Hypersymplectic manifolds and associated geometries

TL;DR

This paper explores the geometric structure of hypersymplectic manifolds with SU(1,1) symmetry, identifying conditions under which they form cones over split 3-Sasakian manifolds and fibrations over para-quaternionic Kähler manifolds, with applications to Nahm-Schmid equations.

Contribution

It introduces an obstruction criterion for hypersymplectic manifolds with SU(1,1) actions and characterizes their geometric structures when the obstruction vanishes, including new examples.

Findings

Hypersymplectic manifolds with vanishing obstruction are metric cones over split 3-Sasakian manifolds.

Proper SU(1,1) actions induce fibrations over para-quaternionic Kähler manifolds.

The moduli space of Nahm-Schmid equations admits a fibration over a para-quaternionic Kähler manifold.

Abstract

We investigate an obstruction for hypersymplectic manifolds equipped with a free, isometric action of SU(1,1). When the obstruction vanishes, we show that the manifold is a metric cone over a split 3-Sasakian manifold. Furthermore, if the action of SU(1,1) is also proper, then the hypersymplectic manifold fibres over a para-quaternionic Kahler manifold. We conclude the article with some examples for which the obstruction vanishes. In particular, we show that the moduli space to Nahm-Schmid equations admits a fibration over a para-quaternionic Kahler manifold.

Equations

$$\mathrm{Sp}(2n,\mathbb{R})\cdot \mathrm{Sp}(2,\mathbb{R})=\mathrm{Sp}(2n,\mathbb{R}){\times }_{\pm 1}\mathrm{Sp}(2,\mathbb{R}).$$

$${\mathbf{i}}^2=-1,{\mathbf{s}}^2=1={\mathbf{t}}^2,\mathbf{i}\cdot \mathbf{s}=\mathbf{t}=-\mathbf{s}\cdot \mathbf{i},\mathbf{i}=\mathbf{t}\cdot \mathbf{s}.$$

$$\mathfrak{su}(1,1)=\text{Span}\left\{\left[\begin{array}{cc}\mathbf{i}& 0\\ 0& -\mathbf{i}\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],\left[\begin{array}{cc}0& \mathbf{i}\\ -\mathbf{i}& 0\end{array}\right]\right\}\cong \mathfrak{Im}(\mathbb{B}).$$

$$\mathcal{H}=\{q\in \mathfrak{Im}(\mathbb{B})|{\Vert q\Vert }^2=\pm 1\}\cong \{(x_1,x_2,x_3)\in {\mathbb{R}}^3|x_1^2-x_2^2-x_3^2=\pm 1\}$$

$$I(q)=q\overline{\mathbf{i}},S(q)=q\mathbf{s},T(q)=q\mathbf{t}.$$

$$\lambda (q)={\mathbb{I}}_q:=q_{1}I+q_{2}S+q_{3}T.$$

$$⟨\alpha ,\beta ⟩:=\mathrm{Re}\left(\beta {\overline{\alpha }}^T\right),\alpha ,\beta \in {\mathbb{B}}^n.$$

$$\mathrm{Sp}(n,\mathbb{B}):=\{A\in M_n(\mathbb{B})|A{\overline{A}}^T=\mathrm{id}\}\cong \mathrm{Sp}(2n,\mathbb{R}),$$

$$\mathfrak{sp}(n,\mathbb{B}):=\{A\in M_n(\mathbb{B})|A+{\overline{A}}^T=0\}.$$

$$(A,\xi )\cdot q⟼Aq\overline{\xi }.$$

$$\xi \cdot ({\mathbb{I}}_q(h))=\xi \cdot (h\overline{q})=h\overline{q}\overline{\xi }=h\overline{\xi }(\xi \overline{q}\overline{\xi })={\mathbb{I}}_{{\text{Ad}}_{{\xi }^{-1}}q}\cdot (\xi \cdot h).$$

$$g_M(IX,IY)=g_M(X,Y),g_M(SX,SY)=-g_M(X,Y)=g_M(TX,TY).$$

$${\omega }_1(X,Y):=g_M(IX,Y),{\omega }_2(X,Y):=g_M(SX,Y),{\omega }_3(X,Y):=g_M(TX,Y).$$

$$(x_1,x_2,x_3)⟼x_{1}I+x_{2}S+x_{3}T.$$

$$(y_1,y_2,y_3)⟼y_{1}I+y_{2}S+y_{3}T.$$

$$\lambda :\mathfrak{sp}(1,\mathbb{B})⟼\mathrm{End}(TM),h⟼{\mathbb{I}}_h:=\lambda (h),h\in \mathfrak{sp}(1,\mathbb{B})=\mathrm{Im}(\mathbb{B}).$$

$$⟨\omega ,h⟩:=g_M({\mathbb{I}}_h\cdot ,\cdot )=h_1{\omega }_1\mathbf{i}+h_2{\omega }_2\mathbf{s}+h_3{\omega }_3\mathbf{t}.$$

$$\Omega ={\omega }_1\wedge {\omega }_1-{\omega }_2\wedge {\omega }_2-{\omega }_3\wedge {\omega }_3.$$

$${\mathrm{St}}_{\Omega }\to \mathrm{Sp}(1,\mathbb{B})/\pm 1\cong {\mathrm{SO}}^{+}(1,2).$$

$$\iota :{\otimes }^p\mathfrak{sp}(1,\mathbb{B}{)}^{\ast }\otimes {\Omega }^q(M)⟶\mathfrak{sp}(1,\mathbb{B}{)}^{\ast }{\otimes }^p\mathfrak{sp}(1,\mathbb{B}{)}^{\ast }\otimes {\Omega }^{q-1}(M),⟨\iota (\alpha ),\xi ⟩={\iota }_{K_{\xi }^M}\alpha$$

$${\mathcal{L}}_{\mathfrak{sp}(1,\mathbb{B})}:{\otimes }^p\mathfrak{sp}(1,\mathbb{B}{)}^{\ast }\otimes {\Omega }^q(M)⟶\mathfrak{sp}(1,\mathbb{B}{)}^{\ast }{\otimes }^p\mathfrak{sp}(1,\mathbb{B}{)}^{\ast }\otimes {\Omega }^q(M),⟨{\mathcal{L}}_{\mathfrak{sp}(1,\mathbb{B})}\alpha ,\xi ⟩={\mathcal{L}}_{K_{\xi }^M}\alpha .$$

$${\mathcal{L}}_{\mathfrak{sp}(1,\mathbb{B})}\omega =2\omega .$$

$$⟨q^{\ast }\omega ,\xi ⟩(V,W)=g_M({\mathbb{I}}_{\xi }(q_{\ast }V),q_{\ast }W)=g_M(q_{\ast }^{-1}{\mathbb{I}}_{\xi }(q_{\ast }V),W)=⟨\omega ,{\text{Ad}}_{q^{-1}}(\xi )⟩(V,W).$$

$$⟨{\mathcal{L}}_{\mathfrak{sp}(1,\mathbb{B})}\omega ,\xi \otimes {\xi }^{\prime }⟩$$

$$[\cdot ,\cdot ]:{\Lambda }^2\mathfrak{sp}(1,\mathbb{B})⟶\mathfrak{sp}(1,\mathbb{B})$$

$$\mathbf{i}\wedge \mathbf{s}⟼[\mathbf{i},\mathbf{s}]=2\mathbf{t},\mathbf{s}\wedge \mathbf{t}⟼[\mathbf{s},\mathbf{t}]=2\overline{\mathbf{i}},\mathbf{t}\wedge \mathbf{i}⟼[\mathbf{t},\mathbf{i}]=2\mathbf{s}.$$

$$⟨\gamma ,\xi \otimes {\xi }^{\prime }⟩=\frac{1}{2}{g}_M({\mathbb{I}}_{\xi }{K}_{{\xi }^{\prime }}^M,\cdot ).$$

$$\mathfrak{sp}(1,\mathbb{B}{)}^{\ast }\otimes \mathfrak{sp}(1,\mathbb{B}{)}^{\ast }=\mathbb{R}\oplus {\Lambda }^2\left(\mathfrak{sp}(1,\mathbb{B}{)}^{\ast }\right)\oplus S_0\left(\mathfrak{sp}(1,\mathbb{B}{)}^{\ast }\right).$$

$${\iota }_{\mathfrak{sp}(1,\mathbb{B})}\omega =({\gamma }_0,{\gamma }_1,{\gamma }_2).$$

$${\mathcal{L}}_{\mathfrak{sp}(1,\mathbb{B})}{\gamma }_1=2{\gamma }_1.$$

Full text

1. Introduction

A hypersymplectic manifold is a $4n$-dimensional pseudo-Riemannian manifold, equipped with a metric of neutral signature $(2n,2n)$, and whose holonomy is contained inside the symplectic group ${\rm Sp}(2n,{\mathbb{R}})$. It can be viewed as a pseudo-Riemannian analogue of hyperKähler manifolds. Hypersymplectic geometry appears naturally in the study of integrable systems [3], string theory [14] - where it is also known by Kleinian geometry - and gauge theory [5, 19]. The terminology “hypersymplectic” is due to Hitchin [13].

A powerful tool for constructing hypersymplectic manifolds is the hypersymplectic quotient construction, which is an adaptation of the Marden-Weinstein construction in symplectic geometry. However, in contrast with the hyperKähler situation, more ofthen than not, the hypersymplectic structure on the quotient, is degenerate [9].

Another way of obtaining hypersymplectic manifolds is via an adaptation of Swann’s bundle construction in hyperKähler geometry [22]. Starting with a quaternionic Kähler manifold of positive scalar curvature, say $N$, Swann’s construction produces a bundle, $\mathcal{U}(N)\rightarrow N$ with a typical fibre ${\mathbb{H}}^{\ast}/{\mathbb{Z}}_{2}$, whose total space carries a hyperKähler structure. It is possible to carry over this construction to the pseudo-Riemannian case [9]. In order to do so, one needs a para-quaternionic Kähler manifold; i.e., a $4n$-dimensional pseudo-Riemannian manifold, whose holonomy is contained inside the group

[TABLE]

They can be thought of as pseudo-Riemannian analogues of quaternionic Kähler manifolds. Starting with a para-quaternionic Kähler manifold $N$, the construction produces a bundle $\mathcal{U}(N)\rightarrow N$, with a typical fibre ${\mathbb{B}}^{\ast}/{\mathbb{Z}}_{2}$, where ${\mathbb{B}}^{\ast}$ is the space of non-zero split quaternions with non-zero norm. The total space $\mathcal{U}(N)$ carries a hypersymplectic structure. Both the para-quaternionic Kähler and hypersymplectic geometries are Einstein. Additionally, the latter is also Ricci-flat. Para-quaternionic Kähler manifolds are characterised by the existence of a closed 4-form, whereas hypersymplectic manifolds are equipped with family of symplectic 2-forms.

In this article, we study a more general picture. Namely, given a hypersymplectic manifold, when does it admit a para-quaternionic Kähler quotient? Our basic observation is that, the total space of a Swann bundle over a para-quaternionic Kähler manifold admits a free, proper, isometric action of ${\rm Sp}(2,{\mathbb{R}})\cong{\rm SU}(1,1)$, which is an analogue of the permuting ${\rm Sp}(1)$-action on hyperKähler manifolds. For such hypersymplectic manifolds, we construct two maps $\rho_{2}:M\rightarrow S^{4}(B)$ - where $B$ is the standard representation of ${\rm SU}(1,1)$ on the vector space of split quaternions ${\mathbb{B}}$ - and $\rho_{0}:M\rightarrow{\mathbb{R}}_{>0}$. If $\rho_{2}$ vanishes, we show that $\rho_{0}$ is a hypersymplectic potential. The level-sets of $\rho_{0}$ carry a split 3-Sasakian structure and the metrics on different level-sets are homothetic. In particular, the hypersymplectic manifold can be thought of as a metric cone over a split 3-Sasakian manifold. Additionally, if the action of ${\rm SU}(1,1)$ is also proper, we show that the quotient of a level-set of $\rho_{0}$, by $\rm{SU}(1,1)$, is a para-quaternionic Kähler manifold.

This approach is analogous to that of Boyer, Galicki and Mann [7] for hyperKähler manifolds with permuting ${\rm Sp}(1)$-action.

Split 3-Sasakian structures were introduced by Swann, Jørgensen and Dancer in [9] and have also been studied by Caldarella and Pastore [8], where they are referred to as “mixed 3-Sasakian structures”. The authors show that any split 3-Sasakian structure is necessarily Einstein.

We give two examples of hypersymplectic manifolds which have the ${\rm SU}(1,1)$ symmetry, with vanishing obstruction. First example is that of hypersymplectic manifolds, obtained via hypersymplectic reduction of flat-space. We show that the Swann-bundle construction commutes with the quotient construction, which produces a family of examples of the theory. The results complement the work of Swann, Jørgensen and Dancer in [9].

The second example is that of the moduli space of Nahm-Schmid equations defined on the interval $[0,1]$ [5]. The solutions to Nahm-Schmid equations exist for all times. As a result, it is possible to define a scaling action on the moduli space of solutions. The moduli space also carries a free, proper, permuting action of ${\rm SU}(1,1)$. Therefore, topologically, it has the structure of a metric cone over a split 3-Sasakian manifold. The quotient of the latter by the ${\rm SU}(1,1)$ action is a paraquaternionic Kähler manifold. In other words, the moduli space can be expressed as the total space of a Swann bundle over a paraquaternionic Kähler manifold.

2. Acknowledgements

The author wishes to express his heartfelt thanks to Prof. Andrew Dancer and Dr. Markus Röser for many helpful discussions and to Prof. Stefan Ivanov for the references [15, 16]. The author also wishes to thank the anonymous referee for many helpful comments, especially on the content in Sub-section 6.2 on Nahm-Schmid equations. This work originated, in part, through discussions with Prof. Nuno Rumão during the program Integrable systems in Mathematics, Condensed Matter and Statistical Physics (Code: ICTS/integrability2018/07) at the International Centre for Theoretical Sciences (ICTS-TIFR). The author wishes to thank the organizers of the program for their support.

3. Brief introduction to split quaternionic geometry

The space of split quaternions is a 4-dimensional vector space ${\mathbb{B}}$, spanned by $(1,\mathrm{\mathbf{i}},\mathbf{s},\mathbf{t})$, satisfying the following relations

[TABLE]

Like quaternions, the vector space of split quaternions comes equipped with a multiplication operation, which gives it a structure of an associative algebra.

The vector space carries a natural inner product defined by $\left\langle p,q\right\rangle_{{\mathbb{B}}}=\operatorname{Re}p\,\overline{q}$, where $p=p_{0}+\mathrm{\mathbf{i}}\,p_{1}+\mathbf{s}\,p_{2}+\mathbf{t}\,p_{3}$ and $\overline{p}=p_{0}-\mathrm{\mathbf{i}}\,p_{1}-\mathbf{s}\,p_{2}-\mathbf{t}\,p_{3}$. Since the metric is not positive definite, it only makes sense to talk about the (isotropic) quadratic form (or the ‘norm-square’) $\left\|p\right\|^{2}:=\left\langle p,p\right\rangle_{{\mathbb{B}}}$, associated with the neutral signature metric. Given two split-quaternions $p$ and $q$, we have $\left\|p\,q\right\|^{2}=\left\|p\right\|^{2}\cdot\left\|q\right\|^{2},$ showing that the quadratic form is multiplicative. Note that if $p$ is a point in $\mathfrak{Im}({\mathbb{B}})$, then $\left\|p\right\|^{2}=p\cdot\overline{p}=-p^{2}=p_{1}^{2}-p_{2}^{2}-p_{3}^{2}\in{\mathbb{R}}$.

Unlike the quaternionic algebra, the split quaternion algebra contains non-trivial zero divisors. Moreover, the elements $\mathrm{\mathbf{i}},\mathbf{s},\mathbf{t}\in\mathfrak{Im}({\mathbb{B}})$ are not the only elements with length $\pm 1$. Elements with norm $1$ are parametrized by the 2-sheeted hyperboloid $x_{1}^{2}-x_{2}^{2}-x_{3}^{2}=1$, while those with norm $-1$ are parametrized by the 1-sheeted hyperboloid $x_{1}^{2}-x_{2}^{2}-x_{3}^{2}=-1$. Any triple $\{\mathrm{\mathbf{i}},\mathbf{s},\mathbf{t}\}$ satisfying (3.1) defines a split-quaternionic structure on ${\mathbb{B}}$.

Let $U=\{q\in{\mathbb{B}}~{}|\left\|q\right\|^{2}\neq 0\}$ be the set of all units in ${\mathbb{B}}$. This is clearly a multiplicative group. The subset of $U$ consisting of all elements $q$ with $\left\|q\right\|^{2}=1$ forms a non-compact topological group ${\rm SU}(1,1)$. This is the special unitary group of all complex $2\times 2$ matrices $g$ that satisfy

1. (1)

Unimodularity, i.e, $\det g=1$ 2. (2)

Pseudo-unitary condition: i.e, $g^{\ast}\,J\,g=J$ where $J=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}$

In particular, any element $g\in{\rm SU}(1,1)$ has the form $g=\begin{bmatrix}\alpha&\beta\\ \beta^{\ast}&\alpha^{\ast}\end{bmatrix}$, where $\alpha$ and $\beta$ are complex numbers subject to the condition $\left\lvert\alpha\right\rvert^{2}-\left\lvert\beta\right\rvert^{2}=1$. The Lie algebra of ${\rm SU}(1,1)$ is 3-dimensional

[TABLE]

Relation with ${\rm SO}(1,2)$:

Consider the 3-dimensional Lorentz group ${\rm SO}(1,2)$. This is the group of transformations of the 3-dimensional Minkowski space ${\mathbb{M}}^{3}$, with determinant $1$, that preserves the quadratic form. The group acts transitively on the 1-sheeted and 2-sheeted hyperboloids and also on the cone $\left\|x\right\|^{2}=0$.

Alternatively, if we consider the pseudo-sphere

[TABLE]

then, the hyperboloids can be thought of as unit spacelike and timelike vectors in the $\mathscr{H}$ and the ${\rm SO}(1,2)$-action on $\mathscr{H}$ is then an analogue of the standard action of ${\rm SO}(3)$ on the 2-sphere $S^{2}$.

The group ${\rm SO}(1,2)$ is disconnected and has two connected components. We denote by ${\rm SO}^{+}(1,2)$ the identity component. Identifying ${\mathbb{M}}^{3}$ with the imaginary split-quaternions $\mathfrak{Im}({\mathbb{B}})$, it is easily seen that the adjoint action of ${\rm SU}(1,1)$ on ${\mathbb{M}}^{3}$ preserves the quadratic form, the pseudo sphere $\mathscr{H}$ and the null-cone. Therefore the linear transformations corresponding to the adjoint action of the elements of ${\rm SU}(1,1)$ belong to the identity component ${\rm SO}^{+}(1,2)$. This gives a homomorphism from ${\rm SU}(1,1)$ to ${\rm SO}^{+}(1,2)$ with kernel $\pm 1$; i.e., ${\rm SU}(1,1)/\pm 1\cong{\rm SO}^{+}(1,2)$, similar to the homomorphism between ${\rm SU}(2)$ and ${\rm SO}(3)$.

3.1. Modules over split quaternions

Consider the left ${\mathbb{B}}$ module ${\mathbb{B}}^{n}\cong{\mathbb{R}}^{4n}$, equipped with the split quaternionic structure $I,S,T$, given by

[TABLE]

Remark 1**.**

Any element $q\in\mathscr{H}$ determines a product or a complex structure on ${\mathbb{B}}^{n}$. To see this, let $\lambda:\,\mathfrak{Im}({\mathbb{B}})\longrightarrow\operatorname{End}\,({\mathbb{B}}^{n})$ denote the algebra homomorphism

[TABLE]

Then $q$ determines a product or a complex structure, depending on whether $\left\|q\right\|^{2}=\mp 1$. In other words, the pseudo-sphere $\mathscr{H}$ parametrizes the complex and the product structures on ${\mathbb{B}}^{n}$.

The module ${\mathbb{B}}^{n}$ inherits the natural inner product

[TABLE]

The automorphism group of ${\mathbb{B}}^{n}$, given by

[TABLE]

is nothing but the automorphism group of the symplectic vector space $({\mathbb{R}}^{2n},\omega_{\scriptscriptstyle{\mathbb{R}}^{2n}})$. The Lie algebra of ${\rm Sp}(n,{\mathbb{B}})$ is given by

[TABLE]

Note that for $n=1$, we have the isomorphism ${\rm Sp}(1,{\mathbb{B}})\cong{\rm SU}(1,1)\cong{\rm SL}(2,{\mathbb{R}})$ and so, we can identify the Lie algebra $\mathfrak{sp}(1,{\mathbb{B}})=\mathfrak{Im}({\mathbb{B}})$.

Consider the action of the group ${\rm Sp}(n,{\mathbb{B}})\times{\rm Sp}(1,{\mathbb{B}})$ on ${\mathbb{B}}^{n}$, given by

[TABLE]

Let $\Lambda_{\mathscr{H}}$ denote the image of $\mathfrak{Im}({\mathbb{B}})$ under the map $\lambda$. It is easy to see that the action of ${\rm Sp}(n,{\mathbb{B}})$ is isometric and the induced action on $\Lambda_{\mathscr{H}}$ preserves $\Lambda_{\mathscr{H}}$, pointwise. On the other hand, the ${\rm Sp}(1,{\mathbb{B}})$-action is isometric, but the induced action on $\Lambda_{\mathscr{H}}$ is, pointwise, nothing but the standard action of ${\rm SO}^{+}(1,2)$ on $\mathscr{H}$. Indeed, for any $\xi\in{\rm Sp}(1,{\mathbb{B}})$,

[TABLE]

4. Hypersymplectic manifolds

Let $(M,g_{\scriptscriptstyle M},I,S,T)$ be a $4n$-dimensional pseudo-Riemannian manifold, endowed with a triple of endomorphisms $I,S,T$, satisfying the split quaternionic relations (3.1) and a metric of neutral signature $(2n,2n)$, that is compatible with the split quaternionic structure

[TABLE]

The split quaternionic structure allows us to define the following 2-forms on $M$

[TABLE]

If each of the above 2-forms are closed, the manifold $M$ is called a hypersymplectic manifold. Using Hitchin’s arguments for the hyperKähler manifolds, one can show that the structures $I,S,T$ are integrable; i.e., they are parallel with respect to the Levi-Civita connection. As a result, the holonomy group of $M$ reduces to ${\rm Sp}(n,{\mathbb{B}})$.

The endomorphisms $S$ and $T$ are called product structures. This is because the integrability of these structures implies that the manifold $M$ locally looks like a product $M^{+}\times M^{-}$, where $\pm$ denotes the eigenvalues $\pm 1$ of $S$ ot $T$ and $TM^{\pm}$ denotes the corresponding eigenspaces. In fact, every element of the 1-sheeted hyperboloid $x_{1}^{2}-x_{2}^{2}-x_{3}^{2}=-1$ determines a product structure as

[TABLE]

Such structures are also known by paracomplex structures in literature. On the other hand, $M$ also has a family of pseudo-Kähler structures, which are parametrized by the 2-sheeted hyperboloid $y_{1}^{2}-y_{2}^{2}-y_{3}^{2}=1$ as

[TABLE]

A pseudo-Kähler structure on a manifold $M$ is a complex structure $I$ on $M$ along with a pseudo-Riemannian metric $g_{\scriptscriptstyle M}$, such that the metric is compatible with $I$ and the 2-form $\omega_{\scriptscriptstyle M}(\cdot,\cdot)=g_{\scriptscriptstyle M}(I(\cdot),\cdot)$ is closed. In other words, a pseudo-Kähler structure is just a pseudo-Riemannian analogue of Kähler structure.

In some cases, it is possible to explicitly construct a family of examples of hypersymplectic manifolds. Ivanov and Zamkovoy [16] constructed a hypersymplectic structure on Kodaira-Thurston (properly elliptic) surfaces. Andrada and Salamon, in [2], show that if there exists a complex product structure on a real Lie algebra $\mathfrak{g}$; i.e., a pair ${I,S}$ of complex structure and a product structure, then, it induces a hypersymplectic structure on the complexification $\mathfrak{g}^{{\mathbb{C}}}$. In [15], Ivanov and Tsanov showed that the manifolds underlying the Lie groups ${\rm SL}(2m-1,R)$ and ${\rm SU}(m,m-1)$ carry a complex product structure which induces a hypersymplectic structure on their complexifications.

For a hypersymplectic manifold $(M,g_{\scriptscriptstyle M},I,S,T)$, let ${\mathcal{I}}\subset\operatorname{End}(TM)$ denote the trivial 3-dimensional sub-bundle spanned by $(I,S,T)$. Any covariantly constant endomorphism ${\mathbb{I}}\in{\mathcal{I}}$ can be thought of as a map with values in $\mathfrak{sp}(1,{\mathbb{B}})^{\ast}$, using the algebra homomorphism

[TABLE]

Similarly, the associated symplectic 2-forms can be clubbed into a single $\mathfrak{sp}(1,{\mathbb{B}})^{\ast}$-valued 2-form as

[TABLE]

Let $\Lambda_{\mathscr{H}}\subset{\mathcal{I}}$ denote the image of $\mathscr{H}\subset\mathfrak{sp}(1,{\mathbb{B}})$ under the map $\lambda$. In particular, $\Lambda_{\mathscr{H}}$ consists of all the product and complex structures on $M$.

4.1. Permuting actions

Consider the fundamental 4-form

[TABLE]

The form is globally defined on $M$. The stabilizer group ${\rm St}_{\Omega}\in\text{Isom}(M,g_{\scriptscriptstyle M})$ of $\Omega$ is a sub-group of the group of isometries that preserves each symplectic 2-form $\omega_{i}$. The induced action of ${\rm St}_{\Omega}$, on $\Lambda_{\mathscr{H}}$, determines the homomorphism

[TABLE]

The kernel of this homomorphism is the group of hypersymplectic isometries, whose induced action on $\Lambda_{\mathscr{H}}$, pointwise, fixes $\Lambda_{\mathscr{H}}$.

Definition 1**.**

An isometric action of the group ${\rm Sp}(1,{\mathbb{B}})$ on a hypersymplectic manifold $M$ is said to be permuting, if the induced action on $\Lambda_{\mathscr{H}}$, is the standard action of ${\rm SO}^{+}(1,2)$ on $\mathscr{H}$. In other words, the action is induced via the epimorphism ${\rm Sp}(1,{\mathbb{B}})\rightarrow{\rm St}_{\Omega}\rightarrow{\rm SO}^{+}(1,2)$.

Henceforth, without loss of generality, we will assume that $M$ admits a free, permuting, effective action of the group ${\rm Sp}(1,{\mathbb{B}})$. The arguments that follow are an adaptation of the representation theoretic arguments in [7, 18].

Let $K^{M}_{\xi}$ denote the fundamental vector field on $M$ corresponding to $\xi\in\mathfrak{sp}(1,{\mathbb{B}})$. Define the following operators:

[TABLE]

and

[TABLE]

Then Cartan’s formula ${\mathcal{L}}_{\mathfrak{sp}(1,{\mathbb{B}})}=d\,\iota_{\mathfrak{sp}(1,{\mathbb{B}})}+\iota_{\mathfrak{sp}(1,{\mathbb{B}})}\,d$ is easily verified.

Lemma 4.1** ([7, 18]).**

For the $\mathfrak{sp}(1,{\mathbb{B}})^{\ast}$-valued 2-form $\omega$ we have

[TABLE]

Proof.

We first verify that $\omega$ is ${\rm Sp}(1,{\mathbb{B}})$-equivariant. Let $q\in{\rm Sp}(1,{\mathbb{B}})$ and $\xi\in{\rm Sp}(1,{\mathbb{B}})$. Then for the vector fields $V$, $W$ on $M$

[TABLE]

Consider $\xi,\xi^{\prime}\in\mathfrak{sp}(1,{\mathbb{B}})$ such that $\xi\xi^{\prime}\neq 0\neq\xi^{\prime}\xi$. Then, using the identity above, we get:

[TABLE]

Note that this implies that ${\mathcal{L}}_{\mathfrak{sp}(1,{\mathbb{B}})}\,\omega$ is a $\Lambda^{2}(\mathfrak{sp}(1,{\mathbb{B}}))^{\ast}$-valued 2-form on $M$. We have the isomorphism

[TABLE]

given by

[TABLE]

This induces an isomorphism of between $\Lambda^{2}\mathfrak{sp}(1,{\mathbb{B}})^{\ast}$ and $\mathfrak{sp}(1,{\mathbb{B}})^{\ast}$. Therefore, we can think of ${\mathcal{L}}_{\mathfrak{sp}(1,{\mathbb{B}})}\,\omega$ as a $\mathfrak{sp}(1,{\mathbb{B}})^{\ast}$-valued 2-form on $M$. It is in this sense that we write the equality in (4.2). A straight forward computation using the isomorphism $[\cdot,\cdot]$ now shows that ${\mathcal{L}}_{\mathfrak{sp}(1,{\mathbb{B}})}\,\omega=2\,\omega$.

∎

Define the 1-form $\gamma=\frac{1}{2}\,\iota_{\mathfrak{sp}(1,{\mathbb{B}})}\,\omega\in\mathfrak{sp}(1,{\mathbb{B}})^{*}\otimes\mathfrak{sp}(1,{\mathbb{B}})^{*}\otimes\Omega^{1}(M)$. More precisely

[TABLE]

The tensor product $\mathfrak{sp}(1,{\mathbb{B}})^{*}\otimes\mathfrak{sp}(1,{\mathbb{B}})^{*}$ splits into a direct sum of sub-representations $S^{2}\left(\mathfrak{sp}(1,{\mathbb{B}})^{*}\right)\oplus\Lambda^{2}\left(\mathfrak{sp}(1,{\mathbb{B}})^{*}\right)$. The symmetric part further decomposes into a direct sum of the trace and the traceless component. Consequently,

[TABLE]

This is the Clebsch-Gordon decomposition. Correspondingly, the 1-form $\gamma$ decomposes into three components

[TABLE]

From Lemma 4.1, it follows that $d\gamma=2\omega$, since $d\omega=0$. However, note that the right hand side belongs to the $\mathfrak{sp}(1,{\mathbb{B}})^{\ast}\cong\Lambda^{2}\left(\mathfrak{sp}(1,{\mathbb{B}})^{*}\right)$. This implies that $d\gamma_{0}=0=d\gamma_{2}$ and $d\gamma_{1}=2\omega$.

Proposition 4.2**.**

Let $M$ be a hypersymplectic manifold with a permuting action of the group ${\rm Sp}(1,{\mathbb{B}})$. Let ${\mathcal{G}}\subset\Lambda^{2}(M)$ denote the trivial sub-bundle spanned by $\omega_{1},\omega_{2},\omega_{3}$. Then, the de-Rham cohomology class of any symplectic 2-form in ${\mathcal{G}}$ vanishes. In particular, $M$ can never be compact.

Lemma 4.3** ([18]).**

The map $\gamma_{1}$ satisfies the following identity

[TABLE]

Proof.

The proof is identical to that of Lemma 4.1. ∎

Following the approach in [18, 21] for the hyperKähler case, we now show that $\gamma_{0}$ and $\gamma_{2}$ are exact. Define $\rho:=\iota_{\mathfrak{sp}(1,{\mathbb{B}})}\,\gamma_{1}\in\Omega^{0}\left(M,\,\mathfrak{sp}(1,{\mathbb{B}})^{\ast}\otimes\mathfrak{sp}(1,{\mathbb{B}})^{\ast}\right)$. Corresponding to the decomposition (4.4), the map $\rho$ has 3 components:

[TABLE]

Denote by Alt, the projection of $\mathfrak{sp}(1,{\mathbb{B}})^{*}\otimes\mathfrak{sp}(1,{\mathbb{B}})^{*}$ to the alternating part $\Lambda^{2}\,\mathfrak{sp}(1,{\mathbb{B}})^{*}$ and by $\text{\rm\bf Sym}_{0}$, the projection of $\mathfrak{sp}(1,{\mathbb{B}})^{*}\otimes\mathfrak{sp}(1,{\mathbb{B}})^{*}$ to the traceless, symmetric part $S^{2}\,(\mathfrak{sp}(1,{\mathbb{B}})^{*})$. Then, the identity (4.5) can be written as

[TABLE]

Therefore we can write

[TABLE]

It follows that

[TABLE]

and

[TABLE]

In particular, $\gamma_{0}$ and $\gamma_{2}$ are exact.

4.1.1. Potentials

Para-Kähler manifold

A para-Kähler manifold $(M,g_{\scriptscriptstyle M})$, is a pseudo-Riemannian manifold, endowed with a metric compatible, parallel, skew-symmetric endomorphism $T\in\operatorname{End}(TM)$ satisfying $T^{2}=1$.

Suppose that $(M,g_{\scriptscriptstyle M},T)$ is a para-Kähler manifold. Let $\omega$ be the para-Kähler 2-form given by

[TABLE]

For any 1-form $\alpha$ on $M$, define $T\,\alpha(v):=-\alpha(Tv)$ for $v\in TM$. A para-Kähler potential is a smooth function $\rho:M\rightarrow{\mathbb{R}}$ such that $\frac{1}{2}\,\,dId\rho=\omega$. Note that any hypersymplectic manifold is also a para-Kähler manifold in many different ways. The para-Kähler structures are parametrized by the 1-sheeted hyperboloid.

Define $\kappa(\xi):=-\iota_{K^{M}_{\xi}}\,\gamma_{1}(\xi)$ for any $\xi\in\mathfrak{sp}(1,{\mathbb{B}})$ such that $\left\|\xi\right\|\neq 0$.

Case 1:  Suppose that $\xi\in\mathfrak{sp}(1,{\mathbb{B}})$ is such that $\xi^{2}=1$. Then the stabilizer of ${\mathbb{I}}_{\xi}$ is a sub-group ${\rm SO}^{+}(1,1)\subset{\rm Sp}(1,{\mathbb{B}})$, consisting of $2\times 2$ real matrices of the form $g=\begin{bmatrix}a&b\\ b&a\end{bmatrix}$ such that $a^{2}-b^{2}=1$. Its Lie algebra is the vector space of real numbers $({\mathbb{R}},+)$. The group ${\rm SO}^{+}(1,1)$ preserves the symplectic $2$-form $\omega_{\xi}$. Moreover, the associated moment map is given by $\kappa(\xi)$. Indeed, this can be seen as follows:

[TABLE]

Here we have used the identity (4.6).

Case 2:  Suppose that $\xi\in\mathfrak{sp}(1,{\mathbb{B}})$ is such that $\xi^{2}=-1$. Then ${\mathbb{I}}_{\xi}$ defines a complex structure. Let ${\rm U}(1)\in{\rm Sp}(1,{\mathbb{B}})$ denote the stabilizer of ${\mathbb{I}}_{\xi}$. Then, by the same argument as above, $\kappa(\xi)$ defines the moment map with respect to the ${\rm U}(1)$-action, for the pseudo-Kähler 2-form $\omega_{\xi}$.

Proposition 4.4**.**

Let $M$ be a hypersymplectic manifold with a permuting ${\rm Sp}(1,{\mathbb{B}})$-action. Let $\xi,\xi^{\prime}\in\mathfrak{sp}(1,{\mathbb{B}})$ with $\xi\xi^{\prime}\neq 0\neq\xi^{\prime}\xi$. Then the following holds

1. (1)

If their squares are $-1$ and they are perpendicular, then, $-\kappa(\xi^{\prime})$ is the pseudo-Kähler potential for the pseudo-Kähler 2-form $\omega_{\xi}$. 2. (2)

If their squares are $-1$ and $1$ respectively, then, $\kappa(\xi^{\prime})$ is the pseudo-Kähler potential for the pseudo-Kähler 2-form $\omega_{\xi}$. 3. (3)

If their squares are $1$ and $-1$ respectively OR both the squares are $1$ and if $\xi$ and $\xi^{\prime}$ are perpendicular, then, $\kappa(\xi^{\prime})$ is a “para Kähler potential” for the symplectic 2-form $\omega_{\xi}$.

Proof.

1. (1)

The proof follows from the following straight-forward computation

[TABLE]

The last equality follows from the fact that $[\xi^{\prime},\,[\xi,\,\xi^{\prime}]]=4\,\xi$. 2. (2)

Proof follows from the following computation, which is a slight variation of the one above

[TABLE]

The last equality follows from the fact that $[\xi^{\prime},\,[\xi,\,\xi^{\prime}]]=-4\,\xi$. Thus $-\kappa(\mathbf{s})$ and $-\kappa(\mathbf{t})$ are Kähler potentials for $\omega_{1}$. 3. (3)

By arguments identical to the ones above, we have

[TABLE]

Therefore $\kappa(\mathbf{s}),\,\kappa(\mathbf{t})$ are para Kähler potentials for $\omega_{3}$ and $\omega_{2}$, respectively.

∎

Define

[TABLE]

Note that if $\left\|\xi\right\|^{2}=0$, then $g_{\scriptscriptstyle M}({\mathcal{X}}(\xi,\xi^{\prime}),\,{\mathcal{X}}(\xi,\xi^{\prime}))=0$. In other words, ${\mathbb{I}}_{\xi}$ maps $K^{M}_{\xi^{\prime}}$ to a null-vector, for any $\xi^{\prime}\in\mathfrak{sp}(1,{\mathbb{B}})$. Owing to the Clebsch-Gordon decomposition (4.3), the map ${\mathcal{X}}$ splits into three parts: ${\mathcal{X}}_{0}$, ${\mathcal{X}}_{1}$, ${\mathcal{X}}_{2}$, given by

[TABLE]

Clearly then, $\gamma_{i}=\frac{1}{2}\,g_{\scriptscriptstyle M}({\mathcal{X}}_{i},\cdot)$ and therefore ${\mathcal{X}}_{i}$ are the gradient vector fields for $\rho_{i}$.

Definition 2** (Hypersymplectic potential).**

Given a hypersymplectic manifold $(M,g_{\scriptscriptstyle M},I,S,T)$, a smooth function $\rho_{0}:M\rightarrow{\mathbb{R}}$ is said to be a hypersymplectic potential if it is simultaneously a potential for $I,S,T$.

Lemma 4.5**.**

Let $M$ be a hypersymplectic manifold with a free, permuting ${\rm Sp}(1,{\mathbb{B}})$-action and assume ${\mathcal{X}}_{2}=0$ (equivalently, $\rho_{2}=0$). Then, $\rho_{0}$ is the hypersymplectic potential and we have

[TABLE]

Moreover, for any $\xi$ with $\left\|\xi\right\|^{2}=\pm 1$, the vector field $\mathcal{X}_{0}$ is independent of $\xi$.

Proof.

Since ${\mathcal{X}}_{2}=0$, it implies that $\rho_{2}$ is constant on connected components. However, since $\rho_{2}$ is ${\rm Sp}(1,{\mathbb{B}})$-equivariant, it must be identically zero on each of the connected components. In particular, $\rho_{2}\equiv 0$. Conversely, if $\rho_{2}\equiv 0$, then it follows that ${\mathcal{X}}_{2}=0$, since ${\mathcal{X}}_{2}=\operatorname{grad}(\rho_{2})$. Consequently, since

[TABLE]

Clearly, if $\left\|\xi\right\|=\pm 1$, the $\mathcal{X}_{0}$ is independent of $\xi$. Since $\mathcal{X}_{0}$ is the gradient vector field of $\rho_{0}$, it follows that $\frac{1}{2}\,g_{\scriptscriptstyle M}(\mathcal{X}_{0},\mathcal{X}_{0})=\rho_{0}$. From Proposition 4.4 we have $\rho_{0}=\kappa(\mathbf{i})=-\kappa(\mathbf{s})=-\kappa(\mathbf{t})$. Thus $\rho_{0}$ is the hypersymplectic potential. In particular, $d\,{\mathbb{I}}_{\xi}\,d\rho_{0}=\epsilon\,\omega_{\xi}$, where $\epsilon=\pm 1$, according to whether $\left\|\xi\right\|^{2}=\mp 1$. ∎

Note**.**

Let $\{\,\xi_{1},\xi_{2},\xi_{3}\,\}$ be the basis of $\mathfrak{sp}(1,{\mathbb{B}})$. If ${\mathcal{X}}_{2}=0$, then, the above Lemma says that

[TABLE]

It is important to note here that since the vector fields $K^{M}_{\xi_{i}}$ generate the free action of ${\rm Sp}(1,{\mathbb{B}})$ on $M$, the norm $g_{\scriptscriptstyle M}(K^{M}_{\xi_{1}},\,K^{M}_{\xi_{1}})$ must be positive, while, the norms $g_{\scriptscriptstyle M}(K^{M}_{\xi_{2}},\,K^{M}_{\xi_{2}})$ and $g_{\scriptscriptstyle M}(K^{M}_{\xi_{3}},\,K^{M}_{\xi_{3}})$ must be negative. Therefore we must have $\rho_{0}=g_{\scriptscriptstyle M}(\mathcal{X}_{0},\mathcal{X}_{0})>0$.

The existence of a hypersymplectic potential on $M$ implies that the metric $g_{\scriptscriptstyle M}$ is incomplete. The remainder of the section is dedicated to proving this and a few other consequences of the vanishing of the map $\rho_{2}$.

Proposition 4.6**.**

Let Let $M$ be a hypersymplectic manifold with a free, permuting ${\rm Sp}(1,{\mathbb{B}})$-action and assume that ${\mathcal{X}}_{2}=0$. Then the following holds

1. (1)

$\gamma_{1}\,=\,\iota_{\mathcal{X}_{0}}\,\omega$** 2. (2)

${\mathcal{L}}_{\mathcal{X}_{0}}\,\gamma_{1}\,=\,2\,\gamma_{1}$** 3. (3)

${\mathcal{L}}_{\mathcal{X}_{0}}\,\omega\,=\,2\,\omega$** 4. (4)

${\mathcal{L}}_{\mathcal{X}_{0}}\,\rho_{0}\,=\,2\,\rho_{0}$** 5. (5)

${\mathcal{L}}_{\mathfrak{sp}(1,{\mathbb{B}})}\,\mathcal{X}_{0}\,=\,0$**

Proof.

First, we make the following observation. Owing to Lemma 4.5, we have

[TABLE]

1. (1)

Recall that $\gamma_{1}=\operatorname{\rm\bf Alt}\,\left(\iota_{\mathfrak{sp}(1,{\mathbb{B}})}\,\omega\right)$. Therefore,

[TABLE]

The last equality can be seen as follows

[TABLE]

Similarly, one can show that

[TABLE]

Therefore, we have $\gamma_{1}\,=\,\iota_{\mathcal{X}_{0}}\,\omega$. 2. (2)

The second claim follows directly from the first one by observing that

[TABLE] 3. (3)

Consider the argument above Lemma 4.1. We have shown that $d\gamma_{1}=2\,\omega$. But from the claim (1), it follows that $d\gamma_{1}={\mathcal{L}}_{\mathcal{X}_{0}}\,\omega$. In conclusion, ${\mathcal{L}}_{\mathcal{X}_{0}}\,\omega=2\,\omega$. 4. (4)

Observe that

[TABLE] 5. (5)

This follows from the invariance of $\rho_{0}$ under the action of ${\rm Sp}(1,{\mathbb{B}})$.

∎

Proposition 4.7**.**

The gradient vector field of the hypersymplectic potential $\rho_{0}$ satisfies

[TABLE]

where $\nabla$ is the Levi-Civita connection of the metric $g_{\scriptscriptstyle M}$.

To show this, we use the following result by Swann

Proposition 4.8** ([22]).**

Let $(N,I,g_{\scriptscriptstyle N})$ be a Kähler manifold and $\nabla$ be the associated Chern connection. A smooth function $\rho_{0}:N\longrightarrow{\mathbb{R}}^{+}$ is a Kähler potential if and only if

[TABLE]

The above statement of the theorem holds even if the metric $g_{\scriptscriptstyle N}$ is a pseudo-Kähler metric.

Proof of Lemma 4.7.

If we regard $M$ as a pseudo-Kähler manifold with respect to the complex structure $I$, then we have shown that $\rho_{0}$ is a Kähler potential for $I$. Then, from (4.7) it follows that

[TABLE]

Now consider

[TABLE]

Therefore,

[TABLE]

In Proposition 4.6, we have shown that ${\mathcal{L}}_{\mathcal{X}_{0}}\,\omega_{1}=2\,\omega_{1}$ and therefore ${\mathcal{L}}_{\mathcal{X}_{0}}\,I=0$. Using this, we have

[TABLE]

Plugging this in the previous equation, we get

[TABLE]

From Proposition 4.8, it follows that $g_{\scriptscriptstyle M}(\nabla_{X}\,\mathcal{X}_{0},\,Y)=g_{\scriptscriptstyle M}(X,\,Y)$, for all $X,Y\in TM$. Therefore, $\nabla\,\mathcal{X}_{0}=\operatorname{id}_{TM}$.

∎

Corollary 4.9**.**

The hypersymplectic potential $\rho_{0}$ satisfies

[TABLE]

There are two consequences of the above corollary. First, the metric on $M$ cannot be complete (see [25]). Second, the metrics from difference level-sets of $\rho_{0}$ are homothetic.

In the section that follows, we will show that the level-sets of the hypersymplectic potential carry a split 3-Sasakian structure.

5. Split 3-Sasakian geometry

We begin this section by introducing $\varepsilon$-Sasakian manifolds. These are the pseudo-Riemannian analogues of Sasakian manifolds with either a complex or a product structure on the leaves of the 1-dimensional foliation. If the leaves of the foliation are endowed with a complex structure ($\varepsilon=-1$), we call it a pseudo-Sasakian structure. If they are endowed with a product structure ($\varepsilon=1$), then we call the manifold as para Sasakian manifold. This is slightly different than the conventional terminologies in the literature. However, it allows for a simultaneous treatment of both the cases. Let ${\mathcal{S}}$ be a smooth manifold, equipped with a $(1,1)$-tensor $\Phi$, a nowhere vanishing vector field $\xi$ and a 1-form $\eta$ metric dual to $\xi$, satisfying the following relations

[TABLE]

If $\varepsilon=-1$, then $({\mathcal{S}},\Phi,\eta,\xi)$ is called an almost-contact manifold and if $\varepsilon=1$, it is called an almost-para contact manifold. To give a simultaneous treatment, we will refer to $({\mathcal{S}},\Phi,\eta,\xi)$ as an $\varepsilon$-almost contact structure. Consider the Nijenuis tensor of $\Phi$, which is a $(2,1)$-tensor, defined as

[TABLE]

The $({\mathcal{S}},\Phi,\eta,\xi)$ is said to be normal, if $N_{\Phi}=d\eta\otimes\xi$. Suppose now that ${\mathcal{S}}$ is endowed with a pseudo-Riemannian metric $g_{\scriptscriptstyle{\mathcal{S}}}$, such that

[TABLE]

where $\tau=g_{\scriptscriptstyle{\mathcal{S}}}(\xi,\xi)$, then, the $\varepsilon$-almost contact structure is said to be an $\varepsilon$-para contact metric structure and the metric $g_{\scriptscriptstyle{\mathcal{S}}}$ is said to be compatible with the $\varepsilon$-para contact structure. Additionally, if the structure is normal, the manifold $({\mathcal{S}},g_{\scriptscriptstyle{\mathcal{S}}},\Phi,\eta,\xi)$ is said to be a $\varepsilon$-Sasakian manifold. In other words, an $\varepsilon$-Sasakian manifold is an $\varepsilon$-almost contact manifold, endowed with a compatible pseudo-Riemannian metric and the structure is normal.

Proposition 5.1**.**

[12*]**

An $\varepsilon$-contact manifold $({\mathcal{S}},g_{\scriptscriptstyle{\mathcal{S}}},\Phi,\eta,\xi)$ is $\varepsilon$-Sasakian if and only if the following is satisfied*

[TABLE]

where $\nabla$ is the Levi-Civita connection of $g_{\scriptscriptstyle{\mathcal{S}}}$.

5.1. Split 3-Sasakian manifolds

Suppose that ${\mathcal{S}}$ is a pseudo-Riemannian manifold of dimension $4n+3$, carrying a metric $g_{\scriptscriptstyle{\mathcal{S}}}$ of signature $(2n+1,2n-2)$. Additionally, suppose that ${\mathcal{S}}$ also carries a triple of orthogonal Killing vector fields $(X_{1},X_{2},X_{3})$, of lengths $1,-1,-1$ respectively, satisfying

[TABLE]

Define the 1-forms $\eta_{i}(Y)=\tau_{i}\,g_{\scriptscriptstyle{\mathcal{S}}}(X_{i},Y)$, where $\tau_{i}=g_{\scriptscriptstyle{\mathcal{S}}}(X_{i},X_{i})$. Then, $\eta_{i}(X_{j})=\delta_{ij}$. Suppose that ${\mathcal{S}}$ is also endowed with endomorphisms $(\Phi_{1},\Phi_{2},\Phi_{3})$, satisfying

[TABLE]

where, $(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})=(1,-1,-1)$. Then, the structure $({\mathcal{S}},g_{\scriptscriptstyle{\mathcal{S}}},\{\Phi_{i},\eta_{i},\xi_{i}\}_{i=1}^{3})$ is said to be an almost split 3-contact structure. Additionally, if

[TABLE]

then the structure is said to be a metric split 3-contact structure.

Definition 3**.**

A metric split 3-contact manifold $({\mathcal{S}},g_{\scriptscriptstyle{\mathcal{S}}},\{\Phi_{i},\eta_{i},X_{i}\}_{i=1}^{3})$ is a split 3-Sasakian manifold if the structures are normal; i.e.,

[TABLE]

Equivalently, from Proposition 5.1, we see that $({\mathcal{S}},g_{\scriptscriptstyle{\mathcal{S}}},\{\Phi_{i},\eta_{i},X_{i}\}_{i=1}^{3})$ is a split 3-Sasakian manifold if

[TABLE]

Note that $({\mathcal{S}},g_{\scriptscriptstyle{\mathcal{S}}},\Phi_{1},\eta_{1},\xi_{1})$ is a pseudo-Sasakian manifold, whereas, for $i=2,3$, $({\mathcal{S}},g_{\scriptscriptstyle{\mathcal{S}}},\Phi_{i},\eta_{i},X_{i})$ is a para-Sasakian manifold.

Theorem 5.2** ([8], Thm. 4.1, 5.7 and Prop. 4.2).**

Let $({\mathcal{S}},g_{\scriptscriptstyle{\mathcal{S}}},\{\Phi_{i},\eta_{i},\xi_{i}\}_{i=1}^{3})$ be a split 3-contact manifold of dimension $4n+3$. Then, $S$ is necessarily a split 3-Sasakian manifold. The metric $g_{\scriptscriptstyle{\mathcal{S}}}$ is Einstein and has a constant scalar curvature equal to $(4n+2)(4n+3)$.

Remark 2**.**

The split 3-Sasakian structure we discuss below is of the type $(1,-1,-1)$; i.e, the norms of the Reeb vector fields generating the split 3-Sasakian structures are $(1,-1,-1)$. The authors in [8] refers to this as the “negative mixed 3-Sasakian structure”.

The simplest example of a split 3-Sasakian manifold is the positive pseudo-sphere in the split quaternionic module ${\mathbb{B}}^{n+1}$

[TABLE]

The manifold $S_{+}$ carries a pseudo-Riemannian metric of signature $(2n-1,\,2n+1)$. Moreover, there is an isometric and transitive action of ${\rm Sp}(1,{\mathbb{B}})$ on $S_{+}$. The Killing vector-fields corresponding to the basis of the 3-dimensional Lie algebra $\mathfrak{sp}(1,{\mathbb{B}})$ and the restricted metric on $S_{+}$ determine a split 3-Sasakian structure on $S_{+}$.

5.2. Level-set of the hypersymplectic potential

Suppose now that $M$ is a hypersymplectic manifold with a free action of ${\rm Sp}(1,{\mathbb{B}})$, such that the obstruction ${\mathcal{X}}_{2}$ vanishes. Then we have a canonically defined hypersymplectic potential on $M$. Consider the level-set ${\mathcal{S}}:=\rho_{0}^{-1}\left(\frac{1}{2}\right)$. Then, ${\mathcal{S}}$ is ${\rm Sp}(1,{\mathbb{B}})$-invariant and the Killing vector fields

[TABLE]

can be thought of as vector fields on ${\mathcal{S}}$. We denote their restriction to ${\mathcal{S}}$ by $K^{{\mathcal{S}}}_{\xi_{i}}$. Let $g_{\scriptscriptstyle{\mathcal{S}}}:=\iota^{\ast}g_{\scriptscriptstyle M}$ denote the restriction of the hypersymplectic metric to the hypersurface ${\mathcal{S}}$. Define

[TABLE]

Note that

[TABLE]

Define the 1-forms

[TABLE]

Theorem 5.3**.**

The manifold $\left({\mathcal{S}},\widehat{g_{\scriptscriptstyle{\mathcal{S}}}},\{\,K^{{\mathcal{S}}}_{\xi_{i}},\,\widehat{\eta}_{i},\,\widehat{\Phi}_{i}\,\}_{i=1}^{3}\right)$ is a split 3-Sasakian manifold.

Proof.

Owing to Theorem 5.2, we need only show that $\left({\mathcal{S}},\widehat{g_{\scriptscriptstyle{\mathcal{S}}}},\{\,K^{{\mathcal{S}}}_{\xi_{i}},\,\widehat{\eta}_{i},\,\widehat{\Phi}_{i}\,\}_{i=1}^{3}\right)$ is a metric split 3-contact manifold. For that, it is enough to show that $(g_{\scriptscriptstyle{\mathcal{S}}},\,K^{{\mathcal{S}}}_{\xi_{2}},\,\widehat{\eta}_{2},\,\widehat{\Phi}_{2})$ is a para contact metric structure. From their definitions, it is clear that

[TABLE]

Therefore we need only show that

[TABLE]

Let $\eta_{i}\,(\cdot)=\,g_{\scriptscriptstyle M}\left(K^{M}_{\xi_{i}},\,\cdot\right)$. Then $\widehat{\eta_{i}}=\iota^{\ast}\eta_{i}$. Observe that for any $X,Y\in T{\mathcal{S}}$,

[TABLE]

We now check the second condition

[TABLE]

for all $X,Y\in T{\mathcal{S}}$. It follows from eq. (4.2) and eq.(5.6) and the fact that external derivative commutes with the pull-back that $d\widehat{\eta_{2}}=\tau_{2}\,\iota^{\ast}\omega_{2}$. This shows that $d\eta_{2}(X,Y)=\tau_{2}\,\widehat{g_{\scriptscriptstyle{\mathcal{S}}}}(\widehat{\Phi}_{2}(X),\,Y)$. Thus $(\widehat{g_{\scriptscriptstyle{\mathcal{S}}}},\,\xi_{2},\,\widehat{\eta}_{2},\,\widehat{\Phi}_{2})$ defines a para contact metric structure on $S$.

Similar arguments show that $(\widehat{g_{\scriptscriptstyle{\mathcal{S}}}},\,\xi_{1},\,\widehat{\eta}_{1},\,\widehat{\Phi}_{1})$ and $(\widehat{g_{\scriptscriptstyle{\mathcal{S}}}},\,\xi_{3},\,\widehat{\eta}_{3},\,\widehat{\Phi}_{3})$ define a pseudo and a para contact metric structure respectively. Moreover, the vector fields $(K^{{\mathcal{S}}}_{\xi_{1}},K^{{\mathcal{S}}}_{\xi_{2}},K^{{\mathcal{S}}}_{\xi_{3}})$ clearly satisfy the split quaternionic relations (3.1). The claim thus follows from the statement of Theorem 5.2. ∎

Note that since the metrics on the level-sets ${\mathcal{S}}_{c}:=\rho_{0}^{-1}(c)$ are homothetic, a split 3-Sasakian structure can be defined on every level-set.

Note**.**

Analogous to the hyperKähler case, it can be easily seen that a metric cone over any split 3-Sasakian manifold is a hypersymplectic manifold, with a free, permuting action of ${\rm Sp}(1,{\mathbb{B}})$ and a hypersymplectic potential. In particular, the obstruction $\rho_{2}$ vanishes. This can be considered as a characterizing property of such hypersymplectic manifolds. In other words, if $\rho_{2}$ vanishes, then the hypersymplectic manifold in consideration, can be written as a metric cone over a split 3-Sasakian manifold, as constructed above.

5.3. Para quaternionic Kähler manifolds

A para-quaternionic Kähler manifold is a, pseudo-Riemannian manifold of dimension $4n$, whose holonomy is contained inside the group

[TABLE]

Equivalently, we say that a manifold $N$ is almost para-quaternionic Kähler manifold if there exists a sub-bundle ${\mathcal{I}}^{\prime}\subset\operatorname{End}(TN)$ which is locally spanned by a triple $(I,S,T)$ satisfying the split quaternionic relations (3.1). For $n>1$, the requirement that the holonomy of $N$ be contained inside ${\rm Sp}(n,{\mathbb{B}})\cdot{\rm Sp}(1,{\mathbb{B}})$ is equivalent to asking the sub-bundle ${\mathcal{I}}^{\prime}$ being preserved by the Levi-Civita connection. If $n\geq 3$, then, this is equivalent to showing that the globally defined 4-form

[TABLE]

is closed. For $n=1$, we additionally require that the manifold be self-dual and Einstein.

Theorem 5.4**.**

[12]** Any para-quaternionic Kähler manifold $(N,g_{\scriptscriptstyle N},\widehat{\Omega})$ is Einstein, provided that the dimension of $N$ is greater than 4.

The representation theoretic argument by S. Salamon [20] can also be adapted to the pseudo-Riemannian setting to prove the above theorem.

A wide range of examples of para-quaternionic Kähler manifolds can be constructed by adapting LeBrun’s construction of quaternionic Kähler manifolds [17], to the pseudo-Riemannian case [9]. Starting with a real analytic manifold $F$ of dimension $2n+1$, endowed with an indefinite metric, it is possible to construct a para-quaternionic manifold of dimension $4n$. Different manifolds which are conformal to $F$ give rise to distinct para-quaternionic Kähler manifolds. We can thus construct a wide variety of para-quaternionic Kähler manifolds of dimension greater than four.

Looking at Berger’s list [4, 9], one can also construct symmetric para-quaternionic Kähler manifolds of the type $G/H$, where $G$ is semi-simple. Symmetric para-quaternionic Kähler manifolds have been completely classified by D. Alekseevsky and V. Cortés [1].

In the hyperKähler case, the quotient of the 3-Sasakian manifold (a level-set of the hyperKähler potential) by the group ${\rm Sp}(1)$ produces a quaternionic Kähler manifold of positive scalar curvature. This, however, cannot be directly carried over to the hypersymplectic situation as the group ${\rm SU}(1,1)$ is non-compact and therefore the quotient may not even be Hausdorff. However, if the ${\rm Sp}(1,{\mathbb{B}})$-action is proper, then, we show that the quotient of the split 3-Sasakian manifold by ${\rm Sp}(1,{\mathbb{B}})$ is a para-quaternionic Kähler manifold. Henceforth, we assume that the ${\rm Sp}(1,{\mathbb{B}})$-action on $M$ proper.

Let $N={\mathcal{S}}/{\rm Sp}(1,{\mathbb{B}})$ be the quotient of the split 3-Sasakian manifold and consider the diagram,

where $\iota$ is the pseudo-Riemannian embedding and the map $\pi$ is the pseudo-Riemannian principal submersion. The normal bundle ${\mathcal{N}}$ is the 1-dimensional vector bundle $\iota^{\ast}\text{span}\{\mathcal{X}_{0}\}\subset\iota^{\ast}TM$. The pull-back bundle $\iota^{\ast}TM$ splits into the direct sum $\iota^{\ast}TM={\mathcal{N}}\oplus\,T{\mathcal{S}}$,

The pull-back metric on ${\mathcal{S}}$ is of signature $(2n-1,2n)$. Further, $T{\mathcal{S}}$ splits into a direct sum ${\mathcal{V}}\oplus{\mathcal{H}}$, where

[TABLE]

In conclusion, the pullback-bundle splits as

[TABLE]

We will call ${\mathcal{H}}$ the “horizontal bundle” of $T{\mathcal{S}}$. Let $\beta_{i}(\cdot,\cdot)=\widehat{g}_{\scriptscriptstyle{\mathcal{S}}}(\widehat{\Phi}_{i}(\cdot),\cdot)$ and $\theta_{i}:=\beta_{i}+\epsilon_{ijk}\,\eta_{j}\wedge\eta_{k}$. Define the 4-form

[TABLE]

Then, $\widehat{\Omega}$ is an ${\rm Sp}(1,{\mathbb{B}})$-invariant, horizontal 4-form and therefore it descends to a 4-form $\Omega_{N}$ on $N$ with $\pi^{\ast}\Omega_{N}=\widehat{\Omega}$.

Observe that since $\rho_{2}=0$, we have that

[TABLE]

where $\tau^{ijk}$ denotes the sign of the permutation $(i,j,k)$. It follows that ${\mathcal{V}}\oplus{\mathcal{N}}$ is invariant under $I,S,T$ and therefore, ${\mathcal{H}}$ is invariant under $I,S,T$. We thus get an ${\rm Sp}(1,{\mathbb{B}})$-invariant almost para-quaternionic Kähler structure on ${\mathcal{H}}$, which descends to $N$. The 4-form associated to the almost para-quaternionic Kähler structure is $\Omega_{N}$. In order to show that the structure is para-quaternionic Kähler, we need to show that the quotient metric $g_{\scriptscriptstyle N}$ has holonomy group contained in ${\rm Sp}(n,{\mathbb{B}})\cdot{\rm Sp}(1,{\mathbb{B}})$. Or equivalently,

[TABLE]

where $\nabla^{N}$ is the Levi-Civita connection of the metric $g_{\scriptscriptstyle N}$. Observe that for any $W,X,Y,Z\in{\mathcal{H}}$,

[TABLE]

The Levi-Civita connection on $M$ induces a connection $\nabla^{{\mathcal{S}}}$ on ${\mathcal{S}}$, which is precisely the Levi-Civita connection of the pull-back metric $\iota^{\ast}g_{\scriptscriptstyle M}=\widehat{g}_{\scriptscriptstyle{\mathcal{S}}}$. It follows that $\nabla^{{\mathcal{S}}}\widehat{\Omega}=0$. From the fact that ${\mathcal{S}}\rightarrow N$ is a pseudo-Riemannian submersion and the standard computation using O’ Neil’s formula ([11], Thm. 3.1), we conclude that the holonomy of the quotient metric is a sub-group of ${\rm Sp}(n,{\mathbb{B}})\cdot{\rm Sp}(1,{\mathbb{B}})$. Thus, $N$ is a para quaternionic Kähler manifold. Note that the signature of $g_{\scriptscriptstyle N}$ is $(2n-2,2n-2)$. To sum-up

Theorem 5.5**.**

Suppose that $M$ is a hypersymplectic manifold of dimension $4n$ with a free and proper permuting action of ${\rm Sp}(1,{\mathbb{B}})$. Assume that the obstruction $\rho_{2}=0$. Then, the quotient of any level set $\rho_{0}^{-1}(c)/{\rm Sp}(1,{\mathbb{B}})$ is a para-quaternionic Kähler manifold of dimension $4n-4$, endowed with a metric of signature $(2n-2,2n-2)$.

The converse of the above statement is also true. Namely, if the quotient $\rho_{0}^{-1}(c)/{\rm Sp}(1,{\mathbb{B}})$ is para-quaternionic Kähler, then, $\rho_{2}=c$. This follows directly from the arguments in proof of Theorem 2.15 of [7] and Lemmas 4.1, 4.3 and 4.5.

Swann bundles on para-quaternionic Kähler manifolds

Going in the the other direction, given a para-quaternionic Kähler manifold, consider its reduced ${\rm Sp}(n,{\mathbb{B}})\cdot{\rm Sp}(1,{\mathbb{B}})$-frame bundle $F$. Then, $\mathscr{S}(N)=F/{\rm Sp}(1,{\mathbb{B}})$ is a principal ${\rm SO}^{+}(1,2)$ bundle. Let $\widetilde{{\mathbb{B}}}$ denote the space of all the zero-divisors in ${\mathbb{B}}$ and let ${\mathbb{B}}^{\ast}:=({\mathbb{B}}\setminus\{0\})\setminus\widetilde{{\mathbb{B}}}$.

The permuting ${\rm Sp}(1,{\mathbb{B}})$-action on ${\mathbb{B}}$, given by $(q,h)\mapsto h\overline{q}$, $h\in{\mathbb{B}}$, $q\in{\rm Sp}(1,{\mathbb{B}})$, descends to an action of ${\rm SO}^{+}(1,2)$ on ${\mathbb{B}}^{\ast}/{\mathbb{Z}}_{2}$. Note that the action is transitive. There is yet another action of ${\rm Sp}(1,{\mathbb{B}})$ on ${\mathbb{B}}^{\ast}$, which is given by left multiplication and which commutes with the first one. It therefore descends to an action of ${\rm SO}^{+}(1,2)$ on ${\mathbb{B}}^{\ast}/{\mathbb{Z}}_{2}$.

Consider the principal bundle

[TABLE]

The total space $\mathcal{U}(N)_{+}$ of the bundle, is a hypersymplectic manifold, with the induced (permuting) action of ${\rm SO}^{+}(1,2)$ on the fibres. This is the positive Swann bundle constructed by Dancer, Jørgensen and Swann [9]. Note that since ${\mathbb{B}}^{\ast}\cong\rm{Sp}(1,{\mathbb{B}})\times({\mathbb{R}}\setminus\{0\})$, the fibre at each point

[TABLE]

Here, we think of ${\mathbb{Z}}_{2}$ as the normal sub-group generated by the element ${(-1,-1)}\in\rm{Sp}(1,{\mathbb{B}})\times({\mathbb{R}}\setminus\{0\})$. The isomorphism is seen quite easily by considering the adjoint action of ${\mathbb{B}}^{\ast}$ on ${\mathbb{B}}$:

[TABLE]

Therefore, the total space of the Swann bundle can alternatively be written as a metric cone

[TABLE]

This shows that $\mathscr{S}(N)$ is a split 3-Sasakian manifold. Note that the hypersymplectic potential on $\mathcal{U}(N)_{+}$ is just $\rho_{0}(s,r)=\frac{1}{2}\,r^{2}$ and the Euler vector field $\mathcal{X}_{0}=r\,\partial/\partial r$.

Using Le Brun’s construction [17], one can construct a para-quaternionic Kähler manifold of dimension $4k$ from a real-analytic, pseudo-Riemannian manifold of dimension $k+1$ [9]. Moreover, different manifolds, conformal to the real-analytic manifold give rise to distinct para-quaternionic Kähler manifold. In this way, we have a plethora of examples of para-quaternionic Kähler manifolds and therefore also of hypersymplectic manifolds with a permuting ${\rm SO}^{+}(1,2)$-action.

6. Examples

6.1. Commutativity of constructions and split quaternionic modules

Many non-trivial examples of hyperKähler manifolds with permuting ${\rm Sp}(1)$-action are obtained as reductions of the flat-space ${\mathbb{H}}^{n}$. Analogously, we use the Marsden-Weinstein construction for constructing non-trivial examples of hypersymplectic manifolds, carrying a permuting ${\rm Sp}(1,{\mathbb{B}})$-permuting action, starting with the flat-space ${\mathbb{B}}^{n}$.

Suppose that $(M,g_{\scriptscriptstyle M},I,S,T)$ is a hypersymplectic manifold, with a free and isometric action of a compact Lie group $G$ that preserves the hypersymplectic structure. Let $\mu:M\rightarrow\mathfrak{g}^{\ast}\otimes\mathfrak{sp}(1,{\mathbb{B}})^{\ast}$ denote a hypersymplectic moment map and assume that [math] is a regular value of the moment map. Let $\mathfrak{O}=\{\,K^{M}_{\xi}\,|\,\xi\in\mathfrak{g}\,\}$. Suppose that the following holds:

1. (1)

$G$ acts freely on $\mu^{-1}(0)$ 2. (2)

Rank $d\mu$ = 3 dim $\mathfrak{g}$ at each point of $M$ 3. (3)

$\mathfrak{O}\cap\mathfrak{O}^{\perp}=\{0\}$.

The above conditions are referred to as conditions (F), (S) and (D) respectively in [9].

Theorem 6.1** (Hypersymplectic reduction, [13]).**

The quotient $M^{\prime}=\mu^{-1}(0)/G$ is a smooth hypersymplectic manifold, with respect to the induced quotient hypersymplectic structure $(I^{\prime},S^{\prime},T^{\prime})$ and the induced quotient metric $g^{\prime}_{\scriptscriptstyle M^{\prime}}$.

Conditions (1) and (2) guarantee a smooth quotient manifold while (3) says that the hypersymplectic structure is non-degenerate. In contrast with the hyperKähler case, (2) does not follow from (1) automatically, owing to the fact that the metric on $M$ is indefinite. In most known examples, (3) is usually never guaranteed. Consequently, the quotients are hypersymplectic manifolds, away from some singular locus. For example, suppose $G=\rm{U}(1)$. Then, (3) is violated precisely along the locus of all those points $p$, where $g_{\scriptscriptstyle M}(K^{M}_{\mathrm{\mathbf{i}}}|_{p},K^{M}_{\mathrm{\mathbf{i}}}|_{p})=0$.

Let $(N,g_{\scriptscriptstyle N},\Omega_{N})$ be a para-quaternionic Kähler manifold. Let $(I,S,T)$ denote the local basis for the paraquaternionic Kähler structure and $\eta_{I},\eta_{S},\eta_{T}$ denote the corresponding local para-Kähler/ pseudo-Kähler 2-forms. Locally, for a Killing vector field $X$, we can define the 1-form

[TABLE]

The form is independent of the the choice of the local basis $(I,S,T)$ and is therefore globally defined.

Theorem 6.2** (Para-quaternionic Kähler reduction, [23] (Thm. 5.2)).**

Let $G$ be a Lie group acting freely and isometrically on a para-quaternionic Kähler manifold $(N,g_{\scriptscriptstyle N},\Omega_{N})$. Assume that the group action preserves $\Omega_{N}$. Then, there exists a unique map $\mu:N\rightarrow\mathfrak{g}^{\ast}\otimes\mathfrak{sp}(1,{\mathbb{B}})$ such that $d\mu=\iota_{\mathfrak{g}}\,\Theta$.

Suppose that $\mu^{-1}(0)$ is a smooth submanifold of $N$, on which $G$ acts freely and properly, so that $N^{\prime}=\mu^{-1}(0)/G$ is a smooth pseudo-Riemannian submersion. Then, $(N^{\prime},g^{\prime}_{\scriptscriptstyle N^{\prime}},\Omega^{\prime}_{N^{\prime}})$ is again a para-quaternionic Kähler manifold, with respect to the induced para-quaternionic Kähler structure $\Omega^{\prime}_{N^{\prime}}$ and the induced metric $g^{\prime}_{\scriptscriptstyle N^{\prime}}$

With the above two theorems at hand, we can directly adapt Swann’s arguments in [22] to the pseudo-Riemannian setting to show that the quotient construction commutes with reduction. Namely,

Theorem 6.3**.**

Let $(N,g_{\scriptscriptstyle N},\Omega_{N})$ be a para-quaternionic Kähler manifold. Suppose that a Lie group $G$ acts isometrically,freely and properly, preserving the para-quaternionic Kähler structure. Then, $G$ induces an isometric action on $\mathcal{U}(N)$, which preserves the hypersymplectic structure on $\mathcal{U}(N)$. Moreover, the hypersymplectic quotient of $\mathcal{U}(N)$ by the $G$ action is the total space of the Swann bundle over the para-quaternionic Kähler quotient of $N$ by $G$.

Let us now consider the split quaternionic module ${\mathbb{B}}^{n+1}$. Let $\widetilde{{\mathbb{B}}}$ denote the sub-space of all the null-vectors in ${\mathbb{B}}^{n+1}$ and consider the sub-space of all the space-like vectors (positive norm) $\left({\mathbb{B}}^{n+1}\right)^{\ast}:=({\mathbb{B}}^{n+1}\setminus\{0\})\setminus\widetilde{{\mathbb{B}}}$. Then $\left({\mathbb{B}}^{n+1}\right)^{\ast}$ is a union of two disjoint spaces of space-like and time-like vectors. Let $\left({\mathbb{B}}^{n+1}\right)^{\ast}_{+}$ denote the sub-space of space-like vectors. We will show that this is the total space of a Swann bundle over a para-quaternionic Kähler manifold. First, observe that $\left({\mathbb{B}}^{n+1}\right)^{\ast}_{+}$ is a hypersymplectic manifold, equipped with a free and proper action of ${\rm Sp}(1,{\mathbb{B}})$, as described in Subsec. 3.1. Moreover, it also carries a homothetic action of ${\mathbb{R}}^{+}$ given by $(r,h)\mapsto r\cdot h$. Clearly, we see that the obstruction $\rho_{2}$ vanishes and the hypersymplectic potential is given by $\rho_{0}(h)=\frac{1}{2}\left\|h\right\|^{2}$.

Consider the positive sphere

[TABLE]

The sphere carries a metric of signature $(2n-1,2n+1)$. Then, clearly, $\left({\mathbb{B}}^{n+1}\right)^{\ast}_{+}$ is topologically a metric cone over $S_{+}$ and so $\left({\mathbb{B}}^{n+1}\right)^{\ast}_{+}\cong S_{+}\times{\mathbb{R}}_{>0}$. The hypersymplectic potential is just $\rho_{0}(s,r)=\frac{1}{2}\,r^{2}$. Therefore, $S_{+}$ is a split 3-Sasakian manifold. The ${\rm Sp}(1,{\mathbb{B}})$-action on $({\mathbb{B}}^{n+1})^{\ast}_{+}$ induces a free, proper and isometric action of ${\rm Sp}(1,{\mathbb{B}})$ on $S_{+}$. By Theorem 5.5, the quotient $S_{+}/{\rm Sp}(1,{\mathbb{B}})$ is a para-quaternionic Kähler manifold, which is nothing but the para-quaternionic projective space

[TABLE]

It follows that $\left({\mathbb{B}}^{n+1}\right)^{\ast}_{+}$ is the total space of the Swann bundle over ${\mathbb{B}}{\mathbb{P}}^{n}$, i.e, ${\mathcal{U}}({\mathbb{B}}{\mathbb{P}}^{n})$. This is the positive Swann bundle described in [9]. Split quaternionic projective spaces have been studied by Blažić [6] and Wolf [24].

Consider an action of a Lie group $G\subset{\rm Sp}(n+1,{\mathbb{B}})$ on ${\mathbb{B}}^{n+1}$, that commutes with the permuting ${\rm Sp}(1,{\mathbb{B}})$-action by right conjugate multiplication. The induced action on $({\mathbb{B}}^{n+1})^{\ast}_{+}$ preserves the hypersymplectic structure and therefore also the three symplectic forms. So, there exist three moment maps, which we combine into a single $G\times{\rm Sp}(1,{\mathbb{B}})$-equivariant map

[TABLE]

Suppose that [math] is a regular value of $\mu$ and $G$ acts freely and properly on the zero level-set of the moment map. Additionally, assume that the metric, restricted to the group orbits in $\mu^{-1}(0)$ is non-degenerate. Hitchin’s work [13] now guarantees that the quotient $\widetilde{{\mathcal{M}}}=\mu^{-1}(0)/G$ is a hypersymplectic manifold. The permuting action of ${\rm Sp}(1,{\mathbb{B}})\times{\mathbb{R}}^{+}$ on $({\mathbb{B}}^{n+1})^{\ast}_{+}$ commutes with the action of $G$ (therefore preserving the zero-level-set of $\mu$) and hence descends to the quotient $\widetilde{{\mathcal{M}}}$. In particular the obstruction $\rho_{2}$ vanishes; i.e., $\widetilde{{\mathcal{M}}}=\mathcal{U}(N)$ for some para-quaternionic Kähler manifold $N$. Since ${\rm Sp}(1,{\mathbb{B}})\times{\mathbb{R}}^{+}$-action commutes with that of $G$, the latter descends to a para-quaternionic Kähler action on ${\mathbb{B}}{\mathbb{P}}^{n}$. By Theorem 6.3, it follows that $N$ is the para-quaternionic Kähler reduction of ${\mathbb{B}}{\mathbb{P}}^{n}$ by $G$.

When $G$ is a compact subgroup of ${\mathbb{T}}^{n+1}$, Dancer and Swann [10] show that the hypersymplectic reduction of ${\mathbb{B}}^{n+1}$ by $G$ is, a hypersymplectic manifold, with a non-trivial de-generacy locus - the set of all the points where the metric is de-generate along the orbits of the $G$-action. For example, when $G={\rm U}(1)$, this is precisely the set of points where $g_{\scriptscriptstyle M}(K^{M},K^{M})=0$, where $K^{M}$ is the fundamental vector field due to ${\rm U}(1)$-action. The ${\rm U}(1)$-action descends to an action on $({\mathbb{B}}^{n+1})^{\ast}_{+}$ and the moment map is just the restriction of $\mu$. From the discussion above, it follows that the reduced manifold is a smooth hypersymplectic manifold, which is the total space of a Swann bundle over the para-quaternionic Kähler reduction of ${\mathbb{B}}{\mathbb{P}}^{n}$ by ${\rm U}(1)$.

6.2. Moduli spaces of Nahm-Schmid equations

We give another set of examples of hypersymplectic manifolds, carrying a permuting ${\rm SU}(1,1)$-action, for which the obstruction $\rho_{2}$ vanishes. Namely, we consider the moduli space of Nahm-Schmid equations. The equations can be interpreted as the zero-level set of infinite-dimensional hypersymplectic moment map for the action of gauge group on the configuration space. Consequently, the moduli space of solutions is an infinite-dimensional hypersymplectic reduction. We refer to [5] for more details.

Nahm-Schmid equations:

Nahm-Schmid equations are pseudo-Riemannian analogues of Nahm’s equations and arise as dimensional reduction of Yang-Mills equations on ${\mathbb{R}}^{2,2}$. Let $G$ be a compact Lie group and $\mathfrak{g}$ denote its Lie algebra. Let $T_{i}:{\mathbb{R}}\rightarrow\mathfrak{g}$ be $C^{1}$-differentiable maps for $i=0,1,2,3$. The Nahm-Schmid equations for $\{T_{i}\}_{i=0}^{3}$ is a system of equations, satisfying

[TABLE]

The equations are invariant under the action of the gauge group ${\mathcal{G}}:=C^{2}({\mathbb{R}},G)$. Moreover, there exists a gauge transformation such that $T_{0}=0$ and therefore we may assume, without loss of generality, that $T_{0}=0$ in the above equations. Using this, we get the reduced Nahm-Schmid equations:

[TABLE]

Let $\left\|\cdot\right\|$ denote the Ad-invariant inner product on the Lie algebra $\mathfrak{g}$.

Proposition 6.4** (Prop. 2.2, [5]).**

Let $(T_{1},T_{2},T_{3})$ be a solution to (6.2). Then,

[TABLE]

Consequently, the solutions exist for all times.

Nahm-Schmid equations on [0,1]

Let $G$ be a compact Lie group and $\mathfrak{g}$ denote its Lie algebra. We will consider the solutions $\{T_{i}\}_{i=0}^{3}$ defined on the interval $[0,1]$.

Let

[TABLE]

We can identify ${\mathcal{A}}\cong C^{1}([0,1],\mathfrak{g})\otimes{\mathbb{B}}$ by mapping:

[TABLE]

The tangent space to ${\mathcal{A}}$ at a point ${\mathcal{T}}$ is given by $T_{{\mathcal{T}}}{\mathcal{A}}=C^{1}([0,1],\mathfrak{g})\otimes{\mathbb{B}}$. There is a split-quaternionic structure on ${\mathcal{A}}$, which is induced by the split-quaternionic structure on ${\mathbb{B}}$, given by:

[TABLE]

The space ${\mathcal{A}}$ is equipped with the indefinite, neutral signature metric, induced from the metric on ${\mathbb{B}}\cong{\mathbb{R}}^{2,2}$ and an ad-invariant metric $\left\langle\cdot,\cdot\right\rangle$ on $G$

[TABLE]

where $a_{00}=a_{11}=1$ and $a_{22}=a_{33}=-1$. The metric is compatible with the split-quaternionic structure $I,S,T$. This endows ${\mathcal{A}}$ with a structure of a flat hypersymplectic manifold. The hypersymplectic structure is preserved by the action of the gauge group ${\mathcal{G}}:=C^{2}([0,1],G)$. Let ${\mathcal{G}}_{00}$ be the normal subgroup of ${\mathcal{G}}$, given by

[TABLE]

Then, the infinite-dimensional moment maps for the action of ${\mathcal{G}}_{00}$ on ${\mathcal{A}}$ are given by

[TABLE]

Writing $\mu_{\scriptscriptstyle{\mathcal{A}}}=\mu_{I}\,\mathrm{\mathbf{i}}+\mu_{S}\,\mathbf{s}+\mu_{T}\,\mathbf{t}$, we see that the solutions to the Nahm-Schmid equations can be interpreted as the zero level-set of the hypersymplectic moment map $\mu_{{\mathcal{A}}}$.

The space ${\mathcal{A}}$ also carries a permuting action of the group ${\rm Sp}(1,{\mathbb{B}})$, which is induced by a permuting action of ${\rm Sp}(1,{\mathbb{B}})$ on ${\mathbb{B}}$. More precisely, the action is given by

[TABLE]

Since the action is induced by the permuting action on ${\mathbb{B}}$, it is isometric, free and proper. Moreover, the action commutes with that of ${\mathcal{G}}_{00}$, thus preserving the zero-level set of $\mu_{{\mathcal{A}}}$. In essence, equations (6.1) are invariant under the action of ${\rm Sp}(1,{\mathbb{B}})$. The induced action on the space of product and complex structures $\Lambda_{\mathscr{H}}$ on ${\mathcal{A}}$ is, pointwise, just the standard action of ${\rm SO}^{+}(1,2)$ on the pseudo-sphere $\mathscr{H}$. Namely, suppose that $F$ denotes either of $I,S$ ot $T$, and $\mathbf{f}$ correspondingly denotes either $\overline{\mathrm{\mathbf{i}}},\mathbf{s}$ or $\mathbf{t}$. Then, the induced action is $(q,F)\mapsto q_{\ast}\,F\,\overline{q_{\ast}}$. Indeed,

[TABLE]

Theorem 6.5** ([5]).**

The gauge group ${\mathcal{G}}_{00}$ acts freely and properly on ${\mathcal{A}}$ and therefore the moduli space

[TABLE]

is a smooth Banach manifold, diffeomorphic to $G\times\mathfrak{g}\times\mathfrak{g}\times\mathfrak{g}\cong G\times(\mathfrak{g}\otimes\mathfrak{Im}({\mathbb{B}}))$.

Note**.**

The moduli space ${\mathcal{M}}$, although a smooth manifold, does not carry a smooth hypersymplectic structure. There exists a degeneracy locus, which is precisely the locus of points ${\mathcal{T}}$ where the metric, when restricted to the tangent space of the ${\mathcal{G}}_{00}$-orbit through ${\mathcal{T}}$, is degenerate. Outside of this degeneracy locus, ${\mathcal{M}}$ carries a smooth hypersymplectic structure. Note that the ${\rm Sp}(1,{\mathbb{B}})$-action and the action of ${\mathcal{G}}_{00}$ on ${\mathcal{A}}$ preserves the metric and therefore also the degeneracy locus.

Proposition 6.6** ([5]).**

The degeneracy locus ${\mathcal{D}}$ is in one-to-one correspondance with those solutions ${\mathcal{T}}=(T_{0},T_{1},T_{2},T_{3})$ to Nahm-Schmid equations (6.1), for which there exists a solution to the following ODE

[TABLE]

Homothetic action of ${\mathbb{R}}^{+}$ and degeneracy locus

We define a homothetic action of ${\mathbb{R}}^{+}$ on ${\mathcal{A}}$ as follows: let $a\in{\mathbb{R}}$ and consider the map

[TABLE]

It is easy to see that the equations (6.1) are invariant under $\phi_{a}$. As a result, $\phi_{a}$ maps solutions on $[0,1]$, to solutions defined on $[0,a^{-1}]$. The homomorphism between the gauge groups

[TABLE]

induces a morphism of algebras ${\rm Lie}\,({\mathcal{G}}_{00})$ and ${\rm Lie}\,(({\mathcal{G}}_{a})_{00})$, sending $\xi(t)\mapsto\xi(at)$.

Lemma 6.7**.**

The map $\phi_{a}$ preserves the degeneracy locus.

Proof.

Suppose that a solution ${\mathcal{T}}$ lies in the degeneracy locus, which we denote by ${\mathcal{D}}$. This implies that (6.5) has a non-trivial solution $\xi(t)$ at ${\mathcal{T}}$. We claim that for any $a\in{\mathbb{R}}^{+}$, $\phi_{a}({\mathcal{T}})$ lies in the degeneracy locus ${\mathcal{D}}_{a^{-1}}$ for the equations defined on $[0,a^{-1}]$. To see this, note that the ODE (6.5) has a non-trivial solution given by $\xi(at)$. In other words, the image $\phi_{a}({\mathcal{D}})\subset{\mathcal{D}}_{a^{-1}}$ for any $a\in{\mathbb{R}}^{+}$. On the other hand, the map $\phi_{a}$ has a smooth inverse given by $\phi_{a^{-1}}$ and a verbatim argument in the other direction shows that $\phi_{a^{-1}}({\mathcal{D}}_{a^{-1}})\subset{\mathcal{D}}$. In conclusion, we see that $\phi_{a}({\mathcal{D}})={\mathcal{D}}_{a^{-1}}$ and thus $\phi_{a}$ maps the degeneracy locus to degeneracy locus. ∎

With this observation at hand, we will now define a homothetic action of ${\mathbb{R}}^{+}$ on ${\mathcal{A}}$. Assume first that $a<1$, so that $a^{-1}>1$. Consider the restriction map

[TABLE]

Clearly, $\beta_{a}$ maps solutions to solutions. Composing $\beta_{a}$ with $\phi_{a}$, we get a map

[TABLE]

Observe that ${\mathcal{T}}(t)\mapsto a\,{\mathcal{T}}\,(at)|_{\,[0,1]}$ is a homothety. To show that $\Phi_{a}$ has a smooth inverse, note that owing to Proposition 6.4, any solution to Nahm-Schmid equations, defined on $[0,1]$, can be uniquely extended to a solution on $[0,a^{-1}]$. This follows from the standard theory of existence and uniqueness of solutions to ODEs. Composing this extension map with $\phi_{a^{-1}}$ gives the inverse to $\Phi_{a}$. In particular, $\Phi_{a}$ is a diffeomorphism.

Since any solution ${\mathcal{T}}$ can be uniquely extended from $[0,1]$ to $[0,a^{-1}]$, and $\{T_{i}\}_{i=0}^{3}$ are continuous, it follows from the standard theory of existence and uniqueness of solutions to second order ODEs that any solution to (6.5) on $[0,1]$, can be extended uniquely to a solution on $[0,a^{-1}]$. In other words, we have $\Phi_{a}({\mathcal{D}})={\mathcal{D}}$.

Now suppose if $a>1$, then $\Phi_{a}$ is given by composing $\phi_{a}$ with the extension map and the inverse is given by composition of restriction map $\beta_{a}$ with $\phi_{a^{-1}}$. The rest of the arguments are verbatim to the ones given above. In conclusion, the map $\Phi_{a}$ determines a homothetic action of ${\mathbb{R}}^{+}$ on ${\mathcal{A}}$, that preserves the space of solutions to Nahm-Schmid equations and also the degeneracy locus. The action commutes with that of ${\mathcal{G}}_{00}$ and ${\rm Sp}(1,{\mathbb{B}})$.

Note**.**

Above lemma implies that the ${\mathbb{R}}^{+}$-orbits of elements in the complement of the degeneracy locus, does not intersect ${\mathcal{D}}$.

Theorem 6.8**.**

The moduli space of solutions to the Nahm-Schmid equations, away from the degeneracy locus ${\mathcal{D}}$, is the total space of a Swann bundle over a paraquaternionic Kähler manifold. In other words,

[TABLE]

Proof.

Consider the open set $\mu^{-1,\ast}_{\scriptscriptstyle{\mathcal{A}}}(0)=\mu^{-1}_{\scriptscriptstyle{\mathcal{A}}}(0)\setminus{\mathcal{D}}$. Then, ${\mathcal{M}}_{0}:=\mu^{-1,\ast}_{\scriptscriptstyle{\mathcal{A}}}(0)/{\mathcal{G}}_{00}$ is a hypersymplectic manifold, which is an open set of ${\mathcal{M}}$. Moreover, since ${\rm Sp}(1,{\mathbb{B}})\times{\mathbb{R}}^{+}$ action preserves $\mu^{-1,\ast}_{\scriptscriptstyle{\mathcal{A}}}(0)$, the action descends to ${\mathcal{M}}_{0}$. As a result, ${\mathcal{M}}_{0}$ is topologically a metric cone over a split 3-Sasakian manifold ${\mathcal{S}}_{0}$

[TABLE]

Since the group ${\rm Sp}(1,{\mathbb{B}})$ acts freely and properly on ${\mathcal{S}}_{0}$, by Theorem 5.5, $N_{0}:={\mathcal{S}}_{0}/{\rm Sp}(1,{\mathbb{B}})$ is a paraquaternionic Kähler manifold. ∎