2.1 Submanifolds via the Calabi–Yau and QK structures

Recall that every hyperkähler manifold is a Kähler manifold in an $S^2$ -family of ways, and given such a choice, it is a Calabi–Yau manifold in an $S^1$ -family of ways. Due to these structures, we may consider the following classes of submanifolds.

Definition 2.1 A submanifold $N^{2k} \subset C^{4n+4}$ is $I_1$ -complex if

$$ \begin{align*}\left. \frac{1}{k!}\omega_1^k\right|_N = \mathrm{vol}_N.\end{align*} $$

That is, if it is calibrated with respect to $\frac {1}{k!}\omega _1^k$ .

It is $I_1$ -anti-complex, or $-I_1$ -complex, if it is calibrated with respect to $-\frac {1}{k!}\omega _1^k$ . Equivalently, if it is $I_1$ -complex when equipped with the opposite orientation.

A submanifold is $\pm I_1$ -complex if and only if its tangent spaces are $I_1$ -invariant:

$$ \begin{align*}I_1(T_xN) = T_xN, \ \ \forall x \in N.\end{align*} $$

The definitions of $I_2$ -complex and $I_3$ -complex are analogous.

Definition 2.2 A submanifold $N \subset C^{4n+4}$ is $\omega _1$ -isotropic if

$$ \begin{align*}\left.\omega_1\right|_N = 0.\end{align*} $$

An $\omega _1$ -isotropic submanifold satisfies $\dim (N) \leq 2n+2$ . An $\omega _1$ -Lagrangian submanifold is an $\omega _1$ -isotropic submanifold of maximal dimension $2n+2$ .

Let $X, Y \in TL$ . Since $\omega _1(X, Y) = g(I_1 X, Y)$ , we see that L is $\omega _1$ -isotropic if and only if $I_1 (TL) \subseteq NL$ . If N has dimension $2n+2$ , then $I_1 (TL) = NL$ if and only if L is $\omega _1$ -Lagrangian. We use these facts repeatedly.

Definition 2.3 Fix $\theta \in [0,2\pi )$ . A $(2n+2)$ -dimensional submanifold $N^{2n+2} \subset C^{4n+4}$ is called $\Upsilon _1$ -special Lagrangian of phase $e^{i \theta }$ if

$$ \begin{align*}\left.\text{Re}(e^{-i\theta}\Upsilon_1)\right|_N = \mathrm{vol}_N.\end{align*} $$

Equivalently [Reference Harvey and Blaine Lawson20, Corollary 1.11], there exists an orientation on $N^{2n+2}$ making it $\Upsilon _1$ -special Lagrangian of phase $e^{i \theta }$ if and only if

$$ \begin{align*} \left.\text{Im}(e^{-i \theta}\Upsilon_1)\right|_N & = 0, & \left.\omega_1\right|_N & = 0. \end{align*} $$

When the phase is left unspecified, we assume it to be $e^{i \theta } = 1$ .

2.2 Submanifolds via the hyperkähler structure

In addition to the submanifolds discussed above, hyperkähler manifolds also admit three more notable classes of submanifolds: the complex isotropic, special isotropic, and generalized Cayley submanifolds. We discuss each of these in turn.

2.2.1 Complex isotropic submanifolds

Definition 2.5 A $2k$ -dimensional submanifold $L^{2k} \subset C^{4n+4}$ is called $I_1$ -complex isotropic if it is both $I_1$ -complex and $\sigma _1$ -isotropic. That is, if

$$ \begin{align*} \left.\frac{1}{k!}\omega_1^k\right|_L & = \mathrm{vol}_L, & \left.\sigma_1\right|_L & = 0. \end{align*} $$

Said another way, L is $I_1$ -complex, $\omega _2$ -isotropic, and $\omega _3$ -isotropic:

$$ \begin{align*} \left.\frac{1}{k!}\omega_1^k\right|_L & = \mathrm{vol}_L, & \left.\omega_2\right|_L & = 0, & \left.\omega_3\right|_L & = 0. \end{align*} $$

An $I_1$ -complex Lagrangian submanifold $L^{2n+2} \subset C^{4n+4}$ is an $I_1$ -complex isotropic submanifold of maximal dimension $2n+2$ . That is, an $I_1$ -complex Lagrangian submanifold is simultaneously $I_1$ -complex, $\omega _2$ -Lagrangian, and $\omega _3$ -Lagrangian. The definitions of $I_2$ - and $I_3$ -complex isotropic (resp. complex Lagrangian) are analogous.

Complex isotropic submanifolds are interesting from several points of view. For example, in algebraic geometry, one often considers holomorphic symplectic manifolds that are fibered by complex Lagrangians, as in [Reference Sawon29]. As another example, Doan and Rezchikov [Reference Doan and Rezchikov11] use complex Lagrangians as part of a hyperkähler Floer theory. In the differential geometry literature, complex isotropic submanifolds have been studied by, for example, Bryant and Harvey [Reference Bryant and Harvey9], Hitchin [Reference Hitchin22], and Grantcharov and Verbitsky [Reference Grantcharov and Verbitsky17].

Proposition 2.6 Let $L^{2k} \subset C^{4n+4}$ be a $2k$ -dimensional submanifold. The following are equivalent:

1. (1) L is $I_1$ -complex, $\omega _2$ -isotropic, and $\omega _3$ -isotropic.

2. (2) L is $I_1$ -complex and $\omega _2$ -isotropic.

Proof One direction is immediate. For the converse, suppose L is $I_1$ -complex and $\omega _2$ -isotropic. Let $X \in TL$ , so that $I_1X \in TL$ , and thus $- I_3 X = I_2(I_1X) \in NL$ . Hence, $I_3X \in NL$ . This shows that L is $\omega _3$ -isotropic.

In the complex Lagrangian case, we can say more:

Proposition 2.7 Let $L^{2n+2} \subset C^{4n+4}$ be a $(2n+2)$ -dimensional submanifold. The following are equivalent:

1. (1) L is $I_1$ -complex, $\omega _2$ -Lagrangian, and $\omega _3$ -Lagrangian.

2. (2) L is $I_1$ -complex and $\omega _2$ -Lagrangian.

3. (3) L is $\omega _2$ -Lagrangian and $\omega _3$ -Lagrangian.

4. (4) L is $I_1$ -complex, $\Upsilon _2$ -special Lagrangian of phase $i^{n+1}$ , and $\Upsilon _3$ -special Lagrangian of phase $1$ .

Proof The equivalence (i) $\iff $ (ii) was observed above. It is clear that (i) $\implies $ (iii). For (iii) $\implies $ (i), suppose that L is $\omega _2$ - and $\omega _3$ -Lagrangian. Let $X \in TL$ , so that $I_3X \in NL$ , and thus $I_1X = I_2(I_3X) \in TL$ . Hence, L is $I_1$ -complex.

It is clear that (iv) $\implies $ (i). For (i) $\implies $ (iv), suppose that L is $I_1$ -complex, $\omega _2$ -Lagrangian, and $\omega _3$ -Lagrangian. Then L satisfies $\left .\frac {1}{(n+1)!}\omega _1^{n+1}\right |_L = \mathrm {vol}_L$ and $\left .\omega _2\right |_L = 0$ and $\left .\omega _3\right |_L = 0$ . Recalling that

$$ \begin{align*} (-i)^{n+1}\Upsilon_2 & = \frac{1}{(n+1)!}(\omega_1 - i\omega_3)^{n+1}, & \Upsilon_3 & = \frac{1}{(n+1)!}(\omega_1 + i\omega_2)^{n+1}, \end{align*} $$

we have

$$ \begin{align*} \left.\text{Re}((-i)^{n+1} \Upsilon_2)\right|_L & = \left.\frac{1}{(n+1)!}\omega_1^{n+1}\right|_L = \mathrm{vol}_L, & \left.\text{Re}( \Upsilon_3)\right|_L = \left.\frac{1}{(n+1)!}\omega_1^{n+1}\right|_L = \mathrm{vol}_L. \end{align*} $$

2.2.2 Special isotropic submanifolds

The following definition is due to Bryant and Harvey [Reference Bryant and Harvey9]. We prove that these forms are calibrations in Theorem A.6 in the Appendix.

Definition 2.8 The special isotropic forms are the $2k$ -forms $\Theta _{I,2k}, \Theta _{J,2k}, \Theta _{K,2k} \in \Omega ^{2k}(C)$ defined by

$$ \begin{align*} \Theta_{I,2k} & = \frac{1}{k!}\text{Re}(\sigma_1^k), & \Theta_{J,2k} & = \frac{1}{k!}\text{Re}(\sigma_2^k), & \Theta_{K,2k} & = \frac{1}{k!}\text{Re}(\sigma_3^k). \end{align*} $$

A $2k$ -dimensional submanifold $N^{2k} \subset C^{4n+4}$ is $\Theta _{I,2k}$ -special isotropic if it is calibrated by $\Theta _{I,k}$ :

$$ \begin{align*}\left.\Theta_{I,2k}\right|_N = \mathrm{vol}_N.\end{align*} $$

The definitions of $\Theta _{J,2k}$ - and $\Theta _{K,2k}$ -special isotropic $2k$ -manifold are analogous.

Let us highlight the cases $2k = 2, 4, 2n+2$ .

Example 2.1

1. (1) For $2k = 2$ , the special isotropic $2$ -forms are

   $$ \begin{align*} \Theta_{I,2} & = \omega_2, & \Theta_{J,2} & = \omega_3, & \Theta_{K,2} & = \omega_1. \end{align*} $$
   In particular, a $\Theta _{I,2}$ -special isotropic $2$ -fold is the same as an $I_2$ -complex $2$ -fold.

2. (2) For $2k = 4$ , the special isotropic $4$ -forms are

   $$ \begin{align*} \Theta_{I,4} & = \frac{1}{2}(\omega_2^2 - \omega_3^2), & \Theta_{J,4} & = \frac{1}{2}(\omega_3^2 - \omega_1^2), & \Theta_{K,4} & = \frac{1}{2}(\omega_1^2 - \omega_2^2). \end{align*} $$
   In particular, if L is an $I_1$ -complex isotropic $4$ -fold, then L is both $-\Theta _{J,4}$ -special isotropic and $\Theta _{K,4}$ -special isotropic.

3. (3) For $2k = 2n+2$ , the special isotropic $(2n+2)$ -forms are

   $$ \begin{align*} \Theta_{I,2n+2} & = \text{Re}(\Upsilon_1), & \Theta_{J, 2n+2} & = \text{Re}(\Upsilon_2), & \Theta_{K, 2n+2} & = \text{Re}(\Upsilon_3). \end{align*} $$
   In particular, a $\Theta _{I, 2n+2}$ -special isotropic $(2n+2)$ -fold is the same as an $\Upsilon _1$ -special Lagrangian, which explains the name “special isotropic.”

At present, it appears that little is known about special isotropic $2k$ -folds in hyperkähler $(4n+4)$ -manifolds when $2 < 2k < 2n+2$ .

2.2.3 Cayley $4$ -folds

The following definition is due to Bryant and Harvey [Reference Bryant and Harvey9], though our sign conventions are opposite to theirs.

Definition 2.9 The generalized Cayley $4$ -forms are the $4$ -forms $\Phi _1, \Phi _2, \Phi _3 \in \Omega ^4(C)$ defined by

$$ \begin{align*} \Phi_{1} & = -\frac{1}{2}\omega_1^2 + \frac{1}{2}\omega_2^2 + \frac{1}{2}\omega_3^2, & \Phi_2 & = \frac{1}{2}\omega_1^2 - \frac{1}{2}\omega_2^2 + \frac{1}{2}\omega_3^2, & \Phi_3 & = \frac{1}{2}\omega_1^2 + \frac{1}{2}\omega_2^2 - \frac{1}{2}\omega_3^2. \end{align*} $$

Note that

(2.2) $$ \begin{align} \Phi_2 & = \ \frac{1}{2}\omega_1^2 - \Theta_{I,4} = \frac{1}{2}\omega_3^2 + \Theta_{K,4} = - \frac{1}{2} \omega_2^2 + \frac{1}{2} (\omega_1^2 + \omega_3^2), \end{align} $$

and similarly for cyclic permutations. A four-dimensional submanifold $N^4 \subset C^{4n+4}$ is $\Phi _2$ -Cayley if it is calibrated by $\Phi _2$ :

$$ \begin{align*}\left.\Phi_2\right|_N = \mathrm{vol}_N.\end{align*} $$

The definitions of $\Phi _1$ -Cayley and $\Phi _3$ -Cayley are analogous.

Remark 2.10 Bryant and Harvey [Reference Bryant and Harvey9, Lemma 2.14] computed that the $\mathrm {SO}(4n+4)$ -stabilizer of the generalized Cayley $4$ -forms in $\mathbb {R}^{4n+4}$ are

$$ \begin{align*}\mathrm{Stab}(\Phi_1) = \begin{cases} \mathrm{Spin}(7), & \text{if } n = 1, \\ \mathrm{Sp}(n+1)\mathrm{O}(2), & \text{if } n \geq 2. \end{cases}\end{align*} $$

This above definition was inspired by $\mathrm {Spin}(7)$ -geometry, as we now recall. If $(X^8, (g, \omega , I, \Upsilon ))$ is a Calabi–Yau $8$ -manifold, where $\omega \in \Omega ^2(X)$ is the Kähler form and $\Upsilon \in \Omega ^4(X;\mathbb {C})$ is the holomorphic volume form, then X inherits a torsion-free $\mathrm {Spin}(7)$ -structure via the following formula:

(2.3) $$ \begin{align} \Phi & = \frac{1}{2}\omega^2 - \text{Re}(\Upsilon). \end{align} $$

The real $4$ -form $\Phi \in \Omega ^4(X)$ is called the Cayley $4$ -form, and a four-dimensional submanifold $N \subset X$ satisfying $\Phi |_N = \mathrm {vol}_N$ is called Cayley. The following fact is well known, but we include a proof for completeness.

Proposition 2.11 Let $(X^8, (g,\omega , I, \Upsilon ))$ be a Calabi–Yau $8$ -manifold, and equip X with its induced $\mathrm {Spin}(7)$ -structure. Let $N^4 \subset X$ be a four-dimensional submanifold.

1. (1) If N is complex, then N is Cayley.

2. (2) If N is special Lagrangian of phase $e^{i\pi } = -1$ , then N is Cayley.

Proof If N is complex, each tangent space $T_x N$ admits a basis of the form $\{ e_1, I e_1, e_2, I e_2 \}$ . Then $v_k = e_k - i I e_k$ is of type $(1,0)$ for $k = 1, 2$ , and $T_x N = e_1 \wedge I e_1 \wedge e_2 \wedge I e_2$ is a multiple of $v_1 \wedge \overline {v_1} \wedge v_2 \wedge \overline {v_2}$ and thus of type $(2,2)$ . Since $\text {Re}(\Upsilon )$ is type $(4,0) + (0,4)$ , it vanishes on $T_x N$ . But $\frac {1}{2} \omega ^2$ restricts to the volume form on $T_x N$ , so by (2.3), N is calibrated by $\Phi $ .

If N is special Lagrangian with phase $-1$ , it is calibrated by $- \text {Re}(\Upsilon )$ . Since it is also Lagrangian, $\frac {1}{2} \omega ^2$ vanishes on N, and thus, again by (2.3), N is calibrated by $\Phi $ .

When the ambient space is hyperkähler, Bryant and Harvey showed that the above fact can be generalized to higher dimensions in the following sense.

Proposition 2.12 ([Reference Bryant and Harvey9, Theorem 8.20])

Let $C^{4n+4}$ be a hyperkähler $(4n+4)$ -manifold. Let $L^4 \subset C^{4n+4}$ be a four-dimensional submanifold. Then:

1. (1) If N is $I_1$ -complex or $I_3$ -complex, then N is $\Phi _2$ -Cayley.

2. (2) If N is $-\Theta _{I,4}$ -special isotropic or $\Theta _{K,4}$ -special isotropic, then N is $\Phi _2$ -Cayley.

3. (3) If N is $I_1$ -complex isotropic, then N is simultaneously $I_1$ -complex, $-\Theta _{J,4}$ -special isotropic, and $\Theta _{K,4}$ -special isotropic, and hence is $\Phi _2$ -Cayley.

Proof Parts (a) and (b) are contained in [Reference Bryant and Harvey9, Theorem 8.20]. It is easy to see from (2.2) that (a) holds. For example, if N is $I_1$ -complex, then $\frac {1}{2} \omega _1^2$ restricts to the volume form, but $- \Theta _{I, 4} = - \mathrm {Re} (\frac {1}{2} \sigma _1^2)$ is of $I_1$ -type $(4,0) + (0,4)$ , and thus vanishes on N since the tangent spaces of N are of $I_1$ -type $(2,2)$ . Part (b) is less obvious, and uses a normal form for the tangent spaces of N. Details are given in [Reference Bryant and Harvey9, Sections 2 and 3]. Part (c) is immediate from the first two.

Remark 2.13 Note that every calibration $\phi \in \Omega ^k(C)$ discussed in this section is stabilized by the Lie group $\mathrm {Sp}(n+1)$ , which acts transitively on the unit sphere in $T_xC \simeq \mathbb {R}^{4n+4}$ . Consequently, at any point $x \in C$ , every unit vector $v \in T_xC$ lies in some $\phi $ -calibrated k-plane.

2.3 Bookkeeping: summary of forms on C

Starting in the next section, we will assume that the hyperkähler manifold $C^{4n+4}$ is a metric cone, say $C = \mathrm {C}(M)$ for some Riemannian $(4n+3)$ -manifold M. Studying the geometry of M and its relationship with C will require the introduction of further tensors and differential forms. So, before continuing, we briefly summarize the tensors and forms already defined on C:

$$ \begin{align*} & g_{\mathrm{C}} & \text{Riemannian metric} \\ & I_1, I_2, I_3 & \text{Complex structures} \\ & \omega_1, \omega_2, \omega_3 & \text{K\"{a}hler 2-forms} \\ & \Upsilon_1, \Upsilon_2, \Upsilon_3 & \text{Complex volume }(2n+2)\text{-forms} \\ & \sigma_1, \sigma_2, \sigma_3 & \text{Complex symplectic 2-forms} \\ & \Theta_{I,2k}, \Theta_{J,2k}, \Theta_{K,2k} & \text{Special isotropic } 2k\text{-forms} \\ & \Phi_1, \Phi_2, \Phi_3 & \text{Cayley 4-forms} \\ & \Lambda & \text{Quaternionic 4-form} \end{align*} $$

3.1 $3$ -Sasakian manifolds as links

Definition 3.1 Let M be an odd-dimensional manifold. An almost contact metric structure on M is a triple $(g_M, \alpha , \mathsf {J})$ consisting of a Riemannian metric $g_M$ , a $1$ -form $\alpha \in \Omega ^1(M)$ , and an endomorphism $\mathsf {J} \in \Gamma (\mathrm {End}(TM))$ satisfying

$$ \begin{align*} \alpha(\mathsf{J}X) & = 0, & \mathsf{J}(A) & = 0, & \mathsf{J}^2|_{\text{Ker}(\alpha)} & = -\mathrm{Id}, & g_M(\mathsf{J}X, \mathsf{J}Y) & = g_M(X,Y) - \alpha(X)\alpha(Y), \end{align*} $$

where $A := \alpha ^\sharp \in \Gamma (TM)$ is the Reeb vector field. It follows that $\alpha (A) = 1$ .

Thus, if M is equipped with an almost contact metric structure, then each tangent space splits as

$$ \begin{align*}T_xM = \mathbb{R} A|_x \oplus \mathrm{Ker}(\alpha|_x).\end{align*} $$

Further, restricting to the hyperplane $\mathrm {Ker}(\alpha |_x) \subset T_xM$ , the endomorphism $\mathsf {J} \colon \mathrm {Ker}(\alpha |_x) \to \mathrm {Ker}(\alpha |_x)$ is a $g_M$ -orthogonal complex structure. Thus, the hyperplane field $\mathrm {Ker}(\alpha ) \subset TM$ is naturally endowed with the Hermitian structure $(g_M, \mathsf {J}, \Omega )$ , where $\Omega := g_M(\mathsf {J} \cdot , \cdot )$ is the corresponding nondegenerate $2$ -form.

Let $M^{4n+3}$ carry an $(\mathrm {Sp}(n) \times 3)$ -structure. We make three remarks. First, for each $p = 1, 2, 3$ , the tangent bundle splits as

(3.1) $$ \begin{align} TM = \mathbb{R} A_p \oplus \mathrm{Ker}(\alpha_p), \end{align} $$

and the hyperplane field $\mathrm {Ker}(\alpha _p) \subset TM$ carries a Hermitian structure $(g_M, \mathsf {J}_p, \Omega _p)$ , where $\Omega _p := g_M(\mathsf {J}_p \cdot , \cdot )$ . In fact, each $\mathrm {Ker}(\alpha _p)$ is also endowed with the complex volume form $\Psi _p \in \Lambda ^{2n+1,0}(\mathrm {Ker}(\alpha _p))$ given by

(3.2) $$ \begin{align} \begin{aligned} \Psi_1 & = (\alpha_2 + i\alpha_3) \wedge \frac{1}{n!}(\Omega_2 + i\Omega_3)^n, \\ \Psi_2 & = (\alpha_3 + i\alpha_1) \wedge \frac{1}{n!}(\Omega_3 + i\Omega_1)^n, \\ \Psi_3 & = (\alpha_1 + i\alpha_2) \wedge \frac{1}{n!}(\Omega_1 + i\Omega_2)^n. \end{aligned} \end{align} $$

Second, considering (3.1) for $p = 1, 2, 3$ simultaneously, we see that the tangent bundle splits further as

(3.3) $$ \begin{align} TM = \widetilde{\mathsf{H}} \oplus \widetilde{\mathsf{V}}, \end{align} $$

where

$$ \begin{align*} \widetilde{\mathsf{H}} & = \mathrm{Ker}(\alpha_1, \alpha_2, \alpha_3), & \widetilde{\mathsf{V}} &= \mathbb{R} A_1 \oplus \mathbb{R} A_2 \oplus \mathbb{R} A_3. \end{align*} $$

Note that the $4n$ -plane field $\widetilde {\mathsf {H}} \subset TM$ is preserved by the three endomorphisms $\mathsf {J}_1, \mathsf {J}_2, \mathsf {J}_3$ . In fact, the restrictions of $\mathsf {J}_1, \mathsf {J}_2, \mathsf {J}_3$ to $\widetilde {\mathsf {H}}$ are $g_M$ -orthogonal complex structures that satisfy the quaternionic relations $\mathsf {J}_1 \mathsf {J}_2 = \mathsf {J}_3$ , etc.

Third, we consider the relationship between the structure on a manifold $(M^{4n+3}, g_M)$ and that of its metric cone

$$ \begin{align*}C^{4n+4} = \mathrm{C}(M) = (\mathbb{R}^+ \times M, g_{\mathrm{C}} = dr^2 + r^2 g_M).\end{align*} $$

In one direction, if $(M, g_M)$ is equipped with a compatible $(\mathrm {Sp}(n) \times 3)$ -structure $(g_M, (\alpha _p), (\mathsf {J}_p))$ , then the $(4n+4)$ -manifold C inherits a Riemannian metric $g_{\mathrm {C}}$ , a triple of $g_{\mathrm {C}}$ -orthogonal almost-complex structures $(I_1, I_2, I_3)$ satisfying $I_1I_2 = I_3$ , etc., and a triple of nondegenerate $2$ -forms $\omega _p$ defined by

$$ \begin{align*} g_{\mathrm{C}} & = dr^2 + r^2 g_M, & \omega_p(X,Y) & = g_{\mathrm{C}}(I_pX,Y), \\ I_p(X) & = \begin{cases} \mathsf{J}_pX - \alpha_p(X)r\partial_r, &\text{if } X \in TM, \\ A_p, & \text{if } X = r\partial_r, \end{cases} \end{align*} $$

where $X,Y \in TC$ . A computation shows that for each $p = 1,2,3$ ,

(3.4) $$ \begin{align} \omega_p = r\,dr \wedge \alpha_p + r^2\Omega_p. \end{align} $$

Altogether, the data $(g_{\mathrm {C}}, (\omega _1, \omega _2, \omega _3), (I_1, I_2, I_3))$ are an almost hyper-Hermitian structure on C.

Conversely, if the metric cone $(C^{4n+4}, g_{\mathrm {C}} = dr^2 + r^2g_M)$ carries an almost hyper-Hermitian structure $(g_{\mathrm {C}}, (\omega _1, \omega _2, \omega _3), (I_1, I_2, I_3))$ that is conical in the sense of Definition A.9, namely that

$$ \begin{align*}\mathscr{L}_{r\partial_r}(\omega_p) = 2\omega_p, \ \ \ p = 1,2,3,\end{align*} $$

then its link $(M, g_M)$ inherits a compatible $(\mathrm {Sp}(n) \times 3)$ -structure $(g_M, (\alpha _p), (\mathsf {J}_p))$ via

$$ \begin{align*} \alpha_p & = (r\partial_r\,\lrcorner\,\omega_p)|_M, & \mathsf{J}_p & = \begin{cases} I_p, & \text{on }\mathrm{Ker}(\alpha_p), \\ 0, & \text{on } \mathbb{R} A_p.\end{cases} \end{align*} $$

This relationship leads to the following definition:

3.1.1 Distinguished forms on $3$ -Sasakian manifolds

For the remainder of this work, $M^{4n+3}$ will denote a $3$ -Sasakian $(4n+3)$ -manifold with $3$ -Sasakian structure $(g_M, (\alpha _1, \alpha _2, \alpha _3), (\mathsf {J}_1, \mathsf {J}_2, \mathsf {J}_3))$ . The induced conical hyperkähler structure on $C^{4n+4} = \mathbb {R}^+ \times M$ will be denoted $(g_{\mathrm {C}}, (\omega _1, \omega _2, \omega _3), (I_1, I_2, I_3))$ . In this section, we record some of the distinguished differential forms on M and compute their exterior derivatives.

To begin, we consider the contact $1$ -forms $\alpha _1, \alpha _2, \alpha _3 \in \Omega ^1(M)$ and the transverse Kähler forms $\Omega _1, \Omega _2, \Omega _3 \in \Omega ^2(M)$ defined by $\Omega _p(X,Y) = g_M(\mathsf {J}_pX, Y)$ . By (3.4), we may compute

$$ \begin{align*} 0 = d\omega_p = d(r \,dr \wedge \alpha_p) + d(r^2 \Omega_p) = r\,dr \wedge (-d\alpha_p + 2 \Omega_p) + r^2 d\Omega_p, \end{align*} $$

which implies that

(3.5) $$ \begin{align} d\alpha_p & = 2\Omega_p, & d\Omega_p & = 0. \end{align} $$

(The first equation in (3.5) shows that each $\alpha _p$ is indeed a contact form. That is, that $\alpha _p \wedge (d\alpha _p)^{2n+1}$ is nowhere zero.)

Next, we decompose the $2$ -forms $\Omega _1, \Omega _2, \Omega _3$ according to the splitting

$$ \begin{align*}\Lambda^2(T^*M) = \Lambda^2(\widetilde{\mathsf{V}}^*) \oplus (\widetilde{\mathsf{V}} \otimes \widetilde{\mathsf{H}}) \oplus \Lambda^2(\widetilde{\mathsf{H}}^*).\end{align*} $$

One can show that each $\Omega _p$ has no component in $\widetilde {\mathsf {V}}^* \otimes \widetilde {\mathsf {H}}^*$ and that the $\Lambda ^2(\widetilde {\mathsf {V}}^*)$ -component of $\Omega _1$ is $\alpha _2 \wedge \alpha _3$ . Letting $\kappa _1, \kappa _2, \kappa _3$ denote the $\Lambda ^2(\widetilde {\mathsf {H}}^*)$ -component of $\Omega _p$ , we arrive at the formulas

(3.6) $$ \begin{align} \Omega_1 & = \alpha_2 \wedge \alpha_3 + \kappa_1, & \Omega_2 & = \alpha_3 \wedge \alpha_1 + \kappa_2, & \Omega_3 & = \alpha_1 \wedge \alpha_2 + \kappa_3. \end{align} $$

Taking d of (3.6) and using (3.5) shows that

(3.7) $$ \begin{align} d\kappa_1 & = 2(\alpha_2 \wedge \Omega_3 - \alpha_3 \wedge \Omega_2) = 2(\alpha_2 \wedge \kappa_3 - \alpha_3 \wedge \kappa_2), \nonumber \\ d\kappa_2 & = 2(\alpha_3 \wedge \Omega_1 - \alpha_1 \wedge \Omega_3) = 2(\alpha_3 \wedge \kappa_1 - \alpha_1 \wedge \kappa_3), \\ d\kappa_3 & = 2(\alpha_1 \wedge \Omega_2 - \alpha_2 \wedge \Omega_1) = 2(\alpha_1 \wedge \kappa_2 - \alpha_2 \wedge \kappa_1). \nonumber \end{align} $$

Finally, recalling the transverse complex volume forms $\Psi _1, \Psi _2, \Psi _3 \in \Omega ^{2n+1}(M; \mathbb {C})$ of (3.2), we compute

(3.8) $$ \begin{align} d\Psi_1 & = \frac{2}{n!}(\Omega_2 + i\Omega_3)^{n+1}, & d\Psi_2 & = \frac{2}{n!}(\Omega_3 + i\Omega_1)^{n+1}, & d\Psi_3 & = \frac{2}{n!}(\Omega_1 + i\Omega_2)^{n+1}. \end{align} $$

To conclude this section, we summarize the relationships between various forms on the hyperkähler cone $C^{4n+4}$ and those on its $3$ -Sasakian link $M^{4n+3}$ .

Proposition 3.4 We have

(3.9) $$ \begin{align} \omega_1 & = r\,dr \wedge \alpha_1 + r^2\,\Omega_1,\end{align} $$

(3.10) $$ \begin{align} \frac{1}{2}\omega_1^2 & = r^3\,dr \wedge (\alpha_1 \wedge \Omega_1) + r^4\,\frac{1}{2}\Omega_1^2, \end{align} $$

(3.11) $$ \begin{align} \frac{1}{k!}\omega_1^k & = r^{2k-1}\,dr \wedge \frac{1}{(k-1)!}(\alpha_1 \wedge \Omega_1^{k-1}) + r^{2k}\,\frac{1}{k!}\Omega_1^k. \end{align} $$

Consequently,

(3.12) $$ \begin{align} \Upsilon_1 & = r^{2n+1}\,dr \wedge \Psi_1 + r^{2n+2}\,\frac{1}{(n+1)!} (\Omega_2 + i\Omega_3)^{n+1}, \\ \Theta_{I,4} & = r^3\,dr \wedge (\alpha_2 \wedge \Omega_2 - \alpha_3 \wedge \Omega_3) + r^4 \frac{1}{2}(\Omega_2^2 - \Omega_3^2), \nonumber \\ \Phi_1 & = r^3\,dr \wedge (-\alpha_1 \wedge \Omega_1 + \alpha_2 \wedge \Omega_2 + \alpha_3 \wedge \Omega_3) + r^4\, \frac{1}{2}(-\Omega_1^2 + \Omega_2^2 + \Omega_3^2), \nonumber \\ \Lambda & = r^3\,dr \wedge \frac{1}{3}(\alpha_1 \wedge \Omega_1 + \alpha_2 \wedge \Omega_2 + \alpha_3 \wedge \Omega_3) + r^4\, \frac{1}{6}(\Omega_1^2 + \Omega_2^2 + \Omega_3^2). \nonumber \end{align} $$

Proof Each of these follows from a straightforward calculation.

3.2 Submanifolds via the Sasaki–Einstein structure

By analogy with our discussion in Sections 2.1 and 2.2, we now consider the various classes of submanifolds of M. We begin with those defined in terms of a Sasaki–Einstein structure.

By Remark 2.13, we can apply Proposition A.1 to (3.11) with k replaced by $k+1$ . We deduce that for $p = 1,2,3$ , the $(2k+1)$ -forms

$$ \begin{align*}\frac{1}{k!}(\alpha_p \wedge \Omega_p^k) \in \Omega^{2k+1}(M)\end{align*} $$

are semi-calibrations. Their calibrated submanifolds are called $I_p$ -CR submanifolds. To be precise:

Next, from formula (3.12) together with Proposition A.1 and Remark 2.13, we observe that for $p = 1,2,3$ and a constant $e^{i \theta } \in S^1$ , the $(2n+1)$ -forms

$$ \begin{align*}\mathrm{Re}(e^{-i\theta}\Psi_p) \in \Omega^{2n+1}(M)\end{align*} $$

are semi-calibrations. Their calibrated submanifolds are called $\Psi _p$ -special Legendrian submanifolds of phase $e^{i \theta }$ . We observe:

3.3 Submanifolds via the $3$ -Sasakian structure

We now turn to those submanifolds of M whose definition requires more than the Sasaki–Einstein structure. Here, we will discuss the CR isotropic, special isotropic, and associative submanifolds.

3.3.2 Special isotropic submanifolds

Definition 3.10 The special isotropic forms on M are the real $(2k-1)$ -forms $\theta _{I,2k-1}, \theta _{J,2k-1}, \theta _{K,2k-1} \in \Omega ^{2k-1}(M)$ defined by

$$ \begin{align*} \theta_{I, 2k-1} & := (r \partial_r\,\lrcorner\,\Theta_{I,2k})|_M, & \theta_{J,2p-1} & := (r \partial_r\,\lrcorner\,\Theta_{J, 2k})|_M, & \theta_{K,2k-1} & := (r \partial_r\,\lrcorner\,\Theta_{K,2k})|_M. \end{align*} $$

In particular, for $2k-1 = 1, 3, 2n+1$ , these are

$$ \begin{align*} \theta_{I,1} & = \alpha_2, \\ \theta_{I,3} & = \alpha_2 \wedge \Omega_2 - \alpha_3 \wedge \Omega_3 = \alpha_2 \wedge \kappa_2 - \alpha_3 \wedge \kappa_3, \\ \theta_{I,2n+1} & = \text{Re}(\Psi_1). \end{align*} $$

By Remark 2.13, Proposition A.1, and Theorem A.6, the special isotropic forms $\theta _{I, 2k-1}, \theta _{J, 2k-1}, \theta _{K,2k-1}$ are semi-calibrations.

Proposition 3.11 Let $L^{2k-1} \subset M^{4n+3}$ be a $(2k-1)$ -dimensional submanifold, $1 \leq k \leq n+1$ . We say L is $\theta _{I,2k-1}$ -special isotropic if either of the following equivalent conditions holds:

1. (1) $\mathrm {C}(L) \subset C$ is $\Theta _{I,2k}$ -special isotropic.

2. (2) L is $\theta _{I,2k-1}$ -special isotropic.

Proof This follows from Remark 2.13 and Proposition A.1.

3.3.3 Associative $3$ -folds

The following definition is due to Bryant and Harvey [Reference Bryant and Harvey9].

Definition 3.12 The generalized associative $3$ -forms are the real $3$ -forms $\phi _1, \phi _2, \phi _3 \in \Omega ^3(M)$ defined by

$$ \begin{align*} \phi_1 & = -\alpha_1 \wedge \Omega_1 + \alpha_2 \wedge \Omega_2 + \alpha_3 \wedge \Omega_3, \\ \phi_2 & = \alpha_1 \wedge \Omega_1 - \alpha_2 \wedge \Omega_2 + \alpha_3 \wedge \Omega_3, \\ \phi_3 & = \alpha_1 \wedge \Omega_1 + \alpha_2 \wedge \Omega_2 - \alpha_3 \wedge \Omega_3. \end{align*} $$

Equivalently,

$$ \begin{align*} \phi_1 & = \alpha_1 \wedge \alpha_2 \wedge \alpha_3 - \alpha_1 \wedge \kappa_1 + \alpha_2 \wedge \kappa_2 + \alpha_3 \wedge \kappa_3, \\ \phi_2 & = \alpha_1 \wedge \alpha_2 \wedge \alpha_3 + \alpha_1 \wedge \kappa_1 - \alpha_2 \wedge \kappa_2 + \alpha_3 \wedge \kappa_3, \\ \phi_3 & = \alpha_1 \wedge \alpha_2 \wedge \alpha_3 + \alpha_1 \wedge \kappa_1 + \alpha_2 \wedge \kappa_2 - \alpha_3 \wedge \kappa_3, \end{align*} $$

where the $\kappa _j$ were defined in (3.6). A three-dimensional submanifold $L^3 \subset M^{4n+3}$ is $\phi _1$ -associative if it is calibrated by $\phi _1$ :

$$ \begin{align*}\left.\phi_1\right|_L = \mathrm{vol}_L.\end{align*} $$

The definitions of $\phi _2$ -associative and $\phi _3$ -associative are analogous.

Observing that

$$ \begin{align*}\Phi_1 = r^3\,dr \wedge \phi_1 + r^4\, \frac{1}{2}(-\Omega_1^2 + \Omega_2^2 + \Omega_3^2),\end{align*} $$

we obtain:

Proposition 3.13 Let $L^3 \subset M^{4n+3}$ be a three-dimensional submanifold. The following are equivalent:

1. (1) $\mathrm {C}(L) \subset C$ is $\Phi _1$ -Cayley.

2. (2) $L \subset M$ is $\phi _1$ -associative.

Proof This follows from Remark 2.13 and Proposition A.1.

Finally, we remark on the relationships between the above submanifolds. Let us recall that a manifold is called Sasaki–Einstein if its cone is Calabi–Yau and that a $7$ -manifold is called nearly parallel $\mathrm {G}_2$ if its cone is a $\mathrm {Spin}(7)$ -manifold. Suppose now that $(Y^7, (g, \alpha , \mathsf {J}, \Psi ))$ is a Sasaki–Einstein $7$ -manifold. It is well known that Y inherits a nearly parallel $\mathrm {G}_2$ -structure by the following formula:

$$ \begin{align*} \phi = \alpha \wedge \Omega - \text{Re}(\Psi). \end{align*} $$

The real $3$ -form $\phi \in \Omega ^3(M)$ is called the associative $3$ -form, and a three-dimensional submanifold $\Sigma ^3 \subset M$ satisfying $\phi |_\Sigma = \mathrm {vol}_\Sigma $ is called associative. The following fact is well known, although we prove a more general result in Proposition 3.15.

Proposition 3.14 Let $(Y^7, (g, \alpha , \mathsf {J}, \Psi ))$ be a Sasaki–Einstein $7$ -manifold, and equip Y with its induced nearly parallel $\mathrm {G}_2$ -structure $\phi $ . Let $L^3 \subset Y$ be a three-dimensional submanifold. Then:

1. (1) If L is CR, then L is associative.

2. (2) If L is special Legendrian of phase $e^{i\pi } = -1$ , then L is associative.

When the ambient space is $3$ -Sasakian, the above fact generalizes to higher dimensions in the following way.

Proposition 3.15 Let $M^{4n+3}$ be a $3$ -Sasakian $(4n+3)$ -manifold. Let $L^3 \subset M$ be a three-dimensional submanifold. Then:

1. (1) If L is $I_1$ -CR or $I_3$ -CR, then L is $\phi _2$ -associative.

2. (2) If L is $-\theta _{I,3}$ -special isotropic or $\theta _{K,3}$ -special isotropic, then L is $\phi _2$ -associative.

3. (3) If L is $I_1$ -CR isotropic, then L is simultaneously $I_1$ -CR, $-\theta _{J,3}$ -special isotropic, and $\theta _{K,3}$ -special isotropic, and hence is $\phi _2$ -associative.

Proof (a) If $L \subset M$ is $I_1$ -CR (resp. $I_3$ -CR), then Proposition 3.5 implies that its cone $\mathrm {C}(L) \subset C$ is $I_1$ -complex (resp. $I_3$ -complex). By Proposition 2.12(a), $\mathrm {C}(L)$ is $\Phi _2$ -Cayley, so by Proposition 3.13, L is $\phi _2$ -associative.

(b) If $L \subset M$ is $-\theta _{I,3}$ -special isotropic (resp. $\theta _{K,3}$ -special isotropic), then Proposition 3.11 implies that its cone $\mathrm {C}(L) \subset C$ is $-\Theta _{I,4}$ -special isotropic (resp. $\Theta _{K,4}$ -special isotropic). By Proposition 2.12(b), $\mathrm {C}(L)$ is $\Phi _2$ -Cayley, so by Proposition 3.13, L is $\phi _2$ -associative.

(c) If L is $I_1$ -CR isotropic, then Proposition 3.8 implies that $\mathrm {C}(L) \subset C$ is $I_1$ -complex isotropic, and the result follows from an argument analogous to those used in parts (a) and (b). Alternatively, if L is $I_1$ -CR isotropic, then by definition, L is $I_1$ -CR, $\alpha _2$ -isotropic, and $\alpha _3$ -isotropic. Recalling that

$$ \begin{align*} \theta_{J,3} & = \alpha_3 \wedge \Omega_3 - \alpha_1 \wedge \Omega_1 & \theta_{K,3} & = \alpha_1 \wedge \Omega_1 - \alpha_2 \wedge \Omega_2, \end{align*} $$

we observe that L is $-\theta _{J,3}$ - and $\theta _{K,3}$ -special isotropic.

Remark 3.16 Where associative $3$ -folds in $3$ -Sasakian manifolds $M^{4n+3}$ are concerned, the case $n = 1$ has received the most attention in light of the connection to $\mathrm {G}_2$ -geometry. Recently, several studies have considered the two one-parameter families of squashed associative $3$ -forms on $M^7$ given by

$$ \begin{align*} -\phi_{1,t}^- & = \alpha_1 \wedge \alpha_2 \wedge \alpha_3 + t^2(- \alpha_1 \wedge \kappa_1 + \alpha_2 \wedge \kappa_2 + \alpha_3 \wedge \kappa_3), \\ \phi_{1,t}^+ & = \alpha_1 \wedge \alpha_2 \wedge \alpha_3 - t^2(\alpha_1 \wedge \kappa_1 + \alpha_2 \wedge \kappa_2 + \alpha_3 \wedge \kappa_3). \end{align*} $$

See, for example, [Reference Ball and Madnick6], [Reference Kennon and Lotay23], or [Reference Lotay and Oliveira24].

3.3.4 Summary

The following table summarizes the relationships discussed above.

With the exception of $\alpha _1$ -Legendrian and $\alpha _1$ -isotropic submanifolds, all of the “link” submanifolds $L \subset M^{4n+3}$ that appear in the table are minimal (i.e., have zero mean curvature), because a calibrated cone is minimal, and the link of a minimal cone is minimal.

3.4 $3$ -Sasakian manifolds as circle bundles

From now on, $3$ -Sasakian $(4n+3)$ -manifolds M are assumed to be compact. Above, we viewed M as the link of a hyperkähler cone C. In this section, we adopt a different perspective, viewing M as the total space of a circle bundle. The starting point is the following result.

Thus, every compact $3$ -Sasakian $(4n+3)$ -manifold M has a natural $S^2$ -family of projections $p_v \colon M^{4n+3} \to Z^{4n+2}$ . For definiteness, we choose to work with $p_1 := p_{(1,0,0)} \colon M \to Z$ , with respect to which $\alpha _1 \in \Omega ^1(M)$ is a connection $1$ -form. On M, the choice of $p_1$ preferences the splitting $TM = \mathbb {R} A_1 \oplus \mathrm {Ker}(\alpha _1)$ . On the hyperkähler cone $C^{4n+4} = \mathrm {C}(M)$ , our choice distinguishes the Kähler structure $(g_{\mathrm {C}}, I_1, \omega _1)$ .

3.4.1 The $3$ -forms $\Gamma _1, \Gamma _2, \Gamma _3$ and $4$ -forms $\Xi _1, \Xi _2, \Xi _3$

We now introduce $\mathbb {C}$ -valued $3$ -forms $\Gamma _1, \Gamma _2, \Gamma _3 \in \Omega ^3(M; \mathbb {C})$ and $\mathbb {R}$ -valued $4$ -forms $\Xi _1, \Xi _2, \Xi _3 \in \Omega ^4(M)$ that will play a key role in understanding the structure on the twistor space Z. These forms do not appear to have been studied before. Recalling the $2$ -forms $\kappa _j$ defined in (3.6), we define

(3.13) $$ \begin{align} \Gamma_1 & = (\alpha_2 - i\alpha_3) \wedge (\kappa_2 + i\kappa_3), \nonumber \\ \Gamma_2 & = (\alpha_3 - i\alpha_1) \wedge (\kappa_3 + i\kappa_1), \\ \Gamma_3 & = (\alpha_1 - i\alpha_2) \wedge (\kappa_1 + i\kappa_2),\nonumber \end{align} $$

and

(3.14) $$ \begin{align} \Xi_1 & = \kappa_2^2 + \kappa_3^2, & \Xi_2 & = \kappa_3^2 + \kappa_1^2, & \Xi_3 & = \kappa_1^2 + \kappa_2^2. \end{align} $$

Note that the real and imaginary parts of $\Gamma _1$ are given by

(3.15) $$ \begin{align} \begin{aligned} \text{Re}(\Gamma_1) & = \alpha_2 \wedge \kappa_2 + \alpha_3 \wedge \kappa_3, \\ \text{Im}(\Gamma_1) & = \alpha_2 \wedge \kappa_3 - \alpha_3 \wedge \kappa_2. \end{aligned} \end{align} $$

Their exterior derivatives are given by:

Proposition 3.18 We have

$$ \begin{align*} d\,\mathrm{Re}(\Gamma_1) & = 2\Xi_1 - 4 \alpha_2 \wedge \alpha_3 \wedge \kappa_1, \\ d\,\mathrm{Im}(\Gamma_1) & = 0, \\ d\Xi_1 & = -4\kappa_1 \wedge \mathrm{Im}(\Gamma_1). \end{align*} $$

Proof This is a straightforward computation using the definitions (3.15) and (3.14) and the exterior derivative formulas (3.5) and (3.7).

Remark 3.19 We remark in passing that one can compute

$$ \begin{align*}\kappa_1^2 + \kappa_2^2 + \kappa_3^2 = \textstyle \frac{1}{2}\,d(\alpha_{123} + \alpha_1 \wedge \kappa_1 + \alpha_2 \wedge \kappa_2 + \alpha_3 \wedge \kappa_3),\end{align*} $$

showing that the natural $4$ -form $\kappa _1^2 + \Xi _1 = \kappa _1^2 + \kappa _2^2 + \kappa _3^2$ is exact.

To clarify the geometric meaning of $\Gamma _1 \in \Omega ^3(M; \mathbb {C})$ , we consider the $2$ -form

$$ \begin{align*}\widetilde{\Omega}_1 := 2\kappa_1 - \alpha_2 \wedge \alpha_3.\end{align*} $$

Using equations (3.5)–(3.7), (3.15), and Proposition 3.18, we derive the identities

$$ \begin{align*} d \widetilde{\Omega}_1 & = 3\,\mathrm{Im}(2\Gamma_1), \\ d\,\mathrm{Re}(2\Gamma_1) & = 2(2\Xi_1 - 4\alpha_2 \wedge \alpha_3 \wedge \kappa_1). \end{align*} $$

When $n = 1$ , in which case $\dim (Z) = 6$ and $\dim (M) = 7$ , there is a coincidence ${\kappa _1^2 = \kappa _2^2 = \kappa _3^2}$ , which implies $\Xi _1 = 2\kappa _1^2$ , and therefore

$$ \begin{align*} d \widetilde{\Omega}_1 & = 3\,\mathrm{Im}(2\Gamma_1), \\ d \,\mathrm{Re}(2\Gamma_1) & = 2\widetilde{\Omega}_1^2, \end{align*} $$

which is familiar from the geometry of nearly Kähler $6$ -manifolds [Reference Reyes Carrión and Salamon26]. So, when $n = 1$ , the forms $\widetilde {\Omega }_1 \in \Omega ^2(M)$ and $2\Gamma _1 \in \Omega ^3(M;\mathbb {C})$ are the pullbacks via $p_1 \colon M^7 \to Z^6$ of the nearly Kähler $2$ -form and complex volume form on Z, respectively.

In Sections 4.1 and 4.2, we will see that aspects of this picture persist in higher dimensions. That is, for any $n \geq 1$ , the $2$ -form $\widetilde {\Omega }_1$ is the pullback of the nearly Kähler $2$ -form, while $2\Gamma _1$ is the pullback of a natural $3$ -form that (together with other geometric data) defines an $\mathrm {Sp}(n)\mathrm {U}(1)$ -structure on Z. When $n = 1$ , the $\mathrm {Sp}(1)\mathrm {U}(1) \cong \mathrm {U}(2)$ -structure on Z induces the familiar $\mathrm {SU}(3)$ -structure, but when $n> 1$ the group $\mathrm {Sp}(n)\mathrm {U}(1)$ is not contained in $\mathrm {SU}(2n+1)$ .

3.4.3 Descent to Z

To conclude this section, we observe that certain differential forms defined on M descend to the twistor space Z via the map $p_1 \colon M \to Z$ . For this, we recall that a k-form $\phi \in \Omega ^k(M)$ is called $p_1$ -semibasic if $\iota _X\phi = 0$ for all $X \in \mathrm {Ker}( (p_1)_*)$ . Since the fibers of $p_1 \colon M \to Z$ are connected, it is a standard fact that a k-form $\phi \in \Omega ^k(M)$ descends to Z if and only if both $\phi $ and $d\phi $ are $p_1$ -semibasic.

Proposition 3.21 Consider the projection $p := p_1 \colon M \to Z$ .

1. (1) There exist $\mathbb {R}$ -valued differential $2$ -forms $\omega _{\mathsf {V}}$ , $\omega _{\mathsf {H}}$ , $\omega _{\mathrm {KE}}, \omega _{\mathrm {NK}} \in \Omega ^2(Z)$ satisfying

   $$ \begin{align*} \alpha_2 \wedge \alpha_3 & = p^*(\omega_{\mathsf{V}}), & \kappa_1 + \alpha_2 \wedge \alpha_3 = \Omega_1 & = p^*(\omega_{\mathrm{KE}}), \\ \kappa_1 & = p^*(\omega_{\mathsf{H}}), & 2\kappa_1 - \alpha_2 \wedge \alpha_3 = \widetilde{\Omega}_1 & = p^*(\omega_{\mathrm{NK}}). \end{align*} $$

2. (2) There exist a $\mathbb {C}$ -valued differential $3$ -form $\gamma \in \Omega ^3(Z; \mathbb {C})$ and an $\mathbb {R}$ -valued differential $4$ -form $\xi \in \Omega ^4(Z)$ satisfying

   $$ \begin{align*} \Gamma_1 & = p^*(\gamma), \\ \Xi_1 & = p^*(\xi). \end{align*} $$

Proof (a) By equations (3.5) and (3.7), we have

$$ \begin{align*} d(\alpha_2 \wedge \alpha_3) & = -2(\alpha_2 \wedge \kappa_3 - \alpha_3 \wedge \kappa_2), & d\kappa_1 & = 2(\alpha_2 \wedge \kappa_3 - \alpha_3 \wedge \kappa_2). \end{align*} $$

Therefore, both $\alpha _2 \wedge \alpha _3$ and $d(\alpha _2 \wedge \alpha _3)$ are $p_1$ -semibasic, and similarly for $\kappa _1$ and $d\kappa _1$ .

(b) By Proposition 3.18, we have

$$ \begin{align*} d\Gamma_1 & = 2(\kappa_2^2 + \kappa_3^2) - 4\alpha_2 \wedge \alpha_3 \wedge \kappa_1, & d\Xi_1 & = -4\kappa_1 \wedge (\alpha_2 \wedge \kappa_3 - \alpha_3 \wedge \kappa_2). \end{align*} $$

Therefore, both $\Gamma _1$ and $d\Gamma _1$ are $p_1$ -semibasic, and similarly for $\Xi _1$ and $d\Xi _1$ .

Remark 3.22 By contrast, one can check that the following forms on M do not descend via $p_1 \colon M \to Z$ to forms on Z:

$$ \begin{align*} & \kappa_2, \kappa_3, & & \Gamma_2, \Gamma_3, & & \alpha_1, \alpha_2, \alpha_3, & & \phi_1, \phi_2, \phi_3, \\ & \Omega_2, \Omega_3, & & \Xi_2, \Xi_3, & & \Psi_1, \Psi_2, \Psi_3. & & \end{align*} $$

Remark 3.23 One must be careful to distinguish the $3$ -form $\Gamma _1 = (\alpha _2 - i\alpha _3) \wedge (\kappa _2 + i\kappa _3)$ from the special isotropic $3$ -form

$$ \begin{align*} \textstyle (r\partial_r\,\lrcorner\,\frac{1}{2}\sigma_1^2)|_M & = (\alpha_2 + i\alpha_3) \wedge (\kappa_2 + i\kappa_3). \end{align*} $$

While $\Gamma _1$ descends to Z, the special isotropic $3$ -form $(r\partial _r\,\lrcorner \,\frac {1}{2}\sigma _1^2)|_M$ does not, because its exterior derivative has $\alpha _1$ terms. Note that for $n = 1$ , the object $(r\partial _r\,\lrcorner \,\frac {1}{2}\sigma _1^2)|_M = \Psi _1$ is a $3$ -form on $M^7$ whose real part calibrates special Legendrian $3$ -folds.

4.1 $\mathrm {Sp}(n)\mathrm {U}(1)$ -structures

Let $Y^{4n+2}$ be a smooth $(4n+2)$ -manifold with $n \geq 1$ .

Definition 4.1 A $(\mathrm {U}(2n) \times \mathrm {U}(1))$ -structure on $Y^{4n+2}$ is an almost-Hermitian structure $(g, J_+, \omega _+)$ together with a distribution of $J_+$ -invariant $4n$ -planes $\mathsf {H} \subset TY$ . Equivalently, it is an almost-Hermitian structure $(g, J_+, \omega _+)$ together with an orthogonal splitting

$$ \begin{align*}TY = \mathsf{H} \oplus \mathsf{V},\end{align*} $$

where $\mathsf {H} \subset TY$ and $\mathsf {V} \subset TY$ are $J_+$ -invariant subbundles with $\mathrm {rank}(\mathsf {H}) = 4n$ and $\mathrm {rank}(\mathsf {V}) = 2$ .

Given a $(\mathrm {U}(2n) \times \mathrm {U}(1))$ -structure $(g, J_+, \omega _+, \mathsf {H})$ , we split $(g, J_+, \omega _+)$ into horizontal and vertical parts as follows:

$$ \begin{align*} g & = g_{\mathsf{H}} + g_{\mathsf{V}}, & J_+ & = \left.J_+\right|_{\mathsf{H}} + \left.J_+\right|_{\mathsf{V}}, & \omega_+ & = \omega_{\mathsf{H}} + \omega_{\mathsf{V}}. \end{align*} $$

Further, we can extend it to a one-parameter family $(g(t), J_+, \omega _+(t), \mathsf {H})$ by defining

$$ \begin{align*} g(t) & = t^2g_{\mathsf{H}} + g_{\mathsf{V}}, & \omega_+(t) & = t^2\omega_{\mathsf{H}} + \omega_{\mathsf{V}}. \end{align*} $$

Moreover, by reversing the orientation of the vertical subbundle $\mathsf {V} \subset TY$ , we obtain a second one-parameter family $(g(t)$ , $J_-$ , $\omega _-(t)$ , $\mathsf {H})$ by defining

$$ \begin{align*} J_- & = \left.J_+\right|_{\mathsf{H}} - \left.J_+\right|_{\mathsf{V}}, & \omega_-(t) & = t^2\omega_{\mathsf{H}} - \omega_{\mathsf{V}}. \end{align*} $$

For calculations on Y, we will need local frames adapted to the geometry of the $(\mathrm {U}(2n) \times \mathrm {U}(1))$ -structure. To be precise:

Definition 4.2 A $(\mathrm {U}(2n) \times \mathrm {U}(1))$ -coframe at $y \in Y$ is a g-orthonormal coframe

$$ \begin{align*}(\rho, \mu) := (\rho_{10}, \rho_{11}, \rho_{12}, \rho_{13}, \ldots, \rho_{n0}, \rho_{n1}, \rho_{n2}, \rho_{n3}, \mu_2, \mu_3) \colon T_yY \to \mathbb{R}^{4n} \times \mathbb{R}^{2}\end{align*} $$

for which

$$ \begin{align*} \left.\omega_{\mathsf{V}}\right|_y & = \mu_2 \wedge \mu_3, & \left.\omega_{\mathsf{H}}\right|_y & = \sum_{j=1}^n (\rho_{j0} \wedge \rho_{j1} + \rho_{j2} \wedge \rho_{j3}). \end{align*} $$

For example, we will soon recall (Theorem 4.6) that every twistor space $Z^{4n+2}$ admits a natural $(\mathrm {U}(2n) \times \mathrm {U}(1))$ -structure. In fact, we will show (Theorem 4.7) that twistor spaces admit an additional piece of data:

Definition 4.3 Let $Y^{4n+2}$ be a $(4n+2)$ -manifold with a $(\mathrm {U}(2n) \times \mathrm {U}(1))$ -structure $(g, J_+, \omega _+, \mathsf {H})$ . A compatible $\mathrm {Sp}(n)\mathrm {U}(1)$ -structure is a complex $3$ -form $\gamma \in \Omega ^3(Y;\mathbb {C})$ with the following property: At each $y \in Y$ , there exists a $(\mathrm {U}(2n) \times \mathrm {U}(1))$ -coframe $(\rho , \mu )$ such that

$$ \begin{align*}\left.\gamma\right|_y = (\mu_2 - i \mu_3) \wedge \sum_{j=1}^n (\rho_{j0} + i \rho_{j1}) \wedge (\rho_{j2} + i\rho_{j3}).\end{align*} $$

Note that if $\gamma $ is a compatible $\mathrm {Sp}(n)\mathrm {U}(1)$ -structure, then $\gamma $ has $J_+$ -type $(2,1)$ and $J_-$ -type $(3,0)$ .

To justify this terminology, we make a digression into linear algebra. Consider the following $\mathrm {Sp}(n)\mathrm {U}(1)$ -representation on $\mathbb {R}^{4n+2}$ . For $(A,\lambda ) \in \mathrm {Sp}(n) \times \mathrm {U}(1)$ and $(h,z) \in \mathbb {H}^n \oplus \mathbb {C}$ , define

(4.1) $$ \begin{align} (A,\lambda) \cdot (h,z) := (Ah \lambda^{-1}, \,\lambda^{-2}z). \end{align} $$

Identify $\mathbb {H}^n \simeq \mathbb {C}^{2n}$ by writing $h = h_1 + jh_2$ with $h_1, h_2 \in \mathbb {C}^n$ . This identification endows $\mathbb {H}^n$ with the complex structure given by right multiplication by i, which in turn yields an embedding $\iota \colon \mathrm {Sp}(n) \to \mathrm {U}(2n)$ . In this way, the representation (4.1) induces an embedding

$$ \begin{align*} \mathrm{Sp}(n)\mathrm{U}(1) & \to \mathrm{U}(2n) \times \mathrm{U}(1) \\ (A,\lambda) & \mapsto (\iota(A)\lambda^{-1}, \, \lambda^{-2}). \end{align*} $$

The image of this map is

(4.2) $$ \begin{align} \{ (B, \nu) \in \mathrm{U}(2n) \times \mathrm{U}(1) \colon \nu^{-1/2}B \in \mathrm{Sp}(n)\}. \end{align} $$

Since $\mathrm {Sp}(n)$ contains the element $-\text {Id}$ , the condition $\nu ^{-1/2} B \in \mathrm {Sp}(n)$ does not depend on the choice of square root.

Let $(e_{10}, e_{11}, e_{12}, e_{13}, \ldots , e_{n0}, e_{n1}, e_{n2}, e_{n3}, f_2, f_3)$ denote the standard basis of $\mathbb {R}^{4n+2}$ , and let $(e^{10}, e^{11}, e^{12}, e^{13}, \ldots , e^{n0}, e^{n1}, e^{n2}, e^{n3}, f^2, f^3)$ denote its dual basis. We identify $\mathbb {R}^{4n+2} \simeq \mathbb {C}^{2n} \oplus \mathbb {C}$ via the complex structure $J_0$ whose Kähler form is

$$ \begin{align*}\omega_0 = f^2 \wedge f^3 + \sum (e^{j0} \wedge e^{j1} + e^{j2} \wedge e^{j3}).\end{align*} $$

Identifying $\mathbb {C}^{2n} \simeq \mathbb {H}^n$ , the standard hyperkähler triple on $\mathbb {H}^n$ is

(4.3) $$ \begin{align} \begin{aligned} \beta_1 & = \sum (e^{j0} \wedge e^{j1} + e^{j2} \wedge e^{j3}), & \beta_2 & = \sum (e^{j0} \wedge e^{j2} - e^{j1} \wedge e^{j3}), \\ \beta_3 & = \sum (e^{j0} \wedge e^{j3} + e^{j1} \wedge e^{j2}). \end{aligned} \end{align} $$

We consider the $3$ -form $\gamma _0 \in \Lambda ^3( (\mathbb {R}^{4n+2})^*)$ given by

$$ \begin{align*}\gamma_0 = (f^2 - if^3) \wedge (\beta_2 + i\beta_3).\end{align*} $$

Then:

Proof Let $(B, \nu ) \in \mathrm {U}(2n) \times \mathrm {U}(1)$ , and set $\tau = f^2 - if^3$ and $\beta = \beta _2 + i\beta _3$ . Since $\tau $ has $J_0$ -type $(0,1)$ and $\beta $ has $J_0$ -type $(2,0)$ , we have

$$ \begin{align*} \nu^*\tau & = \nu^{-1}\tau, & \nu^*\beta & = \nu^2\beta, \end{align*} $$

and hence

$$ \begin{align*} (B,\nu)^*\gamma_0 = (B, \nu)^*(\tau \wedge \beta) & = \nu^*\tau \wedge B^*\beta = \tau \wedge (\nu^{-1/2}B)^*\beta. \end{align*} $$

If $(B,\nu ) \in \mathrm {Sp}(n)\mathrm {U}(1)$ , then $\nu ^{-1/2}B \in \mathrm {Sp}(n)$ by (4.2). Thus, since $\mathrm {Sp}(n)$ stabilizes $\beta _1, \beta _2, \beta _3$ , we get

$$ \begin{align*}(B,\nu)^*\gamma_0 = \tau \wedge \beta = \gamma_0.\end{align*} $$

Conversely, if $(B,\nu ) \in \mathrm {U}(2n) \times \mathrm {U}(1)$ stabilizes $\gamma _0$ , then

$$ \begin{align*}\tau \wedge \beta = \tau \wedge (\nu^{-1/2}B)^*\beta.\end{align*} $$

Contracting both sides with the vector $f_2 + if_3$ implies that $\beta = (\nu ^{-1/2}B)^*\beta $ , so that $\nu ^{-1/2}B \in \mathrm {U}(2n)$ stabilizes $\beta $ . Since the $\mathrm {U}(2n)$ -stabilizer of $\beta $ is $\mathrm {Sp}(n)$ , we deduce that $\nu ^{-1/2}B \in \mathrm {Sp}(n)$ , and hence $(B,\nu ) \in \mathrm {Sp}(n)\mathrm {U}(1)$ .

Example 4.1 The case $n=1$ is particularly special. Let $Y^6$ be a $6$ -manifold with a $(\mathrm {U}(2) \times \mathrm {U}(1))$ -structure $(g, J_+, \omega _+, \mathsf {H})$ . By definition, a compatible $\mathrm {Sp}(1)\mathrm {U}(1)$ -structure is a complex $3$ -form $\gamma \in \Omega ^3(Y;\mathbb {C})$ such that at each $y \in Y$ , there exists a $(\mathrm {U}(2) \times \mathrm {U}(1))$ -coframe $(\rho _0, \rho _1, \rho _2, \rho _3, \mu _2, \mu _3) \colon T_yY \to \mathbb {R}^4 \times \mathbb {R}^2$ for which

$$ \begin{align*}\gamma|_y = (\mu_2 - i\mu_3) \wedge (\rho_0 + i\rho_1) \wedge (\rho_2 + i\rho_3).\end{align*} $$

So, $\gamma $ is a nonvanishing $3$ -form of $J_-$ -type $(3,0)$ satisfying

$$ \begin{align*}-\frac{i}{8}\gamma \wedge \overline{\gamma} = \mu_2 \wedge \mu_3 \wedge \rho_0 \wedge \rho_1 \wedge \rho_2 \wedge \rho_3.\end{align*} $$

As such, $\gamma \in \Omega ^3(Y;\mathbb {C})$ defines an $\mathrm {SU}(3)$ -structure on Y.

Alternatively, the presence of a compatible $\mathrm {SU}(3)$ -structure on $Y^6$ follows abstractly from the following group isomorphism of $\mathrm {Sp}(1)\mathrm {U}(1) \cong \mathrm {U}(2)$ onto a subgroup of $\mathrm {SU}(3)$ . Using $\mathrm {Sp}(1) \cong \mathrm {SU}(2)$ , we have

$$ \begin{align*} \mathrm{Sp}(1)\mathrm{U}(1) = \left\{ \begin{pmatrix} B & 0 \\ 0 & \nu \end{pmatrix} \colon \nu^{-1/2}B \in \mathrm{SU}(2) \right\} & \cong \left\{ \begin{pmatrix} T & 0 \\ 0 & (\det T)^{-1} \end{pmatrix} \colon T \in \mathrm{U}(2) \right\} \leq \mathrm{SU}(3), \\ \begin{pmatrix} B & 0 \\ 0 & \det B \end{pmatrix} & \mapsto \begin{pmatrix} B & 0 \\ 0 & (\det B)^{-1} \end{pmatrix}. \end{align*} $$

(This is a group homomorphism because $\mathrm {U}(1)$ is abelian.) The situation is described by the following diagram:

Remark 4.5 The notation in this remark is that made standard in the monograph of Salamon [Reference Salamon28]. Let $T = \mathsf {H} \oplus \mathsf {V} \simeq \mathbb {R}^{4n+2}$ denote the (real) $\mathrm {Sp}(n)\mathrm {U}(1)$ -representation of (4.1). Let $E \simeq \mathbb {C}^{2n}$ denote the standard complex $\mathrm {Sp}(n)$ -representation, and let ${L \simeq \mathbb {C}}$ denote the standard complex $\mathrm {U}(1)$ -representation. Then, by refining the splitting $\Lambda ^2(T^*) = \Lambda ^2(\mathsf {H}^*) \otimes (\mathsf {H}^* \otimes \mathsf {V}^*) \otimes \Lambda ^2(\mathsf {V}^*)$ , one can decompose the space of real $2$ -forms into $\mathrm {Sp}(n)\mathrm {U}(1)$ -irreducible representations as follows:

$$ \begin{align*} \Lambda^2(\mathsf{H}^*) & \cong \mathbb{R} \omega_{\mathsf{H}} \oplus [\mathrm{Sym}^2(E)] \oplus [\Lambda^2_0(E)] \oplus [\![ L^2 ]\!] \oplus [\![ \Lambda^2_0(E) \otimes L^2 ]\!] \\ \mathsf{H}^* \otimes \mathsf{V}^* & \cong [\![ E \otimes L^3 ]\!] \oplus [\![ E \otimes L ]\!] \\ \Lambda^2(\mathsf{V}^*) & \cong \mathbb{R} \omega_{\mathsf{V}}. \end{align*} $$

Alternatively, by refining the $J_+$ -type splitting $\Lambda ^2(T^*) = [\![\Lambda ^{2,0}]\!] \oplus [\Lambda ^{1,1}]$ , one obtains

$$ \begin{align*} [\![ \Lambda^{2,0} ]\!] & \cong [\![ \Lambda^2_0(E) \otimes L^2 ]\!] \oplus [\![ L^2 ]\!] \oplus [\![ E \otimes L^3 ]\!] \\ [\Lambda^{1,1}] & \cong \mathbb{R}\omega_{\mathsf{V}} \oplus \mathbb{R} \omega_{\mathsf{H}} \oplus [\mathrm{Sym}^2_0(E)] \oplus [\Lambda^2_0(E)] \oplus [\![ E \otimes L ]\!]. \end{align*} $$

4.2 The geometry of twistor spaces

We now return to the study of twistor spaces Z. The following fact is well known:

From now on, the twistor space Z will carry the $(\mathrm {U}(2n) \times \mathrm {U}(1))$ -structure $(g, J_+, \omega _+, \mathsf {H})$ described in the previous proposition. We will write

$$ \begin{align*} (g_{\mathrm{KE}}, J_{\mathrm{KE}}, \omega_{\mathrm{KE}}) & := \left(g(1),\, J_+,\, \omega_+(1)\right), \\ (g_{\mathrm{NK}}, J_{\mathrm{NK}}, \omega_{\mathrm{NK}}) & := (g(\sqrt{2}),\, J_-,\, \omega_-(\sqrt{2})). \end{align*} $$

In particular,

(4.4) $$ \begin{align} \omega_{\mathrm{KE}} & = \omega_{\mathsf{H}} + \omega_{\mathsf{V}}, & \omega_{\mathrm{NK}} & = 2\omega_{\mathsf{H}} - \omega_{\mathsf{V}}. \end{align} $$

We now recover the important observation of Alexandrov [Reference Alexandrov3] that Z naturally admits even more structure:

Proof By Proposition 3.21(b), there exists a unique $3$ -form $\gamma \in \Omega ^3(Z;\mathbb {C})$ satisfying

$$ \begin{align*}p^*(\gamma) = \Gamma_1 = (\alpha_2 - i\alpha_3) \wedge (\kappa_2 + i\kappa_3).\end{align*} $$

This $3$ -form is an $\mathrm {Sp}(n)\mathrm {U}(1)$ -structure.

In Section 5.1, we will give a second proof of Theorem 4.7 from the perspective of quaternionic-Kähler geometry. For now, using Proposition 3.21, we can compute the following exterior derivatives:

$$ \begin{align*} d\omega_{\mathsf{V}} & = -\mathrm{Im}(2\gamma), & d\omega_{\mathrm{KE}} & = 0, & d\,\mathrm{Re}(\gamma) & = 2\xi - 4\omega_{\mathsf{H}} \wedge \omega_{\mathsf{V}}, \\ d\omega_{\mathsf{H}} & = \mathrm{Im}(2\gamma), & d\omega_{\mathrm{NK}} & = 3\,\mathrm{Im}(2\gamma), & d\,\mathrm{Im}(\gamma) & = 0, \\ & & & & d\xi & = -4\omega_{\mathsf{H}} \wedge \mathrm{Im}(\gamma). \end{align*} $$

Example 4.2 When $n = 1$ , there is a coincidence $\xi = 2\omega _{\mathsf {H}}^2$ . Therefore, in this case, using that $\omega _{\mathrm {NK}}^2 = (2\omega _{\mathsf {H}} - \omega _{\mathsf {V}})^2 = 4\omega _{\mathsf {H}}^2 - 4 \omega _{\mathsf {H}} \wedge \omega _{\mathsf {V}} = 2\xi - 4\omega _{\mathsf {H}} \wedge \omega _{\mathsf {V}}$ , we recover the equations

$$ \begin{align*} d\omega_{\mathrm{NK}} & = 3\,\mathrm{Im}(2\gamma), \\ d\,\mathrm{Re}(2\gamma) & = 2\, \omega_{\mathrm{NK}} \wedge \omega_{\mathrm{NK}}, \end{align*} $$

familiar from the theory of nearly Kähler $6$ -manifolds.

4.3 $\mathrm {Re}(\gamma )$ -calibrated $3$ -folds

Let $Z^{4n+2}$ be a twistor space equipped with the $(\mathrm {U}(2n) \times \mathrm {U}(1))$ -structure $(g_{\mathrm {KE}}, J_{\mathrm {KE}}, \omega _{\mathrm {KE}}, \mathsf {H})$ . With respect to this structure, one can consider several classes of submanifolds of Z, such as:

• • $J_{\mathrm {KE}}$ -complex (resp. $J_{\mathrm {NK}}$ -complex) submanifolds.

• • Horizontal submanifolds (i.e. those tangent to $\mathsf {H}$ ).

• • $\omega _{\mathrm {KE}}$ -isotropic (resp. $\omega _{\mathrm {NK}}$ -isotropic) submanifolds.

These submanifolds have been the subject of numerous studies, particularly when $\dim (Z) = 6$ . However, since we have now shown that Z admits a compatible $\mathrm {Sp}(n)\mathrm {U}(1)$ -structure $\gamma \in \Omega ^3(Z; \mathbb {C})$ , twistor spaces also admit a distinguished class of $3$ -folds. In this section, we explore these.

We begin by showing that $\mathrm {Re}(\gamma ) \in \Omega ^3(Z)$ is a semi-calibration, for which we need a preliminary lemma.

Proof It suffices to work at a fixed point $z \in Z$ . Let $(\rho ,\mu )$ be an $\mathrm {Sp}(n)\mathrm {U}(1)$ -coframe at z as in Definition 4.3. We may then write $\gamma |_z = \tau \wedge (\beta _2 + i\beta _3)$ , where

$$ \begin{align*} \tau & = \mu_2 - i\mu_3, & \beta_2 & = \sum_{i=1}^n (\rho_{j0} \wedge \rho_{j2} - \rho_{j1} \wedge \rho_{j3}), & \beta_3 & = \sum_{i=1}^n (\rho_{j0} \wedge \rho_{j3} + \rho_{j1} \wedge \rho_{j2}). \end{align*} $$

Define complex structures $J_2$ and $J_3$ on $\mathsf {H}|_z$ by declaring

$$ \begin{align*} J_2(\rho_{j0}) & = \rho_{j2}, & J_3(\rho_{j0}) & = \rho_{j3}, \\ J_2(\rho_{j1}) & = -\rho_{j3}, & J_3(\rho_{j1}) & = \rho_{j2}, \end{align*} $$

which implies $J_+J_2 = J_3$ and $g(J_2 \cdot , \cdot ) = \beta _2$ and $g(J_3 \cdot , \cdot ) = \beta _3$ . Note that $\tau $ , $\beta _2, \beta _3$ , as well as $J_2, J_3$ , depend on the choice of $\mathrm {Sp}(n)\mathrm {U}(1)$ -frame.

Now, let $v \in \mathsf {H}$ be a horizontal unit vector. Let $w = J_2v$ , so that

$$ \begin{align*} \iota_v\gamma = \iota_v(\tau \wedge (\beta_2 + i\beta_3)) = \tau \wedge \iota_v(\beta_2 + i\beta_3) & = \tau \wedge (g(J_2v, \cdot) + ig(J_+J_2v, \cdot)) \\ & = \tau \wedge (w^\flat - iJ_+ w^\flat) \\ & = (\mu_2 - i \mu_3) \wedge (w^\flat - iJ_- w^\flat) \end{align*} $$

since $J_+ = J_-$ on horizontal vectors. This $2$ -form is decomposable and has $J_-$ -type $(2,0)$ . Moreover, $\{\mu _2, \mu _3, w^\flat , J_- w^\flat \}$ is an orthonormal set. Thus, this $2$ -form is a standard complex volume form, and hence its real part is a semi-calibration.

Proof If $E \in \mathrm {Gr}_3^+(T_zZ)$ is an oriented $3$ -plane at $z \in Z$ , then $\dim (E \cap \mathsf {H}) \geq 1$ , so there exists a unit vector $v \in E \cap \mathsf {H}$ , and we may orthogonally split $E = \mathbb {R} v \oplus E'$ . Then

$$ \begin{align*}(\mathrm{Re} (\gamma)) (E) = ( \iota_v \mathrm{Re}(\gamma))(E') \leq 1\end{align*} $$

by Lemma 4.8, so the comass of $\mathrm {Re} (\gamma )$ is at most $1$ . Now, let v be a horizontal unit vector and let $E'$ be an $\iota _v (\mathrm {Re} (\gamma ))$ -calibrated $2$ -plane, which exists by Lemma 4.8. Then $E = \mathbb {R} v \oplus E'$ is $\mathrm {Re}(\gamma )$ -calibrated, which shows that $\mathrm {Re}(\gamma )$ has comass equal to one. Further, we have seen that an oriented $3$ -plane E is $\mathrm {Re}(\gamma )$ -calibrated if and only if $E'$ is $\iota _v(\mathrm {Re}(\gamma ))$ -calibrated, which proves (a).

Part (b) follows from Remark 4.9. Finally, since $\gamma $ is of $J_-$ -type $(3,0)$ , part (c) follows from Proposition A.5.

Returning to the $3$ -Sasakian manifold $M^{4n+3}$ , we can now establish the following:

4.3.1 A normal form for $\mathrm {Re}(\gamma )$ -calibrated $3$ -planes

We now aim to establish a normal form for $\mathrm {Re}(\gamma )$ -calibrated $3$ -planes in Z. Since the subsequent discussion is a matter of linear algebra, we work in $\mathbb {R}^{4n+2} \simeq \mathbb {H}^n \oplus \mathbb {C}$ . As we have done previously, we let

$$ \begin{align*}(e_{10}, e_{11}, e_{12}, e_{13}, \ldots, e_{n0}, e_{n1}, e_{n2}, e_{n3}, f_2, f_3) \end{align*} $$

denote the standard basis of $\mathbb {R}^{4n+2}$ , let $(e^{10}, e^{11}, \ldots , f^2, f^3)$ denote its dual basis, let $\beta _1, \beta _2, \beta _3$ be the standard hyperkähler triple on $\mathbb {H}^n$ as in (4.3), and consider the $3$ -form $\gamma _0 \in \Lambda ^3((\mathbb {R}^{4n+2})^*)$ given by

$$ \begin{align*}\gamma_0 = (f^2 - if^3) \wedge (\beta_2 + i\beta_3).\end{align*} $$

Now, for $e^{i \theta } \in S^1$ , define the $2$ -plane

(4.5) $$ \begin{align} V_\theta & = \mathrm{span}\left( c_\theta(-f_2 - e_{13}) + s_\theta(- f_3 - e_{12}), s_\theta(-f_2 + e_{13}) + c_\theta (- f_3 + e_{12}) \right)\!. \end{align} $$

In particular, we highlight

(4.6) $$ \begin{align} V_{\frac{\pi}{4}} & = \mathrm{span}\left( f_2 + f_3 + e_{12} + e_{13}, f_2 + f_3 - e_{12} - e_{13} \right). \end{align} $$

Proposition 4.13 Consider the $\mathrm {Sp}(n)\mathrm {U}(1)$ -action on $\mathbb {H}^n \oplus \mathbb {C}$ given in (4.1). Let ${E \subset \mathbb {H}^n \oplus \mathbb {C}}$ be a $\mathrm {Re}(\gamma _0)$ -calibrated $3$ -plane. Then there exist $(A, \lambda ) \in \mathrm {Sp}(n)\mathrm {U}(1)$ and a unique $\theta \in [0, \frac {\pi }{4}]$ such that $(A, \lambda ) \cdot E = \mathbb {R} e_{10} \oplus V_\theta $ . Moreover, the following are equivalent:

1. (1) $\dim (E \cap \mathbb {H}^n) = 2$ .

2. (2) $E = (E \cap \mathbb {H}^n) \oplus (E \cap \mathbb {C})$ .

3. (3) E is $\omega _{\mathrm {KE}}$ -isotropic.

4. (4) $\theta = \frac {\pi }{4}$ .

Proof Let $E \subset \mathbb {H}^n \oplus \mathbb {C}$ be a $\mathrm {Re}(\gamma _0)$ -calibrated $3$ -plane. By Proposition 4.10, there exists a quaternionic line $L \subset \mathbb {H}^n$ for which $E \subset L \oplus \mathbb {C}$ . Since the subgroup $\mathrm {Sp}(n) \leq \mathrm {Sp}(n) \mathrm {U}(1)$ acts transitively on the quaternionic lines of $\mathbb {H}^n$ , there exists $A_0 \in \mathrm {Sp}(n)$ such that $A_0 \cdot L = L_0$ , where $L_0$ is the standard quaternionic line

$$ \begin{align*}L_0 = \mathrm{span}(e_{10}, e_{11}, e_{12}, e_{13}).\end{align*} $$

Thus, $(A_0, 1) \cdot E \subset L_0 \oplus \mathbb {C}$ , so we can without loss of generality suppose that ${E \subset L_0 \oplus \mathbb {C}}$ .

Now, $L_0 \oplus \mathbb {C}$ is a complex $3$ -plane, and the restriction of $\gamma _0$ to $L_0 \oplus \mathbb {C}$ is a complex volume form. Thus, the problem reduces to finding a normal form for special Lagrangian $3$ -planes in a complex $3$ -space with respect to the action of $\mathrm {Sp}(1)\mathrm {U}(1) \cong \mathrm {U}(2)$ . Such a normal form was established in [Reference Aslan5, Proposition 3.2]. (Translating between notations, the $b_1, i b_1, b_2, i b_2, b_3, i b_3$ of [Reference Aslan5] corresponds to our $e_{10}, e_{11}, e_{12}, e_{13}, f_2, - f_3$ .)

For $\theta \in [0, \frac {\pi }{4}]$ , write $W_\theta = \mathbb {R} e_{10} \oplus V_\theta $ . We observe that the conditions (a), (b), and (c) above are invariant under the action of $\mathrm {Sp}(n) \mathrm {U}(1)$ , so it is enough to verify that for $W_{\theta }$ they are equivalent to $\theta = \frac {\pi }{4}$ . If $\theta = \frac {\pi }{4}$ , we have

$$ \begin{align*}W_{\frac{\pi}{4}} = \mathrm{span}(e_{10},\, e_{12} + e_{13}, \, f_2 + f_3) = (W_{\frac{\pi}{4}} \cap \mathbb{H}^n) \oplus (W_{\frac{\pi}{4}} \cap \mathbb{C}),\end{align*} $$

so both (a) and (b) hold. If $\theta \neq \frac {\pi }{4}$ , then one can compute from (4.5) that $\dim (W_\theta \cap \mathbb {H}^n) = 1$ . Since a $\mathrm {Re}(\gamma _0)$ -calibrated $3$ -plane cannot contain any complex lines, we have $\dim (W_\theta \cap \mathbb {C}) < 2$ , and hence

$$ \begin{align*}\dim( (W_\theta \cap \mathbb{H}^n) \oplus (W_\theta \cap \mathbb{C})) = \dim(W_\theta \cap \mathbb{H}^n) + \dim(W_\theta \cap \mathbb{C}) < 3 = \dim(W_\theta),\end{align*} $$

so both (a) and (b) do not hold.

With respect to the above basis, we have $\omega _{\mathrm {KE}} = \beta _1 + f^2 \wedge f^3$ . Letting

$$ \begin{align*}v_2 = c_\theta(-f_2 - e_{13}) + s_\theta(- f_3 - e_{12}), \quad v_3 = s_\theta(-f_2 + e_{13}) + c_\theta (- f_3 + e_{12}),\end{align*} $$

so $V_\theta = \mathrm {span}(v_2, v_3)$ and $W_{\theta } = \mathbb {R} e_{10} \oplus V_{\theta }$ , a computation shows that

$$ \begin{align*}\omega_{\mathrm{KE}}(e_{10}, v_2) = \omega_{\mathrm{KE}}(e_{10}, v_3) = 0, \qquad \omega_{\mathrm{KE}}(v_2, v_3) = 2 (c_\theta^2 - s_\theta^2),\end{align*} $$

so $\left .\omega _{\mathrm {KE}}\right |_{W_{\theta }} = 0$ if and only if $\theta = \frac {\pi }{4}$ .