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New supersymmetric sigma-model duality

We study dualities in off-shell 4D N = 2 supersymmetric sigma-models, using the projective superspace approach. These include (i) duality between the real O(2n) and polar multiplets; and (ii) polar-polar duality. We demonstrate that the dual of any superconformal sigma-model is superconformal. Since N = 2 superconformal sigma-models (for which target spaces are hyperkahler cones) formulated in terms of polar multiplets are naturally associated with Kahler cones (which are target spaces for N = 1 superconformal sigma-models), polar-polar duality generates a transformation between different Kahler cones. In the non-superconformal case, we study implications of polar-polar duality for the sigma-model formulation in terms of N = 1 chiral superfields. In particular, we find the relation between the original hyperkahler potential and its dual. As an application of polar-polar duality, we study self-dual models.


Introduction
Dualities in supersymmetric theories have a long history. In four-dimensional σ-models the duality between scalar and tensor multiplets, e.g., was discussed in N = 1 superspace both for N = 1 and N = 2 models already in [1]. Here we shall be interested in N = 2 supersymmetric σ-models and their dualities. These are best described in projective superspace [2,3] where the N = 2 supersymmetry is manifest. 1 There are several types of dualities (obtained by applying generalized Legendre transformations) for off-shell multiplets in projective superspace. These include (i) duality between so-called real O(2n) and polar multiplets which was introduced in [3] (see also [7]); and (ii) the duality between polar multiplets which was introduced in [8] and also studied in [9]. Of particular interest to us here is the latter, polar-polar duality.
We pay special attention to off-shell N = 2 superconformal σ-models formulated in terms of projective superconformal multiplets [10]. As always, superconformal invariance is of interest in itself. What is more important, the σ-models under consideration can be coupled to N = 2 conformal supergravity [11]. Superconformal σ-model dynamics turns out to require interesting target space geometry. In the component approach, general N = 2 superconformal σ-models were studied in [12,13,14] (see also [15]). Their target spaces are hyperkähler spaces possessing a homothetic conformal Killing vector which is the gradient of a function, and hence an isometric action of SU(2) rotating the complex structures. Such spaces are known as "hyperkähler cones" [14] and they are intimately related to quaternion Kähler manifolds which are 1 See [4,5] for alternative approaches. General N = 2 supersymmetric σ-models in harmonic superspace and their dualities were studied in [6]. Such σ-models do not possess a natural decomposition in terms of standard N = 1 superfields, a property that is desirable for various applications. The existence of such a decomposition is one of the powerful inborn features of N = 2 multiplets in projective superspace. target spaces for N = 2 locally supersymmetric σ-models [16]. Specifically, there exists a one-to-one correspondence [17] (see also [18]) between 4n-dimensional quaternion Kähler manifolds and 4(n + 1)-dimensional hyperkähler cones.
In the projective superspace approach, general off-shell N = 2 superconformal σmodels were studied in [10,19,20], while the superconformal couplings of N = 2 tensor multiplets had appeared already in [2] without a discussion of the conformal properties. The superconformal couplings of N = 2 tensor multiplets were systematically discussed in the component approach in [14].
General off-shell N = 2 superconformal σ-models are associated with Kähler cones [10,19,20]. If the dimension of the σ-model hyperkähler target space is 4n, then the associated Kähler cone (following the terminology of [21,22]) has dimension 2n. As defined in [21,22], a Kähler cone M is a Kähler space possessing a homothetic conformal Killing vector which is the gradient of a function, and therefore holomorphic. If (χ I ,χJ ) are the components of the homothetic conformal Killing vector, and g IJ is the Kähler metric, then ∇ I χ J = δ J I ,∇Īχ J =∂Ī χ J = 0 (1.1a) where K can be chosen to be We can choose local complex coordinates, Φ I , on M in such a way that χ I = Φ I . Then K(Φ I ,ΦJ ) obeys the following homogeneity condition: Any Kähler cone is a cone [21]. If M in the above discussion is hyprekähler, it is called a hyperkähler cone [14]. For the general properties of hyperkähler cones, see [14,21]. As shown in [10,19,20], the target spaces for general off-shell N = 2 superconformal σ-models, which are hyperkähler cones, are locally cotangent bundles over Kähler cones.
As is seen from (1.1b), the function K(Φ,Φ) can be identified with the Kähler potential of M. Kähler cones are target spaces for N = 1 superconformal σ-model, see, e.g., [20] for a detailed discussion. The relationship between the hyperkähler potential in the target space of a N = 2 superconformal σ-model and the associated Kähler cone was elaborated in some detail in [20]. At the level of N = 2 superfields polar-polar duality amounts to a particular diffeomorphism [9]. 2 Here we shall see that the N = 1 interpretation is considerably more interesting. It turns out that polar-polar duality exchanges one Kähler cone with a different (dual) cone. Since any N = 1 superconformal σ-model has a Kähler cone as its target space, we may interpret polar-polar duality as a transformation in the set of N = 1 superconformal σ-models.
We further discuss the interpretation of polar-polar duality for the non-superconformal σ-models in terms of physical N = 1 fields and show that it defines a transformation of certain n-dimensional Kähler spaces to other n-dimensional Kähler spaces.
Finally, as an important application, polar-polar duality allows us to introduce the family of self-dual models.
The paper is organized as follows: In section 2 we recapitulate some salient features of projective superspace and the definition of superconformal projective mutiplets. Section 3 starts our duality discussion by providing a manifestly N = 2 supersymmetric (and, where appropriate, superconformal) description of O(2n)/polar and polar/polar dualities in terms of N = 2 projective superfields. In section 4 we examine these dualities when reduced to N = 1 superspace. One of the main results obtained in sections 3 and 4 is the proof of the fact that the dual of any N = 2 superconformal σ-model is superconformal. Our analysis is deepened and carried out in more detail in section 5 for models with one polar multiplet. This section also contains several important examples. In section 6 we extend the analysis of the previous section to models containing a set of n polar multiplets, again giving several examples. Section 7 contains a discussion of the intriguing possibility of self-dual models in the present setting, while section 8 contains a few concluding comments. We have collected some relevant features of superconformal Killing vectors in Appendix A. Finally, in Appendix B we discuss properties of the tensor multiplet formulation for σ-models with U(1)×U(1) symmetry (5.43). Review material is collated in section 2 and Appendix A.

Superconformal projective multiplets
We start from the algebra of N = 2 spinor covariant derivatives 3 These relations encode an important structure that can be uncovered by introducing an auxiliary isotwistor v i ∈ C 2 \ {0} and defining the following operators: α . Then, the anti-commutation relations (2.1) imply that These identities constitute the integrability conditions for existence of certain constrained N = 2 superfields that live in R 4|8 × CP 1 and are annihilated by D α andD . α . Following [10], a superconformal projective multiplet of weight n, Q (n) (z, v), is a superfield that lives on R 4|8 , is holomorphic with respect to v i on an open domain of CP 1 , and is characterized by the following conditions: (a) it obeys the analyticity constraints (c) it obeys the following N = 2 superconformal transformation law: Here ξ = ξ A (z)D A is a N = 2 superconformal Killing vector, and Λ ij (z) and σ(z) are related to ξ as in eqs. (A.4)-(A.7). In the transformation law (2.5), u i denotes a fixed isotwistor chosen to be arbitrary modulo the condition (v, u) := v i u i = 0. Both Q (n) and δQ (n) are independent of u i . The parameters Σ and Λ (2) obey the identities: Given a superconformal weight-n multiplet Q (n) (v i ), its smile conjugate, 4Q(n) (v i ), is defined by is a superconformal weight-n multiplet, unlike the complex conjugate of Q (n) (v). One can also check that Therefore, if n is even, one can define real isotwistor superfields, Our next goal is to understand how to engineer N = 2 superconformal field theories described by superconformal projective multiplets. Let L (2) be a real superconformal weight-2 multiplet, which is constructed in terms of the dynamical superfields. Associated with L (2) is the superconformal action: . (2.10) Here γ denotes a closed contour in CP 1 , v i (t), parametrized by an evolution parameter t.
The action makes use of the following fourth-order differential operator: Here u i is defined below eq. (2.7), and it is kept fixed along the integration contour. The action can be shown to be invariant under arbitrary infinitesimal N = 2 superconformal transformations [10].
An important property of the action (2.10) is its invariance under projective transformations of the form: where t is the evolution parameter along the contour, and the matrix elements a(t) and b(t) obey the first-order equations: ψ denoting the derivative of a function ψ(t) with respect to t. Equations (2.13) guarantee that the transformed isotwistor u ′ i is t-independent. This invariance allows one to make u i arbitrary modulo the constraint (v, u) = 0, and therefore the action is independent of u i , that is (∂/∂u i )S = 0.
Let ξ K be a superconformal Killing vector obeying the conditions Λ ij (z) = σ(z) = 0 , (2.14) with Λ ij (z) and σ(z) defined in eqs. (A.7) and (A.5), respectively. It is called a N = 2 Killing vector, for the set of all such vectors can be seen to form a superalgebra isomorphic to the N = 2 super-Poincaré algebra. In the super-Poincaré case, the transformation law (2.5) reduces to the universal (weight-independent) form: Suppose we wish to construct an off-shell N = 2 superconformal theory described by a given set of superconformal projective multiplets P (n A ) A (z, v). Then, the corresponding Lagrangian must be an algebraic function of the dynamical superfields, and possess no explicit dependence on the isotwistor v i . Imposing the homogeneity condition s-conformal is a superconformal weight-two projective multiplet. In the more general case of super-Poincaré invariant theories, the Lagrangian may depend explicitly on the isotwistor v i , and must obey the homogeneity condition It is easy to show that the action (2.10) generated by L s-Poincaré is N = 2 supersymmetric. In the super-Poincaré case, the action (2.10) can be shown to be equivalent to that proposed originally in [2].
Without loss of generality, we can assume that the integration contour γ in (2.10) does not pass through the "north pole" v i north ∼ (0, 1) of CP 1 . It is then useful to introduce a complex (inhomogeneous) coordinate ζ in the north chart, C, of CP 1 = C ∪ {∞}: and define projective multiplets in this chart. Given a weight-n projective superfield Q (n) (z, v), we can associate with it a new object Q [n] (z, ζ) defined as The explicit form of Q [n] (z, ζ) depends on the multiplet under consideration, and will be specified below. In terms of Q [n] (z, ζ), the analyticity constraints (2.3) take the form: The Q [n] (z, ζ) can be represented by a Laurent series with Q k (z) some ordinary N = 2 superfields. In accordance with (2.8), the smile conjugate of Q [n] (z, ζ) is defined as follows: Unlike eq. (2.9), we now haveQ A real projective superfield is characterized by the properties: When switching from Q (n) (v) to Q [n] (ζ), the information about the degree of homogeneity, n, remains encoded only in the superconformal transformation law, eq. (2.5). In the super-Poincaré case, the superscript [n] becomes redundant and is usually omitted.
To describe gauge superfields (n = 0) and Lagrange multipliers for various duality transformations, one uses the so-called real tropical multiplet U (2n) (z, v) [3] defined by The superfield U [2n] (z, ζ) is real in the sense of (2.26).

Formulation in N = 2 superspace
The formalism presented in the previous section is convenient for the formulation of manifestly N = 2 supersymmetric duality transformations. The main purpose of this section is to show that the dual of any superconformal field theory is superconformal.

Duality between the real O(2n) and polar multiplets
Consider an off-shell N = 2 supersymmetric σ-model described by an O(2n)-multiplet η (2n) (z, v) and some other projective multiplets Ω ; v) be the Lagrangian of the theory. Note that, in general, L (2) may explicitly depend on the isotwistor v i . In the superconformal case, however, the Lagrangian must be v-independent, The theory under consideration has a dual formulation given by a different Lagrangian L ; v), in which Ξ (2−2n) is an arctic multiplet, andΞ (2−2n) its smile conjugate antarctic multiplet. The dual description is obtained by Legendre transformation. One proceeds by replacing the original system by an auxiliary first-order formulation with Lagrangian where U (2n) is a real tropical multiplet. This model is equivalent to the original one. Indeed, varying the first-order action, S first-order , with respect to Ξ (2−2n) andΞ (2−2n) proves to constrain U (2n) to become a real O(2n) multiplet, and then S first-order reduces to the original action. On the other hand, varying the firstorder action with respect to U (2n) gives 6 Suppose this equation allows us to uniquely express U (2n) as a function of the other variables, that is ; v). Then, the dual Lagrangian is defined by where the vertical stroke on the right indicates that the variable U (2n) should be replaced by its on-shell value ; v).
The duality transformation presented is compatible with N = 2 superconformal invariance. Indeed, suppose the original model is superconformal, and hence its Lagrangian has no explicit v-dependence, ). It leads to the first-order Lagrangian which also has no explicit v-dependence, and therefore generates a superconformal theory.
Integrating out U (2n) does not generate any explicit v-dependence. We conclude that the dual Lagrangian is v-independent, L , and therefore the dual theory is N = 2 superconformal.

Polar-polar duality
A different type of duality can be defined in the case of a nonlinear σ-model in which the dynamical variables include a polar multiplet realized in terms of an arctic superfield Υ (n) (z, v) and its smile conjugate antarctic superfieldΥ (n) (z, v). Along with this polar multiplet, the theory may also describe the dynamics of some other multiplets Ω We denote the corresponding Lagrangian by L (2) The theory under consideration possesses an equivalent first-order formulation generated by where W (n) is complex tropical, and Ξ (2−n) arctic. Let S first-order be the corresponding action. Indeed, it will be shown in subsection 5.1 that the equation of motion for Ξ (2−n) implies that W (n) is a weight-n arctic multiplet, Then, the action S first-order reduces to that generated by ; v). On the other hand, the equations of motion for W (n) andW (n) are: Under rather general assumptions, these algebraic equations can be used to express W (n) andW (n) in terms of the other variables. This leads to the dual Lagrangian: where the vertical stroke on the right indicates that the variables W (n) andW (n) should be replaced by their on-shell values. In the special case n = 1, both the original and dual polar multiplets have the same weight.

Polar-polar duality and superconformal σ-models
We consider a system of interacting weight-one 7 arctic multiplets, Υ +I (z, v), and their smile-conjugates,Υ +Ī (z, v), described by a Lagrangian of the form [10]: Here K(Φ I ,ΦJ ) is a real function of n complex variables Φ I , with I = 1, . . . , n, which obeys the homogeneity condition (1.3). The function K(Φ I ,ΦJ ) can be interpreted as the Kähler potential of a Kähler cone (following the terminology of [21,22]). Of course, this interpretation requires the Kähler metric g IJ := K IJ to be non-singular, To simplify the notation, in this subsection we denote Υ + ≡ Υ (1) .
where we have used the standard the notation: The action with K(Φ I ,ΦJ ) obeying the homogeneity condition (1.3), defines the most general N = 1 superconformal σ-model, see, e.g., [20].
We are interested in the dual formulation for the theory (3.10) which is obtained by performing the polar-polar duality with respect to all the multiplets: where the vertical stroke on the right indicates that the complex tropical superfields W +I and their smile-conjugatesW +Ī should be expressed in terms of weight-one arctic superfields Ξ + I and their smile-conjugatesΞ + I using the following equations of motion: This requires the Kähler potential K(Φ I ,ΦJ ) to obey the condition Making use of the equations (3.15a) and (3.15b), in conjunction with (1.3) and the standard properties of the Legendre transformation, one can show that the dual Lagrangian (3.14) obeys the homogeneity condition: As a result, we can represent where K D (Ψ I ,ΨJ ) is a real analytic function of n complex variables Ψ I , with I = 1, . . . , n, which obeys the homogeneity condition This function can be interpreted to be the Kähler potential of a Kähler cone. For such an interpretation to be consistent, the corresponding Kähler metric g D IJ := K D IJ should be nonsingular, This indeed follows from eqs. (3.11) and (3.16) of which the latter implies We conclude that the N = 2 polar-polar duality transformation induces a transformation in the family of N = 1 superconformal σ-models. Specifically, the σ-model (3.13) turns into It should be emphasized that the above conclusions hold if the duality transformation is applied to all the polar multiplets in the superconformal σ-model (3.10) and (1.3). Had we dualized some of the polar multiplets, we would have ended up with a dual formulation in which the Lagrangian obeys a different homogeneity condition. Specifically, let us split the original set of arctic multiplets, Υ I , into two subsets Υ I = (Υ i , Υ a ), and apply the polar-polar duality to the first subset. Then, we generate a dual Lagrangian multiplet Q [n] (z, ζ) is encoded in its N = 1 projection What is the structure of the N = 1 superfields Q k θ 2 =θ 2 =0 associated with the N = 2 projective multiplet? If the Laurent series (2.23) terminates from below, then the analyticity constraints (2.22) imply that the lowest components Q p and Q p+1 are N = 1 chiral and linear superfields, respectively.
If the Laurent series (2.23) terminates from above, then the analyticity constraints (2.22) imply that the highest components Q q and Q q−1 are N = 1 anti-chiral and anti-linear superfields, respectively.
In the N = 2 supersymmetric action (2.10), the Lagrangian L (2) is a projective multiplet, and therefore it is fully determined by its N = 1 projection L (2) θ 2 =θ 2 =0 . Let us express the action (2.10) in terms of this projection. We recall that the integration contour γ in (2.10) is chosen to lie outside the "north pole" v i north ∼ (0, 1) of CP 1 , which allows us to use the inhomogeneous complex coordinate, ζ, defined by v i = v 1 (1, ζ). Since the action is independent of u i , the latter can be chosen to be u i = (1, 0), such that (v, u) = v 1 = 0. We represent the Lagrangian in the form: It is important to remark that L(z, ζ) is a real projective superfield. Now, a short calculation (see, e.g. [37]) allows us to bring the action (2.10) to the form: Here the integration is carried out over the N = 1 superspace. The action is now formulated entirely in terms of N = 1 superfields. At the same time, by construction, it is off-shell N = 2 supersymmetric.
The main goal of this section is to reformulate the duality transformations, which we presented in section 3. in terms of N = 1 superfields. In what follows, the symbol of N = 1 projection in expressions like (4.7) is omitted.

Polar-polar duality
We turn to a N = 1 formulation for the theory with Lagrangian L (2) (Υ (n) ,Υ (n) , Ω; v) and its dual version considered in subsection 3.2.
The arctic multiplet Υ (n) (v) is represented by the series (2.31), in which the two leading components Υ 0 and Υ 1 are, respectively, chiral and complex linear N = 1 superfields, ; v), we appropriately replace Ω a (z, ζ). The supersymmetric action turns into (4.14) Consider now the dual formulation. Associated with the complex tropical multiplet For its smile-conjugate antarctic multiplet,W (n) (z, v), we get Finally, the arctic superfield Ξ (2−n) (z, v) and its smile-conjugateΞ (2−n) (z, v) will be represented similarly to eqs. (2.31) and (2.32) with the replacement n → 2 − n. The first-order action becomes This formulation of the polar-polar duality coincides with that given in [8,9].

Polar-polar duality and superconformal σ-models
Of special interest for us is the superconformal σ-model defined by eqs. (3.10) and (1.3), for it can be argued to realize general N = 2 superconformal couplings. We represent the weight-one arctic multiplets as Υ +I (v) = v 1 Υ I (ζ), where We recall that the components Υ 2 , Υ 3 , . . . , are complex unconstrained N = 1 superfields. The N = 2 superconformal action turns into The dual formulation is described by the following N = 2 superconformal action: Here the arctic multiplets Ξ I (ζ) is related to Ξ + I (v) by the rule Ξ + I (v) = v 1 Ξ I (ζ), and the structure of Ξ I (ζ) is completely similar to that given in eq. (4.18). The dual Lagrangian is defined by where the tropical superfields W I andWĪ must be unique solutions of the algebraic equations: The Kähler potential K(Φ,Φ) and its dual K D (Ψ,Ψ) correspond, in general, to different Kähler cones. As will be argued in the remainder of this paper, both potentials are encoded in the hyperkähler potential, K(Φ,Φ, Ψ,Ψ), in the target space for the original σ-model (4.19). It will also be shown that the original hyperkähler potential K and its dual K D are related to each other by a holomorphic reparametrization.
To conclude this section, we summarize, without proof, the explicit structure of the hyperkähler potential K(Φ I ,ΦĪ, Ψ J ,ΨJ); the technical details can be found in [20]. It has the form: where the second term obeys the homogeneity condition

Polar-polar duality with a single hypermultiplet
In this section we carry out a more systematic study of the polar-polar duality. In particular, we provide proofs for several statements, specifically eqs. (3.7), (3.8a) and (3.8b), which were taken for granted in our previous consideration. For the sake of simplicity, our discussion is restricted to the case of N = 2 supersymmetric sigma-models described by a single polar multiplet. However, many results can be readily extended to the case of n polar multiplets. All nontrivial N = 2 supersymmetric sigma-models of the type specified (that is, described by one polar multiplet) are non-superconformal, except L =Υ Υ, when Υ(ζ) has weight one.

General analysis
Consider an off-shell N = 2 supersymmetric nonlinear σ-model described by a polar multiplet realized in terms of an arctic superfield Υ(ζ) and its smile conjugateΥ(ζ).
where Υ(ζ) looks like and its smile-conjugateΥ(ζ) has the form We recall that the components Υ 2 , Υ 3 , . . . , in (5.2) are complex unconstrained N = 1 superfields. These superfields appear in the action without derivatives, and therefore they are purely auxiliary.
Before discussing the dual formulation for the theory (5.1), it is worth recalling the explicit structure of the equations of motion, see, e.g., [8,9]. Since the N = 1 superfields Υ 2 , Υ 3 , . . . , in (5.2) are complex unconstrained, their equations of motion δ δΥ n S = 0 , n ≥ 2 (5.4) have the form: This infinite set of nonlinear algebraic equations are equivalent to for some superfield Π(ζ). These equations can be used, in principle, to express the auxiliary superfields Υ n , with n ≥ 2, in terms of the physical superfields Φ and Σ and their conjugates; after that, the explicit form of Π(ζ) can be determined as well. It remains to consider the equations of motion for the physical chiral (Φ := Υ 0 ) and complex linear (Σ := Υ 1 ) superfields. 10 These equations imply that Π(ζ) is an arctic multiplet. We see that the off-shell constraint (Υ(ζ) is arctic) and the equation of motion (Π(ζ) is arctic) have the same superfield type. This is characteristic of duality-covariant theories in which the Bianchi identities and equations of motion have the same functional type (see, e.g., [38] for a review).
Now, let us apply the polar-polar duality transformation to the action (5.1). Following subsection 4.2, we replace (5.1) by the first-order action where W (ζ) is complex tropical, 10 In deriving the equations of motion for Φ and Σ, it is useful to represent Φ =D 2R and Σ =D. αξ . α , for unconstrained superfieldsR andξ . α . and the Lagrange multiplier Ξ(ζ) is arctic, We would like to show that the theory with action (5.7) is equivalent to the original one, eq. (5.1). To vary (5.7) with respect to Ξ(ζ), it is useful (i) to do the contour integrals in the second and third terms on the right of (5.7), as well as (ii) to separate the contributions involving the auxiliary and the physical superfields contained in Ξ(ζ): Since the superfields Ξ n , with n = 2, 3, . . . , are complex unconstrained, their equations of motion are Next, the equations of motion for Ψ and Γ are equivalent to the conditions that W 1 and W 0 are complex linear and chiral, respectively. Our conclusion is thus the following: As a result, the second and third terms in (5.10) drop out, and S first-order reduces to (5.1).
The above derivation of eq. (5.12) can be readily extended to justify the equation (3.7) in the general case.
On the other hand, instead of varying (5.7) with respect to Ξ(ζ), we can first vary S first-order with respect to W (ζ). Since all the components W n in (5.8) are complex unconstrained superfields, we immediately obtain This equation and its smile-conjugate can be used to express W (ζ) in terms of Ξ(ζ),Ξ(ζ) and ζ. As a result, S first-order turns into the dual action Our derivation of eq. (5.13) can be readily generalized to justify the equation (3.8a).

Chiral-linear duality
The N = 2 supersymmetric nonlinear σ-model (5.1) can be formulated solely in terms of the physical superfields Φ, Σ and their conjugates. The equations (5.6) can be used to express all the auxiliary superfields Υ 2 , Υ 3 , . . . (as well as the components Π n in (5.6)) in terms of the physical ones. Then, the action (5.1) turns into the chiral-linear (CL) one The chiral-linear formulation can also be obtained for the dual theory (5.14) following the same rules. This leads to Let us return to the first-order formulation (5.7) for the theory (5.1). This firstorder action is equivalent to (5.10). Consider the equations of motion for the auxiliary superfields Ξ n and W n , where n ≥ 2. The equations of motion for Ξ n , with n ≥ 2, are given by (5.11). The equations of motion for W n , with n ≥ 2, are It is easy to see the order in which these Legendre transformations is performed (say, first carry out (a) and then (b), or vise versa) does not matter. We can also apply single Legendre transformations, specifically:

N = 2 σ-models on cotangent bundles of Kähler manifolds
Let us now consider those σ-models (5.1) in which the Lagrangian has no explicit dependence on ζ, with Υ ′ (ζ) defined in (5.28). The hyperkähler potential in the target space turns out to have the form: The hyperkähler potential must obey the Monge-Ampère equation (see, e.g., [39]) grangian, L = K Υ I ,ΥJ , it was shown in [8,25] that the σ-model target space is (an open domain of the zero section of) the cotangent bundle T * M of a Kähler manifold M for which K Φ I ,ΦJ is the Kähler potential. This supersymmetric σ-model result implies the existence of a hyperkähler structure on T * M, for an arbitrary real-analytic Kähler manifold [8,25]. The latter result was independently established by purely mathematical means [26,27].
the role of base co-vectors. By the definition of the covariant derivative on the Kähler base manifold we realize that (5.34) Using this, the Monge-Ampère equation is equivalent to with R denoting the scalar curvature of M, that is g ΦΦ R = −2∂ΦΓ Φ ΦΦ . In the righthand side of (5.35), the covariant derivatives ∇ Φ and ∇Φ act only on the scalar fields H n appearing in (5.33), that is (5.36) Eq. (5.35) is equivalent to a recursion relation to uniquely compute the coefficients H n . The recursion relation is as follows: To construct the dual formulation, we should consider the first-order action with W (ζ) a complex tropical multiplet, and Ξ(ζ) an arctic superfield. The U(1) symmetry of the original model, eq. (5.28), turns into From (5.38) we read off the dual Lagrangian for some real function K D (Ψ,Ψ). Unlike the original Lagrangian, eq. (5.27), the dual Lagrangian depends, in general, on ζ. Such a dependence disappears in special cases which will be discussed below.

N = 2 σ-models with U(1)×U(1) symmetry
Here we would like to consider a subclass of hypermultiplet models (5.27) which are invariant under two rigid U(1) symmetries: phase transformations and shadow chiral rotations 13 The most general Lagrangian compatible with such symmetries is with L(x) a real analytic function of one real variable. We can interpret K(Φ,Φ) := L(ΦΦ) as the Kähler potential of a two-dimensional space M in canonical (or Kähler normal) complex coordinates, see subsection 6.1.
With the above Lagrangian, the first-order action (5.7) takes the form with i = 1, 2, as part of the R-symmetry group SU(2) × U(1), see [10] for more details. A shadow chiral rotation is a phase transformation of θ α 2 only, with θ α 1 kept unchanged. The N = 1 superspace is identified with the surface θ α 2 = 0.
The dual Lagrangian, L D , and the original one, L, are related to each other as follows where W and its smile-conjugateW are to be expressed via Ξ andΞ using the equations The dual Lagrangian L D can be given a geometric interpretation of the Kähler potential, K D (Ψ,Ψ) := L D (ΨΨ), of a two-dimensional space M with U(1) × U(1) isometry. By construction, the Kähler potential is given in canonical complex coordinates.
It turns out that both the original Kähler potential K(Φ,Φ) and its dual K D (Ψ,Ψ) are encoded in the hyperkähler potential (5.30). Before turning to a detailed justification of this claim, it is worth considering an example.

The Eguchi-Hanson metric and polar-polar duality
As an instructive example of the sigma-models studied in the previous subsection, consider the Kähler potential corresponding to CP 1 : K(Φ,Φ) = ln (1 + ΦΦ) . A short calculation gives for the dual Lagrangian: The dual Lagrangian is associated with the Kähler potential which corresponds to a new Kähler manifold that differs from the two-sphere. This follows from the fact that the Kähler metric for CP 1 , is characterized by a constant curvature, while the Kähler metric generated by the dual Kähler potential (5.53)), is no longer a metric of constant curvature. In addition, the dual Kähler manifold is non-compact, unlike CP 1 .
To get a better understanding of the relationship between the Kähler potential (5.50) and its dual (5.53), consider the hyperkähler potential generated by the N = 2 supersymmetric Lagrangian (5.51). 14 It is The Kähler potential (5.50) and its dual (5.53) can be seen to correspond to two different limits one can define in terms of the hyperkähler potential K(Φ,Φ, Ψ,Ψ): Since these limits are defined in terms of local complex coordinates on T * CP 1 , one might think they are non-geometric. This is not quite true, for the coordinate system used in (5.56) is canonical, and canonical coordinates for Kähler manifolds [28] are intrinsic (they are defined modulo linear holomorphic reparametrizations), see subsection 6.1.

Polar-polar duality with n hypermultiplets
This section generalizes the analysis given in subsections 5.4 and 5.5 to the case of n interacting hypermultiplets.

Canonical coordinates for Kähler manifolds
We start by recalling the concept of canonical coordinates for Kähler manifolds [28]. Given a Kähler manifold M, for any point p 0 ∈ M there exists a neighborhood of p 0 such that holomorphic reparametrizations and Kähler transformations can be used to choose coordinates with origin at p 0 in which the Kähler potential is Such a coordinate system in the Kähler manifold is called canonical. It was first introduced by Bochner [28] and extensively used by Calabi in the 1950s [30]. 15 There still remains freedom to perform linear holomorphic reparametrizations which can be used to set the metric at the origin, p 0 ∈ M, to be g IJ | = δ IJ . The resulting frame is defined modulo linear holomorphic U(n) transformations.
It turns out that the coefficients K I 1 ···ImJ 1 ···Jn | in (6.1) are tensor functions of the Kähler metric g IJ |, the Riemann curvature R IJKL | and its covariant derivatives, all evaluated at the origin. In particular, one finds The functions K (4,3) and K (4,4) are given in [10].
6.2 N = 2 σ-models with U(1)×U(1) symmetry Here we consider a family of Kähler manifolds M with holomorphic U(1) isometry. In canonical coordinates, the corresponding Kähler potential is (6.3) 15 In the modern literature on supersymmetric σ-models, some authors, unaware of the work of [28], refer to the canonical coordinates as a normal gauge [31] or Kähler normal coordinates [32].
The relevant isometry acts as a phase transformation which leaves the Kähler potential invariant. The condition (6.3) is equivalent to All Hermitian symmetric spaces belong to this family of manifolds. Note that eq. (6.5) is weaker than the homogeneity condition (1.3) which corresponds to the superconformal action (4.19). We apply the polar-polar duality transformation to all the arctic multiplets Υ I and their conjugates in (6.6). The dual Lagrangian K D (Ξ I ,ΞJ ), which is defined as in eqs. There is a simple relationship between the Kähler potential K(Φ I ,ΦJ ) and its dual K D (Ψ I ,ΨJ). Let us first discuss the structure of the hyperkähler target space for the N = 2 supersymmetric σ-model (6.6). As argued in [8,25], the σ-model target space is an open domain of the zero section of the cotangent bundle T * M parametrized by complex variables (Φ I , Ψ I ) and their conjugates, with Ψ I a holomorphic one-form at the point (Φ,Φ) of the base space M. The hyperkähler potential for T * M can be chosen as where the function H(Φ,Φ, Ψ,Ψ) can be represented by a Taylor series Here the Taylor coefficients H I 1 ···InJ 1 ···Jn Φ,Φ are some tensor functions of the Kähler metric g IJ Φ,Φ = ∂ I ∂J K(Φ,Φ), the Riemann curvature R IJKL Φ,Φ and its covariant derivatives. One can see from (6.7) that the complex coordinate system (Φ I , Ψ I ) in T * M is canonical.
Next, let us turn to the dual σ-model. Its target space is an open domain of the zero section of the cotangent bundle T * M D parametrized by complex variables (Ψ I , Φ I ) and their conjugates. The hyperkähler potential for T * M D is where H D has a series representation which is similar to that given in (6.8) and is obtained from the latter by the replacement Φ ←→ Ψ (including the replacement of all relevant geometric objects).
Finally, we can apply the chiral-linear duality of subsection 5.2 to show that We can immediately conclude that As a result, we see that the Kähler potential K(Φ,Φ) and its dual K D (Ψ,Ψ) are encoded in the hyperkähler potential K Φ,Φ, Ψ,Ψ .

N = 2 σ-models on cotangent bundles of Hermitian symmetric spaces
Hermitian symmetric spaces form a subclass in the family of Kähler manifolds introduced in the previous subsection. If M is Hermitian symmetric, then This follows from the fact that, for Hermitian symmetric spaces, there exists a closed-form expression for the Kähler potential in the canonical coordinates [36]: Here we have introduced 14) The program of deriving the hyperkähler potential on T * M from the σ-model (6.6), for various Hermitian symmetric spaces, has been carried in a series of papers [8,25,33,34,35]. It has resulted in a universal expression for the hyperkähler potential derived in [36] using the results of [35]. It is given by eq. (6.7), where Here the function F (x) is defined as 16) and the operator R Ψ,Ψ has the form: In the canonical coordinates, the curvature R K I L J is a constant tensor, see, e.g., [36] for more details.
Using eq. (6.11), from (6.15) we can immediately read off the polar-polar dual of the σ-model (6.6) if K(Φ,Φ) is the Kähler potential of a Hermitian symmetric space.

Self-dual hypermultiplet models
The concept of polar-polar duality allows us to introduce self-dual hypermultiplet models. For simplicity, here we consider the case of a single polar hypermultiplet.
The theory with action (5.1) is said to be self-dual if the dual Lagrangian coincides with the original one, L D Υ,Υ; ζ = L Υ,Υ; ζ . (7.1) The simplest example of self-dual systems was given in [8]. It is the free hypermultiplet model Below we construct an infinite family of self-dual nonlinear hypermultiplet models.

The meaning of self-duality
If the off-shell σ-model is self-dual, eq. (7.1), then we also have In accordance with the above consideration, the nonlinear σ-model is N = 2 supersymmetric, and therefore L (CC) Φ,Φ; Ψ,Ψ should be the hyperkähler potential of a hyperkähler manifold M [29]. If the original off-shell action (5.1) is selfdual, then the hyperkähler potential L (CC) Φ,Φ; Ψ,Ψ must be symmetric, eq. (7.6). This means that the target space M must possess a Z 2 symmetry.

Self-duality equation
In this section we consider a simple special class of self-dual models. We assume that the dependence on the polar multiplet is through the combination x = ΥΥ with no explicit ζ-dependence so that the Lagrangian is given by an ordinary function L(x).
Suppose the theory under consideration is self-dual, L D = L. Then the Lagrangian L(x) can be seen to obey the algebraic equation where the variables x and y are related to each other as follows It follows from eqs. (7.9a) and (7.9b) that In particular, any function f (x) with the property that x = f (f (x)) gives a self-dual Lagrangian through (7.10) Self-dual Lagrangians are not rare. Given any function h(x) we can construct a function satisfying the above properties as f (x) = h −1 (−h(x)) (compare with [40]). If the function h(x) is even or odd, the solution is trivial f (x) = ±x, so for nontrivial solutions we need functions h(−x) = ±h(x).
A simple partial solution of the equations (7.9a) and (7.9b) is with g a coupling constant.
To work out the geometry of this self-dual model one could use that models with this particular dependence on the polar multiplet can be dualized to an O(2) multiplet η as although there may appear contour ambiguities [9]. After working out the component content of the dual model one would have to dualize the N = 1 linear superfield of the O(2) multiplet to get the hyperkähler potential. We leave the explicit solution of this model for a future publication.
In theories with more polar multiplets there are other ways to construct self-dual models. For instance if we have two polar multiplets we may start with an action L(Υ 1Υ1 + Υ 2Υ2 ). Making a polar-polar duality on only one of the polar multiplets we create a model which will be self-dual with respect to a duality transformation of all polar multiplets.

Final comments
In this paper, we demonstrated that polar-polar duality generates a transformation between different Kähler cones. This is a new type of duality. It relates Kähler manifolds that are target space geometries for N = 1 σ-models, although the duality is intrinsically N = 2 supersymmetric. These dual Kähler manifolds are both embedded in the hyperkähler target space of the original N = 2 supersymmetric σ-model.
For non-superconformal σ-models, we derived a simple relationship between the hyperkähler potential K and its dual K D , eq. (6.10). It can naturally be extended to the superconformal case.
In order to find a hyperkähler potential corresponding to a N = 2 supersymmetric σ-model formulated in terms of polar multiplets, one has to eliminate the auxiliary N = 1 superfields. This technical problem has been solved for the σ-models on cotangent bundles of Hermitian symmetric spaces. The full solution is given in [35,36]. Polar-polar duality allows us to extend the class of σ-models for which this is possible. As an example, we looked at the Eguchi-Hanson geometry whose known solution allowed us to solve the auxiliary field problem for the dual model (5.52). The duality thus allows for a treatment of more complicated geometries.
We have discussed a family of self-dual models and identified some interesting features. The intrinsic meaning of self-duality as well as the relation between the geometry of the dual models remain to be understood, however. Here an example with the geometry worked out would be of great help.
For more than one polar multiplet, there are more possibilities to construct self-dual models. In particular, we can also consider self-dual superconformal models. Their properties remain to be investigated.

Acknowledgements:
We are happy to thank the organizers of the 30th Winter School "Geometry and Physics" at Srni, Czech Republic, where this project was initiated. SMK and RvU are grateful to the Department of Physics and Astronomy at Uppsala University for hospitality at final stages of this project. The work of SMK and UL was supported in part by the Australian Research Council. SMK also acknowledges partial support from the Australian Academy of Science. The research of UL was supported by VR grant 621-2009-4066. The research of RvU was supported by a Czech Ministry of Education grant No. MSM0021622409.
Projective superspace and its various implications have been frequently discussed with M. Roček over the years. His inspiration and insights are gratefully acknowledged.

A Superconformal Killing vectors
In this appendix we recall salient properties of the N = 2 superconformal Killing vectors, following [41,42].
In N = 2 superspace R 4|8 parametrized by coordinates z A = (x a , θ α i ,θ i . α ), with i = 1, 2, a first-order differential operator The parameters Λ j i defined by correspond to SU(2) transformations. One can readily check the identity and therefore Comparing this with eq. (2.28), we see that Λ (2) (v) := Λ ij v i v j is an O(2) multiplet.
B Tensor multiplet formulation for N = 2 σ-models with U(1)×U(1) symmetry In subsection 5.4, we discussed the polar-polar duality of N = 2 σ-models (5.43). There is a different dual formulation for such theories which is given in terms of an O(2) multiplet η =φ ζ + G − ϕζ (also known as an N = 2 tensor multiplet) and a N = 2 Lagrangian L 2 (η). Here we will elaborate on the structure of such σ-models. 16 We will assume that the contour integral in the corresponding action has been done, where H(ϕφ, G) is the resulting N = 1 Lagrangian.
In [1] it is shown that the Lagrangian H(ϕφ, G) must satisfy the Laplace equation ∂ ϕ ∂φH + ∂ 2 G H = 0. This can be used to find the form of H(ϕφ, G) from the knowledge of only part of it. For instance, using the ansatz where H 0 (G) is the ϕ independent part of the Lagrangian, the Laplace equation gives us the recursion relation H ′′ n (G) = −(n + 1) 2 H n+1 (G) , (B.3) 16 Four-dimensional quaternion Kähler metrics with torus symmetry were studied in [43].
which can be solved given the initial data H 0 (G). The solution is