2.1. Twists and the HK/QK correspondence

The starting point for the HK/QK correspondence is a pseudo-hyper-Kähler manifold $(N,g_N,I_k)$, $k=1,2,3$, endowed with a distinguished vector field.

In order to apply the HK/QK correspondence, we ask that $(N,g_N,I_k)$ comes equipped with a rotating Killing field Z which is Hamiltonian with respect to the two-form $\omega_\mathrm{H}=\omega_1+\mathrm{d}(g_N(Z,-))$, where ω ₁ is the Kähler form associated with I ₁. Furthermore, we will require that the Hamiltonian function, which we denote by $f_\mathrm{H}$, is nowhere-vanishing. We remark that, under these assumptions, Z is automatically Hamiltonian with respect to ω ₁, with Hamiltonian function $f_Z:= f_\mathrm{H}-g_N(Z,Z)$. We will also assume that f_Z is nowhere-vanishing.

Under the assumption that ω ₁ is an integral form, the triple $(Z,\omega_\mathrm{H},f_\mathrm{H})$ forms twist data on N in the sense of A. Swann [Reference Swann29]. In particular, there exists a principal circle bundle $\pi_N\colon P\to N$ with connection η whose curvature is $\omega_\mathrm{H}$, and Z lifts to a vector field Z_P on P that preserves η. Explicitly, we have

(1)\begin{equation} Z_P=\tilde Z+\pi_N^*f_\mathrm{H} X_P \end{equation}

where X_P generates the principal circle action on P and $\tilde Z$ is the η-horizontal lift of Z. The quotient space $\bar N:= P/\langle Z_P\rangle$ is then called the twist of N with respect to the twist data $(Z,\omega_\mathrm{H},f_\mathrm{H})$. Under the assumption that Z and Z_P act freely and properly, it is a smooth manifold. Note that $\bar N$ comes equipped with a distinguished vector field X, obtained by pushing down X_P.

The power of the twist construction lies in the fact that Z-invariant structures on N can be carried over to $\bar N$. For instance, if φ is an invariant function on N, then $\pi^*_N\varphi\in C^\infty(P)$ is invariant under Z_P and therefore induces a well-defined function ψ on $\bar N$, which we will call the twist of φ. Similarly, we can push down the η-horizontal lift $\tilde U$ of an invariant vector field U on N to obtain its twist $V=d \pi_{\bar N}(\tilde U)$ on $\bar N$. Twists of arbitrary tensor fields are then defined by demanding compatibility of twisting with tensor products and contractions. By comparing lifts to P, one can work out explicit formulas for the twist of a tensor field without much effort.

The following Lemmata, whose proofs can be found in [Reference Swann29], follow from such computations:

Returning to the setting of the HK/QK correspondence, let $(N,g_N,I_k,Z)$, $k=1,2,3$, be a pseudo-hyper-Kähler manifold endowed with a rotating Killing field Z, and consider the twist data $(Z,\omega_\mathrm{H},f_\mathrm{H})$ as above.

The quaternionic structure bundle $\langle I_1,I_2,I_3\rangle\subset \operatorname{End}(TN)$ is invariant under Z and therefore induces an almost quaternionic structure $\mathcal{Q}$ on the resulting twist manifold $\bar N$ (with respect to the twist data $(Z,\omega_\mathrm{H},f_\mathrm{H})$ as above). In fact, it is known [Reference Alekseevsky, Cortés, Dyckmanns and Mohaupt3, Reference Macia and Swann25] that $(\bar N,\mathcal{Q})$ carries a compatible quaternionic Kähler metric $g_{\bar N}$. This metric, however, is not the twist of the pseudo-hyper-Kähler metric g_N, but rather of its so-called elementary deformation $g_\mathrm{H}$ [Reference Macia and Swann25]. The elementary deformation $g_\mathrm{H}$ is related to g_N by the formula

(2)\begin{equation} g_\mathrm{H}= \frac{1}{f_Z}g_N|_{(\mathbb{H} Z)^\perp}+\frac{f_\mathrm{H}}{f_Z^2}g_N|_{\mathbb{H} Z} \end{equation}

where $\mathbb{H} Z=\langle Z,I_1 Z,I_2Z,I_3Z\rangle$ denotes the quaternionic span of Z. We say that the quaternionic Kähler manifold $(\bar N,g_{\bar N},\mathcal{Q})$ arises from the HK/QK correspondence applied to N. Since the signature of a metric is preserved by twisting, the quaternionic Kähler metric $g_{\bar N}$ is positive definite if and only if $g_\mathrm{H}$ is. From (2), we see that this may happen even if g_N has signature $(4k,4)$, as long as $f_Z \gt 0$ and $f_\mathrm{H} \lt 0$. This observation is crucial in applying the HK/QK correspondence to the c-map [Reference Alekseevsky, Cortés and Mohaupt4]. Since we are interested primarily in positive-definite quaternionic Kähler metrics, we will from now on assume that $g_\mathrm{H}$ is Riemannian (though our results remain valid, mutatis mutandis, in the indefinite case).

Note that the above construction depends on our choice of Hamiltonian function $f_\mathrm{H}$, which appears explicitly in the expression for the elementary deformation $g_\mathrm{H}$, whose twist is the quaternionic Kähler metric $g_{\bar N}$. Since there is always a one-parameter freedom in choosing a Hamiltonian function, the HK/QK correspondence in fact yields a one-parameter family of quaternionic Kähler metrics on $\bar N$.

We note the following property of quaternionic Kähler manifolds resulting from the HK/QK correspondence:

2.2. The c-map

In order to put the HK/QK correspondence to use, we need a way to construct pseudo-hyper-Kähler manifolds endowed with a rotating Killing field. The theory of special Kähler manifolds provides a plentiful source of examples.

We recall how to construct a pseudo-hyper-Kähler manifold N with rotating Killing field from a CASK manifold $(M,g_M,J_M,\nabla^{\mathrm{SK}},\xi)$ of real dimension $2n+2$. The smooth manifold underlying N is $T^*M$. Now note that the special Kähler connection $\nabla^{\mathrm{SK}}$ induces a splitting $T(T^*M)\cong \pi^*T^*M\oplus \pi^*TM$. With respect to this splitting, we define the tensor fields

(3)\begin{equation} g_N= \begin{pmatrix} g_M & 0 \\ 0 & g^{-1}_M \end{pmatrix} \qquad I_1= \begin{pmatrix} J_M & 0 \\ 0 & J_M^* \end{pmatrix} \qquad I_2= \begin{pmatrix} 0 & -\omega_M^{-1} \\ \omega_M & 0 \end{pmatrix} \qquad I_3=I_1I_2 \end{equation}

where we have omitted pullbacks throughout to simplify notation. It is well-known (see, e.g., [Reference Alekseevsky, Cortés and Devchand2] and references therein) that these tensor fields define a pseudo-hyper-Kähler structure on $N=T^*M$ of quaternionic signature $(n,1)$. This construction is known as the rigid c-map. Moreover, the $\nabla^{\mathrm{SK}}$-horizontal lift of $-J_M\xi$ to N endows it with a canonical rotating Killing field Z [Reference Alekseevsky, Cortés and Mohaupt4].

This construction gives rise to many examples, since there is an abundance of CASK manifolds. Indeed, if $U\subset \mathbb{C}^n$ is any $\mathbb{C}^*$-invariant domain, there is a CASK manifold associated with any generic holomorphic function on U which is homogeneous of degree 2 under the $\mathbb{C}^*$-action, see [Reference Alekseevsky, Cortés and Devchand2] for more detailed statements.

It follows easily from the definitions that the vector field $-J_M\xi$ on a CASK manifold M is Killing and Hamiltonian with negative Hamiltonian function $\frac{1}{2}g_M(\xi,\xi)$, so we may consider the Kähler quotient $M\ \unicode{x2AFD}\ S^1=\mu^{-1}\big(-\frac{1}{2}\big)/S^1$ with respect to the circle action it generates.

Note that the induced Kähler metric on a PSK manifold is always positive-definite.

By this (extrinsic) definition, giving a PSK structure on a manifold is equivalent to giving the corresponding CASK manifold, and the two manifolds can be used interchangeably for our purposes. Thus, we may say that, starting from a PSK manifold, we may pass to the associated CASK manifold and apply the rigid c-map construction and the HK/QK correspondence to obtain a quaternionic Kähler manifold. Thus, we have obtained a method to associate quaternionic Kähler manifolds to PSK manifolds (cf. [Reference Alekseevsky, Cortés, Dyckmanns and Mohaupt3]). This construction recovers what is known as the (supergravity) c-map [Reference Ferrara and Sabharwal16].

Recall that we obtain not just one, but an entire one-parameter family of quaternionic Kähler metrics from a single hyper-Kähler manifold with rotating Killing vector field by varying the corresponding Hamiltonian function. In the case of the c-map, there is a distinguished choice of Hamiltonian. We call the corresponding quaternionic Kähler metric the undeformed c-map metric and denote it by $g_{\bar N}^0$. The quaternionic Kähler metrics that arise from different choices of Hamiltonian function obtained by shifting the distinguished one by a constant $c\in \mathbb{R}$ are known as (one-loop) deformed c-map metrics [Reference Llana, Saueressig and Vandoren24] and denoted by $g_{\bar N}^c$. It turns out that these metrics have good completeness properties only if we restrict to $c\geq 0$ [Reference Cortés, Dyckmanns and Suhr9], which we will do from now on.

6.1. Local classification from the Przanowski–Tod ansatz

The one-parameter family $(\bar N_1,g_1^c)$ is known as the (one-loop) deformed universal hypermultiplet in the physics literature and its members are the only 4D quaternionic Kähler manifolds that arise from the c-map (namely, by applying it to a point). Mathematically, $(\bar N_1,g_1^c)$ is a one-parameter deformation of the Hermitian symmetric space $\mathbb{C} H^2\cong (\bar N_1,g_1^0)$. In this section, we construct further examples belonging to the class described in theorem 3.2. In order to do so, we widen our scope to arbitrary quaternionic Kähler four-manifolds endowed with a Killing field.

Via HK/QK correspondence, any quaternionic Kähler manifold of dimension 4 equipped with a Killing field (locally) arises from a hyper-Kähler four-manifold $(N,g_N)$ with rotating Killing field. Such a hyper-Kähler metric is locally homothetic to a metric satisfying the Boyer–Finley ansatz [Reference Boyer and Finley6]. With respect to local coordinates $(\rho,x,y,t)$, this means that

\begin{equation*} g_N= K(\partial_\rho u)(\mathrm{d} \rho^2 + 2 e^u(\mathrm{d} x^2 +\mathrm{d} y^2)) +\frac{4K}{\partial_\rho u} \bigg(\mathrm{d} t-\frac{1}{2}(\partial_y u\,\mathrm{d} x-\partial_x u\,\mathrm{d} y)\bigg)^2 \end{equation*}

with u a smooth function of ρ, x, y (but not t) satisfying the continuous Toda equation

\begin{equation*} (\partial_x^2+\partial_y^2)u =- 2\partial_\rho^2(e^u). \end{equation*}

The rotating Killing vector field is $\partial_t$ and the invariant Kähler form is

\begin{equation*} \omega_1 =2K\mathrm{d}\rho\wedge \bigg(\mathrm{d} t-\frac{1}{2}(\partial_y u\,\mathrm{d} x-\partial_x u\,\mathrm{d} y)\bigg) +2Ke^u\partial_u\rho\,\mathrm{d} x\wedge \mathrm{d} y. \end{equation*}

The Killing vector field $\partial_t$ is Hamiltonian. Without loss of generality, the Hamiltonian function can be taken to be $f_Z = 2K\rho$ (shifts in ρ can be adsorbed into a redefinition of the function u).

The result of applying the HK/QK correspondence to the Boyer–Finley ansatz is the Przanowski–Tod ansatz [Reference Przanowski27]. Up to constant scaling, any 4D quaternionic Kähler metric admitting a Killing field is locally given by

\begin{equation*} g_{\bar{N}} = \frac{1}{4\rho^2}\left(P\,\mathrm{d} \rho^2 + 2P e^u(\mathrm{d} x^2 +\mathrm{d} y^2) + \frac{1}{P}\,(\mathrm{d} t + \Theta)^2\right) \end{equation*}

where P is a $\partial_t$-invariant smooth function and Θ is a $\partial_t$-invariant one-form satisfying

\begin{equation*} P=K(\rho\,\partial_\rho u -2) \gt 0, \quad \mathrm{d} \Theta =\big(\partial_y P\,\mathrm{d} x - \partial_{x} P\,\mathrm{d} y\big)\wedge \mathrm{d} \rho - 2\partial_\rho (Pe^u)\,\mathrm{d} x\wedge \mathrm{d} y. \end{equation*}

We have normalized $g_{\bar N}$ so that the reduced scalar curvature ν equals $\frac{2}{K}$. In particular, since P is required to be positive, the function $\rho\,\partial_\rho u -2$ must have a definite sign over the domain of definition. This sign is furthermore equal to the sign of K and hence the sign of the scalar curvature.

Note that if we let $\zeta = x + i y$ and f be a holomorphic function of ζ whose derivative is non-vanishing in the domain of definition, then the replacement

(10)\begin{equation} u(\rho,\zeta, \overline \zeta) \mapsto u(\rho,f(\zeta),\overline{f(\zeta)}) - \ln\left\lvert\frac{\mathrm{d} f}{\mathrm{d}\zeta}\right\rvert^2 \end{equation}

doesn’t change the isometry class of the metrics g_N and $g_{\bar N}$. Such a replacement can, therefore, be regarded as a gauge transformation.

Now we can explicitly characterize which of such quaternionic Kähler metrics are conformal to a Kähler metric compatible with $\tilde J_1$ in terms of the function u up to gauge transformations.

Theorem 6.1. A quaternionic Kähler four-manifold which arises from the HK/QK correspondence has the structure $\tilde{J}_1$ integrable if and only if it is locally isometric to a metric of the form

(11)\begin{equation} \begin{aligned} g^{a,b,c} &= -\frac{K}{4\rho^2}\left(\frac{b\rho + 2c}{a\rho^2 + b\rho + c}\,\mathrm{d} \rho^2 + \frac{2(b\rho+2c)|\mathrm{d}\zeta|^2}{(1 + \frac{a}{2}|\zeta|^2)^2}\right.\\ &\quad+\left. \frac{a\rho^2 + b\rho + c}{b\rho + 2c}\left(-\frac{\mathrm{d} t}{K} + \frac{b\,\mathrm{Im}(\zeta \,\mathrm{d}\overline\zeta)}{1 + \frac{a}{2}|\zeta|^2}\right)^2\right) \end{aligned} \end{equation}

where $a,b,c$ are real constants, and ρ and ζ satisfy

(12)\begin{equation} \rho \gt 0,\qquad a\rho^2 + b\rho + c \gt 0, \qquad b\rho + 2c \neq 0,\qquad 1+ \frac{a}{2}|\zeta|^2 \gt 0 \end{equation}

and K is a non-zero constant for which the metric is positive-definite. These metrics arise by applying the Przanowski–Tod ansatz to

(13)\begin{equation} e^u = \frac{a\rho^2 + b\rho + c}{(1 + \frac{a}{2}|\zeta|^2)^2} \end{equation}

up to gauge transformation.

Proof. Any quaternionic Kähler four-manifold which arises from the HK/QK correspondence is locally homothetic to the Przanowski–Tod ansatz, so we restrict to this case. By theorem 5.6 and lemma 3.5, it suffices to show that $\mathrm{d} (g_N(Z,Z))\wedge \mathrm{d} f_Z=0$ if and only if (13) holds. The Hamiltonian f_Z is $4K\rho$ in our case while $g_N(Z,Z)$ can be seen to be $4K/\partial_\rho u$. So, the condition above amounts to saying that the differential of $\partial_\rho u$ is proportional to $\mathrm{d} \rho$, which is to say u is a separable function

\begin{equation*} u = F(\rho) + G(\zeta,\overline\zeta). \end{equation*}

Substituting this into the continuous Toda equation, we obtain

(14)\begin{equation} \partial_\zeta\partial_{\overline\zeta} G = -ae^G, \quad \partial_\rho^2 e^F = 2a. \end{equation}

The equation for F implies that e^F is a quadratic polynomial in ρ with leading coefficient a. That is, its general solutions are given by

(15)\begin{equation} e^F = a\rho^2 + b\rho + c. \end{equation}

Note in particular that the function ρ is nowhere vanishing and so has a definite sign if we assume connectedness. Since the sign can always be absorbed into the constant b, we can assume without loss of generality ρ > 0.

Meanwhile, the equation for G is just the 2D Liouville equation whose general solutions are known to be of the form

(16)\begin{equation} e^G = \frac{4}{(1 + 2a|f(\zeta)|^2)^2}\left\lvert\frac{\mathrm{d} f}{\mathrm{d} \zeta}\right\rvert^2 \end{equation}

where f is some holomorphic function which is non-vanishing in the domain of definition. The freedom to choose f may be absorbed into the gauge transformation (10). In particular, we may set $f(\zeta) = \frac{1}{2}\zeta$. This gives us the required metric. Moreover, we may additionally assume without any loss of generality that $1+ \frac{a}{2}|\zeta|^2 \gt 0$. To see this, note that when $a\ge 0$, this is automatic, and when a < 0, the replacement $\zeta\mapsto \frac{2}{a\zeta}$ leaves e^G (and hence the metric) unchanged but swaps the regions $\pm(1+ \frac{a}{2}|\zeta|^2) \gt 0$.

6.2. Case-by-case analysis

The metrics $g^{a,b,c}$ have previously been studied by Ketov [Reference Ketov21]. There are many ranges for the parameters in which we recover well-known examples of quaternionic Kähler four-manifolds, as can be seen from the following case-by-case analysis of the metrics.

1. (i) When $a=b=0$ we get the real hyperbolic 4-space $\mathrm{SO}(4,1)/\mathrm{O}(4)$ presented as solvable Lie group $\mathbb{R}_{ \gt 0} \lt imes \mathbb{R}^3$ (the Iwasawa subgroup of $\mathrm{SO}_0(4,1)$) with a left-invariant metric.

2. (ii) For a = 0, b ≠ 0, the metrics $g^{a,b,c}$ are isometric to the (one-loop) deformed universal hypermultiplet metrics, i.e., yield a one-parameter deformation of the complex hyperbolic plane $\mathrm{SU}(2,1)/\mathrm{U}(2)$ (c = 0) through quaternionic Kähler metrics. For c ≠ 0, the full isometry group of the deformed universal hypermultiplet is known to be $\mathrm{O}(2) \lt imes \operatorname{Heis}_3$, acting with cohomogeneity 1 [Reference Cortés, Saha and Thung12].

3. (iii) For $a,b \gt 0$, we may perform the following change of coordinates:

   \begin{equation*} \begin{aligned} \rho = \frac{b}{2a}\left(\frac{1}{\varrho^2}-1\right), \quad \zeta = \sqrt{\frac{2}{a}}\,\xi,\quad t = \frac{b K}{a}\,\theta \end{aligned} \end{equation*}
   where $0 \lt \varrho \lt 1$. In these coordinates, after making the identification $\theta\sim \theta + 2\pi$ and smoothly extending to $\zeta =\infty$ and ϱ = 0, the metric $g^{a,b,c}$ becomes the Pedersen metric [Reference Pedersen26] on the open unit ball in $\mathbb{R}^4$:
   \begin{align*} g^{a,b,c}&=\frac{-1}{\nu(1-\varrho^2)^2} \left(\frac{1 + k \varrho^2}{1 + k\varrho^4}\,\mathrm{d}\varrho^2 + \varrho^2(1+ k \varrho^2)(\varsigma^2_1 + \varsigma^2_2) \right. \\ &\quad \left. + \frac{\varrho^2(1 + k \varrho^4)}{1 + k\varrho^2}\varsigma^2_3\right). \end{align*}

   Here, k is given by $k = \frac{4ac}{b^2} -1$. The Pedersen metric is known to be invariant under the standard action of $\mathrm{U}(2)$ on S ³, as confirmed by the appearance of the $\mathrm{SU}(2)$-invariant one-forms $\varsigma_1,\varsigma_2,\varsigma_3$, which are given by

   \begin{equation*} \begin{aligned} \varsigma_1 &=\frac{\mathrm{Re}(e^{i \theta}\mathrm{d}\xi)}{1 + |\xi|^2}, \quad \varsigma_2 = \frac{\mathrm{Im}(e^{i \theta}\mathrm{d}\xi)}{1 + |\xi|^2}, \quad \varsigma_3 = \frac{1}{4}\,\mathrm{d} \theta + \frac{\operatorname{Im}(\xi\,\mathrm{d} \overline\xi)}{1 + |\xi|^2} \end{aligned} \end{equation*}
   with respect to the Hopf parametrization. The combination in which they appear in the metric ensures that it is even $\mathrm{U}(2)$-invariant.

4. (iv) As a limiting case of the above, we have a > 0 and b = 0. If c < 0, we carry out a change of coordinate $\rho = \frac{1}{\rho'}\sqrt{\frac{c}{a}}$ to obtain the following metric on $\mathbb{R}\times \mathbb{R}_{ \gt 0}\times S^2 \cong \mathbb{R}\times (\mathbb{R}^3\setminus \{0\}) \subset \mathbb{R}^4$:

   \begin{equation*} \begin{aligned} g^{a,b,c} &= -\frac{1}{\nu}\left(\frac{\mathrm{d} \rho'^2}{1+\rho'^2} + (1+\rho'^2)\left(\frac{\mathrm{d} t}{2K}\right)^2 + \rho'^2 g_{S^2} \right) \end{aligned} \end{equation*}
   where $g_{S^2}$ is the standard metric on a unit S ². By including a line parametrized by t at $\rho'=0$, this can be extended to the real hyperbolic 4-space metric on $\mathbb{R}^4$.

   On the other hand, if c < 0, we carry out a change of coordinate $\rho = \frac{1}{\rho'}\sqrt{-\frac{c}{a}}$ to obtain the following metric on $\mathbb{R}\times (0,1)\times S^2 \subset \mathbb{R}^4$:

   \begin{equation*} \begin{aligned} g^{a,b,c} &= -\frac{1}{\nu}\left(\frac{\mathrm{d} \rho'^2}{1-\rho'^2} + (1-\rho'^2)\left(\frac{\mathrm{d} t}{2K}\right)^2 + \rho'^2 g_{S^2} \right). \end{aligned} \end{equation*}

   Upon making the identification $t\sim t + 4\pi K$, the first two terms give the standard S ² metric restricted to a hemisphere minus the pole, with $\sin^{-1}(\rho')$ and $\frac{t}{2K}$ being the latitude and longitude respectively. With this identification, the full metric $g^{a,b,c}$ can be extended to S ⁴ to give the standard $\mathrm O(5)$-invariant metric on it.

5. (v) When $b \lt 0 \lt a$ and $2a\rho + b \lt 0$, the same change of coordinates as in case (iii) can be carried out, but now we have ϱ > 1, as can be seen from

   \begin{equation*} \varrho^2 =\frac{b}{2a\rho + b}= \frac{|b|}{|b| - 2a\rho} \gt 1. \end{equation*}

   We are, therefore, dealing with a metric on the complement of the closed unit ball in $\mathbb{R}^4$. The scalar curvature remains negative since

   \begin{equation*} -(b\rho +2c) = (2a\rho + b)\rho - 2(a\rho^2 + b\rho + c) \lt 0. \end{equation*}

   This does not change the fact that the metric is $\mathrm{U}(2)$-invariant, since it is given by the same formal expression as the Pedersen metric.

6. (vi) When $b \lt 0 \lt a$ and $2a\rho + b \gt 0$, we instead set

   \begin{equation*} \begin{aligned} \rho = -\frac{b}{2a}\left(\frac{1}{\varrho^2}+1\right), \quad \zeta = \sqrt{\frac{2}{a}}\,\xi,\quad t = \frac{bK}{a}\,\theta. \end{aligned} \end{equation*}

   The metric now becomes

   \begin{equation*} \begin{aligned} g^{a,b,c}\!=\!\frac{1}{\nu(1+\varrho^2)^2} \left(\frac{1 - k \varrho^2}{1 + k\varrho^4}\,\mathrm{d}\varrho^2 + \varrho^2(1- k \varrho^2)(\varsigma^2_1 \!+\! \varsigma^2_2) + \frac{\varrho^2(1 + k \varrho^4)}{1 - k\varrho^2}\varsigma^2_3\right)\!. \end{aligned} \end{equation*}

   For $k\le0$, the scalar curvature is positive since

   \begin{equation*} -(b\rho+2c)= \frac{-b}{2a}(2a\rho + b + kb) \gt 0. \end{equation*}

   These metrics are $\mathrm{U}(2)$-invariant in general (for the same reason as the Pedersen metric), but for particular values of k we can say more. Setting k = 0 gives the standard $\mathrm{O}(5)$-invariant metric on the complement of a point in S ⁴. If we set $k = -1$ instead (which corresponds to c = 0) we obtain

   \begin{equation*} \begin{aligned} g^{a,b,c}=\frac{1}{\nu(1+\varrho^2)^2} \left(\frac{\mathrm{d}\varrho^2}{1 - \varrho^2} + \varrho^2(1+ \varrho^2)(\varsigma^2_1 + \varsigma^2_2) + \varrho^2(1 - \varrho^2)\varsigma^2_3\right). \end{aligned} \end{equation*}

   This is the $\mathrm{U}(3)$-invariant Fubini–Study metric on the complement of the point $[z_0:z_1:z_2]=[1:0:0]$ in the complex projective plane with

   \begin{equation*} \frac{1}{\varrho^{2}} = \frac{2}{z_1^2+z_2^2} + 1. \end{equation*}

7. (vii) For $a,b \lt 0$, we may perform the following change of coordinates:

   \begin{equation*} \begin{aligned} \rho = \frac{b}{2a}\left(\frac{1}{\varrho^2}-1\right), \quad \zeta = \sqrt{-\frac{2}{a}}\,\xi,\quad t = \frac{bK}{a}\,\theta \end{aligned} \end{equation*}
   where $0 \lt \varrho \lt 1$. In these coordinates, after making the identification $\theta\sim \theta + 2\pi$, the metric $g^{a,b,c}$ becomes
   \begin{align*} g^{a,b,c}&=\frac{-1}{\nu(1-\varrho^2)^2} \left(\frac{1 + k \varrho^2}{1 + k\varrho^4}\,\mathrm{d}\varrho^2 - \varrho^2(1+ k \varrho^2)(\varsigma'^2_1 + \varsigma'^2_2) \right.\\ &\quad \left.+ \frac{\varrho^2(1 + k \varrho^4)}{1 + k\varrho^2}\varsigma'^2_3\right). \end{align*}

   This is formally very similar to the Pedersen metric, except that we now have $\mathrm{SU}(1,1)$-invariant one-forms $\varsigma'_1,\varsigma'_2,\varsigma'_3$ on $B^2\times S^1$ given by

   \begin{equation*} \begin{aligned} \varsigma'_1 &=\frac{\mathrm{Re}(e^{i \theta}\mathrm{d}\xi)}{1 - |\xi|^2}, \quad \varsigma'_2 = \frac{\mathrm{Im}(e^{i \theta}\mathrm{d}\xi)}{1 - |\xi|^2}, \quad \varsigma'_3 = \frac{1}{4}\,\mathrm{d} \theta - \frac{\operatorname{Im}(\xi\,\mathrm{d} \overline\xi)}{1 - |\xi|^2}. \end{aligned} \end{equation*}

   Correspondingly, the metric is invariant under the transitive action of $\mathrm{U}(1,1)$ on $B^2\times S^1$. Note that k is necessarily negative in order for the metric to be definite. This can also be seen from the fact that, were k non-negative, the discriminant of $a\rho^2 + b\rho + c$ would be positive. Since a < 0, this would then imply that $a\rho^2 + b\rho + c$ is nowhere positive.

8. (viii) As a limiting case of the above, we have a < 0 and b = 0. Noting that c > 0 necessarily in order for $a\rho^2 + c$ to be positive, and carrying out the change of variable $\rho = \frac{1}{\rho'}\sqrt{-\frac{c}{a}}$, we get

   \begin{equation*} \begin{aligned} g^{a,b,c} &= -\frac{1}{\nu}\left((1-\rho'^2)\left(\frac{\mathrm{d} t}{2K}\right)^2+\frac{\mathrm{d} \rho'^2}{1-\rho'^2} + \frac{4 \rho'^2|\mathrm{d}\xi|^2}{(1 - |\xi|^2)^2}\right). \end{aligned} \end{equation*}

   Note that as in case (iv) with c < 0, upon making the identification $t\sim t + 4\pi K$, the first two terms give the standard S ² metric restricted to a hemisphere minus the pole, with $\sin^{-1}(\rho')$ and $\frac{t}{2K}$ being the latitude and longitude respectively, while the last term is the disc form of the real hyperbolic plane metric. The full metric $g^{a,b,c}$ is in fact the real hyperbolic 4-space, presented in terms of the fibration over S ² given by sending a point $(x_0,x_1,x_2,x_3,x_4)$ on the hyperboloid $x_0^2 -\sum_{i=1}^4x_i^2 = (-\nu)^{-1/2}$ sitting inside $\mathbb{R}^{1,4}$ to $(\sqrt{x_0^2 - x_1^2 -x_2^2},x_3,x_4)\in S^2\subset \mathbb{R}^3$.

9. (ix) When $a \lt 0 \lt b$ and $2a\rho + b \gt 0$, the same change of coordinates as in case (vii) can be done, but now we have ϱ > 1, as can be seen from

   \begin{equation*} \varrho^2 =\frac{b}{2a\rho + b}= \frac{b}{b - 2|a|\rho} \gt 1. \end{equation*}

   Because the metric is formally identical to the metric from case (vii), it is also $\mathrm{U}(1,1)$-invariant.

10. (x) In the final case, $a \lt 0 \lt b$ and $2a\rho + b \lt 0$, we instead set

    \begin{equation*} \begin{aligned} \rho = -\frac{b}{2a}\left(\frac{1}{\varrho^2}+1\right), \quad \zeta = \sqrt{-\frac{2}{a}}\,\xi,\quad t = \frac{bK}{a}\,\theta. \end{aligned} \end{equation*}

    The metric now becomes

    \begin{equation*} \begin{aligned} g^{a,b,c}\!=\!\frac{1}{\nu(1+\varrho^2)^2} \!\left(\frac{1 - k \varrho^2}{1 + k\varrho^4}\,\mathrm{d}\varrho^2 \!-\! \varrho^2(1- k \varrho^2)(\varsigma'^2_1 \!+\! \varsigma'^2_2) + \frac{\varrho^2(1 + k \varrho^4)}{1 - k\varrho^2}\varsigma'^2_3\right) \end{aligned} \end{equation*}
    which is once again $\mathrm{U}(1,1)$-invariant because of its general form.

A careful inspection of our list shows that, in every case, we have either identified the metric $g^{a,b,c}$ as locally symmetric or given an explicit action of a group of isometries whose orbits are 3D. This means, in the latter case, that the metrics are of cohomogeneity at most one.

It is known that the (one-loop) deformed universal hypermultiplet is of cohomogeneity (exactly) 1 for non-zero values of the deformation parameter. It is, therefore, natural to ask whether this holds for the metrics $g^{a,b,c}$ which are not locally symmetric. We answer this question affirmatively:

Proof. The condition $bc(b^2-4ac)= 0$ implies that either we have b = 0 or c = 0 or $b^2-4ac=0$. In the last case, if we assume that b ≠ 0, this is the same as saying $k:=\frac{4ac}{b^2} -1=0$. From the case-by-case analysis in the previous subsection, we see that these yield locally symmetric spaces.

To see that the cohomogeneity of the metric is at least 1 when $bc(b^2-4ac)\neq 0$, we note that its curvature norm

(17)\begin{equation} \mathrm{tr}(\mathscr{R}^2) = 6\nu^2\left(1+b^2(b^2-4ac)^2\left(\frac{\rho}{b\rho + 2c}\right)^6\right) \end{equation}

is an injective function of ρ > 0 whenever $b(b^2-4ac)\neq 0$ and c ≠ 0.

Though we have already given explicit actions of groups of isometries whose orbits are of codimension 1 in every case, we can provide a direct proof that such a group must exist. When $b(b^2-4ac)\neq 0$, in the notation of theorem 6.1, the level sets of ρ admit four Killing fields which span a 3D distribution (at every point). Their explicit expressions are the following:

\begin{equation*} \begin{aligned} &\mathrm{Re}\bigg(\partial_\zeta + \frac{a}{2}\,\overline{\zeta}^2\partial_{\overline\zeta} - \frac{i}{2}\,Kb\overline \zeta \partial_t\bigg),\qquad -a\,\mathrm{Im}(\zeta\partial_\zeta) -\frac{1}{2}\,Kb\partial_t ,\\ &\mathrm{Im}\bigg(\partial_\zeta + \frac{a}{2}\,\overline{\zeta}^2\partial_{\overline\zeta} - \frac{i}{2}\,Kb\overline \zeta \partial_t\bigg),\qquad \phantom{-a\,} \mathrm{Im}(\zeta\partial_\zeta). \end{aligned} \end{equation*}

These are generators of $\mathfrak{o}(2) \lt imes\mathfrak{heis}_3(\mathbb R)$ if a = 0, $\mathfrak{u}(2)$ if a > 0, and $\mathfrak{u}(1,1)$ if a < 0, in complete agreement with our conclusions from the case-by-case analysis.

The attentive reader may also have noticed that the examples with positive scalar curvature are incomplete, and well-known spaces like S ⁴ or $\mathbb{C}\mathrm{P}^2$ are recovered only upon adding points. This is no accident: A quaternionic Kähler manifold of positive scalar curvature that arises from the HK/QK correspondence is necessarily incomplete. Indeed, it is known that no complete quaternionic Kähler manifold of positive scalar curvature carries an almost complex structure compatible with the quaternionic structure [Reference Alekseevsky, Marchiavafa and Pontecorvo5] or in fact, with the exception of the Grassmannians of complex two planes, any almost complex structure at all [Reference Gauduchon, Moroianu and Semmelmann18]. Proposition 2.5, on the other hand, shows that any example that arises from the HK/QK correspondence admits an (integrable) complex structure compatible with the quaternionic structure.

Using the expression (17) for the curvature norm, we can rule out the possibility of finding a smooth completion in many cases:

Proof. First of all, by the previous proposition, the condition $bc(b^2-4ac)\neq 0$ implies that the metric $g^{a,b,c}$ is not locally symmetric.

If there is a solution to $b\rho + 2c \lt 0$ in the closure of the allowed range of the coordinate ρ, then, because the condition is open, there is a solution $b\rho + 2c \lt 0$ in the allowed range of the coordinate ρ. As the sign of the scalar curvature is the opposite of the sign of $b\rho + 2c$, the scalar curvature is positive. It is well-known that there are no complete quaternionic Kähler metrics of dimension 4 of positive scalar curvature which are not locally symmetric [Reference Friedrich and Kurke17, Reference Hitchin20]. Thus, $g^{a,b,c}$ cannot be extended to a complete metric.

On the other hand, if there is a solution to $b\rho + 2c = 0$ in the closure of the allowed range of the coordinate ρ, we can find a ρ ₀ in the neighbourhood of $-\frac{2c}{b}$ such that the half-open interval between ρ ₀ and $-\frac{2c}{b}$ (with ρ ₀ included) is contained in the allowed range of the coordinate ρ. At $\rho = {-\frac{2c}{b}}$, the polynomial $a\rho^2 + b\rho + c$ takes the value $\frac{c}{b^2}(4ac -b^2)\neq 0$. As a result, we have

\begin{equation*} \int_{\rho_0}^{-\frac{2c}{b}}\sqrt{\frac{b\rho + 2c}{a\rho^2 + b\rho + c}}\frac{\mathrm{d} \rho}{\rho} \lt \infty. \end{equation*}

By (17), $\rho = {-\frac{2c}{b}}$ is a curvature singularity. It follows from the finiteness of the above integral that the curvature singularity is a finite distance away from $\rho =\rho_0$. So, $g^{a,b,c}$ cannot be extended to a complete metric in this case either.