Geometry of SU(3) Manifolds By

Home , G2 manifold, Twistor space

Geometry of SU(3) Manifolds by

Feng Xu

Department of Mathematics Duke University

Date:

Approved:

Robert L. Bryant, Advisor

William Allard

Hubert Bray

Mark Stern

Dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2008 Abstract (Diﬀerential Geometry)

Geometry of SU(3) Manifolds

Feng Xu

Department of Mathematics Duke University

Date:

Approved:

Robert L. Bryant, Advisor

William Allard

Hubert Bray

Mark Stern

An abstract of a dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2008 Copyright c 2008 by Feng Xu All rights reserved Abstract

I study the differential geometry of 6-manifolds endowed with various SU(3) structures from three perspectives. The first is special Lagrangian geometry; The second is pseudo-Hermitian-Yang-Mills connections or, more generally, ω-anti-self dual instantons; The third is pseudo-holomorphic curves. For the first perspective, I am interested in the interplay between SU(3)-structures and their special Lagrangian submanifolds. More precisely, I study SU(3)-structures which locally support as ‘nice’ special Lagrangian geometry as Calabi-Yau 3-folds do. Roughly speaking, this means that there should be a local special Lagrangian submanifold tangent to any special Lagrangian 3-plane. I call these SU(3)-structures admissible. By employing Cartan-Kählermachinery, I show that locally such admissible SU(3)-structures are abundant and much more general than local Calabi-Yau structures. However, the moduli space of the compact special Lagrangian submanifolds is not so well-behaved in an admissible SU(3)-manifold as in the Calabi-Yau case. For this reason, I narrow attention to nearly Calabi-Yau manifolds, for which the special Lagrangian moduli space is smooth. I compute the local generality of nearly Calabi-Yau structures and find that they are still much more general than Calabi-Yau structures. I also discuss the relationship between nearly Calabi-Yau and half-flat SU(3)-structures. To construct complete or compact admissible examples, I study the twistor spaces of Riemannian 4-manifolds. It turns out that twistor spaces over self-dual Einstein 4-manifolds provide admissible and nearly Calabi-Yau manifolds. I also construct some explicit special Lagrangian examples in nearly Kähler CP3 and the twistor space of H4. For the second perspective, we are mainly interested in pseudo-Hermitian-Yang-

iv Mills connections on nearly Kählersix manifolds. Pseudo-Hermitian-Yang-Mills connections were introduced by R. Bryant in [4] to generalize Hermitian-Yang-Mills concept in Kähler geometry to almost complex geometry. If the SU(3)-structure is nearly Kähler,I show that pseudo-Hermitian-Yang-Mills connections (or, more generally, ω-anti-self-dual instantons) enjoy many nice properties. For example, they satisfy the Yang-Mills equation and thus removable singularity results hold for such connections. Moreover, they are critical points of a Chern-Simons functional. I derive a Weitzenböck formula for the deformation and discuss some of its application. I construct some explicit examples that display interesting singularities. For the third perspective, I study pseudo-holomorphic curves in nearly Kähler CP3. I construct a one-to-one correspondence between null torsion curves in the

3 3 nearly K¨ahler CP and contact curves in the K¨ahler CP (considered as a complex contact manifold). From this, I derive a Weierstrass formula for all null torsion curves by employing a result of R. Bryant in [9]. In this way, I classify all pseudo- holomorphic curves of genus 0.

v Contents

Abstract iv

Acknowledgements viii

Introduction1

0.1 SU(3)-structure on vector spaces...... 1

0.2 SU(3)-structures on 6-manifolds...... 3

0.2.1 G-structure...... 3

0.2.2 SU(3)-structures...... 5

1 Special Lagrangian and SU(3)-Structures9

1.1 Special Lagrangian geometry and admissible SU(3) structures....9

1.1.1 Admissible SU(3)-structures...... 10

1.1.2 First examples...... 12

1.1.3 Generalities...... 14

1.2 Examples from twistor spaces of Riemannian four-manifolds..... 22

1.2.1 Four dimensional Riemannian geometry...... 23

1.2.2 Twistor spaces of self-dual Einstein 4-manifolds...... 26

1.3 Complete special Lagrangian examples...... 27

1.3.1 An example in J (S4) = CP3 ...... 28

1.3.2 An example in J (H4)...... 32

1.4 Compact special Lagrangian submanifolds in nearly Calabi-Yau manifolds...... 36

1.4.1 Deformations of compact special Lagrangian 3-folds...... 37

1.4.2 Obstructions to the existence of compact SL 3-folds...... 39

vi 2 Instantons on Nearly K¨ahler6-Manifolds 42

2.1 Some linear algebra in 6 and 7 dimensions...... 44

2.1.1 Dimension 6...... 45

2.1.2 Dimension 7...... 57

2.2 Anti-self-dual instantons on nearly K¨ahler6-manifolds and G2-cones. 58 2.2.1 Nearly K¨ahler6-manifolds...... 59

2.2.2 G2-cones over a nearly K¨ahler6-manifold...... 61 2.2.3 ω anti-self-dual instantons...... 62

2.2.4 Ω-anti-self-dual instantons on the G2-cone...... 69 2.3 A Weitzenb¨ock formula...... 71

2.3.1 The general formula...... 71

2.3.2 Deformation of ω-anti-self-dual instantons...... 74

2.4 SO(4)-invariant examples...... 81

2.4.1 A dense open subset U of S6 ...... 82

2.4.2 Bundle constructions and SO(4)-invariant connections.... 83

2.4.3 SO(4)-invariant instantons...... 88

3 Pseudo-Holomorphic Curves in Nearly K¨ahlerCP3 93

3.1 Structure equations, projective spaces, and the ﬂag manifold..... 93

3.2 Pseudo-holomorphic curves in CP3 ...... 98

4 Summary 106

Bibliography 107

Biography 110

vii Acknowledgements

It is a great pleasure to thank my advisor, Robert Bryant. for spending numer- ous hours sharing his knowledge of Differential Geometry with me. Without his encouragements, this thesis would never have been. I thank Mark Stern, Chad Schoen, Les Saper and many other professors at Duke who taught me various mathematical courses. I have gained useful knowledge from these courses toward this thesis. I enjoyed many conversations with Shu Dai. I thank him for this and for his help with mailing my stuffs when I was away from Duke. I thank my mathematical brothers, Abe Smith and Conrad Hengebach for many interesting discussions both on mathematics and on life. I am grateful for the hospitality of MSRI and Berkeley where I have worked during the academic year 2007-2008. Berkeley is also a great place to know many new friends, Jia Yu and Xinwen Zhu, Liyan Lin and Guoliang Wu, Jing Xiong and Haijian Shi and above all, my girl friend, Yun Long. It is never forgettable that so many friends back at Duke have made Durham life fascinating. I enjoyed many nice dinners and poker games with Shuyi Wanng, Junjie Zhang, Huiyan Sang, Ping Zhou and Jianwei Li. I also thank Chenghui Cai and Tingting Chen for help and encouragements to overcome much difficulty in life. I thank Yun Long for her love and encouragement especially when I was so frus- trated in job hunting. She has made Berkeley a home for me. Finally, this thesis would not be possible were it not for the love and encouragement provided by my parents, Shusheng Xu and Kai Zhang, and my brother Jian Xu.

viii Introduction

0.1 SU(3)-structure on vector spaces

Let V be a 6-dimensional vector space. Let (Ω, φ) ∈ Λ2V ∗ ⊕ Λ3V ∗ be a pair of forms with Ω nondegenerate. Clearly the full linear group GL(V ) acts on such pairs by pullback. It is interesting to ask what the orbits and stabilizer groups are like. This problem was solved first by Banos in [3]. Later on, Bryant gave a simplified proof in [4]. There are two ways to this problem in the literature. One is to normalize Ω first and then consider the orbits of φ under the symplectic group Sp(Ω). Note that under

3 ∗ ∗ 3 ∗ the action of Sp(Ω), Λ V decomposes into two irreducible pieces: Ω ∧ V and Λ0V of eﬀective forms annihilated by wedge product by Ω. Since Sp(V ) acts transitively on V ∗ \{0},Ω ∧ V ∗ has only two obvious orbits. The diﬃcult part is the orbits in

3 ∗ Λ0V . This approach was adopted in [3]. The other is just the opposite, namely to classify φ orbits under GL(V ) and then to normalize Ω under diﬀerent stabilizer groups of φ. The second turns out to be simpler and was adopted in [4]. We cite the result as follows

Proposition 0.1.1 ([3],[4]). Suppose (ω, Ψ) ∈ Λ2V ∗ ⊕ Λ3V ∗. Assume ω is nondegenerate and Ψ is primitive (i.e., ω ∧ Ψ = 0). Then under the action of GL(V ), the pair (ω, φ) can be normalized to one of the following with the corresponding stabilizer group G(Ω, φ):

1 Ω φ G(Ω, φ)

e12 + e25 + e36 µ(e123 + e456) SL(3, R)

e14 + e25 + e36 µ(e123 − e156 − e246 − e345) SU(3)

e14 + e25 + e36 µ(e123 − e156 + e246 + e345) SU(1, 2)

e14 + e25 + e36 e156 + e264 + e345 R5 × SO(3)

e14 + e25 + e36 e156 − e264 − e345 R5 × SO(1, 2)

e14 + e25 + e36 e123 ± e345

e14 + e25 + e36 e123

e14 + e25 + e36 0 where µ > 0 and we have denoted e12 = e1 ∧ e2 and e123 = e1 ∧ e2 ∧ e3, etc.

In the list of Proposition 0.1.1, the only compact stabilizer group is SU(3), the case we are mainly interested in. We assume µ = 1. By introducing a complex basis

ω1 = e1 + ie4, ω2 = e2 + ie5, ω3 = e3 + ie6 we rewrite i Ω = (ω1 ∧ ω1 + ω2 ∧ ω2 + ω3 ∧ ω3) 2 and φ = ReΨ

with Ψ = ω1 ∧ ω2 ∧ ω3.

Thus, by specifying Ω and φ (or Ψ), we ﬁx an SU(3) structure on V such that ωi are special unitary complex basis. It is easy to verify the normalization condition

1 i ω3 = Ψ ∧ Ψ. 8 6

2 0.2 SU(3)-structures on 6-manifolds

We brieﬂy review G-structures and then focus on SU(3)-structures. A good reference for general theory of G-structures is [5].

0.2.1 G-structure

Let V = Rn be the n-dimensional vector space of column vectors. The linear transformation group GL(V ) may be considered as invertible n × n matrices acting by multiplication on vectors. Suppose that M is an n-dimensional smooth manifold. Let x : F → M be the total coframe bundle of M. Explicitly,

' F = {(x, u): u : TxM −→ V }.

It is a principal GL(V )-bundle over M with the right action

(x, u) ◦ g = (x, g−1u) for g ∈ GL(V ). On F, there is a tautological 1-form ω deﬁned by

ω(x,u)(V ) = u(x∗(V ))

for any V ∈ T(x,u)F. Clearly, ω vanishes on all vertical vectors, so the n components of ω form a basis for the semi-basic 1-forms. It also satisﬁes the GL(V )-equivariance property, g∗ω = g−1ω for all g ∈ GL(V ). Moreover, it has an interesting reproducing property: For any local section s of F, s∗ω = s. This follows from the deﬁnition. Let G be a Lie subgroup of GL(V ).

Deﬁnition 0.2.1. A G-structure on M is a reduction of the total coframe bundle F to a principal G-subbundle F.

3 We still denote by ω the tautological 1-form restricted to F. Then ω is G- equivariant. Pick a connection θ on F (by the general theory of principal bundles, one always exists). From the equivariance of ω, we have the ﬁrst structure equation

1 dω = −θ ∧ ω + T (ω ∧ ω) (1) 2

where ω ∧ ω is a Λ2V -valued 2-form and T is Λ2V ∗ ⊗ V -valued. Due to the G- equivariance of both ω and θ, T is also G-equivariant. In other words, T deﬁnes

2 ∗ a section of the vector bundle F ×G (Λ V ⊗ V ). We usually call T the apparent torsion. Now any other connection θˆ on F diﬀers from θ by a G-equivariant semibasic

∗ ˆ g = Lie(G)-valued 1-form, i.e., a section a of F ×G (V × g). The tensor T in (1) with θ replaced by θˆ changes from T to

Tˆ(ω ∧ ω) = T (ω ∧ ω) + 2a ∧ ω. (2)

The discussion so far is illustrated by the following sequence of G-modules

0 → g(1) → g ⊗ V ∗ −→δ V ⊗ Λ2V ∗ → H(0,2)(g) → 0. (3)

Here, g is regarded as a subspace of gl(V ) = V ⊗ V ∗. The map δ skew-symmetrizes the two V ∗ factors in g ⊗ V ∗ ⊂ V ⊗ V ∗ ⊗ V ∗. The kernel g(1) of δ is called the ﬁrst prolongation of g and the cokernel is denoted by H(0,2)(g). By associating the various spaces in the sequence with the principal bundle F, we get an exact sequence

∗ ˆ of vector bundles. We see that a is a section of F ×G (g ⊗ V ) and T , T are sections

2 ∗ of F ×G (Λ V ⊗ V ). The equation (2) says that a diﬀerent choice of connection

∗ only changes T by the δ image of a ∈ F ⊗G (g ⊗ V ). Thus, its equivalence class

0,2 [T ] ∈ F ×G H (g) is independent of the connection chosen. The tensor [T ] is called the intrinsic torsion of the G-structure F. For this reason, H(0,2)(g) is sometimes

4 called the essential torsion space. The first prolongation g(1) measures the freedom we have in choosing the connections θ once we fix a representative T of the torsion [T ]. If all finite-dimensional representations of G are completely reducible, we can identify H(0,2)(g) as a (not necessarily unique) subspace of V ⊗ Λ2V ∗. Then we can always modify θ so that the apparent torsion T falls in H(0,2)(g). This is called absorbing non-essential torsion. If, further, g(1) = 0, e.g., when G is a subgroup of O(g) for some nondegenerate symmetric bi-linear form g, then the connection θ will be unique once we require the torsion is fixed inside V ⊗ Λ2V ∗. The connection is canonical in the sense that it is preserved by diffeomorphisms.

Example 0.2.2. Let g be a Riemannian metric on M and let F be the orthogonal coframe bundle of g. Then F is an O(n)-structure on M. In the sequence (3), when G is replaced by O(n), both g(1) and H(0,2)(g) vanish. Thus the sequence degenerates

∗ ' 2 ∗ to δo : o(n) ⊗ V −→ V ⊗ Λ V . It follows that there exists on F a unique connection θ such that T = 0. The connection is usually called the Levi-Civita connection.

0.2.2 SU(3)-structures

From now on, let us assume n = 6 and G = SU(3). An SU(3) structure on a 6- manifold M is equivalent to specifying a pair of forms (Ω, φ) ∈ Λ2T ∗M ⊕ Λ3T ∗M

i 6 such that, at each point x ∈ M, there exists a basis of 1-forms {e }i=1 so that

14 25 36 123 156 246 345 Ωx = e + e + e , φx = e − e − e − e .

To see this, suppose ﬁrst π : F → M is an SU(3)-structure. For any (x, u) ∈ F, let

∗ ∗ Ωx = u (Ω0), φx = u (φ0)

5 √ −1 with Ω0 = 2 (dz1 ∧ dz1 + dz2 ∧ dz2 + dz3 ∧ dz3) and φ0 = Re(dz1 ∧ dz2 ∧ dz3) are 6 3 −1 forms on R ' C . Since (x, u) ◦ g = (x, g u) for all g ∈ SU(3), Ωx and φx are independent of u. Conversely, suppose such Ω and φ exist. Let

∗ ∗ F = {(x, u) ∈ F : u Ω0 = Ωx, u φ0 = φx}.

By Proposition 0.1.1, F is an SU(3) principal bundle, i.e., an SU(3)-structure on M. Specially adapted for SU(3) structures, we will use complex tautological 1-forms. Deﬁne

∗ ωi|(x,u) = u (dzi), i = 1, 2, 3

3 where dzi is the standard i-th complex coordinate on C . Then it follows from the above discussion that √ −1 π∗Ω = (ω ∧ ω + ω ∧ ω + ω ∧ ω ), π∗φ = Re(ω ∧ ω ∧ ω ). 2 1 1 2 2 3 3 1 2 3

Zeroth order invariants: complex volume form, metric, and almost complex structure

The existence of an SU(3)-structure on M usually requires topological conditions, e.g., certain characteristic classes vanish. We do not pursue this but are more interested in the geometric consequences of an SU(3)-structure. We first discuss invariants of zeroth order, i.e., without differentiating the defining forms Ω and φ. Besides Ω and φ themselves, another obvious one is the complex 3-form Ψ which,

when pulled back to F by π, has the form ω1 ∧ ω2 ∧ ω3. Since SU(3) is compact, an SU(3) structure determines a metric g on M, with the property

∗ π g = ω1 ◦ ω1 + ω2 ◦ ω2 + ω3 ◦ ω3.

Since SU(3) is a subgroup of GL(3, C), an SU(3)-structure determines an almost

−1 complex structure J on M. At each point x, J = u ◦ J0 ◦ u : TxM → TxM where

3 J0 is the standard complex structure on C . It is clearly independent of the coframe 6 (x, u) ∈ Fx chosen. With respect to J, Ψ is of type (3, 0). We call it the complex volume form.

First order invariants: connection and torsion

Zeroth order invariants do not distinguish SU(3) structures. To go further, we need study the ﬁrst structure equations. First let us take a closer look at the sequence (3). As mentioned before, su(3)(1) = 0. Moreover,

0 → su(3) ⊗ V ∗ −→δ V ⊗ Λ2V ∗ → H(0,2)(su(3)) → 0 ¯ k ↑ δo δo 0 → su(3) ⊗ V ∗ → so(6) ⊗ V ∗ → so(6)/su(3) ⊗ V ∗ → 0

¯ is a commutative diagram where δo is introduced in Example 0.2.2 and δo is induced ¯ from it. Since δo is an SU(3) isomorphism, so is δo. The second exact sequence has an obvious splitting by identifying so(6)/su(3) as the orthogonal complement su(3)⊥ in so(6) with respect to the Killing metric. In this way, we obtain a splitting of the

(0,2) ⊥ ∗ ﬁrst exact sequence by identifying H (su(3)) as δo-image of su(3) ⊗ V . This splitting uniquely determines an su(3) connection θ on F by requiring its apparent

⊥ ∗ torsion lie entirely in δo(su(3) ⊗ V ).

One way to think about θ is to relate it to the Levi-Civita connection θ0 on the orthogonal coframe bundle F · O(n). Restricting θ0 to F,

θ0 = θ ⊕ (−τ) according to so(6) = su(3) ⊕ su(3)⊥. The su(3)⊥ component −τ is clearly semibasic

⊥ ∗ and hence takes values in su(3) ⊗ V . Since the Levi-Civita connection θ0 is torsion free, the apparent torsion of θ is δo(τ). For later use, we need to write the components of torsion more explicitly. First for complexiﬁed tautological 1-form ω, the ﬁrst structure equation reads

ω ω d = −θ ∧ , ω 0 ω 7 √ ! θ + −1 µ β 3 √ θ0 = −1 β θ − 3 µ

where θ + θt = tr(θ) = 0, µ is real and β is complex and skew-symmetric. On F, µ and β are semibasic and hence are linear combinations of ω and ω. Based on this, we expand the structure equations out and rearrange terms to get

√ 1 1 −1 dω = −θ ∧ ω + S ω ∧ ω + N ω ∧ ω + (λ¯ω + λ¯ω ) ∧ ω . (4) i ij j 2 ij jkl k l 2 ij jkl k l 3 k k k k i

The quantities S, N and λ change tensorially along the fiber and thus are well-defined tensors over M. These are called the torsion of the SU(3)-structure F. For example, N is just the famous Nijenhuis tensor and it vanishes if and only if the almost structure J implied by the SU(3)-structure is integrable. If all torsion vanishes, the connection θ coincides with the Levi-Civita connection θ0. Thus the holonomy of the Riemannian metric is contained in SU(3). In other words, the SU(3)-structure is Calabi-Yau. These tensors also show up in the covariant differentiation of the defining forms Ω and φ. In fact, one can show that another characterization of Calabi-Yau is dΩ = dΨ = 0. For our later use, we compute the differential of Ω and ψ = Im(Ψ):

√ √ −1 −1 π∗(dΩ) = (N Ψ − N Ψ) + (S ω ∧ ω ∧ ω − S ω ∧ ω ∧ ω ); (5) 2 ii ii 4 il ljk i j k il ljk i j k

and √ −1 1 π∗dΨ = − (N − N )ω ∧ ω ∧ ω ∧ ω + (λ ω ∧ Ψ + λ ω ∧ Ψ). (6) 8 ijk lpq il li p q j k 2 l l l l

8 1

Special Lagrangian and SU(3)-Structures

1.1 Special Lagrangian geometry and admissible SU(3) structures

A submanifold L3 ⊂ M is called special Lagrangian if it is φ = ReΨ calibrated, i.e., if L∗(φ) is the volume form. The form φ is a calibration if it is closed. In this case L is minimal. Assume L is orientable. Then L is special Lagrangian with one of its orientations if and only if Ω|L = ψ|L = 0 where ψ = ImΨ. In other words it is an integral manifold of the diﬀerential ideal I generated algebraically by Ω and ψ. A generic SU(3)-structure will not admit any special Lagrangian submanifolds at all, even locally. For example, if

dΩ ≡ aφ mod (Ω, ψ) for some non-vanishing function a, then any special Lagrangian submanifold has to annihilate dΩ and hence φ. Thus no special Lagrangian submanifold exists. We are interested to know which SU(3)-structures support as many local special Lagrangian submanifolds as the ﬂat C3 does. If the SU(3)-structure is real analytic, so is the ideal I. Now if dΩ ∈ I, then we may invoke the Cartan-K¨ahlertheorem to show

9 that I is involutive and has the same Cartan characters as the ideal in C3. Here we need not care about whether or not d(ImΨ) is in I because d(ImΨ) is of degree 3 + 1 and hence vanishes automatically on any 3-dimensional submanifold.

1.1.1 Admissible SU(3)-structures

We make the following deﬁnition.

Deﬁnition 1.1.1. (Admissible SU(3)-structures) An SU(3)-structure (Ω, Ψ) on M is called admissible if there exist a 1-form θ and a real function a such that

dΩ = θ ∧ Ω + aψ. (1.1)

Suppose condition (1.1) is satisﬁed with

θ = uiωi + uiωi.

Then it also holds that √ √ −1 −1 dΩ = (u δ − u δ )ω ∧ ω ∧ ω − (u δ − u δ )ω ∧ ω ∧ ω + aψ. 4 i jk j ik i j k 4 i kj j ki i j k

Comparing with (5), we get

ljkSil = ujδik − ukδij

and a = Nii. We summarize the discussion so far as follows:

Lemma 1.1.2. Suppose (M, Ω, Ψ) is analytic. The ideal I is involutive and every Ω-Lagrangian analytic 2-submanifold in M 6 can be thickened uniquely to a special Lagrangian submanifold if there exists a (necessarily unique) connection α so that

1 dω = −α ∧ ω + β ∧ ω + N ω ∧ ω (1.2) i ij j i 2 ij jkl k l

where β is a complex 1-form and the trace of the Nijenhuis tensor, Nii, is real.

10 Remark 1.1.3. It is easy to see that the condition (1.2) in Lemma 1.1.2 essentially gives an equivalent deﬁnition of admissible SU(3)-structures. For this reason, we will also call an SU(3)-structure satisfying (1.2)admissible.

Remark 1.1.4. The same result for C3 was shown by Harvey and Lawson in [21]. Lemma 1.1.2 says an admissible SU(3)-manifold supports as nice a local special Lagrangian geometry as C3 does. This is the best situation one can hope.

Out of these we pick a class of special interest and call it nearly Calabi-Yau.

Deﬁnition 1.1.5 (Nearly Calabi-Yau). An SU(3)-structure (M 6, Ω, Ψ) is called nearly Calabi-Yau if for some real function U,

dΩ = d(eU ImΨ) = 0. √ From (6), nearly Calabi-Yau condition amounts to Nij −Nji = 0 and −1λ¯l = U¯l

∗ where we have denoted π (dU) = U¯lωl + U¯lωl. In terms of structure equations we have

Proposition 1.1.6. An SU(3)-structure is nearly Calabi-Yau for a real funciton U if and only if there exists a (necessarily unique) connection α so that

1 1 dω = −α ∧ ω + N ω ∧ ω + (U¯ω − U¯ω ) ∧ ω , (1.3) i ij j 2 ij jkl k l 3 k k k k i

where dU = Uk¯ωk + Uk¯ωk, Nij − Nji = 0 and tr(N) = Nii = 0.

In other words, the Nijenhuis tensor is Hermitian symmetric and trace free.

Remark 1.1.7. This deﬁnition is motivated by two reasons. First, as we will see shortly, it generalizes the concept of almost Calabi-Yau in that the underlying almost complex structure may not be integrable. Second, the moduli space of special Lagrangian submanifolds of a nearly Calabi-Yau manifold is well-behaved. We will prove this in §1.4.

11 Remark 1.1.8 (The case U = 0). When the defining function vanishes identically, a nearly Calabi-Yau structure is also half-flat. A half-flat SU(3)-structure is defined so that Ω ∧ dΩ = dImΨ = 0. This is in general not admissible. However, it has been used by string physicists to study heterotic string compactifications for a long time. In this context, half-flat manifolds are as important as Calabi-Yau structures. Mathematically, it has been studied by S. Chiossi and S. Salamon in its relation with

G2-structures. They showed that this structure behaves well under Hitchin’s flow equation [22]. Half-flat structures were also studied in [28]. In spite of much interest, few non-Calabi-Yau examples are known. In the next subsection, we will analyze the local existence of half-flat nearly Calabi-Yau. We will see that local half-flat nearly Calabi-Yau structures are much more general than Calabi-Yau. In §1.2 we will construct a class of complete or even compact half-flat nearly Calabi-Yau but non-Calabi-Yau examples.

1.1.2 First examples Calabi-Yau

Calabi-Yau is clearly nearly Calabi-Yau. It is the case of most interest so far.

Almost Calabi-Yau

A more general SU(3)-structure is called almost Calabi-Yau in [23]. It is also considered by R. Bryant [10] and E. Goldstein [17]. An almost Calabi-Yau manifold is a K¨ahlermanifold with a nowhere vanishing holomorphic 3-form Φˆ speciﬁed. Let B be the unique function such that

1√ 1 −1Ψ ∧ Ψ = B2Ω3. 8 6

Φ It is clear that (Ω, B ,B) deﬁnes a nearly Calabi-Yau structure on M. Thus, an almost Calabi-Yau structures is nearly Calabi-Yau.

12 Many interesting almost Calabi-Yau examples are provided by degree 5 smooth varieties in CP 4. It is well-known that the canonical line bundle of such a variety is trivial. Thus a nowhere vanishing holomorphic 3-form exists. However, the induced Fubini-Study metric is not Calabi-Yau in general. These provide many almost Calabi- Yau but non-Calabi-Yau examples.

Nearly K¨ahler

The fundamental forms Ω and Ψ satisfy

dΩ = 3ImΨ

and dΨ = 2Ω2.

It is well-known that the underlying metric is Einstein by [16]. It also follows that such structures are real analytic, in, say, coordinates harmonic for the metric. The ideal I is clearly diﬀerentially closed. Hence the almost special Lagrangian geometry of nearly K¨ahler6 manifolds is well-behaved locally. The underlying almost complex structure structure is non-integrable. In fact,

the Nijenhuis tensor Nij is the identity matrix. Nearly K¨ahlerbut non-Calabi-Yau Examples include S6 with the standard metric and almost complex structure, S3×S3, the ﬂag manifold SU(3)/T 2 and the projective space CP3 (with an unusual almost complex structure, however).

Remark 1.1.9. It should be cautioned that a nearly Calabi-Yau manifold, unless it is Calabi-Yau, is NOT nearly K¨ahler.

13 1.1.3 Generalities

Calabi-Yau and nearly K¨ahlerprovide ﬁrst examples of admissible SU(3)-structures. However, we are about to show that they are only a ‘closed’ subset of the moduli of local admissible SU(3)-structures. For this, we need to study the local generality of admissible SU(3)-structures. Let p : F → M 6 be the total coframe bundle of M 6. Thus F is a principal

6 GL(6, R)-bundle over M . The ﬁber Fx over x consists of the linear isomorphisms

6 2 u : TxM → R . We consider the quotient F/SU(3) which is 34(= 6 + 6 − 8) dimensional and projects onto M with ﬁbers diﬀeomorphic to GL(6, R)/SU(3). An SU(3)-structure over M may be regarded as a section of p : F/SU(3) → M 6. In fact,

if P is an SU(3)-structure, then P is a subbundle of F and every ﬁber Px determines

6 a unique SU(3) orbit of Fx and thus a section of F/SU(3) → M . Conversely, if σ : M → F/SU(3) is such a section, let P be the preimage of σ(M) under the projection F → F/SU(3). Then P is the needed SU(3)-structure.

2 3 Remark 1.1.10. There is a more concrete realization of F/SU(3). Let Λ+M ⊕Λ+M be the subbundle of Λ2M ⊕ Λ3M consiting of the pairs of positive forms (ρ2, ρ3) in

6 2 ∗ the sense that there exists a linear isomorphism u : TxM → R such that ρ = u Ω0

3 ∗ 2 3 and ρ = u Re(Ψ0). We clearly have a projection F → Λ+M ⊕ Λ+M. Since the

isotropy group of (Ω0, ReΨ0) is SU(3) ⊂ GL(6, R), this is indeed a principal SU(3)- bundle. For similar discussions concerning G2-structures and Spin(7)-structures, consult Bryant’s work [7]. In fact, the work directly inspired the discussion in this section.

We will write structures equations for F. For notational conventions on linear algebra see Introduction. Let (ω1, ω2, ω3, ω1, ω2, ω3) denote complexiﬁed tautological

∗ 1-forms. Thus, for example, ω1u = u (dz1) ◦ p∗. We ﬁx a gl(6, R)-valued connection

14 form √ ! α + −1 µ + κ β 3 √ −1 β α − 3 µ + κ

on F where α takes value in su(3), β is gl(3, C)-valued, µ is a real form, and κ satisﬁes κt = κ.

We have the following structure equations on F:

√ ! ω α + −1 µ + κ β ω d = − 3 √ ∧ . (1.4) ω −1 ω β α − 3 µ + κ

√ −1 Note that the forms Ω = 2 (ω1 ∧ ω1 + ω2 ∧ ω2 + ω3 ∧ ω3) and Ψ = ω ∧ ω2 ∧ ω3 are SU(3)-invariant on F, so they descend to F/SU(3). We use the same letters to denote the forms on F/SU(3) and let ψ = ImΨ and φ = ReΨ. We will use Cartan- K¨ahlermachinery to study local admissible SU(3)-structures and nearly Calabi-Yau structures. For background material on exterior diﬀerential systems, see the standard text [12].

Generalities of admissible SU(3)-structures

We introduce a new manifold M = (F ×C3)/SU(3)×R, where SU(3) acts on C3 in

3 the obvious way. We use (u1, u2¯, u3¯) as the coordinate on C and a as the coordinate

on R. M is a vector bundle of rank 7 over F/SU(3). Let θ = u¯iωi + u¯iωi. Then θ is another well-defined differential form on M besides Ω and Ψ. On M define a differential ideal

I = hΠ3 = dΩ − θ ∧ Ω − aψidiff = hΠ3 = dΩ − θ ∧ Ω − aψ, Π4 = dθ ∧ Ω + (da − aθ) ∧ ψ + adψialg.

We are interested in 6-dimensional (local) integral manifolds of this ideal which are also local sections of M → M 6. Such a section pulls back Ω and Ψ to M

15 which satisfies the condition (1.1) and thus defines an admissible SU(3)-structure. √ A section satisfies −1Ψ ∧ Ψ 6= 0. Conversely a 6-dimensional submanifold of M √ on which −1Ψ ∧ Ψ 6= 0 is locally a section. Hence we will consider the integral √ manifolds of I with the independence condition −1Ψ ∧ Ψ 6= 0.

√ Theorem 1.1.11. The diﬀerential system (I, −1Ψ ∧ Ψ 6= 0) on the dense open set M\{a = 0} is involutive with Cartan characters

(s0, s1, s2, s3, s4, s5, s6) = (0, 0, 1, 3, 6, 10, 15).

Proof. Since the system contains no forms of degree 2 or less, we have c0 = c1 = 0.

On the other hand, c1 = 1 and c6 = 35. We need to compute the remaining three

3 characters c3, c4 and c5. For this we pass up to F×C ×R. For eﬀective computations we set Da = da − aθ,

and √ −1 Du¯ = du¯ − u¯α ¯ − u¯κ ¯ − u¯µ − u¯β . i i j ji j ji 3 i j ji

Relative to the projection F × C3 × R → M, the forms ω, κ, β, µ, Da, Du form a basis for semibasic 1-forms. In terms of these forms we have

√ √ −1 −1 Π3 = − 2 κi¯j ∧ ωj ∧ ωi + 2 κi¯j ∧ ωj ∧ ωi √ √ −1 −1 − 4 (βij − βji) ∧ ωj ∧ ωi + 4 (βij − βji) ∧ ωj ∧ ωi √ −1 − 2 (u¯iωi + u¯iωi) ∧ ωj ∧ ωj √ −1 − 2 a(ω1 ∧ ω2 ∧ ω3 − ω1 ∧ ω2 ∧ ω3)

16 and √ −1 Π4 = 2 (Du¯i ∧ ωi + Du¯i ∧ ωi) ∧ ωj ∧ ωj √ −1 + 2 (Da − aκi¯i) ∧ (ω1 ∧ ω2 ∧ ω3 − ω1 ∧ ω2 ∧ ω3)

a − 2 µ ∧ (ω1 ∧ ω2 ∧ ω3 + ω1 ∧ ω2 ∧ ω3) √ −1 − 4 aijkβil ∧ ωl ∧ ωj ∧ ωk √ −1 + 4 aijkβil ∧ ωl ∧ ωj ∧ ωk.

A six dimensional subspace E6 of the tangent plane on which Ψ ∧ Ψ 6= 0 is deﬁned by the following relations

 κ = A ω + A ¯ ω ;  ij ijk k jik k    β = B ω + C ω ,  ij ijk k ijk k   Du¯i = Uijωj + Uijωj, (1.5)     Da = a¯iωi + a¯iωi,    µ = b¯iωi + b¯iωi,

where Aijk,Bijk,Cijj,Uij,Uij and a¯i, b¯i are free parameters. In order that E6 be an

integral element of I, it must annihilate Π3 and Π4. This amounts to the following equations on the parameters in (1.5),

 ijkBijk = a,     2(Ak¯ji − Ai¯jk) + Cik¯j − Cki¯j + u¯iδ¯jk − uk¯δ¯ji = 0,    √ √  −1(U + a¯ − C ¯) − a(b¯ − −1A ) = 0,  ij jik k iki k iik  (1.6) −(U δ + U δ − U δ − U δ )  ij kl lk ji lj ki ik jl     +(Ukl¯ δj¯i − U¯jlδk¯i − Uki¯ δj¯l + U¯jiδk¯l)     −apilBpkj + apilBpjk + apjkBpli − apjkBpil = 0. 

17 By inspection, we have 35 = (2 + 3 × 3 × 2 + 3 × 2 + 3 × 3) linearly independent aﬃne equations in (1.6) (note that the last equations are real, while the others are complex) . The solution space is smooth, even where a = 0. We pick E5 =

span{e1, e2, e3, e4, e5} ⊂ E6 where e1 is dual to Re(ω1) and e4 is dual to Im(ω1), etc. First note that 5 5 c ≤ + = 20. 5 2 3

We will show that the equality holds, i.e., the polar equations of E5 has the largest possible rank 20. It then follows that if we pick any ﬂag E3 ⊂ E4 ⊂ E5 we have

3 3 c = + = 4 3 2 3 and 4 4 c = + = 10. 4 2 3

Since c0 + c1 + c2 + c3 + c4 + c5 = 1 + 4 + 10 + 20 = 35 we apply Cartan’s Test to ﬁnish the proof.

The veriﬁcation that the polar equations of E5 have rank 20 is a lengthy linear algebra exercise. First, by translating κ, β, Du, Da, µ we may assume these forms

vanish on E6 since Π3 and Π4 are aﬃne linear in these forms. Now the rank of

polar equations are the number of linearly independent forms in {(ei ∧ ej)yΠ3, (ei ∧

ej ∧ ek)yΠ4}. We omit the messy details but only point out the following facts (an unsatisﬁed reader may consult the computations in the proof of Theorem (1.1.14)

and make necessary modiﬁcations by himself). The forms {(ei ∧ ej)yΠ3} pick out 10 linearly independent forms from linear combinations of Re(κij), Im(κi¯j), Re(βij −βji)

and Im(βij − βji). The 6 forms (ei ∧ ei+3 ∧ ej)yΠ4 pick out real and imaginary parts 2 S of {Du¯j + linear combinations of β}j=1 {Re(Du3¯ − aβ21), Re(Du3¯ + aβ12)} . The

4 forms (ei ∧ ej ∧ ek)yΠ4 with i ∈ {1, 4}, j ∈ {2, 5} and k = 3 give us non-degenerate 18 P linear combinations of the forms Da−a κii, aµ, aRe(βii) and aIm(βii). These three classes of equations are clearly independent from each other if we assume a 6= 0.

Remark 1.1.12. There does not exist any regular ﬂag over the locus {a = 0} for

simple reasons. When a = 0, only 7 independent forms Du¯i and Da in Π4 could contribute to the polar equations. Thus c5 ≤ 17.

Remark 1.1.13. The last nonzero character is s6 = 15. Modulo diffeomorphisms, which depend on 6 functions of 6 variables, we still have 9 functions of 6 variables of local generality of admissible SU(3)-structures. Both local Calabi-Yau and nearly Kählerstructures depend on 2 functions of 5 variables. Thus local admissible SU(3)- structures are much more general than Calabi-Yau and nearly Kähler.

Generality of nearly Calabi-Yau

Nearly Calabi-Yau is a subclass of admissible SU(3)-structures. One would expect nearly Calabi-Yau to be less general than an admissible SU(3)-structure. We will show this is indeed the case. Now the diﬀerential system I is deﬁned on F/SU(3)×R and generated algebraically by the 3-form dΩ and the 4-form Υ = dψ + dU ∧ ψ with √ the independence condition −1Ψ ∧ Ψ. This system is better-behaved than I for admissible SU(3)-structures in that it is involutive on the whole F/SU(3) × R.

Theorem 1.1.14. The diﬀerential system I on F/SU(3) × R is involutive with

Cartan characters (s0, s1, s2, s3, s4, s5, s6) = (0, 0, 1, 3, 6, 9, 10).

Proof. Since the system contains no forms of degree 2 or less, we have c0 = c1 = 0.

Moreover, it is easy to see c2 = 1. To use Cartan’s Test, we need compute the other

3 characters c3, c4 and c5 and the codimension of the space of 6-dimensional integral elements. For this we pass up to F where

√ √ −1 −1 dΩ = − 2 κi¯jωj ∧ ωi + 2 κi¯jωj ∧ ωi √ √ −1 −1 − 2 βij ∧ ωj ∧ ωi + 2 βij ∧ ωj ∧ ωi, 19 and

1 √ Υ = dψ + dU ∧ ψ = − 2 [µ + −1(κii − dU)] ∧ ω1 ∧ ω2 ∧ ω3

1 √ − 2 [µ − −1(κii − dU)] ∧ ω1 ∧ ω2 ∧ ω3 √ √ −1 −1 − 4 ijkβil ∧ ωl ∧ ωj ∧ ωk + 4 ijkβil ∧ ωl ∧ ωj ∧ ωk.

√ A 6-dimensional integral element E6 on which −1Ψ ∧ Ψ 6= 0 is parametrized the equations similar to (1.5) for κ, β, µ and dU,

 κ = A ω + A ¯ ω ;  ij ijk k jik k    β = B ω + C ω ,  ij ijk k ijk k (1.7)   µ = b¯iωi + b¯iωi,    dU = u¯iωi + u¯iωi

but now the quantities A, B, u and b satisfy the following equations

 B = 0  ijk ijk    2A − 2A + C − C = 0  kji ijk ikj kij √ (1.8)  C − A + u¯ − −1b¯ = 0  iki iik k k    −pilBpkj + pilBpjk + pjkBpli − pjkBpil = 0.

The last equations are real while the others are all complex. Moreover, the last equations imply the imaginary part of the ﬁrst equation. Thus the total rank of

these linear equations is 1 + 3 × 3 × 2 + 3 × 2 + 3 × 3 = 34. The forms Re(ωi) and

Im(ωi) restrict to E6 to be a dual basis. Let {e1, e2, e3, e4, e5, e6} be the basis of E6 for

which e1 is dual to Re(ω1) and e4 is dual to Im(ω1), etc. Again by translating we may

assume A = B = b = C = u = 0. Let E3 = span{e1, e2, e3}, E4 = span{e1, e2, e3, e4},

and E5 = span{e1, e2, e3, e4, e5}.

20 The polar space of E3 consists of vectors annihilating the following 1-forms   (e1 ∧ e2)ydΩ = −2Im(κ12¯) − Im(β12 − β21)    (e ∧ e ) dΩ = −2Im(κ ¯) − Im(β − β )  1 3 y 13 13 31 . (1.9) (e2 ∧ e3) dΩ = −2Im(κ ¯) − Im(β23 − β32)  y 23   (e1 ∧ e2 ∧ e3)yΥ = µ + Im(β11 + β22 + β33) 

Consequently c3 = 4 and s3 = 3.

The polar space of E4 consists of vectors annihilating the forms in (1.9) as well as the following forms   (e1 ∧ e4)ydΩ = −2κ11¯    (e2 ∧ e4)ydΩ = −2Re(κ12¯) − Re(β12 − β21)     (e ∧ e ) dΩ = −2Re(κ ¯) − Re(β − β )  3 4 y 13 13 31 . (1.10)  (e1 ∧ e2 ∧ e4)yΥ = −2Re(β31)     (e1 ∧ e3 ∧ e4)yΥ = 2Re(β21)   P  (e2 ∧ e3 ∧ e4)yΥ = − i κii + dU + Re(−β11 + β22 + β33)

These forms are independent among themselves and also independent from forms in

(1.9). Thus s4 = 6. The polar space for E5 consists of vector annihilating forms in (1.9), (1.10) as well as the following forms   (e ∧ e ) dΩ = −2Re(κ ¯) + Re(β − β )  1 5 y 12 12 21     (e2 ∧ e5)ydΩ = −2κ22¯     (e3 ∧ e5)ydΩ = −2Re(κ32¯) + Re(β32 − β23)     (e4 ∧ e5)ydΩ = −2Im(κ12¯) + Im(β12 − β21)   (e ∧ e ∧ e ) Υ = −2Re(β )  1 2 5 y 32 . (1.11) (e ∧ e ∧ e ) Υ = − P κ + dU + Re(−β + β − β )  1 3 5 y i ii 11 22 33     (e1 ∧ e4 ∧ e5)yΥ = 2Im(β31)     (e2 ∧ e3 ∧ e5)yΥ = −2Re(β12)     (e2 ∧ e4 ∧ e5)yΥ = 2Im(β32)    (e3 ∧ e4 ∧ e5)yΥ = −µ + Im(β11 + β22 − β33)

Note that

(e2 ∧ e4)ydΩ − (e1 ∧ e5)ydΩ − (e2 ∧ e3 ∧ e5)yΥ − (e1 ∧ e3 ∧ e4)yΥ = 0.

No other relations exist among the forms in (1.9), (1.10) and (1.11). Thus s5 = 9

and s6 = 10. Since 6s0 + 5s1 + 4s2 + 3s3 + 2s4 + s5 = 34, Cartan’s test is satisﬁed and the proof is complete.

21 Thus, modulo diﬀeomorphisms, local nearly Calabi-Yau structures depend on 4 functions of 6 variables.

Remark 1.1.15 (Generality of half-flat nearly Calabi-Yau structures). The EDS for half-flat nearly Calabi-Yau structures is defined on F/SU(3) and algebraically generated by dΩ and dImΨ. Similar analysis shows that this system is also involutive with Cartan characters

(s0, s1, s2, s3, s4, s5, s6) = (0, 0, 1, 3, 6, 9, 9).

Thus, modulo diﬀeomorphisms, these structures depend on 3 functions of 6 variables locally. They are much more general than Calabi-Yau structures. It would be interesting to also analyze the half ﬂat system generated by Ω ∧ dΩ and dImΨ.

Remark 1.1.16 (Generality of almost Calabi-Yau structues). The EDS for almost Calabi-Yau structures is deﬁned on F/SU(3) × R and generated algebraically by dΩ and dΨ + dU ∧ Ψ. We conjecture this system is involutive with the last non-zero

Cartan character to be s6 = 7. We leave this for the interested reader to investigate.

1.2 Examples from twistor spaces of Riemannian four-manifolds

There has been an extensive literature on twistor theory. Suppose (M 4, ds2) is a Riemannian 4-manifold. A twistor at x ∈ M is an orthogonal complex structure

2 ∗ 2 2 j : TxM → TxM, j = −1 and j (dsx) = dsx. The space of twistors at points of M forms a smooth manifold J called twistor space of M. It is well-known that J has an almost complex structure. It is moreover complex if M has constant sectional curvatures. However, we will not use this usual almost complex structure in this paper. Instead, we will ‘reverse’ the almost complex structure on the ﬁbers and obtain an SU(3)-structure on J . By doing so, we will lose the possible integrability of the almost complex structures in some cases.

22 1.2.1 Four dimensional Riemannian geometry

We formulate Riemannian geometry of four-manifolds in moving frames. Let π :

F → M be the oriented orthonormal coframe bundle over M. Thus Fx consists of

4 4 orientation preserving isometries u : TxM → R . Let η be the R -valued tautological form on F. By the fundamental theorem of Riemannian geometry, there exists a unique so(4, R)-valued one-form θ so that

dη = −θ ∧ η. √ √ Denote ω1 = η1 + −1η3 and ω2 = η2 + −1η4. We write the structrure equation as

    ω1 ω1  ω2  α β  ω2  d   = − ij ij ∧   (1.12)  ω1  βij αij  ω1  ω2 ω2 where αt + α = 0 and βt + β = 0. The Riemannian curvature is of course

α β α β α β R = d + ∧ . β α β α β α

This is a so(4, R)-valued 2-form. Corresponding to the decomposition so(4, R) =

su(2)+ ⊕ su(2)− we decompose R = R+ + R−, where

α0 0 α0 0 α0 0 R+ = d + ∧ 0 α0 0 α0 0 α0

and

1 1 1 2 tr(α)I β 2 tr(α)I β 2 tr(α)I β R− = d 1 + 1 ∧ 1 . β 2 tr(α)I β 2 tr(α)I β 2 tr(α)I

1 for which α0 = α − 2 tr(α)I takes value in su(2). We are mainly interested in R− so we examine this part more carefully. Write

0 ω β = 3 , −ω3 0 23 1 (R ) = dtr(α) + ω ∧ ω , − 1 2 3 3 and

(R−)2 = dω3 − tr(α) ∧ ω3.

The forms √ −1 Θ = ω ∧ ω , Θ = ω ∧ ω , Θ = (ω ∧ ω + ω ∧ ω ) 1 1 2 2 1 2 3 2 1 1 2 2 give a basis for anti-self dual complex forms at x, while √ −1 Σ = ω ∧ ω , Σ = ω ∧ ω , Σ = (ω ∧ ω − ω ∧ ω ) 1 1 2 2 1 2 3 2 1 1 2 2 form a basis for self dual forms at x. Since R− is semibasic, √ √ (R−)1 = AΘ1 − AΘ2 + −1aΘ3 + BΣ1 − BΣ2 + −1bΣ3, and

(R−)2 = C1Θ1 + C2Θ2 + C3Θ3 + D1Σ1 + D2Σ2 + D3Σ3 where A, B, Ci,Di are complex and a, b are real. For our purposes, we view R− as a (2, 2) tensor. Using ds2, we write R as

R− = 2(R−)1 ⊗ (E1¯ ∧ E1¯ + E2¯ ∧ E2¯) + 2(R−)2 ⊗ E1¯ ∧ E2¯ + 2(R−)2 ⊗ E1¯ ∧ E2¯ √ ∗ ∗ ∗ = 2 −1(R−)1 ⊗ Θ3 + 2(R−)2 ⊗ Θ2 + 2(R−)2 ⊗ Θ1 where, by abuse of notation, we use E¯i to denote the tangent vector dual to ωi. In

2 2 2 2 this way we may regard R− as a linear map R− :Λ− → Λ = Λ− ⊕ Λ+. Relative to the basis Θ and Σ we write the matrix representation

 √  C2 C1 √−1A  C1 C2 − −1A     C3 C3 −a  R−(Θ1, Θ2, Θ3) = 2(Θ1, Θ2, Θ3, Σ1, Σ2, Σ3)  √  .  D2 D1 −1B   √   D1 D2 − −1B  D3 D3 −b 24 − s It is well-known that R− decomposes as Z + W + 12 Id (see [6], p. 51) where Z is the traceless Ricci curvature, W − is the anti-self-dual part of the Weyl curvature and s is the scalar curvature. In our notation, Z is represented by the matrix

 √  D2 D1 √−1B 2  D1 D2 − −1B  , D3 D3 −b

s = 8(C2 + C2 − a), and W − is represented by √  C C −1A  2 1 √ 2 2 C C − −1A − (C + C − a)I.  1 2  3 2 2 C3 C3 −a

The metric with W − = 0 is called self-dual. If in addition, ds2 is Einstein, then s is necessarily constant. In our notations,

Proposition 1.2.1. The metric ds2 is self-dual and Einstein if and only if b = A =

s B = C1 = C3 = D1 = D2 = D3 = 0 and C2 = C2 = −a = 24 . In this case, a part of the structure equation simpliﬁes greatly

 ω   ω   ω ∧ ω  1 α 0 1 2 3 d  ω2  = − ∧  ω2  +  ω3 ∧ ω1  . (1.13) 0 −tr(α) s ω3 ω3 24 ω1 ∧ ω2

Self-dual Einstein metrics will play important roles in our following constructions. There are not many compact examples with s ≥ 0 due to the classiﬁcation by Hitchin (see [6], p 376):

Theorem 1.2.2. Let M be be self-dual Einstein manifold. Then (1) If s > 0, M is isometric to S4 or CP 2 with their canonical metrics. (2) If s = 0, M is either ﬂat or its universal covering is a K3 surface with the Calabi-Yau metric.

25 The proof uses Bochner Technique, which, however does not work well when s < 0. No similar results are available for self-dual Einstein metrics with negative scalar curvature.

1.2.2 Twistor spaces of self-dual Einstein 4-manifolds √ 4 We ﬁx a complex structure J0 on R by requiring dz1 = dx1 + −1dx3 and dz2 = √ dx2 + −1dx4 be complex linear. We deﬁne a map j : F → J as follows

−1 j(u) = u ◦ J0 ◦ u.

Since SO(4) acts transitively on the orthogonal complex structures on R4 and the

isotropy group of J0 is U(2), j makes F a principal U(2)-bundle over J . This deﬁnes a U(2)-structure on J . It in turn determines an SU(3)-structure on J by the standard embedding of U(2) into SU(3). Relative to j, β becomes semi-basic.

The almost structure on J determined by this SU(3)-strucure is such that ω1, ω2 and ω3 are complex linear. Let us now concentrate on self-dual Einstein manifolds. The equations in (1.13) are the ﬁrst structure equations on J . It clearly satisﬁes the condition of Lemma 1.1.2.

Lemma 1.2.3. The twistor space of a self-dual Einstein 4-manifold carries an admissible SU(3)-structure.

s > 0

In this case, we scale the metric so that s = 24. Now the structure equation (1.13) indicates that the twistor space is actually nearly Kähler. By the aforementioned Hitchin’s result, the only two possibilities are M = S4 and M = CP2. The corresponding twistor spaces are two familiar nearly Kähler examples, CP3 and the flag manifold SU(3)/T 2.

26 s = 0

Again, by Hitchin’s result we have two examples: one is the ﬂat case, the other is

K3 surfaces.

s < 0

This is the most interesting case in many respects. We scale the metric to make s = −48. Now the structure equation (1.13) reads

 ω   ω   ω ∧ ω  1 α 0 1 2 3 d ω = − ∧ ω + ω ∧ ω . (1.14)  2  0 −tr(α)  2   3 1  ω3 ω3 −2ω1 ∧ ω2

The only torsion is the Nijenhuis tensor, in local unitary basis,

 1 0 0  N =  0 1 0  . 0 0 −2

Comparing with (1.3) (with U = 0 being understood), we have

Theorem 1.2.4. The twistor space of a self-dual Einstein manifold of negative scalar curvature is half-ﬂat nearly Calabi-Yau but non-Calabi-Yau.

The simplest example of this category is, of course, the twistor space of the hyperbolic space H4. Compact examples can be obtained from the quotients of H4 by certain discrete isometry groups. It remains open to construct complete or compact nearly Calabi-Yau manifolds that are not half-ﬂat.

1.3 Complete special Lagrangian examples

In this section we will construct some complete special Lagrangian submanifolds in the twistor spaces J (S4) = CP3 and J (H4) considered in the previous section. Our

27 method is based on the following observation due to Robert Bryant [8]. Suppose on an SU(3) manifold (M, Ω, Ψ), there is a real structure, i.e., an involution c such that

c∗Ω = −Ω, c∗Ψ = Ψ,

and the set Nc of points ﬁxed under c is a smooth submanifold. Then it is easy to see that Nc, with one of its two possible orientations, is a special Lagrangian submanifold of M . Thus our major task is to construct such involutions for J (S4) = CP3 and J (H4).

1.3.1 An example in J (S4) = CP3

We need a more explicit description of the twistor fibration T : CP3 → S4. We follow the discussion in [9]. However, as aforementioned, we will use a different almost complex structure on CP3. Let H denote the real division algebra of quaternions. An element of H can be written uniquely as q = z + jw where z, w ∈ C and j ∈ H satisfies

j2 = −1, zj = jz¯ for all z ∈ C. The quaternion multiplication is thus given by

(z1 + jz2)(z3 + jz4) = z1z3 − z2z4 + j(z2z3 + z1z4). (1.15)

We deﬁne an involution C : H → H by C(z1 + jz2) = z1 + jz2. It can be easily checked via the product rule (1.15) that this is in fact an algebra automorphism, i.e., C(pq) = C(p)C(q) for p, q ∈ H. We regard C as subalgebra of H and give H the structure of a complex vector

2 space by letting C act on the right. We let H denote the space of pairs (q1, q2)

2 where qi ∈ H. We will make H into a quaternion vector space by letting H act on the right

(q1, q2)q = (q1q, q2q). 28 This automatically makes H2 into a complex vector space of dimension 4. In fact,

4 regarding C as the space of 4-tuples (z1, z2, z3, z4), we make the explicit identiﬁcation

(z1, z2, z3, z4) ∼ (z1 + jz2, z3 + jz4). (1.16)

This specific isomorphism is the one we will always mean when we write C4 = H2. If v ∈ H2\(0, 0) is given, let vC and vH denote, respectively, the complex line and the quaternion line spanned by v. As is well-known, HP1, the space of quaternion lines in H2, is isometric to S4. For this reason, we will speak interchangeably of S4 and HP1. The assignment vC → vH is exactly the twistor mapping T : CP3 → HP1. The fibres of T are CP1’s. Thus, we have a fibration

CP1 → CP3 ↓ (1.17) HP1

This is the famous twistor ﬁbration. In order to study its geometry more thoroughly, we will now introduce the structure equations of H2. First we endow H2 with a quaternion inner product h, i : H2 × H2 → H deﬁned by

h(q1, q2), (p1, p2)i =q ¯1p1 +q ¯2p2.

We have identities

hv, wqi = hv, wi q, hv, wi = hw, vi , hvq, wi =q ¯hv, wi .

Via the identiﬁcation (1.16),

h(q , q ), (p , p )i = z w + z w + z w + z w 1 2 1 2 1 1 2 2 3 3 4 4 (1.18) +j(z1w2 − z2w1 + z3w4 − z4w3)

for q1 = z1 + jz2, q2 = z3 + jz4, p1 = w1 + jw2, p2 = w3 + jw4. In other words,

h, i essentially consists of two parts: one is the standard Hermitian product dz1 ◦dz1 +

···+dz4 ◦dz4; the other is the standard complex symplectic form dz1 ∧dz2 +dz3 ∧dz4. 29 2 Let F denote the space of pairs f = (e1, e2) with ei ∈ H satisfying

he1, e1i = he2, e2i = 1, he1, e2i = 0.

2 7 8 We regard ei(f) as functions on F with values in H . Clearly e1(F) = S ⊂ E = H2. It is well-known that F may be canonically identiﬁed with Sp(2) up to a left

a translation in Sp(2). There are unique quaternion-valued 1-forms {φb } so that

b dea = ebφa, (1.19)

a a c dφb + φc ∧ φb = 0, (1.20)

and

a b φb + φa = 0. (1.21)

3 We deﬁne a map F → CP by sending (e1, e2) to the complex line spanned by

e1. We will denote this map by j by a slight abuse of notation. The composition π = T ◦ j is actually a spin structure on S4. In fact the oriented coframe bundle F of S4 may be identiﬁed with SO(5) up to a left translation in SO(5). Thus F double covers F as Sp(2) double covers SO(5). We now write structure equations for the map j. First we immediately see that j gives F an S1 × S3-structure over CP3 where we have identiﬁed S1 with the unit complex numbers and S3 with the unit quaternions. The action is given by

f(z, q) = (e1, e2)(z, q) = (e1z, e2q) where z ∈ S1 and q ∈ S3. If we set

1 1 " ω1 ω2 # φ φ iρ1 + jω3 − √ + j √ 1 2 = 2 2 φ2 φ2 √ω1 + j √ω2 iρ + jτ 1 2 2 2 2

30 where ρ1 and ρ2 are real 1-forms while ω1, ω2, ω3 and τ are complex valued, we may rewrite one part of the structure equation (3.3) relative to the S1 × S3-structure on CP3 as

        ω1 i(ρ2 − ρ1) −τ¯ 0 ω1 ω2 ∧ ω3 d  ω2  = −  τ −i(ρ1 + ρ2) 0  ∧  ω2  +  ω3 ∧ ω1  . ω3 0 0 2iρ1 ω3 ω1 ∧ ω2 (1.22)

3 The nearly Kählerstructure on CP is defined by setting ω1, ω2 and ω3 to be complex linear. Via the algebra automorphism C we define an involution on H2 by (p, q) 7→ (C(p),C(q)). We denote this map still by C. This map in turn induces an involution on F, still denoted C, by C(e1, e2) = (C(e1),C(e2)). From (1.18) we see that the defining equations for F are preserved and the involution is well-defined. The map C further descends to an involution c on CP3 as well as an involutionc ¯ on S4 by

e1C 7→ C(e1)C and e1H 7→ C(e1)H respectively. We have the following commutative diagram

F −→C F ↓ ↓ CP3 −→c CP3 . ↓ ↓ HP1 −→c¯ HP1

Apply the automorphism to the structure equations (3.2) and we get

b dC(ea) = C(eb)C(φa).

Thus in particular we have on F

∗ C ωi = ωi for i = 1, 2, 3. Consequently

C∗j∗Ω = −j∗Ω,C∗j∗Ψ = j∗Ψ.

31 Since jC = cj and j∗ is injective, we have on CP3

c∗Ω = −Ω, c∗Ψ = Ψ.

Thus by the general principle the ﬁxed set of c is a special Lagrangian submanifold of CP3. Moreover it is easy to see that this locus is just the usual RP3.

3 Theorem 1.3.1. The real projective space RP = {[x1 : x2 : x3 : x4]: xi ∈ R} ⊂ CP3 is a special Lagrangian submanifold of the nearly K¨ahler CP3.

The twistor map T : CP3 → HP1 restricted to the real projective space RP3 now looks like

[x1 : x2 : x3 : x4] 7→ [x1 + jx2 : x3 + jx4].

Thus the image is a CP1 ⊂ HP1 and T is the Hopf ﬁbration

S1 → RP3 ↓ . CP1

A dual construction for J (H4) will follow.

1.3.2 An example in J (H4)

Let H2 and the involution C be as before. But now we endow H2 with a (1, 1) quaternion inner product h, i : H2 × H2 → H deﬁned by

h(q1, q2), (p1, p2)i = q1p1 − q2p2.

We still have the identities

hv, wqi = hv, wi q, hv, wi = hw, vi , hvq, wi =q ¯hv, wi .

Via the identiﬁcation (1.16),

h(q , q ), (p , p )i = z w + z w − z w − z w 1 2 1 2 1 1 2 2 3 3 4 4 (1.23) +j(z1w2 + z2w1 − z3w4 − z4w3) 32 for q1 = z1 +jz2, q2 = z3 +jz4, p1 = w1 +jw2, p2 = w3 +jw4. In other words, h, i

essentially consists of two parts: one is the (2, 2)-Hermitian product dz1 ◦ dz1 + dz2 ◦ dz2 −dz3 ◦dz3 −dz4 ◦dz4; the other is a complex symplectic form dz1 ∧dz2 −dz3 ∧dz4. Denote the pseudo-sphere in H2 by

ΨS7 = {(p, q) ∈ H2 : pp − qq = 1}.

This is a connected non-compact smooth hypersurface in H2. The group S3 acts on ΨS7 by (p, q) · r = (pr, qr)

where r ∈ S3 is a unit quaternion number. This action is clearly free. Thus the quotient space ΨHP1 = ΨS7/S3 is smooth. Indeed ΨHP1 = H4. Similarly if we regard S1 as a subgroup of S3 consisting of unit complex numbers, the quotient space ΨCP3 = ΨS7/S1 is smooth. The clearly well-defined map T :ΨCP3 → H4 is exactly the twistor fibration of H4. We have the following commutative diagram of fibrations S1 ,→ ΨS7 ↓ CP1 ,→ ΨCP3 . ↓ ΨHP1

2 Let F denote the space of pairs f = (e1, e2) with ei ∈ H satisfying

he1, e1i = 1, he2, e2i = −1, he1, e2i = 0.

2 7 (4,4) We regard ei(f) as functions on F with values in H . Clearly e1(F) = ΨS ⊂E = H2. It is well-known that F maybe canonically identiﬁed with Sp(1, 1) up to a left translation in Sp(1, 1), where

1 0 1 0 Sp(1, 1) = {A ∈ gl(2, H): A At = .} 0 −1 0 −1 33 a There are unique quaternion-valued 1-forms {φb } so that

b dea = ebφa, (1.24)

a a c dφb + φc ∧ φb = 0, (1.25)

and 1 0 1 0 φ + φt = 0. (1.26) 0 −1 0 −1

3 1 We have a canonical map F → ΨCP by sending (e1, e2) to the coset e1 · S . We will denote this map by j by a slight abuse of notation. The composition π = T ◦ j is actually a spin structure on H4. In fact, the oriented coframe bundle F of H4 may be identiﬁed with SO0(4, 1), the identity component of SO(4, 1), up to a left translation in SO0(4, 1). Thus F double covers F as Sp(1, 1) double covers SO0(4, 1) (see Harvey [20], p. 272 for the isomorphism Sp(1, 1) ∼= Spin0(4, 1) where he used the notation HU(1, 1) for Sp(1, 1)). We now write structure equations for the map j. First we immediately see that j gives F an S1 × S3-structure over ΨCP3 where we have identiﬁed S1 with the unit complex numbers and S3 with the unit quaternions. The action is given by

f(z, q) = (e1, e2) · (z, q) = (e1z, e2q) where z ∈ S1 and q ∈ S3. If we set

1 1 φ1 φ2 iρ1 + jω3 ω1 − jω2 2 2 = φ1 φ2 ω1 + jω2 iρ2 + jτ

where ρ1 and ρ2 are real 1-forms while ω1, ω2, ω3 and τ are complex valued, we may rewrite one part of the structure equation (3.3) relative to the nearly Calabi-Yau

34 structure on ΨCP3 as

        ω1 i(ρ2 − ρ1) −τ¯ 0 ω1 ω2 ∧ ω3 d  ω2  = −  τ −i(ρ1 + ρ2) 0  ∧  ω2  +  ω3 ∧ ω1  . ω3 0 0 2iρ1 ω3 −2ω1 ∧ ω2 (1.27)

3 The nearly Calabi-Yau structure on ΨCP is deﬁned by taking ω1, ω2 and ω3 to be complex linear. This involution C on H2 induces an involution on F, still denoted by C, by

C(e1, e2) = (C(e1),C(e2)). From (1.23), we see that the deﬁning equations for F are preserved and the involution is well-deﬁned. The map C further descends to an

3 4 1 1 involution c on ΨCP as well as an involutionc ¯ on H by e1 · S 7→ C(e1) · S and

3 3 e1 · S 7→ C(e1) · S repectively. We have the following commutative diagram

F −→C F ↓ ↓ ΨCP3 −→c ΨCP3 . ↓ ↓ H4 −→c¯ H4

Apply the automorphism to the structure equations (1.24) and we get

b dC(ea) = C(eb)C(φa).

Thus in particular we have on F

∗ C ωi = ωi for i = 1, 2, 3. Consequently

C∗j∗Ω = −j∗Ω,C∗j∗Ψ = j∗Ψ.

Since jC = cj and j∗ is injective, we have on ΨCP3

c∗Ω = −Ω, c∗Ψ = Ψ.

35 Thus by the general principle the ﬁxed set of c is a special Lagrangian submanifold of ΨCP3. It is easy to see that this manifold is the pseudo-projective 3-space ΨRP3,

3 4 2 deﬁned as the quotient of the pseudo 3-sphere ΨS = {(x1, x2, x3, x4) ∈ R : x1 +

2 2 2 x2 − x3 − x4 = 1} by Z2.

Theorem 1.3.2. The real pseudo-projective space ΨRP3 ⊂ ΨCP3 is a special La- grangian submanifold of the nearly Calabi-Yau ΨCP3.

The image under T of this pseudo-sphere is easily seen to be the hyperbolic 2-space H2 ⊂ H4. Thus we have the following ﬁbration

S1 → ΨRP3 ↓ . H2

1.4 Compact special Lagrangian submanifolds in nearly Calabi-Yau manifolds

We discuss compact special Lagrangian submanifolds in nearly Calabi-Yau manifolds. We answer two questions:

1. Let N be a compact special Lagrangian 3-fold in a ﬁxed nearly Calabi-Yau

manifold (M, Ω, Ψ,U). Let MN be the moduli space of special Lagrangian deformations of N, that is, the connected component of the set of special

Lagrangian 3-folds containing N. What can we say about MN ? Is it a smooth manifold? What is its dimension?

2. Let {(M, Ωt, Ψt,Ut): t ∈ (−, )} be a smooth 1-parameter family of nearly

Calabi-Yau manifolds. Suppose N0 is an SL-3−fold. Under what conditions

can we extend N0 to a smooth family of special Lagrangian 3-folds Nt in

(M, Ωt, Ψt,Ut)?

36 These questions concern the deformations of special Lagrangian 3-folds and obstructions to their existence respectively. In the Calabi-Yau case, the ﬁrst question is answered by R. McLean in [24], and the second is answered by D. Joyce in [23]. Moreover, [23] also answers these questions for almost Calabi-Yau manifolds. We show that their proofs generalize to nearly Calabi-Yau manifolds. The argument uses a rescaling trick, communicated to me by R. Bryant. I would like to thank him for this.

1.4.1 Deformations of compact special Lagrangian 3-folds

We have the following result similar to one in [24]

Theorem 1.4.1. Let (M, Ω, Ψ,U) be a nearly Calabi-Yau 3-fold, and N a compact special Lagrangian 3-fold in M. Then the moduli space MN of special Lagrangian deformations of N is a smooth manifold of dimension b1(N), the ﬁrst Betti number of N.

Proof. We emphasize the diﬀerence from the arguments in [24]. Let νN be the normal

bundle of N. Nearby submanifolds of M can be viewed as small sections of νN . Thus we consider the following map between Banach spaces

1,α 1,α 1 1,α 2 F : C (νN ) → dC (Λ (N)) × dC (Λ (N))

deﬁned by

∗ ∗ U V 7→ (expV (Ω), expV (e ImΨ)).

As in [24], it is easy to see from standard Hodge theory and elliptic regularity that this map is well-deﬁned. The kernel of F consists of sections V on whose exp-image Ω and eU ImΨ vanishes. Thus exp(V ) is a special Lagrangian submanifold.

37 Now the diﬀerential F∗ of F at 0 may computed in a way similar to [24],

U F∗(V ) = (LV Ω|N , LV (e ImΨ)|N )

U = (d(V yΩ), d(e V yImΨ)).

∗ Note that the map V 7→ v = V yΩ gives a bundle isomorphism between νN and T N. Via this correspondence V yImΨ = − ∗ v, where it should be cautioned that ∗ is

deﬁned by the induced metric on N. Thus F∗ may be equivalently viewed as a map

C1,α(Λ1(N) → dC1,α(Λ1(N)) × dC1,α(Λ2(N))

by v 7→ (dv, −d ∗ (eU v)).

Here we see the major diﬀerence. The second term is no longer −d ∗ v as in the Calabi-Yau situation. To overcome this diﬃculty, we rescale the induced metric by a proper factor (in terms of U) so that the new Hodge star operator ∗ˆ = eU ∗. This is clearly possible. Then the map is

v 7→ (dv, −d∗ˆv).

By Hodge Theory, the map F∗ is surjective and the kernel consists of harmonic 1-forms (with the rescaled metric being understood). The proof is ﬁnished as in [24] by employing the Implicit Function Theorem for smooth maps between Banach spaces .

Remark 1.4.2 (S. Salur’s result). A similar deformation theorem was proved in [25] for slightly diﬀerently deﬁned special Lagrangian submanifolds in a more general class of sympletic manifolds.

38 1.4.2 Obstructions to the existence of compact SL 3-folds

We address Question 2 above. Let {(M, Ωt, Ψt,Ut)} be a smooth 1-parameter family of nearly Calabi-Yau manifolds. Suppose N0 is a special Lagrangian 3-fold of

(M, Ω0, Ψ0,U0) and Nt is an extension. Then we can view Nt as a family of em-

∗ ∗ Ut beddings of it : N0 → M such that it (Ωt) = it (e ImΨt) = 0. Since the coho-

∗ ∗ Ut ∗ mology classes [is(Ωt)] and [is(e ImΨt)] do not vary with s, we have [i0(Ωt)] =

∗ Ut [i0(e ImΨt)] = 0. Thus, a necessary condition for such an extension of N0 to exist is

Ut [Ωt|N0 ] = [e ImΨt|N0 ] = 0.

Actually this is also suﬃcient:

Theorem 1.4.3. Let {(M, Ωt, Ψt,Ut): t ∈ (−, )} be a smooth 1-parameter family of nearly Calabi-Yau 3-folds. Let N0 be a compact SL 3-fold in (M, Ω0, Ψ0), and suppose

2 Ut 3 [Ωt|N0 ] = 0 in H (N0, R) and [e ImΨt|N0 ] = 0 in H (N0, R) for all t ∈ (−, ). Then

N0 extends to a smooth 1-parameter family {Nt : t ∈ (−δ, δ)} for some 0 < δ ≤ and Nt is a compact SL 3-fold in (M, Ωt, Ψt).

Again, the proof combines the rescaling trick and the argument for the Calabi- Yau case as in [23]. However, since the details are not readily available, we write them down.

Proof. Let νN0 be the normal bundle of N0 in (M, Ω0, Ψ0). Denote by exp the

1,α exponential map of (M, Ω0, Ψ0). For a vector bundle E over N0, we use C (E) and C0,α(E) to denote the sections of E of class C1,α and C0,α respectively. We deﬁne a map

1,α 1,α 1 1,α 2 F : C (νN0 ) × (−, ) → dC (Λ (N0)) × dC (Λ (N0)) by

∗ ∗ Ut F (V, t) = (expV (Ωt), expV (e ImΨt)). 39 We need show this is well-deﬁned. The maps F (sV, t)(0 ≤ s ≤ 1) provide a homotopy

Ut ∗ between F (0, t) = (Ωt|N0 , e ImΨ|N0 ) and F (V, t). Since [Ωt|N0 ] = 0, [expV (Ωt)] = 0. ∗ Thus, expV (Ωt) = dτ for some τ. Moreover, by the standard Hodge theory the 2,α 1,α ∗ form τ can be chosen to be in C because V is C and so is expV (Ωt). A similar

∗ Ut 1,α 2 argument shows that expV (e ImΨt) lies in dC (Λ (N0)). Now we compute the tangent map of F at the point (0, 0),

1,α 1,α 1 1,α 2 F∗ : R × C (νN0 ) → dC (Λ (N0)) × dC (Λ (N0)).

First ∂ d ∗ ∗ F∗( ∂t , 0) = dt |t=0,V =0(expV Ωt, expV ImΨt)

0 U 0 = (Ω |N0 , Im(e Ψ) |N0 ). where d d Ω0 = | Ω , (eU Ψ)0 = | (eUt Ψ ). dt t=0 t dt t=0 t Second

d ∗ ∗ U0 F∗(0,V ) = ds |s=0(expsV Ω0, expsV (e ImΨ))

U0 = (LV Ω0|N0 , LV (e ImΨ0|N0 ))

U0 U0 = ((V ydΩ0 + d(V yΩ0))|N0 , (V yd(e ImΨ0) + d(e V yImΨ0))|N0 )

U0 = (d(V yΩ0)|N0 , d(e V yImΨ0)|N0 ) where LV is the Lie derivative in the V direction and the Cartan formula is used. Actually, in order to take the Lie derivative, one must extend the normal vector ﬁeld V to an open neighborhood ﬁrst. It is easy to see the result is independent of this extension.

Note that the mapping V 7→ v = V yΩ0 gives a bundle isomorphism between ∗ T N0 and νN0 . Translated via this correspondence V yImΨ0 = − ∗ v as is shown in [24], where the Hodge star ∗ is deﬁned by the induced metric. As before, we rescale

40 U0 the metric by a conformal factor so that ∗ˆ = e ∗. By Hodge theory, F∗(0,V ) runs

1,α 1 1,α 2 over every element in dC (Λ (N0)) × dC (Λ )(N0). Thus F∗ is surjective. We can also compute the kernel

∂ F −1(0, 0) = {(r ,V ): rΩ˙ | = −dv, rIm(Ψ)˙ | = d∗ˆv, r ∈ R}, ∗ ∂t N0 N0

−1 1 Hodge theory, F∗ (0, 0) is nonempty and ﬁnite-dimensional, with dimension b (N0)+ 1. By the Implicit Function Theorem for smooth maps between Banach spaces,

−1 −1 F (0, 0) is a smooth manifold with its tangent space at (0, 0) equal to F∗ (0, 0).

1,α −1 The C (νN0 ) components of elements of F (0, 0) are in fact smooth sections by the elliptic regularity theorem. Note that the projection map t restricted to F −1(0, 0) is nondegenerate at (0, 0). Thus the manifold F −1(0, 0) is a local smooth ﬁbration

over (−, ). Pick a local section (t, Vt) of such a local ﬁbration where −δ ≤ t ≤ δ for some 0 < δ ≤ . Then N = (N ) are the desired 1-parameter family of smooth t expVt 0 special Lagrangian manifolds.

Remark 1.4.4 (on the proof). Strictly speaking, the domain of F is not a Banach space because of the (−, ) part. This minor difficulty can be overcome by either using a cut-off function of t or reparametrizing t by a diffeomorphism between (−, ) and R preserving 0.

41 2

Instantons on Nearly K¨ahler6-Manifolds

The notion of anti-self-dual instantons plays an important role in Donaldson’s theory of 4-manifolds ([14]). This concept has been generalized to higher dimensions (e.g., [15] and [27]). To motivate the generalization, we first recall the 4-dimensional theory. Suppose M is an oriented 4-dimensional Riemannian 4-manifold. It is well known that the space of 2-forms splits into self-dual and anti-self-dual parts, corresponding respectively to ±1-eigenspaces of Hodge ∗ operator. A connection A on a certain principal bundle over M is said to be an anti-self-dual instanton if its curvature F , when viewed as a vector-bundle valued two-form, satisfies ∗F = −F . Of course, this definition does not generalize directly to higher dimensions. If, moreover, M is almost Hermitian, i.e., endowed with an almost complex structure compatible with the Riemannian structure, we can formulate the notion in another way. This is based on the observation that anti-self-dual 2-forms are exactly ω-trace free (1, 1)-forms. Thus, in the almost Hermitian case, we can equally define anti-self-dual instantons to be those connections A satisfying

2,0 F = trωF = 0. (2.1)

42 The latter description obviously allows generalizations to higher dimensional almost Hermitian manifolds. We will also call connections satisfying (2.1) pseudo- Hermitian-Yang-Mills by slight abuse of terminology (compare [4], for example). When the dimension is 6, we can formulate (2.1) in yet another way. Notice that the operator ∗(ω ∧ ·) maps the space of two forms into itself. It can also be shown that the space of ω-trace free (1, 1)-forms is exactly the −1 eigenspace of ∗(ω ∧ ·). Thus, we can rewrite the equation (2.1) as

ω ∧ F = − ∗ F. (2.2)

For this reason, we also call pseudo-Hermitian-Yang-Mills connections ω-anti-self- dual instantons. Now, (2.2) makes sense in even more general contexts. Suppose that M is endowed with an n − 4 form Ω. Then the operator ∗(Ω ∧ ·) maps 2-forms into 2-forms. We can deﬁne Ω-anti-self-dual instantons to be those connections A whose curvatures F satisﬁes Ω ∧ F = − ∗ F. (2.3)

This definition behaves the best when M has a special structure such as SU(3), G2 or Spin(7). In this situation, Ω is naturally defined, i.e., Ω is the Kählerform for an

SU(3)-structure, the defining 3-form for a G2-structure, or the defining 4-form for a Spin(7)-structure. However, even when Ω is parallel, (2.3) is in general overdetermined. It is natural to ask when (2.3) has solutions, even locally, and how general they are. In dimension 6, R. Bryant showed in [4] that there is a large class of almost Hermi- tian structures, called quasi-integrable, for which the differential system for pseudo- Hermitian-Yang-Mills SU(n)-connections is involutive. Thus the theory behaves well in quasi-integrable case. It is interesting to ask under what conditions other instanton differential systems will be involutive.

43 In this chapter, we are mainly interested in ω-anti-self-dual instantons on a nearly

Kähler6-manifold and Ω-anti-self-dual instantons on its G2-cone. We first show that ω-anti-self-dual instantons are automatically Yang-Mills, i.e., are critical points of the Yang-Mills functional. We prove the involutivity of the ω-anti-self-dual instanton system. We construct a Chern-Simons type functional on nearly Kähler6-manifold. This is an R-valued functional, rather than R/Z-valued as in 3-manifold case. We show that its critical connections are exactly the ω-anti-self-dual instantons. We compute its gradient flow and discuss its relation with Ω-instantons on the G2-cone. Second, we derive a Weitzenböck formula for an elliptic operator on nearly Kähler manifolds and apply it to study deformations of ω-anti-self-dual instantons. Finally, we construct a class of instantons on S6 and R7 that display interesting singularities.

2.1 Some linear algebra in 6 and 7 dimensions

In this section, we clarify notational convention of inner product spaces in 6 and 7 dimensions with emphasis on representation theory of SU(3) and G2. The interplay between Hodge star operations will be important in later discussions.

n Suppose V is an n-dimensional oriented inner product space and let {ei}i=1 be a oriented orthonormal basis. The inner product on V induces an inner product h, i

∗ on its dual V with the dual basis denoted by {dxi}. By taking the convention that

∗ ∗ {dxi1 ∧ · · · ∧ dxik } be orthonormal, we make Λ V an inner product space. We deﬁne Hodge star ∗ on Λ∗V ∗ by the following rule. Let φ ∈ Λ∗V ∗ and its Hodge star ∗φ is determined by

∗ φ ∧ ψ = hφ, ψivolV . (2.4)

∗ ∗ for any ψ ∈ Λ V where volV = dx1 ∧ · · · ∧ dxn is the volume form on V .

44 Remark 2.1.1. Through the inner product, we identify vectors and 1-forms. We will not distinguish between them. Thus for example, an linear operator deﬁned on vectors may be thought of as an operator on 1-forms. No confusion should be caused.

2.1.1 Dimension 6

In dimension 6, we suppose further that V is endowed with a complex structure and a complex volume form Ψ. The complex structure coupled with the inner product

1 3 determines a symplectic form ω on V . We normalize these quantities so that 6 ω =

i ∗ 8 Ψ ∧ Ψ = volV . It is now natural to complexify V and its various exterior powers.

∗ ∗ ∗ ∗ Denote VC the space of complex linear forms on V . Then V ⊗ C = VC ⊕ VC. We extend the inner product and Hodge star operation complex linearly to V ⊗ C. √ 3 ∗ We pick an orthonormal basis {dxi, dyi}i=1 for V such that dzi = dxi + −1dyi is complex linear and that √ −1 ω = (dz ∧ dz + dz ∧ dz + dz ∧ dz ), Ψ = dz ∧ dz ∧ dz . 2 1 1 2 2 3 3 1 2 3

SU(3)-representations

The subgroup of SO(6) preserving both ω and ψ is the special unitary group SU(3). Under the action of SU(3), Λ∗V ∗ ⊗ C may be decomposed into irreducible pieces

∗ ∗ ∗ V ⊗ C = VC ⊕ VC

2 ∗ 2 ∗ 2 ∗ (1,1) Λ V ⊗ C = ∧ VC ⊕ ∧ VC ⊕ C · ω ⊕ V

3 ∗ (2,0) (0,2) ∗ ∗ Λ V ⊗ C = C · Ψ ⊕ C · Ψ ⊕ V ⊕ V ⊕ VC ∧ ω ⊕ VC ∧ ω

4 ∗ ∗ ∗ 2 (1,1) Λ V ⊗ C = VC ∧ Ψ ⊕ VC ∧ Ψ ⊕ Cω ⊕ VC ∧ ω

5 ∗ 2 ∗ 2 Λ V ⊗ C = VC ∧ ω ⊕ VC ∧ ω ,

where V (1,1) denotes the representation of the highest weight (1, 1), which consists of

(0,2) 2 ∗ (1, 1)-forms whose inner product with ω is zero, V ' sym VC is the representation 45 of the highest weight (0, 2) and V (0,2) ' V (2,0). The decomposition of 2-forms and 4- forms will be the most important for us. Note that the wedge product with ω gives an isomorphism between the irreducible pieces in Λ2 and Λ4 as outlined above. Another isomorphism is given by Hodge star. These two isomorphisms will be fundamental in the deﬁnition of anti-self-dual instantons later, so we examine their relation carefully below.

Hodge star

It is easy to compute that √ −1 ∗(dz ∧ dz ) = dz ∧ dz ∧ dz ∧ dz 1 2 2 1 2 3 3 √ −1 ∗(dz ∧ dz ) = dz ∧ dz ∧ dz ∧ dz 2 3 2 1 2 3 1 and √ −1 ∗(dz ∧ dz ) = dz ∧ dz ∧ dz ∧ dz . 3 1 2 1 2 3 2 Also i ω ∧ dz ∧ dz = dz ∧ dz ∧ dz ∧ dz 1 2 2 1 2 3 3

and similarly for dz2 ∧ dz3, dz3 ∧ dz1. Thus we have

∗ α = ω ∧ α (2.5)

2 ∗ 2 ∗ for any α ∈ ∧ VC ⊕ ∧ VC. Moreover, 1 ∗ ω = ω2. (2.6) 2 On the other hand, √ −1 ∗(dz ∧ dz ) = − dz ∧ dz ∧ dz ∧ dz = −ω ∧ (dz ∧ dz ). 1 2 2 1 2 3 3 1 2 46 More generally, we have ∗ α = −ω ∧ α (2.7)

for any α ∈ V (1,1). To conclude, the irreducible (real) SU(3)-modules in Λ2V ∗ are indexed by the eigenvalues of the operator ∗(ω∧) (note ∗2 = 1 on 2-forms).

∗ 2 ∗ The other chain of isomorphic SU(3)-representations consists of VC, ∧ VC and various Hodge star images. Again, there are many isomorphisms among these spaces given by compositions of Hodge star, wedge product with the Ψ and with ω. We exploit some of them. First, we compute that

√ −1 ∗(dz ) = dz ∧ dz ∧ dz ∧ dz ∧ dz , 3 4 1 2 3 1 2

and thus, √ −1 ∗ (dz ∧ dz ∧ dz ∧ dz ∧ dz ) = −dz . 4 1 2 3 1 2 3

On the other hand √ −1 ImΨ ∧ dz ∧ dz = (dz ∧ dz ∧ dz ∧ dz ∧ dz ). 1 2 2 1 2 3 1 2

Thus we have

√ ImΨ ∧ ∗(ImΨ ∧ dz1 ∧ dz2) = − −1dz1 ∧ dz2 ∧ dz3 ∧ dz3.

It is easy to see

√ −1 ω ∧ dz ∧ dz = − dz ∧ dz ∧ dz ∧ dz 1 2 2 1 2 3 3

and thus

ImΨ ∧ ∗(ImΨ ∧ dz1 ∧ dz2) = 2ω ∧ dz1 ∧ dz2. 47 2 ∗ Because ∧ VC is an irreducible SU(3)-representation, it must hold that

ImΨ ∧ ∗(ImΨ ∧ α) = 2ω ∧ α. (2.8)

Some linear operators

We use the SU(3)-representation theory to describe several useful linear operators. Some of them are standard, but we hope to ﬁx notation.

∗ k ∗ k−1 ∗ First, we describe y. For any 1-form v ∈ V , vy :Λ V → Λ V is deﬁned as

X i−1 vy(α1 ∧ · · · αk) = (−1) hv, αiiα1 ∧ · · · αˆi ∧ · · · ∧ αk i

where hi is the inner product. Note that this is adjoint to the wedge product in the sense that

hvyα, βi = hα, v ∧ βi.

∗ ∗ We extend y complex linearly to V ⊗ C and ΛC. Something must be cautioned. For instance

dz1ydz1 = 0.

∗ ∗ ∗ Next, we use y to identify Λ V inside so(V ) by

β : α 7→ αyβ.

The inverse map is given by for any A ∈ so(V ∗)

1 X A :7→ ω ∧ A(ω ) 2 i i

where ωi is an orthonormal basis.

∗ ∗ Now, if a linear map commutes with the complex structure on V , i.e., maps VC to itself, then it is easy to see that viewed as a 2-form, A lies in the space Λ1,1. In fact, the corresponding 2-form is given by

1 dz ∧ A(dz ). 2 i i 48 Since Ψ is SU(3)-invariant, any linear combination r of maps v 7→ vyRe(Ψ) and ∗ 2 ∗ v 7→ vyImΨ gives an SU(3)-equivariant map V → ∧ V . The image r(v) may be viewed as a map V ∗ → V ∗. Skewsymmetrizing r(v) gives a map ∧2V ∗ → ∧2V ∗

r(v): α ∧ β 7→ r(v)(α) ∧ β + α ∧ r(v)(β).

We still denote the map by r(v). From SU(3)-equivariance of r, we see that

r(g(v))(α) = g(r(v)(g−1α)), (2.9)

for any g ∈ SU(3), v ∈ V and α ∈ ∧2V ∗. We deﬁne

1 1 π(2,0)(β) = 4 hβ, dz1 ∧ dz2idz1 ∧ dz2 + 4 hβ, dz1 ∧ dz2idz1 ∧ dz2

1 1 + 4 hβ, dz1 ∧ dz3idz1 ∧ dz3 + 4 hβ, dz1 ∧ dz3idz1 ∧ dz3 (2.10)

1 1 + 4 hβ, dz2 ∧ dz3idz2 ∧ dz3 + 4 hβ, dz2 ∧ dz3idz2 ∧ dz3,

dually

√ 1 1 − −1π(0,2)(β) = 4 hβ, dz1 ∧ dz2idz1 ∧ dz2 − 4 hβ, dz1 ∧ dz2idz1 ∧ dz2

1 1 + 4 hβ, dz1 ∧ dz3idz1 ∧ dz3 − 4 hβ, dz1 ∧ dz3idz1 ∧ dz3 (2.11)

1 1 + 4 hβ, dz2 ∧ dz3idz2 ∧ dz3 − 4 hβ, dz2 ∧ dz3idz2 ∧ dz3.

and

1 π (β) = hβ, ωiω (2.12) ω 3 where the bracket is the complex extension of the inner product. Note that both π(2,0) and π(0,2) are real operators. Also deﬁne the projection onto ω-trace free 2-forms

1,1 π0 = I − π(2,0) − πω. (2.13)

49 Note that π(2,0) are identity on forms of type (2, 0) and type (0, 2). While π(0,2) is √ √ multiplication by −1 on (2, 0) forms and − −1 on (0, 2) forms. Both of them are clearly SU(3) equivariant. In fact, if we think of the diagonal elements in Λ2V ∗⊕Λ2V ∗ as a real representation of SU(3), the space of SU(3) equivariant homomorphisms is real 2-dimensional, spanned by π(2,0) and π(0,2). They satisfy the relation

2 2 π(2,0) = π(2,0), π(0,2) = −π(2,0). (2.14)

In particular, π(2,0) is a projection but π(0,2) is not. Denote

P = λπ(2,0) + µπω (2.15) where λ and µ are real constants. Clearly, P is a real operator and commutes with

2 2 2 the action of SU(3). Moreover, P = λ π(2,0) + µ πω.

6 Let {vi}i=1 be a orthonormal basis of V and ωi dual basis. We deﬁne a map by

X 2 B(α) = ωiy[r(vi),P ](α) (2.16) i where α ∈ ∧2V ∗. Note that the deﬁnition of B does not depend on the choice of the orthonormal basis. We have the following result concerning the B.

Proposition 2.1.2. The operator B factors through a (possibly complex) linear combination of π(2,0) and π(0,2).

Proof. For any α ∈ ∧2V ∗ and g ∈ SU(3), we have

P 2 B(α) = i ωiy[r(vi),P ](gα)

P −1 2 = i ωiyg([r(g (vi)),P ](α))

P −1 −1 2 = g( i g (ωi)y([r(g (vi)),P ](α))

= g(B(α)). 50 where the second equality is due to (2.9) as well as the commutativity of P and SU(3), the third is because g(vyα) = g(v)yg(α) and the last is because of the independence of orthonormal coframes in the deﬁnition of B. So B gives a SU(3)-equivariant map from ∧2V ∗ → V ∗. Since as a SU(3)-space, ∧2V ∗ contains only a copy of the

∗ 2 ∗ 2 ∗ irreducible representation isomorphic to V , namely, (∧ VC ⊕∧ VC)R, we know from

Schur’s Lemma, B must factor through a linear combination of π(2,0) and π(0,2).

Next, for each i, j, we consider the operator on V ∗,

2 L(ωi, ωj)(α) = ωi ∧ (ωjyα) + ωiyP (ωj ∧ α). (2.17)

We have the following result concerning L. √ √ Proposition 2.1.3. Let λ = 2 and µ = 3 and thus √ √ P = 2π(2,0) + 3πω. (2.18)

Then the operator L satisﬁes the Cliﬀord relations, i.e.,

L(ωi, ωj) + L(ωj, ωi) = 2δij.

2 6 6 Moreover, we deﬁne an operator M : ∧ R → End(R ) by linearly extending L(ωi, ωj) for i 6= j. Then M is an SU(3) equivariant map from Λ2V ∗ to V ⊗ V ∗. In fact, we have

1,1 M(β)(v) = vy(−2π0 β + πωβ).

Proof. Since L is real, it suﬃces to prove the proposition for (1, 0) forms. Without

loss of generality, we check for L(dx1, dx1), L(dx1, dy1) and L(dx1, dx2). Let αi = √ ∗ dxi + −1dyi. These form a basis for VC. Then

2 L(dx1, dx1)(α1) = dx1 ∧ (dx1yα1) + dx1yP (dx1 ∧ α1)

2 2 1 = dx1 + dx1y(λ π(2,0) + µ πω)( 2 dz1 ∧ dz1) √ −1 2 = dx1 + dx1y 3 µ ω √ = dx1 + −1dy1 51 because µ2 = 3. For i = 2, 3

2 L(dx1, dx1)(αi) = dx1 ∧ (dx1yαi) + dx1yP (dx1 ∧ αi)

2 2 1 = 0 + dx1y(λ π(2,0) + µ πω)( 2 (dz1 + dz1) ∧ dzi)

2 1 = dx1yλ 2 dz1 ∧ dzi

λ2 = 2 dzi

= αi because λ2 = 2. This proves the ﬁrst equality.

Now consider L(dx1, dy1). We compute

2 L(dx1, dy1)(α1) = dx1 ∧ (dy1yα1) + dx1yP (dy1 ∧ α1) √ 2 2 √−1 = −1dx1 + dy1y(λ π(2,0) + µ πω)( 2 −1 dz1 ∧ dz1)

√ µ2 √ = −1dx1 + dx1y 3 −1ω √ 1 2 = −1dx1 − 3 µ dx1yω √ = −1dx1 − dy1 √ = −1(dz1)

and

2 L(dy1, dx1)(α1) = dy1 ∧ (dx1ydz1) + dy1yP (dx1 ∧ dz1)

2 2 1 = dy1 + dy1y(λ π(2,0) + µ πω)( 2 dz1 ∧ dz1)

2 1 = dy1 + dy1yµ πω( 2 dz1 ∧ dz1) √ 2 −1 = dy1 + dy1yµ 3 ω √ = dy√1 − −1dx1√ = − −1(dx1 + −1dy1). 52 Thus

L(dx1, dy1)(α1) + L(dy1, dx1)(α1) = 0.

Also,

2 L(dx1, dy1)(α2) = dx1 ∧ (dy1ydz2) + dx1yP (dy1 ∧ α2)

2 2 √1 = 0 + dx1y(λ π(2,0) + µ πω)( 2 −1 (dz1 ∧ dz2 − dz1 ∧ dz2)) √ = − −1dx1y(dz1 ∧ dz2) √ = − −1dz2

and 2 L(dy1, dx1)(α2) = dy1 ∧ (dx1ydz2) + dy1yP (dx1 ∧ dz2)

2 1 = dy1y(λ π2,0)( 2 dz1 ∧ dz2) √ = −1dz2.

Thus

L(dx1, dy1)(α2) + L(dy1, dx1)(α2) = 0.

Similarly

L(dx1, dy1)(α3) + L(dy1, dx1)(α3) = 0.

Next we consider L(dx1, dx2).

2 L(dx1, dx2)(dz1) = dx1 ∧ (dx2ydz1) + dx1yP (dx2 ∧ dz1)

2 2 1 1 = 0 + dx1y(λ π(2,0) + µ πω)( 2 dz2 ∧ dz1 + 2 dz2 ∧ dz1)

2 1 = dx1yλ π(2,0)( 2 dz2 ∧ dz1)

= dx1y(dz2 ∧ dz1)

= −dz2

53 and 2 L(dx2, dx1)(dz1) = dx2 ∧ (dx1ydz1) + dx2yP (dx1 ∧ dz1)

2 2 1 = dx2 + dx2y(λ π(2,0) + µ πω)( 2 dz1 ∧ dz1)

2 1 = dx2 + µ πω( 2 dz1 ∧ dz1) √ = dx2 + −1dx2yω √ = dx2 + −1dy2 = dz2

Thus

L(dx1, dx2)(dz1) + L(dx2, dx1)(dz1) = 0.

Similarly

L(dx1, dx2)(dz2) + L(dx2, dx1)(dz2) = 0.

Moreover,

2 L(dx1, dx2)(dz3) = dx1 ∧ (dx2ydz3) + dx1yP (dx2 ∧ dz3)

2 2 1 1 = 0 + dx1y(λ π(2,0) + µ πω)( 2 dz2 ∧ dz3 + 2 dz2 ∧ dz3)

2 1 = dx1yλ π(2,0) 2 dz2 ∧ dz3

= 0. and 2 L(dx2, dx1)(dz3) = dx2 ∧ (dx1ydz3) + dx2yP (dx1ydz3)

2 1 = 0 + dx2yλ 2 dz1 ∧ dz3

= 0.

Thus

L(dx1, dx2)(dz3) + L(dx2, dx1)(dz3) = 0.

So far we have proved that

L(dx1, dx1) = 1 54 and

L(dx1, dy1) + L(dy1, dx1) = L(dx1, dx2) + L(dx2, dx1) = 0

. By symmetry and the linearity of L we see that

L(dxi, dxi) = L(dyi, dyi) = 1

L(dxi, dxj) + L(dxj, dxi) = 0, i 6= j and

L(dxi, dyj) + L(dyj, dxi) = 0

. For instance, in order to show

L(dx1, dy2) + L(dy2, dx1) = 0

we replace dx2 by dy2 and dy2 by −dx2. Then it follows from the calculation on

L(dx1, dx2).

Now for arbitray orthonormal basis ωi, the Cliﬀord relations follow from the fact that the orthogonal transformations act transitively on coframes. If they hold for a particular coframe, they hold for all. Suppose g ∈ SU(3). We have for any 1-form α,

2 L(ωi, ωj)(gα) = ωi ∧ ωjyg(α) + ωiyP (ωj ∧ g(α))

−1 −1 −1 2 −1 = g[(g ωi) ∧ (g ωj)yα + (g ωi)yP (g (ωj) ∧ α)]

−1 −1 = gL(g ωi, g ωj)(α).

Now by the deﬁnition of M we have for any α = aiωi and β = bjωj,

1 M(α, β) = (a b − a b )L(ω , ω ). 2 i j j i i j

55 Thus,

1 M(gα, gβ)(v) = 2 (aibj − ajbi)L(gωi, gωj)(v)

1 −1 = 2 (aibj − ajbi)gL(ωi, ωj)g (v)

= gM(α, β)g−1(v)

i.e., M is SU(3) equivariant. Note from the above computations, M(β) maps (1, 0) forms to (1, 0) forms for any two-form β. Moreover, since P 2 is self-adjoint, M(β) also preserves the inner product. Thus M(β), when identiﬁed as a two-form, takes value in Λ1,1. Combined with the SU(3)-equivariance, M gives a SU(3)-equivariant map from Λ2V ∗ to Λ1,1. Since

1,1 2 1,1 both of the two irreducible components Λ0 and Rω are real, HomSU(3)(Λ , Λ ) is real 2-dimensional. In other words, there exist two constants a, b so that

1,1 M(β) = aπ0 (β) + bπω(β).

It is a matter of computing examples to determine the constants.

If we take β = dx1 ∧ dy1, then by the convention described above,

1 M(β) = (dz ∧ M(β)(dz ) + dz ∧ M(β)(dz ) + dz ∧ M(β)(dz )) 2 1 1 2 2 3 3 √ −1 = (dz ∧ dz − dz ∧ dz − dz ∧ dz ) 2 1 1 2 2 3 3

= −(dx1 ∧ dy1 − dx2 ∧ dy2 − dx3 ∧ dy3).

On the other hand 1 π (dx ∧ dy ) = ω. ω 1 1 3 and 1 π1,1(dx ∧ dy ) = (2dx ∧ dy − dx ∧ dy − dx ∧ dy ). 0 1 1 3 1 1 2 2 3 3 Consequently a = −2, b = 1.

56 Thus M is of the desired form.

2.1.2 Dimension 7

Now we assume that dimV = 7 and we pick an oriented orthonormal basis for V ∗ denoted by {dx1, dy1, dx2, dy2, dx3, dy3, du}. For later use, let √ dzi = dxi + −1dyi

and deﬁne ω and Ψ as in §2.1. We introduce a special three form

Ω = du ∧ ω + ImΨ = du ∧ (dx ∧ dy + dx ∧ dy + dx ∧ dy ) 1 1 2 2 3 3 (2.19) +dx1 ∧ dx2 ∧ dy3 − dy1 ∧ dy2 ∧ dy3 +dy1 ∧ dx2 ∧ dx3 + dx1 ∧ dy2 ∧ dx3.

Due to [7], it is now well-known that the exceptional Lie group G2 may be deﬁned as the stabilizers of Ω. For this reason, we call Ω the fundamental 3-form. We embed R6 considered in the last section into V to be the hyperplane du = 0. We also let

SU(3) act on V by identity on the line dxi = dyi = 0 and the standard action on

du = 0. Clearly, SU(3) preserves Ω, so it embeds into G2 as a Lie subgroup.

G2-representations

A good resource on this part is [11]. We recall some basic facts. The standard V ∗ is irreducible with the highest weight (1, 0). The most important part for us is Λ2V ∗. It decomposes as the sum of two irreducible pieces V (1,0) ⊕ V (1,1) where V (a,b) is the

(1,0) irreducible representation of G2 with the highest weight (a, b). The subspace V

∗ (1,1) is 7-dimensional, consisting of 2-forms vyΩ for any v ∈ V . The other one V is

isomorphic to the Lie algebra g2.

5 ∗ 2 The space Λ V is isomorphic to Λ as G2-modules either by wedge product with Ω or by the Hodge star operation. Again the interplay between these two isomorphisms will be important in deﬁning anti-self-dual instantons in dimesion 7.

57 Hodge star

We only consider the Hodge star on Λ2. Now we may compute that

1 ∗ (α) = Ω ∧ (α) (2.20) 2

for all α ∈ V (1,0) ⊂ Λ2 and that

∗ α = −Ω ∧ α (2.21)

(1,0) for α ∈ g2. These may be checked for special forms (e. g., α = duyΩ ∈ V and

α = dz1 ∧ dz2 ∈ su(3) ⊂ g2). Then, since these spaces are irreducible and both * and

Ω∧ commutes with G2 action, we know these relations must be true for the whole spaces. Thus, these irreducible subspaces are indexed by the eigenvalues of ∗(Ω∧).

2.2 Anti-self-dual instantons on nearly K¨ahler 6-manifolds and G2- cones

Let G be a compact Lie group. Suppose Xn is a smooth manifold endowed with an (n-4)-form Υ (for our purposes, X = M is nearly K¨ahlerand Υ is the (1, 1)-form ω,

or X = N has G2 holonomy and Υ is the fundamental 3-form Ω). Suppose also Υ is a (n-4)-form on M and P is a principal G-bundle over X. A connection A on P is

called Υ-instanton if its curvature FA satisﬁes

Υ ∧ FA = − ∗X FA. (2.22)

Remark 2.2.1. When G is a unitary group, our definition is different from the one used in [27] (see Remark 1 in its §1.2, however). When G is a special unitary group, these two definitions coincide. This is the group we will use mostly.

58 Note that if Υ is closed, an Υ-instanton A is Yang-Mills, i.e., it satisﬁes the Euler-Lagrange equation of the Yang-Mills functional, since

dA ∗X F = −dΥ ∧ F ± Υ ∧ dAF = 0

because of the Bianchi identity. Thus, an Ω-instanton on a manifold with holonomy

in G2 is Yang-Mills since Ω is closed. Remarkably, as we will show later, when X = M is nearly K¨ahler,although ω is not closed, an ω-anti-self-dual instanton is still Yang-Mills.

2.2.1 Nearly K¨ahler6-manifolds

In this subsection, we collect basic facts about nearly Kähler6-manifolds. The concept was first introduced and studied by A. Gray in [18]. Later on, N. Hitchin [22] found that it is a critical point of a diffeomorphism invariant functional and thus put it in a more natural context. An SU(3) structure on a 6-manifold M is a reduction of the total coframe bundle to an SU(3) subbundle. It may be specified by a real two-form ω of type (1, 1) and

1 3 i a (3, 0)-form Ψ normalized so that 6 ω = 8 Ψ ∧ Ψ. A nearly K¨ahlerstructure is an SU(3)-structure for which

dω = 3cImΨ, dΨ = 2cω2. (2.23)

for some real constant c. When c = 0, the underlying almost complex structure is integrable. In fact, M is Calabi-Yau. When c 6= 0, by scaling the metric, we can always assume c = 1. In this situation, M is usually called strictly nearly K¨ahler. In this chapter, we assume from now on that c = 1 and we speak of this as nearly K¨ahler without danger of confusion.

59 Structure equations

Let αi, i = 1, ··· , 3 be a local special unitary coframe, i.e., αi is complex linear and √ −1 ω = (α ∧ α + α ∧ α + α ∧ α ), Ψ = α ∧ α ∧ α . 2 1 1 2 2 3 3 1 2 3

There exists a unique su(3)-valued 1-form (κi¯j) so that

dαi = −κij ∧ αj + ijkαj ∧ αk (2.24) where summation is understood when repeated barred and unbarred indices appear. Diﬀerentiate this and we get the curvature of κ:

1 dκ + κ ∧ κ = (3α ∧ α − δ α ∧ α ) + K α ∧ α , (2.25) ij ik kj 4 i j ij l l ijpq q p where Kijpq = Kpjiq = Kiqpj = Kjiqp and Kiipq = 0. It follows from the structure equations that κ is a pseudo-Hermitian-Yang-Mills connection on the complex tangent bundle of M. Compact nearly Kählerexamples include the standard S6, the flag manifold SU(3)/T 2, S3 × S3, and CP3 (with an unusual almost complex structure). All these examples are homogeneous. On the other hand, it remains open to find nonhomogeneous compact examples.

6 Example 2.2.2 (G2-invariant S ). The standard G2 invariant almost complex structure on S6 is perhaps the best known non-integrable almost complex structure. As

6 a subgroup of SO(7), G2 acts transitively on S and the stabilizer of any point is

6 isomorphic to SU(3) ⊂ G2. Thus, G2 preserves an SU(3)-structure on S . This

SU(3) structure is in fact nearly K¨ahler. Using Maurer-Cartan forms on G2, we write the nearly K¨ahlerstructure equations

dαi = −κij ∧ αj + ijkαj ∧ αk (2.26)

1 dκ + κ ∧ κ = (3α ∧ α − δ α ∧ α ). (2.27) ij ik kj 4 i j ij l l

60 2.2.2 G2-cones over a nearly K¨ahler6-manifold

7 A G2-structure on a 7-manifold N is a reduction of the total coframe bundle to

7 a G2-subbundle. Suppose N has such a G2 structure. Then, on N, there exists a fundamental 3-form Ω characterized by the property that at each point x, there

7 ∗ exists a linear isomorphism u : TxN → R so that Ωx = u (Ω0). Conversely, given

such a fundamental 3-form Ω on N, the set of such linear isomorphisms forms a G2 subbundle of the total coframe bundle and thus deﬁnes a G2-structure on N.

Associated with any G2-structure, N has a metric g. The Levi-Civita connection of g has its holonomy group contained in G2 if and only if dΩ = d(∗Ω) = 0. R.

Bryant constructed the ﬁrst metric with holonomy G2 [7]. It was the cone metric

2 5 over R+ × SU(3)/T . It is now well-known that if M is nearly K¨ahlerwith the

6 metric gM , the cone metric on N = R+ × M deﬁned as

2 2 gN = dt + t gM

has holonomy in G2. The fundamental 3-form is

Ω = t2dt ∧ ω + t3ImΨ.

Such conical G2-singularities were used by string physicists recently to construct string models with chiral matter ﬁelds (see [2], [1]). For us, the case M = S6 is especially important. Then the cone has a removable singularity and in fact N = R7.

7 When studying anti-self-dual instantons on manifolds with G2 holonomy, R plays the natural role of an inﬁnitesimal model.

Hodge star on 2-forms

Suppose ωi(i = 1, ··· , 6) is an oriented local orthonormal coframe for M. Then, dt, tωi(i = 1, ··· , 6) form an oriented local orthonormal coframe for the cone N.

61 Denote ∗M and ∗N Hodge star operations on M and N respectively. It is easy to show that

2 t dt ∧ ∗M (ωi ∧ ωj) = ∗N (ωi ∧ ωj) and

4 ∗N (dt ∧ ωi) = t ∗M (ωi).

∂ Consequently, if a 2-form α on N satisﬁes ∂t yα = 0, its Hodge star may be computed by

2 ∗N α = t dt ∧ ∗M (α), (2.28)

∂ and if α = dt ∧ β with ∂t yβ = 0,

4 ∗N (dt ∧ β) = t ∗M (β), (2.29)

where we extend ∗M linearly across functions on N. The formula (2.28), (2.29) will be important below.

2.2.3 ω anti-self-dual instantons

If the underlying manifold is almost Hermitian with the K¨ahlerform ω, we may decompose the curvature as

F = F 2,0 + F 2,0 + (F ◦)1,1 + Hω, where F 2,0 is of type (2, 0) and (F ◦)1,1 is of type (1, 1) but with zero ω-trace. Now the ω-anti-self-dual instanton condition (2.22) implies F 2,0 = H = 0.

Remark 2.2.3. In the case G is a special unitary group, the above argument implies that an ω-instanton is the same as a pseudo-Hermitian-Yang-Mills connection on the canonically associated complex vector bundle.

If, moreover, we are working on a nearly K¨ahlermanifold, this condition may be simpliﬁed.

62 Lemma 2.2.4. Suppose A is a connection on nearly K¨ahler M 6 and F is its curvature. The following are equivalent:

a. F ∧ ImΨ = 0.

b. F ∧ Ψ = 0.

c. F ∧ ReΨ = 0.

d. A is an ω-anti-self-dual instanton.

Consequently, if F is of type (1, 1), A is an ω-anti-self-dual instanton.

Proof. 1. a=⇒b. We write F = F 2,0 + F 2,0 + (F ◦)1,1 + Hω. Then F ∧ ImΨ = 0 gives F ∧ (Ψ − Ψ) = 0,

i.e., F 2,0 ∧ Ψ = 0.

It follows then that F ∧ Ψ = 0

and hence F ∧ Ψ = 0.

2. b=⇒c is obvious.

3. c=⇒d. As mentioned before, A is ω-anti-self-dual if and only if F 2,0 = 0 and H = 0 in the above decomposition. Now

1 1 F ∧ ReΨ = F ∧ (Ψ + Ψ) = F 2,0 ∧ Ψ + F 2,0 ∧ Ψ. 2 2 63 Thus c gives F 2,0 = 0. Diﬀerentiating c gives

0 = dA(F ∧ ReΨ)

= dAF ∧ ReΨ + F ∧ dReΨ

= 0 + 2F ∧ ω2

= 2Hω3

where the last equality uses (2.23) and the Bianchi identity. Hence

H = 0.

4. d=⇒a is obvious.

This lemma says that we could have deﬁned an ω-anti-self-dual as F 2,0 = 0. This reduces the indeterminacy and will be useful later when we construct concrete examples.

Remark 2.2.5. The same result holds for a more general class of almost complex manifolds, called strictly quasi-integrable in [4]. We leave it for the reader to carry out the details. In fact, this has already been observed in [4] for unitary instantons.

Generality

We now address the problem of the involutivity problem of the instanton equations. First, the instanton equations may be rephrased as

F ∧ Ψ = 0 (2.30) and F ∧ ω2 = 0. (2.31)

64 Since the problem is local, we assume that the bundle is trivial, and the connection is simply a g-valued 1-form A. The diﬀerential system we need to analyze is

I = hF ∧ Ψ,F ∧ ω2i,

1 deﬁned on M × g where F = dA + 2 [A, A]. We have

Lemma 2.2.6. The system I is involutive with Cartan characters

(s0, s1, s2, s3, s4, s5, s6) = (0, 0, 0, 0, 2d, 3d, d). where d = dim G.

Proof. Note that

d(F ∧ Ψ) = [A, F ] ∧ Ψ + F ∧ dΨ ≡ 0, mod I. because of Bianchi identity and the nearly K¨ahlercondition dΨ = 2ω2. It is now routine to check the system is involutive with displayed characters.

Some remarks are in order.

Remark 2.2.7. More generally, Lemma 2.2.6 holds for similarly deﬁned ω-instantons on a quasi-integrable U(3)-structure (see [4] for the deﬁnition). We leave the details for the interested readers. When G = U(r) is a unitary group, it is treated in [4].

Remark 2.2.8. For nearly K¨ahler M, we could have used the diﬀerential system hF ∧ ImΨi by Lemma 2.2.4. The reader can check that this is involutive with the last Cartan character also equal to d. The advantage of the original system is that it applies to more general almost complex manifolds.

Remark 2.2.9. The last character is d, due to the fact that gauge transformations depend on d functions of 6 variables and that instanton equations are gauge-invariant. We leave for the interested reader to impose a symmetry breaking condition.

65 Instantons are Yang-Mills

Now we compute

dA ∗M F = −dω ∧ F − ω ∧ dAF = −3ImΨ ∧ F = 0

because F is of type (1, 1).

Proposition 2.2.10. An ω-instanton on a nearly K¨ahler6-manifold is Yang-Mills.

A consequence is some removable singularity results for instantons on nearly K¨ahler6-manifolds.

Corollary 2.2.11. Suppose that all representations of π1(M) → G are trivial and that E is a trivial smooth bundle over M. Assume that A is a ω-instanton on E with a closed singular set S whose n − 4 Hausdorﬀ measure is locally ﬁnite. Then there exists = (G, M) such that if

k FA k∞≤ ,

then the singularity of A is removable.

Corollary 2.2.12. Suppose that all representations of π1(M) → G are trivial and that E is a trivial smooth G bundle over M. Assume that A is a ω-instanton on E whose singular set is a closed smooth submanifold of codimension at least 4. Then there exists = (G, M) such that if

k FA k 6 ≤ , L 2 (M) then the singularity of A is removable.

Both are proved by employing the results in [29].

66 Instantons as critical points of a Chern-Simons functional

Consider the functional Z 2 CS(A) = tr(FA) ∧ ω. (2.32) M

On a Kählermanifold, since ω is closed, CS is a topological constant. However, on a nearly Kählermanifold, this gives more interesting information. It is easy to compute that the first variation of CS is

Z δCS = 2 tr(FA ∧ dAδA) ∧ ω. M

Integration by parts gives

Z δCS = 2 tr[dA(FA ∧ ω) ∧ δA]. M

Thus the Euler-Lagrange equation for CS is

dA(FA ∧ ω) = 0.

Using Bianchi Identity, we see this is equivalent to

F ∧ ImΨ = 0. (2.33)

It follows from Lemma 2.2.4 that

Proposition 2.2.13. An ω-anti-self-dual instanton is equivalent to a critical connection of the Chern-Simons functional CS.

This makes it possible to use variational methods to study ω-anti-self-dual instantons on nearly K¨ahler6-manifolds. It also follows that the gradient ﬂow of CS takes the form

d A = ∗ (F ∧ ImΨ). (2.34) dt M

67 Remark 2.2.14. To illustrate, we assume that the principal bundle under consideration is topologically trivial. Using dω = 3ImΨ and transgression formula, it can be shown that up to a constant

Z 1 CS(A) = tr(F ∧ A − A ∧ A ∧ A) ∧ ImΨ. (2.35) M 3

Here we regard G as a matrix Lie group. This formulation is more similar to the Chern-Simons functional on 3-manifolds.

Next we compute the second variation Q of CS. Suppose that A(s, t) (for small s, t) are a two parameter family of connections such that A = A(0, 0) is an instanton.

∂A ∂A Let a = ∂s |s=0,t=0, b = ∂t |s=0,t=0.Then by deﬁnition

∂2 Q(a, b) = | CS(A) ∂s∂t s=0,t=0

We have essentially computed that

∂ Z ∂ CS(A) = −6 tr(FA ∧ ImΨ ∧ A(s, t)). ∂t M ∂t

Thus the second derivative is (remember FA ∧ ImΨ = 0)

Z ∂A ∂A Q(a, b) = −6 tr(dA( |s=0,t=0) ∧ ImΨ ∧ |s=0,t=0) (2.36) M ∂s ∂t Z = −6 tr(dAa ∧ ImΨ ∧ b) (2.37) M

Clearly, this is a symmetric bilinear form. The null space of Q consists of a so that

dAa ∧ ImΨ = 0.

This implies dAa is of type (1, 1) and hence

dAa ∧ Ψ = 0. 68 Diﬀerentiating once and using Bianchi Identity gives one more equation

2 dAa ∧ ω = 0.

This is exactly the inﬁnitesimal deformation of the instanton equation.

2.2.4 Ω-anti-self-dual instantons on the G2-cone

We investigate the relation between ω-anti-self-dual instantons on M and Ω-instantons on N. First, note that any principal G-bundle over N is isomorphic to a bundle P × R+ → M × R+ for a G-bundle P over M. Thus, without loss of generality, we assume that the G-bundle we are working on is a pull-back from M and we use the same letter P to denote these two bundles. Suppose A is an Ω-instanton. A priori, A involves a dt-term a · dt. However, we may perform a gauge transformation A 7→ g−1Ag + g−1dg to eliminate the dt-term. It is easy to see that we can simply take g as a solution to the diﬀerential equation

g−1agdt + g−1dg = 0.

Thus, we assume that A has no dt-term. We regard A as a family of connections ˙ d on P parametrized by t and denote A = dt A. Now the curvature may be computed

1 F N = dA + [A, A] = dt ∧ A˙ + F M , 2

M 1 where F = dM A + 2 [A, A]. The Ω-instanton condition with the formulae (2.28) and (2.29) gives

M t ∗M α = −ImΨ ∧ F and

M M ω ∧ F − tImΨ ∧ α = − ∗M F .

69 M M M We denote the (1, 1)-part (with coeﬃcients depending on t) of F by F0 and F1 =

M M F − F0 . By type decomposition in the above two equations we have

˙ M t ∗M A = −ImΨ ∧ F1 ,

M M ω ∧ F0 = − ∗M F0 , (2.38) and M ˙ M ω ∧ F1 − tImΨ ∧ A = − ∗M F1 .

By taking Hodge star of both sides, we see that the ﬁrst equation is equivalent to

˙ M tA = ∗(ImΨ ∧ F1 ). (2.39)

Combining (2.8) and (2.5) we see that the last equation is implied by (2.39). The equation (2.38) looks very much like the ω-anti-self-dual instanton equation

M on M. The only problem is that F0 is not necessarily the curvature of a well-defined connection. The equation (2.39) is exactly the gradient flow of the Chern-Simons functional CS. It would be interesting to analyze this equation coupled with (2.38). The first natural question is whether we could evolve through (2.39) in the class of ω-anti- self-dual instantons on M to get a Ω-anti-self-dual instanton on N. Unfortunately, this is impossible. An ω-instanton has its curvature of type (1, 1). If A(t) stays

d ω-anti-self-dual for all t, the evolution equation (2.39) will imply that dt A = 0, i.e., A is constant in t. On the other hand, if A is constant in t and ω-anti-self-dual, it is Ω-anti-self-dual when pulled back to the cone N. These give a class of special solutions.

Lemma 2.2.15. Suppose A is an ω-anti-self-dual connection on the nearly K¨ahler

6-manifold M and extend it to the G2-cone N by constant in t. Then A is a Ω-anti- self-dual connection on N.

70 Remark 2.2.16. When M = S6, in order that the principal bundle extend through the origin in R7, P has to be trivial over M. Even when this is true, the extended Ω-anti-self-dual connection on R7 \{0} described in the above Lemma does not necessarily extend through origin. It is interesting to ask under what condition this singularity is removable after a gauge transformation.

2.3 A Weitzenb¨ock formula

In this section, we derive a Weitzenb¨ock formula for nearly K¨ahler6-manifolds and describe its application to the deformation of ω-anti-self-dual instantons.

2.3.1 The general formula

Let E be a vector bundle over M. Suppose E is equipped with a metric and a metric-

compatible connection A. Suppose also that the curvature FA is an ω-instanton. Consider the following complex

dA ∗ P dA (2,0) ∗ 0 → Γ(E) −→ Γ(E ⊗ T M) −→ Γ(E ⊗ (Λ T M)R ⊕ Rω),

where the operator dA is induced from d and the connection A and P is deﬁned in §2.1.1 the projection onto the orthogonal complement of ω-trace free (1, 1)-forms. This complex is elliptic at the middle term. It could be extended to an elliptic complex, but we will not need the full sequence. The 0th cohomology group consists of parallel sections of E. We are mainly interested in the 1st cohomology group. A well-known result in Hodge Theory states that this group can be represented by harmonic sections, i.e., the kernel of the elliptic operator

∗ ∗ ∗ ∗ ∗ 2 ∆A = (dA + P dA) (dA + P dA) = dAdA + dAP dA

71 As usual, we will compare ∆A with a certain rough Laplacian of a connection. Note that, on E⊗T ∗M, there are several connections, e.g., A, coupled with the su(3)- connection on T ∗M, denoted by Dˆ as well as A with the Levi-Civita connection, denoted by D. After many trials, we choose D. However, Dˆ will be useful.

6 Suppose x ∈ M is a ﬁxed point. Let {ei}i=1 be a local orthonormal frames centered at x whose covariant derivatives with respect to the Levi-Civita connection vanish at x. Let {ωi} be the coframe. The Hodge Laplacian may be computed

∗ ∗ 2 ∆A = dAdA + dAP dA

P6 P6 = ( i=1 ωi ∧ Dei ) ◦ (− j=1 ωjyDej )

P6 2 P6 −( i=1 ωiyDei ) ◦ P ◦ ( j=1 ωj ∧ Dej )

P6 P6 = ( i=1 ωi ∧ Dei ) ◦ (− j=1 ωjyDej )

P6 2 P6 −( i=1 ωiy ◦ P ◦ Dei ) ◦ ( j=1 ωj ∧ Dej )

P6 2 P6 −( i=1 ωiy ◦ [Dei ,P ]) ◦ ( j=1 ωj ∧ Dej )

P6 2 = − i,j=1(ωi ∧ ◦ωjy + ωiy ◦ P ◦ ωj∧)Dei Dej

P6 2 −( i=1 ωiy ◦ [Dei ,P ]) ◦ dA

P6 2 = − i=1(ωi ∧ ◦ωiy + ωiy ◦ P ◦ ωi∧)Dei Dei

P 2 − i6=j(ωi ∧ ◦ωjy + ωiy ◦ P ◦ ωj∧)Dei Dej

P6 2 −( i=1 ωiy ◦ [Dei ,P ]) ◦ dA

P6 = − i=1 Dei Dei P − i

P6 2 −( i=1 ωiy ◦ [Dei ,P ]) ◦ dA, where the operator M is deﬁned in §2.1.1.

72 Recall Dei|x = 0. Thus at x,

6 X ∗ − Dei Dei = D D i=1

the rough Laplacian. For the same reason, Dei Dej − Dej Dei is the curvature on

∗ E T M ⊗ E. The curvature has two parts R ⊗ IdE + IdT ∗M ⊗ F where R is the

1 Riemannian curvature of M. We write R = 4 Rklijωl ∧ ωk ⊗ ωi ∧ ωj. Given a 1-form α and two vectors X and Y

1 D D α − D D α − D α = R ω ∧ ω (X,Y )α (ω ∧ ω ). X Y Y X [X,Y ] 4 klij i j y l k

P6 2 Now consider the term in the formula i=1 ωiy ◦ [Dei ,P ]. Note that the su(3)- ˆ 2 ˆ 2 connection D commutes with P , i.e., [Dei ,P ] = 0 for any ei. Moreover, the dif- ˆ ference r(ei) = Dei − Dei is exactly the su(3)-torsion up to a constant. Here, the nearly Kählerstructure plays the central role. By definition, this torsion r is covari- antly constant with respect to the su(3)-connection. Hence, r satisfies (2.9) and the operator

6 6 X 2 X 2 ωiy ◦ [Dei ,P ] = ωiy ◦ [r(ei),P ] = B i=1 i=1 factors through a linear combination of π(2,0) and π(0,2) according to (2.1.2). In summary, we have the following Weitzenb¨ock formula.

∗ ∆A = ∇A∇A (2.40)

−B ◦ dA (2.41)

1 X 1 X − M(ω ∧ ω ) ◦ R ⊗ Id − M(ω ∧ ω ) ⊗ F . (2.42) 2 i j ij E 2 i j ij i,j ij

A routine consequence of this formula is the following:

73 Lemma 2.3.1. Suppose M is a compact nearly Kähler6-manifold. Suppose the curvature (2.42) is non-negative as an operator on T ∗M ⊗ E. Then the first cohomology group is at most 6·rankE-dimensional. If, moreover, the curvature is positive somewhere, the first cohomology group vanishes.

Proof. The key observation is that for any harmonic section s of the elliptic sequence representing an element in the ﬁrst cohomology group, we have

BdA(s) = 0.

The rest of the proof parallels the argument in usual Bochner Technique.

Remark 2.3.2. It is not diﬃcult to work out the explicit formula for the curvature term (2.42) using SU(3)-representation theory. We will discuss this for M = S6 and leave the general case as an exercise for the interested reader.

2.3.2 Deformation of ω-anti-self-dual instantons

Suppose P is a principal G-bundle with G a compact Lie group. As said before, a connection A on P is an ω-anti-self-dual instanton if and only if its curvature F satisﬁes P (F ) = 0. (2.43)

Unless G is Abelian, this equation is nonlinear in A. Moreover, it is invariant under the action of the gauge transformations of P. The linearization of (2.43) at an ω-anti-self-dual instanton A is given by

P dAα = 0

∗ for α ∈ T M ⊗P×G g. Of course, one would like to divide by the inﬁnitesimal gauge transformation since (2.43) is gauge-invariant. These inﬁnitesimal gauge transfor-

∗ mations are given by the image of dA : P ×G g → T M ⊗ P ×G g. Thus, in fact, the

74 essential inﬁnitesimal deformations of the ω-anti-self-dual instanton A correspond to the elements of the ﬁrst cohomology group of the following sequence

dA ∗ P dA (2,0) ∗ 0 → Γ(E) −→ Γ(E ⊗ T M) −→ Γ(E ⊗ (Λ T M)R ⊕ Rω) → 0, where E = P ×G g. It follows that, all discussion in the previous section applies to instanton deformations. In particular, if the curvature of A is small enough (with a bound depending only on the base manifold M), then A is rigid, i.e., allows no deformation. We will illustrate this by analyzing S6.

Applications to S6

For S6, the Riemannian curvature simpliﬁes greatly

1 R = ω ∧ ω ⊗ ω ∧ ω . 2 j i i j

It may be computed that

1 P 1 P 4 M(ωi ∧ ωj)(ω1y(ωj ∧ ωi)) = 4 M(ωi ∧ ωj)(δ1jωi − δ1iωj)

1 P6 = 4 ( i=1 M(ωi ∧ ω1)(ωi)

P6 − j=1 M(ω1 ∧ ωj)(ωj)

5 = − 2 ω1.

By symmetry, it holds that

1 X 5 M(ω ∧ ω )(α (ω ∧ ω )) = − α 4 i j y j i 2 for any one-form α. Thus the ﬁrst curvature term in the Weitzenbock formula

1 X 5 − M(ω ∧ ω ) ◦ R ⊗ Id = . 2 i j ij E 2 i,j

An easy consequence is

75 Theorem 2.3.3. A ﬂat ω-instanton on S6 is rigid.

As another application, we consider the su(3)-connection on the standard struc-

6 ture bundle G2 → S . We need to describe the su(3) connection a bit. Recall the connection 1-form κi¯j in (2.26). Through this, the connection on (1, 0)-forms is

DX αi = −αjκij(X) for any vector ﬁeld X. Correspondingly the curvature is give by

DX DY − DY DX − D[X,Y ](αi) = −αj(dκij + κik ∧ κkj)(X,Y ).

Thus the action of F on (1, 0) forms is given by

αi 7→ −αj ⊗ (dκij + κik ∧ κkj).

More generally,

1 F : α 7→ α − (α ∧ α ) ⊗ (dκ + κ ∧ κ ). y 2 i j ij ik kj

1 Denote Ωij = dκij + κik ∧ κkj = 4 (3αi ∧ αj − δijαl ∧ αl). For each k, l, Fkl in the curvature term is 1 F : α 7→ α − (α ∧ α )Ω (e , e ). kl y 2 i j ij k l

Then the second curvature term in this context when E = T 1,0 is

1 − M(ω ∧ ω )(v) ⊗ F (α) 2 k l kl 1 1 = − M(ω ∧ ω )(v) ⊗ α (− )(α ∧ α )Ω (e , e ) 2 k l y 2 i j ij k l 1 = M(Ω )(v) ⊗ α (α ∧ α ) 4 ij y i j 1 = − v Ω ⊗ α (α ∧ α ) 2 y ij y i j 76 1,1 because Ωij is in Λ0 . This is not exactly what we want when we study the deforma-

tion of su(3) connection on G2. However, all we need is to replace α by a section of

1,1 adG2 ' Λ0 where the identification has been defined before. The action of αi ∧ αj will be the Lie bracket whose meaning should be clear via the aforementioned identification. Denote 1 B(s, t) = h− M(ω , ω ) ⊗ F (s), ti 2 i j ij

∗ where s, t are sections of T ⊗ adG2 . Note that if s = φ ⊗ X and t = ψ ⊗ Y we have

1 B(s, t) = − hM(ω ∧ ω )(φ) ⊗ [F ,X], ψ ⊗ Y i 2 i j ij 1 = − hM(ω ∧ ω )(φ), ψih[F ,X],Y i 2 i j ij 1 = − hM(ω ∧ ω ), φ ∧ ψihF , [X,Y ]]i 2 i j ij 1 = h− M(ω ∧ ω ) ⊗ F , [s, t]i. 2 i j ij where we view M(ωi ∧ ωj) as a 2-form. Then since F is SU(3)-invariant, B is a SU(3)-invariant symmetric bilinear form on R6 ⊗ su(3). We study the space of SU(3) invariant symmetric bilinear forms on R6 ⊗ u(3). One candidate is obvious, the SU(3) invariant inner product, denoted by B0. For others we apply representation theory of SU(3). Then complexiﬁed

6 ∼ 3 3 representation (R ⊗ su(3)) ⊗ C = (C ⊕ C ) ⊗C sl3(C) decomposes as

(V (1,0) ⊕ V (1,0)) ⊕ V (2,0) ⊕ V (2,0) ⊕ V (2,1) ⊕ V (2,1),

where V (a,b) denotes the irreducible complex representation of SU(3) with the highest

weight (a, b). The representation V (a,b) is real (i.e., V (a,b) ∼= V (a,b)) if and only if a = b. Thus the original R6 ⊗ u(3) decomposes as

(1,0) (2,0) (2,1) (V )R ⊕ (V )R ⊕ (V )R, 77 where VR means the real representation by forgetting the complex structure of V . The irreducible pieces are 6, 12, 30 dimensional respectively. One of them is known as V (1,0) = C3. Coupled with the standard inner product, every SU(3)-invariant bilinear form on R6 ⊗ u(3) will give rise to a SU(3)-invariant endomorphism. The space of such endomorphisms is 6 dimensional, 2 for each irreducible component. Of the two independent bilinear forms on every irreducible component, one can be taken as the SU(3)-invariant inner product and the other is symplectic. Thus the space of SU(3)-invariant symmetric bilinear forms is 3-dimensional, represented by linear combinations of inner products of various components. We will construct a basis for this 3-dimensional space. We already have one– the inner product of the whole space B0. To construct two more, we need more information about the irreducible components. First consider the map

6 6 T1 : R ⊗ su(3) → R deﬁned by v ⊗ α 7→ vyα. This map is clearly SU(3)-equivariant so B1(u, v) = 6 hT1u, T1vi is clearly SU(3)-invariant. Moreover, since R ⊗ su(3) contains only one

6 (2,0) copy of R and T1 is nonzero, by Schur’s Lemma, T1 and thus B1 is zero on (V )R⊕

(2,1) (V )R.

For later estimate, we need the right inverse of T1. Deﬁne the operatore

6 6 S1 : R → R ⊗ su(3) by 3 X 3 X v → α ⊗ π1,1(α ∧ v) + α ⊗ π1,1(α ∧ v). 16 i 0 i 16 i 0 i i i

6 It is clearly SU(3) equivariant. Since R is irreducible, S1 maps onto the irreducible

(1,0) 6 components VR ∈ R ⊗ su(3).

78 Then the composition T1 ◦S1 must be a linear combination of Id and J(the almost complex structure). However, it may be computed that

3 S (α ) = (α ⊗ π1,1(α ∧ α ) + α ⊗ π1,1(α ∧ α ) + α ⊗ π1,1(α ∧ α ) 1 1 16 1 0 1 1 2 0 2 1 3 0 3 1 3 1 = (α ⊗ (2α ∧ α + α ∧ α + α ∧ α ) + α ⊗ α ∧ α + α ⊗ α ∧ α ) 16 1 3 1 1 2 2 3 3 2 2 1 3 3 1

Thus T1S1(α1) = α1 and hence T1S1 = Id. Meanwhile, it is easy to compute that

3 B (S (α),S (α)) = B (S (α),S (α)). (2.44) 0 1 1 4 1 1 1

Second consider the map

6 3 6 6 ⊥ T2 : R ⊗ su(3) → ∧ R → (R ∧ ω) . deﬁned by v ⊗α 7→ v ∧α followed by the projection onto the orghogonal complement

6 of R ∧ ω. Deﬁne B2(u, v) = hT2u, T2vi. Then T2 is SU(3) equivariant and B2 is

SU(3) invariant. The image of T2 lies in the space of type (2, 1) + (1, 2) forms.

We also need the partial inverse of T2. Deﬁne

1 S : ψ 7→ (α ⊗ π1,1(α ψ) + α ⊗ π1,1(α ψ)). 2 4 i 0 iy i 0 iy

It is clearly SU(3) equivariant. The image under S2 of (2, 1)+(1, 2) forms orthogonal to R6 ∧ ω is V (2,0). It is easy to compute

1 S (α ∧ α ∧ α ) = (2α ⊗ α ∧ α − 2α ⊗ α ∧ α ). 2 1 2 3 4 1 2 3 2 1 3

Consequently, T2S2(α1 ∧ α2 ∧ α3) = α1 ∧ α2 ∧ α3. Thus T2S2 = 1. It is also easy to verify that

1 B (S (ψ),S (ψ)) = B (S (ψ),S (ψ)). (2.45) 0 2 2 2 2 2 2 79 On the other hand, it may be computed that

T2S1 = 0,T1S2 = 0. (2.46)

The 3 symmetric bilinear forms B0,B1,B2 are clearly linearly independent. Thus there exist constants λi such that B = λ0B0 + λ1B1 + λ2B2. We will compute examples to determine these constants. Set √ u1 = α1 ⊗ −1(2α1 ∧ α1 − α2 ∧ α2 − α3 ∧ α3),

u2 = α1 ⊗ (α2 ∧ α3 + α2 ∧ α3). and

u3 = α1 ⊗ α1 ∧ α2.

It is easy to see that [u1, u1] = [u2, u2] = 0. Thus

4 0 = B(u , u ) = λ B (u , u ) + λ B (u , u ) = (λ + λ )B (u , u ), 1 1 0 0 1 1 1 1 1 1 0 3 1 0 1 1

0 = B(u2, u2) = λ0B0(u2, u2) + λ2B2(u2, u2) = (λ0 + 2λ2)B0(u2, u2), and

B(u3, u3) = λ0B0(u3, u3).

Hence 3 1 λ = − λ , λ = − λ 1 4 0 2 2 0 and

B(u3, u3) λ0 = . B0(u3, u3)

1 The curvature F = − 8 (3αi ∧ αj − δijαl ∧ αl) ⊗C αi ∧ αj. Thus

B(u3, u3) = hF, [u3, u3]i = hF, α1 ∧ α1 ⊗ (−2α1 ∧ α1 + 2α2 ∧ α2)i = 12. 80 3 Thus λ0 = 2 . Consequently,

3 3 1 B = (B − B − B ). 2 0 4 1 2 2

Lemma 2.3.4. B ≥ 0.

6 ˆ Proof. Let ψ ∈ R ⊗ su(3) be real. Write ψ = S1T1(ψ) + S2T2(ψ) + ψ. Note that ˆ (2,1) ψ ∈ ker T1 ∩ ker T2. Thus, in fact ψ ∈ VR . These three diﬀerent components are thus pairwise perpendicular, since they lie in diﬀerent irreducible pieces. It follows that 2 B(ψ, ψ) = B (ψ,ˆ ψˆ). 3 0

The contribution from the second curvature term is nonnegative. All together the curvature part is strictly positive. To summarize, we have the following result.

6 Theorem 2.3.5. The su(3)-connection on G2 → S in (2.26) is a rigid SU(3) instanton.

2.4 SO(4)-invariant examples

We construct cohomogeneity one SU(2)(S3) anti-self-dual instantons (equivalent to pseudo-Hermitian-Yang-Mills here) on S6. The idea is to impose symmetries to reduce the instanton equations to ODEs. We regard SU(2) = S3 as the set of unit quaternions whose Lie algebra is the tangent space at 1 consisting of imaginary quaternions for which we use I, J, K to denote the standard basis for imaginary quaternions. A remark on the notation is necessary. Throughout this section, we √ use −1 to represent complex numbers to avoid confusion with quaternions. It

81 should be cautioned that when complex numbers are regarded as coeﬃcients in the complexiﬁed Lie algebra, they commute with I, J, K rather than following the usual rule of multiplication with quaternions. Hopefully, this will be clear from context.

2.4.1 A dense open subset U of S6

6 2 3 2 3 π More precisely, U = S \ (S ∪ S ) is parametrized by S × S × (0, 2 ) as

(x, y, t) 7→ v = (x cos t, y sin t))

where we think of x ∈ S3 ⊂ R4 as a unit 4-vector and y ∈ S2 ⊂ R3 as a unit 3-vector.

π 6 Actually, if we extend the map to the closed interval [0, 2 ], we cover the whole S . 6 π Reverse the picture and we get a map t : S → [0, 2 ] which is roughly the distance function from the totally geodesic pseudo-holomorphic S2 = {t = 0}. A generic level

2 3 π 3 set is a scaled S × S and {t = 2 } is a totally geodesic, special Lagrangian S . For later use, S3 × S2 = S3 × S3/S1 as a homogeneous space via (p, q) ∼ (pz, qz) for (p, q) ∈ S3 × S3 and z ∈ S1.

3 3 π Composing this quotient with the map (x, y, t) 7→ v, we have a map S ×S ×(0, 2 ) → U ⊂ S6 by (p, q) 7→ (pIp cos t, qp−1 sin t).

Denote by ω = ω1I + ω2J + ω3K and ψ = ψ1I + ψ2J + ψ3K the left-invariant

3 Maurer-Cartan forms on the two copies of S , respectively. Then, dt, ω2, ω3, ψ2, ψ3

3 3 π and τ = ω1−ψ1 form a basis of semibasic 1-forms for the projection S ×S ×(0, 2 ) → U. We use this to describe the nearly K¨ahlerstructures on U induced from S6.

6 Recall that the G2-invariant almost complex structure J on TvS is given by the left Cayley multiplication by v when we we regard both v and tangent vectors as

Cayley numbers in R8 = O. In other words,

J : dv 7→ v · dv. (2.47)

82 The standard metric and J determines the K¨ahler2-form ω = hJdv, dvi. Using (2.47) and Cayley-Dickson rule of Cayley multiplication, we can establish the following

J(dt) = sin tτ,

2 2 2 J(2 cos tω3) = (2 cos t − sin t)ω2 + sin tψ2,

2 2 2 J(−2 cos tω2) = (2 cos t − sin t)ω3 + sin tψ3.

The K¨ahlerform ω is determined by

−ω = hv, Jvi = 2 sin tψ1 ∧ dt − 2 sin tω1 ∧ dt

2 2 +2 cos t(9 cos t − 5)ω3 ∧ ω2 + 6 sin t cos tω3 ∧ ψ2

2 2 −6 cos t sin tω2 ∧ ψ3 + 2 sin t cos tψ2 ∧ ψ3.

2.4.2 Bundle constructions and SO(4)-invariant connections S3-bundles

We now describe the principal S3-bundles on which to construct instantons. First,

3 3 π 3 2 π 1 note that S × S × (0, 2 ) → S × S × (0, 2 ) in §2.4.1 is a principal S -bundle. The principal S3-bundles are obtained by extending the structure group through the group homomorphisms z 7→ zl

for z ∈ S1. More explicitly, denote

π B = S3 × S3 × S3 × (0, )/ ∼ l 2

where (p, q, r, t) ∼ (pz, qz, rz−l, t)

83 3 3 3 π π 1 for any (p, q, r, t) ∈ S × S × S × (0, 2 ), t ∈ (0, 2 ) and z ∈ S . The structure group 3 S acts on Bl by [p, q, r, t] 7→ [p, q, rg, t] for any g ∈ S3. Clearly this is well-deﬁned. Then the projection

[p, q, r, t] 7→ (pIp−1 cos t, qp−1 sin t)

3 makes Bl a principal S -bundle over U.

Remark 2.4.1 (on the symmetry of Bl). Note that if we let [g1, g2] ∈ SO(4) =

3 3 S × S /Z2 act on Bl by

[p, q, r, t] → [g1p, g2q, r, t] and on S6 by (x, y) 7→ (pap−1, qbp−1), this action commutes with the bundle projection. In other words, the principal bundle

6 Bl over U has an SO(4)-symmetry. It is well-known that the action on S is induced from the embedding of SO(4) into G2 and has cohomogeneity 1. We will construct SO(4)-invariant instantons, i.e., instantons of cohomogeneity one.

Remark 2.4.2 (on the topology of Bl). A priori, Bl is only deﬁned on U. However,

3 2 note that Bl is actually the pullback of a S -bundle from S obtained by extending

3 3 the structure group of a Hopf circle bundle. Since π1(S ) is trivial, every S -bundle

2 over S must be trivial. As a consequence, Bl is also trivial. In other words, it is

3 possible to make gauge transformations so that Bl ∼ U × S . Thus this bundle has natural extension to the whole S6, and, for later use, to the whole R7. The former description has the advantage that it makes the SO(4)-symmetry clear.

Remark 2.4.3 (on the numbers l). A priori, this construction only makes sense for integer l. However, we will see that it is more interesting if we think of l as real valued.

84 3 3 3 π We will carry out computations on S ×S ×S ×(0, 2 ). We will continue with the notation in §2.4.1 on the left-invariant forms on the ﬁrst two copies of S3. However, for the last S3, we need use the right- invariant Maurer-Cartan form drr−1 = β =

−1 −1 β1I + β2J + β3K. The left invariant Maurer-Cartan form is r dr = r βr. Of course, the following Maurer-Cartan equations hold

dω = −ω ∧ ω,

dψ = −ψ ∧ ψ, and dβ = β ∧ β.

More explicitly

dω1 = −2ω2 ∧ ω3, dω2 = −2ω3 ∧ ω1, dω3 = −2ω1 ∧ ω2,

similarly for ψi and

dβ1 = 2β2 ∧ β3, dβ2 = 2β3 ∧ β1, dβ3 = 2β1 ∧ β2.

3 3 3 π The space of semibasic 1-forms for the projection S × S × S × (0, 2 ) is spanned by dt, ω2, ω3, ψ2, ψ3, β2, β3, ω1 − ψ1 and lψ1 + β1.

Invariant connections

3 Now suppose A is an SO(4)-invariant connection on Bl. We pull back A to S ×

3 3 π S × S × (0, 2 ) and denote it by the same letter. Then, since A is semibasic with 3 3 3 π respect to the projection S × S × S × (0, 2 ) → Bl, we can write

A = A0τ + A1(lψ1 + β1) + A2ω2 + A3ω3 + B2ψ2 + B3ψ3 + C2β2 + C3β3 + B0dt

3 with Ai,Bi,Ci valued in Lie(S ). Since A is SO(4)-invariant and the 1-forms listed are also SO(4)-invariant, the coeﬃcients do not depend on (p, q), i.e., they are functions only in t and r. Moreover, A has to satisfy the following properties:

85 3 3 3 3 3 π 1. A must be right S -equivariant where we let S act on S × S × S × (0, 2 ) 3 and Bl by right multiplication on the last S factor.

2. A restricts to the last S3 factor to be the Maurer-Cartan left invariant form r−1βr.

3. The diﬀerential dA must be semibasic.

We investigate the consequences of these conditions.

1. Since all the forms listed in A are S3 right-invariant, this condition is equivalent to

−1 −1 −1 Ai(t, r) = r A(t, 1)r, Bi(t, r) = r Bi(t, 1)r, Ci = r Ci(t, 1)r.

To save notation, we will, from now on, write

−1 A = r (A0τ +A1(lψ1 +β1)+A2ω2 +A3ω3 +B2ψ2 +B3ψ3 +C2β2 +C3β3 +B0dt)r

where Ai,Bi,Ci are functions of t.

2. This condition says that

A1 = I,C2 = J, C3 = K.

Thus we may further reduce A to

−1 −1 A = r (A0τ + Ilψ1 + A2ω2 + A3ω3 + B2ψ2 + B3ψ3 + B0dt)r + r βr

3. It can be computed from Maurer-Cartan equations that

−1 rdAr ≡ −l[B0,I]ψ1 ∧ dt − l[A0,I]ψ1 ∧ τ −(l[A2,I] + 2A3)ψ1 ∧ ω2 − (l[A3,I] + 2A2)ψ ∧ ω3 −(l[B2,I] + 2B3)ψ1 ∧ ψ2 − (l[B3,I] + 2B2)ψ1 ∧ ψ3

86 mod semibasic 2-forms. Thus this condition is equivalent to the following algebraic equations

l[A0,I] = 0, l[B0,I] = 0,

l[B2,I] + 2B3 = 0, l[B3,I] + 2B2 = 0, (2.48)

l[A2,I] + 2A3 = 0, l[A3,I] + 2A2 = 0.

Hence, we solve the algebraic equations (2.48). We divide the solutions into several cases according to diﬀerent values of l.

1. Case l = 0. We have B2 = B3 = A2 = A3 = 0 but (2.48) puts no restrictions

on A0 and B0. Therefore A is reduced to

−1 −1 A = r (A0τ + B0dt)r + r βr.

2. Case l = 1. We have

A0 = a0I,B0 = b0I,

A2 = u1J + u2K,A3 = −u2J + u1K,

B2 = v1J + v2K,B3 = −v2J + v1K,

for a0, b0, ui, vi functions of t.

3. Case l = −1. We have

A0 = a0I,B0 = b0I,

A2 = u1J + u2K,A3 = u2J − u1K,

B2 = v1J + v2K,B3 = v2J − v1K,

for a0, b0, ui, vi functions of t.

4. Case l 6= 0, ±1. We have

A0 = a0I,B0 = b0I,

A2 = A3 = B2 = B3 = 0.

87 2.4.3 SO(4)-invariant instantons

Now we take instanton conditions into consideration. As mentioned before, A is an ω-anti-self-dual instanton if and only if its curvature F satisﬁes

2,0 F = trωF = 0.

It is easily seen that, restricted to U, this equivalent to

F ∧ σ0 ∧ σ1 ∧ σ2 = 0, (2.49)

and F ∧ ω2 = 0. (2.50)

According to Lemma 2.2.4,(2.50) is implied by (2.49), so we only care about (2.49). This simplifies the problem greatly. We consider four different cases according to the four different values of l in the last section.

l = 0

It may be computed that

−1 ˙ F = r {(A0 + [B0,A0])dt ∧ τ − 2A0(ω2 ∧ ω3 − ψ2 ∧ ψ3)}r,

˙ d where A0 = dt A0. The equation (2.49) gives

√ √ 2 32 2A cos t sin tω3 ∧ ω2 ∧ ψ2 ∧ ψ3 ∧ (dt − −1 sin tτ) = 0.

The only solution is A0 = 0, which is the trivial connection

A = r−1dr.

This is a case of little interest.

88 l 6= 0, ±1

It may be computed that

−1 F = {a˙0dt ∧ τ − 2a0(ω2 ∧ ω3 − ψ2 ∧ ψ3) − 2lψ2 ∧ ψ3}r Ir.

The equation (2.49) gives

2 2 8a0 cos t + l − 9l cos t = 0.

It is solved by l 9 cos2 t − 1 a = . 0 8 cos2 t

For safety, one can check that, in fact, a0 also satisﬁes the equation (2.50) which, in this case, is

2 2 −4a0 + 5l + 8 cos ta0 − 9 cos tl + 2 sin t cos ta˙0.

3 3 3 π We arrive at the corresponding instanton, pulled back to S × S × S × (0, 2 ),

1 9 cos2 t − 1 A = r−1lIr τ + ψ + bdt + r−1dr. (2.51) 8 cos2 t 1

Theorem 2.4.4. (2.51) deﬁnes for each l ∈ Z a singular Hermitian-Yang-Mill connection on S6.

Remark 2.4.5 (on singularity). The coordinate system is not extendable through

2 3 π the submanifolds S = {t = 0} and S = {t = 2 }. However, the connection A has π π diﬀerent behavior when t approaches 0 and 2 . When t → 2 , the curvature F blows up. However, for t = 0, the connection is bounded. It might be possible to remove the singularity by (2.2.11), we can extend the connection to the locus t = 0. In other words, this might be a singularity due to unwise choice of coordinates, rather than a singularity of the instanton A itself.

89 Remark 2.4.6 (on reducibility). A cautious reader may have noticed that, A has its holonomy in S1, so it is reducible. If we restrict the connection to the generic level sets of t, we obtain the standard Hopf connection up to a constant.

Remark 2.4.7 (on b). Note that b is not essential. We could have applied a gauge transformation in the t direction to A at the beginning to remove the dt component. The same remark applies to the next subsection. l = ±1

We only deal with the case l = 1. The other case is similar. According to Case 2 in §5.2.2, the curvature is computed to be

−1 rF r =a ˙0Idt ∧ τ − 2Iψ2 ∧ ψ3 +(u ˙1J +u ˙2K)dt ∧ ω2 + (−u˙2J +u ˙1K)dt ∧ ω3 +(v ˙1J +v ˙2K)dt ∧ ψ2 + (−v˙2J +v ˙1K)dt ∧ ψ3 −2a0I(ω2 ∧ ω3 − ψ2 ∧ ψ3) −2(u1J + u2K)ω3 ∧ τ − 2(−u2J + u1K)τ ∧ ω2 +2a0(u1K − u2J)τ ∧ ω2 + 2a0(−u2K − u1J)τ ∧ ω3 +2a0(v1K − v2J)τ ∧ ψ2 + 2a0(−v2K − v1J)τ ∧ ψ3 2 2 2 2 +2(u1 + u2)Iω2 ∧ ω3 + 2(v1 + v2)Iψ2 ∧ ψ3 +2(u1v2 − u2v1)I(ω2 ∧ ψ2 + ω3 ∧ ψ3) +2(u1v1 + u2v2)I(ω2 ∧ ψ3 − ω3 ∧ ψ2)

d where, again, ˙ means dt . A tedious computation shows that the equation (2.49) amounts to the following

2 3 sin t(1 − 3 cos t)v ˙1 − sin tu˙1 + 4a cos tv1 = 0,

2 3 sin t(1 − 3 cos t)v ˙2 − sin tu˙2 + 4a cos tv2 = 0,

2 2 sin t cos tv˙1 + u1(1 − a) sin t + a(3 cos t − 1)v1 = 0,

2 2 sin t cos tv˙2 + u2(1 − a) sin t + a(3 cos t − 1)v2 = 0,

u1v2 = u2v1

90 2 2 2 2 2 −9 cos t + 1 + 8a cos t − sin t(u1 + u2)

2 2 2 2 +(9 cos t − 1)(v1 + v2) + (6 cos t − 2)(u1v1 + u2v2) = 0.

We may assume that

u2 = λu1, v2 = λv1 with λ necessarily constant. It can be shown that by a substitution like (u1, v1) 7→ √ 2 1 + λ (u1, v1), we may simply assume that v2 = u2 = 0. The system reduces to

2 3 sin t(1 − 3 cos t)v ˙1 − sin tu˙1 + 4a cos tv1 = 0,

2 2 sin t cos tv˙1 + u1(1 − a) sin t + a(3 cos t − 1)v1 = 0,

2 2 2 2 2 2 2 −9 cos t + 1 + 8a cos t − sin tu1 + (9 cos t − 1)v1 + (6 cos t − 2)u1v1 = 0 which is now determined and thus solvable. It is easy to see that any solution must be of the form

u1 = U(sin t), v1 = V (sin t), a = W (sin t), where the functions U(x),V (x) and W (x) deﬁned on [0, 1] satisfy

d d x(−2 + 3x2) V − x3 U + 4WV = 0 dx dx

d x(1 − x2) V + x2U(1 − W ) + (2 − 3x2)WV = 0 dx

−8 + 9x2 + 8(1 − x2)W − x2U 2 + (8 − 9x2)V 2 + (4 − 6x2)UV = 0.

We rewrite the ODEs as

d x(1 − x2) V = −x2U(1 − W ) − (2 − 3x2)WV (2.52) dx d x3(1 − x2) U = x2(2 − 3x2)U(1 − W ) + (8 − 16x2 + 9x4) (2.53) dx −8 + 9x2 + 8(1 − x2)W − x2U 2 + (8 − 9x2)V 2 + (4 − 6x2)UV = 0. (2.54)

91 It is clear that the system (2.52), (2.53), (2.54) has many solutions which have possible singularities along x = 0 and x = 1.

Theorem 2.4.8. Each solution of the ode system (2.52), (2.53), (2.54) and a real number λ determine a unique Hermitian-Yang-Mills connection on the trivial SU(2) bundle over S6, with possible singularities along submanifolds S2 and S3.

Remark 2.4.9. It is interesting to ask whether we could apply Corollary (2.2.11) or (2.2.12) to remove the possible singularities along S2. This is doable by analyzing the singular behavior of the above ODE system along x = 0.

92 3

Pseudo-Holomorphic Curves in Nearly K¨ahler CP3

3.1 Structure equations, projective spaces, and the ﬂag manifold

In this section we collect some facts needed in the next section and formulate them in terms of the moving frame. Let H denote the real division algebra of quaternions. An element of H can be written uniquely as q = z + jw where z, w ∈ C and j ∈ H satisﬁes j2 = −1, zj = jz¯ for all z ∈ C. In this way we regard C as subalgebra of H and give H the structure of a complex vector space by letting C act on the right. We let H2 denote the space

2 of pairs (q1, q2) where qi ∈ H. We will make H into a quarternion vector space by letting H act on the right

(q1, q2)q = (q1q, q2q).

This automatically makes H2 into a complex vector space of dimension 4. In fact,

4 regarding C as the space of 4-tuples (z1, z2, z3, z4) we make the explicit identiﬁcation

(z1, z2, z3, z4) ∼ (z1 + jz2, z3 + jz4).

93 This specific isomorphism is the one we will always mean when we write C4 = H2. If v ∈ H2 \(0, 0) is given, let vC and vH denote respectively the complex line and the quaternion line spanned by v. The assignment vC → vH is a well-defined mapping T : CP3 → HP1. The fibers of T are CP1’s. So we have a fibration

CP1 → CP3 ↓ (3.1) HP1

This is the famous twistor ﬁbration. In order to study its geometry more thoroughly, we will now introduce the structure equations of H2. First, we endow H2 with a quaternion inner product h, i : H2 × H2 → H deﬁned by

h(q1, q2), (p1, p2)i =q ¯1p1 +q ¯2p2.

We have identities

hv, wqi = hv, wi q, hv, wi = hw, vi , hvq, wi =q ¯hv, wi .

Moreover, Re h, i is a positive deﬁnite inner product that gives H2 the structure of

8 2 a Euclidean space E . Let F denote the space of pairs f = (e1, e2) with ei ∈ H satisfying

he1, e1i = he2, e2i = 1, he1, e2i = 0.

2 7 8 2 We regard ei(f) as functions on F with values in H . Clearly, e1(F) = S ⊂ E = H . It is well known that F maybe canonically identiﬁed with Sp(2) up to a left translation

a in Sp(2). There are unique quaternion-valued 1-forms {φb } so that

b dea = ebφa, (3.2)

a a c dφb + φc ∧ φb = 0, (3.3)

94 and

a b φb + φa = 0. (3.4)

3 3 We deﬁne two canonical maps C1 : F → CP and C2 : F → CP by sending f ∈ F to the complex lines spanned by e1(f) and e2(f) respectively. Recall that we have

3 3 denoted the K¨ahlerprojective space by CP and the nearly K¨ahlerone by CP whose structure will be explicitly described below. We are mainly interested in CP3.

3 However, CP will play an important role. We now write structure equations for C1 1 3 and C2. First, we immediately see that C1 gives F the structure of an S ×S bundle over CP3, where we have identiﬁed S1 with the unit complex numbers and S3 with the unit quaternions. The action is given by

f(z, q) = (e1, e2)(z, q) = (e1z, e2q), where z ∈ S1 and q ∈ S3. If we set

1 1 " ω1 ω2 # φ φ iρ1 + jω3 − √ + j √ 1 2 = 2 2 φ2 φ2 √ω1 + j √ω2 iρ + jτ 1 2 2 2 2

where ρ1 and ρ2 are real 1-forms while ω1, ω2, ω3 and τ are complex-valued, we may rewrite one part of the structure equation (3.3) relative to the S1 × S3 structure on CP3 as

        ω1 i(ρ2 − ρ1) −τ¯ 0 ω1 ω2 ∧ ω3 d  ω2  = −  τ −i(ρ1 + ρ2) 0 ∧ ω2 + ω3 ∧ ω1  . (3.5) ω3 0 0 2iρ1 ω3 ω1 ∧ ω2

3 This in particular deﬁnes a nearly K¨ahlerstructure on CP by requiring ω1, ω2 and

ω3 to be of type (1, 0) (note that this almost complex structure is nonintegrable, thus diﬀerent from the usual integrable one). We denote

κ κ ¯ i(ρ − ρ ) −τ¯ 11 12 = 2 1 κ21¯ κ22¯ τ −i(ρ1 + ρ2) 95 and κ33¯ = 2iρ1 in the usual notation of a connection. Then the other part of the structure equation (3.3) may be written as the curvature of this nearly K¨ahlerstructure

κ ¯ κ ¯ κ ¯ κ ¯ κ ¯ κ ¯ d 11 12 + 11 12 ∧ 11 12 κ21¯ κ22¯ κ21¯ κ22¯ κ21¯ κ22¯

ω ∧ ω − ω ∧ ω ω ∧ ω = 1 1 3 3 1 2 , ω2 ∧ ω1 ω2 ∧ ω2 − ω3 ∧ ω3

as well as

dκ33¯ = −(ω1 ∧ ω1 + ω2 ∧ ω2 − 2ω3 ∧ ω3). (3.6)

1 3 In an exactly analogous fashion, C2 gives F a structure of an S × S bundle over

3 CP with the action now given by

(e1, e2)(q, z) = (e1q, e2z),

1 3 where z ∈ S and q ∈ S . However, ω1, ω2, κ12¯ and their complex conjugates become

3 semibasic and ω3 is not. The usual Kählerstructure on CP is defined by requiring ω ω √1 , √2 and κ ¯ to be of type (1, 0) and unitary. Relative to this Kählerstructure, 2 2 21 we may rewrite part of the structure equations as

ω1  ω2  ω1  √  −κ ¯ ω3 − √  √  2 11 2 2 ω2 ω1 ω2 d √ = −  −ω3 κ ¯ √  √ . (3.7)  2  22 2  2   ω ω  √2 − √1 κ ¯ − κ ¯ κ21¯ 2 2 22 11 κ21¯

We will also need some properties of the ﬂag manifold Fl = F/(U(1)×U(1)). Equiv-

alently, Fl consists of pairs of complex lines ([e1], [e2]) with he1, e2i = 0. Of course

1 1 F deﬁnes a natural S × S structure on Fl for which the forms ω1, ω2, ω3, κ21¯ and their complex conjugates are semibasic. Moreover, we have a double ﬁbration of Fl

96 over the two projective spaces:

F ↓ Fl (3.8) .& 3 3 CP CP

We denote the first fibration by Π1 and the second fibration by Π2. Explicitly Πa

(a = 1, 2) sends ([e1], [e2]) ∈ Fl to the complex line [ea]. By requiring ω1, ω2, ω3, κ21¯ to be complex linear we deﬁne an almost complex structure on Fl. It is easy to check from the structure equations that this almost complex structure is integrable and Π2 is thus a holomorphic projection. Finally, there are various complex vector bundles associated with F that will be important. First, on CP3, there are two obvious complex bundles, the tautological bundle and the trivial rank 4 bundle C4. We view as the subbundle of C4 spanned by e1 and denote the quotient bundle by Q. Using the obvious Hermitian product,

4 we identify Q as a subbundle of C locally spanned by e1j, e2 and e2j. Note that e1j itself spans a well-deﬁned line bundle, which is isomorphic to ∗. Denote the quotient Q/∗ by Q˜ which, again, may be regarded as a subbunle of C4 locally spanned by

3 3 1 3 e2, e2j. We write T CP the complex tangent bundle of CP . The S ×S determines a splitting T CP3 = H ⊕ V, where H has rank 2 and V has rank 1. We call H the horizontal part and V the vertical part relative to the ﬁbration 3.1. One can show V is isomorphic to 2 as a Hermitian line bundle by locally identifying √1 e ⊗ e with the complex tangent 2 1 1 ∗ ˜ vector dual to the (1, 0) form ω1, denoted by f1¯. Similarly H is isomorphic to ⊗ Q

1 ∗ 1 ∗ with √ e ⊗e identiﬁed with the tangent vector f¯ dual to ω and √ e ⊗e j identiﬁed 2 1 2 2 2 2 1 2

97 ˜ with f3¯ dual to ω3. Pulled back to Fl by Π1, the bundles Q splits as

∗ ˜ ∗ Π1Q = ˜⊕ ˜ ,

∗ where ˜ is locally spanned by e2 and ˜ denotes its dual, locally spanned by e2j. Of

∗ course Π1H splits correspondingly as

∗ ∗ ∗ ∗ Π1H = ⊗ ˜⊕ ⊗ ˜ .

3 Similar constructions apply to CP . We will only point out diﬀerences and some 3 relations. The tautological bundle on CP becomes ˜ when pulled back to Fl. The complex tangent bundle also splits as a sum of a vertical part V and a horizontal 3 part H. The vertical part is isomorphic to (˜∗)2 compared with the CP case because of the reversed almost complex structure. The horizontal part, when pulled back by

∗ ∗ ∗ Π2 to Fl, is isomorphic to ˜ ⊗ ⊕ ˜ ⊗ . Note this splitting shares a common factor

∗ with Π1H, which will become important later. In the various isomorphisms, we no longer need the √1 to make them Hermitian. Moreover, since the almost complex 2

3 structure on CP is integrable, many of these bundles have holomorphic structures. 3 Among them, the dual of the vertical tangent bundle of CP , which we denote by ∗ ∗ V , is particularly important. Locally, V is spanned by κ21¯ as a subbundle of the 3 complex cotangent bundle of CP . We have the following result due to Bryant [9]

Lemma 3.1.1. The bundle V∗ is isomorphic to ˜2 as a Hermitian holomorphic line 3 bundle. Moreover, it induces a holomorphic contact structure on CP .

The integrals of this holomorphic contact system were thoroughly investigated in [9] (see Section 3).

3.2 Pseudo-holomorphic curves in CP3

Let M 2 be a connected Riemann surface. A map X : M 2 → CP3 is called a pseudo- holomorphic curve if X is nonconstant and the diﬀerential of X commutes with the

98 2 2 2 almost complex structures. We let x : FX → M , VX → M and HX → M be the pullback bundles of F → CP3, V → CP3, and H → CP3 respectively. Thus, for instance, we have

2 FX = {(x, f) ∈ M × F|X(x) = C1(f)}.

1 3 2 Of course, FX is an S × S bundle over M and VX and HX are Hermitian complex

bundles of rank 1 and 2 respectively. Moreover, the natural map FX → F pulls back

various quantities on F, which we still denote by the same letters. For example, f1¯, f2¯

now denote functions on FX valued in HX . The structure equations (3.5), (3.6) and

2 (3.6) still hold, on FX now. Also for functions and sections with domains in M , we

∗ 2 will pull these back up via x to FX . For example, any section s : M → HX can be

written in the form s = f1¯s1 +f2¯s2 where si are complex functions on FX . Using this

convention, the pullback of κ induces connections on HX and VX compatible with

∗ 2 the Hermitian structures. Namely ∇ : Γ(HX ) → Γ(HX ⊗ T M ) is given by

∇(f¯isi) = f¯i ⊗ (dsi + κi¯jsj).

Since we are working over a Riemann surface, it is well-known that there are unique

holomorphic structures on HX and VX compatible with these connections. From now on we will regard these two bundles as holomorphic Hermitian vector bundles over M 2.

2 Another thing to notice is that {ωi} are semi-basic with respect to x : FX → M . Moreover, they are of type (1, 0) since dX is complex linear. Set

I1 = f1¯ ⊗ ω1 + f2¯ ⊗ ω2, I2 = f3¯ ⊗ ω3.

∗ 2 ∗ 2 It is clear that I1 and I2 are well deﬁned sections of HX ⊗ T M and V ⊗ T M respectively where T ∗M 2 is the holomorphic line bundle of (1, 0) forms on M 2.

Lemma 3.2.1. The sections I1 and I2 are holomorphic. Moreover, I1 and I2 only

2 vanish at isolated points unless X(M ) is horizontal ( when I2 vanishes identically) 99 2 or vertical (when I1 vanishes identically and thus X(M ) is an open set of a ﬁber CP1 in (3.1)).

Proof. We only show I1 is holomorphic and leave I2 for the reader. Choose a uni-

−1 formizing parameter z on a neighborhood of x0 ∈ M. In a neighborhood of x (x0), there exist functions ai so that ωi = aidz. It follows that ωi ∧ ωj = 0, so we have dωi = −κi¯j ∧ ωj. This translates to (dai + κi¯jaj) ∧ dz = 0 so there exist bi so that

dai + κi¯jaj = bidz.

Thus, when we compute ∂I1 we have

¯ 0,1 ∂I1 = (∇(fiai) ⊗ dz) 0,1 = f¯i ⊗ dz ⊗ (dai + κi¯jaj) 0,1 = f¯i ⊗ dz ⊗ (bidz) = 0,

so I1 is holomorphic. Moreover, by complex analysis, if I1 or I2 vanishes at a sequence of points with an accumulation, the section has to be identically 0 since M 2 is connected.

Remark 3.2.2. It is clear that I1 and I2 are just horizontal and vertical parts of the

evaluation map X∗(TM) → TX .

We will call a curve with I1 = 0 (I2 = 0) vertical (horizontal). Of course vertical curves are just the ﬁbers CP1 of T . To study horizontal curves it does no harm to reverse the almost complex structure on the ﬁber of T . This new complex structure is integrable and actually equivalent to the usual complex structure on the 3 projective space. The horizontal bundle H turns out to be a holomorphic contact structure under the usual complex structure. The integral curves of this contact system are

100 thoroughly described in [9]. We therefore have a good understanding of horizontal pseudo-holomorphic curves in CP3.

We now assume I1 is not identically 0. There exists a holomorphic line bundle

∗ L ⊂ HX so that I1 is a nonzero section of L ⊗ T M. We let R1 be the ramiﬁcation divisor of I1. That is, X R1 = ordp(I1)p.

p:I1(p)=0

R1 is obviously eﬀective, and we have

L = TM ⊗ [R1].

Similarly if I2 does not vanish identically let R2 be the ramiﬁcation divisor of I2.

Then R2 is eﬀective and

VX = TM ⊗ [R2].

(1) Now we adapt frames in accordance with the general theory. We let FX be the (1) subbundle of pairs (x, f) with f2¯ ∈ Lx. Then FX is a U(1) × U(1) bundle over M. The canonical connection on L is described as follows: If s : M → L is a section,

(1) then s = f2¯s2 for some function s2 on FX . Then

∇s = f2¯ ⊗ (ds2 + κ22¯s2).

Similarly the quotient bundle NX = HX /L has a natural holomorphic Hermitian

(1) structure. Let (f1¯): FX → NX be the function f1¯ followed by the projection (1) HX → NX . If s : M → NX is any section, then s = (f1¯)s1 for s1 on FX and we have

∇s = (f1¯) ⊗ (ds1 + κ11¯s1).

∗ (1) Note since I1 has values in L⊗T M, we must have ω1 = 0 on FX . If we diﬀerentiate this using structure equations (3.5) we have

dω1 = −κ12 ∧ ω2 = 0. 101 It follows that κ12¯ is of type (1, 0).

Lemma 3.2.3. Let II = (f1¯) ⊗ f2 ⊗ κ12¯ where f2 is the dual of f2¯. Then II is a

∗ ∗ holomorphic section of NX ⊗ L ⊗ T M.

Proof. Since κ12¯ is of type (1, 0), there exists b locally such that κ12 = bdz. The

(1) structure equations (3.6) pulled back to F gives dκ12¯ = −κ11¯ ∧ κ12¯ − κ12¯ ∧ κ22¯ +

ω1 ∧ ω2 = −(κ11¯ − κ22¯) ∧ κ12¯. This translates into

(db + (κ11¯ − κ22¯)b) ∧ dz = 0.

The rest follows exactly as in Lemma 3.2.1.

We say a curve has null-torsion if II = 0. Since ∧2H ⊗ V ∼= C we have

∼ NX ⊗ L ⊗ V = C.

If II is not identically 0, we deﬁne the planar divisor by

X P = ordp(II)p. p:II(p)=0

In this case, we have

NX = [P ] ⊗ L ⊗ TM.

Theorem 3.2.4. Let M = CP1. Then any complex curve X : M → CP3 either is one of the vertical ﬁbers or horizontal or has null-torsion.

Proof. Assume both I1 and I2 are not identically 0. We must show that II vanishes identically. If not, we have, for R1,R2,P ≥ 0,

VX = [R2] ⊗ TM,L = [R1] ⊗ TM,NX = [P ] ⊗ L ⊗ TM,

∼ which implies, since NX ⊗ L ⊗ V = C, that

3 ∼ (TM) ⊗ [2R1 + P + R2] = C,

thus degT M ≤ 0, but degT M = 2 when M = CP1.

102 Remark 3.2.5. The computation in this theorem actually shows that if M 2 has

3 genus g, then any pseudo-holomorphic curve X : M → CP with none of I1, I2 and II vanishing identically must satisfy

6(g − 1) = 2deg(R1) + deg(R2) + deg(P ).

This puts severe restrictions on the bundles L, VX and NX . For example, if g = 1, so that M is elliptic, then a pseudo-holomorphic curve X : M → CP3 must satisfy

2 R1 = R2 = P = 0, so that VX = TM, L = TM and NX = (TM) .

2 3 If the pseudo-holomorphic curve X : M → CP has I1 6= 0, we have a lift of X ˆ 2 to a map X : M → Fl deﬁned by x 7→ (X(x),NX (x) ⊗ X(x)). Some clariﬁcation

may be necessary. The bundle NX can be viewed canonically as a subbundle of X∗(Q˜ ⊗ ∗) ⊂ C4 ⊗ X∗(∗). By tensoring with X∗ and canonically identifying

∗ ∗ 4 ⊗ = C we see that NX (x) ⊗ X ()(x) = NX (x) ⊗ X(x) is a complex line in C . It is easy to see that this line is Hermitian orthogonal to X(x) ⊂ and X(x)j ⊂ ∗

ˆ ˆ ∗ and thus X is well-deﬁned. Moreover, X has null torsion iﬀ X (κ21¯) = 0. Composed 3 ˆ ˆ 2 3 with Π2 : Fl → CP , X induces a map Y = Π2 ◦ X : M → CP .

Theorem 3.2.6. The assignment X 7→ Y establishes a 1 − 1 correspondence between null-torsion pseudo-holomorphic curves in CP3 and nonconstant holomorphic

3 integrals of the holomorphic contact system V∗ on CP .

Proof. It is clear from the structure equations (3.7) that Y is an integral of V∗ if 3 X has null torsion. Conversely, if Y : M 2 → CP is a nonconstant holomorphic 3 integral of V∗, there exists a unique line bundle L ⊂ H ⊂ T CP which contains ˆ 2 Y∗TM. We lift Y to a map Y : M → Fl by x 7→ ((H/L)(x) ⊗ Y (x),Y (x)). ˆ 3 We deﬁne the corresponding map X = Π1 ◦ Y : M → CP . It is clear from the structure equations(3.5) that such an X has null-torsion. We next show that if we start with null-torsion curve X : M 2 → CP3 and run through the procedure

103 X → Y → X of the above constructions, we arrive at the original curve. In fact the

∗ frame adaptations we made before show that we can arrange {ea} so that Π1LX (x)

is spanned by √1 e j ⊗ e∗ and Π∗N (x) is spanned by √1 e ⊗ e∗. Thus by deﬁnition 2 2 1 1 X 2 2 1 ∗ Y (x) = [e2]. Since ω1 = 0, Π2L is spanned by e1j by the structure equations. ∗ ˆ Therefore Π2(H/L) = [e1] from which we see Π1(Y (x)) = X(x). We omit the proof that if we start with Y and run the procedure of constructions Y → X → Y we get Y back.

As mentioned before, a powerful construction the integrals of the holomorphic

contact system V∗ was provided in [9] (see Section 3). Of course, there are corresponding results about null-torsion pseudo-holomorphic curves in CP3. We omit the details for most of translation work and only mention some consequences.

Theorem 3.2.7. Let M be a compact Riemann surface. There always exists a pseudo-holomorphic embedding M → CP3 with null torsion.

This is the translation of Theorem G in [9]. A horizontal pseudo-holomorphic curve X : M 2 → CP3 with null torsion cor-

3 1 responds to Y : M 2 → CP which is simply a projective line CP (see the remark in [9] following Theorem F). In particular, if M 6= S2 and let X : M → CP3 be null-torsion, X is neither vertical or horizontal. On the other hand, it is easy to

3 construct a horizontal rational curve Y : S2 → CP which is not a complex line using the “Weierstrass formula” in Theorem F in [9]. The corresponding X will be neither vertical nor horizontal. We state these as the following

Corollary 3.2.8. For any Riemann surface M, there exists a pseudo-holomorphic curve X : M → CP3 which is neither vertical nor horizontal.

104 A rational pseudo-holomorphic curve is either vertical or horizontal or has null torsion. Both horizontal and null-torsion curves are reduced to integrals of the holo-

morphic contact system V∗ by Theorem 2.2. By the result in [9], Section 2, such an integral represents a lift of a minimal 2-sphere in S4. Thus the space of nonvertical

3 rational curves in CP can be regarded as the union of 2 copies of the space of minimal 2-spheres in S4. These two copies have a nonempty intersection, corresponding to geodesic 2-spheres.

105 4

Summary

In this summary, we discuss further research directions. It is an open problem to construct nonhomogeneous nearly K¨ahler6-manifolds.

One way might be to consider the submanifolds in a G2 manifold. Since Joyce has constructed many compact manifolds with G2 holonomy, it is natural to ask what kind of SU(3) structures their submanifolds could have. Note that the normal bundle essentially gives the almost complex structure. So conditions on Nijenhaus tensor translates into conditions on normal bundle. Thus the question is largely on submanifolds with special normal bundle, a subject studied extensively from integrable system point of view. Anti-self-dual connections on nearly K¨ahler manifolds have much left to explore. It is natural to ask whether the Chern-Simons functional is Morse-Smale. If this is true, it is natural to ask what kind of invariants can be built out of the study of Morse theory on the moduli. The discussion in this thesis may provide some hint.

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109 Biography

Feng Xu was born in JiangSu, China. He spent his undergraduate time in Zhejiang University, HangZhou. After that, he spent 3 years in Tsinghua University, Beijing to get a Master’s degree in Mathematics. From 2003 to 2008, he was a Ph. D candidate at Duke University. He will get his Ph. D in Mathematics in May, 2008.

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