An analogue of the Gibbons-Hawking Ansatz for quaternionic K\"ahler spaces
Radu A. Ionas
TL;DR
This paper extends the Gibbons-Hawking Ansatz to higher-dimensional quaternionic K"ahler spaces with symmetries, providing a new explicit construction method and applications to twistor space completions in mathematical physics.
Contribution
It introduces a Gibbons-Hawking-like description for quaternionic K"ahler spaces with symmetries, generalizing four-dimensional constructions to higher dimensions.
Findings
Provides a new geometric framework for quaternionic K"ahler spaces.
Explicitly constructs equivariant twistor space completions.
Connects to the local c-map in string theory.
Abstract
We show that the geometry of -dimensional quaternionic K\"ahler spaces with a locally free -action admits a Gibbons-Hawking-like description based on the Galicki-Lawson notion of quaternionic K\"ahler moment map. This generalizes to higher dimensions a four-dimensional construction, due to Calderbank and Pedersen, of self-dual Einstein manifolds with two linearly independent commuting Killing vector fields. As an application, we use this new Ansatz to give an explicit equivariant completion of the twistor space construction of the local c-map proposed by Ro\v{c}ek, Vafa and Vandoren.
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TopicsGeometric Analysis and Curvature Flows · Black Holes and Theoretical Physics · Algebraic Geometry and Number Theory
An analogue of the Gibbons–Hawking Ansatz for quaternionic Kähler spaces
Radu A. Ionaş
Abstract.
We show that the geometry of -dimensional quaternionic Kähler spaces with a locally free -action admits a Gibbons–Hawking-like description based on the Galicki–Lawson notion of quaternionic Kähler moment map. This generalizes to higher dimensions a four-dimensional construction, due to Calderbank and Pedersen, of self-dual Einstein manifolds with two linearly independent commuting Killing vector fields. As an application, we use this new Ansatz to give an explicit equivariant completion of the twistor space construction of the local c-map proposed by Roček, Vafa and Vandoren.
Contents
-
3 Analogue of the extended Gibbons–Hawking Ansatz for quaternionic Kähler metrics
-
3.2 Extended Gibbons–Hawking spaces with a hyperkähler cone structure
0. Introduction
The Gibbons–Hawking Ansatz [27] is a method of constructing four-dimensional hyperkähler spaces with a tri-Hamiltonian locally free -action in terms of the solutions of a Bogomolny-type differential equation whose consistency condition is a Laplace equation. It is one of the fundamental constructions in hyperkähler geometry, with vast applications ranging from the construction of gravitational instantons to serving as an asymptotic model or some other type of limit for various geometries and moduli spaces. The original method of Gibbons and Hawking was generalized in [41] by Pedersen and Poon to -dimensional hyperkähler spaces with a tri-Hamiltonian locally free -action; in what follows we will refer to such spaces as extended Gibbons–Hawking spaces. A key feature of these constructions is the fact that the rank of the action is such that the orbit parameters of the action together with the components of its associated hyperkähler moment map provide a complete set of coordinates for the hyperkähler space. Geometrically, the hyperkähler structure is defined on the total space of an -bundle over an open subset of the space , where it is given explicitly in terms of a Higgs field and -connection 1-form satisfying a generalized form of the Bogomolny equation. This construction is general: any -dimensional hyperkähler space with a locally free tri-Hamiltonian -action can be proven to arise locally in this way.
In this paper we show that a close analogue of this Ansatz exists as well in quaternionic Kähler setting. Recall that hyperkähler and quaternionic Kähler manifolds are -dimensional Riemannian manifolds with the holonomy groups of their Levi-Civita connections given by and , respectively, or subgroups thereof (here, we extend these definitions to include the pseudo-Riemannian variants of these geometries resulting from replacing the unitary quaternionic group with any one of its non-compact versions; also, for , the holonomy definition is rather too unrestrictive in the quaternionic Kähler case, and one usually discards it: the natural analogues of quaternionic Kähler manifolds in four dimensions are self-dual Einstein manifolds). Both types of geometries are Einstein. Hyperkähler geometry, however, is essentially distinguished from more general quaternionic Kähler one by having zero scalar curvature. This distinction notwithstanding, in specific circumstances it is nonetheless useful to think of the former as the zero scalar curvature limit of the latter. Another, less obvious way in which the two geometries are related was discovered by Swann, who showed in [46] that over any quaternionic Kähler manifold one can construct a quaternionic bundle with fiber , where denotes the multiplicative group of non-zero quaternions, the total space of which carries a natural hyperkähler structure as well as a homothetic vector field: that is to say, a hyperkähler cone structure. The power of Swann’s result resides in its transformative potential: it gives a canonical way to radically reformulate a quaternionic Kähler geometry problem as a hyperkähler geometry one.
And this is precisely the path that we take in our search for a quaternionic Kähler analogue of the extended Gibbons–Hawking Ansatz: rather than ask the question directly about quaternionic Kähler spaces, we ask instead the corresponding question about hyperkähler cones. Namely, we ask under what conditions an extended Gibbons–Hawking space possesses a hyperkähler cone structure. For any extended Gibbons–Hawking space of real dimension (this dimension is chosen for later convenience), the base of its -fibration is, as we have stated above, an open subset of the space , which can be viewed as the space of configurations of distinguishable points in . On this space, one can always define a natural quaternionic action by considering the rigid rotations and simultaneous scalings of such configurations which leave a given point, say, the origin, fixed. These transformations form indeed a quaternionic group since . So then, a more specific interrogation would be when can this -action on the base of the -fibration be lifted to a genuine hyperkähler cone structure on the total space, that is, on the extended Gibbons–Hawking space? This question is answered in Proposition 14, which states that a necessary and sufficient condition for that to happen is for the Higgs field of the extended Gibbons–Hawking space to be invariant at rigid rotations and scale with a certain weight under simultaneous scalings of the configurations. What remains now is to match this description of the hyperkähler cone, in which the -bundle structure is manifest, to the Swann description, centered instead on the -bundle structure. This is facilitated by a certain condensed quaternionic reformulation of the extended Gibbons–Hawking formulas that we introduce in § 3.1.3. It ultimately yields an explicit description of the geometry of the base of the Swann bundle, which is in this case a -dimensional quaternionic Kähler geometry with an inherited locally free isometric -action.
The emerging picture is summarized in intrinsic terms in subsection 3.5. Its main features are as follows: The quaternionic Kähler structure is defined on the total space of an -bundle over the space of inequivalent (with respect to rigid rotations and simultaneous scalings) non-degenerate configurations of distinguishable points in , which we denote by (and define precisely in subsection 3.3). A natural set of coordinates is given by the orbit parameters of the -action together with this action’s associated Galicki–Lawson-type moment maps, multiplied by a scale. The latter can also be understood as inhomogeneous local coordinates on . The quaternionic Kähler metric, 2-forms and connection 1-forms are given explicitly in these coordinates in terms of a reduced Higgs field and -connection 1-form satisfying a set of partial differential equations on similar to the Bogomolny equations from the hyperkähler case. These field equations admit also a dual formulation, a remarkable consequence of which is that the local quaternionic Kähler structure turns out to be completely determined by a single real-valued function on satisfying a set of linear partial differential constraints. This function is closely related to the hyperkähler potential of the Swann bundle. (Swann bundles have the distinctive feature among hyperkähler spaces that all three elements of a defining triplet of 2-forms — see e.g. Theorem 2 — can be derived, as Kähler forms with respect to their corresponding complex structures, from the same Kähler potential.) Very importantly, like in the hyperkähler case, this picture is general: any -dimensional quaternionic Kähler space with a locally free isometric -action arises locally in this way.
The main body of the paper is organized into five sections. In section 1 we give a non-Riemannian characterization of both hyperkähler and quaternionic Kähler manifolds, in a spirit similar to that of exterior differential systems [12]. That is, in each case we formulate a criterion in which the metric is not among the fundamental objects defining the geometry but, rather, is a derived, composite object. The building blocks of either geometry may be considered instead to be triplets of non-degenerate and pointwise linearly independent 2-forms: globally defined in the hyperkähler case, and locally defined in the quaternionic Kähler one. Such a triplet is required to satisfy two conditions: an algebraic condition, and an exterior differential one (Theorems 2 and 3). In the hyperkähler case the latter condition is due to Hitchin [30] and requires that the three 2-forms be symplectic. In the quaternionic Kähler case a corresponding condition was found by Alekseevsky, Bonan and Marchiafava [2]. These conditions do not constrain the triplets uniquely. In the hyperkähler case one has in fact a half 3-sphere’s worth of equivalent triplets to choose from (the triplets can be rotated by transformations, whose group manifold is topologically ). In the quaternionic Kähler case, on the other hand, one regards the triplets as local frames for an -bundle. The proofs we give to these criterions are new; we base them in both cases on the successive application of two key identities, which we collect together in Lemma 1. We show also that Alekseevsky et al.’s condition can be equivalently replaced by a Hermitian–Einstein-type condition for the curvature 2-form of the associated principal -bundle, which takes as well the form of an exterior differential condition. This gives us the very useful equivalent quaternionic Kähler criterion of Theorem 4. In the remainder of the section we describe how certain symmetry conditions — namely, the Killing condition and what we call the homothetic Euler condition, which is the basic symmetry of hyperkähler cones — fit into this framework. In spite of the perhaps unusual point of view, the material discussed in this preliminary section is for the most part known, and readers accustomed to the basic concepts of hyperkähler and quaternionic Kähler geometry and interested mainly in the headline subject of the paper can safely skip ahead, to return maybe only for occasional references.
In section 2 we review in detail Swann’s quaternionic bundle construction. We clarify among other things the role of the hypercomplex structures on the quaternionic fiber, and review how Killing symmetries fit into this picture as well.
In section 3, after many preparations, we finally address in full force the issue of finding a quaternionic Kähler analogue of the extended Gibbons–Hawking Ansatz, following the strategy exposed above. In the last part of the section we then specialize to four dimensions (i.e. ). The space is isomorphic to the complex upper half-plane, and we find that our formulas yield in this case precisely the description given by Calderbank and Pedersen in [13] to the four-dimensional avatars of the quaternionic Kähler spaces we consider, which are self-dual Einstein spaces with two linearly independent commuting Killing vector fields. In this sense, therefore, our results can be thought of as a natural higher-dimensional generalization of those of Calderbank and Pedersen.
In section 4 we translate the description of hyperkähler cones from the Gibbons–Hawking-type framework used up until now to the language of the Legendre transform approach of Lindström and Roček [38, 32]. This is useful particularly when one wants to solve the field equations by means of twistor methods.
In section 5, as an application of our newfound Ansatz, we give a thorough and explicit account of the twistor construction of a quaternionic Kähler metric found in [23] by Ferrara and Sabharwal, in connection with the so-called local c-map construction from string theory and supergravity. The twistor approach to this metric was initiated in [43, 44] by Roček, Vafa and Vandoren, and developed further in [40]. The idea is that, if one goes from the Ferrara–Sabharwal space six dimensions higher to the twistor space (in the sense of [32]) of its Swann bundle, there the quaternionic Kähler metric information is encoded in a single — and simple — holomorphic symplectic gluing function. The challenge becomes then to show that the quaternionic Kähler metric can be retrieved from this holomorphic data. In a first stage, by making use of the general methods associated to the Legendre transform approach, one can come down two dimensions from the twistor space and determine from this function the geometry of the Swann bundle. This step has been generally already understood in the literature. The new results of section 3 can then be used to descend a further four dimensions to retrieve explicitly and in an equivariant manner the Ferrara–Sabharwal metric. This exercise clarifies a number of issues and cements the basis of the twistor-theoretic understanding of this important construction.
1. Hyperkähler vs. quaternionic Kähler geometry
1.1. General aspects
1.1.1.
In Berger’s list of special holonomy manifolds, hyperkähler and quaternionic Kähler manifolds are -dimensional Riemannian manifolds with the holonomy group of their Levi-Civita connection a subgroup of the unitary quaternionic group , respectively the group .111 An explicit embedding of as a Lie subgroup of , the holonomy group of a general orientable -dimensional Riemannian manifold, is given as follows: regard as and consider the linear endomorphisms on the latter given by the simultaneous left and right actions for any , quaternionic unitary matrix , and unit quaternion ; the summation convention over repeated indices is understood. This preserves the quaternionic norm on , and so the corresponding Euclidean norm on , which thus exhibits it as an element of , and in fact of . The -quotient comes from the invariance under a simultaneous change of sign of both and . Note also that for , is a maximal subgroup of [29]. Here we will distinguish between the two notions and assume, moreover, that hyperkähler manifolds are not subsummed to the quaternionic Kähler ones. There exist also pseudo-Riemannian versions of these manifolds which are obtained by replacing the group with one of its non-compact versions with . In these notes we will assume that the notions of hyperkähler and quaternionic Kähler manifolds include these variations as well. In four dimensions (that is, for ), due to the isomorphism the above definition of quaternionic Kähler manifolds is rather unrestrictive as it encompasses all orientable four-manifolds. In this case the holonomy condition is usually replaced by a curvature condition, and the natural four-dimensional analogues of quaternionic Kähler manifolds are considered to be the self-dual Einstein manifolds. In fact, one can show that quaternionic Kähler manifolds are always Einstein [1], while hyperkähler ones are Einstein with vanishing scalar curvature — that is, Ricci-flat.
The holonomy definition can be shown to be equivalent in the quaternionic Kähler case to the existence of a three-dimensional subbundle of the bundle of endomorphisms of the tangent bundle which is locally spanned by three almost complex structures , , , forming the algebra of imaginary quaternions (), and is preserved by the Levi-Civita connection . That is to say, there exist locally defined 1-forms , , such that
[TABLE]
for any vector field , where the indices run over the values , denotes the anti-symmetric Levi-Civita symbol and the numerical factor is conventional. In four dimensions, the can be viewed as the self-dual part of the spin connection.
In the hyperkähler case, on the other hand, the holonomy definition is equivalent to the existence of a globally defined triplet of almost complex structures , , forming the algebra of imaginary quaternions, each element of which is individually preserved by the Levi-Civita connection:
[TABLE]
for any vector field . In contrast to the quaternionic Kähler setting, one can now show that the almost complex structures are in fact integrable and thus give rise to genuine complex structures.
In either case, the metric can always be chosen to be simultaneously Hermitian with respect to each , and by combining it with them one can form three 2-forms , for any , which are defined locally in the quaternionic Kähler setting and globally in the hyperkähler one. In four dimensions, these form a frame of the bundle of self-dual 2-forms. The above covariant differentiation properties readily translate for the 2-forms into
[TABLE]
in the quaternionic Kähler case and
[TABLE]
in the hyperkähler one, for an arbitrary vector field . Since the Levi-Civita connection is symmetric, these formulas imply further that
[TABLE]
in the first case and
[TABLE]
in the second, where we now view and as differential forms. Note, however, that the reverse implication does not trivially hold.
Assuming the dimension to be finite and the metric non-degenerate, since each is invertible (with inverse ), it follows that each , viewed as an element of , is also invertible, with inverse . A simple argument based on the quaternionic algebra shows then that we can write
[TABLE]
1.1.2.
In this approach, the three 2-forms are composite objects constructed from the metric and the almost complex structures. In what follows we will present an alternative approach in which this relationship is reversed, and they rather than the latter are regarded as the fundamental building objects of the geometry. Our discussion will emphasize in particular the fact that the two geometries inform each other and can be treated in parallel, and that, although we have made a special point of distinguishing between them, it is often convenient in practice to think of hyperkähler geometry as a limit case of quaternionic Kähler geometry, when the hallmark three-dimensional subbundle of the latter becomes trivial and the scalar curvature vanishes.
The proofs of the subsequent two theorems rely crucially on the following technical lemma:
Lemma 1**.**
Let be a manifold with an almost complex structure — that is, an endomorphism of the tangent bundle such that — and let
[TABLE]
for arbitrary vector fields be the associated Nijenhuis tensor. A real-valued 2-form on is
- a)
of type with respect to if and only if
[TABLE]
for any . In this case the following identity holds:
[TABLE]
where is the Levi-Civita connection corresponding to the compatible metric .
- b)
of mixed type with respect to if and only if
[TABLE]
for any . (The complex-valued 2-forms and are then of pure and type with respect to , respectively.) In this case we have the identity
[TABLE]
Proof.
Part is essentially Proposition 4.6 from ch. IX, sec. 4, p. 148 of reference [36] (the numerical factors differ due to differing normalization conventions for differential forms). Let us reproduce here the proof briefly. On one hand, we transform the last term on the right-hand side of the identity we want to prove as follows
[TABLE]
and then apply for each resulting term the Koszul formula for the Levi-Civita connection
[TABLE]
On the other hand, for the remaining two terms on the right-hand side we use the identity
[TABLE]
The result follows then by isolating the expression on the left-hand side and then cancelling the remaining terms by making use of the relationship between and , and the properties which follow from it.
The formula at part can be verified in a relatively more straightforward manner by applying the identity (12) to both the and the terms, and then systematically replacing in the resulting expression with by using some version of the defining relation for the former. ∎
With this lemma in hand we can now return to our initial discussion. On the hyperkähler side we have the following criterion, which is in essence a version of Lemma 6.8 of reference [30]:
Theorem 2**.**
A finite-dimensional manifold is hyperkähler if and only if it possesses a triplet of non-degenerate and pointwise linearly independent 2-forms , , satisfying simultaneously:
- •
an algebraic condition*: if we define*
[TABLE]
then we have .
- •
an exterior differential condition*: the 2-forms are symplectic, i.e.,*
[TABLE]
Proof.
Observe that although the algebraic condition formally requires only that , , be almost complex structures, it fully implies in fact that they satisfy the entire imaginary quaternionic algebra. Indeed, since they square to minus the identity we have
[TABLE]
from which we get immediately that .
Let us define now the following sections of or, equivalently, rank-2 contravariant tensors:
[TABLE]
Since the ’s are all antisymmetric, by comparing for example the third expression for with the fourth one for we see that , where the superscript denotes transposition. Similarly, and , from which it follows immediately that a symmetric rank-2 contravariant tensor. In other words, the algebraic condition also implies that we can define a metric on by
[TABLE]
for any . The minus signs have been chosen in order to ensure compatibility with our original definition of the 2-forms . Such a metric is automatically Hermitian with respect to each (this is possible only when ), and its signature is in general of type , with , so the manifold is in general pseudo-Riemannian.
Remark now that in order to prove that the manifold carries a hyperkähler structure it suffices to prove that each is covariantly constant with respect to the Levi-Civita connection of the metric . The covariant constancy of the follows then right away.
This can be shown in two steps. First, note that based on the quaternionic algebra we can write successively , and since the last expression is antisymmetric at the exchange of and it follows that is of mixed type with respect to . By part of Lemma 1 we then have
[TABLE]
As and are by assumption closed, the right-hand side of this equation vanishes, and since moreover is non-degenerate it follows that the Nijenhuis tensor of vanishes. By the Newlander–Nirenberg theorem, is then an integrable complex structure.
Second, as is symmetric at the exchange of and , is of type with respect to . By part of Lemma 1 we can then write
[TABLE]
The vanishing of the Nijenhuis tensor for together with the closure of imply then that is covariantly constant with respect to the Levi-Civita connection. The covariant constancy of and , as well as the integrability of and , follow by cyclically permuting the indices in the argument. ∎
Remark*.*
The triplets of 2-forms defining a given hyperkähler metric are not unique. To be more precise, the conditions of the theorem are invariant under the following two types of constant linear transformations:
for any non-vanishing constant . The corresponding hyperkähler metric rescales then from to .
- 2)
for any constant rotation matrix . These transformations leave the metric unchanged. Thus, for any triplet of 2-forms satisfying the conditions of the theorem one automatically has a half 3-sphere’s worth of such triplets giving the same hyperkähler metric (recall that, topologically, ).
Out of these, the only invariance which is perhaps not immediately obvious is that of the algebraic condition at rotations. The considerations in the first part of the proof make it clear that the algebraic condition is equivalent to the existence of a triplet and a metric such that . Then, since transformations preserve both the -symbol and the -symbol (the and in , respectively), this relation is invariant under simultaneous rotations and while stays unchanged. That is to say, for each rotated set of ’s there exists a rotated set of ’s satisfying the same algebraic condition and determining the same metric.
On the quaternionic Kähler side, on the other hand, we have the following version of Proposition 3 from reference [2]:
Theorem 3**.**
A finite-dimensional manifold is quaternionic Kähler if and only if it possesses a rank-3 -subbundle of the bundle of non-degenerate 2-forms on and an open covering of on each element of which one can pick a local frame , , for satisfying simultaneously
- •
an algebraic condition*: if we define*
[TABLE]
then we have .
- •
an exterior differential condition*: there exist locally defined 1-forms , , such that*
[TABLE]
Proof.
The structure of the proof mirrors closely that of the previous theorem. In particular, by virtually the same arguments as before the algebraic condition implies that the almost complex structures , , satisfy the imaginary quaternionic algebra, and that, moreover, a tri-Hermitian metric can be defined by means of the same equation (17).
The essential difference with respect to the hyperkähler case is that these quantities are now defined locally, and we need to check in addition that they give rise to the requisite global structures. For that, let us consider two arbitrary overlapping open sets . The two corresponding local frames and of the vector bundle are related on the intersection by means of a transition function which is a local rotation: . Adjoining to the symbol of each locally defined quantity its domain of definition we have then by essentially the same argument as in the Remark following the previous theorem . By considering the off-diagonal terms in this relation we can see immediately that we must have , that is, the almost complex structures are local sections — and in fact, frames — of an associated -subbundle of . On the other had, from the diagonal terms we get that , that is, the metric is globally defined.
To prove that the manifold carries a quaternionic Käher structure it suffices to show that the action of the Levi-Civita covariant derivative with respect to the metric on the 2-forms is of the form (3). The corresponding covariant differentiation property (1) of the almost complex structures follows then immediately.
By the same reasoning as in the hyperkähler case, and are of mixed respectively pure type with respect to . So then by parts and of Lemma 1 the same identities (18) and (19) hold now, too. From the first one, by inserting the expressions for and given by the current exterior differential condition we retrieve, after a purely algebraic computation, the following form for the Nijenhuis tensor of :
[TABLE]
Plugging then this into the second identity together with the remaining expression for gives us after another series of algebraic manipulations that
[TABLE]
Permuting the indices cyclically in this argument yields the rest of the components of the desired property (3). ∎
Remark*.*
Quaternionic Käher manifolds are particular examples of quaternionic manifolds (for a detailed review see e.g. [5]). These are manifolds which carry a three-dimensional subbundle of with fibers spanned at each point by a basis whose elements satisfy the algebra of imaginary quaternions, and which, in addition, admit a torsion-free connection preserving this structure. If it exists, such a connection, called an Oproiu connection, is not unique. Their so-called structure tensor takes the form
[TABLE]
where the triplet of 1-forms are the Oproiu connection 1-forms.
From the above formula for the Nijenuis tensor for and its cyclic permutations one can verify that in the quaternionic Kähler case the structure tensor is indeed of this form, with
[TABLE]
(Here we use the convention that the act on vector fields from the left and dually, on 1-forms, from the right.) Note that these can be rewritten as
[TABLE]
which essentially means (see e.g. [9]) that in the quaternionic Kähler case the Oproiu connection is equivalent to the Levi-Civita one.
The following alternative version of the criterion enunciated in Theorem 3 also holds:
Theorem 4**.**
The exterior differential condition of Theorem 3 can be replaced with the following condition:
- •
The principal -bundle associated to the bundle admits a connection with curvature 2-form equal, up to a non-vanishing constant, to . That is, there exist locally defined connection 1-forms and a constant such that
[TABLE]
Proof.
The fact that this condition implies the second condition of Theorem 3 is straightforward: the latter is a Bianchi-type consistency condition for the former. The converse implication follows, for , from a result first shown by Alekseevsky, see e.g. part of Theorem 5.7 in [5]. For , the condition above is equivalent to the Einstein equation. In either case, one can identify
[TABLE]
where is the scalar curvature of the metric . ( is called reduced scalar curvature, and the extra factor with respect to the definition in [5] is a matter of convention stemming from our unusual normalization of the connection 1-forms.) ∎
1.2. Killing vector fields
Next, we want to explore how symmetry conditions, and in particular the Killing condition, translate in the framework in which triplets of 2-forms rather than the metric and the (almost) complex structures play the fundamental role. In order to address this question, we need to recall first some background material culminating with a general theorem due to B. Kostant concerning Killing vector fields on Riemannian manifolds with special holonomy.
1.2.1.
Let be a Riemannian manifold with metric . By definition, we call an endomorphism skew-symmetric if
[TABLE]
for any pair . Skew-symmetric endomorphisms from are in one-to-one correspondence with 2-forms on by way of the metric. On the space of such endomorphisms one can define a natural inner product by
[TABLE]
This is in general non-degenerate and, in Riemannian signature, positive definite. Moreover, a choice of orthonormal basis on the tangent space at a point gives a vector space isometry , with the latter space endowed with the standard Euclidean metric. This determines in turn an isomorphism between and , the Lie algebra of skew-symmetric endomorphisms of .
1.2.2.
For any vector field let us define the operator measuring the difference between the Lie derivative and the Levi-Civita covariant derivative along . Also, for any two vector fields , let be the usual Riemannian curvature operator of the Levi-Civita connection. As derivations acting linearly on any tensor field on and vanishing on functions, both of these operators are linear algebraic rather than differential in nature and are represented by tensors. The tensors are in either case endomorphisms of the tangent bundle . For , this is , the covariant derivative of , which one may view as the endomorphism of defined by for any . For , it is the Riemann curvature endomorphism . So for example, on functions, vector fields and endomorphisms of the tangent bundle we have, respectively,
[TABLE]
where the square brackets denote commutators. What is more, for all , whereas if and only if is a Killing vector field.
In fact, in regard to Killing vector fields we have, moreover, the following properties (see e.g. Proposition 2.6 in ch. VI, sec. 2 of [35]):
Lemma 5**.**
Let be a Riemannian manifold.
If is a Killing vector field on then
[TABLE]
for any vector field .
- 2)
If both and are Killing vector fields on then we have
[TABLE]
Proof.
The relations follow immediately from the definitions upon using the fact that the Lie derivative with respect to a Killing vector field commutes with the Levi-Civita connection. Formally, this can be expressed as follows: if is a Killing vector field, then for any vector field . ∎
Note that in terms of the associated skew-symmetric endomorphisms the operatorial relation (32) can be equivalently written as
[TABLE]
This is sometimes known as Kostant’s formula.
1.2.3.
The isometries of a Riemannian manifold form a Lie group (Myers–Steenrod). On another hand, we have the notion of holonomy group. Given a connected Riemannian manifold , its holonomy group at a point is by definition the set of linear endomorphisms of the tangent space obtained by parallel transporting tangent vectors with respect to the Levi-Civita connection around piecewise smooth loops based at . This set has naturally the structure of a group. The restricted holonomy group at is the subgroup coming from contractible loops. The latter is a connected Lie subgroup of the group of orthogonal transformations of . By choosing an orthonormal basis for we may identify it with a Lie subgroup of . This subgroup depends on both the choice of basis and of base point , but its conjugacy class depends on neither. The possible holonomy groups of simply-connected irreducible Riemannian manifolds have been classified by M. Berger.
The relation between the holonomy group and the (continuous part of the) isometry group of a Riemannian manifold is described by the following result:
Theorem 6** (Kostant [37]).**
Let be a Riemannian manifold and be the Lie algebra of the restricted holonomy group at a point . If is a Killing vector field on , then the skew-symmetric endomorphism of the tangent space determined by belongs to the normalizer of in . Moreover, if is orientable and compact then this belongs simply to .
Proof.
As a skew-symmetric endomorphism of the tangent bundle, can be decomposed uniquely into two orthogonal components with respect to the inner product (30), one belonging to and the other to its orthogonal complement in :
[TABLE]
Suppose then we act with for some vector field simultaneously on both sides of this equation. One the left-hand side we obtain, by way of the property (34), , which, by the Ambrose-Singer theorem, belongs to . On the other hand, since the inner product is preserved by parallel transport, one can argue that and for all . The inner product being non-degenerate, the intersection is trivial, and so it follows that one must have . This means in particular that is invariant at parallel transport around a closed loop and thus belongs to the centralizer of in . The first part of the theorem follows then immediately.
For further details as well as the proof of the fact that when is orientable and compact vanishes we direct the interested reader to the original paper [37], or the review in ch. VI of [35]. ∎
1.2.4.
Let us return now to the particular case of quaternionic Kähler and hyperkähler manifolds, for which . Remark that we have now at our disposal three more skew-symmetric endomorphisms of the tangent bundle, namely , , . Since is isomorphic in either case, for any point , to the fundamental representation of the Lie algebra , it inherits this one’s structure. Thus, it naturally splits into three subspaces isomorphic to the and Lie subalgebras of , and their complement in , respectively. The second subspace is spanned by , , and, moreover, any element from the first subspace commutes with any element from the second. What is more, the three subspaces are mutually orthogonal with respect to the inner product (30).
These algebraic properties can be seen explicitly for instance within the so-called – formalism of [45]. Let us recall briefly its main aspects. For quaternionic Kähler manifolds one can associate in principle a locally defined vector bundle to each representation of the holonomy group . In particular, the complexified cotangent bundle splits locally into a tensor product of two such vector bundles corresponding to the standard complex representations of rank and of and , respectively. For , in four dimensions, where the holonomy definition is side-stepped, one takes instead and to be the two spinor bundles of . The bundles and come equipped with natural symplectic forms — constant anti-symmetric sections and . In the hyperkähler case, which in this context is most profitably viewed as a quaternionic Kähler limit case, the same split structure exists, except that the bundle is now trivial.
Any -module decomposes into direct sums of irreducible -modules, which are tensor products of even total number of and -modules. In particular, this is the case for the bundle . Any 2-form thus decomposes as
[TABLE]
where, in a local – frame, we have
[TABLE]
Here, and denote symmetric respectively anti-symmetric tensor products, and denotes the kernel of the contraction of with the inverse of the symplectic form (this kernel is trivial for ). Note that the quaternionic Kähler/hyperkähler 2-forms are of pure type, and they provide a frame for the respective subbundle of .
On another hand, in the same – frame the metric factorizes locally as . Recalling then that any skew-symmetric endomorphism of is related to a corresponding 2-form by , the statements above, and in particular the orthogonality claim, can be verified directly through simple algebraic calculations.
1.2.5.
The curvature tensor also admits an irreducible – decomposition [45, Theorem 3.1]. In what follows, though, we will only need the fact that the curvature endomorphism of a quaternionic Kähler manifold satisfies the commutation property
[TABLE]
For hyperkähler manifolds the right-hand side vanishes, consistent with taking the limit to . In the quaternionic Kähler case this is essentially the consistency condition for the differential equation (1), with the Einstein condition (27) taken into account. In the hyperkähler case it is simply the consistency condition for the corresponding equation (2). In either case its structure can be easily understood by purely algebraic arguments from the Ambrose–Singer theorem.
Note in particular that if we multiply this equality from either side with an and then take the trace of the result we obtain — using successively the cyclicity property of the trace, the quaternionic algebra, and the fact that the trace of any element from vanishes — the following consequence:
[TABLE]
1.2.6.
For the remainder of this section we will assume that the manifolds are irreducible, by which we mean that their holonomy groups are assumed to be equal to rather than included in in the quaternionic Kähler case (for ), and in the hyperkähler one.
If the quaternionic Kähler or hyperkähler manifold possesses a Killing vector field , then by Theorem 6 its covariant derivative belongs to the normalizer of the Lie subalgebra respectively in , which one can easily see is in both cases . With the notations above, this means that , and so
[TABLE]
with the two remaining components both orthogonal and commuting. In four dimensions this decomposition is automatic for any skew-symmetric endomorphism of , so in this case one need not resort to holonomy arguments. The second component, in particular, is a linear superposition
[TABLE]
with coefficients , where the minus sign is conventionally chosen for later convenience. The generators satisfy the orthogonality relations and , and thus we may identify
[TABLE]
1.2.7.
These coefficients play an important role in the description of Killing symmetries. In the quaternionic Kähler case, switching to an -vector notation, we have:
Theorem 7** (Galicki, Lawson [25, 26]).**
*Consider a quaternionic Kähler manifold with associated principal -bundle , and assume that has a continuous group of isometries , with Lie algebra . Then a vector field , that is is Killing, if and only if there exists a local section of the tensor product bundle over such that 222 The symbol denotes the usual -vector cross product. *
[TABLE]
or, equivalently, such that
[TABLE]
Explicitly, this section is given by the formula for any and it is
nowhere vanishing
the unique solution to the quaternionic-Kähler moment map equation (44)
-equivariant
and, for any , we have
[TABLE]
Proof.
Suppose is a Killing vector field on . Then by the rules laid out in the list (31) we have
[TABLE]
The component of drops out because it commutes with the generators, and for the remaining component we use the representation (41). Converting then the almost complex structures into 2-forms by means of the metric, which is preserved by the action of , gives us promptly the rotation property (43).
Conversely, let us assume that we have a vector field whose Lie action rotates the quaternionic Kähler 2-forms, that is for some locally defined triplet of functions . Using then any one of the expressions (16) of the metric in terms of the quaternionic Kähler 2-forms one can easily check that this implies that is Killing, regardless of the particular form of the functions .
Suppose now again that is Killing. If we act with for some arbitrary on the expression (42) for we obtain, by applying the Leibniz rule and then the properties (1), (34) and (39), that . The quaternionic Kähler moment map equation (44) is simply a rewriting of this equation in the language of differential forms.
Conversely, let us assume that there exists a vector field satisfying the quaternionic Kähler moment map equation for some locally defined functions . Then by acting on this equation with an exterior differential and making use of the Einstein property (27) one can show that the rotation property (43) is implied. By the argument above, this implies further that must be Killing.
The non-vanishing of the section follows directly from the quaternionic Kähler moment map equation. Should it vanish at some point, the equation would imply that would be a null vector for the 2-forms at that point, which would contradict their non-degeneracy.
For an argument that the solution to the quaternionic Kähler moment map equation is unique we refer the reader to reference [26].
Remark*.*
Note that, unlike the rest of the arguments in the proof, each of the arguments in the last three paragraphs fails in the limit when and thus cannot be extended to the hyperkähler case.
Assume now that we have two Killing vector fields . Using the representation (33) for the curvature operator, and then the formulas (46) and (38), we can write
[TABLE]
with defined as in the theorem.
Finally, since acts on scalars as , from the quaternionic Kähler moment map equation it follows that , showing that the sections transform indeed equivariantly under the infinitesimal action of . ∎
1.2.8.
In the hyperkähler limit the coefficients lose the moment map interpretation and become constant.
Theorem 8**.**
Let be a hyperkähler manifold having a continuous group of isometries , with Lie algebra . Then a vector field , that is is Killing, if and only if there exists a constant map such that
[TABLE]
Explicitly, the map is given by the formula and, for any , we have
[TABLE]
Moreover, if is compact, vanishes entirely.
Proof.
The proof in the quaternionic Kähler case carries over in the limit when and vanish, apart from the three instances identified in the intermediary Remark. In particular, equation (44) now reads simply , implying that the functions are constant in hyperkähler setting (note also that rather than being local sections, they are now globally defined). This equation no longer has the form of a moment map condition, nor does it imply anymore the rotation property. As for the remaining equations, (43) and (45), they take in this limit, with less dramatic consequences, the forms (48) and (49), respectively. Finally, the last statement of Theorem 6 implies that, when is compact, for any . ∎
Killing vector fields for which vanishes are known as tri-Hamiltonian (or tri-holomorphic), and the ones for which this is not the case as rotational (or permuting). (Note that the two notions do not depend on the particular frame chosen in the space of hyperkähler symplectic 2-forms.) So the theorem states that Killing vector fields on hyperkähler manifolds can only be either of tri-Hamiltonian or of rotational type. This generalizes a result proved by other means in four dimensions by Boyer and Finley [11]. In particular, when the manifold is compact, Killing vector fields, should they exist, are necessarily tri-Hamiltonian.
1.3. Homothetic Euler vector fields
We end this section with an examination of a different type of symmetry, conformal in nature rather than Killing, which plays a central role in Swann’s celebrated quaternionic bundle construction, reviewed in the next section.
Consider a Riemannian or pseudo-Riemannian manifold with Levi-Civita connection . We call a vector field on homothetic Euler if
[TABLE]
for any vector field . In the notation of subsection 1.2 this is equivalent to requiring that be equal to the identity endomorphism of . The terminology is justified by the fact that a manifold possessing such a vector field is automatically a metric cone [28], in which context the vector field is known as an Euler vector field.
A vector field is homothetic Euler if and only if it is homothetic and hypersurface-orthogonal. In fact, we have the following list of equivalent characterizations:
Proposition 9**.**
Let be a Riemannian or pseudo-Riemannian manifold with metric and corresponding Levi-Civita connection . Then the following conditions are equivalent:
- a)
* possesses a homothetic Euler vector field .*
- b)
There exists a function on such that .
- c)
There exists a function on such that its gradient vector field with respect to the metric, , satisfies the homothetic property .
If is assumed in addition to be hyperkähler, then we can add to this list the conditions
- d)
There exists a function on which is simultaneously a Kähler potential for every hyperkähler Kähler 2-form with respect to its corresponding complex structure, i.e.
[TABLE]
- e)
There exist four vector fields and a function on such that
[TABLE]
Remark*.*
A hyperkähler manifold with this structure is known as a hyperkähler cone. Due to the property the function is called in this context a hyperkähler potential. Homothetic Euler vector fields were first introduced in a physical setting in relation to theories of hypermultiplets in four spacetime dimensions invariant under superconformal symmetries by de Wit, Kleijn and Vandoren in [19] (see also [20]). The characterizations and precede this definition and are due to Swann [46]. Through the property , involving exterior algebra conditions on the hyperkähler symplectic 2-forms, we add to this collection of characterizations one which is closest in spirit to the approach that we have developed in these notes.
Proof.
We begin with the case when is simply Riemannian or pseudo-Riemannian, with metric and Levi-Civita connection . Let us recall, first, that for any we have the identity . And, second, that for any function , its Hessian tensor is given by for any , where represents now the gradient vector field of with respect to the metric; this formula holds in fact for any metric-preserving affine connection.
Assuming the condition holds, define the function
[TABLE]
Then, for any vector field we have , which is to say, is the gradient of with respect to the metric. The first and second identities above imply then immediately the conditions and , respectively.
Let in this case be, by definition, the gradient vector field of the function with respect to the metric. The implication follows then promptly from the Hessian formula above.
The homothetic condition implies on one hand, by way of the first Levi-Civita identity above, that for any . On another hand, from the symmetry of the Hessian tensor it follows, by way of the second identity, that . One gets then that for any , which then by the fact that the metric is non-degenerate implies the condition .
Let us assume now that the manifold is, moreover, hyperkähler.
We have in this case , for any . Since the hyperkähler complex structures are preserved individually by the Levi-Civita connection we can replace the last term (ignoring the sign) with . In differential form language, the result can be written rather more nicely in the form . As the Levi-Civita connection is torsion-free, which translates into an index symmetry for the Christoffel symbols, we can eventually replace in this expression with the usual exterior derivative, .
Given the potential function on , define the vector fields as the gradients of with respect to the metric respectively the three hyperkähler symplectic forms, as follows:
[TABLE]
By using the quaternionic properties of the hyperkähler complex structures we get then that . Cyclically permuting the indices in this last argument yields a set of formulas which, together with the definitions of the four vector fields, can be assembled in the form of the first condition (52). To obtain the second condition (52), on the other hand, note that we have . Then .
Consider, say, the following off-diagonal component of the first condition (52): . From this it follows that \smash{X_{1}=\iota_{X}\omega_{3}\hskip 1.0pt\omega_{\mathrlap{2}}{}^{\raisebox{0.3pt}{\scriptstyle-1}}=\omega_{\mathrlap{2}}{}^{\raisebox{0.3pt}{\scriptstyle-1}}\omega_{3}X=-I_{1}X}, where in the last step we made use of the second expression for in (15). Cyclically permuting the indices in this argument gives us
[TABLE]
Substituting this back into the condition we get then from the diagonal components that . On another hand, the second condition (52) implies immediately by way of any one of the expressions (16) for the metric that . This concludes the proof of the chain of equivalences. ∎
Hyperkähler cones are naturally foliated by four-dimensional leaves. More precisely, we have:
Proposition 10**.**
Let be a hyperkähler cone. Then the four vector fields at point of Proposition 9 generate the algebra
[TABLE]
with homothetic Euler and Killing vector fields of rotating type.
Proof.
Acting with an exterior derivative on the first condition (52) and using the closure of the to pass to Lie derivatives by way of Cartan’s homotopy formula we get that
[TABLE]
This means that each one of the vector fields generates a rotational action on and is therefore Killing. We then have
[TABLE]
Since the are non-degenerate, this is equivalent to the first algebraic condition (56). The remaining condition (56) can be derived by a similar argument. ∎
2. Swann bundles
In [46] Swann introduced a powerful tool for the study of quaternionic Kähler manifolds by showing that over each such manifold one can construct a quaternionic bundle whose total space carries a hyperkähler cone structure encoding the quaternionic Kähler geometry of the base. This allows one to describe the geometry of manifolds which in general have no integrable complex structures, apart perhaps from accidental ones inessential to their quaternionic Kähler structure, in terms of the hyperkähler geometry of a space of four real dimensions higher possessing a wealth of integrable complex structures. Before we proceed to review the details of Swann’s construction it is useful to recall a few facts about (a quotient of) the multiplicative group of non-zero quaternions, , which plays a key role in this story.
2.1. Two structures on the group
2.1.1.
Consider an embedding of the group of unitary quaternions into . Clearly, every non-zero quaternion can be uniquely represented as a positive real number, its absolute value, times a unit quaternion; this gives a natural isomorphism
[TABLE]
Thus, has the structure of a Lie group, with Lie algebra generated by the standard quaternionic basis , , , . In what follows we will denote the elements of this basis uniformly with , , , , respectively — or, simply, . The indices will be assumed to run from 0 to 3 while the indices to run, as until now, from 1 to 3, and the summation convention over repeated indices will continue to be implied.
The adjoint representation of is four-dimensional and defined by , for any . It satisfies the properties
[TABLE]
The first one follows immediately from the definition. Taking in it in particular and , one can see that the adjoint representation descends to a representation of the rotation group , whose double cover is . This is reducible: one can write it in block-diagonal form as the direct sum of the one-dimensional trivial representation and the three-dimensional irreducible representation of . On the other hand, by the associativity of quaternionic multiplication and the multiplicative property of the norm, , which is to say, adjoint transformations preserve the quaternionic norm. They preserve hence also the associated inner product, and the second relation is the expression of this orthogonality property.
The Lie algebra (*i.e.* quaternion)-valued left and right-invariant Cartan–Maurer 1-forms of are given by
[TABLE]
The left and right-invariant Cartan–Maurer 1-forms are related to each other by an adjoint transformation up to a minus sign: \smash{\sigma_{\mathrlap{a}}{}^{L}=-\hskip 0.5ptR_{ab}(q^{-1})\hskip 0.5pt\sigma_{\mathrlap{b}}{}^{\raisebox{0.3pt}{\scriptstyle R}}}. Note that in particular we have
[TABLE]
Dual to the Cartan–Maurer 1-forms one has the left and right-invariant vector fields of the Lie group . The duality requirement with respect to the corresponding set of invariant 1-forms determines them completely. They form two commuting copies of the Lie algebra , i.e.
[TABLE]
which one may think of as the dual Cartan–Maurer equations, and the two sets of vector fields are related by \smash{\ell_{\mathrlap{a}}{}^{L}=-\hskip 0.5ptR_{ab}(q^{-1})\hskip 1.0pt\ell_{\mathrlap{b}}{}^{\raisebox{0.3pt}{\scriptstyle R}}}. In particular, we have
[TABLE]
where , , , are the components of the quaternion .
2.1.2.
In addition to the Lie group structure, carries also two hypercomplex structures. That is, its tangent bundle is equipped with two natural actions of the algebra of imaginary quaternions, with the three generators defining integrable complex structures. Viewing complex structures in their dual guise as endomorphisms of the cotangent bundle, and in accordance with our convention that complex structures act from the right on 1-forms, these actions are induced by right and left quaternionic multiplication on a quaternionic coframe of as follows:
[TABLE]
They are respectively left and right-invariant, and the two sets of hypercomplex generators commute with each other.
Let for any be the inner product induced on the space of quaternions by the quaternionic norm. Observe that in the standard quaternionic basis this takes the Euclidean form and, moreover, that in terms of it the alternating quaternionic products of any two quaternions admit the representations
[TABLE]
where the overhead bar indicates the operation of quaternionic conjugation. One can then verify that in the alternative coframes provided by the Cartan–Maurer 1-forms and their respective dual vector frames the generators of the two hypercomplex structures may be expressed as
[TABLE]
By resorting to the orthogonality property of the adjoint representation and the left-right transition formulas one can also recast these in the form
[TABLE]
2.1.3.
Finally, note that both the Lie group structure and the two hypercomplex structures of considered above descend on the quotient of with respect to the -action generated by . In this case we have
[TABLE]
This is the conformal special orthogonal group in three dimensions.
2.2. The Swann bundle of a quaternion Kähler manifold
2.2.1.
Let be a quaternionic Kähler manifold, with a choice of local frame , , for its -bundle of 2-forms and a principal connection with local connection 1-forms , , on the associated principal bundle . For the considerations which follow it will be convenient to regard these two triplets as the components of two imaginary quaternion-valued differential forms, and . In this quaternionic formalism then the Einstein condition (27) of Theorem 4 can be cast in the form
[TABLE]
where the wedge symbol denotes here the natural extension of the operation of exterior product to quaternion-valued differential forms.
By definition, the Swann bundle over the quaternion Kähler manifold is the associated bundle
[TABLE]
On its total space one assembles the following -valued 2-form:
[TABLE]
Although this is made up of local quantities, it is globally defined. Indeed, consider an open covering of and two overlapping open sets . On the intersection the corresponding local frames of are related by an transition function, that is, , with being either one of the two opposite-sign unit quaternions which determine the rotation matrix through the double cover map (recall that ). In quaternionic notation this transition relation reads \omega_{V}=u_{VU}\hskip 1.0pt\hskip 0.5pt\omega_{U}\hskip 0.5ptu_{VU}{}^{\mathllap{\raisebox{1.0pt}{\scriptstyle-1}\hskip 1.5pt}}. On another hand, the transition relation for the connection 1-form is \theta_{V}=u_{VU}\hskip 1.0pt\theta_{U}u_{VU}{}^{\mathllap{\raisebox{1.0pt}{\scriptstyle-1}\hskip 1.5pt}}+u_{VU}\hskip 0.5ptdu_{VU}{}^{\mathllap{\raisebox{1.0pt}{\scriptstyle-1}\hskip 1.5pt}}. Thus, if the quaternionic fiber coordinates patch up according either to the rule or to , then we have , which then by the arbitrariness in the choice of and implies that is globally defined on . Note also that in either case , so the function is globally defined on as well.
In fact, there are more globally defined quantities one can consider. To see that, let \smash{\sigma_{aU}{}^{\raisebox{1.0pt}{\scriptstyle L}}\hskip 4.0pt} and \smash{\sigma_{aU}{}^{\raisebox{1.0pt}{\scriptstyle R}}\hskip 4.0pt} for any be the pull-backs by the local trivialization map of the left and right-invariant Cartan–Maurer 1-forms from to the fibers above — and then let \smash{\ell_{aU}{}^{\mathllap{\raisebox{1.0pt}{\scriptstyle L}\hskip 5.5pt}}} and \smash{\ell_{aU}{}^{\mathllap{\raisebox{1.0pt}{\scriptstyle R}\hskip 5.2pt}}} be their respective dual vector fields. Using the above patching rules for the quaternionic fiber coordinates one can then check that on any intersection we have \ell_{aV}{}^{\mathllap{\raisebox{1.0pt}{\scriptstyle L}\hskip 5.5pt}}=\ell_{aU}{}^{\mathllap{\raisebox{1.0pt}{\scriptstyle L}\hskip 5.5pt}} and \ell_{aV}{}^{\mathllap{\raisebox{1.0pt}{\scriptstyle R}\hskip 5.5pt}}=R_{ab}(u_{VU})\hskip 0.5pt\ell_{b\hskip 0.5ptU}{}^{\mathllap{\raisebox{1.0pt}{\scriptstyle R}\hskip 4.7pt}}. Thus, in particular, the left-invariant vertical vector fields are also globally defined, as one would expect to be the case for a bundle associated to a right principal bundle.
Let , , be the real components of . The rationale behind the above definition is provided by the following result:
Theorem 11** (Swann [46]).**
Let be a quaternionic Kähler manifold. The bundle endowed with the 2-forms , , constructed as above is a hyperkähler cone with homothetic Euler vector field and hyperkähler potential .
Proof.
The fact that the three 2-forms determine a hyperkähler structure on can be seen fairly quickly by checking that they satisfy the conditions of Theorem 4. Thus, one one hand, the verification of the differential condition of the theorem is considerably facilitated if one observes that Swann’s formula (73) can be equivalently rearranged in the form
[TABLE]
By virtue of the Einstein condition (71) is then clearly locally exact and hence closed. On the other hand, for the algebraic condition, note that the same formula (73) can also be equivalently rewritten as
[TABLE]
or, in components, with notations introduced in the previous subsection,
[TABLE]
where it is understood that . The last expression, in particular, displays the as block-diagonally split along the horizontal and vertical subspaces determined locally by the fiber structure. So then, in order to show that they satisfy the algebraic condition of Theorem 4, it suffices to verify that their horizontal and vertical components satisfy it separately. (It is worth noting also that the twisting prefactor can can be safely ignored since the algebraic condition is preserved by simultaneous rescalings and rotations, see the Remark following Theorem 2.) This is indeed ensured, on the horizontal side, by the fact that the quaternionic Kähler 2-forms satisfy themselves the algebraic condition, and on the vertical side, by the fact that the map gives a four-dimensional real matrix representation of the standard imaginary quaternions.
Regarding the cone structure, note that Swann bundles possess a natural - action induced by right quaternionic multiplication in the fiber: taking with constant results in and . The -quotient reflects the invariance of these transformations under . The infinitesimal generators of this action are the left-invariant vertical vector fields, and we have
[TABLE]
which then by part of Proposition 9 means that the bundle has a hyperkähler cone structure with hyperkähler potential and homothetic Euler vector field . ∎
Observe moreover, in line with our stance of regarding the metric as a composite object, that from the formulas (73) and (75) for the hyperkähler symplectic forms one can work out for the hyperkähler metric on the expressions
[TABLE]
respectively. From either one of these it is clear that if the signature of the quaternionic Kähler metric is , then the signature of Swann’s hyperkähler metric is either if , or if .
Likewise, one can also work out the corresponding hyperkähler complex structures on to obtain
[TABLE]
where the \smash{\mathscr{I}_{\mathrlap{i}}{}^{\hskip 1.0pt\raisebox{0.4pt}{\scriptstyle H}}} represent the horizontal lifts to of the almost complex structures associated to the triplet of 2-forms on . (One-forms on can be lifted to by pullback, while the horizontal lift of a vector field on to is given by \smash{\mathscr{X}^{H}=\mathscr{X}+(\iota_{\mathscr{X}}\theta_{i})\hskip 0.5pt\ell_{\mathrlap{i}}{}^{\raisebox{0.3pt}{\scriptstyle R}}}; elements of such as the are then lifted in accordance with the tensor product rule.) This shows that the complex structures , too, split block-diagonally along the local fibration structure into horizontal and vertical components, with the first ones induced by the quaternionic structure on the quaternionic Käher base and twisted by a fiber coordinate-dependent rotation, and the second ones induced by the left-invariant hypercomplex structure on the fiber (see in particular the first formula (69)). This is reminiscent of the way in which the integrable complex structure on the twistor space of a hyperkähler manifold was constructed in [32].
The previous theorem admits the following converse, which we give here without proof and refer the reader to Theorem 5.9 in [46]:
Theorem 12** (Swann [46]).**
Any hyperkähler cone with positive hyperkähler potential, whose four canonical vector fields generate a locally free -action is locally homothetic to a Swann bundle.
Note that when the metric of a hyperkähler cone is positive definite, the formula (53) makes it clear that its hyperkähler potential can always be chosen to be strictly positive. In indefinite signature, on the other hand, the hyperkähler potential can always be made at least locally positive, if it is not already so, by flipping the signature of the metric.
2.2.2.
Let us consider now how Killing symmetries fit into this construction. The main result in this regard is the following:
Theorem 13**.**
Let be a quaternionic Kähler manifold with Swann bundle . Then any Killing vector field on can be lifted to a tri-Hamiltonian vector field on which commutes with the - action, and conversely, any tri-Hamiltonian vector field on which commutes with the - action descends to a Killing vector field on . If is the quaternionic Kähler moment map of , we have
[TABLE]
Moreover, there exists a choice of hyperkähler moment map for such that the two moment maps are related by
[TABLE]
Both the descent and the lift maps preserve the Lie bracket.
Remark*.*
In equation (82) we have deliberately blurred the line between -vectors and imaginary quaternions in view of the isomorphism . That is, despite employing the traditional -vector notation, we clearly regard the two moment maps as -valued functions. In what follows we will institutionalize this practice and often resort to this notational ambivalence when the context allows for an unequivocal interpretation.
Proof.
In what follows we denote, for conciseness, the -valued vertical 1-form by , with and designating its real and quaternionic imaginary components, respectively. Let us begin then by stating the following lemma: for any vector field on preserving the hyperkähler potential — that is, such that , or equivalently, such that — we have
[TABLE]
This follows by writing the formula (75) in the form and using the identity
[TABLE]
valid for any triplet of functions .
Assume now that is some Killing vector field on . Let be its quaternionic Kähler moment map and define on the vector field to be the sum of the horizontal lift of to and a vertical component as in equation (81). Observe then on one hand that , thus defined, commutes automatically with the generators of the -action on . On another hand we have and , and so the formula (83) applies and gives us immediately
[TABLE]
The first term on the right-hand side vanishes by the Galicki–Lawson moment map equation for , leaving behind a hyperkähler moment map equation for , which is thus also tri-Hamiltonian, with hyperkähler moment map (82).
Conversely, let us consider on a tri-Hamiltonian vector field commuting with the canonical - action. Like any vector field on , locally this can be decomposed with respect to the Swann fibration structure into horizontal and vertical components, , with the horizontal lift of a vector and the coefficients locally defined functions on . The requirement that commute with the generators of the - action forces the coefficients to be independent of the fiber coordinates. Moreover, the condition that be tri-Hamiltonian implies that it must preserve the hyperkähler potential up to a constant additive shift (for this, note that tri-Hamiltonian vector fields are automatically tri-holomorphic; the result then follows by applying Lemma 20 for each hyperkähler complex structure). Since , to avoid contradiction we must have . If we rename the non-vanishing coefficients , then is formally of the form (81). The formula (83) gives us again just as above the equation (85), from which we now conclude that
[TABLE]
By assumption this must vanish, which implies that must satisfy the Galicki–Lawson equation and is thus a quaternionic Kähler moment map for . By Theorem 7, is then necessarily a Killing vector field.
Finally, given two pairs of vector fields and each of which satisfy the conditions of the theorem, one can verify explicitly that \smash{[X,Y]=[\mathscr{X},\mathscr{Y}]^{H}+\nu_{[\mathscr{X},\mathscr{Y}]}{}_{i}\hskip 1.0pt\ell_{\mathrlap{i}}{}^{\raisebox{0.3pt}{\scriptstyle R}}}. This proves the last statement of the theorem. ∎
Remark*.*
Similarly to the hyperkähler quotient construction [32], the Galicki–Lawson generalized moment map can be used to define a quaternionic Kähler quotient construction by which a quaternionic Kähler space with continuous isometries is reduced to another quaternionic Kähler space in which the isometries are divided out [25, 26]. These quotient constructions are compatible with Swann bundles in the sense that the associated Swann bundle of the quaternionic Kähler quotient of by an isometric Lie group action is isomorphic to the hyperkähler quotient at zero moment map level of the Swann bundle of by the corresponding lifted action [46, Theorem 4.6].
3. Analogue of the extended Gibbons–Hawking Ansatz for quaternionic Kähler metrics
3.1. The extended Gibbons–Hawking Ansatz
3.1.1.
The Gibbons–Hawking Ansatz [27] allows one to construct four-dimensional hyperkähler metrics with a tri-Hamiltonian -action out of the solutions of a set of partial differential monopole equations defined on an open subset of . This construction was extended by Pedersen and Poon in [41] (see also [32]) to -dimensional hyperkähler metrics with a local tri-Hamiltonian -action, and in what follows we will refer to this generalization as the extended Gibbons–Hawking Ansatz.
Consider a principal -bundle over an open subset equipped with a principal connection with Lie algebra-valued curvature 2-form , where denotes an index set of cardinality , and a section of the symmetric second tensor power of the associated adjoint bundle, the Higgs field, such that . Regarding as the space of configurations of distinguishable points in , one may coordinatize it globally with -vectors . Let denote the corresponding coordinate frame and assume that the following system of partial differential equations generalizing the abelian Bogomolny monopole equation in holds on :
[TABLE]
The action of the linear Hodge-like star operators is defined by , and summation over the repeated index is understood. Consistency with the Bianchi identity for the curvature 2-form requires that the components of the Higgs field satisfy a set of second-order differential constraints generalizing the Laplace equation in .
To a solution of these equations one then associates on the total space of the -bundle the triplet of 2-forms
[TABLE]
expressed here in a local trivialization, with coordinates on the fibers and a local connection 1-form on the base with curvature . The two extended Bogomolny equations guarantee then that , thus defined, is closed. What is more (see an argument below), its components also satisfy the algebraic condition of Theorem 2. Therefore, by this theorem, they define on a hyperkähler structure, with hyperkähler metric
[TABLE]
where denotes the matrix inverse of . The vertical vector fields generate a tri-Hamiltonian -action with hyperkähler moment maps . It is a signature characteristic of the Gibbons–Hawking setup and its higher-dimensional extensions that the rank of the action is just large enough for the group orbit parameters together with the moment map functions to provide a complete parametrization of the space.
3.1.2.
Conversely, any -dimensional hyperkähler manifold with a free local tri-Hamiltonian -action arises locally in this way. This follows immediately from Proposition 2.1 in [10].
3.1.3.
These formulas admit an equivalent formulation which brings to the fore their quaternionic rather than their -bundle structure. Consider the local coframe of the cotangent bundle of given by the real components of the quaternionic-valued 1-forms
[TABLE]
where in accordance with our established practice we regard -vectors as imaginary quaternions. Assuming the same natural definitions as before for the exterior and tensor products of quaternion-valued forms, we can then recast the above formulas for the triplet of 2-forms and metric in the following compact form:
[TABLE]
Here we view again as an imaginary quaternion-valued 2-form. Note by the way that this formula for yields immediately, with the notations from § 2.1.2, the further expression
[TABLE]
In this coframe the algebraic condition of Theorem 2 is straightforward to check if one recalls that the map gives a four-dimensional representation of the standard basis of imaginary quaternions.
The second extended Bogomolny equation can be replaced in this framework by the differential condition
[TABLE]
Another, more interesting, reformulation of the extended Bogomolny equations emerges if one considers instead the dual frame to the above quaternionic coframe of . This is given explicitly by the vertical vector fields together with the horizontal lifts to of the coordinate frame on the base (the horizontal lift map is for any ). Then the two extended Bogomolny equations (87) are respectively equivalent to the following two sets of commutator relations
[TABLE]
with a structure reminiscent of that of the self-dual Yang-Mills (SDYM) equations. For it is indeed well known that the Bogomolny equations emerge via reduction of the SDYM equations from four to three dimensions, and this formulation makes that origin transparent.
3.2. Extended Gibbons–Hawking spaces with a hyperkähler cone structure
3.2.1.
Let us investigate now under what conditions an extended Gibbons–Hawking hyperkähler space possesses a hyperkähler cone structure compatible with its -bundle structure. The first thing to observe in this regard is that on the base of any extended Gibbons–Hawking fibration one has a natural action of the group , namely the one generated by the vector fields
[TABLE]
satisfying the algebra
[TABLE]
Remark*.*
The structure constants here differ from the ones we have encountered so far, such as the ones in equations (56) or (64), by a factor of 2. This discrepancy can be removed by a simple rescaling of the definitions of the generators and is a matter of convention and convenience rather than of any profound significance. In practice, one needs to be alert to this normalization issue when using here the results of section 1.3 and scale accordingly.
In the three-dimensional configuration-of-points picture for , and are the generators of rigid rotations respectively simultaneous scalings of configurations. Borrowing a term from the classical mechanics of rigid bodies we will refer to such transformations as collective transformations.
Inserting the generating vector fields of collective transformations into the second Bogomolny equation (87) and then using the first one to manipulate the result, we get that
[TABLE]
where the index runs as usual from 0 to 3, \smash{x_{\mathrlap{a}}{}^{\raisebox{0.3pt}{\scriptstyle{\scriptscriptstyle J}}}} are the components of the imaginary quaternions associated to the -vectors (with, therefore, ), and denote the components of the 1-forms
[TABLE]
Note, incidentally — and the relevance of this observation will soon become apparent — that by resorting further to the first Bogomolny equation we can show that
[TABLE]
and then that
[TABLE]
Formula (97) suggests that, when approaching the question of whether the above -action on the base can be lifted to a full hyperkähler cone structure on , we should consider not just the horizontal lifts of the generators but rather the lifts
[TABLE]
containing additional vertical components. Using nothing but the definitions of the quantities involved and no field equations — yet, one can then show by a rather long but otherwise straightforward calculation that these vector fields satisfy the identities
[TABLE]
Note in particular that any other choice of lifting prescription would spoil the very convenient and simple tensorial structure of the right-hand side of the first equality. By deploying now in the first identity the relation (99) and in the second one the component of the relation (97), both consequences of the field equations, we can eventually cast them in the form
[TABLE]
with the same triplets of 1-forms that we have encountered above. These two formulas should be understood as expressing the obstructions to the criterion of Proposition 9 (with the normalization adjustments stipulated in the Remark above) being satisfied by the lifts of the collective transformations generators . We stress that all the equations that we have written down so far hold in full generality for any extended Gibbons–Hawking space.
We are now ready to prove the following criterion:
Proposition 14**.**
The collective -action on the base of an extended Gibbons–Hawking hyperkähler space can be lifted to a hyperkähler cone structure on if and only if the Higgs field satisfies the linear differential constraints
[TABLE]
that is, if and only if it is invariant at rigid rotations and homogeneous of degree in the variables. In this case the hyperkähler potential is given by
[TABLE]
Proof.
By the criterion criterion of Proposition 9, the lifts generate a hyperkähler cone structure on if and only if there exists a function such that
[TABLE]
When the constraints (104) hold, the 1-forms vanish, and then the formulas (103) make it readily clear that this is indeed the case, with defined as above. This proves the converse implication.
For the direct implication, note that if the vector fields do generate a hyperkähler cone structure then the coefficient of on the right-hand side of the first equation (103) must be exact, and therefore closed, and the last term on the right-hand side of the second equation (103) must vanish. The first condition entails that must be closed, which then by the formula (100) requires that must vanish. In view of the definition of , this immediately implies the first constraint (104). Together with this, the second condition implies then right away the remaining constraint (104), which concludes the proof. ∎
When the constraints (104) are satisfied, , and from the relation (97) we have
[TABLE]
That is, under the Lie action of the generators of the hyperkähler cone structure the 1-forms shift by exact terms. But as connection 1-forms these are defined only up to exact terms, so in principle the shifts can be absorbed into their definitions. Equivalently put, one can always choose representative elements of their gauge equivalence classes which satisfy the gauge-fixing conditions
[TABLE]
and are thus fully invariant under the action of the generators. Note that these conditions do not fix the representative elements completely but only up to exact differentials of rotation and scaling-invariant functions. In this gauge we have simply , that is, the generators of the hyperkähler cone structure coincide with the generators of the collective -action on the base. From now on we will assume this to always be the case.
3.2.2.
Remarkably, the conditions of Proposition 14 entail that the corresponding hyperkähler metrics are determined by a single function, which is not so surprisingly the hyperkähler potential. In particular, the differential conditions on the Higgs field can be exchanged in this case with a dual set of differential conditions on the hyperkähler potential.
Proposition 15**.**
Consider the following two assumptions holding on a open subset of :
- a)
There exists an symmetric matrix-valued function on whose elements satisfy
the first-order differential equations
the symmetry constraints and . 2. b)
There exists a function on satisfying
the second-order differential equations
the symmetry constraints and .
Then the two assumptions are equivalent, with the direct and reverse implication maps given respectively by
[TABLE]
Proof.
The proof is an exercise in three-dimensional differential vector calculus. Consider first the implication . Acting with the operatorial identity
[TABLE]
on and summing over the indices , using the first-order differential equations to permute indices and then the symmetry constraints to simplify or eliminate terms, we obtain that
[TABLE]
with the first equation (109) taken as the definition of . This relation implies immediately the symmetry constraints on , and by a more circuitous route involving similar manipulations as above, the second equation (109) and the second-order differential constraints on .
The reverse implication follows rather more straightforwardly by acting on the function with the operatorial identities
[TABLE]
and taking the second equation (109) as the definition of . These yield respectively the first-order differential equation for and the first equation (109). The remaining symmetry constraints on are fairly easy then to verify directly. ∎
3.2.3.
Theorem 12 tells us that, assuming that the -action is locally free, such a space as defined by the conditions of Proposition 14 is, modulo a possible sign redefinition of the metric, at least locally homothetic to a Swann bundle. Our main objective in what follows is to determine explicitly, in intrinsic terms, the quaternionic Kähler metric on the base of this Swann bundle, including the differential constraints on its building blocks induced by the Bogomolny equations. Notice that the base has quaternionic dimension one less than the quaternionic dimension of , and that, on another hand, by Theorem 13 the -action on descends to it with no alteration in rank. Thus, if we take , then the quaternionic Kähler metric we end up with will be a -dimensional one with a locally free isometric -action.
It is natural to assume in these circumstances that the subset of is closed under the action generated by and . On , the corresponding induced action can be represented by means of quaternions as follows:
[TABLE]
The invariance of the coordinates is predicated on the gauge-fixing conditions (108) being enforced. In accordance with the above choice for the dimension the index will be assumed to run from 0 to .
Simply formulated, our strategy will be to match the symmetry-centered extended Gibbons–Hawking description of against the quaternionic structure-centered Swann bundle one, and to extract in the process the quaternionic Kähler metric. For this, these transformation rules will be key. A nice geometrical interpretation is afforded if we introduce a certain auxiliary space which will play for the quaternionic Kähler space a role similar to the one that plays in the extended Gibbons–Hawking hyperkähler case. This and some other useful related concepts will be defined first in the next subsection, before we embark in a more head-on derivation of what in effect will be a quaternionic Kähler analogue of the extended Gibbons–Hawking Ansatz.
3.3. The space
3.3.1.
As we have remarked before, a point in can be viewed as a configuration of distinguishable points in with position vectors . On this space one can define a natural action of the group generated by the vector fields and defined above, which acts on configurations either by simultaneously rescaling all their position vectors by a common factor or by rigidly rotating them around the origin of . Quaternions allow us to express these two actions together concisely as
[TABLE]
We shall call a configuration degenerate if its stabilizer under the -action is non-trivial. The set of degenerate configurations comprises the null configuration (that is, the configuration will all vectors vanishing), which is a fixed point for the action and is thus stabilized by the whole group, and all the configurations for which all non-vanishing position vectors are collinear (we include in this category also the configurations with a single non-vanishing position vector), which are stabilized by an subgroup of the subgroup of . On the set of non-degenerate configurations, the complement of in , the -action acts freely. In analogy with the classical projective spaces we define the quotient space
[TABLE]
This has .
As in the case of projective spaces, on it we can define open atlases consisting of a finite number of inhomogeneous coordinate charts. In practice, to build such an atlas one needs to cut a number of open slices across the -orbits in the space of non-degenerate configurations in such a way that their projections onto form an open covering. Each slice is in essence a uniform prescription giving us a representative configuration for every orbit intersecting it.
Concretely, the data required to define an inhomogeneous coordinate atlas on consists of a choice of half-plane in with boundary line passing through the origin, together with a choice of orientation. Note that the condition that a configuration be non-degenerate can be equivalently stated as the requirement that the configuration have at least one pair of non-vanishing and non-collinear position vectors. To define a slice, choose two position vectors , with \text{{\smallI_{1}}}<\text{{\smallI_{2}}}, and consider the subset of all the non-degenerate configurations for which these two position vectors are non-vanishing and non-collinear. Observe that this is closed under the -action. Any configuration from can be transformed through a rigid rotation around the origin of into a configuration with the first position vector lying on the boundary line and the second position vector lying in the interior of the chosen half-plane. A little consideration reveals that there are actually two ways in which this can be achieved, related by a rotation, and we select between them by requiring that the cross product of the two vectors — taken in the given order — be parallel, as opposed to anti-parallel, to the normal vector to the half-plane (hence the need for orientation data). Furthermore, through a simultaneous rescaling, the length of the first vector can always be adjusted to have a given fixed value.
Example** (part I).**
If we fix in the half-plane bounded by the -axis and containing the vector , with normal vector , and single out the pair of vectors , , then any configuration with these two vectors non-vanishing and non-collinear — that is, in our notation, from — can be transformed as above into a configuration of the form
[TABLE]
with . Clearly, the vector product points in the preferred normal direction.
The restricted configurations constructed in this way have, in kinematic terms, four frozen degrees of freedom and, crucially, are in one-to-one correspondence with the orbits contained in , for which they can be thus considered to be representative configurations. Together they define an open slice transversal to these orbits, and their position vectors, which depend on parameters, provide a local inhomogeneous coordinate chart on the projection of to .
The position vectors of any configuration from can be uniquely represented as
[TABLE]
for some and position vectors of a restricted configuration. Following the projective space analogy one may think of and as global homogeneous respectively local inhomogeneous coordinates on .
Example** (part II).**
In the example above one can show that, in agreement with uniqueness, this relation can indeed be inverted to yield
[TABLE]
The components are manifestly invariant under rigid rotations and simultaneous scalings, which reinforces the argument that they are, when they are not constant, natural candidates for the role of coordinates on (the projection of to) the quotient space .
Finally, by considering all the possible choices of two position vectors we can construct in this way, using each time the same oriented half-plane, an entire open atlas on with inhomogeneous coordinate charts.
3.3.2.
The extended Bogomolny field equations (87) are formulated in terms of the derivatives , which, for our next purpose, it is useful to think of as defining a flat, torsion-free affine connection on the tangent bundle of . There exists a notion of Lie derivative of an affine connection with respect to a vector field, see e.g. the definition in [8]; in particular, a vector field whose Lie derivative preserves the connection is called an affine vector field. In our case one can show, for instance by applying the criterion of Proposition 1.18 in op. cit., that the generating vector fields of the -action on are affine vector fields with respect to this trivial affine connection. As we shall see later on, Bogomolny equations associated to hyperkähler cones can be reduced from an open subset of to one of , which in practice is achieved, in the terminology that we have introduced earlier, by expressing them in inhomogeneous coordinates. The formulation of the reduced equations involves sections of certain vector bundles associated to the natural principal -bundle over and covariant derivatives of these sections induced by this flat, torsion-free affine connection on . In the remainder of this subsection we want to show through a few examples what properties such sections have, and how to construct these induced covariant derivatives explicitly for a generic choice of inhomogeneous coordinates.
Consider the two coordinate systems that we have defined on the space in the course of § 3.3.1: the homogeneous global coordinates , and the -adapted local coordinates . Recalling the differential identity (84), the relation (117) between them yields
[TABLE]
where the local vector fields and quaternionic-valued coefficients depend exclusively on the homogeneous coordinates and are completely fixed by their choice. Concretely, by matching for instance the expressions of the exterior derivative on in the two coordinate systems we obtain the following set of conditions which these must satisfy, and which suffice to determine them:
[TABLE]
Summation over repeated indices is as usual understood.
Example** (part III).**
Let us see explicitly how this works for the particular choice of inhomogeneous coordinates considered above. The coefficients , on one hand, are determined by the two conditions on the first line: the first condition implies that for all pairs of indices \text{{\smallI}},i such that ; this leaves us with unknown components, which are then fixed by the 16 linear equations of the second condition. Thus, for the choice (116) of inhomogeneous coordinates the solution for these is
[TABLE]
The second line of constraints, on the other hand, can be solved in terms of the first one to yield
[TABLE]
where we assume the convention that only the terms for which the derivatives are well defined are considered.
3.3.3.
Note now that in the coordinate system the generators of the -action act strictly on the -coordinates and take the form
[TABLE]
3.3.4.
In order to formulate our results later on we will need to consider two basic types of vector bundles over or subsets thereof : on one hand, an -bundle, and on the other, vector bundles associated to the natural principal -bundle.
[TABLE]
The latter bundles can be classified in accordance with the finite-dimensional representations of their structure group . Since this is isomorphic to , specifying its representations means specifying the representations of its two simple factors. The representations of , the multiplicative group of strictly positive real numbers, are characterized by their scaling weight , an integer number. On the other hand, the only representations of the group that we will be concerned with here will be the trivial and the vector representations, which we denote with reference to their dimensions by and respectively.
3.3.5.
To develop a sense of the properties that sections of such bundles have, it is useful to consider two basic examples corresponding to representations with one factor trivial and one not.
Thus, suppose we have a section of the bundle locally defined on the open set of coordinatized by the inhomogeneous coordinates . We can lift this section to a function
[TABLE]
on (an open subset of) , and then using the expression (125) for the generators verify that this satisfies
[TABLE]
That is, sections of correspond to functions on invariant at rigid rotations and homogeneous of degree in the homogeneous coordinates. Conversely, such functions always descend to sections of . On another hand, by resorting to the relation (121) we get that
[TABLE]
where by definition
[TABLE]
This is the connection induced on the bundle over by the flat connection on . It maps sections of to sections of . A simple counting argument shows that its components cannot be all be independent since there are four more of them than the dimension of . One has to have therefore four constraints, which we shall call section type constraints. This follows indeed from the two conditions on the right in (122). However, an easier way to arrive at them is to use equivariance to strip off the -dependence from the symmetry properties (128), a process we will refer to as reduction. We obtain in this way
[TABLE]
Suppose now that we have instead a section of the bundle , again, locally defined on the open set of coordinatized by the inhomogeneous coordinates . We lift this to a triplet of functions
[TABLE]
or in quaternionic notation, , on a corresponding subset of . Proceeding in a similar manner as above one can check that these satisfy
[TABLE]
Conversely, functions on with these transformation properties under rigid rotations and simultaneous rescalings give rise to sections of over . Using the derivative formula (121) we then get
[TABLE]
with
[TABLE]
This connection, mapping sections of to sections of , satisfies as well section type constraints, which can be argued as before to be given by the reduction of the symmetry properties (133) and thus read
[TABLE]
This discussion extends naturally to sections of more general vector bundles over constructed out of these two basic types in accordance with the rules of the tensor product. On each of these bundles the trivial connection on induces a connection , which takes its sections to sections of the bundle obtained by tensoring with . Having at face count four more components than the dimension of , the form an overcomplete set: while the two conditions on the left in (122) ensure that they always satisfy the completeness relation , the ones on the right duly imply the requisite number of what we have called section type constraints, whose precise form depends on the bundle in question. Moreover, it is easy to see that the flatness of the inducing connection implies that the commutators vanish. Note also that the inhomogeneous coordinates on themselves can be regarded as local sections of the bundle , their natural lifts to being the homogeneous coordinates \smash{x_{\mathrlap{i}}{}^{\raisebox{0.3pt}{{\scriptscriptstyle I}}}=|q|^{2}R_{ij}(q^{-1})\hskip 0.5pt\rho_{\mathrlap{j}}{}^{\raisebox{0.3pt}{{\scriptscriptstyle I}}}}, and we have \smash{\nabla_{{\scriptscriptstyle I}i}\rho_{\mathrlap{j}}{}^{\raisebox{0.3pt}{{\scriptscriptstyle J}}}=\delta_{ij}\hskip 1.0pt\delta^{{\scriptscriptstyle J}}_{{\scriptscriptstyle I}}}. An upshot of these considerations is that the covariant calculus on emulates formally the regular differential vector calculus on , with the added feature that one can now make use of section type constraints.
3.4. The reduction
3.4.1.
This technical detour complete, let us now return to pick up and continue the discussion that we left off at the end of subsection 3.2.
The subset S of , which we are assuming to be closed under the collective -action on the latter space, can also be assumed to be in fact a subset of , in concordance with the requirement that the corresponding action on be locally free. Let be the projection of to through the quotient map. The structure of the action (113) implies then that the -bundle over descends to an -bundle over , the total space of which is locally isomorphic to the base of , viewed now as a Swann bundle. This is expressed by the commutative diagram
[TABLE]
To each of the two fibration structures on we can associate an adapted system of coordinates. For the -fibration this is of course the extended Gibbons–Hawking one, given by and . If, however, in this coordinate set we replace what we have called in the previous subsection the homogeneous coordinates with the inhomogeneous ones and the quaternionic coordinates , both of which are now only locally defined, we obtain an alternative system of coordinates which is adapted to the -fibration structure, with and coordinatizing the base and the quaternionic fiber. The two coordinate systems and their component counts are summarized in Table 1.
[TABLE]
Our goal in what follows will be to re-express the hyperkähler 2-forms and metric on as well as the extended Bogomolny field equations in terms of the -fibration-adapted coordinates. Let us begin with the last ones.
3.4.2. The reduction of the first set of extended Bogomolny equations
The symmetry constraints (104) of Proposition 14 mean that the Higgs field determines, in the language coined above, a section of the bundle over such that in -fibration-adapted coordinates we have
[TABLE]
By the considerations in § 3.3.5 it is straightforward to see then that the reduction of the first set of extended Bogomolny equations (87) yields the conditions
[TABLE]
One should not forget also that, as a section of the bundle , the reduced Higgs field automatically satisfies the section type constraints
[TABLE]
3.4.3. The reduction of the second set of extended Bogomolny equations
In -fibration-adapted coordinates the generators of the -action on act along the fibers and take the form (125). Exploiting this fact, one can argue that in these coordinates connection 1-forms satisfying the gauge-fixing conditions (108) must be of the form
[TABLE]
with the components supported in , although still allowed in principle — for now — to depend on the -coordinates.
On another hand, from the decomposition relations (119) and (137), by way of the formula (129), we get that
[TABLE]
If we distribute the wedge product, then use the reduced Bogomolny equation (138) in tandem with the section type constraints (139) to eliminate or simplify terms, and eventually the Cartan–Maurer equation for to consolidate the result, this becomes
[TABLE]
Recalling that , the second set of extended Bogomolny equations (87) implies then the conditions
[TABLE]
This shows in particular that the 1-forms can be chosen in fact so as not to depend on the -coordinates at all, and can be thus thought of as bona fide connection 1-forms on the -bundle induced by the connection 1-forms on the -bundle .
3.4.4. The reduction of the hyperkähler 2-forms and metric
At this point, it is opportune to switch temporarily to a quaternionic formalism. The operative observation here is that the key decomposition relations (117) (or, rather, its differential version (119)), (137) and (140) can be pieced together into the following quaternionic decomposition formula for the quaternionic-valued coframe elements (90) of :
[TABLE]
where are the imaginary quaternionic avatars of the vectors and, by definition,
[TABLE]
Note that these form in turn a complete set of quaternionic-valued coframe elements for .
Substituting then this expression for the elements together with the formula (137) for the Higgs field into the quaternionic formulas (91) for the hyperkähler 2-forms and metric on gives us after a series of rather straightforward algebraic manipulations the Swann-like expressions
[TABLE]
where we have denoted , which we assume to be non-vanishing,
[TABLE]
and
[TABLE]
The two formulas (146) differ from the standard Swann bundle ones (75) and (79) in two respects: first, by the presence of the overall factor , and second, and perhaps less obviously, since obscured by the notation, by the fact that is not an imaginary quaternion-valued 1-form but rather a full quaternion-valued one; in other words, by the fact that its real part is non-vanishing. However, notice that this can be cast in the form
[TABLE]
The first expression follows directly from the definition. To arrive at the second one, let us observe first that can be viewed as originating from the reduction of the hyperkähler potential , more precisely, that we have . The symmetry constraints of part of Proposition 15 which the latter satisfies imply then that is a section of the line bundle . This means in particular that the covariant derivatives are well-defined. In light of these facts the relation (111) can be easily reduced to yield
[TABLE]
From this, the second expression for follows then immediately. Let us record also while at this that for the remaining imaginary/vectorial part of we obtain directly, without any further argument,
[TABLE]
The final expression (149) for shows that the departure from the Swann canonical forms is rectifiable: the overall -factor and the real component of in the formulas (146) can both be removed simultaneously through a rescaling
[TABLE]
of the fiber coordinates. When this must be supplemented by a sign redefinition , and consequently . None of these maneuvers, however, affects the expressions for , or the imaginary part of , which can now be legitimately interpreted as quaternionic Kähler 2-forms, metric and connection 1-forms, respectively.
3.4.5. A direct check
In fact, since we are now in possession of explicit intrinsic expressions for these, we can verify directly that they satisfy the requisite quaternionic Kähler properties without resorting to the extrinsic device of the Swann bundle. For this, let us observe first that similarly to the extended Gibbons–Hawking hyperkähler case (see § 3.1.3), the reduced Bogomolny equation (143) can be replaced in the quaternionic formalism with the equivalent condition
[TABLE]
Using this along with the other reduced Bogomolny equation (138) and the section type constraints (139), we can then show through a somewhat involved and rather technical exercise in differential form-valued vector calculus that the following two differentiation formulas
[TABLE]
hold. From these and the above definition of we eventually infer that
[TABLE]
Note that the real component of drops out identically from this equality leaving behind only its imaginary quaternionic part, and so this is precisely the Einstein condition (27) of the quaternionic Kähler criterion laid out in Theorem 4.
3.4.6. Alternative expressions
Let us return now to the quaternionic formulas (148) for the quaternionic Kähler 2-forms and metric, and observe that they can be alternatively recast through a purely algebraic effort in the — also quaternionic — form
[TABLE]
with given by the expression (147). These are essentially the quaternionic Kähler analogues of the hyperkähler formulas (91).
From them, we can transition back to a vectorial formalism by writing
[TABLE]
which then, upon substitution into the two formulas above, gives us the equivalent — and also most explicit — expressions
[TABLE]
The 1-forms and are given in turn by the formulas (149) and (151). In this formulation, in particular, it is straightforward to see that the vector fields generate an isometric -action with Galicki–Lawson moment map
[TABLE]
In the next subsection we compile for easy reference and present in an intrinsic geometric language the quaternionic Kähler construction which emerged from these considerations, in a form mirroring our earlier presentation of the extended Gibbons–Hawking hyperkähler construction.
3.5. The quaternionic Kähler Ansatz
Consider a principal -bundle over an open subset of and, alongside it, the restrictions to of the associated -bundles over defined in subsection 3.3, which we denote with the same symbols.
[TABLE]
Let denote the connections induced on the latter bundles by the trivial connection on .
In this setup we want to formulate two sets of field equations. For the first one let us assume that we have either **
- I a.
a section of the bundle over satisfying the first-order differential constraints
[TABLE]
or, alternatively,**
- I b.
a section of the line bundle over satisfying the second-order differential constraints
[TABLE]
For which one of these conditions we opt is immaterial because
Lemma 16**.**
The two conditions are equivalent, with the maps between them given, respectively, by
[TABLE]
This follows immediately from Proposition 15 by reduction. One can in fact give a completely intrinsic proof of this result simply by mimicking the proof of Proposition 15 in covariant calculus language and noticing that the symmetry constraints there show up here in the guise of section type constraints.
On the other hand, for the second set of field equations let us assume that we have, in addition, **
- II.
a principal connection on with curvature 2-form determined by the Higgs field through the following Bogomolny-like condition:
[TABLE]
Based on the previous considerations we can now state the following
Theorem 17**.**
Let or, equivalently, be a set of solutions to the above field equations, and assume that we can adjust the open subset of on which they are defined in such a way as to have both and everywhere on it. On the total space of the -bundle we associate to this solution the triplet of 1-forms
[TABLE]
and triplet of 2-forms
[TABLE]
possessing an isometric locally free -action generated by the vector fields , with corresponding Galicki–Lawson moment map images .
Conversely, any -dimensional quaternionic Kähler manifold with an isometric locally free -action arises locally in this way.
The direct claim encapsulates the results of the previous subsection. Our arguments there have yielded in fact three more equivalent sets of formulas for the quaternionic Kähler connection 1-forms, 2-forms and metric, which we list by reference number in Table 2.
[TABLE]
For the converse statement, if we assume that we have a -dimensional quaternionic Kähler space with an isometric locally free -action, then by Theorem 13 this action lifts to an -action on the -dimensional Swann bundle . Bielawski’s Proposition 2.1 from [10] guarantees then that the hyperkähler structure on is given locally by an extended Gibbons–Hawking construction. In particular, corresponding to these actions, the quaternionic Kähler moment maps are lifted to hyperkähler moment maps in accordance with the rule (82), and so we conclude that the canonical -action on acts on the base of the extended Gibbons–Hawking fibration by what we have called collective transformations. We find ourselves therefore within the bounds of Proposition 14, which means that we can repeat ad litteram the reduction procedure of subsection § 3.4 to obtain an explicit local description of the quaternionic Kähler structure on in the precise same category as the one above, proving thus the generality of this construction.
The theorem encompasses the range of both definite and indefinite signature metrics. The construction it describes is in a very clear sense a quaternionic Kähler analogue of the extended Gibbons–Hawking hyperkähler Ansatz. The role of hyperkähler moment map is played here by Galicki and Lawson’s concept of quaternionic Kähler moment map; however, unlike in the Gibbons–Hawking case, the moment map images do not define coordinates (for which purpose they would be too many) but, rather, they are related to these through a scaling factor. Another, more surprising departure from the extended Gibbons–Hawking case is the existence of a dual description implying that the quaternionic Kähler structure is, quite remarkably, completely determined locally in all dimensions by a single real-valued potential satisfying a set of linear differential constraints. The linearity, in particular, means that in a very definite sense one can superpose two such quaternionic Kähler structures to obtain a third.
3.6. The case of four dimensions
We end this section with a closer look at the four-dimensional (i.e. ) specialization of this construction. In this case, is the four-dimensional avatar of a quaternionic Kähler space — namely, a self-dual Einstein space with non-vanishing scalar curvature — possessing two linearly independent commuting Killing vector fields. The local geometry of this class of spaces was described explicitly by Calderbank and Pedersen in [13] in terms of an eigenfunction of the Laplacian on the Poincaré half-plane, following an intrinsic, four-dimensional analysis based on previous work by Joyce [34], Ward [48] and Tod [47], among others. We will now show that our results reproduce theirs in this dimension, while also offering a few additional bonuses. In this sense, therefore, the construction of Theorem 17 may be viewed as a higher-dimensional generalization of the Calderbank–Pedersen construction — similarly, although much less trivially, to the way in which, in hyperkähler context, the extended Gibbons–Hawking construction of [32, 41] can be viewed as a higher-dimensional generalization of the original four-dimensional Gibbons–Hawking one.
First, let us observe that for the space is isomorphic to the upper-half plane and can be covered by a single inhomogeneous coordinate chart. We choose the inhomogeneous coordinates as in the Example in § 3.3.1 to be given by
[TABLE]
with , where, for simplicity, we drop the now inessential upper index 1 from the notation of the components of the second vector. As argued subsequently in § 3.3.2, a choice of inhomogeneous coordinates fixes completely the form of the covariant connections induced on bundles by the trivial connection on, in this case, . To determine these, we need both the connection coefficients , which are already listed in (123), and the operators , which in this case can be worked out to be
[TABLE]
Note that for this choice of inhomogeneous coordinates the only non-vanishing contribution on the right-hand side of the second reduced Bogomolny equation (165) comes possibly from the term . However, this covariant derivative vanishes, too, and the equation becomes simply
[TABLE]
This means that in four dimensions we can consistently set in the formulas, which are locally defined, .
In fact, a simple inspection shows that for this choice of inhomogeneous coordinates any covariant derivative with of no matter what section vanishes. The second-order differential constraints (163) reduce therefore to a single one, which reads
[TABLE]
with the indices indicating derivatives. To obtain this result it is important to recall that is a section of the bundle , and so is a section of the bundle . The calculation of covariant derivatives of both such sections has been described in detail in § 3.3.5. The constraint can be equivalently recast in the form
[TABLE]
where is the Laplacian corresponding to the Poincaré metric
[TABLE]
on the upper half-plane. We retrieve thus a first key result of Calderbank and Pedersen.
Let us define now the following 1-forms supported on the -fibers
[TABLE]
and similarly fiberwise acting commuting vector fields
[TABLE]
The third component vanishes in each case. The remaining components, , and , , form a local coframe and frame, respectively, for the -fibers. By resorting to the relation (150), we obtain then that
[TABLE]
Again, all components along the third direction vanish, leaving a two-dimensional non-vanishing square block. Since in this case the Higgs field matrix is also two-dimensional, this relation can be used to determine it explicitly through a straightforward exercise in linear algebra. One obtains the same result, although at a greater expense of calculational effort, from the second relation (164), upon use of the constraint (172), which serves to eliminate the second derivatives of .
Knowledge of the Higgs field and principal -connection allows us now to work out explicitly the geometric structure on given by Theorem 17. Upon substitution, the formula (151) yields, on one hand,
[TABLE]
To formulate the remaining results it is useful to define the quaternionic-valued 1-form
[TABLE]
where is the standard symplectic form on the -fibers and are the same thing as . The formulas (158) and (159) reduce then, on the other hand, following an extended calculation, to the simple expressions
[TABLE]
One can easily verify that this metric coincides indeed, up to a function redefinition, with the one found by Calderbank and Pedersen in [13] or, rather, with the version given by formula (2.4) from [14]. The structure it describes is that of an -fibration over the Poincaré upper half-plane, conformally rescaled. In particular, this form makes it clear that, as observed in [13], if we assume that is positive definite, the sign of the scalar curvature is dictated by the sign of the factor .
4. Hyperkähler cones via the Legendre transform construction
4.1. The Legendre transform construction
Translating into geometric terms insights derived from a supersymmetric quantum field-theoretic problem [38], Hitchin, Karlhede, Lindström and Roček introduced in [32] a method of constructing hyperkähler metrics of extended Gibbons–Hawking type known as the Legendre transform construction. In this approach, the metric information is stored in a single real-valued function defined on some open subset of , satisfying the linear second-order differential constraints
[TABLE]
This function is associated to a choice of direction in , which we take here to be the direction of the -axis. Accordingly, it is convenient to view the space as and work with the complex linear combinations , , of the real coordinates . In terms of these, the differential constraints take the form
[TABLE]
The hyperkähler geometric structure is then extracted from based on the following two properties:
The (flipped-sign) Legendre transform
[TABLE]
of the function with respect to the -variables, where one assumes that the Legendre constraints can be implicitly solved to give unequivocally , yields a Kähler potential for the hyperkähler symplectic form along the chosen direction, together with a complete set of coordinates and holomorphic with respect to the corresponding complex structure .
- 2)
These coordinates are, simultaneously, complex Darboux coordinates for the transversal complex symplectic form, that is,
[TABLE]
The two conditions suffice to determine all three hyperkähler symplectic forms , , , and hence also the hyperkähler metric . And indeed, one can verify that these are of extended Gibbons–Hawking type, with
[TABLE]
The differential constraints (182) guarantee moreover that these connection 1-forms and Higgs field satisfy as required the extended Bogomolny field equations.
This construction is general, in the sense that any hyperkähler manifold of extended Gibbons–Hawking type is locally given by such a Legendre transform construction for some potential (see Proposition 2.1 in [10]).
4.2. Hyperkähler cone structure conditions
4.2.1.
Let us work out now how hyperkähler cone structures are described in this approach, or, in other words, how the conditions of subsection 3.2 translate in terms of the potential .
Consider the following two complex linear combinations of the generators of the collective -action on the space :
[TABLE]
On the domain of definition of , the differential constraints (182) imply the relations
[TABLE]
Observe now, on one hand, that from the expression (185) for the Higgs field we have that
[TABLE]
where in the last step we have used the commutation property . Note that both equalities hold in full generality, for any hyperkähler space of extended Gibbons–Hawking type. In view of these considerations the criterion of Proposition 14 can then be equivalently rephrased as follows:
Proposition 18**.**
The collective -action on the base of an extended Gibbons–Hawking hyperkähler space with potential can be lifted to a hyperkähler cone structure on if and only if satisfies the condition
[TABLE]
An immediate corollary is then that a sufficient set of constraints for that to happen is given by [33]
[TABLE]
On another hand, from the Legendre transform approach expression for the connection 1-forms we obtain the following identities
[TABLE]
where, of course, . We stress that these are indeed identities, with no constraints whatsoever being needed for their derivation. In the light of formula (187) it is clear then that
Lemma 19**.**
The connection 1-forms (186) satisfy the gauge-fixing condition (108) if and only if there exist shifts such that
[TABLE]
for all values of .
4.2.2.
Let us assume now that the function satisfies the linear differential constraints (193). One can then check that with these conditions its Legendre transform is both invariant at collective rotations and scales with weight under the action of . That is to say, the Legendre transform gives us in this case not just a Kähler potential, but in fact a hyperkähler potential. Accordingly, we write
[TABLE]
As we have discussed at the end of § 3.2.1, when the gauge-fixing condition (108) holds, the collective -action on lifts trivially to the hyperkähler cone as its standard quaternionic action. In our case it can be shown that its (complexified) generators assume in holomorphic coordinates the form
[TABLE]
4.2.3.
So far, what we have done was simply to rephrase the description of a hyperkähler cone structure on an extended Gibbons–Hawking space in the language of the Legendre transform approach. Ultimately, though, we are interested in determining the quaternionic Kähler metric on the base of the hyperkähler cone. To achieve that, we combine the results of this and the previous section into the following procedure:
Given a function satisfying the differential conditions (182), usually although not necessarily in the form of a contour integral, one should first test whether it satisfies also the conditions of Proposition 18. In case it does, one is then guaranteed that the ensuing metric through the Legendre transform construction will be of hyperkähler cone type.
- 2.
The second partial derivatives of with respect to the -variables give the Higgs field via the formula (185).
- 3.
More challengingly, one must also solve the differential conditions of Lemma 19 to find shift functions for which the connection 1-forms satisfy the gauge-fixing condition (108). As we shall see in the next example, depending on the particularities of the case, additional symmetry considerations might need to be taken into account as well.
- 4.
Once the Higgs field and the appropriately gauge-fixed connection 1-forms on the hyperkähler cone are determined, one should then substitute in them the change of variables (117) for some choice of restricted configuration . The outcomes should be amenable to the form (137) and (140), respectively, from which one can then read off the reduced Higgs field and induced connection 1-forms . This is effectively a quotienting procedure which takes us from the hyperkähler cone down to its base.
- 5.
The Ansatz of Theorem 17, or any one of the equivalent Ansätze listed in Table 2, gives us eventually the quaternionic Kähler metric and structure on the base.
5. An example: the local c-map
As an application of this procedure and, more generally, of the ideas that we have developed in this article, we present an explicit construction of a quaternionic Kähler metric known as the Ferrara–Sabharwal metric [23]. This metric emerged in physics from the construction known as the local c-map (or, sometimes, the projective or supergravity c-map) [15], a map derived from a duality of the moduli spaces of type IIA and type IIB string theories, which yields a quaternionic Kähler manifold of real dimension for each projective special Kähler manifold of real dimension . The approach we pursue was initiated by Roček, Vafa and Vandoren in [43, 44]. The idea is as follows: the -dimensional quaternionic Kähler manifold has an associated -dimensional Swann bundle which, in turn, as a hyperkähler space, has a -dimensional twistor space. The twistor space, these authors posit, is characterized as a complex space by a certain holomorphic gluing function (in physics terms, this is related to a projective superspace lagrangian density). The challenge is then to extract the metric from this holomorphic function by coming down first two real dimensions, to describe the geometry of the Swann bundle, and then another four real dimensions, to the quaternionic Kähler base. Here we fill in a number of critical details in the arguments of [43, 44], which were further developed in [40], and use our previous considerations to perform what these papers refer to as a superconformal quotient — i.e. the descent from the hyperkähler cone to the quaternionic Kähler base — explicitly as well as equivariantly.
5.0.1.
In close analogy with the Legendre transform construction of the rigid c-map metric (also known as the affine c-map or semi-flat metric), the -potential of the Swann bundle over the quaternionic Kähler image of the local c-map was shown in [43, 44] to be given by the following contour integral:
[TABLE]
where
- •
the variables are so-called tropical components of sections of an line bundle over the twistor space of the Swann bundle, that is,
[TABLE]
- •
is a holomorphic function called prepotential, assumed to be homogeneous of degree two in its variables;
- •
the integration contour wraps around the two roots of : anti-clockwise around for the first term, and clockwise around for the second one, where
[TABLE]
with , correspond to antipodally opposite points on the twistor Riemann sphere.
The integrand can be understood as a twistor space holomorphic symplectic gluing function [32]. Its holomorphic dependence on sections guarantees that the contour integral satisfies automatically the generalized Laplace equations (182). Moreover, the specific manner in which it depends on these — namely, the fact that it only depends on implicitly through the variables and the fact that it scales with weight at a scaling of the ’s — implies that the integral satisfies the two constraints (193), and it is thus indeed, as claimed, the -potential of a hyperkähler cone.
The contour integral can be computed explicitly with Cauchy’s residue theorem. Using the homogeneity property of the prepotential we obtain
[TABLE]
where, by definition,
[TABLE]
Note that the twistor sections take antipodally conjugated variables to complex conjugated ones. Here and throughout this section we use the following index notation conventions:
[TABLE]
5.0.2.
Before we continue, we pause for a moment to list for future reference a number of properties that these variables satisfy. First, direct evaluation gives us the expression
[TABLE]
This can be equivalently recast in the vectorial form
[TABLE]
The last one, in particular, makes it clear that ratios of ’s are invariant under collective transformations and thus descend naturally to functions on the base of the hyperkähler cone.
In order to present the next property in a concise manner it will be convenient to introduce a number of temporarily notations. Thus, if we denote the parameters of the -bundle sections by , , such that we have (this is sometimes called the complex spherical basis), then in terms of these the various derivatives of can be summarized in the formula
[TABLE]
In view of the imminent application of Lemma 19 it would be useful to have a list of elementary functions belonging to the intersection of kernels from the condition (195). And indeed, such a list exists and reads as follows:
[TABLE]
For the first three elements membership in the list follows immediately from a cursory survey of the formulas (188). For the last two, however, this is a little less obvious. One can verify it directly, which is rather tedious, or by observing successively that
[TABLE]
The remaining gap in the verification can then be bridged by a very simple calculation.
Lastly, let us also make a note of the following identity:
[TABLE]
The manifestly invariant expression on the right-hand side, the same one which shows up in the differentiation formula (206), will play an important role in the considerations to come.
5.0.3.
In order to compute the Legendre transform of one needs to compute first its partial derivatives with respect to the variables . The components with of the differentiation formula (206) give us promptly, on one hand,
[TABLE]
This last relation can be used to determine . With the help of these formulas one can then show that the Legendre transform of yields the hyperkähler potential
[TABLE]
where it is understood that refers to .
One calls dualization the result of replacing the prepotential function in the definition of with (minus) its Legendre transform. In the ensuing formulas, this operation results in the replacement of with , of with , and of the latter with ; similar statements hold also for the complex conjugated quantities. On the other hand, the zero-indexed variables such as , , are inert under dualization.
The first expression above for the hyperkähler potential is the one best suited for an analysis of its dualization properties. From it, one can see immediately that the hyperkähler potential is self-dual. This strongly suggests that one might be able to lift dualization to a symmetry of the metric, which, we shall see, will indeed be the case.
The second expression of the hyperkähler potential brings instead to the fore its collective transformation properties. Note that since the function is homogeneous of degree zero, and since ratios of ’s are invariant under collective transformations, it follows that so is the Hessian . It is then evident that the hyperkähler potential is, as required, invariant under collective rotations and scales with weight under the action of .
5.0.4.
Following [40], let us consider on the vector fields
[TABLE]
where c.c. means the complex conjugate of the preceding expression. They are manifestly real-holomorphic with respect to the hyperkähler complex structure and, moreover, satisfy the following graded Heisenberg algebra:
[TABLE]
Observe now that, out of these, the vector fields and are entirely vertical with respect to the -fibration structure. On the other hand, the vector fields and have non-trivial horizontal components given by the horizontal lifts of the vector fields
[TABLE]
Corresponding statements hold for the complex conjugated variables as well.
In light of these facts one can then readily infer using the expression (213) for the hyperkähler potential that this is preserved by all of the generators of the graded Heisenberg algebra:
[TABLE]
Let us recall now the following general result:
Lemma 20**.**
A real-holomorphic vector field on a Kähler manifold preserves the Kähler form if and only if it preserves locally Kähler potentials up to pluriharmonic functions.
A hyperkähler manifold can be viewed as a Kähler manifold with respect to any one of its hyperkähler complex structures. In our case, each one of the vector fields , , and satisfies the conditions of this Lemma when one regards the hyperkähler space as a Kähler space with respect to the complex structure , with Kähler form . Indeed, they are real-holomorphic with respect to and, as we have shown, preserve the hyperkähler potential, which is in particular a Kähler potential for . By the Lemma, they all must then preserve as well the Kähler form . Moreover, a straightforward check using their defining formulas shows that they preserve also both the transversal symplectic form (184) and its complex conjugate. We are then entitled to conclude that , , and are all tri-Hamiltonian vector fields.
What is more, they also commute with the generators of the standard quaternionic action on the hyperkähler cone. This can be seen using the holomorphic coordinate expressions (215) and (197), while taking also into account, once again, the invariance properties (220) of the hyperkähler potential function . By Theorem 13 they descend therefore to the quaternionic Kähler base.
These considerations suggest that in holomorphic coordinates the dualization map takes the form
[TABLE]
Indeed, this represents a holomorphic symplectomorphism with respect to the complex symplectic form , which preserves moreover the graded Heisenberg algebra by mapping to and the latter to while keeping and unchanged. Notice also that the double dual of a non-zero-indexed coordinate returns minus that coordinate, whereas the double dual of a zero-indexed one returns it trivially, without any sign change.
5.0.5.
In order to be able to compute explicitly the connection 1-forms we need to choose a set of shift functions for which the conditions of Lemma 19 are satisfied. This is perhaps the most difficult step in the construction of a hyperkähler cone in the Legendre transform approach. Fortunately, in our case we can use the duality symmetry as a guide. In other words, rather than solve these conditions directly, in what follows we shall exploit duality to find a convenient set of such functions, and then verify a posteriori that they provide indeed a solution to the required conditions.
Note to begin with, that in view of the expression (211), the Legendre transform construction formula (187) takes in this case for non-zero values of the index the form
[TABLE]
Dualization then yields
[TABLE]
where the second equality is the real part of the identity (210). This shows that if we take
[TABLE]
then we get [40]
[TABLE]
which is manifestly invariant under collective transformations, in agreement with the expectation that dualization should commute with the hyperkähler cone quaternionic action. For this choice the initial formula becomes
[TABLE]
that is, precisely the dual of the identity (210). Since the functions are holomorphic and homogeneous of degree 1, then in light of the list (207) it is clear that these ’s satisfy the condition of Lemma 19. This check validates the above choice of shift functions .
On another hand, let us observe that the relation (212) can be equivalently stated as
[TABLE]
By noticing then that the function is anti-self-dual, i.e. that , we can infer from this that . This suggests that we choose a shift by setting
[TABLE]
which is thus real and, moreover, explicitly self-dual. This is the same as having [40]
[TABLE]
Each elementary component of the expression on the right-hand side belongs to the set , which then by way of the Leibniz rule implies that the same is true for . From this formula one can read off the shift , which can then be shown to satisfy
[TABLE]
A rather unusual feature of this choice of shift, imposed on us by the requirements of manifest duality, is that it has an explicit linear dependence on the -variables.
5.0.6.
So far we have used two coordinate systems on : the extended Gibbons–Hawking coordinates and (with a closely related complexified variant given by , , , ), and the Legendre transform-related holomorphic coordinates and . The first system is adapted to the -fibration structure. The second one is adapted to the transversal complex symplectic structure which is holomorphic with respect to . For the current construction, however, it is useful to introduce yet a third system of coordinates, adapted to its duality symmetry this time, given by
[TABLE]
The generators of the Heisenberg algebra take in these coordinates the form
[TABLE]
and, moreover, the grading generator scales the non-zero-indexed coordinates with weight and the zero-indexed ones with weight , except for , which it scales with weight .
The advantage of this coordinate system becomes apparent when one tries to compute the connection 1-forms . For the shift choices (224) and (230), from the Legendre transform formula (186) we obtain following a rather laborious computation the relatively simple expressions
[TABLE]
where, for simplicity, we have used the notations and .
5.0.7.
The Higgs field, on the other hand, can be computed with significantly less effort. By taking derivatives with respect to and of the equation (212) we obtain, respectively,
[TABLE]
(recall the index notation convention (201)!). Moreover, from the preceding expression for the derivative we can easily compute the second derivative . Together with the relations above this gives us
[TABLE]
where, by definition,
[TABLE]
5.0.8.
We are now ready to perform the reduction and descend from the hyperkähler cone down to its quaternionic Kähler base. In practice, this step consists of substituting into the formulas for the Higgs field and connection 1-forms the change of variables (117) for some choice of restricted configuration with position vectors , and then reading off from the result the reduced Higgs field and reduced connection 1-forms .
In the case of the Higgs field, this is rather straightforward. Recall, on one hand, that the variables and, as argued before, the functions are invariant at collective transformations, and hence at the particular collective transformation represented by the equation (117). On another hand, from the multiplicative property of the quaternionic norm we have , and from the representation (214) for the hyperkähler potential, . These last two properties entail in particular that is also invariant at collective transformations, and so, like and , descends naturally on the base of the hyperkähler cone. It is obvious then that if we substitute into the above Higgs field matrix the relation (117) we find as expected from general arguments that this is of the form (137), with the reduced Higgs field matrix given by
[TABLE]
Aside from the overall factor, this matrix is precisely the same as the one in the equation (235); however, here we have introduced the shorthand notations , , a.s.o. Let us also record here for subsequent use that the inverse of this matrix is
[TABLE]
with representing the matrix inverse of .
In the case of the connection 1-forms , if we consider a restricted configuration with
[TABLE]
then the change of variables (117) yields, after a rather more strenuous calculation, a result of the form (140), with the reduced connection 1-forms given by
[TABLE]
5.0.9.
Let us specialize now further to the particular choice of restricted configuration considered in (116). For this we then have, via the formulas (225) and (205),
[TABLE]
suggesting we should define also the variables
[TABLE]
These are all complex, with the exception of , which is real and positive. The reality condition is important for the coordinate count: in what follows we will coordinatize the quaternionic Kähler space by means of the variables , , and , which would not be possible if the total number of their real components were not equal to .
Note that since the function is homogeneous of degree zero, one can argue that , which then allows us to pass from the variables to the ones. With obvious notations, the hyperkähler potential formula (214) gives us in this case
[TABLE]
This satisfies, incidentally, the differential identity
[TABLE]
a consequence of the holomorphicity property of the prepotential.
5.0.10.
Knowledge of the reduced Higgs field and connection 1-forms allows us to finally compute the quaternionic Kähler metric explicitly. To this end, observe first that from the expression (151) for the connection 1-forms and with the above definitions we obtain in this case
[TABLE]
The metric itself can be most conveniently worked out from the version (148) of the metric Ansatz, in which one can make use of the above formulas for the connection 1-forms. In the end, using among other things the identity (244), we arrive as predicted in [43, 44] at the Ferrara–Sabharwal metric [23]
[TABLE]
where
[TABLE]
is a complex -torus metric with period matrix [21]
[TABLE]
5.0.11.
To make the junction with the usual formulation found in the literature we need to perform one final change of coordinates. Letting
[TABLE]
we replace the subset of coordinates with the complex inhomogeneous coordinates plus . The projective special Kähler metric can be understood then as the Kähler metric corresponding to the Kähler potential
[TABLE]
with as complex holomorphic coordinates. (A more geometric definition of projective special Kähler metrics can be found in [24, 3].) The formulas are straightforward to re-express in the new coordinates due to their homogeneity properties. We leave the details to the reader as an exercise.
The -dimensional Ferrara–Sabharwal metric has isometries descending from the corresponding tri-Hamiltonian isometries on the hyperkähler cone and satisfying the same graded Heisenberg algebra (216). In the above inhomogeneous coordinate frame their generating vector fields take the form [22]
[TABLE]
Finally, let us also mention for the sake of completeness the well-known fact that if the prepotential function is such that has Lorentzian signature , then on the open complex domain given by the condition , where we have , both and are negative definite (the first statement is straightforward, the second one was shown in [18] by an ingenious argument based on the so-called inverse Cauchy-Schwarz inequality), and therefore is negative definite. In other words, in this case one can choose the metric to be positive definite, and this metric will have a negative scalar curvature.
5.0.12.
The local c-map has been studied extensively in both the physics and mathematics literature. In particular, a different and very interesting geometric construction, starting from the rigid c-map and based on the so-called hyperkähler/quaternionic Kähler correspondence, has been given in [4] (see also the preceding work of [31] and [6]). Global aspects of the Ferrara–Sabharwal metric have been considered in [17, 39], and completeness issues have been discussed in [16, 17]. Further considerations such as quantum perturbative and instanton corrections have been explored in [7, 42, 6].
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