribute in the rigid ungauged limit. Afterwards, we will include the covariant terms associated with the gauged isometries.

Rigid ungauged terms
The first line of (5.2) we denote as T 0.1 . It can be rewritten separately in terms of arctic and antarctic multiplets as Now let us apply the equations (4.6) for the components of the arctic multiplet. Taking only those terms that survive in the rigid ungauged limit, we find The antarctic term is found by replacing Υ I+ →ΥĪ + , Γ + I →Γ + I and z −− 1 → z −− 2 . We will use the symbol ∼ to denote the terms we are examining at each stage of the calculation.
The above expression is valid off-shell, that is, without assuming that Γ + I is an arctic superfield. If we did not make this assumption, we could begin to derive it now. The above expression is the only place where the arctic component field P I(−3) appears, and so it acts as a Lagrange multiplier enforcing that the lowest component of Γ + I is arctic. We could proceed in this way, rederiving all of the auxiliary equations of motion, but it would be quite involved. Because we already know their content -the lower components of Γ + I must be arctic -it is easier to simply assume them without comment. Proceeding in this way, we see that in the expression (5.4), the terms Γ + I P I(−3) and Aα αI− A αα − I drop out under the contour integral as they are purely arctic expressions -that is, they are of the form for some field-dependent coefficients C n , and these vanish under the contour integral, Similarly, their antarctic conjugates, which take the form vanish as well. The remaining terms can be rearranged to The total derivative can be discarded in the rigid case. Now we exploit the on-shell conditions (4.12) for A αα I− and A αα − I . Using the definition (3.9b) of g ab as well as the relations (We will occasionally abbreviate the antarctic term as "a.t.") Next, we consider the second line of T 0 . We have The leading term is again purely arctic and vanishes under the contour integral. The remaining terms can be rewritten In the rigid case, we can discard the total derivative. Using the relations (4.11), which leads to At this stage we have recovered the kinetic terms for both scalars and fermions, along with some extra terms involving two fermions and a spacetime derivative of φ µ that should not be present in the final action. The only other contribution involving such a term is found in the fifth line of T 0 : for both arctic and antarctic multiplets, the terms in question can be written where S cbb is given by We want to show that this expression vanishes. The second and fourth terms of S cbb are proportional to ∇ [c g b]b , which vanishes. With the useful identity the remaining terms can be rewritten as The first set of terms involves ∇ c g bb , which vanishes. The second set of terms is purely antarctic when divided by v 1+ v 2+ and so vanishes under the contour integral. Thus S cbb does indeed drop out. This leaves us with The only term we must still reconstruct is the four fermion term involving the hyperkähler curvature. It should be found by including the remaining terms in the third, fourth and sixth lines of T 0 . The third line is purely arctic or antarctic in the absence of gauged isometries so we may ignore it. The fourth line gives The sixth line is Combining all terms, we have The contribution to ∆ ++ bab from the antarctic fields is purely antarctic when divided by (v 1+ v 2+ ) 2 , so it drops out. The remaining terms contribute to R ++ abāb as The second line vanishes under the contour integral since is purely arctic when divided by (v 1+ v 2+ ) 2 . The first line simplifies in the same way after commuting ∇ā to act on the arctic superfields. This leaves The final result for the rigid ungauged contributes to T 0 , including the previously discarded total covariant derivatives, is We will return to this expression later when we address the supergravity contributions. Now let us remain with the rigid case and simplify. The contour integral is completely trivial since only the prefactor varies along C, This leads (dropping the total covariant derivative) to for the component Lagrangian. This is the general form of the hypermultiplet action in holomorphic coordinates φ a in a rigid background in the absence of gauged isometries.

Rigid gauged terms
Our next step is to collect the terms corresponding to gauged isometries. We return to the expression (5.2) and collect the terms involving the scalar field W r , the gaugino λ r αi and the auxiliary field Y rij of the vector multiplet.
Let us begin with terms involving Y rij . These come only from the first term of T 0 , which gives after identifying J I+ r Γ + I ≡ D ++ r as the N = 2 moment map. The argument of D −− is holomorphic away from the poles, so it can be discarded under the contour integral. We are left with Now let's collect all terms involving the chiral gaugino λ r αi . These come only from the first two lines of T 0 and give Recalling that D − α D ++ r = Ω + r α = v + i Ω i r α is holomorphic, the argument of D −− is clearly holomorphic so it can be discarded. The remaining terms can be rewritten as Now let's address all terms involving W r or its conjugate. Those from the first two lines of T 0 give Now let us combine these terms with those from T 0.3 , which can be rewritten as The terms in T 0.3 independent of z −− 1 vanish under the contour integral as they are purely arctic. The rest, using (4.13), give The only other term containing explicit W r orW r contributions comes from T 0.4 , which is given by Combining all terms that go as W r ζ a ζ b , we find (after some simplifications) The first line is purely antarctic and the second is purely arctic, so the contour integral kills these terms. Then the full contribution from gauged isometries yields Using the explicit relation (4.24) for D ij D ij r , performing the contour integral, and combining with the result (5.30) of the previous section, we find the full rigid Lagrangian upon eliminating the auxiliary fields associated with the N = 1 chiral multiplets φ a . 18 The superpotential contribution couples the chiral superfield W r of the N = 2 vector multiplet to the chiral component Λ r of the N = 2 moment map. The remaining component of the moment map is contained implicitly within the Kähler potential.

The component action in curved projective superspace
Having derived the correct component action in the rigid limit, we must now include all of the effects of supergravity. The component action from curved projective superspace is where 2) As discussed in [16], it is always possible to complexify the auxiliary SU(2) manifold to SL(2, C) and then choose a contour where u i is constant. The resulting formulation of projective superspace is exactly that given in [17][18][19][20] and is clearly advantageous for evaluating component actions: the integral involving L ++ automatically vanishes. However, we will take an even larger shortcut which avoids the need to make any such choice. Because we are dealing with a hyperkähler cone, we can write where Γ + I andΓ + I are given by (3.2). If we were to take all of the component fields to be on-shell, then Γ + I would be an arctic multiplet to all orders in its θ expansion. But then the superspace Lagrangian would be built out of the sum of an arctic and an antarctic superfield and such Lagrangians vanish even in curved space. The proof of this statement is quite simple (see [19] for an equivalent argument in SU(2) superspace) and we will review it in appendix B.
Of course, what we seek to do is to put only the auxiliary fields on-shell, so that the above argument holds only up through θ 2 . This means that we can use (6.4) to evaluate all terms except the leading ones in L −− . If we write the action S as we will find that all contributions to the component action arise solely from T 0 and T 1 .
There will be additional remainder terms within T 0 and T 1 that involve only the combi-nation Γ + I Υ I+ and its conjugate; when combined with S rest , which also depends solely on this combination, all these terms will turn out to vanish.
One final point: to avoid confusing the CKV χ µ of the target space with the spinor field χ αi of the conformal supergravity multiplet, we will from now on always arrange to lower the target space index of the CKV so that we deal instead with K µ = (K a , Kā) ≡ χ µ or we will rewrite it as A i a = χ µ f µi a .
6.1 Remaining evaluation of T 0 The first term T 0 is the most complicated as it must generate all the interactions present even in the rigid supersymmetric theory. For this reason, we have already performed the majority of its evaluation in the previous section. We again can decompose T 0 as in (5.2), replacing gauged covariant derivatives with supergravity covariant derivatives. Taking all the terms that contributed in the rigid limit, we found up to terms involving covariant supergravity fields. The reader may check that all of the steps followed in the rigid case to derive this expression follow equally well here. Now we must include the covariant supergravity fields. We begin with all terms involving the auxiliary D. These are given only by the first line of T 0 which becomes where we have used the relation (4.25) for the hyperkähler potential K. The argument of D −− is not holomorphic so it does not automatically vanish. For now, let us set it aside. Next, we take all terms involving W αβ . These are given only by the second line of T 0 , leading to Finally, we collect terms involvingχ iα . These are only in the first two lines, giving Again, the argument of D −− is not holomorphic, and we will set it aside. Now we construct the final expression for T 0 . It will be more convenient to rewrite where ∇ a carries the target space connection. (This formula is valid when K describes a hyperkähler cone.) After this rewriting, we find In the final line, we have collected a total covariant derivative, with and a contour derivative with Before moving on to evaluate the first set T 1 of gravitino terms, we must address what to do with ∇α α B −− αα . Because the vector derivative contains a number of connectionsincluding supersymmetry, S-supersymmetry, and SU(2) which are quite non-trivial -it does not vanish identically and must be separately analyzed.

Simplification of the total covariant derivative
Let us denote the total covariant derivative by T 0.TD = ∇α α B −− αα . This can be rewritten as . (6.14) In the above expression we have appearing each of the connections within ∇ αα . In the last line is an actual total derivative (which can be discarded) and a torsion term that arises from the gravitino dependence of the spin connection. Before attacking this, let us recall the guiding principle discussed at the beginning of this section. The superfield Γ + I is arctic to order θ 2 , so any expressions involving at most two spinor derivatives can be treated more simply than terms involving three or more. In light of this, we choose to separate the first line of T 0.TD , which involves up to three spinor derivatives. For convenience, we break it into two terms depending on, respectively, the spin-1/2 and spin-3/2 pieces of the gravitino. We will analyze these shortly, but first let us discuss what happens to the remaining terms. Denote these by T 0.TD|rest . Assuming that Γ + I is an arctic superfield to this order, B −− αα can be written as We emphasize that this equation holds only if we do not apply any further spinor derivatives. Now when we insert this expression into T 0.TD|rest , the last term of B −− αα will vanish as it leads to a total contour derivative of a holomorphic quantity. This means that the contribution of T 0.TD to the contour integral will be The important feature of T 0.TD|rest is that it depends only on the arctic combination Γ + I Υ I+ and its antarctic conjugate. For now we can set aside these and focus on the terms first order in the gravitino.

Gravitino terms
Now we must deal with terms involving a single explicit gravitino field. There are three such expressions: two of them, T 0.TD|Q1/2 and T 0.TD|Q3/2 , arise from the total covariant derivative ∇α α B −− αα , and one of them T 1 , arises from the original component action. First we will need a number of formulae for the spinor derivatives of B −− αα . There are a number of ways to potentially organize the resulting expression. We wish to construct as much as possible expressions involving the arctic combination Γ + I Υ I+ , the hyperkähler potential K and its derivatives, and various pullbacks of the hyperkähler two-form Ω ++ . To accomplish this, it is convenient to rewrite B −− αα into the equivalent form (6.21) The first line naturally involves only the arctic combination Γ + I Υ I+ while the second line is manifestly antisymmetric in Γ + I and Υ I+ and lends itself to reconstructing pullbacks of Ω ++ . Keeping in mind relations such as one can show that Using also results such as These results lead to a simple result for the spin-3/2 piece The D −− term can be discarded since its argument is holomorphic. For the spin-1/2 piece, it is natural to consider it in concert with T 1 . The combination can be written The first and second terms involve a common expression (∇ − ) 2 (Γ + I∇ − α Υ I+ ) given by The coefficient of this term in (6.27) involves ψ αα− α − z −− 1 ψα α+ α = ψ αα 1 α /v 1+ and so the first line of (6.28) contributes a purely arctic expression to (6.27) that can be discarded under the contour integral. This had to be the case as otherwise we would need to impose the explicit form for Ξ Iα− , which involves the field equation for the physical fermion.
It is convenient to relabel the tensor auxiliary field as T − ab = 4(σ ab ) α β W β α , in accordance with tensor calculus conventions, where T − ab ≡ T ab ij ε ij . (Note that this changes the definition of self-duality used in [57] so as to agree with component conventions.) Also, a number of terms can be rewritten to involve the fields A i a , with the result Two useful checks can be made. First, in the rigid supersymmetric limit, this matches the component Lagrangian constructed from (5.42). Second, the Lagrangian must be S-supersymmetric. To check this, note that the parenthetical terms in the third and fifth lines, which are the only ones multiplying explicit gravitinos, must vanish under Ssupersymmetry. It is straightforward to check that the remaining terms all cancel against each other. This result can be compared with eq. (5.4) of [27], where standard superconformal tensor calculus conventions (see e.g. [58]) were used. The relation between those conventions and ours for conformal supergravity are spelled out in [57]; for example, one must swap the locations of SU(2) indices, taking care to observe that tensor calculus conventions employ ε 12 = ε 12 = 1 while we use ǫ 12 = ǫ 21 = 1. This amounts to the exchange of ε ij → −ǫ ij and ε ij → ǫ ij . The target space conventions of [27] also differ in several ways from those we have used. For example, their Sp(n) indices (ᾱ, α) correspond to our (a,ā), and we use a different normalization for the fermions. For the component fields of the sigma model, one must exchange The target space geometric quantities are related as The components of the vector multiplet are also defined slightly differently: Other quantities can be derived from these relations and the equations given in [27].

Conclusion
Our goal in this paper was quite specific: to recover the component action of a hyperkähler cone coupled to conformal supergravity given in [27] using projective superspace methods. Our approach was based on the formal solution of the equations (3.1), corresponding to the requirement that the dual multiplets Γ + I andΓ + I were respectively arctic and antarctic, as advocated in [22,46]. We have not attempted to actually solve these equations for any specific models -the formal solutions were sufficient to yield the component action -but it should be mentioned that many specific cases do admit solutions. A major class involve O(2n) hypermultiplets and the generalized Legendre transform construction [10,11,23,59]: here the number of auxiliary fields is finite from the outset. Another major class involve target spaces that are cotangent bundles of Hermitian symmetric spaces [46,50,[60][61][62][63][64] (see also [44] for a pedagogical discussion).
As discussed in [27,32], the component action (6.38) does not directly yield the general action of canonically normalized supergravity coupled to matter. One must fix the Weyl gauge and the SU(2) R gauge: this effectively eliminates four degrees of freedom from the hypermultiplet manifold, reducing the 4n-dimensional hyperkähler cone to a 4(n − 1)dimensional quaternion-Kähler manifold. In addition, one must add another matter sector (e.g. vector multiplets parametrizing a special Kähler manifold) to yield a consistent equation of motion for the auxiliary D. This can all be understood at the component level, but what about in superspace?

(7.2)
Imposing the dilatation+U(1) R gauge φ 0 = e K/6 leads to the standard formulation of an N = 1 supergravity-matter system (including a superpotential is straightforward) in the so-called Kähler superspace [65] with a simple Lagrangian The function K is absorbed within the superspace structure and the original Kähler transformations become associated with an effective U(1) R symmetry of the component fields. This Kähler superspace is extremely useful: for example, higher derivative interactions can easily be incorporated in a Kähler-covariant way. It would be extremely interesting to construct the N = 2 analogue of the above superspace geometry: a proposal along the lines discussed above has already been made in [17].
This problem takes a particularly elegant form if there exists an additional U(1) isometry that separately rotates the arctics and antarctics. Then it is always possible to perform a duality transformation, exchanging one polar multiplet for a tensor multiplet G ++ , giving the supergravity-matter action [20,66] − 1 2π C dτ d 4 x d 4 θ + E −− − G ++ log(G ++ /Υ + 0Υ + 0 ) + G ++ K(Υ,Υ) (7.4) where K(Υ,Υ) is a real function of weight-zero arctics Υ and antarcticsΥ. (The arctic Υ + 0 drops out of the component action but is necessary to keep the argument of the logarithm dimensionless.) This is the natural N = 2 generalization of the N = 1 new minimal supergravity action coupling the tensor multiplet compensator to a Kähler potential. The component Lagrangian of (7.4) gives the general supergravity-matter system (after gaugefixing) involving a quaternion-Kähler target space arising from a hyperkähler cone with a triholomorphic U (1) isometry. An interesting open question would be to address hypermultiplet couplings in the presence of various higher derivative terms. Higher derivative tensor multiplet actions were discussed at the component level in [67]; their lift to projective superspace (and generalization to include O(2n) multiplets) was addressed in [68]. In principle, one should be able to apply the latter construction to include higher derivative actions for off-shell polar multiplets. However, the elimination of the auxiliaries in the presence of such terms becomes (even formally) a formidable process. Nevertheless, superspace would seem to be the natural mechanism for constructing such terms in a systematic way. It would be intriguing to explore these issues further.

A Vector multiplet conventions and supersymmetry transformations
The introduction of a gauge connection to the curved projective superspace [16] is quite straightforward and can be found in [18,20]. We introduce a superspace connection A M = A M r X r , with X r the formal gauge generator which acts on a field Ψ in a representation T r as X r Ψ = T r Ψ. Note that the algebra of the operators possesses a different sign from the algebra of the matrices, For a compact gauge group, T r are anti-Hermitian matrices, and so are frequently defined with an additional factor of i to make them Hermitian. For a non-linear sigma model with gauged isometries, the scalars φ µ = (φ a ,φā) transform as X r φ µ = J r while the component one-form A m r and two-form F mn r are given by the projections of the corresponding superspace quantities A m r and F mn r . The supersymmetry and Ssupersymmetry transformations are given by δA m r = i(ξ i σ mλir ) − ǫ ij (ξ i ψ mj )W r + h.c. , δW r = 2 ξ i λ ir , δλ r αi = (σ ab ξ i ) α F ab r + 1 8 T − abW r − Y i j ξ αj + 1 2 ξ αiW s W t f ts r + i∇ αβ W rξβj − 2 η αi W r , δY r ij = 2iξ αi ∇ ααλα j r − ξ αi λ j α sW t f ts r − 2iξα i ∇α α λ j α r −ξα iλαj s W t f ts r . (A.11) The transformation law of the gaugino involves the supercovariant curvature tensor F ab r , whose component form is given by B Vanishing of a pure arctic action and the remainder terms in eq. (6.33) We begin this appendix by reviewing an important lemma: the projective superspace action vanishes if Λ ++ is a purely arctic (or antarctic) superfield. A nearly identical statement was established in [19] for the choice Λ ++ = G ++ Λ for G ++ a tensor multiplet and Λ an arctic (or antarctic) superfield; the proof is exactly the same so let's briefly review it. As discussed in [16], it is possible to complexify the auxiliary manifold SU(2) to SL(2, C), taking v i+ = v i and v − i = u i /(v, u). This modifies the component Lagrangian only by a total derivative. In particular, we may choose u i to be a fixed coordinate so long as (v, u) = 0 along the contour C. This has the benefit of eliminating the second integral in (6.1), and so significantly simplifies the evaluation of component actions. This approach is exactly the formulation of projective superspace presented in [17][18][19][20].
Let us suppose now that Λ ++ is arctic. Following [19], we may choose u i = (1, 0), so ζ). Examining the component Lagrangian (6.2) with F ++ = Λ ++ , it is immediately apparent that L −− is a purely arctic expression. That is, after factoring out a common factor of 1/(v 1+ ) 2 , the remaining terms are all free of singularities at ζ = 0. Such a Lagrangian exactly vanishes under the contour integral. This is the most direct proof, but not the only one. One could also remain with the real SU(2) manifold and explicitly analyze the component reduction of (B.1). This would require defining the component fields of Λ ++ as we did for Υ I+ in section 4.1 and then proceeding to massage each term of the component action meticulously until everything vanished. This approach is clearly much more laborious.
It turns out that this more difficult approach actually has an excellent use. If we recall the situation in section 6, we argued that the projective superspace Lagrangian that occur, we will show here how the cancellation occurs when all fermionic terms are turned off. The relevant bosonic terms remaining in L −− are Those contributing to L ++ are given by which can be rewritten as The combination of L −− and L ++ is a total contour derivative and so can be discarded. Similar cancellations occur with the fermionic terms, but these require significantly more work.