A Class of Locally Inhomogeneous Complete Quaternionic Kähler Manifolds

We prove that the one-loop deformation of any quaternionic Kähler manifold in the class of c-map spaces is locally inhomogeneous. As a corollary, we obtain that the full isometry group of the one-loop deformation of any homogeneous c-map space has precisely cohomogeneity one.


Introduction
It is by now well known that the supergravity c-map [FS90] and its one-loop deformation [RSV06] can be used to construct a wealth of complete quaternionic Kähler manifolds of negative scalar curvature, see [ACDM15] for a geometric construction of the relevant metrics and [CHM12,CDS17] for some completeness theorems.The local geometry of any c-map space can be encoded in a holomorphic function subject to a (non-holomorphic) non-degeneracy condition on its two-jet, the so-called holomorphic prepotential of special Kähler geometry.In [CST21,CST22] it was shown that the one-loop deformation of one of the classical series of symmetric quaternionic Kähler manifolds of negative scalar curvature has precisely cohomogeneity one.
The purpose of this paper is to show that the one-loop deformation of any c-map space is locally inhomogeneous, see Theorem 3.9.As a consequence, we prove that the one-loop deformation of any homogeneous quaternionic Kähler manifold of negative scalar curvature, with exception of the quaternionic hyperbolic space (that is not a c-map space), has precisely cohomogeneity one, see Corollary 3.10.Our arguments rely on a general formula for the curvature tensor of any (possibly indefinite) hyper-Kähler manifold obtained from the rigid c-map, see Theorem 3.3, as well as on special properties of the curvature tensor in the case of rigid c-map spaces associated with conical affine special Kähler manifolds, see Proposition 3.5.The above information obtained for the hyper-Kähler manifolds in the image of the rigid c-map is transferred to crucial information about the curvature tensor of the quaternionic Kähler manifolds in the image of the one-loop deformed supergravity c-map in Proposition 3.7.Finally, this information is used in Proposition 3.8 to show that the quaternionic Kähler manifold has a non-constant scalar-valued curvature invariant when the deformation parameter c is positive.This essentially implies the claimed inhomogeneity and cohomogeneity results.
• g is negative-definite on the span of ξ and Jξ, and positive-definite on its orthogonal complement, • Dξ = ∇ξ = id, where D denotes the Levi-Civita associated to g.
Remark 2.3.As D(Jξ) = JDξ = J is skew-symmetric with respect to the metric g, it follows that Jξ is a Killing vector field.Similarly, we see that Jξ is Hamiltonian.
We will only consider connected CASK manifolds, so in the following all the manifolds involved will be connected.Moreover, we will always assume that the vector fields ξ and Jξ generate a principal C × -action.The quotient manifold M does then inherit a (positivedefinite) Kähler metric ḡ and ( M, ḡ) is called a projective special Kähler manifold.The metric is obtained by Kähler reduction exploiting the fact that Jξ is a Hamiltonian Killing vector field.
Let us define the tensor field

4
In ∇-affine coordinates q i , this is given by The following three results are well known in special Kähler geometry [Fre99,ACD02].
Lemma 2.4.The tensor S on an ASK (M, g, J, ∇) manifold satisfies the following properties: for all vector fields X, Y, Z ∈ Γ(T M ).
Part (a) follows from the fact that (∇ X J)Z − (∇ X J)Z = (d ∇ J)(X, Z) = 0. Part (b) follows from the fact that g is symmetric.
Note that in particular this means that ∇g is fully symmetric.
Proposition 2.5.Let (M, g, J, ∇) be an ASK manifold.Then we have D − ∇ = 1 2 S, where D is the Levi-Civita connection of g and S = g −1 ∇g.
Proof.Let D := ∇+ 1 2 S. As the Levi-Civita connection is the unique torsion-free connection preserving the metric g, the result will follow if we can show that D is torsion-free and metric.
Since ∇ is torsion-free and S X Y = S Y X, it follows that D is also torsion-free.Now let us check that D is metric.For X, Y, Z ∈ Γ(T M ) we have where in the last step we have used that ∇g is fully symmetric.
Corollary 2.6.Let (M, g, J, ∇, ξ) be a CASK manifold.Then for all X ∈ Γ(T M ), we have Proof.The first equation follows from 1 2 S X ξ = D X ξ − ∇ X ξ = 0 and that S X Y = S Y X for all X, Y ∈ Γ(T M ).For the second one, we note where we have used 1 2 S = D − ∇ = − 1 2 J∇J in the penultimate equality.
Using the flat torsion-free connection ∇, we can identify where π : N = T * M −→ M is the canonical projection, T V N = ker(dπ) is the vertical distribution and T H N is the horizontal distribution defined by ∇.In particular, given a vector field X ∈ Γ(T N ), we will think of its horizontal component X H as a section of π * T M and its vertical component X V as a section of π * T * M .Using these identifications, we have In the case that M is CASK, the rigid c-map space N = T * M enjoys some additional properties.
Proposition 2.7 ([ACM13, Proposition 2]).Let (M, g, J, ∇, ξ) be a CASK manifold and define on the associated rigid c-map space (N = T * M, g N , I 1 , I 2 , I 3 ) the following data: , where Jξ denotes the horizontal lift of Jξ with respect to ∇.Then Note that Proposition 2.7 implies that Z is a rotating Killing vector field for the pseudohyper-Kähler manifold N .This means that it is a Killing vector field that preserves one of the Kähler structures I 1 and rotates the other two into each other.
It will also be convenient for later purposes to introduce some notation.We denote by HZ = span{Z, I 1 Z, I 2 Z, I 3 Z} the distribution generated by the quaternionic span of Z, then T N = HZ ⊕ (HZ) ⊥ . (2) The vector fields Z, I 1 Z ∈ Γ(T N ) are horizontal and the vector fields I 2 Z, I 3 Z ∈ Γ(T N ) are vertical, with respect to the decomposition T N = T H N ⊕ T V N .

HK/QK correspondence, supergravity c-map, and twist construction
Suppose we are given a pseudo-hyper-Kähler manifold (N, g N , I 1 , I 2 , I 3 ) that admits HK/QK data i.e. a tuple • Z is a rotating Killing vector field preserving ω 1 (assume for simplicity that Z generates a free circle action), • ω ) is nowhere vanishing.Note that there is a freedom of adding a constant to the Hamiltonian functions f c Z and f c H , so long as the shifted Hamiltonian functions are still nowhere vanishing.This is reflected in the superscript c in f c Z and f c H .Given the above data, it was shown in [ACM13] generalising [Hay08] that we can construct a quaternionic pseudo-Kähler manifold ( N c , g c N , Q c ) of non-zero scalar curvature with a circle action such that the given pseudo-hyper-Kähler manifold N may be retrieved as a hyper-Kähler reduction of the Swann bundle of N c by a lift of the circle action at a non-zero level set.This construction is known as the HK/QK correspondence.
The signature of the resulting quaternionic pseudo-Kähler manifold and the sign of its scalar curvature depend on the signature of the pseudo-hyper-Kähler manifold N and the signs of the functions f c Z and f c H .The cases when one obtains a (positive-definite) quaternionic Kähler metric were specified in [ACM13] and include the case of quaternionic Kähler metrics of positive scalar curvature considered by Haydys, who started with a (positive-definite) hyper-Kähler metric.In the following theorem we focus on the cases which yield a positive definite metric of negative scalar curvature, of relevance to the present paper.
Then there is a lift of the circle action on N generated by Z to P × H × , so that its quotient M by the lifted action carries a conical pseudo-hyper-Kähler structure with hyper-Kähler reduction (N, g N , I 1 , I 2 , I 3 ).The conical pseudo-hyper-Kähler manifold M is the Swann bundle of a (positive-definite) quaternionic Kähler manifold Note that explicit expressions for all of the above data, including the quaternionic Kähler metric, are obtained in [Hay08,ACM13], [ACDM15, Theorem 2].We have however omitted these in the statement of the theorem to avoid redundancy, since we will describe the metric below using the language of Swann's twist construction.
Moreover, since there is the freedom of adding a constant term to the Hamiltonian function quaternionic Kähler manifolds of fixed scalar curvature associated to a pseudo-hyper-Kähler manifold.
In particular, we know by Proposition 2.7 that the result of applying the rigid c-map to a CASK manifold (M, g, J, ∇, ξ) fulfills the necessary conditions required for applying the HK/QK correspondence.In view of the results of [ACDM15], the composition of these two constructions with the choice of Hamiltonian functions , where f Z and f H are as in Proposition 2.7, will be called the supergravity c-map in this paper.
Note that the case c = 0 is distinguished and is called the undeformed supergravity cmap (corresponding to [FS90]), while the remaining cases are collectively referred to as the deformed supergravity c-map (corresponding to [RSV06]).It was shown in [CHM12,CDS17], under appropriate assumptions 2 on the CASK manifold, that the quaternionic Kähler metrics ( N c , g c N , Q c ) are complete if and only if c ≥ 0. We will therefore be assuming c ≥ 0 henceforth, although our methods work more generally.Note that, for a fixed CASK manifold, all the manifolds ( N c , g c N ) in the above family for different values of c > 0 are locally isometric (see [CDS17, Proposition 10]).For a discussion of further global properties of the deformed supergravity c-map, see [MS22].
The HK/QK correspondence, as described above, was interpreted as an instance of an even more general construction called the twist construction in [MS14].Roughly speaking, this construction, introduced earlier by Swann, takes as input a manifold N with twist data, i.e. a triple (ω, Z, f ) consisting of • an integral closed two-form ω, • a vector field Z generating a circle action which is Hamiltonian with respect to ω, • a choice of nowhere vanishing Hamiltonian function f , and gives as output a new manifold N with a circle action (and in fact, "dual" twist data, but this will not be important for our purposes).Furthermore, it also gives a bijective correspondence called H-relatedness between tensor fields of the same type on N and N which are invariant under the respective circle actions.In particular, if two functions f ∈ C ∞ (N ) and f ∈ C ∞ ( N ) are invariant under the respective circle actions and H-related, then they are either both constant or both non-constant.
We refer the reader to [Swa10, MS14, CST21, CST22] for the details, and only summarise some of the conclusions obtained from this perspective that we will be relying on for our results.
Note that HK/QK data on a pseudo-hyper-Kähler manifold N automatically give rise to twist data (ω H , Z, f c H ) on N .In fact, Macia and Swann prove the following.
Theorem 2.9 ([MS14, Theorem 1]).Let (N, g N , I 1 , I 2 , I 3 ) be a pseudo-hyper-Kähler manifold equipped with HK/QK data (Z, ω 1 , ω H , f c Z , f c H ). Then the quaternionic Kähler manifold ( N c , g c N , Q c ) given by the HK/QK correspondence is obtained by performing the 2 For instance, the assumption for c = 0 is simply that the underlying projective special Kähler manifold is complete, see [CHM12, Theorem 5] for details.For c > 0 the assumptions are specified in [CDS17, Theorems 13 and 27].
twist construction with respect to twist data (ω H , Z, f c H ). In particular, Q c is H-related to span{I 1 , I 2 , I 3 } and g c N is H-related to the metric where K is a non-zero constant of the same sign as f c Z .
Taking K to have the same sign as f c Z gives a quaternionic Kähler metric g c N that is positivedefinite whenever the given pseudo-hyper-Kähler metric g N is positive-definite when restricted to (HZ) ⊥ .The reduced scalar curvature of g c N is then given by ν = − 1 8K .Thus, the sign of f c Z determines the sign of the scalar curvature (they are opposite) while the choice of constant K determines its magnitude.In particular, for the supergravity c-map, f c Z is taken to be positive, so we may set K = 1.This gives us a positive-definite supergravity c-map metric of reduced scalar curvature − 1 8 .
The twist construction was furthermore used in [CST22] to obtain a tensor Rm ∈ Γ (T * N ) ⊗4 on the pseudo-hyper-Kähler manifold N to which the (lowered) Riemann curvature Rm N ∈ Γ (T * N c ) ⊗4 of the quaternionic Kähler metric g c N is H-related.In order to state the result, we will first need to introduce some notation.
(i) We define the Kulkarni-Nomizu map for arbitrary vector fields A, B, C, X.
Theorem 2.11 ([CST22, Theorem 3.4]).Let (N, g N , I 1 , I 2 , I 3 ) be a pseudo-hyper-Kähler manifold equipped with HK/QK data (Z, ω 1 , ω H , f c Z , f c H ) and let ( N c , g c N , Q c ) be the quaternionic Kähler manifold given by the HK/QK correspondence.Then the (lowered) Riemann curvature Rm N of the metric g c N is H-related to the tensor where Rm HK and Rm HP are defined to be Note that (3) reflects a refinement of the Alekseevsky decomposition of the curvature of a quaternionic Kähler manifold of reduced scalar curvature − 1 8 arising from the HK/QK correspondence.The first two terms on the right corresponds to the part of hyper-Kähler type, while the last term corresponds to − 1 8 times the curvature of the quaternionic projective space of unit reduced scalar curvature.In particular, it follows that both Rm N and Rm HK are separately g c H -orthogonal to Rm HP .As an application of Theorem 2.11, we see that the norm of the curvature tensor Rm N of the metric g c N on the quaternionic Kähler side is not constant if the norm of Rm on the pseudohyper-Kähler side is not constant.We will indeed proceed by specialising this argument to the case of the deformed supergravity c-map in the next section.

Curvature of special Kähler manifolds
We now proceed to compute the curvature of ASK manifolds.Proposition 3.1.Let (M, g, J, ω, ∇) be an affine special Kähler manifold.Then the curvature where we have used that ∇ is flat.On the other hand, we have by expressing the final result in a coordinate independent way using only the above intrinsic identifications and basic properties of ASK manifolds (such as the complete symmetry of ∇g).
Note that the remaining components of the Riemann curvature follow from the above by symmetries of the curvature tensor and that ∇ 2 g coincides with ∇S, where S is the totally symmetric (0, 3)-tensor which corresponds to the (1, 2)-tensor S.
Remark 3.4.If the ASK manifold M satisfies that ∇ coincides with the Levi-Civita connection, then Rm M = 0 (since ∇ is flat by definition) and hence Rm N = 0 by Theorem 3.3.
In the case where the ASK manifold M is furthermore CASK, we can say something additional.
Proposition 3.5.Let (M, g, J, ∇, ξ) be a CASK manifold and (N = T * M, g N , I 1 , I 2 , I 3 ) the pseudo-hyper-Kähler manifold given by the rigid c-map.Then the curvature tensor Rm N is a section of the subbundle where we are using the isomorphism T * N ∼ = (HZ) * ⊕ ((HZ) ⊥ ) * corresponding to (2) and ∨ denotes the symmetric tensor product.In particular, Rm N (A, B, C, X) = 0 if at least two of the vectors A, B, C, X belong to HZ.
Proof.We have seen that the curvature of N is completely determined by tensors on the base M .Under the identifications T H p N ∼ = T π(p) M and T V p N ∼ = T * π(p) M the horizontal vector fields Z, I 1 Z on N are identified with the vector fields −Jξ, ξ on M , and the vertical vector fields I 2 Z, I 3 Z with the one-forms ξ ♭ , (−Jξ) ♭ (with the convention ω = g(J•, •)).Every term in Theorem 3.3 can be expressed in terms of the tensor S. From Corollary 2.6 we know that S vanishes on ξ and Jξ, therefore all the curvature elements are zero taking into account that S and ∇ 2 g are totally symmetric.In fact, the total symmetry of S = ∇g was stated in Lemma 2.4 and implies that of ∇S = ∇ 2 g using that ∇ 2 A,B g = ∇ 2 B,A g since ∇ is flat.

Norm of the curvature tensor
Let us start by defining what is a locally homogeneous manifold.
Definition 3.6.A Riemannian manifold (M, g) of dimension n is called locally homogeneous if for all p ∈ M there exist n Killing vector fields defined in a neighborhood of p which are linearly independent at p.
Note that a function on a connected locally homogeneous Riemannian manifold which is invariant under any locally defined isometry is necessarily constant.
We will now finally show that the curvature norm of the (deformed) local c-map metric g c N associated to a CASK manifold M is not constant on the manifold N c unless c = 0. We had already argued in Section 2.3 that this is equivalent to showing that the norm Rm 2 g c H of Rm with respect to g c H , is not constant on the rigid c-map space N = T * M .In order to compute this norm, we work in a g N -orthonormal frame {e i , ǫ µ } of T N that is adapted to the quaternionic distribution HZ.This means that e i span the distribution HZ and ǫ µ span the orthogonal complement (HZ) ⊥ .
In terms of this frame, the norm of an abstract (0, 4)-curvature tensor C with respect to the metric g c H is given by where ĝc H := (g c H ) −1 ⊗4 denotes the metric on the bundle (T * N ) ⊗4 induced by g c H . Let us now specialise Theorem 2.11 to the case of the deformed supergravity c-map.Since the decomposition between the hyper-Kähler part and the projective quaternionic space part is orthogonal, we have The final term 1 64 Rm HP 2 g c H is a constant depending only on the dimension.Meanwhile the remaining terms can be computed to be As a consequence, we obtain our main theorem.
Theorem 3.9.Let (M, g, J, ∇, ξ) be a CASK manifold and ( N c , g c N , Q c ) the quaternionic Kähler manifold given by the deformed supergravity c-map.Then, for any c > 0, ( N c , g c N ) is not locally homogeneous.
By the results of [CST21, CRT21, MS22], given a CASK manifold (M, g, J, ω, ∇, ξ) of real dimension 2n with automorphism group Aut(M ), by which we mean that Aut(M ) is the group of isometries of (M, g) preserving the full CASK data, the associated supergravity c-map spaces ( N c , g c N , Q c ) are isometrically acted on by the group Aut(M ) ⋉ Heis 2n+1 , provided that the underlying projective special Kähler manifold ( M, g M ) is simply connected 3 .In particular, when Aut(M ) acts transitively on the underlying projective special Kähler manifold M , we obtain an action of Aut(M ) ⋉ Heis 2n+1 that is transitive on the level sets of the norm of the quaternionic moment map associated to the circle action on N .Thus, as a corollary of our main theorem, we have the following generalisation of a result in [CST22] concerning the supergravity c-map space associated to a flat CASK manifold.
Corollary 3.10.Let (M, g, J, ω, ∇, ξ) be a CASK manifold fibering over a simply connected projective special Kähler manifold ( M, g M ).Assume that its automorphism group Aut(M ) acts transitively on M .Then the associated supergravity c-map space ( N c , g c N , Q c ) is a complete quaternionic Kähler manifold of cohomogeneity exactly one when c > 0.
Proof.Since the Riemannian manifold ( M, g M ) is complete, the corresponding undeformed c-map space ( N, g N ) = ( N 0 , g 0 N ) is complete in virtue of [CHM12, Theorem 10].As a first step, we will show that ( N, g N ) is not only complete but is in fact homogeneous.
The group of isometries Aut( M ) ⊂ Isom( M ) induced by Aut(M ) extends canonically to a group of isometries of ( N, g N ).This is stated in [CDJL12, Proposition 26] for CASK domains but holds in general as a consequence of [CHM12, Lemma 4].It can be also seen as a special case (c = 0) of the results of [CST21, CRT21, MS22] mentioned above.The group Aut( M ) acts transitively on the base of the fiber bundle N → M mapping fibers to fibers.In addition, there is a fiber-preserving isometric action of the solvable Iwasawa subgroup G 2n+2 of SU(1, n+1) on N | Ū [CHM12, Theorem 5] for every domain U ⊂ M , which is isomorphic to a CASK domain, where Ū denotes the image of U under the projection M → M .(Recall that every CASK manifold is locally isomorphic to a CASK domain.)Note that dim G 2n+2 = 2n + 2, where dim M = 2n − 2.This solvable group action on N | Ū is simply transitive on each fiber.In particular, for every such Ū there is a Lie algebra g Ū ∼ = g 2n+2 = Lie(G 2n+2 ) of Killing fields of N | Ū transitive on each fiber.Moreover, g Ū can be identified with the space of parallel sections over Ū of a flat symplectic vector bundle over M (with Lie algebras as fibers), compare [CHM12, Theorem 9].Since M is simply connected the above vector bundle has a global parallel frame.Thus we obtain a globally defined Lie algebra of Killing fields g ∼ = g 2n+2 of N , which restricts to g Ū on the domain N | Ū ⊂ N .Since N is complete, there is a corresponding Lie group G acting on N , which together with Aut( M ) generates a transitive group of isometries of N .Now that we know that ( N, g N ) is a homogeneous quaternionic Kähler manifold of negative scalar curvature, we can apply the following arguments to show that it belongs to the class 3 The assumption of simply connectedness can be dropped if M is a CASK domain.
of Alekseevsky spaces.First of all, in virtue of the resolution of the Alekseevsky conjecture about the structure of homogeneous Einstein manifolds of negative scalar curvature by Böhm and Lafuente [BL21], we know that ( N, g N ) admits a simply transitive solvable Lie group of isometries.Then, by a result of Lauret [Lau10], it is a standard Einstein solvmanifold in the sense of Heber [Heb98].Finally, by [Heb98, Theorem B] such a manifold admits a simply transitive completely solvable group of isometries.Quaternionic Kähler manifolds with that property were classified by Alekseevsky [Ale75,Cor96].
We claim that the one-loop deformation of any c-map space which is an Alekseevsky space is complete if the deformation parameter c is positive (for c = 0 it holds by homogeneity).First we note that all of the Alekseevsky spaces with exception of the quaternionic hyperbolic spaces and the Hermitian symmetric spaces of non-compact type dual to complex Grassmannians of 2-planes can be represented as q-map spaces [dWP92], a special class of c-map spaces.By [CDS17, Theorem 27] the one-loop deformation of a complete q-map space is complete if c > 0. In particular, the one-loop deformed Alekseevsky q-map spaces with c > 0 are complete.Furthermore, the Hermitian symmetric Alekseevsky spaces were shown to have regular boundary behavior, implying the completeness of their one-loop deformation for c > 0 [CDS17, Example 14 and Theorem 13].
Finally, we are left with the quaternionic hyperbolic spaces HH n .We claim that these cannot be represented as c-map spaces and hence cannot occur in our setting.This can be seen by looking at totally geodesic Kähler submanifolds compatible with the quaternionic structure.Thanks to [AM01] we know that the maximal possible dimension of a Kähler submanifold compatible with the quaternionic structure of a quaternionic Kähler manifold of dimension 4n is 2n.In the case of HH n the only totally geodesic Kähler submanifolds of (real) dimension 2n compatible with the quaternionic structure are the complex hyperbolic subspaces CH n (up to isometries of the ambient space).On the other hand, any c-map space of dimension 4n has a totally geodesic Kähler submanifold compatible with the quaternionic structure of the form CH 1 × M , where M is the underlying projective special Kähler manifold of dimension 2n−2.In fact, the submanifold CH 1 × M ⊂ N is obtained as the fixed point set of the isometric involution expressed in standard fiber coordinates (ρ, φ, ζi , ζ i ), i = 1, . . ., 2n, [CHM12] by (ρ, φ, ζi , ζ i ) → (ρ, φ, − ζi , −ζ i ).Since CH n is irreducible, we see that HH n is not a c-map space if n > 1.The case n = 1 is also excluded, since the c-map space associated with a projective special Kähler manifold reduced to a point is CH 2 (belonging to the Hermitian symmetric series) and not HH 1 .This finishes the proof of the completeness of ( N c , g c N ) for c > 0. Now the corollary follows from the fact that ( N c , g c N ) has a group of isometries acting with cohomogeneity one but no such group acting with cohomogeneity 0.