CAYLEY GRASSMANNIAN AND DEFORMATIONS IN COMPLEX G2 MANIFOLDS

By

Üstün Yıldırım

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Mathematics – Doctor of Philosophy

2018

ABSTRACT

CAYLEY GRASSMANNIAN AND DEFORMATIONS IN COMPLEX G2 MANIFOLDS

By

Üstün Yıldırım

Geometric objects related to the exceptional lie groups G2 and Spin„7” have become increasingly
popular in the recent years. Especially so after Bryant’s (and others’) work which showed the
existence of riemannian manifolds with holonomy group equal to one of these groups [Bry87].
However, not much attention is given to the complex manifestations of these objects. This thesis
consists of two parts which fills some of these gaps.

In the first part of this thesis, we investigate the Cayley Grassmannian (over C) which is the
set of four-planes that are closed under a three-fold cross product in C8. We define a torus action
on the Cayley Grassmannian. Using this action, we prove that the minimal compactification is a
singular variety. We also show that the singular locus is smooth and has the same cohomology ring
as that of CP5. Furthermore, we identify the singular locus with a quotient of GC
2 by a parabolic
subgroup.

In the second part of this thesis, we introduce the notion of (almost) GC

2 -manifolds with
compatible symplectic structures. Further, we describe “complexification” procedures for a G2
manifold M (cid:26) MC. As an application we show that isotropic deformations of an associative
submanifold Y of a G2 manifold inside of its complexification MC is given by Seiberg-Witten type
equations.

Copyright by
ÜSTÜN YILDIRIM
2018

To the memory of my father Halil Yıldırım.

iv

ACKNOWLEDGEMENTS

I am deeply grateful to my advisor Selman Akbulut for all his support, encouragement, and guidance.
The second part of this thesis is a joint work with him. I consider myself extremely lucky to have
worked with him.

I would also like to thank Mahir Bilen Can for many fruitful discussions, sharing his insights
with me and his three week long summer school lectures on Lie algebras and their Representations
which lay the groundwork for the first part of this thesis. Many thanks go to my committee members
Casim Abbas, Igor Rapinchuk and Thomas Walpuski.

Last but not least I would like to thank my family and my wife İrem Gökçe Yıldırım for their

endless support and help during my Ph.D. education.

v

TABLE OF CONTENTS

LIST OF FIGURES . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

CHAPTER 1

INTRODUCTION .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

.

.

.

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.

.

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CHAPTER 2 PRELIMINERIES .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .

4
2.1 Octonions . .
4
2.2 Calibration forms and calibrated planes
9
2.3 Groups G2 and Spin„7” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
CHAPTER 3 CAYLEY GRASSMANNIAN . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Charts of Gr„4; O” . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Torus fixed points and the minimal compactification . . . . . . . . . . . . . . . . . 15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Singular locus .
CHAPTER 4 DEFORMATIONS IN COMPLEX G2 MANIFOLDS . . . . . . . . . . . . 22
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.1.1 Non-degenerate three-forms in seven-space . . . . . . . . . . . . . . . . . 22
4.1.2 The complexification of a G2-space . . . . . . . . . . . . . . . . . . . . . 23
4.1.3 Compatible structures on a GC
2 -space . . . . . . . . . . . . . . . . . . . . 24
2 manifolds and complexification of a G2 manifold . . . . . . . . . . . . . . . . 27
Isotropic associative submanifolds and their deformations . . . . . . . . . . . . . . 29

4.1 Linear algebra .

4.2 GC
4.3

.

.

.

.

.

.

.

.

.

BIBLIOGRAPHY . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

vi

LIST OF FIGURES

Figure 2.1: Multiplication table for octonions . . . . . . . . . . . . . . . . . . . . . . . . .

5

vii

CHAPTER 1

INTRODUCTION

The existence of Riemannian manifolds with exceptional holonomy groups (G2 or Spin„7”) is shown
by Bryant in 1987 [Bry87]. Later, Bryant and Salamon found complete examples [BS89] and Joyce
found compact ones [Joy00]. These advancements naturally motivated many questions among
which is how to distinguish two such manifolds. The main methods that may yield to invariants are
gauge theoretic approach of counting instantons, counting minimal submanifolds or possibly some
combination of the two [DT98, Wal13].

We say that a seven (resp. eight) manifold M has a G2 (resp. Spin„7”) structure if the manifold
is equipped with a special three (resp. four) form ϕ (resp. (cid:8)). These are calibration forms in the
sense of [HL82]. So, they define calibrated submanifolds called associative submanifolds in the
case of G2 and Cayley submanifolds in the case of Spin„7”. These calibration forms naturally define
subvarieties, called associative Grassmannian and Cayley Grassmannian, of Gr„3;7” and Gr„4;8”,
respectively. We can explain these more concretely as follows. The form ϕ (resp. (cid:8)) determines
a unique Riemannian metric g on the manifold M. Then, by setting g„u (cid:2) v; w” = ϕ„u; v; w” (resp.
g„u (cid:2) v (cid:2) w; z” = (cid:8)„u; v; w; z”) one can define a two (resp.
three)-fold cross product operation
on T M. The associative (resp. Cayley)-submanifolds are three (resp. four)-submanifolds whose
tangent spaces are closed under the corresponding cross product operation. Fix a single tangent
space TpM, then a three (resp. four)-plane in TpM is called an associative (resp. Cayley)-plane
if it is closed under the corresponding cross product operation. The set of all associative (resp.
Cayley)-planes is called the associative (resp. Cayley) Grassmannian.

The associative and Cayley Grassmannians are the homogeneous spaces G2(cid:157)SO„4” and Spin„7”(cid:157)K
(where K = „SU„2” (cid:2) SU„2” (cid:2) SU„2””(cid:157)f(cid:6)1g) respectively, and they have been studied in [HL82,
SW10, AK16]. Over real numbers, they are compact spaces. Although over complex numbers they
are not compact spaces, they admit natural compactifications. Some natural compactifications of
the associative Grassmannian have been studied in [AC15].

1

The first part of this thesis is an investigation of the (compactified) Cayley Grassmannian (over
the complex numbers). We study the Cayley Grassmannian using a torus action which has finitely
many fixed points. First, we introduce octonions and multiple cross products. Then, using three-
fold cross product, we define Cayley planes precisely and describe an equivalent formulation. The
latter formulation allows one to express Cayley planes as solutions to some polynomial equations (in
P„(cid:3)4C8” via Plücker embedding). We call the closure of this variety the minimal compactification
of the Cayley Grassmannian. We show that the variety is singular, the singular locus is smooth and
it is a cohomology P5. Further, we identify the singular locus as a quotient of GC
2 by a parabolic
subgroup P2.

The second part of this thesis is a joint work with Akbulut [AY18].

2 manifolds. An (almost) GC

(cid:26) GL„14; R”. In other words, its frame bundle admits a GC

In this part, the main
objects of our study are (almost) GC
2 manifold is a (real) 14-manifold
whose structure group is GC
2 subbundle
2
and one such subbundle is fixed. We provide examples of such manifolds through two different
constructions. They are described as complexification procedures starting with a G2 manifold.
Given a G2 manifold „M; φ”, we consider the cotangent bundle T(cid:3)M as a complexification of M
2 structures „φC; BC; Ω” on it in two different ways. Furthermore, we describe
and construct GC
a compatibility condition for GC
2 structures and symplectic structures. In both complexification
procedures, we obtain GC
2 manifolds with symplectic structures. However, in one of the procedures
the symplectic form is not necessarily closed. On the other hand, we use the canonical symplectic
structure !can on T(cid:3)M in the other construction.

It is possible to extend the notion of associative submanifolds to the complex case as well.
Further, having a compatible symplectic structure allows one to define a new type of special
submanifold which we call isotropic associative submanifolds. These are (real) three dimensional
submanifolds which are isotropic with respect to the symplectic form ! and satisfy an associativity
condition defined using φC. As an application, we investigate infinitesimal deformations of isotropic
associative submanifolds and relate them to Seiberg-Witten type equations.

The organization of the thesis is as follows. In Chapter 2 we collect the background material

2

common to both parts of the thesis. Chapter 3 is devoted to Cayley Grassmannian and Chapter 4 is
devoted to GC

2 manifolds.

3

CHAPTER 2

PRELIMINERIES

In this preparatory chapter, the background material necessary for both parts of the thesis is given.

2.1 Octonions

In this section, we define an octonion algebra and various cross products. All the constructions
In fact, they can be generalized to other fields.

of this chapter can be done over R or over C.
However, that is beyond the scope of this thesis.

Definition 2.1.1 ([SV13]). A composition algebra C over a field k is an algebra over k with identity
element and a nondegenerate quadratic form N such that

N„uv” = N„u”N„v”

for u; v 2 C. The quadratic form N is often referred to as the norm on C, and the associated bilinear
form B„(cid:1);(cid:1)” is called the inner product.

A four dimensional composition algebra is called a quaternion algebra, and an eight dimensional

composition algebra is called an octonion algebra.

A specific example of an octonion algebra O (over R) is given as follows. Let S = f1;i;j;k;l;li;lj;lkg
be an orthonormal basis for (the vector space) O. For each (oriented) line (or the circle) in Figure 2.1
from x to y to z, set

x y = z = (cid:0)yx;

yz = x = (cid:0)zy;

zx = y = (cid:0)xz;

and

x2 = y2 = z2 = (cid:0)1:

Note that the subspace H generated by f1;i;j;kg is closed under multiplication and therefore, it is
a subalgebra. To ease notation later on, we set e0 = 1; e1 = i; e2 = j; e3 = k; e4 = l; e5 = li; e6 = lj
and e7 = lk. We also set ei1:::ik = ei1 ^(cid:1) (cid:1) (cid:1)^ eik where
is the dual basis of feig.The octonions
are non-associative but they are alternative, i.e., the subalgebra generated by any two elements is

{

}

ei

4

Figure 2.1: Multiplication table for octonions

associative. We denote the projection map from O to the span of 1 by Re, and projection to the
orthogonal complement 1? by Im. This allows us to define an involution

u 7! u = Re„u” (cid:0) Im„u”:

The bilinear form B associated to N can be expressed as B„u; v” = Re„uv”. So N„u” = B„u; u” =
Re„uu”.

A key fact one can verify on the basis elements is

Note that (2.1) implies

that is uu 2 Re„O”. Thus,

uv = v u:

uu = uu

(2.1)

N„uv” = Re„uvuv” = Re„v„uu”v” = Re„uu”Re„vv” = N„u”N„v”

proving directly that the above multiplication table defines an eight-dimensional composition
algebra (also proving H is a quaternion algebra.) We may complexify O to get a complex octonion
algebra which we denote by OC or if the field is clear from the context simply by O again. (Here,
we complexify B by extending it as a complex bilinear form on both entries and N is extended so
that it is the (complex) quadratic form associated to B.)

5

ijklljlkliNext, we would like to define various cross product operations using octonions but, first, we

justify their name.
Definition 2.1.2. Let „V; B” be a vector space with a (non-degenerate) symmetric bilinear form. A
multilinear map L : Vr ! V is called an r-fold cross product if

N„L„v1; : : : ; vn”” = N„v1 ^ (cid:1) (cid:1) (cid:1) ^ vn”

with the induced norm on (cid:3)nV and

B„L„v1; : : : ; vn”; vi” = 0

for all i:

(2.2)

(2.3)

Remark 2.1.3. If L is an alternating multilinear map, then it is enough to check (2.2) on orthogonal
vectors in which case (2.2) becomes

N„L„v1; : : : ; vn”” = N„v1” : : : N„vn”:

(2.4)

Remark 2.1.4. The usual cross product operation on R3 is naturally a (two-fold) cross product
according to this definition.

In [BG67], Brown and Gray proved that an r-fold cross product exists on an n-dimensional

vector space only in the following cases:

1. n is even, r = 1
2. n is arbitrary, r = n (cid:0) 1

3. n = 7, r = 2

4. n = 8, r = 3.

We say that the last two cases are exceptional as they occur only in specific dimensions. Below, we
give concrete description of the exceptional two-fold and three-fold cross products using octonions.

6

Then, we introduce a “four-fold cross product” operation on O. Although it is not a cross product
according to Definition 2.1.2, it is conventionally called so [HL82, SW10]. A two-fold cross
product (or simply a cross product) can be defined as the restriction of octonionic multiplication to
the imaginary part, Im„O”:
Definition 2.1.5. For u; v 2 Im„O”, let

u (cid:2) v = Im„uv”:

To be able to prove this is a cross product operation, the following lemma is needed.

Lemma 2.1.6. For u; v; v

′ 2 O,

N„u”B„v; v

′” = B„uv; uv

′” = B„vu; v

′

u”:

(2.5)

(2.6)

In particular, for unit u, (left or right) multiplication by u is an orthogonal transformation of O.
Proof. Since B„v; v

′” = 1
2

N„u”B„v; v

„N„v + v
′” =

(
(

′” (cid:0) N„v” (cid:0) N„v
1
N„u”N„v + v
2
1
N„uv + uv
2

′””, we have
′” (cid:0) N„u”N„v” (cid:0) N„u”N„v
′” (cid:0) N„uv” (cid:0) N„uv

′”)

′”)

=

= B„uv; uv

′”:

The second equality can be proved similarly.
Proposition 2.1.7. The map „u; v” 7! u (cid:2) v = Im„uv” is a two-fold cross product on Im„O”.
Proof. Since uu 2 Re„O”, u (cid:2) u = Im„uu” = (cid:0)Im„uu” = 0. So, u (cid:2) v is an alternating map. By
Remark 2.1.3, we may assume u; v 2 Im„O” are orthogonal, that is, B„u; v” = 0. Then, by Lemma
2.1.6 we get

□

0 = N„u”B„u; v”
= (cid:0)B„uu; uv”
= (cid:0)B„N„u”; uv”:

7

Thus, uv 2 Im„O”. This gives us

N„u (cid:2) v” = N„Im„uv””

= N„uv”
= N„u”N„v”:

To prove (2.3), we once again use Lemma 2.1.6.

B„u (cid:2) v; u” = B„Im„uv”; u”

= B„uv (cid:0) Re„uv”; u”
= B„uv; u” (cid:0) B„Re„uv”; u”
= N„u”B„v;1”
= 0

since u and v are orthogonal to 1.

□

Next, following [SW10] we define a three-fold cross product and a four-fold “cross product” as

follows:
Definition 2.1.8. For u; v; w 2 O, let

u (cid:2) v (cid:2) w =

1
2

„„uv”w (cid:0) „wv”u” :

(2.7)

Definition 2.1.9. For u; v; w; x 2 O, let

x (cid:2) u (cid:2) v (cid:2) w = (cid:0)1
4

»„x (cid:2) u (cid:2) v”w (cid:0) „w (cid:2) x (cid:2) u”v + „v (cid:2) w (cid:2) x”u (cid:0) „u (cid:2) v (cid:2) w”x… :

(2.8)

Remark 2.1.10. In fact, the four-fold cross product operation (2.8) does not satisfy (2.3) but it is
conventionally called cross product [HL82, SW10]. However, it is alternating and satisfies (2.2).
Hence, for orthogonal vectors x; u; v; and w we have

N„x (cid:2) u (cid:2) v (cid:2) w” = N„x”N„u”N„v”N„w”:

(2.9)

8

2.2 Calibration forms and calibrated planes

In this section, we define two calibration forms in the sense of [HL82] and calibrated planes
associated to these forms that are relevant to this thesis. The base field is again either R or C for
this section. Reader may consult [HL82] for the general theory of calibrated geometries.

A k-form ! 2 (cid:3)kV(cid:3) is called a calibration form if for every orthonormal set of vectors
fv1; : : : ; vkg we have j!„v1; : : : ; vn”j (cid:20) 1. Given a k-plane (cid:24) generated by an orthonormal basis
fv1; : : : ; vkg, (cid:24) is called calibrated if !„v1; : : : ; vk” = (cid:6)1. Using the two-fold (resp.
three-fold)
cross product, we define a calibration three-form (resp. four-form) ϕ (resp. (cid:8)) called associative
(resp. Cayley) calibration on Im„O” (resp. O) as follows:
Definition 2.2.1. For u; v; w 2 Im„O”, let

and for x; u; v; w 2 O, let

ϕ„u; v; w” = B„u; v (cid:2) w”

(cid:8)„x; u; v; w” = B„x; u (cid:2) v (cid:2) w”:

(2.10)

(2.11)

For a proof of the following proposition see [SW10].

Proposition 2.2.2. The equations (2.10) and (2.11) define calibration forms and they satisfy

ϕ„u; v; w” = Re„u (cid:2) v (cid:2) w”

(2.12)

and

(cid:8)„x; u; v; w” = Re„x (cid:2) u (cid:2) v (cid:2) w”:

(2.13)
By (2.4) and (2.12), it is clear that ϕ„u; v; w” = (cid:6)1 if and only if Im„u (cid:2) v (cid:2) w” = 0 for
orthonormal u; v; w 2 ImO. Similarly, by (2.9) and (2.13), it is clear that (cid:8)„x; u; v; w” = (cid:6)1 if and
only if (cid:4)„x; u; v; w” := Im„x (cid:2) u (cid:2) v (cid:2) w” = 0 for orthonormal x; u; v; w 2 O.

four-plane) (cid:24) generated by orthonormal fu; v; wg (resp.
Definition 2.2.3. A three-plane (resp.
fx; u; v; wg) is called an associative (resp. Cayley) plane if ϕ„u; v; w” = (cid:6)1 (resp. (cid:8)„x; u; v; w” = (cid:6)1)

9

or, equivalently, Im„u (cid:2) v (cid:2) w” = 0 (resp. (cid:4)„x; u; v; w” = 0). The set of all associative (resp.
Cayley) planes is called the associative (resp. Cayley) Grassmannian. We denote the associative
Grassmannian by Gr„ϕ” and the Cayley Grassmannian by Gr„(cid:8)”.

Remark 2.2.4. Over C, not every three-plane (resp. four-plane) is generated by an orthonormal
basis (with respect to B). Strictly speaking, this is why the associative (resp. Cayley) Grassmannian
is not compact when defined over C.

It is helpful to express ϕ, (cid:8) and (cid:4) in coordinates. The associative calibration form is given by

ϕ = e123 (cid:0) e145 (cid:0) e167 (cid:0) e246 + e257 (cid:0) e347 (cid:0) e356

the Cayley calibration form is given by

(cid:8) = e0123 (cid:0) e0145 (cid:0) e0167 (cid:0) e0246 + e0257 (cid:0) e0347 (cid:0) e0356
(cid:0)e1247 (cid:0) e1256 + e1346 (cid:0) e1357 (cid:0) e2345 (cid:0) e2367 + e4567

and the imaginary part of the four-fold cross product is given by

e1

)

)
)
)
)
)
)

e2

e3

e4

e5

e6

e7:

(2.14)

(2.15)

(

(
(
(
(
(
(

(cid:4) =

+

+

+

+

+

+

(cid:0)e0247 (cid:0) e0256 + e0346 (cid:0) e0357 + e1246 (cid:0) e1257 + e1347 + e1356
+e0147 + e0156 (cid:0) e0345 (cid:0) e0367 (cid:0) e1245 (cid:0) e1267 + e2347 + e2356
(cid:0)e0146 + e0157 + e0245 + e0267 (cid:0) e1345 (cid:0) e1367 (cid:0) e2346 + e2357
(cid:0)e0127 + e0136 (cid:0) e0235 + e0567 + e1234 (cid:0) e1467 + e2457 (cid:0) e3456
(cid:0)e0126 (cid:0) e0137 + e0234 (cid:0) e0467 + e1235 (cid:0) e1567 + e2456 + e3457
+e0125 (cid:0) e0134 (cid:0) e0237 + e0457 + e1236 (cid:0) e1456 (cid:0) e2567 + e3467
+e0124 + e0135 + e0236 (cid:0) e0456 + e1237 (cid:0) e1457 (cid:0) e2467 (cid:0) e3567

Using these expressions, one can immediately see that

where (cid:3) is the Hodge star operator.

(cid:8) = e0 ^ ϕ + (cid:3)ϕ

10

2.3 Groups G2 and Spin„7”

In this section, we define groups G2 and Spin„7” (over R and C). Definitions we give (following
Bryant [Bry87]) are somewhat unusual but better suited for our purposes. Then, we describe a
(maximal) torus of Spin„7” and three subgroups of Spin„7” each of which is isomorphic to SL„2; C”
(over C and SU„2” over R). By restriction, we also get a (maximal) torus of G2 and subgroups that
are isomorphic to SL„2; C” (or SU(2) depending on the field).

Definition 2.3.1.

1. Spin„7; R” is the stabilizer of (cid:8) in SO„8; R”,

2. Spin„7; C” is the identity component of the stabilizer of (cid:8) in SO„8; C”,
3. G2 is the stabilizer of ϕ in SO„7; R”, and

2 is the stabilizer of ϕ in SO„7; C”.

4. GC
Given A 2 GC

a subgroup of Spin„7; C” using (2.15).

2 , we can extend it linearly so that it fixes 1 2 O. Then, it is easy to see that GC
2 is

By definition, an element of Spin(7,C) acts on O preserving orthonormality and the values of
(cid:8). Thus, it takes a Cayley plane to a Cayley plane. In other words, it defines an action on the
Cayley Grassmannian Gr„(cid:8)”.

Consider the following matrix

L(cid:21) =

where P(cid:21) = (cid:21)+(cid:21)(cid:0)1

; M(cid:21) = (cid:21)(cid:0)(cid:21)(cid:0)1

2

2

eigenvalues (cid:21) and (cid:21)(cid:0)1 with eigenvectors'››«1

i

P(cid:21)

iM(cid:21)

'››« P(cid:21) (cid:0)iM(cid:21)
“fifi‹
'››« 1
“fifi‹

and

(cid:0)i

“fifi‹

; and (cid:21) 2 C(cid:3). Note that its determinant is 1.

In fact, it has

respectively.

11

Considering L(cid:21) as a block matrix, we define the following 8(cid:2)8 matrices A(cid:21) = L(cid:21)(cid:8) L(cid:21)(cid:8) L(cid:21)(cid:8) L(cid:21);
B(cid:22) = L(cid:22) (cid:8) L(cid:22)(cid:0)1 (cid:8) I4 and C(cid:13) = I4 (cid:8) L(cid:13) (cid:8) L(cid:13)(cid:0)1 and view them as transformations of O with respect
to the standard basis feig where In is the n (cid:2) n identity matrix.
Lemma 2.3.2. The image of h : „C(cid:3)”3 !SL(8,C) defined by
h„(cid:21); (cid:22); (cid:13)” = A(cid:21)B(cid:22)C(cid:13)

is a maximal torus T of Spin(7,C).

Proof. It is easy to prove that L(cid:21)L(cid:22) = L(cid:21)(cid:22) and if L(cid:21) = I2 then (cid:21) = 1. It follows that A(cid:21); B(cid:22) and
C(cid:13) commute with each other. Hence, h is a well defined homomorphism. Furthermore, the kernel
of h is given by f(cid:6)„1;1;1”g and thus, the image T (cid:27) „C(cid:3)”3(cid:157)Z2 is isomorphic to „C(cid:3)”3. Since the
rank of Spin(7,C) is 3, we only need to show that T (cid:26) Spin(7,C).

A simple computation shows that „L(cid:21)”(cid:0)1 = L(cid:21)(cid:0)1 = LT

implies T (cid:26) SO„8; C”. Finally, we need to show that for M 2 T, M(cid:3)
direct computation with the help of a software.

(cid:21) . In other words, L(cid:21) 2 SO„2; C” which
(cid:8) = (cid:8). We verify this by a
□
We identify SL„2; C” as the subgroup of the multiplicative group of H with N = 1. More

explicitly, u 2 H is identified with the matrix Au given by

u = a1 + bi + cj + dk 7! Au =

b + ic
Note that Au : H ! C2(cid:2)2 is a linear isomorphism and it satisfies

'››«a (cid:0) id (cid:0)b + ic

a + id

“fifi‹ :

Au Av = Auv:

Moreover, N„u” = det„Au”. Hence, „C2(cid:2)2;det” is a quaternion algebra isomorphic to „H; N” via
„u 7! Au”. Thus, SL„2; C” can be identified with the unit sphere of H, i.e., fv 2 H j N„v” = 1g.

Proposition 2.3.3. There are three SL(2,C) actions on O which preserve B and (cid:8). To describe
these actions we express O as a direct sum O = H (cid:8) lH. Let v = „x; y” 2 O.

12

(cid:0)1; y”
(cid:0)1”

1. g (cid:1) v = „xg
2. g (cid:1) v = „x; yg
3. g (cid:1) v = „gx; gy”

Proof. Since g 2 SL(2,C) is identified with an element of H with norm 1, multiplication by g is an
orthogonal transformation by Lemma 2.1.6. Thus, B is preserved in all three actions.

To show that (cid:8) is also preserved, we instead look at the corresponding action of the Lie algebra
sl„2; C” (cid:27) Im„H” = ⟨i;j;k⟩. It is enough to show that i (cid:1) (cid:8) = j (cid:1) (cid:8) = k (cid:1) (cid:8) = 0. We verify this by a
□
direct computation for all three actions.

Remark 2.3.4. Note that all three actions are faithful and thus, provide three different embeddings
of SL„2; C” into Spin(7,C). Furthermore, we can define an action of the group „SL„2; C””3 on O by

„a; b; c” (cid:1) „x; y” = „cxa

(cid:0)1; cyb

(cid:0)1”

(2.16)

for a; b; c 2 SL„2; C” and „x; y” 2 O. The kernel of this action is f(cid:6)„1;1;1”g.

13

CHAPTER 3

CAYLEY GRASSMANNIAN

In this chapter we investigate a natural compactification of the Cayley Grassmannian over C. We
show that this compactification is singular, its singular set is a cohomology CP5 and we identify
the singular locus as a quotient of GC
3.1 Charts of Gr„4; O”

2 by one of its parabolic subgroups.

We would like to be able to do some computations in order to learn more about the Cayley Grass-
mannian. For this purpose, we recall some elementary facts about the charts of a Grassmannian in
this section.

Set pi jkl = ei jkl so they are coordinate functions on (cid:3)4O and recall that Gr„4; O” can be thought

pi1i2i3 j1 pj2 j3 j4 j5 = pi1i2i3 j2 pj1 j3 j4 j5

of as a subvariety of P„(cid:3)4C8” cut out by the Plücker relations (see [KL72]):
(cid:0) pi1i2i3 j3 pj1 j2 j4 j5 + pi1i2i3 j4 pj1 j2 j3 j5

(cid:0) pi1i2i3 j5 pj1 j2 j3 j4
(3.1)
Since Gr„(cid:8)” lives inside Gr„4; O”, Gr„(cid:8)” is also a subvariety of P„(cid:3)4O”. More precisely, Gr„(cid:8)” lie
in the intersection of Gr„4; O” and the zero locus of (cid:4). By (2.14), the zero locus of (cid:4) is given by
these seven linear equations:

f1 := (cid:0)p0247 (cid:0) p0256 + p0346 (cid:0) p0357 + p1246 (cid:0) p1257 + p1347 + p1356 = 0
f2 := +p0147 + p0156 (cid:0) p0345 (cid:0) p0367 (cid:0) p1245 (cid:0) p1267 + p2347 + p2356 = 0
f3 := (cid:0)p0146 + p0157 + p0245 + p0267 (cid:0) p1345 (cid:0) p1367 (cid:0) p2346 + p2357 = 0
f4 := (cid:0)p0127 + p0136 (cid:0) p0235 + p0567 + p1234 (cid:0) p1467 + p2457 (cid:0) p3456 = 0
f5 := (cid:0)p0126 (cid:0) p0137 + p0234 (cid:0) p0467 + p1235 (cid:0) p1567 + p2456 + p3457 = 0
f6 := +p0125 (cid:0) p0134 (cid:0) p0237 + p0457 + p1236 (cid:0) p1456 (cid:0) p2567 + p3467 = 0
f7 := +p0124 + p0135 + p0236 (cid:0) p0456 + p1237 (cid:0) p1457 (cid:0) p2467 (cid:0) p3567 = 0:

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

14

We call the Zariski closure of this variety the minimal compactification of the Cayley Grassmannian
and denote it by Xmin. It is at least 12-dimensional as it contains the Cayley Grassmannian which
is 12-dimensional [HL82]. Later, we shall see that Xmin is indeed 12-dimensional (see Theorem
3.2.3).

{

}

Once we choose a chart Ustun =

x 2 P„(cid:3)4O” j pstun„x” , 0

, we use the following notation

for local coordinates (suppressing the indices s; t; u; n).

qi jkl =

pi jkl
pstun

:

For example, over U0123, using Plücker relations (3.1), we have

p4567
p0123

=

p0456
p0123

p1237
p0123

(cid:0) p1456
p0123

p0237
p0123

+

p2456
p0123

p0137
p0123

(cid:0) p3456
p0123

p0127
p0123

or, more concisely,

q4567 = q0456q1237 (cid:0) q1456q0237 + q2456q0137 (cid:0) q3456q0127:

(3.9)

(3.10)

After fixing a chart Ustun, one can show that any coordinate function can be expressed only in
terms of the variables qi jkl with jfi; j; k; lg\fs; t; u; ngj = 3 by using (3.1) (repeatedly if necessary).
There are exactly 16 such variables corresponding to the fact that dim„Gr„4; O”” = 16 and they give
us the charts of Gr(4,O), see [KL72, AC15].

3.2 Torus fixed points and the minimal compactification

Recall that Spin„7; C” acts on Gr„(cid:8)” and we have described a specific torus of Spin„7; C”
earlier. By restricting the action to the torus T, we would like to obtain more information about
Xmin = Gr„(cid:8)”. In this section, we prove some facts about this torus action.

From our discussion in Section 2.3 for L(cid:21), it is easy to find eigenvalues and eigenvectors for

h„(cid:21); (cid:22); (cid:13)”. They are given in the following table.

15

eigenvalue

eigenvector

(cid:21)(cid:22)

(cid:21)(cid:0)1 (cid:22)(cid:0)1
(cid:21)(cid:22)(cid:0)1
(cid:21)(cid:0)1 (cid:22)

(cid:21)(cid:13)

(cid:21)(cid:0)1(cid:13)(cid:0)1
(cid:21)(cid:13)(cid:0)1
(cid:21)(cid:0)1(cid:13)

1 + ii
1 (cid:0) ii
j + ik
j (cid:0) ik
l + ili
l (cid:0) ili
lj + ilk
lj (cid:0) ilk

(3.11)

We setee0 = 1 + ii;ee1 = 1 (cid:0) ii;ee2 = j + ik;ee3 = j (cid:0) ik;ee4 = l + ili;ee5 = l (cid:0) ili;ee6 = lj + ilk; and
ee7 = lj (cid:0) ilk. We also seteepqrs =eep ^eeq ^eer ^ees and consider the action of h„(cid:21); (cid:22); (cid:13)” on (cid:3)4O
(or its projectivization). If we denote byeppqrs the transformed Plücker coordinates, the equations

(3.2)-(3.8) can be rewritten as

ef1
ef2
ef3
ef4
ef5
ef6
ef7

:= ep0257 (cid:0)ep1346 = 0
:= ep0146 (cid:0)ep0157 (cid:0)ep0245 (cid:0)ep0267 +ep1345 +ep1367 +ep2346 (cid:0)ep2357 = 0
:= ep0146 +ep0157 (cid:0)ep0245 (cid:0)ep0267 (cid:0)ep1345 (cid:0)ep1367 +ep2346 +ep2357 = 0
:= ep0127 (cid:0)ep0136 +ep0235 (cid:0)ep0567 (cid:0)ep1234 +ep1467 (cid:0)ep2457 +ep3456 = 0
:= ep0127 +ep0136 (cid:0)ep0235 +ep0567 (cid:0)ep1234 +ep1467 (cid:0)ep2457 (cid:0)ep3456 = 0
:= ep0125 (cid:0)ep0134 (cid:0)ep0237 +ep0457 +ep1236 (cid:0)ep1456 (cid:0)ep2567 +ep3467 = 0
:= ep0125 +ep0134 +ep0237 (cid:0)ep0457 +ep1236 (cid:0)ep1456 (cid:0)ep2567 (cid:0)ep3467 = 0

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

16

Theorem 3.2.1. The following eigenvectors of h„(cid:21); (cid:22); (cid:13)” lie in Xmin.

ee0123;ee0145;ee0167;ee2345;ee2367;ee4567

eigenvector

eigenvalue

1
(cid:13)(cid:0)2
(cid:13)(cid:0)2(cid:21)(cid:0)2
(cid:13)(cid:0)2(cid:21)2
(cid:13)(cid:0)2 (cid:22)(cid:0)2
(cid:13)(cid:0)2 (cid:22)2

(cid:13)(cid:0)1(cid:21)(cid:0)2 (cid:22)(cid:0)1
(cid:13)(cid:0)1(cid:21)(cid:0)2 (cid:22)
(cid:13)(cid:0)1(cid:21)2 (cid:22)(cid:0)1
(cid:13)(cid:0)1(cid:21)2 (cid:22)
(cid:13)(cid:21)(cid:0)2 (cid:22)(cid:0)1
(cid:13)(cid:21)(cid:0)2 (cid:22)
(cid:13)(cid:21)2 (cid:22)(cid:0)1
(cid:13)(cid:21)2 (cid:22)

(cid:13)2
(cid:13)2(cid:21)(cid:0)2
(cid:13)2(cid:21)2
(cid:13)2 (cid:22)(cid:0)2
(cid:13)2 (cid:22)2
(cid:21)(cid:0)4

ee0156;ee2356
ee1356
ee0256
ee1256
ee0356
ee1235;ee1567
ee0135;ee3567
ee0126;ee2456
ee0236;ee0456
ee1237;ee1457
ee0137;ee3457
ee0124;ee2467
ee0234;ee0467
ee0147;ee2347
ee1347
ee0247
ee1247
ee0347
ee1357

17

eigenvalue
(cid:21)(cid:0)2 (cid:22)(cid:0)2
(cid:21)(cid:0)2 (cid:22)2
(cid:21)2 (cid:22)(cid:0)2
(cid:21)2 (cid:22)2

(cid:21)4
(cid:22)(cid:0)2
(cid:22)2

eigenvector

ee1257
ee0357
ee1246
ee0346
ee0246
ee1245;ee1267
ee0345;ee0367

Proof. Once we evaluate all the eigenvectors on the transformed defining equations (3.12)-(3.18),
□
we see that exactly the above list of vectors satisfy them.

Theorem 3.2.2. The fixed point set XT

min of the maximal torus action is only the above set of points.

Proof. We only need to verify that in eigenspaces of dimension greater than one, there are no other
fixed points. We prove this for the eigenspace associated to the eigenvalue 1. Let

ex =ec0123ee0123 +ec0145ee0145 +ec0167ee0167 +ec2345ee2345 +ec2367ee2367 +ec4567ee4567 2 Xmin
be a torus fixed point that is different fromee0123;ee0145;ee0167;ee2345;ee2367; andee4567. Then, at
least two of the coordinatesecI; andecJ are nonzero for index sets I , J. Let I = fi1i2i3 j1g, and
J = f j2 j3 j4 j5g. Ifex 2 Xmin (cid:26) Gr(4,O) than it has to satisfy the Plücker relation (3.1)
eci1i2i3 j1
ecj1 j2 j4 j5 +eci1i2i3 j4
(cid:0)eci1i2i3 j5
ecj1 j2 j3 j4
This means thatex does not belong to Gr„4; O” and hence,ex < Xmin.

Now, the left hand side of this relation is nonzero by choice. However, the right hand side has to
be zero as the index sets of eigenvectors associated to eigenvalue 1 differ by at least two elements.

ecj1 j2 j3 j5

ecj2 j3 j4 j5 =eci1i2i3 j2

ecj1 j3 j4 j5

(cid:0)eci1i2i3 j3

:

The same argument also works for all the other eigenspaces of dimension greater than one. □

18

Theorem 3.2.3. Xmin is a singular variety of dimension 12. Moreover, among the torus fixed points
listed in Theorem 3.2.1, all but the following six of them are smooth points.

ee0246;ee0347;ee0356;ee1247;ee1256;ee1357

Proof. We need to analyze neighborhoods of fixed points by using affine charts. We start with the

fixed point m =ee0123, which lies on the open charteU0123 as its origin. Recall that Xmin is cut-out on
eU0123 by the vanishing of the seven linear equations (3.12)-(3.18). At first, it may seem that it is nec-
essary to express these equations in local variableseq0124;eq0125;eq0126;eq0127;eq0134;eq0135;eq0136;eq0137;
eq0234;eq0235;eq0236;eq0237;eq1234;eq1235;eq1236; andeq1237. However, the Plücker relations (in local

coordinates) replace linear terms with higher order terms (see, for example, (3.10)) and we compute
Jacobian at the origin. So, there will be no contribution to Jacobian matrix from other variables.
The Jacobian matrix at m is given by

'››››››››››››››››››«

“fifififififififififififififififififi‹

eJ0123 =

0
0
0
1

0
0
0
0
0

0 0
0
0 0 0 0
0 0
0
0 0 0 0
0
0 0
0 0 0 0
0 (cid:0)1 0 0
0 0 0 1
0 0 0 1
0
0 1 0 0 (cid:0)1 0
0
0 1 0 0

0
0
0
0
0 0 (cid:0)1 0
0 0
0 0

0
0

0
0
0

1
0
0

0
0
0

0 0 0
0
0 0 0
0
0
0 0 0
0 (cid:0)1 0 0 0
0 (cid:0)1 0 0 0
0 1 0
0 1 0

0
0

0 (cid:0)1
1
0

1

which is of rank four. So, it is at most 12-dimensional. However, it contains the 12-dimensional
Cayley Grassmannian. So, it must be 12-dimensional, that is, codimension four in Gr(4,O). Hence,

ee0123 is a smooth point of Xmin.

We repeat this computation for the other points and see that their Jacobian matrices are all rank
□

four, except for the six points we have listed above. Thus, they are all smooth points of Xmin.

3.3 Singular locus

Next, we turn our attention to the singular locus (cid:6) := Sing(Xmin) of Xmin. The torus action
on Xmin restricts to (cid:6). By Theorem 3.2.3, we know that there are six points in (cid:6)T. We quote the

19

following lemma from [BBCM02].
Lemma 3.3.1. If Y (cid:26) P„V” is a projective T-variety, then YT contains at least dim Y + 1 points.

Therefore, we have the following corollary.

Corollary 3.3.2. The singular locus (cid:6) is at most five-dimensional.

As we did with Xmin, we can check whether (cid:6) is singular or not, by using Jacobian criterion on

these six torus fixed points (cid:6)T.

{

eU0246 =

}

:

Theorem 3.3.3. The singular locus (cid:6) is smooth and five-dimensional.

x 2 P„(cid:3)4O” jep0246„x” , 0

of Xmin in Gr„4; O” is four, (cid:6) is locally the vanishing locus of the equations (3.12)-(3.18) localized

Proof. We start with the pointee0246. This point lies at the origin of the chart
On this chart the local variables areeq0124;eq0126;eq0146;eq0234;eq0236;eq0245;eq0247;eq0256;eq0267;
eq0346;eq0456;eq0467;eq1246;eq2346;eq2456;andeq2467 whereeqi jkl =epi jkl(cid:157)ep0246. Since the codimension
to eU0246 and all 4 (cid:2) 4 minors of the Jacobian of these localized equations. These equations by
eq1246; eq0346; eq0267 (cid:0)eq2346; eq0256; eq0247; eq0245 (cid:0)eq2346; eq0236 (cid:0)eq0456;
eq0234 (cid:0)eq0467; eq0146 (cid:0)eq2346; eq0126 (cid:0)eq2456;
So, (cid:6) is just cut out by some hyperplanes ineU0246\Gr(4,O). Therefore, it is clearly smooth at the

themselves do not generate a radical ideal, so we take the radical ideal generated by those equations
with the help of a software called Singular. It turns out the ideal is generated by

and eq0124 (cid:0)eq2467:

origin and of dimension five. We repeat this computation for the other fixed points and see they are
□
all smooth points of (cid:6). Hence, (cid:6) is smooth and five-dimensional.

20

Theorem 3.3.4. The singular locus (cid:6) has the same cohomology ring (over Q) as CP5.

Proof. By Theorem 3.3.3, we see that (cid:6) is a smooth projective variety and hence, it is Kähler.
Thus, 2ith Betti number is at least one for i = 0; : : : ;5. So, the sum of its Betti numbers is at least
six. On the other hand, there is a torus action on (cid:6) (induced from T-action on Xmin) with exactly
six fixed points. Thus, by Białynicki-Birula decomposition [BB73], the sum of Betti numbers is
□
exactly six.

Next, we restrict the action of Spin(7,C) on (cid:6) to the subgroup GC

is given by T \ GC

2 . A maximal torus for GC
2
2 and h denote the Cartan subalgebra

⊕((cid:8)(cid:11)2S+g(cid:11)

)

2 . Let g denote the Lie algebra of GC

has the highest weight and let ∆ = f(cid:11)1; (cid:11)2g be the set of simple roots corresponding to this
choice where (cid:11)2 is the longer root. Let Pi be the parabolic subgroup of GC
2 whose Lie algebra is
where g(cid:11) = fX 2 g j »H; X… = (cid:11)„H”X for all H 2 hg. A straight-forward,
g(cid:0)(cid:11)i

corresponding to our choice of maximal torus. Choose a set of positive roots S+ so thatee0246
⊕
albeit lengthy, calculation shows that the stabilizer subgroup ofee0246 is P2. Hence, we get the
2 acts on the singular locus (cid:6) and the stabilizer group ofee0246 is P2. So, by

following theorem.

Theorem 3.3.5. GC
dimensional reasons, (cid:6) = GC
2

(cid:157)P2.

h

21

CHAPTER 4

DEFORMATIONS IN COMPLEX G2 MANIFOLDS

2 manifolds. These are manifolds with a fixed GC

This chapter is based on a joint work with Akbulut [AY18]. The main object of this chapter are
(almost) GC
2 subbundle of their frame bundle. The
constructions in this chapter are first described in a linear algebraic setting. Then, we extend them
to a global setting. More specifically, compatibility of symplectic and GC
2 structures are introduced
which allows us to define the notion of isotropic associative submanifolds. Examples of GC
2
manifolds are provided by “complexifying” any (almost) G2 manifold. In fact, this complexification
also admits a compatible symplectic structure. Therefore, we can consider isotropic associative
submanifolds of the complexification. Finally, we discuss the (infitesimal) deformations of such
submanifolds.

4.1 Linear algebra

4.1.1 Non-degenerate three-forms in seven-space
Let „V; Ω” be an oriented seven-dimensional vector space over C.
Definition 4.1.1. An alternating three form φ 2 (cid:3)3V(cid:3) is called non-degenerate if for every pair of
linearly independent vectors „u; v” there exists w 2 V such that

φ„u; v; w” , 0:

Example 4.1.2. The associative calibration ϕ is a non-degenerate three-form on Im„O”.

We define a symmetric bilinear form B using the equation

„(cid:19)„u”φ” ^ „(cid:19)„v”φ” ^ φ = 6B„u; v”Ω:

(4.1)

(4.2)

For the rest of the discussion, we will only consider non-degenerate φ which defines a non-
degenerate B.

22

Note that if we scale Ω by (cid:21), we scale B by (cid:21)(cid:0)1. Furthermore, B induces a norm on (cid:3)nV(cid:3).
So by scaling Ω by a positive constant, we may require that the norm of Ω is of magnitude 1. We
will implicitly assume this for the rest of the thesis. Also, to simplify our notations and discussions
later on, we abbreviate the following quadruple „V; φ; Ω; B” satisfying (4.2) as GC
2 -(vector) space.

Remark 4.1.3. One can also define real G2-spaces in a similar manner. In fact, over R, φ determines
both a metric and a volume form uniquely. In that case the metric need not be positive definite.
The 3-form φ is called positive if the metric is positive definite.

4.1.2 The complexification of a G2-space

In this section, we exhibit the linear version of some constructions starting with a real 7-dimensional
vector space with a positive 3-form φ. Although it is possible to do a similar construction with
any non-degenerate 3-form, in this section and for the rest of the thesis, we will focus on positive
φ (see Remark 4.1.3). Recall that φ determines a (real) G2-space „V; φ; Ω; g”. Let VC = V (cid:8) iV.
Furthermore, we can extend all of the structures complex linearly and the equation (4.2) continues
to hold. This implies that the complexified three-form is still non-degenerate. Therefore, we get a
(complex) G2-space „VC; φC; ΩC; gC” where we extend each form complex linearly in every entry.

We can also extend g as a hermitian form h. Explicitly, we define

h„x + iy; z + iw” = g„x; z” + g„y; w” + i „g„y; z” (cid:0) g„x; w”” :

Then, the real part of h is a positive definite metric and the imaginary part is a symplectic form !
on VC.

If V is a half dimensional subspace of W with an almost complex structure J such that V (cid:8) JV =
W, we could use J in place of i in the above construction. This flexibility will be important later
on.

23

4.1.3 Compatible structures on a GC

2 -space

Kähler geometry is often said to be at the intersection of Riemannian geometry, symplectic geometry
and complex geometry because it comes with these three structures that are compatible with each
other. Moreover, any (compatible) two of those structures determines the third one. At the group
level, we can state this as follows

GL„n; C” \ O„2n” = O„2n” \ Sp„2n” = Sp„2n” \ GL„n; C” = U„n”;

(4.3)

see [MS17]. Our construction (see subsection 4.1.2) of a positive-definite metric g, a symplectic
form ! and a (complex) non-degenerate three-form φC from a given (real) non-degenerate three-
form φ allows us to talk about compatibility between these structures related to G2 geometry. In
this section, we describe this relation for a complex 7-dimensional vector space „V; J”.
Definition 4.1.4. We say that the triple „g; !; φC” is compatible if there is a real 7 dimensional
subspace (cid:3) of V and a positive φ on (cid:3) (determining a metric g

′ on (cid:3)) such that

1. V = (cid:3) (cid:8) J(cid:3) =: (cid:3)C

2. φC is the complex linear extension of φ
′.

3. g + i! is the hermitian extension of g

In this case, we say they are induced from „(cid:3); φ; J”.

Proposition 4.1.5.

\ U„7” = G2

GC
2

Proof. It is clear from the definition that G2 (cid:26) GC

2

\ U„7”.
For the converse, first note that U„7” \ O„7; C” = O„7; R” since a matrix whose inverse is both
\U„7” (cid:26) O„7; R”, since

2 . Since G2 (cid:26) O„7; R” (cid:26) U„7”, G2 (cid:26) GC

its conjugate transpose and transpose, must be a real matrix. Therefore, GC
2

24

(cid:26) O„7; C”. So, the intersection consist of real 7 (cid:2) 7 matrices preserving φC. In particular, they

GC
2
preserve φ and we get

\ U„7” = G2:

GC
2

□

Now, using (4.3) and Proposition 4.1.5, it is easy to see that we have

\ O„14” = GC
2

GC
2

\ Sp„14” = G2:

We will need the following technical lemma later.

Lemma 4.1.6. Given a symplectic form ! on R14, a Lagrangian subspace (cid:3) and a positive 3-form
′; !′; φC”
φ on (cid:3), let J„!; (cid:3); φ” be the space of almost complex structures J such that the triple „g
induced from „(cid:3); φ; J” satisfies

1. ! = !′
′j(cid:3) = g

2. g

Then, J„!; (cid:3); φ” is contractible.

The proof of this lemma will follow from the next lemma.

Lemma 4.1.7. Given a symplectic form ! on R2n, a Lagrangian subspace (cid:3) and a metric g on (cid:3),
let J„!; (cid:3); g” be the space of almost complex structures compatible with ! and g„x; y” = w„x; J y”
for x; y 2 (cid:3). Then, J„!; (cid:3); g” is contractible.

Remark 4.1.8. Lemma 4.1.7 says that the set of almost complex structures compatible with a given
symplectic form and a fixed metric on some Lagrangian subspace is contractible.
Proof. First, we choose an orthonormal basis feig for (cid:3) and extend it to !-standard basis fei; fig.
i=1 ei ^ f i. We think of J 2 J„!; (cid:3); g” as an 2n (cid:2) 2n matrix with respect to this basis.
So, !0 =
Note that J 2 J„!; (cid:3); g” if and only if

∑n

25

1. J2 = (cid:0)I2n,

'››« 0 (cid:0)In

0

“fifi‹

2. JTJ2nJ = J2n where J2n =

'››« In B

BT C

In

“fifi‹ is symmetric positive definite.

3. (cid:0)J2nJ =

Let P = (cid:0)J2nJ. Note that

PTJ2nP = (cid:0)JTJ2nJ2nJ2nJ

= JTJ2nJ

This implies C = I + BBT. Define the path Pt =

t = Pt. Next, we

check if Pt is a symplectic matrix.

tB

tBT

= J2n:

I + t2BBT

'››« In
'››« In
'››« 0 (cid:0)In
“fifi‹
'››«(cid:0)tBT (cid:0)In (cid:0) t2BBT
“fifi‹

tB

In

In

“fifi‹. Clearly, PT
“fifi‹
“fifi‹

tBT In + t2BBT

tB

0

“fifi‹

tB In + t2BBT

tBT

'››«In
'››«In
'››« 0 (cid:0)In

In

0

=

=

“fifi‹

tBT

tB In + t2BBT

Therefore, Pt is invertible for all t. Since it is always symmetric and at t = 0 (or t = 1) it is positive
definite, Pt is positive definite for all t. Hence, J2nPt is a path in J„!; (cid:3); g” from J2n to J. Clearly,
□
the path depends continuously on J.
Proof of Lemma 4.1.6. The first two properties imply that J„!; (cid:3); φ” = J„!; (cid:3); g” where g is the
metric induced from φ on (cid:3). Thus, Lemma 4.1.7 shows that it is contractible. The third property
□
is trivially satisfied by definition of complex linear extension.

26

4.2 GC

2 manifolds and complexification of a G2 manifold

In this chapter we define (almost) GC

2 manifolds and provide examples of them. The examples
are obtained by complexifying a G2 manifold „M; ϕ”.
In fact, two different complexification
procedures are described. The advantage of the first procedure is that an almost complex structure,
a metric and a symplectic form on the complexification can be written explicitly. However, the
symplectic form is not necessarily closed. In the second procedure one obtains a closed symplectic
form at the cost of losing some control over the corresponding almost complex structure and metric.

Definition 4.2.1. A (real) 14-dimensional manifold M is called an (almost) GC
frame bundle admits a reduction to a principal GC

2 -bundle.

2 -manifold if its

Proposition 4.2.2. A GC

2 -manifold M naturally has the following structures

• an almost complex structure J 2 (cid:0)„M; End„T M””
• a C-linear three-form φ 2 Ω3„M; C”
• a C-linear seven-form Ω 2 Ω7„M; C”
• a symmetric bilinear form B 2 (cid:0)„M; S2„T M” (cid:10) C”
• two signature „n; n” pseudo-Riemannian metrics g1 = ReB and g2 = ImB.

Proof. Since GC
a GC

2 -frame.

2 preserves each one of these structures, one may pull them back onto M by using
□

Next, we reformulate the above definition. Since GC

2 is the stabilizer of ϕ in SO„ImO” (by

Definition 2.3.1), one may also use the following definition of (almost) GC
Definition 4.2.3. A (real) 14 dimensional manifold „M; J; φ; Ω; B” with an almost complex structure
J, a C-multilinear three form φ, a C-multilinear seven-form Ω and a symmetric C-bilinear form
2 -manifold if for every m 2 M, there is an R-linear isomorphism
B is called an (almost) GC
„TmM; J; φ; Ω” (cid:27) „Im„O”;i; φ0; Ω0”.

2 -manifolds.

27

Next, we describe examples of GC

We start with a usual G2 manifold and construct two different GC
cotangent bundle.

2 manifolds with compatible (almost) symplectic structures.
2 manifold structures on its

Our first construction is as follows. Let „M; φ” be a (real) 7-dimensional G2 manifold. Recall

that M is naturally equipped with a Riemannian metric g and a volume form Ω satisfying

(cid:19)„u”φ ^ (cid:19)„v”φ ^ φ = 6g„u; v”Ω:

(4.4)

We can think of the Levi Civita connection on the cotangent bundle as a horizontal distribution and
hence, it induces the isomorphism

(4.5)
p M and p 2 M. To define an almost complex structure on TT(cid:3)M, we view the metric

where (cid:11) 2 T(cid:3)
as a vector bundle isomorphism g : T M ! T(cid:3)M and we set

T(cid:11)T

(cid:3)

M (cid:27) TpM (cid:8) T

(cid:3)
p M

J„X + (cid:12)” = (cid:0)g

(cid:0)1„(cid:12)” + g„X”

(4.6)

for „X; (cid:12)” 2 TpM (cid:8) T(cid:3)

p M = T(cid:11)T(cid:3)M. Clearly, J2 = (cid:0)ITT(cid:3)M.

Next, we “extend φ complex linearly” to TT(cid:3)M, i.e. we define φC to be the unique C-valued

3-form satisfying

1. φC„X;Y; Z” = φ„X;Y; Z” and
2. φC„J„X”;Y; Z” = iφ„X;Y; Z”

for horizontal vectors X;Y; Z; where we identify TpM with horizontal part of T(cid:11)T(cid:3)M using (4.5).
Similarly, we extend g and Ω complex linearly and we denote the complexifications by B and ΩC,
respectively. Then, from (4.4), we immediately get

(cid:19)„(cid:24)”φC ^ (cid:19)„"”φC ^ φC = 6B„(cid:24); "”ΩC

(4.7)
for (cid:24); " 2 TT(cid:3)M. Note that B is non-degenerate and ΩC is a non-vanishing complex volume form.
Therefore, by (4.7), φC is non-degenerate. We extend g as a hermitian form h as well. So, Reh is a

28

positive definite metric and ! = Imh is an almost symplectic form on T(cid:3)M. More explicitly,

!„X + (cid:11);Y + (cid:12)” = (cid:11)„Y” (cid:0) (cid:12)„X”:

From the construction it is clear that φC is compatible with !.

In the above example, the symplectic form we obtained is not necessarily closed. Our next
example is a similar construction but the symplectic form we obtain at the end is the canonical
symplectic form on T(cid:3)M. We obtain this result at the cost of losing some control of the almost
complex structure.

Again, we start with a (real) 7-dimensional G2 manifold „M; φ” and we think of g as an isomor-
phism between T M and T(cid:3)M. Using this isomorphism, we think of φ as an element of (cid:0)„(cid:3)3T M”.
Therefore, „T(cid:3)
p M; φ” is a G2-space. The vertical subspace of T(cid:11)T(cid:3)M is canonically defined and
isomorphic to T(cid:3)
(cid:25)„(cid:11)”M. The vertical subbundle defines a Lagrangian 7-plane distribution on
„T(cid:3)M; !can”. The space of compatible almost complex structures on „T(cid:3)
(cid:25)„(cid:11)”M; φ; !can”
is contractible by Lemma 4.1.6. Therefore, one can find a global almost complex structure J such
that the complexification of „(cid:3); φ” with respect to J gives us a compatible triple „!can; φC; g”.
Compatibility here means compatibility at every point in the sense of subsection 4.1.3.

(cid:11)T M; (cid:3) = T(cid:3)

4.3 Isotropic associative submanifolds and their deformations

(cid:12)(cid:12)

(cid:12)(cid:12)

X

Recall the following two definitions that are well-known in the literature. A submanifold X
of a symplectic manifold „N; !” is called isotropic if !
= 0. Also, a three-submanifold Y of a
G2 manifold „M; φ” is called associative if the restriction φ
Y is the riemannian volume form on
Y. In this section, we define what we call isotropic associative submanifolds of a GC
2 manifold
with a compatible symplectic structure. The definition is a little subtle. The natural notion of
associative submanifold of a GC
2 manifold is, strictly speaking, a complex three-submanifold but
complex submanifolds are not isotropic with respect to a compatible !. Instead we consider the
“real part” of an associative submanifold. The notion of real part can be made precise using the
symplectic form.

29

Definition 4.3.1. Let L be a (real) 3-dimensional subspace of ImO = C7. We call L isotropic
associative if
(cid:17) 0,

(cid:12)(cid:12)

L

1. !
2. BjLC is non-degenerate.
3. ϕ„u; v; w” = (cid:6)1 for u; v; w 2 LC orthonormal (with respect to B)

where LC = L (cid:8) iL. We denote the space of all isotropic associative planes by I φ
3

Moreover, let Y be a (real) 3-dimensional submanifold of a GC

(cid:26) GrR„3;14”.
2 -manifold M. We call Y isotropic

associative submanifold if TpY is an isotropic associative plane in TpM for every p.

Note that an associative submanifold Y of a G2 manifold M, naturally sits as an isotropic
associative submanifold in the zero section of T(cid:3)M. We consider the infinitesimal deformations
of Y in which Y stays isotropic associative. We obtain Seiberg-Witten type equations from these
deformations.

We denote the normal bundle of Y in M (resp. T(cid:3)M) by (cid:23)RY (resp. (cid:23)CY) and set V =

(cid:23)RY (cid:8) J(cid:23)RY. Then we have the following decomposition
(cid:23)CY = JTY (cid:8) V:

(4.8)
Let (cid:27)t : Y ! T(cid:3)M be a one parameter family of embeddings. Without loss of generality, we
may assume that (cid:219)(cid:27)0 is a section of (cid:0)„(cid:23)CY”. Let f 2 (cid:0)„JTY”, v 2 (cid:0)„V” with (cid:17) := f + v = (cid:219)(cid:27)0. Also,

leteG := Gr„3;TT(cid:3)M” ! T(cid:3)M denote the Grassmann 3-plane bundle over T(cid:3)M. We can lift the
embedding Y ,! T(cid:3)M to Y ,! eG using the Gauss map. Then, the infinitesimal deformation of Y

by (cid:17) induces an infinitesimal deformation of the lift as in [AS08].

For a tangent space L = TxY = ⟨e1; e2; e3⟩, infinitesimal deformation is given by

(cid:219)L =

ei (cid:10) L(cid:17)„ei” 2 TL

eG:

3∑

i=1

So, the conditions for Y to stay isotropic associative are given by

30

ei (cid:2) Lv„ei” = 0

∑

1.
2. (cid:27)(cid:3)

t ! = 0

[AS08, AY18].

Using the Levi-Civita connection ∇ of „T(cid:3)M; g”, we define a Dirac type operator

Note that in the role of Clifford multiplication we are using the cross product operation. So, the
first condition can be expressed as

(4.9)

∑

„v”:

ei (cid:2) ∇ei

(cid:157)DA0 : Ω0„(cid:23)CY” ! Ω0„(cid:23)CY”
(cid:157)DA0
„v” =
∑
∑
∑

ei (cid:2) Lv„ei”
∑
ei (cid:2) „∇vei (cid:0) ∇ei v”
ei (cid:2) ∇vei (cid:0)

ei (cid:2) ∇ei v

=

0 =

We set the perturbation parameter a„v” = (cid:0)∑

=

ei (cid:2) ∇vei. So, the last equation becomes
„v” + a„v” = 0

(4.10)

(cid:157)DA„v” = (cid:157)DA0

where A = A0 + a.

∑

For the isotropy condition, we choose a standard coordinate chart „qi; pi” for the symplectic
dqi ^ dpi where „qi” are coordinates on the base space and „pi” are fiber
form so that ! =
„x1; x2; x3” =
t = (cid:27)i
directions. Write (cid:27)i
t
pj„(cid:27)t„x1; x2; x3”” for 8 (cid:20) j (cid:20) 14 where „x1; x2; x3” are local coordinates on Y. Note that (possibly
after reparametrization) we may assume that „(cid:27)1
” = „x1; x2; x3”. Furthermore, since the
image of (cid:27)0 lies in the 0-section of T(cid:3)M, we may also assume (cid:27) j

„x1; x2; x3” = qi„(cid:27)t„x1; x2; x3”” for 1 (cid:20) i (cid:20) 7 and (cid:27) j

0 = 0 for 8 (cid:20) j (cid:20) 14.

t = (cid:27) j
t

t ; (cid:27)3
t

t ; (cid:27)2

31

During the deformation Y stays isotropic if (cid:27)(cid:3)

7∑
3∑

j=1

j=1

@
@qi

(cid:27)t(cid:3) @
@xi

=

=

=

we have

@(cid:27) j
t
@xi

@
@q j

@
@pj

t ! = 0. Since
@(cid:27) j+7
t
@xi
@(cid:27) j
t
@xi

7∑

+

+

7∑

(cid:14) j
i

+

@
@q j

7∑

j=4

+

j=4
@(cid:27) j
t
@xi

@
@q j

@
@q j

7∑

+

j=1

@(cid:27) j+7
t
@xi

@
@pj

j=1
@(cid:27) j+7
t
@xi

@
@pj

;

”; (cid:27)t(cid:3)„ @
@x j

0 = !„(cid:27)t(cid:3)„ @
@xi
(cid:0) @(cid:27) j+7
t
@xi

@(cid:27)i+7
t
@x j

+

=

””

7∑

k=4

@(cid:27)k
t
@xi

@(cid:27)k+7
t
@x j

(cid:0) @(cid:27)k
t
@x j

@(cid:27)k+7

t
@xi

:

(4.11)

Note that the last equation is of the form da = (cid:0)q„ 1 (cid:10)  2” where a is a 1-form on Y given by

a = (cid:27)8

t dx1 + (cid:27)9

t dx2 + (cid:27)10

t dx3;

 1 and  2 are spinors living as sections of Ω1„(cid:23)RY” and Ω1„J(cid:23)RY” given by

3∑
3∑

i=1

j=1

 1 =

 2 =

„(cid:27)4

t ; : : : ; (cid:27)7

t ”dxi;

„(cid:27)11
t

; : : : ; (cid:27)14

t

”dx j;

@
@xi

@
@x j

and q is a bilinear map given by

q„ 1 (cid:10)  2” =  1 (cid:2)  2

here the cross product is taken in the 1-form parts with metric identification.

32

BIBLIOGRAPHY

33

BIBLIOGRAPHY

[AC15]

[AK16]

[AS08]

[AY18]

[BB73]

Selman Akbulut and Mahir Bilen Can. Complex G2 and associative grassmannian.
arXiv preprint arXiv:1512.03191, 2015.
Selman Akbulut and Mustafa Kalafat. Algebraic topology of manifolds. Expositiones
Mathematicae, 34(1):106–129, 2016.
Selman Akbulut and Sema Salur. Deformations in G2 manifolds. Advances in Math-
ematics, 217(5):2130–2140, 2008.
Selman Akbulut and Ustun Yildirim. Complex G2 manifolds.
arXiv:1804.09951, 2018.
A. Bialynicki-Birula. Some theorems on actions of algebraic groups. Annals of
Mathematics, 98(3):480–497, 1973.

arXiv preprint

[BBCM02] Andrzej Białynicki-Birula, James B Carrell, and William M McGovern. Algebraic
quotients torus actions and cohomology the adjoint representation and the adjoint
action. Springer, 2002.
Robert B Brown and Alfred Gray. Vector cross products. Commentarii Mathematici
Helvetici, 42(1):222–236, 1967.
Robert L Bryant. Metrics with exceptional holonomy. Annals of mathematics, pages
525–576, 1987.

[Bry87]

[BG67]

[BS89]

[DT98]

[HL82]

[Joy00]

[KL72]

[MS17]

[SV13]

Robert L Bryant and Simon Salamon. On the construction of some complete metrics
with exceptional holonomy. Duke Math. J, 58(3):829–850, 1989.
Simon Donaldson and Richard Thomas. Gauge theory in higher dimensions. The
geometric universe (Oxford, 1996), pages 31–47, 1998.
Reese Harvey and H Blaine Lawson. Calibrated geometries. Acta Mathematica,
148(1):47–157, 1982.
Dominic D Joyce. Compact manifolds with special holonomy. Oxford University Press
on Demand, 2000.
S. L. Kleiman and Dan Laksov. Schubert calculus. The American Mathematical
Monthly, 79(10):1061–1082, 1972.
Dusa McDuff and Dietmar Salamon. Introduction to symplectic topology. Oxford
University Press, 2017.
Tonny A Springer and Ferdinand D Veldkamp. Octonions, Jordan algebras and
exceptional groups. Springer, 2013.

34

[SW10]

Dietmar A Salamon and Thomas Walpuski. Notes on the octonions. arXiv preprint
arXiv:1005.2820, 2010.

[Wal13]

Thomas Walpuski. Gauge theory on g2–manifolds. 2013.

35