The c-map on groups

TL;DR

This paper investigates the intrinsic properties of homogeneous projective special Kähler structures on groups, providing new defining equations and demonstrating their relation to quaternionic Kähler structures via the c-map.

Contribution

It introduces an intrinsic, computation-free definition of homogeneous projective special Kähler structures on groups and explores their connection to quaternionic Kähler structures through the c-map.

Findings

Intrinsic defining equations for homogeneous projective special Kähler structures

The c-map transforms these structures into left-invariant quaternionic Kähler structures on Lie groups

Emphasis on integrability conditions associated with these structures

Abstract

We study the projective special Kaehler condition on groups, providing an intrinsic definition of homogeneous projective special Kaehler that includes the previously known examples. We give intrinsic defining equations that may be used without resorting to computations in the special cone, and emphasise certain associated integrability equations. The definition is shown to have the property that the image of such structures under the c-map is necessarily a left-invariant quaternionic Kaehler structure on a Lie group.

Equations

$$g_S={\theta }^T\theta =a^{T}a+b^{T}b\text{and}{\omega }_S=\frac{1}{2}{\theta }^T\wedge J\theta =a^T\wedge b.$$

$$d\theta =-{\omega }_{\mathrm{LC}(S)}\wedge \theta ,$$

$${\omega }_{\mathrm{LC}(S)}=\left(\begin{array}{cc}\mu & \lambda \\ -\lambda & \mu \end{array}\right),$$

$${\mu }^T=-\mu ,{\lambda }^T=\lambda .$$

$$d\left(\begin{array}{c}a\\ b\end{array}\right)=-\left(\begin{array}{cc}\mu & \lambda \\ -\lambda & \mu \end{array}\right)\wedge \left(\begin{array}{c}a\\ b\end{array}\right).$$

$${\Omega }_S=d{\omega }_{\mathrm{LC}(S)}+{\omega }_{\mathrm{LC}(S)}\wedge {\omega }_{\mathrm{LC}(S)}=\left(\begin{array}{cc}\mathrm{M}& \Lambda \\ -\Lambda & \mathrm{M}\end{array}\right),$$

$$\mathrm{M}=d\mu +\mu \wedge \mu -\lambda \wedge \lambda \text{and}\Lambda =d\lambda +\mu \wedge \lambda +\lambda \wedge \mu .$$

$$d\phi =2{\pi }^{\ast }{\omega }_S.$$

$$C={\mathbb{R}}_{>0}\times C_0.$$

$$g_C={\hat{a}}^T\hat{a}+{\hat{b}}^T\hat{b}-{\hat{\phi }}^2-{\hat{\psi }}^2,{\omega }_C={\hat{a}}^T\wedge \hat{b}-\hat{\phi }\wedge \hat{\psi }.$$

$$\mathbf{G}=\left(\begin{array}{cc}{\mathrm{id}}_{2n}& 0\\ 0& -{\mathrm{id}}_2\end{array}\right)\text{and}\mathbf{i}=\left(\begin{array}{cc}{\mathbf{i}}_{2n}& 0\\ 0& {\mathbf{i}}_2\end{array}\right)=\left(\begin{array}{cccc}0& {\mathrm{id}}_n& 0& 0\\ -{\mathrm{id}}_n& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right).$$

$$d\hat{\psi }=0,d\hat{\phi }=\frac{1}{t}(\hat{\psi }\wedge \hat{\phi }+2{\hat{a}}^T\wedge \hat{b}),$$

$$d\left(\begin{array}{c}\hat{a}\\ \hat{b}\end{array}\right)=\frac{1}{t}\left(\begin{array}{cc}\hat{\psi }{\mathrm{id}}_n-\hat{\mu }& -\hat{\lambda }\\ \hat{\lambda }& \hat{\psi }{\mathrm{id}}_n-\hat{\mu }\end{array}\right)\wedge \left(\begin{array}{c}\hat{a}\\ \hat{b}\end{array}\right),$$

$${\omega }_{\mathrm{LC}}=\left(\begin{array}{cccc}\tilde{\mu }& \phi {\mathrm{id}}_n+\tilde{\lambda }& \tilde{b}& \tilde{a}\\ -\phi {\mathrm{id}}_n-\tilde{\lambda }& \tilde{\mu }& -\tilde{a}& \tilde{b}\\ {\tilde{b}}^T& -{\tilde{a}}^T& 0& \phi \\ {\tilde{a}}^T& {\tilde{b}}^T& -\phi & 0\end{array}\right)=\left(\begin{array}{cc}\phi {\mathbf{i}}_{2n}+{\tilde{\omega }}_{\mathrm{LC}(S)}& \begin{array}{cc}{\mathbf{i}}_{2n}{\tilde{\theta }}_S& {\tilde{\theta }}_S\end{array}\\ \begin{array}{c}-{{\tilde{\theta }}_S}^T{\mathbf{i}}_{2n}\\ {\tilde{\theta }}_S^T\end{array}& \phi {\mathbf{i}}_2\end{array}\right).$$

$$\begin{array}{rl}\eta =\left(\begin{array}{cccc}u& v& 0& 0\\ v& -u& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)& \in C^{\infty }(M)\otimes {\pi }^{\ast }{\Omega }^1(S,M_{2n+2}(\mathbb{R}))\\ & \subset {\Omega }^1(C,M_{2n+2}(\mathbb{R})),\end{array}$$

$$u\wedge \hat{a}+v\wedge \hat{b}=0=v\wedge \hat{a}-u\wedge \hat{b}.$$

$$\eta =\left(\begin{array}{ccc}u& v& c\\ d& e& f\\ g& h& k\end{array}\right),$$

$$e=-u,d=v,f=-c{\mathbf{i}}_2,h={\mathbf{i}}_{2}g,{\mathbf{i}}_{2}k=-k{\mathbf{i}}_2.$$

$$f=-c{\mathbf{i}}_2=\left(\begin{array}{cc}q& -p\end{array}\right).$$

$$h={\mathbf{i}}_{2}g=\left(\begin{array}{c}m\\ -\ell \end{array}\right).$$

$$\eta =\left(\begin{array}{cccc}u& v& p& q\\ v& -u& q& -p\\ \ell & m& x& y\\ m& -\ell & y& -x\end{array}\right),$$

$$u^T=u,v^T=v,m=-q^T,\ell =-p^T.$$

$$\eta =\left(\begin{array}{cccc}u& v& p& q\\ v& -u& q& -p\\ -p^T& -q^T& x& y\\ -q^T& p^T& y& -x\end{array}\right),u^T=u,v^T=v.$$

$$u\wedge \hat{a}+v\wedge \hat{b}+p\wedge \hat{\phi }+q\wedge \hat{\psi }$$

$$v\wedge \hat{a}-u\wedge \hat{b}+q\wedge \hat{\phi }-p\wedge \hat{\psi }$$

$$-p^T\wedge \hat{a}-q^T\wedge \hat{b}+x\wedge \hat{\phi }+y\wedge \hat{\psi }$$

$$-q^T\wedge \hat{a}+p^T\wedge \hat{b}+y\wedge \hat{\phi }-x\wedge \hat{\psi }$$

$$u=(u_j^i),\text{with}{u}_j^i=u_{ajk}^i{\hat{a}}^k+u_{bj\ell }^i{\hat{b}}^{\ell },\text{for}1⩽i,j,k,\ell ⩽n,$$

$$u_{ajk}^i=-v_{bjk}^i,u_{bjk}^i=v_{ajk}^i,$$

$$u_j^i=-v_{bjk}^i{\hat{a}}^k+v_{ajk}^i{\hat{b}}^k.$$

$$u=Jv.$$

$$v_{ajk}^i=v_{aki}^j=v_{aij}^k=v_{akj}^i=v_{aki}^j=v_{aji}^k,$$

Full text

1 Introduction

In the search for manifolds with special, or even exceptional, holonomy developments in theoretical physics have provided a fruitful ground for examples. In particular, the study of T-duality between type IIA and type IIB superstring theories from the point of view of the low energy effective Lagrangians for $D=4$, $N=2$ supergravity has given insight in the relation between Kähler geometry and hyperKähler or quaternionic Kähler geometry, through a mechanism known as the c-map originally introduced in Cecotti, Ferrara, Girardello [6], Ferrara and Sabharwal [15].

For supersymmetric field theories without gravity, supersymmetry is regarded as a global symmetry and the moduli space of scalar fields of vector multiplets is (affine) special Kähler [10, 16, 19, 22], while the geometry of the moduli space of scalar fields in the hypermultiplets is of hyperKähler type. When supersymmetry is imposed as a local symmetry, thus in the context of supergravity, the geometries of the above moduli spaces of scalar fields become projective special Kähler and quaternionic Kähler, respectively, see [3, 4, 5, 11, 12]. The quaternionic Kähler nature of the hypermultiplet metric was first described in Ferrara and Sabharwal [15]. In this context the “rigid c-map” associates to every special Kähler manifold of complex dimension $n$ a dual hyperKähler manifold of quaternionic dimension $n$, and the “local c-map” associates a quaternionic Kähler manifold of quaternionic dimension $n+1$ to each projective special Kähler manifold of complex dimension $n$.

Although the (local) c-map has its origins in supergravity, it has substantial mathematical interest through the work of de Wit and Van Proeyen [13] where it was used to correct Alekseevsky’s classification [1] of quaternionic Kähler manifolds admitting a transitive completely solvable group of isometries. Recently, mathematical descriptions of the c-map in general have been given in [2] and [20]. The former shows that the local formulas derived by Ferrara & Sabharwal [15] are indeed obtainable by appropriate conification procedures; the latter provides a geometric approach to the global geometry of the c-map via the twist construction and elementary deformations.

The first applications of the c-map were to group manifolds, and papers such as [6, 13, 14] provide several tables of resulting structures. However, the precise mathematical motivation for the classes of examples covered remains unclear, and from a mathematical point of view assumptions derived from supergravity may not necessarily be relevant for the mathematical applications. Indeed all groups obtained are completely solvable, but it is an open conjecture of Alekseevsky whether all homogeneous quaternionic Kähler metrics of negative scalar curvature are left-invariant structures on completely solvable groups.

The purpose of the current paper is to provide a first step towards understanding what constraints the geometric c-map in [20] may impose. The initial data is a group manifold carrying an invariant projective special Kähler structure. However, traditionally the definition of projective special Kähler [16] is specified via the geometry of an auxiliary cone, rather than intrinsically, and is not immediately clear which structures should be regarded as homogeneous. We thus start with a left-invariant Kähler structure on a Lie group $S$, and work through the conditions that this admits a projective special Kähler structure. In the first instance we pass to the cone $C$ and study the standard equations there. We find a certain of integrability conditions enable one to quickly get certain results about projective special Kähler manifolds that are Kähler products. Thereafter we show that the assumption that the Kähler form of $S$ is exact ensures the defining objects descend well from $C$ to $S$, and give us both a reasonable definition of homogeneous structure and an intrinsic formulation of the projective special Kähler condition directly on $S$. We illustrate how these equations and the associated integrability conditions may be used in a four-dimensional example. Finally, we demonstrate the reasonableness of our definition by proving that the c-map applied to a homogeneous structure on a group always yields a group manifold with left-invariant quaternionic Kähler structure.

While the focus of this paper is on group manifolds, its worth noting that mathematically the c-map on inhomogeneous data is known to construct previously unknown complete inhomogeneous quaternionic Kähler [7, 8, 9].

As we were finalising this manuscript, Mauro Mantegazza kindly sent us a copy of [21]. There he obtains the intrinsic equations for projective special Kähler manifolds in general, even when the Kähler form is not exact, and provides various global results. That paper also uses the characterisation to show that the homogeneous examples in real dimension four are exactly the two cases we consider in this paper, and in particular the exactness condition is necessarily satisfied.

2 The special Kähler conditions

Projective special Kähler manifolds $S$ are best defined and understood via their cones $C$, cf. [16]. In this section, we will start with a left-invariant Kähler structure on a group manifold $S$ and use the associated cone to derive the relevant equations in a left-invariant frame. This will follow the general picture described in [20].

Suppose $S$ is a Lie group with Lie algebra $\operatorname{\mathfrak{s}}$, and that this Lie group carries a left-invariant Kähler structure with complex structure $J$, metric $g_{S}$ and Kähler form $\omega_{S}=g_{S}(J\,\cdot\,,\,\cdot\,)$. Choose an orthonormal basis $\{X_{1},\dots,X_{n},JX_{1},$ $\dots,JX_{n}\}$ for $\mathfrak{s}$, and write $A_{i}=X_{i}$, $B_{i}=JX_{i}$, for $i=1,\dots,n$. Denote by $\{a^{i},b^{i}:1\leqslant i\leqslant n\}$ the corresponding dual basis of left-invariant one-forms. The complex structure acts on $\mathfrak{s}^{*}$ with $Ja^{i}=b^{i}$.

In what follows it will be often useful to resort to matrix notation. Therefore we introduce $\mathbb{R}^{n}$-valued one-forms $a=(a^{i})$, $b=(b^{i})$, and the $\mathbb{R}^{2n}$-valued coframe $\theta=(a,b)$. The metric and Kähler forms are

[TABLE]

The connection one-form $\omega_{{\mathrm{LC}(S)}}$ of the Levi-Civita connection of $g_{S}$ is the skew-symmetric matrix determined by the structural equations

[TABLE]

As the structure is Kähler, we have that $\omega_{{\mathrm{LC}(S)}}$ takes values in $\operatorname{\mathfrak{u}}(n)\leqslant\operatorname{\mathfrak{so}}(2n)$, so we may write

[TABLE]

with $\mu=(\mu^{i}_{j})$, $\lambda=(\lambda^{i}_{j})$ $n\times n$-matrices of one-forms satisfying the following symmetries

[TABLE]

The structural equation is thus

[TABLE]

The curvature of $S$ now has the form

[TABLE]

where

[TABLE]

To introduce the projective special Kähler conditions, we need to assume that $(S,2\omega_{S})$ is Hodge, meaning that there is a circle bundle $\pi\colon C_{0}\to S$ with connection one-form $\varphi$ such that

[TABLE]

We write $\pi^{*}\omega_{S}=\tilde{a}^{T}\wedge\tilde{b}$, where $\tilde{a}=\pi^{*}a$, $\tilde{b}=\pi^{*}b$, and let $X$ be the vector field generating the circle action in the fibres. In particular, $\varphi(X)=1$, $L_{X}\varphi=0$, and $\pi_{*}X=0$. Then, the (complex) cone $C$ over $S$ is defined to be

[TABLE]

Let $t$ be the standard coordinate on $\mathbb{R}_{>0}$ and put $\hat{a}=t\tilde{a}$, $\hat{b}=t\tilde{b}$, $\hat{\varphi}=t\varphi$, $\hat{\psi}=dt$. Then $C$ carries a pseudo-Kähler structure with metric and Kähler form given by

[TABLE]

We denote its complex structure by $J$ and note that the conic symmetry $X$ satisfies $JX=t\partial_{t}$ and $g_{C}(X,X)=-t^{2}$.

Then $(\hat{a},\hat{b},\hat{\varphi},\hat{\psi})$ is a unitary coframe for $C$ and we put $\theta_{C}=(\hat{a},\hat{b},\hat{\varphi},\hat{\psi})^{T}$. The Levi-Civita connection of $C$ is uniquely determined by $d\theta_{C}=-\omega_{{\mathrm{LC}}}\wedge\theta_{C}$ together with $\omega_{{\mathrm{LC}}}^{T}\mathbf{G}+\mathbf{G}\omega_{{\mathrm{LC}}}=0$ and $\mathbf{i}\omega_{{\mathrm{LC}}}=\omega_{{\mathrm{LC}}}\mathbf{i}$, where

[TABLE]

Using

[TABLE]

one checks that

[TABLE]

The conditions that $(S,g_{S},\omega_{S})$ be projective special Kähler are that its cone $(C,g_{C},\omega_{C},X)$ is special Kähler with $X$ a conic symmetry. More precisely this means that $C$ admits a torsion-free flat symplectic connection $\nabla$ with $(\nabla_{A}J)B=(\nabla_{B}J)A$, for all $A,B$, and such that the symmetry $X$ satisfies $\nabla X=-J$. Note that our construction of $C$ already ensures that $(g_{C},\omega_{C},J)$ is pseudo-Kähler, that $X$ is non-null and that $\nabla^{{\mathrm{LC}}}X=-J$.

The conditions on $\nabla$ were carefully analysed in [20], summarised there in the proof of Proposition 6.3, giving the following: writing the connection one-form for $\nabla$ in the coframe $\theta_{C}$ as $\omega_{\nabla}$, the conic special Kähler conditions on the pseudo-Kähler manifold $C$ are equivalent to the existence of a matrix-valued one-form $\eta=\omega_{\nabla}-\omega_{{\mathrm{LC}}}$ such that

1. (i)

$\Omega_{\nabla}=d\omega_{\nabla}+\omega_{\nabla}\wedge\omega_{\nabla}=0$ (flat), 2. (ii)

$\eta\wedge\theta=0$ (torsion-free), 3. (iii)

$\mathbf{i}_{2n+2}\eta=-\eta\mathbf{i}_{2n+2}$ (special symplectic), 4. (iv)

$\eta^{T}\mathbf{G}=-\mathbf{G}\eta$ (special symplectic), 5. (v)

$X\mathbin{\lrcorner}\eta=0$ (conic), 6. (vi)

$JX\mathbin{\lrcorner}\eta=0$ (conic).

Lemma 2.1**.**

For $S$ of real dimension $2n$, the difference $\eta=\omega_{\nabla}-\omega_{{\mathrm{LC}}}$of the special and Levi-Civita connections on $C$ is given by a matrix of one forms with the following structure:

[TABLE]

where $u$ and $v$ take values in symmetric $n\times n$ matrices and satisfy

[TABLE]

Proof.

Let us write $\eta$ in block form:

[TABLE]

where $u,v,d,e\in M_{n}(\mathbb{R})$, $c,f,g^{T},h^{T}\in M_{n,2}(\mathbb{R})$, and $k\in M_{2}(\mathbb{R})$.

Then, first special symplectic condition (iii) implies:

[TABLE]

Writing $c\in M_{n,2}$ as two columns $c=\begin{pmatrix}p&q\end{pmatrix}$, $p,q\in M_{n,1}(\mathbb{R})$, we have

[TABLE]

Analogously, writing $g\in M_{2,n}$ as two rows, named $\ell,m\in M_{1,n}(\mathbb{R})$, we have

[TABLE]

Finally, $\mathbf{i}_{2}k=-k\mathbf{i}_{2}$ implies that $k$ is a symmetric traceless matrix, thus leading to

[TABLE]

for some scalar-valued one-forms $x$, $y$.

The second symmetry to be exploited is (iv), which gives

[TABLE]

Thus, we obtain

[TABLE]

Next, the torsion-free condition (ii) gives two vector, and two scalar equations

[TABLE]

However, the conic conditions (v) and (vi) imply that each entry of $\eta$ pointwise lies in the space of the components of $\hat{a}$ and $\hat{b}$. Thus (2.4) and (2.5) imply that $p=0=q$, and then (2.4) and (2.5) show that $x=0=y$. We thus have the claimed equation (2.3). ∎

Remark 2.2*.*

Resorting to index notation, and expanding in the one-forms $\{\hat{a}^{i},\hat{b}^{i},\hat{\varphi},\hat{\psi}\}$, we can write

[TABLE]

and similar expressions for $v$. Considering the $\hat{a}^{i}\wedge\hat{b}^{j}$-terms in the torsion-free condition (2.3) then gives

[TABLE]

hence

[TABLE]

So using the complex structure, we have

[TABLE]

Furthermore, the symmetry of matrices $u$ and $v$ together with the relations (2.8) gives that the coefficients $v^{i}_{ajk}$ and $v^{i}_{bjk}$ are totally symmetric under permutation of all indices:

[TABLE]

and therefore determine a symmetric three-tensor on $C$. This is the standard holomorphic three-tensor associated to special complex geometry, cf. [16, 19].

The one-form for the special connection $\omega_{\nabla}=\omega_{{\mathrm{LC}}}+\eta$ is now

[TABLE]

Using the torsion-free equations (2.3), one finds that the curvature of the special connection is given by

[TABLE]

where

[TABLE]

As the special conditions requires $\Omega_{\nabla}=0$, the next result summarizes the situation.

Proposition 2.3**.**

$S$* is projective special Kähler if and only if on the cone $C$ there is a one-form $\eta$ as in Lemma 2.1 so that the expressions (2.12)–(2.15) satisfy $T=U=V=W=0$.*

Remark 2.4*.*

The special connection reduces to the Levi-Civita connection of the cone exactly when $u=v=0$. In this situation, (2.3) is satisfied and $U=0=V$. What remains are the equations $T=0=W$, now determine the Kähler curvature of $S$:

[TABLE]

Thus

[TABLE]

which is the curvature tensor of complex hyperbolic space $\mathbb{C}H(n)$ with holomorphic sectional curvature $-1$. In this case, we have

[TABLE]

where $a_{R}=(a^{2},\dots,a^{n})$, etc.

Example 2.5*.*

The case the complex hyperbolic line, i.e. $S=\mathbb{C}H(1)$, with arbitrary (negative) holomorphic sectional curvature, was discussed in detail in [20]. One has $\mu=0$ and $\lambda=-cb$ for some $c\in\mathbb{R}$, where the holomorphic sectional curvature is $-c^{2}/4$. It was shown that there are only solutions to the projective special Kähler equations in the cases when the holomorphic sectional curvature is $-1$ or $-1/3$.

3 Integrability equations

To understand the special geometry better, let us consider the integrability conditions related to the torsion-free condition and the vanishing of $T$, $U$, $V$, and $W$. Differentiating $U=0$, and substituting for $du$ and $dv$ from $U=0=V$, we get

[TABLE]

after substituting the expressions for $du$ and $dv$ from $U=0=V$. Similarly, differentiating $V=0$ gives

[TABLE]

It is tempting to substitute for $\tilde{\mathrm{M}}$ and $\tilde{\Lambda}$ using $T=0=W$, but after applying the torsion-free condition (2.3) this yields no information. Thus these equations are consequences of the torsion-free relation and the vanishing of $T$ and $W$. However, these equations can provide useful constraints on $u$ and $v$ as we will see below. Similar considerations show that the system of equations $T=0=U=V=W$, (2.3), together with the lifts of (2.1), (2.2), and the differential of (2.2),

[TABLE]

in the variables $\tilde{\mu},\tilde{\lambda},\tilde{\mathrm{M}},\tilde{\Lambda},u,v$ is closed under exterior derivatives.

4 Flat factors

Following [17], it is reasonable to study the situation when the Kähler group $S$ is a product $\widetilde{S}=S^{0}\times R$ with $S^{0}$ flat and Kähler. Note that statements in [18] modify those of the previous reference, and the correctness of those results is not clear.

If $S=S^{0}\times R$ with $S^{0}$ flat and Kähler, then we can split $a=(a_{0},a_{R})^{T}$ etc. and write

[TABLE]

and so on, with $u_{-}=(u_{+})^{T}$. It follows that

[TABLE]

Now, suppressing wedge signs, (3.1) reads

[TABLE]

The $(0,0)$-component of this equation gives

[TABLE]

which for $\dim_{\mathbb{R}}S>2$, implies $v_{0}=0$. Similarly, from (3.2) we get $u_{0}=0$. The symmetries from the torsion-free conditions, (2.9), (2.10) and (2.11), now imply that $u_{+}$ and $v_{+}$ have no $\tilde{a}_{0}$- or $\tilde{b}_{0}$-components. Now the $+$-component of (4.1) only contains $\tilde{a}_{0}$ and $\tilde{b}_{0}$ terms in $\pi^{*}\omega_{S}$, so $4\pi^{*}\omega_{0}\wedge v_{+}=0$, giving $v_{+}=0$, and $v_{-}=v_{+}^{T}=0$; similarly, $u_{+}=0=u_{-}$.

The $(0,0)$-component of $W=0$, is $\tilde{a}_{0}\wedge\tilde{b}_{0}-\tilde{b}_{0}\wedge\tilde{a}_{0}+2\pi^{*}\omega_{S}\mathop{}\!\mathrm{id}_{m}=0$, for $\dim_{\mathbb{C}}S^{0}=m$. But $\omega_{S}=\omega_{0}+\omega_{R}$, so if $\dim R>0$, we get a contradiction.

If $\dim R=0$, then we have $\dim_{\mathbb{R}}S=2$ and $S=S^{0}$. But then $S^{0}$ is Abelian, $\lambda=0=\mu$ and the vanishing of $U$ and $V$ (2.13) and (2.14), imply $u=0=v$. Finally $W=0$, gives $4\tilde{a}_{0}\wedge\tilde{b}_{0}=0$, a contradiction.

Thus we have proved:

Proposition 4.1**.**

Suppose $S$ is a Lie group with a left-invariant Kähler structure that extends to a (not necessarily left-invariant) projective special Kähler structure. Then the de Rham decomposition of the universal cover $\widetilde{S}$ has no flat Kähler factor. ∎

5 Products

Suppose the universal cover of $S$ is a product of Kähler groups, $\widetilde{S}=S_{1}\times S_{2}$. Then splitting $a=(a_{1},a_{2})$ etc., we have

[TABLE]

Writing

[TABLE]

etc., (3.1) has components

[TABLE]

Let us use $V^{a,b}$ to denote the bundle $\pi^{*}(\Lambda^{a}T^{*}S_{1}\wedge\Lambda^{b}T^{*}S_{2})$. The terms in (5.1) in $V^{1,2}+V^{0,3}$ are $4\pi^{*}\omega_{S_{2}}\wedge v_{1}$, so if $\dim_{\mathbb{R}}S_{2}>2$, we have $v_{1}=0$. Similarly, using (3.2), we get $u_{1}=0$. Now, under this condition, the total symmetry (2.10) and (2.11) implies $u_{+}$ and $v_{+}$ consist of one-forms in $V^{0,1}$.

Similarly, if we also have $\dim_{\mathbb{R}}S_{1}>2$, then $v_{2}=0=u_{2}$, and $u_{\pm}=0=v_{\pm}$, i.e. $u=0=v$. But then $S=\mathbb{C}H(n)$ which is not a product. Thus a product structure has at least one factor of real dimension $2$.

Proposition 5.1**.**

Suppose the universal cover $\widetilde{S}$ is product of three or more Kähler factors. Then there are exactly three factors and $\widetilde{S}=\mathbb{C}H(1)\times\mathbb{C}H(1)\times\mathbb{C}H(1)$, each with holomorphic section curvature $-1$.

Proof.

Write $\widetilde{S}=S_{A}\times S_{B}\times S_{C}$, with each factor of non-zero dimension. Then $\dim_{\mathbb{R}}(S_{A}\times S_{B})$ is strictly greater than $2$, so $\dim_{\mathbb{R}}S_{C}=2$. Grouping in different ways yields $\dim_{\mathbb{R}}S_{A}=2=\dim_{\mathbb{R}}S_{B}$. By Proposition 4.1, these factors are not flat, so are each isomorphic to $\mathbb{C}H(1)$. In particular, $M_{i}=0$ and $\Lambda_{i}=r_{i}\omega_{S_{i}}$, for $i=A,B,C$, with $r_{i}\in\mathbb{R}\setminus\{0\}$.

Writing

[TABLE]

equation (3.1) gives $\pi^{*}(2r_{A}\omega_{S_{A}}+4\omega_{S})v_{A}=0$, implying $v_{A}=0$, and so all diagonal entries in $u$ and $v$ are zero. For the off-diagonal terms, we have $\pi^{*}((r_{A}+4)\omega_{S_{A}}+(r_{B}+4)\omega_{S_{B}}+\omega_{S_{C}})v_{AB}=0$. But $u$ and $v$ are not identically zero, so $r_{i}=-4$ for all $i$, and $v_{AB}\in\pi^{*}T^{*}S_{C}$, etc. Total symmetry of the $u$ and $v$ implies $u_{AB}=u^{\prime}\tilde{a}_{C}+u^{\prime\prime}\tilde{b}_{C}$, $v_{AB}=u^{\prime\prime}\tilde{a}_{C}-u^{\prime}\tilde{b}_{C}$, and cyclically, for some smooth functions $u^{\prime},u^{\prime\prime}$ on the cone. The equations $T=0=W$ only impose the constraint $(u^{\prime})^{2}+(u^{\prime\prime})^{2}=1$, so $u^{\prime}=\cos(s)$, $u^{\prime\prime}=\sin(s)$ for some smooth local function $s$. Equations (2.13) and (2.14), then give $ds=2(\varphi+\tilde{b}_{A}+\tilde{b}_{B}+\tilde{b}_{C})$, which is an exact form, so $s$ is globally defined. ∎

6 Rotations and intrinsic equations

Let $S$ be as in Proposition 4.1. Consider its cone $C$. The conic vector field $X$ satisfies $X\mathbin{\lrcorner}\varphi=1$, $L_{X}\varphi=0$, and $X\mathbin{\lrcorner}u=0=X\mathbin{\lrcorner}v$. Thus the vanishing of $U$ and $V$ gives

[TABLE]

In particular, the matrices of one-forms $u$ and $v$ are not invariant under the conic symmetry and they do not descend to $S$ even though they vanish on $X$ and $JX$.

Suppose the curvature of the circle bundle $C_{0}\to S$ is exact: $d\varphi=2\pi^{*}\omega_{S}=2\pi^{*}d\kappa$ for some $\kappa\in\Omega^{1}(S)$. Then $\varphi^{\prime}=\varphi-2\pi^{*}\kappa$ is a flat connection on $C_{0}$, so the pull-back $C^{\prime}_{0}\to\widetilde{S}$ to the universal cover is a trivial circle bundle. Let $C^{\prime}=\mathbb{R}_{>0}\times C^{\prime}_{0}$ be the special Kähler cone of $\widetilde{S}$ and choose a trivialisation of $C^{\prime}_{0}\cong S^{1}\times\widetilde{S}$, writing points of $S^{1}$ as $e^{\mathbf{i}\tau}$, then $X=\partial_{\tau}$ and $\varphi=d\tau+2\pi^{*}\kappa$.

Consider new matrices of one-forms $P$, $Q$ obtained by rotating the pair $u$, $v$ through some angle $z$, that is

[TABLE]

Then, we find

[TABLE]

with a similar expressions for Lie derivatives of $Q$. Putting $z=2\tau$ we get $L_{X}P=L_{X}Q=0$ and hence that $P$ and $Q$ are basic. Then $P=\pi^{*}p$ and $Q=\pi^{*}q$, for some one-forms on $S$ with values in $n\times n$ symmetric matrices. Furthermore, this essentially the only choice: if $xu+yv$ is basic and nowhere zero, with $x$ and $y$ smooth functions on $C^{\prime}$, then there is a trivialisation of $C_{0}^{\prime}$ such that $x=r\cos(2\tau+s)$ and $y=r\sin(2\tau+s)$, with $r>0$ the pull-back of a smooth function on $\widetilde{S}$ and with $s$ a real constant.

Definition 6.1**.**

Let $S$ be a Kähler group with $\omega_{S}=d\kappa$ for some left-invariant form $\kappa$. A compatible projective special Kähler structure on $S$ is homogeneous if $P$ and $Q$ above are pull-backs of left-invariant matrix-valued one-forms $p$ and $q$ on $S$.

Note that all group manifold examples of projective special Kähler structures in the literature satisfy this definition.

We may now rewrite equations (2.12)–(2.15) in terms of $P,Q$ instead of $u,v$. We will see that the resulting equations are determined by $p,q$ on $S$. First, notice that since $u=Jv$, we have $P=JQ$. Inverting the equations defining $P$, $Q$, gives

[TABLE]

Substituting this in the torsion-free condition (2.3) we see the corresponding equation in $P,Q$ and get that it is equivalent to

[TABLE]

Next, although $u$ and $v$ depend on the angle function $z$, we have

[TABLE]

Thus both sides of these equations are $X$-invariant. We now see that the vanishing $T,W$ is equivalent to

[TABLE]

on $S$, where $\mathrm{M}_{\mathbb{C}H}$ and $\Lambda_{\mathbb{C}H}$ are the blocks of the curvature of $\mathbb{C}H(n)$ given in (2.16). On the other hand the expressions for $U$, $V$ depend on the angle function $z$. More precisely,

[TABLE]

Therefore, vanishing of $U$, $V$ is equivalent to the relations

[TABLE]

on $S$.

Proposition 6.2**.**

A simply-connected Kähler group $S$ of dimension $2n$ with exact Kähler form $\omega_{S}=d\kappa$ admits a compatible projective special Kähler structure if and only if there are matrix-valued one-forms $p,q\in\Omega^{1}(S,M_{n}(\mathbb{R}))$ on $S$ satisfying $p^{T}=p$, $q^{T}=q$, $p=Jq$, the torsion-free condition (6.1), the equations (6.2), (6.3), (6.4), and (6.5).

This structure is homogeneous projective special Kähler if and only if $p$ and $q$ can be chosen left-invariant. ∎

Remark 6.3*.*

The function $z$, or correspondingly the parameter $\tau$, is only defined up to a global additive constant. Therefore any choice of $(p,q)$ can be replaced by a rotated version $R_{s}(p,q)=(p\cos s+q\sin s,-p\sin s+q\cos s)$ for any constant $s$. The projective special conditions (6.1), (6.2), (6.3), (6.4) and (6.5), are equivalent to the same system for $R_{s}(p,q)$.

Remark 6.4*.*

The discussion of Remark 2.2, implies that the torsion-free condition is equivalent to total symmetry conditions of the form (2.10)–(2.11) for the $a,b$-coefficients of $p$ and $q$.

Remark 6.5*.*

The integrability equations of §3 are equivalent to the pair of equations

[TABLE]

on $S$. These have the advantage of not involving the one-form $\kappa$.

7 A four-dimensional example

In this section we consider the Kähler products $\widetilde{S}=\mathbb{C}H(1)\times\mathbb{C}H(1)$. We choose our $a,b$ compatible with the product, so that $da_{1}=0=da_{2}$ and $db_{i}=c_{i}a_{i}\wedge b_{i}$, $i=1,2$, with $c_{i}\neq 0$. Replacing $(a_{i},b_{i})$ by $(-a_{i},-b_{i})$ if necessary, we may ensure that $c_{i}>0$. Then $\omega_{S}=a_{1}\wedge b_{1}+a_{2}\wedge b_{2}=d(b_{1}/c_{1}+b_{2}/c_{2})$ is exact with left-invariant primitive, so we may apply the equations of §6 with the splittings of §5. We have $\mathrm{M}_{i}=0$, $\Lambda_{i}=r_{i}\omega_{S_{i}}$, $r_{i}=-c_{i}^{2}<0$. Moreover, $p,q$ are $M_{2}(\mathbb{R})$-valued one-forms. Hence, $p_{1},p_{\pm},p_{2},q_{1},q_{\pm},q_{2}$ are scalar one-forms. The symmetries $p^{T}=p$, $q^{T}=q$, and $p=Jq$ imply that $p$ and $q$ are determined by, e.g., $q_{1},q_{+},q_{2}$.

Comparing with the discussion in §5, consider the bundles $W^{a,b}=\Lambda^{a}T^{*}S_{1}\wedge\Lambda^{b}T^{*}S_{2}$. The integrability equation (6.6) gives three equations

[TABLE]

The last term of (7.1) implies $q_{1}\in W^{0,1}$; similarly, the first term of (7.3) gives $q_{2}\in W^{1,0}$. By the total symmetry of Remark 6.4, we see that $q_{1}=0=q_{2}$ implies $q=0$; then $p=Jq=0$ too, which is a contradiction with (6.2). Thus, without loss of generality, we may assume that $q_{1}$ is non-zero in $W^{0,1}$. The first part of (7.1), then implies $r_{1}=-2$. Now considering equation (7.2), the first coefficient is non-zero, so $q_{+}\in W^{1,0}$; then the second term forces $r_{2}=-4$; so the holomorphic sectional curvatures of the two factors are $-1/2$ and $-1$. Equation (7.3) now implies $q_{2}=0$.

Let us now consider the equations (6.2), (6.3), (6.4) and (6.5) Firstly (6.2) reduces to

[TABLE]

Equation (6.3) gives three relations

[TABLE]

As $q_{+}$ and $p_{+}=Jq_{+}$ are span $T^{*}S_{1}$, we may use Remark 6.3 to take $q_{+}=ra_{1}$ with $r>0$. It follows that $p_{+}=rb_{1}$ and (7.4) implies $r=1$. Total symmetry now gives $q_{1}=a_{2}$, $p_{1}=b_{2}$. Thus

[TABLE]

It remains to consider equations (6.4) and (6.5). We have $db_{1}=\sqrt{2}a_{1}\wedge b_{1}$, $db_{2}=2a_{2}\wedge b_{2}$, and putting this into (6.4) and (6.5) gives $2\kappa=-\sqrt{2}b_{1}-2b_{2}$, so we indeed have $d\kappa=\omega_{S}$.

8 Twists and the quaternionic Kähler metric

Given a projective special Kähler manifold $S$, with special Kähler cone $C$ carrying the conic isometry $X$, the c-map corresponds to the twist of the cotangent bundle $H$ of $C$. The lift of the conic isometry leads to a rotating symmetry on $H$ which generates the Abelian action needed to twist.

According to the general theory developed in [20], the pseudo-hyperKähler metric on $H=T^{*}C$ is given in terms of the coframes as

[TABLE]

where $\hat{\gamma}=(\hat{a},\hat{b},\hat{\varphi},\hat{\psi})$ is the coframe of $C$ given in §2 and $\hat{\delta}=(\hat{A},\hat{B},\hat{\Phi},\hat{\Psi})$ is a corresponding coframe on each $T_{x}^{*}C$ satisfying $d\hat{\delta}=-\hat{\delta}\wedge\omega_{\nabla}$. The triple of Kähler two-forms is then given by

[TABLE]

The conic symmetry $X$ lifts to a vector field $\tilde{X}$ on $H$ preserving $g_{H}$ and $\omega_{I}$, but with $L_{\tilde{X}}\omega_{J}=\omega_{K}$.

To obtain a quaternionic Kähler metric one first considers the positive definite metric on $H$ given by

[TABLE]

this called an “elementary deformation” of $g_{H}$ in [20]. Now one builds a principal circle bundle $P\to H$ with principal generator $Y$ and curvature

[TABLE]

and constructs the twist $Q$ of $H$ as $Q=P/(\tilde{X}-t^{2}Y/2)$, the factor $-t^{2}/2$ being the twist function satisfying the condition $d(-t^{2}/2)=\tilde{X}\mathbin{\lrcorner}F$. Tensors on $H$ invariant under $\tilde{X}$ may now be transferred to $Q$, and the exterior differential on $Q$ corresponds to the operation $d_{Q}$ on $H$ given

[TABLE]

for invariant forms $\beta$. The general result of [20] is that metric on $Q$ induced by $g_{N}$ is quaternionic Kähler.

Since the coframe $\gamma=(\tilde{a},\tilde{b},\varphi,\tilde{\psi})$ is $X$-invariant, the twisted differentials are directly computed leading to

[TABLE]

In the case that $S$ is a Lie group with $a$, $b$, and $d\varphi=2\omega_{S}$ left-invariant, these results are constant coefficient with respect to $\gamma$. The coframe $\tilde{\delta}=\hat{\delta}/t=(\tilde{A}^{T},\tilde{B}^{T},\tilde{\Phi},\tilde{\Psi})$ is not $\tilde{X}$-invariant, but

[TABLE]

When the Kähler form is exact $\omega_{S}=d\kappa$ with $\kappa$ left-invariant and $S$ is simply-connected, we may choose a principle parameter $\tau$ for $C_{0}$ as in §6 and define $\delta=\tilde{\delta}\exp(\mathbf{i}\tau)$. Then $X\tau=1$ implies

[TABLE]

We may now apply the twisted differential $d_{Q}$ to the coframe $\delta$ to obtain

[TABLE]

In the previous expression all non-$\delta$ terms are constant coefficient except possibly $\eta\exp(2\mathbf{i}\tau)$. But in the notation of §6, using Lemma 2.1 we have

[TABLE]

which is constant coefficient if and only if $P=\pi^{*}p$ and $Q=\pi^{*}q$ with $p,q$ left-invariant. Therefore,

Theorem 8.1**.**

The twist of $T^{*}C$ where $C$ is the special Kähler cone of an invariant projective special Kähler structure on $S$ of dimension $2n$ is a homogeneous quaternionic Kähler manifold of dimension $4n+4$. ∎

For the examples we have provided in the paper, the resulting quaternionic Kähler manifolds are already known, see for example [14, Table 2, p. 499], and the solvable group is a subgroup of larger isometry group. The examples may be checked in the same way in [20, §6], where we verified that the two structures on $\mathbb{C}H(1)$, Example 2.5, yield $\mathrm{U}(2,2)/(\mathrm{U}(2)\times\mathrm{U}(2))$ and $G_{2}^{*}/\mathrm{SO}(4)$. Similarly, the flat cone of Remark 2.4 gives $\mathrm{U}(n,2)/(\mathrm{U}(n)\times\mathrm{U}(2))$; the product $\mathbb{C}H(1)^{2}$, §7, yields $\mathrm{SO}(3,4)/(\mathrm{SO}(3)\times\mathrm{SO}(4))$ and $\mathbb{C}H(1)^{3}$, Proposition 5.1, produces $\mathrm{SO}(4,4)/(\mathrm{SO}(4)\times\mathrm{SO}(4))$.

Acknowledgements

OM would like to thank the hospitality of the Centre for Quantum Geometry of Moduli Spaces, and Aarhus University, where part of this work was developed. His work was partially supported by the MINECO-FEDER project MTM2016-77093-P. AFS was partially supported by the Danish Council for Independent Research | Natural Sciences project DFF - 6108-00358.