On certain classes of $Sp(2,R)$ symmetric $G_2$ structures

TL;DR

This paper constructs two distinct families of $G_2$ structures with $Sp(2,R)$ symmetry in seven dimensions, distinguished by their torsion properties and underlying homogeneous spaces, expanding understanding of special geometric structures related to split real forms.

Contribution

It introduces two new families of $G_2$ structures with $Sp(2,R)$ symmetry, characterized by different torsion conditions and realized on different homogeneous spaces.

Findings

First family has $ au_2 eq 0$, second has $ au_1= au_2=0$

Each family is associated with a different homogeneous space of $Sp(2,R)$

The structures are related to split real form of $G_2$ and root diagram distinctions

Abstract

We find two different families of $Sp(2,R)$ symmetric $G_2$ structures in seven dimensions. These are $G_2$ structures with $G_2$ being the split real form of the simple exceptional complex Lie group $G_2$. The first family has $\tau_2\equiv 0$, while the second family has $\tau_1\equiv\tau_2\equiv 0$. The families are different in the sense that the first one lives on a homogoneous space $Sp(2,R)/SL(2,R)_l$, and the second one lives on a homogeneous space $Sp(2,R)/Sl(2,R)_s$. Here $SL(2,R)_l$ is an $SL(2,R)$ corresponding to the $\mathfrak{sl}(2,R)$ related to the long roots in the root diagram of $\mathfrak{sp}(2,R)$, and $SL(2,R)_s$ is an $SL(2,R)$ corresponding to the $\mathfrak{sl}(2,R)$ related to the short roots in the root diagram of $\mathfrak{sp}(2,R)$.

Equations

$$E=(E^{\alpha }{}_{\beta })=\left(\begin{array}{cccc}{a}_5& a_7& a_9& 2a_{10}\\ -a_4& a_6& a_8& a_9\\ a_2& a_3& -a_6& -a_7\\ -2a_1& a_2& a_4& -a_5\end{array}\right),$$

$$E_I=\frac{\partial E}{\partial a_I},I=1,2,\dots 10,$$

$$\mathfrak{sl}(2,\mathbb{R}{)}_l={\mathrm{Span}}_{\mathbb{R}}(E_1,E_5,E_{10}),$$

$$\mathfrak{sl}(2,\mathbb{R}{)}_s={\mathrm{Span}}_{\mathbb{R}}(E_2,E_5+E_6,E_9).$$

$$K=\tfrac{1}{12}K_{IJ}E^{I}\odot E^{J}=-4{E^{1}}\odot{E^{10}}+2{E^{2}}\odot{E^{9}}+{E^{3}}\odot{E^{8}}-2{E^{4}}\odot{E^{7}}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E^{5}}\odot{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E^{5}}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E^{6}}\odot{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E^{6}},$$

$$e_1=E_2,e_2=E_3,e_3=E_4,e_4=E_6,e_5=E_7,e_6=E_8,e_7=E_9,e_8=E_1,e_9=E_5,e_{10}=E_{10}.$$

$$\mathrm{d}{e}^1=$$

$$\mathrm{d}{e}^2=$$

$$\mathrm{d}{e}^3=$$

$$\mathrm{d}{e}^4=$$

$$\mathrm{d}{e}^5=$$

$$\mathrm{d}{e}^6=$$

$$\mathrm{d}{e}^7=$$

$$\mathrm{d}{e}^8=$$

$$\mathrm{d}{e}^9=$$

$$\mathrm{d}{e}^{10}=$$

$$K=\frac{1}{12}{c}^I{}_{JK}{c}^K{}_{LI}{e}^J\odot e^L=(e^4{)}^2-2e^3\odot e^5+e^2\odot e^6+2e^1\odot e^7+(e^9{)}^2-4e^8\odot e^{10}.$$

$${\mathcal{L}}_{X}g=0\mathrm{for}\mathrm{all}X\mathrm{in}{D}_l.$$

$$g=g_{22}(e^2{)}^2+2g_{24}{e}^2\odot e^4+g_{44}(e^4{)}^2+2g_{35}(e^3\odot e^5-e^1\odot e^7)+2g_{26}{e}^2\odot e^6+2g_{46}{e}^4\odot e^6+g_{66}(e^6{)}^2.$$

$$g_K=(e^4{)}^2-2e^3\odot e^5+e^2\odot e^6+2e^1\odot e^7,$$

$${\mathcal{L}}_X\varphi =0\mathrm{for}\mathrm{all}X\mathrm{in}{D}_l.$$

$$\varphi =f{e}^{125}+a(e^{235}-e^{127})+p{e}^{145}+q(e^{147}+e^{345})+s{e}^{156}+t(e^{356}-e^{167})+h{e}^{237}+b{e}^{246}+r{e}^{347}+u{e}^{367}.$$

$$(X\text{makebox}[6.00006pt][r]-\text{makebox}[3.99994pt][l]|\varphi )\wedge (X\text{makebox}[6.00006pt][r]-\text{makebox}[3.99994pt][l]|\varphi )\wedge \varphi =3g(X,Y)\mathrm{vol}(g),\forall X,Y\in \mathrm{T}M.$$

$$(e_{\mu }\text{makebox}[6.00006pt][r]-\text{makebox}[3.99994pt][l]|\varphi )\wedge (e_{\nu }\text{makebox}[6.00006pt][r]-\text{makebox}[3.99994pt][l]|\varphi )\wedge \varphi =3g_K(e_{\mu },e_{\nu })e^1\wedge e^2\wedge e^3\wedge e^4\wedge e^5\wedge e^6\wedge e^7,$$

$$b=\frac{1}{2},f=\frac{ap}{1-q},h=\frac{a(q-1)}{p},r=\frac{q^2-1}{p},s=\frac{p(1-q)}{4a},t=\frac{1-q^2}{4a},u=\frac{(q^2-1)(q+1)}{4ap}.$$

$$g_K=(e^4{)}^2-2e^3\odot e^5+e^2\odot e^6+2e^1\odot e^7,$$

$$\varphi =$$

$$\frac{1-q^2}{4a}(e^{356}-e^{167})+\frac{a(q-1)}{p}{e}^{237}+\frac{1}{2}{e}^{246}+\frac{q^2-1}{p}{e}^{347}+\frac{(q^2-1)(q+1)}{4ap}{e}^{367}.$$

$$\mathrm{d}\varphi =$$

$$\mathrm{d}\ast \varphi =$$

$$\ast \lambda (e_{{\mu }_1},\dots ,e_{{\mu }_{7-p}})\mathrm{vol}(g)=\lambda \wedge g(e_{{\mu }_1})\wedge \cdots \wedge g(e_{{\mu }_{7-p}}),X\text{makebox}[6.00006pt][r]-\text{makebox}[3.99994pt][l]|g(e_{\mu })=g(e_{\mu },X).$$

$$g_K=(e^4{)}^2-2e^3\odot e^5+e^2\odot e^6+2e^1\odot e^7.$$

$$\varphi =$$

$$\frac{1-q^2}{4a}(e^{356}-e^{167})+\frac{a(q-1)}{p}{e}^{237}+\frac{1}{2}{e}^{246}+\frac{q^2-1}{p}{e}^{347}+\frac{(q^2-1)(q+1)}{4ap}{e}^{367}.$$

$${\tau }_0=$$

$${\tau }_1=$$

$${\tau }_2=$$

Full text

Support: This work was supported by the Polish National Science Centre (NCN) via the grant number 2018/29/B/ST1/02583.

1. Introduction: a question of Maciej Dunajski

Recently, together with C. D. Hill [5], we uncovered an $\mathbf{Sp}(2,\mathbb{R})$ symmetry of the nonholonomic kinematics of a car. I talked about this at the Abel Symposium in Ålesund, Norway, this June. After my talk Maciej Dunajski, intrigued by the root diagram of $\mathfrak{sp}(2,\mathbb{R})$ which appeared at the talk, asked me if using it I can see a $G_{2}$ structure on a 7-dimensional homogeneous space $M=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})$.

My immediate answer was: ‘I can think about it, but I have to know which of the $\mathbf{SL}(2,\mathbb{R})$ subgroups of $\mathbf{Sp}(2,\mathbb{R})$ I shall use to built $M$’. The reason for the ‘but’ word in my answer was that there are at least two $\mathbf{SL}(2,\mathbb{R})$ subgroups of $\mathbf{Sp}(2,\mathbb{R})$, which lie quite differently in there. One can see them in the root diagram above: the first $\mathbf{SL}(2,\mathbb{R})$ corresponds to the long roots, as e.g. $E_{1}$ and $E_{10}$, whereas the second one corresponds to the short roots, as e.g. $E_{2}$ and $E_{9}$. Since Maciej never told me which $\mathbf{SL}(2,\mathbb{R})$ he wants, I decided to consider both of them, and to determine what kind of $G_{2}$ structures one can associate with each choice of subgroups, separately.

I emphasize that in the below considerations I will use the split real form of the simple exceptional Lie group $G_{2}$. Therefore the corresponding $G_{2}$ structure metrics will not be riemannian. They will have signature $(3,4)$.

2. The Lie algebra $\mathfrak{sp}(2,\mathbb{R})$

The Lie algebra $\mathfrak{sp}(2,\mathbb{R})$ is given by the $4\times 4$ real matrices

[TABLE]

where the coefficients $a_{I}$, $I=1,2,\dots 10$, are real constants. The commutator in $\mathfrak{sp}(2,\mathbb{R})$ is the usual commutator [E,E^{\prime}]=E\hbox to2.5pt{\hss\cdot\hss}E^{\prime}-E^{\prime}\hbox to2.5pt{\hss\cdot\hss}E of two matrices $E$ and $E^{\prime}$. We start with the following basis $(E_{I})$,

[TABLE]

in $\mathfrak{sp}(2,\mathbb{R})$.

In this basis, modulo the antisymmetry, we have the following nonvanishing commutators: [{E_{1}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{5}}]=2{E_{1}}, $[{E_{1}},{E_{7}}]={-2E_{2}}$, $[{E_{1}},{E_{9}}]={-2E_{4}}$, [{E_{1}},{E_{10}}]={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}4E_{5}}, $[{E_{2}},{E_{4}}]={E_{1}}$, [{E_{2}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{5}}]={E_{2}}, [{E_{2}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{6}}]={E_{2}}, $[{E_{2}},{E_{7}}]={2E_{3}}$, $[{E_{2}},{E_{8}}]={E_{4}}$, [{E_{2}},{E_{9}}]={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-E_{5}-E_{6}}, $[{E_{2}},{E_{10}}]={-2E_{7}}$, $[{E_{3}},{E_{4}}]={-E_{2}}$, [{E_{3}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{6}}]={2E_{3}}, [{E_{3}},{E_{8}}]={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-E_{6}}, $[{E_{3}},{E_{9}}]={-E_{7}}$, [{E_{4}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{5}}]={E_{4}}, [{E_{4}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{6}}]={-E_{4}}, [{E_{4}},{E_{7}}]={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{5}-E_{6}}, $[{E_{4}},{E_{9}}]={-2E_{8}}$, $[{E_{4}},{E_{10}}]={-2E_{9}}$, [{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{5}},{E_{7}}]={E_{7}}, [{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{5}},{E_{9}}]={E_{9}}, [{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{5}},{E_{10}}]={2E_{10}}, [{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{6}},{E_{7}}]={-E_{7}}, [{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{6}},{E_{8}}]={2E_{8}}, [{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E_{6}},{E_{9}}]={E_{9}}, $[{E_{7}},{E_{8}}]={E_{9}}$, $[{E_{7}},{E_{9}}]={E_{10}}$.

We see that there are at least two $\mathfrak{sl}(2,\mathbb{R})$ Lie algebras here. The first one is

[TABLE]

and the second is

[TABLE]

The reason for distinguishing these two is as follows:

The eight 1-dimensional vector subspaces $\mathfrak{g}_{I}=\mathrm{Span}(E_{I})$, $I=1,2,3,4,7,8,9,10$, of $\mathfrak{sp}(2,\mathbb{R})$ are the root spaces of this Lie algebra. They correspond to the Cartan subalgebra of $\mathfrak{sp}(2,\mathbb{R})$ given by $\mathfrak{h}=\mathrm{Span}(E_{5},E_{6})$. It follows that the pairs $(E_{I},E_{J})$ of the root vectors, such that $I+J=11$, $I,J\neq 5,6$, correspond to the opposite roots of $\mathfrak{sl}(2,\mathbb{R})$. Knowing the Killing form for $\mathfrak{sl}(2,\mathbb{R})$, which in the basis $(E_{I})$, and its dual basis $(E^{I})$, E_{I}\raisebox{-1.50696pt}{\makebox[6.00006pt][r]{\scriptsize-}}\raisebox{1.07639pt}{\makebox[3.99994pt][l]{\tiny|}}E^{J}=\delta^{J}{}_{I}, is

[TABLE]

one can see that the roots corresponding to the root vectors $(E_{1},E_{10})$ and $(E_{3},E_{8})$ are long, and the roots cooresponding to the root vectors $(E_{2},E_{9})$ and $(E_{4},E_{7})$ are short. Thus the Lie algebra $\mathfrak{sl}(2,\mathbb{R})_{l}$ corresponding to the root vectors $(E_{1},E_{10})$, and in turn to the long roots, lies quite different in $\mathfrak{sp}(2,\mathbb{R})$ than the Lie algebra $\mathfrak{sl}(2,\mathbb{R})_{s}$ corresponding to the short roots associated with the root vectors $(E_{2},E_{9})$.

3. $G_{2}$ structures on $\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{l}$

3.1. Compatible pairs $(g,\phi)$ on $M_{l}$

To consider the homogeneous space $M_{l}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{l}$ it is convenient to change the basis $(E_{I})$ in $\mathbf{Sp}(2,\mathbb{R})$ to a new one, $(e_{I})$, in which the last three vectors span $\mathbf{SL}(2,\mathbb{R})_{l}$. Thus we take:

[TABLE]

If now, one considers $(e_{I})$ as the basis of the Lie algebra of invariant vector fields on the Lie group $\mathbf{Sp}(2,\mathbb{R})$ then the dual basis $(e^{I})$, e_{I}\raisebox{-1.50696pt}{\makebox[6.00006pt][r]{\scriptsize-}}\raisebox{1.07639pt}{\makebox[3.99994pt][l]{\tiny|}}e^{J}=\delta^{J}{}_{I}, of the left invariant forms on $\mathbf{Sp}(2,\mathbb{R})$ satisfies:

[TABLE]

Here we used the usual formula relating the structure constants $c^{I}{}_{JK}$, from $[e_{J},e_{K}]=c^{I}{}_{JK}e_{I}$, to the differentials of the Maurer-Cartan forms $(e^{I})$, ${\rm d}e^{I}=-\tfrac{1}{2}c^{I}{}_{JK}e^{J}\wedge e^{K}$.

In this basis the Killing form on $\mathbf{Sp}(2,\mathbb{R})$ is

[TABLE]

Here we have used the notation $e^{I}\odot e^{J}=\tfrac{1}{2}(e^{I}\otimes e^{J}+e^{J}\otimes e^{I})$, $(e^{I})^{2}=e^{I}\odot e^{I}$.

Looking at the equations (3.1) one sees that $\mathbf{Sp}(2,\mathbb{R})$ has the structure of the principal $\mathbf{SL}(2,\mathbb{R})$ fiber bundle $\mathbf{SL}(2,\mathbb{R})_{l}\to\mathbf{Sp}(2,\mathbb{R})\to M_{l}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{l}$ over the homogeneous space $M_{l}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{l}$.

Indeed, the 3-dimensional distribution $D_{l}$, generated by the vector fields $X$ on $\mathbf{Sp}(2,\mathbb{R})$ annihilating the span of the 1-forms $(e^{1},e^{2},\dots,e^{7})$, is integrable, ${\rm d}e^{\mu}\wedge e^{1}\wedge e^{2}\dots\wedge e^{7}\equiv 0$, $\mu=1,2\dots,7$, so that we have a well defined 7-dimensional leaf space $M_{l}$ of the corresponding foliation. Moreover, the Maurer-Cartan equations (3.1), restricted to a leaf defined by $(e^{1},e^{2},\dots,e^{7})\equiv 0$, reduce to ${\rm d}e^{8}=-2e^{8}\wedge e^{9}$, ${\rm d}e^{9}=-4e^{8}\wedge e^{10}$, ${\rm d}e^{10}=-2e^{9}\wedge e^{10}$, showing that each leaf can be identified with the Lie group $\mathbf{SL}(2,\mathbb{R})_{l}$. Thus the projection $\mathbf{Sp}(2,\mathbb{R})\to M_{l}$ from the Lie group $\mathbf{Sp}(2,\mathbb{R})$ to the leaf space $M_{l}$ is the projection to the homogeneous space $M_{l}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{l}$.

From now on, in this Section, I will use Greek indices $\mu,\nu$, etc., to run from 1 to 7. They number the first seven basis elements in the bases $(e_{I})$ and $(e^{I})$.

Now, I look for all bilinear symmetric forms $g=g_{\mu\nu}e^{\mu}\odot e^{\nu}$ on $\mathbf{Sp}(2,\mathbb{R})$, with constant coefficients $g_{\mu\nu}=g_{\nu\mu}$, which are constant along the leaves of the foliation defined by $D_{l}$. Technically, I search for those $g$ whose Lie derivative with respect to any vector field $X$ from $D_{l}$ vanishes,

[TABLE]

I have the following proposition:

Proposition 3.1**.**

The most general $g=g_{\mu\nu}e^{\mu}\odot e^{\nu}$ satisfying condition (3.2) is

[TABLE]

Thus I have a 7-parameter family of bilinear forms on $\mathbf{Sp}(2,\mathbb{R})$ that descend to well defined pseudoriemannian metrics on the leaf space $M_{l}$. Note that, the restriction of the Killing form $K$ to the space where $(e^{8},e^{9},e^{10})\equiv 0$ is in this family. This corresponds to $g_{22}=g_{24}=g_{46}=0$ and $g_{44}=2g_{26}=-g_{35}=1$.

Since the aim of my note is not to be exhaustive, but rather to show how to produce $G_{2}$ structures on $\mathbf{Sp}(2,\mathbb{R})$ homogeneous spaces, from now on I will restirict myself to only one $\mathbf{SL}(2,\mathbb{R})_{l}$ invariant bilinear form $g$ on $\mathbf{Sp}(2,\mathbb{R})$, namely to

[TABLE]

coming from the restriction of the Killing form. It follows from the Proposition 3.1 that this form is a well defined $(3,4)$ signature metric on the quotient space $M_{l}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{l}$.

I now look for the 3-forms $\phi=\tfrac{1}{6}\phi_{\mu\nu\rho}e^{\mu}\wedge e^{\nu}\wedge e^{\rho}$ on $\mathbf{Sp}(2,\mathbb{R})$ that are constant along the leaves of the distribution $D_{l}$, i.e. such that

[TABLE]

Then, I have the following proposition.

Proposition 3.2**.**

There is a 10-parameter family of 3-forms $\phi=\tfrac{1}{6}\phi_{\mu\nu\rho}e^{\mu}\wedge e^{\nu}\wedge e^{\rho}$ on $\mathbf{Sp}(2,\mathbb{R})$ which satisfy condition (3.4). The general formula for them is:

[TABLE]

Here $e^{\mu\nu\rho}=e^{\mu}\wedge e^{\nu}\wedge e^{\rho}$, and $a$, $b$, $f$, $h$, $p$, $q$, $r$, $s$, $t$ and $u$ are real constants.

Thus there is a 10-parameter family of 3-forms $\phi$ that descends from $\mathbf{Sp}(2,\mathbb{R})$ to the $\mathbf{Sp}(2,\mathbb{R})$ homogeneous space $M_{l}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{l}$, and is well defined there.

Now, I introduce an important notion of compatibility of a pair $(g,\phi)$ where $g$ is a metric, and $\phi$ is a 3-form on a 7-dimensional oriented manifold $M$. The pair $(g,\phi)$ on $M$ is compatible if and only if

[TABLE]

Here $\mathrm{vol}(g)$ is a volume form on $M$ related to the metric $g$.

Restricting, as I did, to the $\mathbf{Sp}(2,\mathbb{R})$ invariant metric $g_{K}$ on $M_{l}$ as in (3.3), I now ask which of the 3-forms $\phi$ from Proposition 3.2 are compatible with the metric (3.3). In other words, I now look for the constants $a$, $b$, $f$, $h$, $p$, $q$, $r$, $s$, $t$ and $u$ such that

[TABLE]

for $g=g_{K}$ given in (3.3).

I have the following proposition.

Proposition 3.3**.**

The general solution to the equations (3.5) is given by

[TABLE]

This leads to the following corollary.

Corollary 3.4**.**

The most general pair $(g_{K},\phi)$ on $M_{l}$ compatible with the $\mathbf{Sp}(2,\mathbb{R})$ invariant metric

[TABLE]

coming from the Killing form in $\mathbf{Sp}(2,\mathbb{R})$, is a 3-parameter family with $\phi$ given by:

[TABLE]

Here $a\neq 0$, $p\neq 0$, $q\neq 1$ are free parameters, and $e^{\mu\nu\rho}=e^{\mu}\wedge e^{\nu}\wedge e^{\rho}$ as before.

3.2. $G_{2}$ structures in general

Compatible pairs $(g,\phi)$ on 7-dimensional manifolds are interesting since they give examples of $G_{2}$ structures [2]. In general, a $G_{2}$ structure consists of a compatible pair $(g,\phi)$ of a metric $g$ and a 3-form $\phi$ on a 7-dimensional manifold $M$. It is in addition assumedthat the 3-form $\phi$ is generic, meaning that at every point of $M$ it lies in one of the two open orbits of the natural action of $\mathbf{GL}(7,\mathbb{R})$ on 3-forms in $\mathbb{R}^{7}$. The simple exceptional Lie group $G_{2}$ appears here as the common stabilizer in $\mathbf{GL}(7,\mathbb{R})$ of both $g$ and $\phi$.

It follows (from compatibility) that the $G_{2}$ structures can have metrics $g$ of only two signatures: the riemannian ones and $(3,4)$ signature ones. If the signature of $g$ is riemannian, the corresponding $G_{2}$ structure is related to the compact real form of the simple exceptional complex Lie group $G_{2}$, and in the $(3,4)$ signature case the corresponding $G_{2}$ structure is related to the noncompact (split) real form of the complex group $G_{2}$. In this sense our Corollary 3.4 provides a 3-parameter family of split real form $G_{2}$ structures on $M_{l}$.

$G_{2}$ structures can be classified according to their torsion [1, 2]. Making the long story short we say that every $G_{2}$ structure $(g,\phi)$ on $M$ defines four forms $\tau_{0}$, $\tau_{1}$, $\tau_{2}$ and $\tau_{3}$ such that

[TABLE]

where $*$ is the Hodge dual which is defined on $p$-forms $\lambda$ by

[TABLE]

As it is visible from equations (3.6), which we call Bryant’s [1, 2] equations in the following, each $\tau_{i}$, $i=0,1,2,3$, is an $i$-form on $M$. Since there is a natural action of the group $G_{2}$ in $\mathbb{R}^{7}$, which induces its action on forms, it is further required that the 3-form $\tau_{3}$ has values in the 27-dimensional irreducible representation $\bigwedge^{3}_{27}$ of this group, the 2-form $\tau_{2}$ has values in the 14-dimensional irreducible representation $\bigwedge^{2}_{14}$, and the 1-form has values in the 7-dimensional irreducible representation $\bigwedge^{1}_{7}$. We add that the space of 1-forms $\bigwedge^{1}$ is irreducible, $\bigwedge^{1}=\bigwedge^{1}_{7}$, and that the $G_{2}$ irreducible decompositions of the spaces of 2- and 3-forms look like $\bigwedge^{2}=\bigwedge^{2}_{7}\oplus\bigwedge^{2}_{14}$ and $\bigwedge^{3}=\bigwedge^{3}_{1}\oplus\bigwedge^{3}_{7}\oplus\bigwedge^{3}_{27}$. By the Hodge duality, the decomposition of $\bigwedge^{4}$ onto $G_{2}$ irreducible components is similar as this for $\bigwedge^{3}$. Here we use the convention that the lower index $i$ in $\bigwedge^{p}_{i}$ denotes the dimension of the corresponding representation. We further mention that the 7-dimensional representations $\bigwedge^{1}_{7}$, $\bigwedge^{2}_{7}$ and $\bigwedge^{3}_{7}$ are all $G_{2}$ equivalent. Also, one can see that e.g. $\textstyle{\bigwedge^{3}_{27}}=\{\alpha\in\bigwedge^{3}\,\,\,\mathrm{s.t.}\,\,\,\alpha\wedge\phi=0\,\,\&\,\,\alpha\wedge*\phi=0\}$.

3.3. All $\mathbf{Sp}(2,\mathbb{R})$ symmetric $G_{2}$ structures on $M_{l}$ with the metric coming from the Killing form

The below Theorem characterizes the $G_{2}$ structures corresponding to compatible pairs $(g_{K},\phi)$ from Corollary 3.4.

Theorem 3.5**.**

Let $g_{K}$ be the $(3,4)$ signature metric on $M_{l}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{l}$ arising as the restriction of the Killing form $K$ from $\mathbf{Sp}(2,\mathbb{R})$ to $M_{l}$,

[TABLE]

Then the most general $G_{2}$ structure associated with such $g_{K}$ is a 3-parameter family $(g_{K},\phi)$ with the 3-form

[TABLE]

For this structure the torsions $\tau_{\mu}$ solving the Bryant’s equations (3.6) are:

[TABLE]

[TABLE]

where, as usual $e^{\mu\nu}=e^{\mu}\wedge e^{\nu}$ and $e^{\mu\nu\rho}=e^{\mu}\wedge e^{\nu}\wedge e^{\rho}$.

Thus the 3-parameter family of $G_{2}$ structures on $M_{l}$ described in this Theorem have the entire 14-dimensional torsion $\tau_{2}=0$. This means that all these $G_{2}$ structures are integrable in the terminology of [3, 4], or what is the same, this means that they all have the totally skew symmetric torsion.

4. $G_{2}$ structures on $\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{s}$

Now we consider the homogeneous space $M_{s}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{s}$. Since $\mathbf{SL}(2,\mathbb{R})$ is spanned by $E_{2},E_{5}+E_{6},E_{9}$ it is convenient to put these vectors at the end of the new basis of the Lie algebra $\mathbf{Sp}(2,\mathbb{R})$. We choose this new basis $(f_{I})$ in $\mathbf{Sp}(2,\mathbb{R})$ as:

[TABLE]

If now, one considers $(f_{I})$ as the basis of the Lie algebra of invariant vector fields on the Lie group $\mathbf{Sp}(2,\mathbb{R})$ then the dual basis $(f^{I})$, f_{I}\raisebox{-1.50696pt}{\makebox[6.00006pt][r]{\scriptsize-}}\raisebox{1.07639pt}{\makebox[3.99994pt][l]{\tiny|}}f^{J}=\delta^{J}{}_{I}, of the left invariant forms on $\mathbf{Sp}(2,\mathbb{R})$ satisfies:

[TABLE]

In this basis the Killing form on $\mathbf{Sp}(2,\mathbb{R})$ is

[TABLE]

where as usual the structure constants $c^{I}{}_{JK}$ are defined by $[f_{I},f_{J}]=c^{K}{}_{IJ}f_{K}$.

Using the same arguments, as in the case of $M_{l}$, we again see that $\mathbf{Sp}(2,\mathbb{R})$ has the structure of the principal $\mathbf{SL}(2,\mathbb{R})$ fiber bundle $\mathbf{SL}(2,\mathbb{R})_{s}\to\mathbf{Sp}(2,\mathbb{R})\to M_{s}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{s}$ over the homogeneous space $M_{s}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{s}$. In particular we have a foliation of $\mathbf{Sp}(2,\mathbb{R})$ by integral leaves of an integrable distribution $D_{s}$ spanned by the annihilator of the forms $(f^{1},f^{2},\dots,f^{7})$. As before, also in this Section, we will use Greek indices $\mu,\nu$, etc., to run from 1 to 7. They now number the first seven basis elements in the bases $(f_{I})$ and $(f^{I})$.

Repeating the procedure from the previous Sections, I now search for all bilinear symmetric forms $g=g_{\mu\nu}f^{\mu}\odot f^{\nu}$ on $\mathbf{Sp}(2,\mathbb{R})$, with constant coefficients $g_{\mu\nu}=g_{\nu\mu}$, whose Lie derivative with respect to any vector field $X$ from $D_{l}$ vanishes,

[TABLE]

I have the following proposition.

Proposition 4.1**.**

The most general $g=g_{\mu\nu}f^{\mu}\odot f^{\nu}$ satisfying condition (4.2) is

[TABLE]

Thus, this time, I only have a 4-parameter family of bilinear forms on $\mathbf{Sp}(2,\mathbb{R})$ that descend to well defined pseudoriemannian metrics on the leaf space $M_{s}$. Note that, the restriction of the Killing form $K$ to the space where $(f^{8},f^{9},f^{10})\equiv 0$ is in this family. This corresponds to $g_{33}=g_{55}=0$ and $g_{44}=2$, $g_{26}=1/2$.

Again for simplicity reasons, I will solve the problem of finding $\mathbf{Sp}(2,\mathbb{R})$ invariant $G_{2}$ structures on $M_{s}$ restricting to only those pairs $(g,\phi)$ for which $g=g_{K}$, where

[TABLE]

i.e. I only will consider one metric, the one coming from the restriction of the Killing form of $\mathbf{Sp}(2,\mathbb{R})$ to $M_{s}$. It is a well defined $(3,4)$ signature metric on the quotient space $M_{s}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{s}$.

I now look for the 3-forms $\phi=\tfrac{1}{6}\phi_{\mu\nu\rho}f^{\mu}\wedge f^{\nu}\wedge f^{\rho}$ on $\mathbf{Sp}(2,\mathbb{R})$ which are such that

[TABLE]

I have the following proposition.

Proposition 4.2**.**

There is precisely a 5-parameter family of 3-forms $\phi=\tfrac{1}{6}\phi_{\mu\nu\rho}f^{\mu}\wedge f^{\nu}\wedge f^{\rho}$ on $\mathbf{Sp}(2,\mathbb{R})$ which satisfies condition (4.4). The general formula for $\phi$ is:

[TABLE]

Here $f^{\mu\nu\rho}=f^{\mu}\wedge f^{\nu}\wedge f^{\rho}$, and $a$, $b$, $q$, $h$ and $p$ are real constants.

Solving for all 3-forms $\phi$ from this 5-parameter family that are compatible, as in (3.5), with the metric $g_{K}$ from (4.3), I arrive at the following proposition.

Proposition 4.3**.**

The general solution to the equations (3.5) is given by

[TABLE]

This leads to the following corollary.

Corollary 4.4**.**

The most general pair $(g_{K},\phi)$ on $M_{s}$ compatible with the $\mathbf{Sp}(2,\mathbb{R})$ invariant metric

[TABLE]

coming from the Killing form in $\mathbf{Sp}(2,\mathbb{R})$, is a 1-parameter family with $\phi$ given by:

[TABLE]

Here $q\neq 0$ is a free parameter, and $f^{\mu\nu\rho}=f^{\mu}\wedge f^{\nu}\wedge f^{\rho}$ as before.

4.1. All $\mathbf{Sp}(2,\mathbb{R})$ symmetric $G_{2}$ structures on $M_{s}$ with the metric coming from the Killing form

Now I characterize the $G_{2}$ structures corresponding to compatible pairs $(g_{K},\phi)$ from Corollary 4.4.

Theorem 4.5**.**

Let $g_{K}$ be the $(3,4)$ signature metric on $M_{s}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{s}$ arising as the restriction of the Killing form $K$ from $\mathbf{Sp}(2,\mathbb{R})$ to $M_{s}$,

[TABLE]

Then the most general $G_{2}$ structure associated with such $g_{K}$ is a 1-parameter family $(g_{K},\phi)$ with the 3-form

[TABLE]

For this structure

[TABLE]

i.e. the torsions

[TABLE]

The rest of the torsions solving Bryant’s equations (3.6) are:

[TABLE]

where, as usual $f^{\mu\nu\rho}=f^{\mu}\wedge f^{\nu}\wedge f^{\rho}$; $q\neq 0$.

So on $M_{s}=\mathbf{Sp}(2,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_{s}$ there exists a 1-parameter family of the above $G_{2}$ structures which is coclosed. Therefore, in particular, it is integrable

I note that formally I can also obtain coclosed $G_{2}$ structures on $M_{l}$, using the Theorem 3.5. It is enough to take $p=2a$ in the solutions of this Theorem. The question if in the resulting 2-parameter family of the coclosed $G_{2}$ structures there is a 1-parameter subfamily equivalent to the structures I have on $M_{s}$ via Theorem 4.5 needs further investigation. However, I doubt that the answer to this question is positive, since it is visible from the root diagram for $\mathbf{Sp}(2,\mathbb{R})$ that the spaces $M_{l}$ and $M_{s}$ are geometrically quite different. Indeed, apart from the $\mathbf{Sp}(2,\mathbb{R})$ invariant $G_{2}$ structures, which I have just introduced in this note, the spaces $M_{l}$ and $M_{s}$ have quite different additional $\mathbf{Sp}(2,\mathbb{R})$ invariant structures. A short look at the root diagram on page 1 of this note, shows that $M_{l}$ has two well defined $\mathbf{Sp}(2,\mathbb{R})$ invariant rank 3-distributions, corresponding to the pushforwards from $\mathbf{Sp}(2,\mathbb{R})$ to $M_{l}$ of the vector spaces $D_{l1}=\mathrm{Span}_{\mathbb{R}}(E_{2},E_{3},E_{7})$ and $D_{l2}=\mathrm{Span}_{\mathbb{R}}(E_{4},E_{8},E_{9})$. Likewise $M_{s}$, in addition to the discussed $G_{2}$ structures, has also a well defined pair of $\mathbf{Sp}(2,\mathbb{R})$ invariant rank 3-distributions, corresponding to the pushforwards from $\mathbf{Sp}(2,\mathbb{R})$ to $M_{s}$ of the vector spaces $D_{s1}=\mathrm{Span}_{\mathbb{R}}(E_{1},E_{4},E_{8})$ and $D_{s2}=\mathrm{Span}_{\mathbb{R}}(E_{3},E_{7},E_{10})$. The problem is that these two sets of pairs of $\mathbf{Sp}(2,\mathbb{R})$ invariant distributions are quite different. The distributions on $M_{l}$ have constant growth vector $(2,3)$, while the distributions on $M_{s}$ are integrable. These pairs of distributions constitute an immanent ingredient of the geometry on the corresponding spaces $M_{l}$ and $M_{s}$ and, since they are diffeomorphically nonequivalent, they make the $G_{2}$ geometries there quite diferent. I believe that this fact makes the $G_{2}$ structures obtained on $M_{l}$ and $M_{s}$ really nonequivalent.