Mathematical and physical aspects of the quaternion-Kahler

manifold that arises in type II A string compactification on

rigid Calabi-Yau manifolds

Yves van Gennip

June 24, 2004

1

This paper is set by LATEX.Cover illustration: a two dimensional impression of the geometrical situation discussed in section5.7.

2

Contents

1 Introduction 4

2 Preliminary mathematics 62.1 Some differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Almost complex structures and holonomy groups . . . . . . . . . . . . . . . . . . . 11

3 Type II A string theory compactification 183.1 String theory and compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Starting assumptions from type II A string theory . . . . . . . . . . . . . . . . . . 193.3 Expanding the massless spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Compactifying the action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 The manifold SU(1, 2)/U(2) 344.1 Why study SU(1, 2)/U(2)? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Some matrix groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.4 U(2) as subgroup of SU(1, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.5 Bases for u(2) and su(1, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5.1 The Lie algebra of U(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5.2 The Lie algebra of SU(1, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.6 Homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.6.1 Homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.6.2 The homogeneous space SU(1, 2)/U(2) on Lie algebra level . . . . . . . . . 49

5 Geometry and isometries 505.1 Two important geometrical pictures . . . . . . . . . . . . . . . . . . . . . . . . . . 505.2 The (S,C)-coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3 SU(1, 2) as isometry group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.4 Generators of isometries in physics literature . . . . . . . . . . . . . . . . . . . . . 545.5 Special boundary points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.6 The complex upper half plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.7 The dilaton and the Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . 615.8 The Bergman metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.9 The quaternion-Kahler structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.9.1 Quaternions and geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.9.2 Trying to understand the quaternion-Kahler structure . . . . . . . . . . . . 76

A Transparancies 80

3

Chapter 1

Introduction

This paper I have written as part of my final year research project at Universiteit Utrecht (UtrechtUniversity) to complete my TWIN Natuur- en Wiskunde (TWIN Physics and Mathematics) ed-ucation at the Faculteit Natuur- en Sterrenkunde (Faculty of Physics and Astronomy) and theFaculteit Wiskunde en Informatica (Faculty of Mathematics and Computer Science) of UtrechtUniversity. Before I turn to the contents of this paper I would like to use this space to thankmy supervisors, dr. Jan Stienstra of the Mathematisch Instituut (Department of Mathematics) ofUtrecht University and dr. Stefan J.G. Vandoren of the Spinoza Institute and Institute of Theo-retical Physics, both of Utrecht University. They have enthousiastically aided me where necessaryover the course of the last couple of months (September 2003 - June 2004) during which I workedon my research and this paper.

The paper deals with physics as well as mathematics. I will begin with some preliminarymathematics in chapter 2. Following that, chapter 3 deals with the phyiscs, in particular a smallaspect of type II A string theory. From the underlying theory (which I will not discuss in depthin this paper) it follows that type II A string theory has to be defined in ten dimensions. Sincein “real everyday life” we only see four dimensions (three space dimensions and time) we wouldlike to hide of six dimensions. This can be done by the process called compactification, which canbe thought of, as is often done, as wrapping up six dimensions to very small distances, so thatwe are left with four “macroscopic” dimensions. In chapter 3 I will compactify type II A stringtheory on a rigid Calabi-Yau manifold and show that this leads to a so called sigma model in fourdimensions. From this sigma model a metric can be extracted as I will show. This is the mainphysical input in this paper: compactifying the string theory and ending up with a metric. Therest of this paper is concerned with the manifold SU(1, 2)/U(2)1 which is the manifold to whichthe metric we found naturally belongs.

My approach is not to start from scratch looking for a manifold for the metric we found, butuse the suggestion that SU(1, 2)/U(2) is the object I am looking for. This means that chapters4 and 5 are devoted to studying this manifold. In chapter 4 the group theoretical aspects of thismanifold will be scrutinized. The theory of Lie groups and Lie algebras will play an important rolein that chapter. The way in which U(2) is a subgroup of SU(1, 2) will be studied as well as howthis affects the quotient space SU(1, 2)/U(2). Moreover, chapter 4 lays some important groundwork for chapter 5.

Chapter 5 deals with the geometrical aspects of the manifold. It will be shown that we canview our manifold as the space of complex numbers with negative self inner product with respectto diverse bilinear forms. This will in particular lead to the geometrical interpretation in terms ofthe so called (S,C)-coordinates as well as the Bergman coordinates on a complex open ball. Inthis chapter I will also show that SU(1, 2) is an isometry group of the metric we found in chapter3 and what this has to do with the aforementioned geometrical points of view. Furthermore the

1In this paper I will often use the phrase “the manifold SU(1, 2)/U(2)”. Actually, since there are a lot of waysto embed U(2) into SU(1, 2), this is not an accurate terminology. For the sake of simplicity of notation I will usethis terminology though and in each occurence it will be clear from the context what is exactly meant by it.

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special role of the Heisenberg group and the dilaton will be made clear and finally I will saysomething about the quaternionic structure on the manifold. This final section, 5.9, will containresults that are interesting in their own right, but will not give a complete geometrical discriptionof the quaternion-Kahler structure. Hopefully it does give results of interest though to futureresearchers in this area.

In appendix A I have added the text of the transparancies I have used during my undergraduatethesis talk about this paper.

My main goal in this paper is to study the process of compactification on a rigid Calabi-Yaumanifold, the metric that we get as a result of that process and the manifold that correspondsnaturally to that metric. This situation is particularly interesting for physics as well as mathe-matics due to the idea of mirror symmetry . This says that type II A string theory compactified ona Calabi-Yau manifold gives the same theory in four dimensions as type II B theory compactifiedon a mirror Calabi-Yau manifold. By mirror manifold is meant that the hodge numbers are inter-changed: h1,1 for the one manifold is h1,2 for the mirror manifold and vice versa. One interestingcomplication arises when type II A theory is compactified on a rigid Calabi-Yau manifold, since inthat case h1,2 = 0, so using mirror symmetry this gives that h1,1 = 0 on the mirror manifold. Thismeans that this mirror manifold cannot be a Kahler manifold (in particular it cannot be Calabi-Yau), since there cannot exist a Kahler two form. Another interesting aspect of mirror symmetryis that in most computer generated examples it turns out that if type II A theory compactifiedon a non-rigid Calabi-Yau manifold M is equal to type II B theory compactified on a manifoldN , then also type II A theory compactified on N is equal to type II B theory compactified on M .Studying type II A compactification on rigid Calabi-Yau manifolds can perhaps tell us somethingmore about the situation were M is rigid. This paper will not deal with mirror symmetry, butperhaps the information I will give about type II A compactification on rigid Calabi-Yau manifoldsprovides useful information for further studies on that subject.

Finally a remark about the layout I have used. I end proofs with a 2, remarks with a andnotes on notation with a ♦.

Yves van GennipUtrecht, The NetherlandsJune 2004

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Chapter 2

Preliminary mathematics

In this chapter I will introduce some mathematical background needed to understand the physicsin chapter 3 (and even some that will be needed in later chapters). These mathematics can bedivided in two categories. Firstly I need some differential geometry. A lot of this enters into thephysics I will be using via quantum field theory. For more information on this subject see forexample [47]. Secondly I want to know what a Calabi-Yau manifold is, since we will compactifythe type II A string theory on such a manifold in chapter 3. This and related subjects can befound in the second part of this chapter. You will also find some more differential geometry there.This section is by no means meant to provide an extensive overview of differential geometry anda basic knowledge of this subject is assumed throughout this paper. Some useful books on thissubject are e.g. [22], [27], [44] and [52].

2.1 Some differential geometry

Notation: When I write N I will mean the natural numbers not including 0. ♦

Definition 2.1.1 (Complexification) Let V be a real vector space. Then the complexificationof V is the real linear space:

VC := V ⊗R C. (2.1)

Multiplication by complex scalars is defined by:

(∑

j

vj ⊗ zj)λ =∑

j

vj ⊗ (zjλ). (2.2)

An R-linear map f : V →W complexifies to a C-linear map f : VC →WC by:

f(∑

j

vj ⊗ zj) =∑

j

f(vj)⊗ zj . (2.3)

Definition 2.1.2 (Fibre bundle) Let F,E and B be topological spaces. Then a fibre bundlewith fibres F is a continuous map:

f : E → B, (2.4)

with the property that there is an open covering of B such that for each set U in this coveringthere is a homeomorphism (i.e. a continuous function with continuous inverse):

φU : U × F → f−1(U), (2.5)

such that f φU = π1|U , where π1|U is the projection on the first coordinate restricted to U :

π1|U : U × F → U : (u, v) 7→ u. (2.6)

6

B is called the base manifold of the fibre bundle and E the total space. A pair (U, φU ) as aboveis called a chart or a local trivialization for the bundle. For every pair of charts (U1, φ1), (U2, φ2)we have the homeomorphism (called transition map):

φ−11 φ2 : (U1 ∩ U2 × F ) −→ (U1 ∩ U2 × F ) : (x, v) 7→ (x, g(x, v)), (2.7)

for some continuous map g : (U1 ∩ U2 × F ) −→ F which, for every x ∈ U1 ∩ U2, gives a homeo-morphism g(x,−) : F −→ F .

One creates special bundles by putting extra conditions (like differentiability or linearity) on thetransition functions, i.e. on the functions g(x,−) for every x; cf. the definition of vector bundlebelow.

Bundles can also completely be specified by their transition maps (the g(x,−) for every x); cf.the definition of (co)tangent bundle below.

Definition 2.1.3 (Section of a fibre bundle) Use the notation of definition2.1.2. Then a section of E is a differentiable map:

s : B → E, (2.8)

such π1|U s is the identity on B (where U is such that s(b) ∈ U × F ).

Notation: If X is a manifold, then I will denote the tangent space at x ∈ X by TxX . Thedual vectorspace of real linear functions TxX → R I will denote by T ∗

xX . In general I will denotethe dual space of a vectorspace V by V ∗. ♦

Remark: If we have a fibre bundle where the fibres are vector spaces and the transitionfunctions are linear, it is called a vector bundle. The real dimension of the fibres is called the rankof the vectorbundle.In a abuse of terminology the total space sometimes is called the fibre bundleor vector bundle. The most important example of that for this paper is the tangent bundle of amanifold X :

TX := (x, v)| x ∈ X, v ∈ TxX , (2.9)

or put differently, TM is the disjoint union of the vectorspaces TxX for x ∈ X . The transitionfunctions on the tangent bundle are the transposed inverses of the derivatives of the transitionfunctions of the charts on X (on the cotangent bundle T ∗X the transition functions are thederivatives of the transition functions of the charts on X).

Remark: When talking about the continuity or differentiability of functions or fields (seedefinition 2.1.5) on a manifoldX the following is meant. Locally there are charts onX to identifyXlocally with Rn or Cn for a certain n ∈ N. In this way functions and fields on X can locally be seenas functions and fields on R

n or Cn. Then the usual conditions for continuity and differentiability

hold.

Definition 2.1.4 (Tensor product of two fibre bundles) In the notation ofdefinition 2.1.2 let:

f : E → B and f : E → B, (2.10)

be two fibre bundles over the same base manifold B. I will distinguish the objects regarding thefirst and second bundle by adding a ˆ to those of the latter bundle. Then define a new fibre bundle:

f ⊗ f : E ⊗ E → B, (2.11)

with charts (U ⊗ U , φ|U ⊗ φ∣

∣

∣

U) and a projection function π1|U ⊗ π1|U .

Definition 2.1.5 (Tensors and fields) Let V be a real linear space, then a real (q, r)-tensor onV is a map:

⊗qV ∗ ⊗r V → R. (2.12)

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Let X be a manifold and x ∈ X. Then a real (q, r)-tensor at x is a map:

⊗qT ∗xX ⊗r TxX → R. (2.13)

By Tqr,x I will denote the set of all (q, r)-tensors at x ∈ X.

Let k ∈ N ∪ ∞. Let X be a manifold, then a real Ck scalar field on X is a Ck function:

φ : X → R. (2.14)

A Ck vector field on X is a Ck map:v : X → TX, (2.15)

such that for every x ∈ X we have that v(x) ∈ x × TxX. If we write v(x) = (x,A(x)), thenoften A is also called a vectorfield. I will also use this convention. A Ck (q, r)-tensor field on Xis a Ck map:

t : X → ⊗qT ∗X ⊗r TX, (2.16)

for fixed q and r, such that t(x) ∈ x × Tqr,x. If we write t(x) = (x,B(x)), then often B isalso called a (q, r)-tensorfield. I too will use this terminology. We see that a vectorfield is a(1, 0)-tensorfield. By Tqr I will denote all C∞ (q, r)-tensorfields on X.

Remark: Complexification of real tensor(field)s goes according to definition 2.1.1. Remark: If I do not specify the differentiability of the fields I use, I will always mean smooth

(i.e. C∞) fields. Remark: Note that a vector field on a manifold X is a section of the tangent bundle TX of

X .

Definition 2.1.6 (q-form) Let V be a real linear space, then a real q-form on V is a completelyantisymmetric real (q, 0)-tensor on V . Let X be a manifold and x ∈ X. Then a real q-form at xis a completely antisymmetric real (q, 0)-tensor at x. By Ωq

x I will denote the vector space of allq-forms at x ∈ X and by Ωq the space of all smooth q-forms (i.e. completely antisymmetric C∞

(q, 0)-tensorfields) on X.

Remark: Complexification of real forms goes according to definition 2.1.1. Notation: Assume the notation of definition 2.1.5. Let ∂

∂xν ν , 1 ≤ ν ≤ dimX , be a basis ofTxX , for a given x ∈ X and let dxµµ, 1 ≤ µ ≤ dimX , be the dual basis of T ∗

xX . Then:

A(x) =

dimX∑

µ=1

Aµ(x)dxµ (2.17)

and:

B(x) =

dimX∑

ν1=1

· · ·dimX∑

νr=1

dimX∑

µ1=1

· · ·dimX∑

µq=1

Bν1···νrµ1···µq

(x)∂

∂xν1∧ · · · ∧ ∂

∂xνr∧ dxµ1 · · · dxµq . (2.18)

To shorten the notation I will use the Einstein summation convention in the following unlessstated otherwise, which says that in a given expression you must sum over the indices that appearas both subscript and superscript. The previous two expressions become in this convention:

A(x) = Aµ(x)dxµ (2.19)

and:

B(x) = Bν1···νrµ1···µq

(x)∂

∂xν1∧ · · · ∧ ∂

∂xνr∧ dxµ1 · · · dxµq . (2.20)

Physicists tend to work more often with this notation with indices than mathematicians, e.g.physicists would call Aµ the vectorfield. With the above remarks we will be able to keep track of

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the different notations. Note however that for fixed indices we have that Aµ(x) and Bν1···νrµ1···µq

(x)are functions from X to C. The x-dependence however will often be suppressed in the notationto make it less cumbersome, e.g. Aµ(x) becomes Aµ. ♦

Notation: Often, when dealing with symmetric tensors, I will use the notation dxdy insteadof 1

2 (dx⊗ dy + dy ⊗ dx), where ⊗ is the tensor product. For dxdx I will write dx2. ♦A useful object in calculations is the Kronecker delta.

Definition 2.1.7 (Kronecker delta) The Kronecker delta is defined as follows:

δji :=

1 if i = j;0 if i 6= j.

(2.21)

Definition 2.1.8 (Metric on a manifold) Let k ∈ N and let X be a manifold. Then a Ck

metric on X or Riemannian structure on X is a Ck map g which assigns to each x ∈ X an innerproduct (i.e. a positive definite symmetric bilinear form) gx on TxX such that g varies continuouslyif x is varied. The pair (X, g) is called a Riemannian manifold (often if a Riemannian structureon X exists, then X is called a Riemannian manifold). A Ck pseudo-Riemannian structure onX is a Ck map g which assigns to each x ∈ X a non-degenerate symmetric bilinear form gx onTxX such that g varies continuously if x is varied. If this is the case (X, g) (or X) is called apseudo-Riemannian manifold. I will also use the terminolgy metric in this case (e.g. Minkowskimetric).

Lemma 2.1.9 (The (anti)symmetry of ∧) Let α be a p-form and β a q-form. Then:

α ∧ β = (−1)pqβ ∧ α. (2.22)

Proof: This is a very well known fact. See e.g. [22]. 2

Next I will give some definitions and rules of calculation for working with the Levi-Civitaepsilon symbol and tensor. I will follow the conventions from [13] (pp. 51-54) here.

Definition 2.1.10 (The antisymmetric Levi-Civita epsilon symbol) Let n ∈ N and let Xbe an n-dimensional orientable manifold with local real coordinates (xi)i. Let dx1∧· · ·∧dxn be thepositive orientation locally. Then define the Levi-Civita epsilon symbol via the following equation:

εj1···jndxj1 ∧ · · · ∧ dxjn := dx1 ∧ · · · ∧ dxn, (2.23)

where j1, · · · , jn ∈ 1, · · · , n and I have not used the Einstein summation convention (so thereis no summation over j1, · · · jn). We see that this gives

εj1···jn :=

1 if j1 · · · jn is an even permutation of 1, 2, · · · , n;−1 if j1 · · · jn is an odd permutation of 1, 2, · · · , n;0 otherwise,

(2.24)

as long as the j1, · · · , jn refer to the local coordinates xi.

When we introduce new local coordinates in the situation of definition 2.1.10, the epsilonsymbol can take other values than −1, 0 and 1 as shown in the next lemma for an explicit situationwe will encounter later.

Lemma 2.1.11 (ε on three complex coordinates in six dimensions) Let X in definition2.1.10 be such that locally we can define complex coordinates (so in particular the dimension n mustbe even). Let (xi)i be locally defined real coordinates and let dx1∧· · ·∧dxn be the positive orientationlocally. Take εj1···jn as in definition 2.1.10 and define complex coordinates zj := xj + ixj+

n2 with

corresponding complex conjugates z j := xj − ixj+ n2 . Now let n = 6. Then:

ε123123 = − i8

=: − i8ε123ε123, (2.25)

where the values for permutations of the indices can be found by using the complete antisymmetryof ε.

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Proof: We know from definition 2.1.10 that ε123123 is defined such that:

ε123123dz1 ∧ dz2 ∧ dz3 ∧ dz1 ∧ dz2 ∧ dz3 = dx1 ∧ dx2 ∧ dx3 ∧ dx4 ∧ dx5 ∧ dx6. (2.26)

Inserting zj := xj + ixj+32 (with j ∈ 1, 2, 3), shows that:

dz1 ∧ dz2 ∧ dz3 ∧ dz1 ∧ dz2 ∧ dz3 = 8idx1 ∧ dx2 ∧ dx3 ∧ dx4 ∧ dx5 ∧ dx6, (2.27)

and thus the desired result follows. 2

The next thing to do is to go from the epsilon symbol epsilon tensor on an orientable manifold.

Definition 2.1.12 (The epsilon tensor on an orientable manifold) Let X be an orientablemanifold with metric g, then the Levi-Civita epsilon tensor is, in terms of the epsilon symbol fromdefinition 2.1.10:

εj1j2···jn =√

|detg|εj1j2···jn , (2.28)

εj1j2···jn = sgn(g)1

√

|detg|εj1j2···jn , (2.29)

where sgn(g) = sgn(detg).

Corollary 2.1.13 (ε on three complex coordinates in six dimensions) In the situation oflemma 2.1.11 and definition 2.1.12 we have that:

ε123123 = − i8ε123ε123. (2.30)

Proof: This follows from lemma 2.1.11 and definition 2.1.12. 2

Definition 2.1.14 (Hodge star operator) Let X be an orientable manifold of dimension mwith metric g. Then, for every 0 ≤ r ≤ m, the Hodge star operator ∗ is a linear isomorphism:

∗ : Ωr → Ωm−r, (2.31)

defined on basis elements by

∗(dxµ1 ∧ dxµ2 ∧ · · · ∧ dxµr ) =1

(m− r)! εµ1µ2···µr

νr+1νr+2···νmdxνr+1 ∧ · · · ∧ dxνm . (2.32)

For an explanation of the epsilon tensor see definition 2.1.12. For the components of an r-tensorT in m dimensions this means:

(∗T )j1j2···jr =1

(m− r)! εj1j2···jrjr+1···jmTjr+1···jm , (2.33)

(∗T )j1j2···jr =1

(m− r)! εj1j2···jrjr+1···jmTjr+1···jm . (2.34)

Lemma 2.1.15 (Twice the Hodge star) If α is a p-form on an n-dimensional orientable Rie-mannian manifold, then:

∗(∗α) = (−1)p(n−p)α. (2.35)

If α is a p-form on an n-dimensional pseudo-Riemannian manifold that is not Riemannian (likee.g. the Lorentzian Minkowski space), then:

∗(∗α) = −(−1)p(n−p)α. (2.36)

Proof: The proof of this lemma can be found in e.g. [22], [44] or [52] (vol I, p. 120). 2

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Definition 2.1.16 (Invariant volume element) Let X be an orientable manifold of dimensiond, then the invariant volume element of X, volX , is defined as:

volX := ±√

|detg|dx1 ∧ · · · ∧ dxd = ∗1. (2.37)

The ± depends on what is chosen as the positive and what as the negative orientation of X.

Lemma 2.1.17 (Symmetry of ∧∗) Let α and β be two q-forms. Then

α ∧ ∗β = β ∧ ∗α. (2.38)

Proof: Using the definition of the Hodge star operator from definition 2.1.14 and the fact that αand β are forms of the same order q, this follows if for example one writes out the components ofα ∧ ∗β in local coordinates. 2

Definition 2.1.18 (The codifferential operator) Let X be a d-dimensional manifold with aglobal scalar product (., .) for q-forms (for any q). Let d be the differential operator, then thecodifferential operator, d∗, is the map from Ωq to Ωq−1 defined to be such that for a (q − 1)-formα and a q-form β it holds that:

(dα, β) = (α, d∗β). (2.39)

As can be read in e.g. [22] this means for a q-form β on a Riemannian manifold:

d∗β := (−1)d(q+1)+1 ∗ d ∗ β (2.40)

and for β on a pseudo-Riemannian manifold:

d∗β := (−1)d(q+1) ∗ d ∗ β. (2.41)

For the definitions of (pseudo-)Riemannian manifolds see definition 2.1.8.

2.2 Almost complex structures and holonomy groups

Definition 2.2.1 (Almost complex structure on a vector space) Let V be a vector space.An almost complex structure on V is an endomorpism, J , on V such that J 2 = −IdV .

Definition 2.2.2 (Almost complex structure on a manifold) Let X be amanifold. An almost complex structure on X is an endomorpism, J , on the tangent bundle TXsuch that J2 = −IdTX .

Remark: Note that a manifold X with an almost complex structure is not necessarily acomplex manifold . I will not go into detail here about the differences, but, as is explained in e.g.[44], a manifold with an almost complex structure is a complex manifold if and only if the almostcomplex structure is integrable or equivalently if the so called Nijenhuis tensor vanishes. Moreinformation on these concepts can be found in e.g. [44] or [51] (note that [51] uses the term torsioninstead of Nijenhuis tensor. This is related to how the action of the almost complex structure onTxX changes when one varies the point x ∈ X .

Lemma 2.2.3 (Expanding with respect to the almost complex structure) Let V be a realvector space with an almost complex structure J . Then we can expand VC as follows:

VC = V + ⊕ V −, (2.42)

whereV ± := v ∈ VC|Jv = ±iv . (2.43)

Proof: Since J2 = −IdV the eigenvalues λ of J must satisfy λ2 = −1, so they are ±i. Theexpansion now follows. 2

11

Definition 2.2.4 (Hermitian metric) If a metric g satisfies:

g(JA, JB) = g(A,B), (2.44)

for an almost complex structure J it is called Hermitian with respect to J .

Definition 2.2.5 (Hermitian manifold) Let X be a manifold with a nalmost complex structuredefined on it and g a metric on X. Then (X, g) -or X, if the metric used is clear from the context-is a Hermitian manifold if and only if g is a Hermitian metric.

Definition 2.2.6 (Kahler form) If J is an almost complex structure and g a Hermitian metric,then the two form ωJ defined as:

ωJ(A,B) := g(A, JB), (2.45)

is called the Kahler form of J .

Definition 2.2.7 (Kahler manifold) Let (X, g) be a Hermitian manifold with almost complexstructure J and Kahler form ωJ . Then it is a Kahler manifold if and only if ωJ is closed (i.e.dωJ = 0). If (X, g) is a Kahler manifold, then g is called the Kahler metric. If the metric is clearfrom the context, often X is called the Kahler manifold.

Remark: Not all complex manifolds admit Kahler metrics, see e.g. [44]. Definition 2.2.8 ((r, s)-form) Let ω := αC be the complexification of a real q-form α on acomplex linear space VC. Then, for r and s such that q = r+ s, ω is called a (r, s)-form or a formof bidegree (r, s) if and only if ω(v1, · · · , vq) = 0 unless r of the vi are in V + and s of the vi arein V −, where V ± is as in lemma 2.2.3.

Definition 2.2.9 (Dolbeault operators) Let ω be a differentiable (r, s)-form, then dω is an(r+ s+1)-form. Now define the Dolbeault operators, ∂ and ∂, by defining ∂ω to be the (r+1, s)-form part of dω and ∂ω to be the (r, s+ 1)-form part of dω. Thus by definition we have:

d = ∂ + ∂. (2.46)

Corollary 2.2.10 (Hermitian and Kahler metrics and Kahler potential) Let X be an n-dimensional complex manifold with metric g and local coordinates (xi)i. In local complex coordi-nates zj = xj + ix

n2 +j and z j = xj − ixn

2 +j (where 1 ≤ j ≤ n2 ), we can write:

g =

n∑

i,j=1

gijdzi ⊗ dzj +

n∑

i,j=1

gijdzi ⊗ dzj +

n∑

i,j=1

gijdzi ⊗ dz j +

n∑

i,j=1

gijdzi ⊗ dz j . (2.47)

To get rid of the cumbersome notation with the sum signs I will use throughout this corollary theEinstein summation convention. The almost complex structure J acting on the real vectorspacesTxX (for x ∈ X) with bases

∂∂x1 · · · ∂

∂xn

is defined to be

J∂

∂xj=

∂

∂xn2 +j

if j ≤ n2 and (2.48)

J∂

∂xj= − ∂

∂xn2 +j

if j > n2 . (2.49)

If and only if g is a Hermitian metric with respect to J then:

gij = gij = 0. (2.50)

If and only if g is also a Kahler metric with respect to J , then in addition to the previous propertythere exist a locally defined function K, called the Kahler potential, such that:

gij =∂2K

∂zi∂z j. (2.51)

The Kahler form is:ωJ = igijdz

i ∧ dz j . (2.52)

12

Proof: If we construct the basis of the complexified vector spaces TxXC we get

∂

∂zj=

∂

∂xj+ i

∂

∂xn2 +j

;

∂

∂z j=

∂

∂xj− i ∂

∂xn2 +j

. (2.53)

We see that:

J∂

∂zj= −i ∂

∂zj;

J∂

∂z j= i

∂

∂z j. (2.54)

Note that this is compatible with the statement in lemma 2.2.3. If g is Hermitian with respect toJ , then:

gij = g(∂

∂zi,∂

∂zj)

= g(J∂

∂zi, J

∂

∂zj)

= g(−i ∂∂zi

,−i ∂∂zj

)

= −g( ∂

∂zi,∂

∂zj)

= −gij , (2.55)

thus gij = 0. In the second identity I used the Hermiticity of g and in the fourth identity I usedits bilinearity. It will be clear now that we can do a similar calculation for gij , but this argumentdoes not supply us with conditions on gij , since:

g(J∂

∂zi, J

∂

∂z j) = g(−i ∂

∂zi, i

∂

∂z j)

= g(∂

∂zi,∂

∂z j). (2.56)

We see that the reverse statement also follows: if gij = gij = 0, then g is Hermitian. We canconstruct the Kahler form as in definition 2.2.6 (since we already know that only the gij terms ofthe metric are non-zero, we only have to investigate these mixed terms):

ωJ(∂

∂zi,∂

∂z j) = g(

∂

∂zi, J

∂

∂z j)

= (∂

∂zi, i

∂

∂z j)

= igij ; (2.57)

ωJ(∂

∂z j,∂

∂zi) = g(

∂

∂z j, J

∂

∂zi)

= g(∂

∂z j,−i ∂

∂zi)

= −igji. (2.58)

Using the symmetry gij = gji and the relation between the wedge product ∧ and the tensorproduct ⊗ (see e.g. [22]), we get:

ωJ = igijdzi ⊗ dz j − igjidz j ⊗ dzi

= igijdzi ∧ dz j . (2.59)

13

If the metric is Kahler, then the Kahler form is closed: dωJ = 0. This gives:

dωJ = (∂ + ∂)igijdzi ∧ dz j , (2.60)

with ∂ and ∂ the Dolbeault operators from definition 2.2.9. Written out we get

∂gij =∂gij∂zl

dzl;

∂gij =∂gij

∂z ldz l. (2.61)

We thus see that equation (2.60) implies (due to the antisymmetry of the wedge product) for allindices i, l, j and k we have:

∂gij∂zl

=∂glj∂zi

; (2.62)

∂gij

∂zk=∂gik∂z j

. (2.63)

From this we conclude that locally we have a function K such that:

gij =∂2K

∂zi∂z j. (2.64)

Conversely we see that if g obeys equation (2.64) for a locally defined function K, then it is Kahler.2

Next I will define the important concepts of covariant derivative and connection. I will notexplore the background and necessity of these concepts. More information on these things can befound in e.g. [13] or [22].

Definition 2.2.11 (Covariant derivative and connection) Let X be a manifold. Then a co-variant derivative is a map ∇, defined on

⊕

r,s Tsr, which sends a (r, s)-tensorfield to a (r, s+ 1)-tensorfield and satisfies the following properties for T, S (r, s)-tensorfields and φ a scalar field:

∇(T + S) = ∇(T ) +∇(S),

∇(T ⊗ S) = ∇(T )⊗ S + T ⊗∇(S),

∇µ(

Tν1···νi−1λνi+1···νr

ρ1···ρj−1λρj+1 ···ρs

)

= ∇(

Tν1···νi−1λνi+1···νr

µρ1···ρj−1λρj+1 ···ρs

)

,

∇µφ =∂φ

∂xµ, (2.65)

where the last two equations are written out on coordinates x. ∇µ is then covariant differentiationalong the µth vector field in a frame of vector fields (i.e. a local collection of dimX linearlyindependent vector fields). Covariant differentiation along a general vector field V µ ∂

∂xµ is then∇V := V µ∇µ. In these coordinates the covariant derivative of a vector field V is:

∇µV ν =∂V ν

∂xµ+ ΓνµλV

λ, (2.66)

where Γνµλ := (Γµ)ν

λare the connection coefficients or the entries in the connection matrices Γµ.

For the covariant derivative of a one-form field ω we have:

∇µων =∂ων∂xµ

− Γλµνωλ. (2.67)

Notice the sign change with respect to the vector case. This can now be straightforwardly generalizedto the covariant derivative of a general tensor as done in e.g. [13] and [22].

In general the choice of connection is not unique. Beware that sometimes in the literaturethe covariant derivative as a whole is called the connection. I will not do so. Another oftenencountered statement (especially in physics literature) is that an object is covariantly constant.By this is meant that the covariant derivative of the object is zero.

14

Corollary 2.2.12 (Covariant derivative of an endomorphism) Let X be amanifold, let J be an endomorphism on the vectorfields on X and let ∂

∂tbe an, at least locally

defined, vectorfield on X. If we then want to define the covariant derivative of J along thevectorfield ∂

∂t, ∂∂tJ , such that it is compatibel with the covariant derivative of vectorfields and the

Jacobi identity as described in definition 2.2.11, then we have to define:

∂

∂tJ := ∇

(

∂

∂t

)

J = ∇(

∂

∂t

)

J − J ∇(

∂

∂t

)

. (2.68)

If the connection matrix corresponding to ∇(

∂∂t

)

is given by A, we see that we get the map:

∇ : J 7→ AJ − JA. (2.69)

Proof: Let v be a tangent vector. Then the Jacobi identity when computing the derivative of Jvgives:

∂

∂t(Jv) =

(

∂

∂tJ

)

v + J ∂v∂t. (2.70)

Rewriting this as:(

∂

∂tJ

)

v =∂

∂t(Jv)− J ∂v

∂t(2.71)

leads to the desired result. 2

Definition 2.2.13 (Torsion tensor) Let X be a manifold with connection Γ, then the torsiontensor T is defined as follows:

T λµν = Γλµν − Γλνµ. (2.72)

Note that if the connection is symmetric in the lower indices the torsion vanishes. In this case wecall Γ a torsion free connection.

We can equivalently see torsion as a map that assigns to two vector fields V,W a third one:

T (V,W ) = ∇VW −∇WV − [V,W ]. (2.73)

Definition 2.2.14 (Levi-Cevita connection) Let X be a manifold with metric g, then theLevi-Cevita connection is the unique torsion free connection which is compatible with the met-ric, i.e. for which

∇(g) = 0 (2.74)

holds. This connection is also known as the Christoffel connection or the Riemannian connec-tion. The corresponding connection coefficients are known as the Christoffel symbols and can beexpressed written out on coordinates x in terms of the metric (as shown in e.g. [13]) as:

Γσµν =1

2gσρ

(

∂gνρ∂xµ

+∂gρµ∂xν

− ∂gµν∂xρ

)

. (2.75)

Remember that gµν are components of the metric, so gνµ are components of the inverse of themetric.

Corollary 2.2.15 (The Levi-Cevita connection on Kahler manifolds) Let X be a Kahlermanifold with metric g, then with respect to complex coordinates z and z the only non-zero com-ponents of the Levi-Cevita connection are:

Γljk = gls∂gks∂zj

,

Γljk

= gls∂gks∂z j

. (2.76)

Proof: This follows from equations (2.62), (2.63) and (2.75). 2

15

Definition 2.2.16 (Curvature tensor) Let X be a manifold with a connection Γ. Then theRiemann tensor or curvature tensor assigns to three vector fields U, V,W a fourth vector field:

R(U, V )W = ∇U∇VW −∇V∇UW −∇[U,V ]W. (2.77)

As shown in e.g. [13] written out in coordinates we get:

Rρσµν =∂

∂xµΓρνσ −

∂

∂xνΓρµσ + ΓρµλΓ

λνσ − ΓρνλΓ

λµσ . (2.78)

Note that Rρσµν is antisymmetric in µ and ν.

A more geometrical interpretation of the curvature tensor (of which I will not prove here thatit leads to the same tensor as in definition 2.2.16) is to parallel transport a vector V along theclosed parallelogram spanned by the vectors A and B with infinitesimal lengths δa and δb. Aftercompleting the loop V is changed into V + δV , where δV is related to the curvature as:

δV ρ = (δa)(δb)AνBµRρσµνVσ. (2.79)

A bit more information on this view van be found in [13].There are two often encountered objects that can be derived from the curvature.

Definition 2.2.17 (Ricci tensor and Ricci scalar) Let X be a manifold with metric g a con-nection Γ and corresponding curvature R. Then by contracting we define the Ricci tensor:

Rµν := Rλµλν , (2.80)

and the Ricci scalar:R := Rµµ = gµνRµν . (2.81)

If the Ricci tensor is zero, the manifold X is called Ricci flat.

For the next definition the notion of parrallel transport from differential geometry is assumedknown. For information on parallel transport, its relation with the covariant derivative and relateddifferential geometry see e.g. [13], [22], [29] or [33].

Definition 2.2.18 (Holonomy group) Let X be a manifold, let x ∈ X and let c(t) be a closedloop that starts and ends in x. Let Pc(t) : TxX → TxX be the map that maps a vector v ∈ TxXon the vector Pc(t)v ∈ TxX by parallel transporting v along c(t). The set of these transformations

is the holonomy group at x, H(x). This is indeed a group since Pid = id, Pc(−t) = P−1c(t) and

Pdc(t) = Pd(t) Pc(t).Furthermore, if X is arcwise connected we have that for every x, y ∈ X we can define a

curve a(t) from x to y which induces by parallel transport a map τa(t) : TxX → TyX. Then

H(y) = τ−1a(t)H(x)τa(t), thus in this case H(x) and H(y) are isomorph. We can then speak of the

holonomy group of the manifold.

In definition 2.2.19 and lemma 2.2.20 that follow some knowledge of Lie groups and Lie algebrasis assumed. In sections 4.3 and 4.5.1 these subjects are treated in greater detail.

Definition 2.2.19 (G-connection) Let X be a manifold with connection Γ and let G be a Liegroup with an action on X. Then Γ is called a G-connection if the curvature corresponding to Γis an element of the Lie algebra g1. We see from the form of the curvature (i.e. derivatives andcommutators of connection matrices, see definition 2.2.16) that this is the case if and only if theconnection matrices are in g.

1Note that this makes sense in the following way. Because G acts on X, the Lie algebra g acts on the tangentbundle TX. The curvature is a 2-form that takes values in the endomorphims bundle on the tangent bundle TX.So these values can also be in the Lie algebra g

16

Lemma 2.2.20 (Holonomy and connection) Let X be a manifold with holonomy group G.Then on X there is a G-connection.

Proof: I will only sketch the proof here. For more details I refer to e.g. [41]. We see fromequation (2.79) that after parallel transport along an infinitesimal loop a vector V changes into anew vector of the form V (1 + εR) where ε is an infinitesimal parameter and R is the curvature.So R is an element of the Lie algebra of the holonomy group. 2

Simply connected Riemannian manifolds can be classified by way of their holonomy groups.This is known as Berger’s classification or Berger’s list . For more information on this classificationsee e.g. [10]. I will now define different types of manifolds by the properties of their holonomygroups. First I’ll introduce a handy and often encountered notation.

Definition 2.2.21 (Berger’s list) Let d, n ∈ N. Let X be a simply connected Riemannian man-ifold (that is also irreducible and nonsymmetric, see e.g. [35]) of dimension d. Then is X is oneof the following:

• For every Riemannian manifold the holonomy group of X is contained in SO(d). We caneven take this as definition of Riemannian manifold, but it also follows from our previousdefinition of Riemannian manifold in definition 2.1.8 together with the fact that paralleltransport conserves inner products.

• If d = 2n and the holonomy group of X is contained in U(n) the manifold is called a Kahlermanifold. I will not show here that this agrees with our definition 2.2.7. Also without proof Istate that Kahler manifolds are complex, so there is a covariantly constant complex structure.

• If d = 2n and the holonomy group of X is contained in SU(n) the manifold is called aCalabi-Yau manifold. Usually the notation CYn is used to denote X in this case. I willfollow this convention. Since a Calabi-Yau manifold is also Kahler, it is also complex.

• If d = 4n and the holonomy group of X is contained in Sp(n,H) = Sp(4n,R) = USp(2n,C) :=U(2n) ∩ Sp(2n,C), then X is called a hyper Kahler manifold. Without proof I state that onthis manifold there exist three covariantly constant complex structures.

• If d = 4n and the holonomy group of X is contained in Sp(1,H)·Sp(n,H) = SU(2) ·Sp(n,H)2

where the Sp(1,H)-connection must be non-trivial, then X is called a quaternionic-Kahlermanifold3. Such a manifold has three almost complex structures which are not covariantlyconstant.

• Another possibility is that d = 7 and the holonomy group of X is contained in the exceptionalgroup G2.

• The final possibility is that d = 8 and the holonomy group of X is contained in Spin(7).

More information about the groups in Berger’s list (except G2 and Spin(7)) can be found insection 4.2. The cases where the holonomy group is contained in G2 or Spin(7) are added forcompleteness, but we will not encounter those cases in this paper. For more information on theexceptional groups, see e.g. [23]. In literature often the following terminology is used. A CY1

manifold is called an elliptic curve and a CY2 manifold is called a K3 surface. The two typesof manifolds that are most important for us in this paper are the Calabi-Yau manifolds and thequaternion-Kahler manifolds.

2Sp(1, H) · Sp(n, H) := Sp(1, H) × Sp(n, H)/

(−IdSp(1,H),−IdSp(n,H)), (IdSp(1,H), IdSp(n,H))

and an analogousexpression holds for SU(2) · Sp(n, H).

3[51] gives a different definition of a quaternion-Kahler manifold, where he distinguishes the cases n > 1 andn = 1. In the latter case the manifold is called quaternionic-Kahler if it is Einstein and self-dual. I will not go intofurther details here, nor will I prove if this definition is equivalent to the one in Berger’s list.

17

Chapter 3

Type II A string theory

compactification

3.1 String theory and compactification

This chapter deals with string theory. String theory is a theory in development and accordingto popular lore the most promising candidate to unify Einstein’s theory of general relativity andquantum mechanics (or quantum field theory) in one theory. Although I will not go into thebasics of string theory in this paper (good introductions can be found in e.g. [24] and [31]), myapproach will be such that background knowledge about string theory is not needed to followmy calculations, though it will certainly help to understand the physical relevance of my paper.Without going into much detail I will say a few words about string theory and in particularstring theory type II A, which I will be discussing. This short introduction will necessarily give asomewhat simplified view. Note that I do assume some knowledge of general relativity (see e.g.[13], [29] or [33]), quantum mechanics (see e.g. [48]) and quantum field theory (see e.g. [47]).

The primary notion of string theory is that at the basic level we cannot describe Naturewith point particles, but we need one dimensional strings (although in recent years also higherdimensional objects called branes have been studied). One way of setting up string theory is tobegin with an ideal classical string. Ideal in the sense that we are dealing with a one-dimensionalobject, whereas a non-ideal macroscopic physical string, like e.g. a violin string, also has somestructure in the two dimensions transversal to the string. By solving the differential equationsfor a vibrating string for both open and closed strings we can get a description of the string interms of a moving centre of gravity and vibration modes of the string. These modes of vibrationare divided in left moving modes and right moving modes and depending on if we are studyingan open string or a closed one, we have to impose different boundary conditions on these modes.One important difference between open and closed strings is that in closed strings the left andright moving modes are independent whereas for open strings they are related because there isreflection of waves at the endpoints of an open string. After the classical solution is found thestring is quantized. That means that location and impulse of the string’s centre of gravity as wellas the vibration modes are turned into operators, much the same like is done in ordinary quantummechanics with the location and momentum. The resulting operators can then be used to createstring states by acting with them on a vacuum state. Different string states will give rise to themultitude of different fields (or particles) we have in field theory. Now to fully understand whatit means to be talking about type II A string theory and what the difference is with type I openand closed string theory, type II B string theory and heterotic string theory, we need concepts likesupersymmetry, GSO projection (named after Gliozzi, Scherk and Olive) and the related conceptof chirality, but for this short introduction it is enough to know that these concepts refer to thebehaviour of string states when acted on by certain operators. Now type II A string theory isbased on closed strings where the left and right moving string modes have opposite chiralities,

18

whereas for example type II B is based on closed strings where those string modes have equalchiralities.

A deeper and more detailed overview of the basics of string theory goes beyond the intentionof this paper. Interested readers will find more information in e.g. the before-mentioned [24] and[31]. All information that is necessary to understand my calculations will be provided in thispaper. The necessary results from type II string theory needed will be given in section 3.2.

One characteristic of type II A string theory is that it is defined in ten dimensions. Since weare aware of only four dimensions in daily life, we want to hide of six dimensions. This is done bycompactifying the theory. In this chapter I will compactify the massless sector of type II A stringtheory on a six dimensional rigid Calabi-Yau manifold. By this I mean that we will take a lookat the massless spectrum of the theory (fields that describe massless string states) and see that apart of them lives on a six dimensional rigid Calabi-Yau manifold and a part on four dimensionalMinkowski space. Compactifying the theory now means truncating the parts that live on the rigidCalabi-Yau space (see section 3.3) and finding a four dimensional effective action for the fieldsliving in the four dimensional space. (see section 3.4). As we will see in this chapter this can bedone and we will be left with a four dimensional sigma model action, which in turn means thatwe are left with a metric on a yet unknown four dimensional manifold, as explained in section 3.5.

This metric we will find will be the main result of this chapter. The next chapters are devotedto the study of the manifold on which this metric can naturally be placed.

3.2 Starting assumptions from type II A string theory

In this section I will present some assumptions from which I will start my calculations. Theseassumptions follow from string theory, type II A string theory in particular. As such they arenot assumptions in that there is no theory to back them up, but they are assumptions for thepurpose of this paper, because the underlying theory from which these results follow will not beexplained here. An understanding of the underlying string theory may help in understandingthe physical relevance of the problem, but is not necessary for an understanding of this paper.Some background knowledge of theoretical physics, quantum field theory in particular, is assumedthough. For more information on string theory see e.g. [24], [31] and [45].

Throughout this paper I will set ~ = 1.

Definition 3.2.1 (Massless spectrum) Let H be the Hamiltonian of a given system on a man-ifold X. Then the massless spectrum consists of all solutions F (fields on X) of:

HF = M2F, (3.1)

with M2 = 0. Note that the fields F have to be at least twice differentiable. In what followsI will assume that there are locally neighbourhoods on which fields in the massless spectrum areinvertible.

Remark: HF = M2F are the equations of motion that follow from the action of the theory.

Definition 3.2.2 (Four dimensional space-time) For four dimensionalMinkowski space-time I will use the following notation:

R1,3 := R× R

3 = R4, (3.2)

with a metric with respect to the standard basis of R4

g =

−1 0 0 00 1 0 00 0 1 00 0 0 1

. (3.3)

19

The first coordinate of R1,3 is called the time coordinate and the other three coordinates are thespace coordinates.

Notation: By XIIA I will denote the ten dimensional space on which type II A string theoryis defined. The coordinates on XIIA I will denote by w = (x, y), where x is the coordinate onR1,3 and y is the real coordinate on CY3 (see assumption 3.2.6). If I have to label each specificcoordinate I will use the following convention. Greek letters with a hat will denote coordinates onXIIA, e.g. wµ with µ = 0, · · · , 9. Greek letters without a hat will denote coordinates on R1,3, e.g.xµ with µ = 0, · · · , 3. Latin letters are used for coordinates on CY3, e.g. yi with i = 4, · · · , 9. SinceCY3 is a Kahler manifold with vanishing Nijenhuis tensor we can also use complex coordinates onCY3. These I will denote by z and z (the conjugate coordinates). The indices will be Latin letters,overlined in the case of conjugate coordinates, e.g. (za, za) with a = 1, 2, 3 and a = 1, 2, 3. ♦

Remark: As a consequence of the notation introduced above we can write

• a 1-form A on CY3 as a sum of a (1, 0)-form and a (0, 1)-form: Aidyi = Aadz

a +Aadza;

• a 2-form on CY3 as a sum of a (2, 0)-form, a (1, 1)-form and a (0, 2)-form: Aijdyi ∧ dyj =

Aabdza ∧ dzb +Aabdz

a ∧ dzb +Aabdza ∧ dzb and

• a 3-form on CY3 as a sum of a (3, 0)-form, a (2, 1)-form, a (1, 2)-form and a (0, 3)-form.

It should be clear now that a generic q-form can be written as a sum of all (r, s)-forms such thatr + s = q.

In the following assumption the terminology Neveu-Schwarz and Ramond appears. Thesephrases refer to the boundary conditions you impose on the left moving and right moving modesin your string. Neveu-Schwarz means antiperiodic boundary conditions, whereas Ramond meansperiodic ones. The exact meaning of these terms is not important for the present discours, so formore information I will refer to e.g. [24] or [31].

Assumption 3.2.3 (II A massless bosonic spectrum in ten dimensions)The massless bosonic spectrum of type II A string theory on XIIA consists of a Neveu-Schwarz-Neveu-Schwarz (NS-NS) sector and a Ramond-Ramond (R-R) sector. In the NS-NS sector wehave:

• a symmetric (2, 0)-tensorfield gµν , via gauge fixing related to the graviton;

• an antisymmetric (2, 0)-tensorfield Bµν , called the NS 2-form, with corresponding field

strength H(3)10d := 16dB

(2)10d and

• a real scalar field φ, called the dilaton.

In the RR sector we have:

• a vector field A(1) with corresponding field strength, the antisymmetric (2, 0)-tensorfieldF (2) := dA(1) and

• an antisymmetric (3, 0)-tensorfield A(3) with corresponding field strength, the antisymmetric(4, 0)-tensorfield F (4) := dA(3).

The R-NS and NS-R sectors give fermions. See e.g. [31].

We will also need an action for these fields of the massless bosonic spectrum.Remark: In physics there is the familiar concept of the action of a system as the space-time

integral over the Lagrangian of a system. Most of the times if the word action is used it willbe clear if I mean this physical action or the mathematical notion of a group action that will beintroduced in definition 4.6.1, but in cases of doubt or confusion I will explicitly state which actionI mean.

Notation: In the following I will often use√g for

√

|detg|. Remember that g is the metricin the ten dimensional space, unless stated otherwise. Furthermore, where confusion can arise, Iwill distinguish between the Hodge star operator acting in R1,3 (∗(4)), in CY3 (∗(6)) and in XIIA

(∗(10)). ♦

20

Assumption 3.2.4 (Bosonic action in ten dimensions) The bosonic action in ten dimen-sions for type II A string theory is (with ∗ = ∗(10)):

S10[g, φ, F (2), F (4), B(2)10d] = 1

α′4(2π)7

∫ √gR d4x d6y

− 12α′4(2π)7

∫

(

e−φH(3)10d ∧ ∗H

(3)10d + dφ ∧ ∗dφ

+e3φ2 F (2) ∧ ∗F (2) + e

φ2 F (4) ∧ ∗F (4)

+B(2)10d ∧ F (4) ∧ F (4)

)

, (3.4)

where:F (4) = dA(3) −A(1) ∧H(3)

10d, (3.5)

and α′ is a constant related to the string tension (see e.g. [7], [31] or [45]). The first term in theaction is the ten dimensional Einstein-Hilbert action.

Assumption 3.2.5 (Background fields and fluctuations) The solutions to theEuler-Lagrange equations derived from the ten dimensional action are the so called backgroundfields. If we allow fluctuations away from these classical solutions, we can write a field F asF = F 0 + δF , where δF is the fluctuation. For all the fields from the massless spectrum ofassumption 3.2.3, with the exception of the metric the background fields are zero. So for these fieldsF = δF . The assumption for the background field g0 for the metric is discussed in assumption3.2.6.

Assumption 3.2.6 (Type II A ten dimensional background) Type II Astring theory is formulated on a ten dimensional manifold XIIA. The solutions of the Euler-Lagrange equations (see e.g. [47]) from the ten dimensional action that will be given in assumption3.2.4 give that the (classical) ten dimensional metric must satisfy the Einstein equations. Sincewe chose to set all the background fields (except for the metric) to zero in assumption 3.2.5, thismeans the Ricci tensor must be zero (see definition 2.2.17). An assumption compatible with thisdemand is to assume that the ten dimensional metric is a metric on R

1,3×CY3. We then considerfluctuations on this metric such that the resulting manifold is a fibre bundle over R1,3 with fibresisomorphic to CY3. This CY3 can be chosen to be rigid. For a definition of rigidity see definition3.3.6.

Remark: The choice to compactify on a rigid CY3 manifold is not without reason. Firstof all if we want the four dimensional space to be the Ricci flat Minkowski space, then the sixdimensional space must also be Ricci flat. This still leaves some options open. The next argumentinvolves supersymmetry. For those familiar with the terminology, compactifying on CY3 givesN = 2 supersymmetry, where for example compactifying on a six dimensional torus T 6 would giveN = 8 supersymmetry and compactifying on aK3×T 2 manifold would giveN = 4 supersymmetry(where N is the real dimension of the space of Killing spinors on the manifold, i.e. solutions toDirac’s equation ∇ψ = 0). The reason to choose for the CY3 manifold is because we want N tobe as small as possible, since we observe no supersymmetry in Nature. N = 0 would be ideal, butthis case is too hard to tackle at the moment. Finally a rigid Calabi-Yau is chosen for the sake ofsimplicity. This minimalizes namely the number of hypermultiplets you find after compactification.

3.3 Expanding the massless spectrum

My goal is to compactify this theory so that we are left with a theory on the four dimensionalspace-time. What is meant by this is that we want to get rid of the components on CY3 of thefields of the massless spectrum and only keep the components of these fields on R1,3. So what wehave to do first is find out which parts of these fields live on CY3 and which parts live on R1,3.To do this we will need some more mathematics. I begin by defining the Laplace operator and

21

stating without proof a result from the theory of differential equations about eigenfunctions ofthis operator.

Definition 3.3.1 (The Laplace operator) LetX be a manifold with metric g and charts (Ui, φi)and write y = φi(x) for x ∈ Ui. Then the Laplace operator on X acting on functions (0-forms) isdefined as follows:

∆X :=1√detg

∂

∂yi

√

detg gij∂

∂yj. (3.6)

Note that gij is the inverse of the metric gij , i.e. g = gijdxi ∧ dxj and gijg

jl = δli. In general theLaplace operator for q-forms is:

∆X : Ωq → Ωq : ω 7→ (dd∗ + d∗d)ω, (3.7)

where d∗ is the codifferential operator from definition 2.1.18. Since d2 = 0 and d∗d∗ = 0 we canalso write ∆X = (d+ d∗)2.

Lemma 3.3.2 (Basis of eigenfunctions of the Laplace operator) Let, for every m, the func-tion:

fm : CY3 → C, (3.8)

be an eigenfunction of the Laplace operator ∆CY3 on CY3. Let ξ be a function from XIIA to C.Then ξ can be expanded on a basis of the eigenfunctions as follows:

ξ(x, y) =∑

m

ζm(x)fm(y), (3.9)

where, for every m, ζm is a function from R1,3 to C. A similar result holds for eigenforms.

Assumption 3.3.3 (Harmonic forms) When expanding a q-form on a basis of eigenforms ofthe Laplace operator on CY3 as explained in lemma 3.3.2, we will, to simplify the calculation, onlyuse the eigenforms with eigenvalue zero, i.e. the harmonic forms.

Definition 3.3.4 (Hodge numbers) The Hodge number hr,s of a manifold X is the complexdimension of the space of all (r, s)-forms A that satisfy ∆XA = 0, where ∆X is the Laplaceoperator from definition 3.3.1.

The following lemma is a known result from algebraic geometry that I will give here withoutproof.

Lemma 3.3.5 (The Hodge numbers of CY3) The Hodge numbers of CY3 are

h0,0 = 1; (3.10)

h1,0 = h0,1 = h2,0 = h0,2 = 0; (3.11)

h3,0 = h0,3 = 1; (3.12)

h2,1 = h1,2. (3.13)

The specific values of h1,1 and h2,1 depend on the specific choice of the CY3 manifold.

Proof: I will not give a proof here, except for the case of h0,0. From the connectedness of CY3 itfollows that if ∆CY3A = 0 for A a (0, 0)-form, thus a function, then A is a constant. 2

Definition 3.3.6 (Rigid CY3 manifold) A CY3 manifold is called rigid if h2,1 = h1,2 = 0,where h2,1 and h1,2 are the Hodge numbers from lemma 3.3.5.

Notation: From lemma 3.3.5 we see that we can choose the following bases:

• a basis for (0, 0)-forms, 1;

22

• a basis for (1, 1)-forms, V A = V Aabdza ∧ dzb, where A = 1, · · · , h1,1 labels the h1,1 different

basis-(1, 1)-forms;

• a basis for (3, 0)-forms, Ω;

• a basis for (2, 1)-forms, Φα = Φαabcdza ∧ dzb ∧ dcc, where α = 1, · · · , h2,1 labels the h2,1

different basis-(2, 1)-forms and

• a combination of two previous bases for notational convenience: dαab = i‖Ω‖2 Ωlka Φαlkb, where

‖Ω‖2 = 13!ΩabcΩ

abc; see [11].

♦It’s important to remember that the above choices for bases are just that, choices. Other

choices for bases, e.g. multiples of the above choices, will lead to other results when expandingfields on these bases. For example the fields φ and C we will encounter in theorem 3.3.7 dependon these choices, or to put it in other words these fields reflect the choice of bases.

Notation: The metric tensor in ten (and later on in four) dimensions will as usual be used to

raise and lower indices. For example Ωdea := Ωabcgbdgce. ♦

The way I will get the massless bosonic spectrum in four dimensions is now as follows. I willrecognize in the ten dimensional fields q-forms on CY3 for different q ∈ N ∪ 0. These forms Iwill expand on a basis of harmonic q-forms, as explained in assumption 3.3.3. The coefficients inthis expansions will be my fields on Minkowski space.

Theorem 3.3.7 (II A massless bosonic spectrum in four dimensions) After compactifica-tion from ten to four dimensions on a CY3 manifold the massless bosonic spectrum of type II Astring theory consists of the following supersymmetry multiplets:

• a graviton gµν , a graviphoton Aµ, a cosmological constant Aµνρ;

• a tensor multiplet φ, Bµν , C, C;

• h1,2 hypermultiplets of complex structure moduli bα, Bα, bα, B

αand

• h1,1 vectormultiplets of Kahler moduli MA, AAµ , bA.

I will here not elaborate on all the names given to the fields in the spectrum or on the categoriesover which the fields are divided..

Proof: The (0, 0)-form dilaton can be expanded as follows: φ(x, y) = φ(x)1. This gives us onefield φ in four dimensions. This field is real, since the dilaton in ten dimensions is a real field.Since the Hodge numbers tell us that the space of (0, 0)-forms on CY3 is one dimensional every(0, 0)-form on CY3 (thus a field with all indices in R1,3) will give one field in four dimensions onR1,3, exactly like φ(x, y) gives φ(x).

The antisymmetric (2, 0)-tensorfield Bµν can be expanded as:

Bµνdxµ ∧ dxν = Bµνdx

µ ∧ dxν +Bµidxµ ∧ dyi

+Biµdyi ∧ dxµ + Bijdy

i ∧ dyj

= Bµνdxµ ∧ dxν + 2Bµidx

µ ∧ dyi +Bijdyi ∧ dyj . (3.14)

The last equality follows from the antisymmetry of Bµi. The term Bµνdxµ ∧ dxν is a 2-form on

R1,3, so this is already like we want it (it is a (0, 0)-form on CY3 and thus, analogously to whatwas done for φ, it will give one field Bµν(x) on R1,3). The term 2Bµidx

µ ∧ dyi is a 1-form onCY3 (due to the dyi) so we can try to write it out on a basis of 1-forms on CY3. But from lemma3.3.5 we see that h1,0 = h0,1 = 0 so there are no 1-forms on CY3. Consequently this term doesnot contribute. The last term, Bijdy

i ∧ dyj , is a 2-form on CY3, so a priori this can consist of(2, 0)-forms, (1, 1)-forms and (0, 2)-forms, but since we see from lemma 3.3.5 that h2,0 = h0,2 = 0

23

only the (1, 1)-forms contribute. So we can expand this term on the basis V A introduced aboveinto

∑

A bA(x)V A. So we end up with:

Bµνdxµ ∧ dxν = Bµνdx

µ ∧ dxν +∑

A

bAV A. (3.15)

So from the NS 2-form we get two fields in four dimensions: Bµν and bA.Next we will take a look at the vector field A(1):

Aµdxµ = Aµdx

µ +Aidyi

= Aµdxµ. (3.16)

The last equality follows from lemma 3.3.5, because from this lemma we see that h0,1 = h1,0 = 0.So A(1) gives us one field in four dimensions: Aµ.

Now lets take a look at the metric. I will not discuss this in the utmost detail. For moreinformation see [11] paragraph 2.2.

gµνdxµdxν = gµνdx

µdxν + gµidxµdyi

+giµdyidxµ + gijdy

idyj

= gµνdxµdxν + gijdy

idyj , (3.17)

where the last equation again follows from the fact from lemma 3.3.5 that h1,0 = h0,1 = 0. Thefirst term we are left with has all indices in R

1,3. For the term we are left with we look not at themetric itself, but at fluctuations of the metric:

gµν = g0µν + δgµν , (3.18)

where g0µν is a background metric on XIIA as explained in assumption 3.2.6 and δgµν is the

variation of the metric. If we go from real to complex coordinates we find with the chain rule:

dyi = dza∂yi

∂za+ dzb

∂yi

∂zb; (3.19)

gij can be written (see [11]) as:

gijdyidyj = gabdz

adzb + gabdzadzb + gabdz

adzb + gabdzadzb. (3.20)

Since gab = gab and gab = gba we don’t need to discuss all these cases separately. From [11] we seethat:

δgab =∑

A

MA(x)V Aab

(3.21)

δgab =i

‖Ω‖2 bα(x)Ωcda Φαcdb = bα(x)dαab. (3.22)

Equation (3.21) follows since δgab is a 1-form in both the holomorphic (a) as the anti holomorphic

(b) indices and for 1-forms there is no distinction between symmetry and antisymmetry, so for theexpansion of δgab we can use the basis V A. Equation (3.22) follows from the equality:

gabΩb

cd = bαΦαcda, (3.23)

and equation (3.18). Note that neither δgab, nor dαab are antisymmetric forms. We conclude that

from the graviton we get four new fields in four dimensions: gµν , bα, b

αand MA.

Lastly we will expand the (3, 0)-tensorfield A(3):

Aµνρdxµ ∧ dxν ∧ dxρ = Aµνρdx

µdxνdxρ +Aµijdxµ ∧ dyi ∧ dyj

+Aijkdyi ∧ dyj ∧ dyk. (3.24)

24

I already left the term with Aµνi (and permutations of these indices) out because as we have seenbefore there are no 1-forms on CY3. The term with Aµνρ is already in the desired form. Theterm with the 2-form can be expanded on the basis of (1, 1)-forms, because, as we have seen,h2,0 = h0,2 = 0:

Aµij (x)dxµ ∧ dyi ∧ dyj =

∑

A

AA(x)V A. (3.25)

Finally the term with the 3-form can be expanded in (3, 0)-forms, (1, 2)-forms, (2, 1)-forms and(0, 3)-forms:

Aijk(x)dyi ∧ dyj ∧ dyk = C(x)Ω +Bα(x)Φα +B

α(x)Φα + C(x)Ω. (3.26)

Thus from A(3) we get six new fields in four dimensions: Aµνρ, AAµ , C, C,B

α and Bα.

All these fields together give the fourteen fields from the theorem. Like I said in the theoremI will not elaborate on the names of the fields and the categories over which they are dividedhere, but we see that the hypermultiplet consists of (2, 1)-forms and (1, 2)-forms and the vectormultiplet consists of (1, 1)-forms. 2

Corollary 3.3.8 (Less hypermultiplets on a rigid CY3) Let CY3 be a rigid

Calabi-Yau manifold. Then the hypermultiplets bα, Bα, bα, B

αfrom theorem 3.3.7 vanish.

Proof: This follows immediately from the facts that a rigid Calabi-Yau has h1,2 = h2,1 = 0(lemma 3.3.5) and that the hypermultiplets consist only of (2, 1)-forms and (1, 2)-forms. 2

Remark: In the following we will be only interested in the tensor multiplet φ, Bµν , C, C fromtheorem 3.3.7. These fields together (or better: the fields φ, σ, ϕ and χ I will derive from thesefields in definition 3.4.10 and corollary 3.4.11) are also called the universal hypermultiplet .

Before I turn to the compactification of the ten dimensional action I will give one more usefuldefinition.

Definition 3.3.9 (The field strength of B(2) in four dimensions) In four dimensions thefield strength of B(2), where B(2) := Bµνdx

µ ∧ dxν from theorem 3.3.7 is defined to be H (3) :=16dB(2).

In the next section I will perform the integration over the Calabi-Yau part of the ten dimen-sional action of assumption 3.2.4 to get the four dimensional action for the tensor multiplet oftheorem 3.3.7.

3.4 Compactifying the action

In the previous section we found the fields of the massless spectrum in four dimensions. Toknow the physics these fields describe we have to know their action in four dimensions. Like Iremarked in the previous section, I will only deal with the universal hypermultiplet in this paper.In this section I will calculate the effective low energy action for the universal hypermultiplet byintegrating out that part of the ten dimensional action that takes place on the CY3 manifold. Anarticle on this same subject is [11] Before I can get started on integrating the action however, Iwill first give some useful definitions and results.

Notation: In the following, unless stated otherwise, g will be the metric on XIIA and g willbe the metric on CY3. ♦

Definition 3.4.1 (The (anti) holomorphic 3-form Ω on CY3) From lemma 3.3.5 follows theexistence of a holomorphic 3-form on CY3. Moreover, up to a constant this form is unique. I willuse the notation Ω for this form as said before. Taking its complex conjugate we get an antiholomorphic 3-form on CY3 Ω, which is also unique up to a constant factor. More informationabout the constant prefactors follows after the next definition.

25

Note that the existence of a holomorphic 3-form on CY3 assures that this manifold is orientableand so I can define a volume form.

Definition 3.4.2 (The volume form on CY3) I will use the following definition for the volumeform on CY3:

volCY3 := Ω ∧ ∗(6)Ω. (3.27)

Remark: Actually the volume form is defined as ωJ ∧ ωJ ∧ ωJ , where ωJ is the Kahler formon CY3, but remember that Ω is unique up to a constant factor. I now take this factor to be suchthat my definition of the volume form in definition 3.4.2 agrees with the definition according tothe Kahler form.

Assumption 3.4.3 (The signature of the metric on CY3)

sgn(g) = −1. (3.28)

This sign is relevant with respect to definition 2.1.12.Notation: I will use this notation for the volume of CY3:

VolCY3 :=

∫

CY3

volCY3 . (3.29)

♦

Lemma 3.4.4 (Properties of the (anti) holomorphic 3-form on CY3) Thefollowing identities hold for the (3, 0)-form Ω and the (0, 3)-form Ω that I introduced as bases forthe (3, 0)-forms, respectively the (0, 3)-forms, on CY3 (with ∗ = ∗(6)):

∗Ω =1

8iΩ (3.30)

∗Ω = −1

8iΩ (3.31)

∫

CY3

Ω ∧ Ω = 8iVolCY3 (3.32)

∫

CY3

Ω ∧ ∗Ω = VolCY3 (3.33)

Ω ∧ ∗Ω = 0 (3.34)

Ω ∧ ∗Ω = 0. (3.35)

Proof: Equation (3.33) follows from definition 3.4.2. Since the Hodge star maps a holomorphic3-form to a holomorphic 3-form and an anti holomorphic 3-form to an anti holomorphic 3-form, wesee that equations (3.34) and (3.35) hold due to lemma 2.1.9. Moreover the factor ± 1

8 i in equations(3.30) and (3.31) follows from corollary 2.1.13. With these results finally equation (3.32) follows.2

Theorem 3.4.5 (Action for the tensor multiplet in four dimensions) With g the metricon a rigid CY3 and H(3) = 16dB(2) we get for the action of the tensor multiplet in four dimensions:

S4[φ,B(2), C, C ] = − VolCY3

2α′4(2π)7

∫

R1,3

dφ ∧ ∗(4)dφ+ e−φH(3) ∧ ∗(4)H(3)

+2eφ2 dC ∧ ∗(4)dC −

i

2H(3) ∧ C←→d C. (3.36)

Proof: Since I am only interested in the action for the tensor multiplet of theorem 3.3.7 here, Iwill only consider those parts of the ten dimensional action from theorem 3.2.4 that contain oneor more of the fields φ,Bµν , C and C. From theorem 3.3.7 we see that these fields come from the

26

ten dimensional fields φ,B(2)10d and A(3). So I will set A(1) = 0, which means that in the notation

of assumption 3.2.4 we have that F (4) = F (4). Remember furthermore that H(3)10d = 16dB

(2)10d

and F (4) = dA(3). Concretely φ(x, y) in ten dimensions gives φ(x) in four dimensions, B(2)10d

gives B(2) (and thus H(3)10d gives H(3)) and A(3) gives CΩ + CΩ (and consequently F (4) gives

d(CΩ + CΩ) = dC ∧ Ω + dC ∧ Ω). With this information we can turn our attention to the tendimensional action.

∫

XIIA

dφ ∧ ∗(10)dφ (3.37)

will give∫

XIIA

dφ(x)dφ(x)√g d4x d6y = VolCY3

∫

R1,3

dφ ∧ ∗(4)dφ. (3.38)

The term∫

XIIA

e3φ2 F (2) ∧ ∗(10)F (2) (3.39)

contains none of the relevant fields and thus contributes nothing to the action we are looking for.

∫

XIIA

e−φH(3)10d ∧ ∗(10)H

(3)10d (3.40)

gives for the fields we are interested in:

∫

XIIA

e−φ(x)H(3)(x)H(3)(x)√g d4x d6y = VolCY3

∫

R1,3

e−φH(3) ∧ ∗(4)H(3). (3.41)

To calculate∫

XIIA

eφ2 F (4) ∧ ∗(10)F (4) (3.42)

I will first find a handy expression for the relevant fields in F (4) ∧ ∗(10)F (4):

F (4) ∧ ∗(10)F (4) gives (dC ∧ Ω + dC ∧ Ω) ∧ ∗(10)(dC ∧ Ω + dC ∧ Ω)

= (dC ∧ Ω) ∧ ∗(10)(dC ∧ Ω) + (dC ∧ Ω) ∧ ∗(10)(dC ∧ Ω)

+(dC ∧ Ω) ∧ ∗(10)(dC ∧ Ω) + (dC ∧ Ω) ∧ ∗(10)(dC ∧ Ω)

= dC ∧ Ω ∧ ∗(4)dC ∧ ∗(6)Ω + dC ∧ Ω ∧ ∗(4)dC ∧ ∗(6)Ω+dC ∧ Ω ∧ ∗(4)dC ∧ ∗(6)Ω + dC ∧ Ω ∧ ∗(4)dC ∧ ∗(6)Ω

= (dC ∧ ∗(4)dC) ∧ (Ω ∧ ∗(6)Ω) + (dC ∧ ∗(4)dC) ∧ (Ω ∧ ∗(6)Ω)

= 2(dC ∧ ∗(4)dC) ∧ (Ω ∧ ∗(6)Ω). (3.43)

For the second identity I used that ∗(10)(dC ∧ Ω) = ∗(4)dC ∧ ∗(6)Ω, while in the third identity Iused lemmas 2.1.9 and 3.4.4 and in the fourth identity lemma 2.1.17. So equation (3.42) gives

2

∫

XIIA

eφ2 dC ∧ ∗(4)dC ∧ Ω ∧ ∗(6)Ω

= 2VolCY3

∫

R1,3

eφ2 dC ∧ ∗(4)dC, (3.44)

where in the last line I used lemma 3.4.4. The last term to look at (we do not consider theEinstein-Hilbert term1) is:

∫

XIIA

B(2)10d ∧ F (4) ∧ F (4). (3.45)

1There is an Einstein-Hilbert term in the four dimensional action, which gives the equations of motion of themetric on the four dimensional space, but I will not consider this term.

27

The first step is to partially integrate this expression, where I use the assumption that the fields

B(2)10d and A(3) vanish on the boundary (where infinity is not excluded) of XIIA:

∫

XIIA

B(2)10d ∧ F (4) ∧ F (4) =

∫

XIIA

B(2)10d ∧ dA(3) ∧ dA(3)

= −∫

XIIA

d(

B(2)10d ∧ dA(3)

)

∧ A(3)

= −∫

XIIA

dB(2)10d ∧ dA(3) ∧ A(3)

= −∫

XIIA

1

16H

(3)10d ∧ dA(3) ∧ A(3). (3.46)

For the third identity I used that d2A(3) = 0. Now

dA(3) ∧ A(3) gives d(CΩ + CΩ) ∧ (CΩ + CΩ)

= (dC ∧ Ω + dC ∧ Ω) ∧ (CΩ + CΩ)

= dC ∧ Ω ∧ CΩ + dC ∧ Ω ∧ CΩ

= (CdC − CdC) ∧ Ω ∧ Ω

= C←→d C ∧ Ω ∧ Ω, (3.47)

where in the second and third identities I used lemma 2.1.9 and in the last line C←→d C is just

shorthand notation for CdC − (dC)C. Thus equation (3.45) gives

−∫

XIIA

1

16H(3) ∧ C←→d C ∧ Ω ∧ Ω = − i

2VolCY3

∫

R1,3

H(3) ∧ C←→d C. (3.48)

Now we have integrated all terms of assumption 3.2.4. Inserting our results in the ten dimensionalaction we get the desired result. 2

The action we have derived above depends on three scalar fields and one 2-form. I would liketo construct a physically equivalent action that depends on four scalar forms (and nothing else)instead. The next lemma provides.

Lemma 3.4.6 (Classically equivalent action for the B(2)-action) Define for notational con-

venience q := − VolCY3

2α′4(2π)7 . In this lemma ∗ = ∗(4). Then the following two actions describe classi-

cally the same physics:

S(4)

B(2) [φ,B(2), C, C ] = q

∫

R1,3

e−φH(3) ∧ ∗H(3) − i

2H(3) ∧ C←→d C (3.49)

S(4)σ [φ,D,C, C ] = q

∫

R1,3

eφ(

i

4C←→d C +

1

2dD

)

∧ ∗(

i

4C←→d C +

1

2dD

)

,

(3.50)

where the relation between B(2) and D consist of the following two relations for H (3):

H(3) = 16dB(2)locally (3.51)

H(3) =1

2eφ ∗

(

i

2C←→d C + dD

)

. (3.52)

Proof: The proof goes along the following lines. First I will write down a third action thatdepends on four scalar fields φ,C, C,D and one 3-form H(3), where I regard all these fields asbeing independent of each other. This action I will construct by adding to the action of equation(3.49) a Lagrange mutiplier term that couples H (3) to D. Next I will use the Euler-Lagrange fieldequations (see e.g. [47]; these equations follow from the variational principle) that follow from

28

this new action to get both actions in the lemma. This proves that both these actions describeclassically (because I used the solutions to the field equations to derive the actions) the samephysics. Now I will show this reasoning in more detail.

Define a new action:

S[φ,H(3), C, C,D] = q

∫

R1,3

e−φH(3) ∧ ∗H(3)

− i2H(3) ∧ C←→d C −H(3) ∧ dD, (3.53)

where I take all the fields to be independent of each other. To get the equation of motion (Euler-Lagrange equation) for D I first have to partially integrate the last term in the action. Assumingthat the boundary terms vanish this gives:

−∫

R1,3

H(3) ∧ dD =

∫

R1,3

dH(3) ∧D. (3.54)

The variational principle for D thus gives:

dH(3) = 0, (3.55)

which assures that locally there is a 2-form B(2), such that H(3) = 16dB(2). The equations ofmotion for H(3) give:

2e−φ ∗H(3) − i

2C←→d C − dD = 0, (3.56)

which results, with the use of equation (2.36) in:

H(3) =1

2eφ ∗

(

i

2C←→d C + dD

)

. (3.57)

We see that if we insert the equation of motion for D, equation (3.55), into the action (3.53), weget back our original action (3.49). If on the other hand we insert the equation of motion for H (3),equation (3.57), into the action (3.53), we get action (3.50) which can easily be checked by writingit out and using lemmas 2.1.17 and 2.1.9. 2

Corollary 3.4.7 (The four dimensional action for four scalar fields) Thefour dimensional action for the scalar fields φ, σ, C and C is (where ∗ = ∗(4)):

S4[φ, σ, C, C ] = − VolCY3

2α′4(2π)7

∫

R1,3

dφ ∧ ∗dφ+ 2eφ2 dC ∧ ∗dC

+eφ(

i

4C←→d C +

1

2dD

)

∧ ∗(

i

4C←→d C +

1

2dD

)

. (3.58)

Proof: This follows by replacing action (3.49) with action (3.50) in our action from theorem 3.4.5.2

I will now make some field redefinitions to rewrite the action in a more convenient form.

Definition 3.4.8 (Some field redefinitions) Denote the fields I used above by the subscriptold and the redefined fields by the subscript new, then:

Cold = 2√

2Cnew ; (3.59)

φold = 4φnew; (3.60)

Dold = 8Dnew. (3.61)

Since I will only use the new fields from now on, I will lose the subscripts.

29

Corollary 3.4.9 (The four dimensional action of the redefined fields)The four dimensional action for the scalar fields φ, σ, C and C is (where ∗ = ∗(4)):

S4[φ, σ, C, C ] = −16VolCY3

2α′4(2π)7

∫

R1,3

dφ ∧ ∗dφ+ e2φdC ∧ ∗dC

+e4φ(

i

2C←→d C + dD

)

∧ ∗(

i

2C←→d C + dD

)

. (3.62)

Proof: This follows simply by inserting the field redefinitions from definition 3.4.8. 2

For some applications it is convenient to write this action in terms of other fields. Two of thesedifferent forms of this action I will now give. The first one will give the action in terms of fourreal scalar fields instead of two real and two complex fields. The second form will show explicitlythat we can extract a Kahler metric from this action. To accomplish the first goal I will rewritethe four scalar fields in terms of four real scalar fields.

Definition 3.4.10 (Some new real scalar fields) Define the real fields χ and ϕ as√

2 timesthe real and imaginary parts of C, such that:

C =1√2(χ+ iϕ)

C =1√2(χ− iϕ). (3.63)

Furthermore I introduce the field σ:

σ = −D − 1

2ϕχ. (3.64)

Now I can rewrite the action from corollary 3.4.9 in terms of four real scalar fields.

Corollary 3.4.11 (The four dimensional action for four real scalar fields) The fourdimensional action for the real scalar fields φ, σ, χ and ϕ is (where ∗ = ∗(4)):

S4[φ, σ, χ, ϕ] = −16VolCY3

2α′4(2π)7

∫

R1,3

dφ ∧ ∗dφ+ e2φ(dχ ∧ ∗dχ+ dϕ ∧ ∗dϕ)

+e4φ (χdϕ+ dσ) ∧ ∗ (χdϕ+ dσ) . (3.65)

Proof: To get this action I rewrote C and C in terms of χ and ϕ using definition 3.4.10 and usedthat:

dC ∧ ∗dC =1

2(dχ ∧ ∗dχ+ dϕ ∧ ∗dϕ) (3.66)

C←→d C = i(χdϕ− ϕdχ)

= i(χdϕ− (d(χϕ) − χdϕ))

= i(2χdϕ− d(χϕ). (3.67)

Thusi

2C←→d C + dD = −(χdϕ+ dσ). (3.68)

Inserting these results into the action of equation (3.62) gives the desired result. 2

Instead of rewriting the action in terms of four real scalar fields, I can also rewrite the actionin terms of two complex fields in such a way that its Kahlerian character will be clear. To thisend I introduce a new complex field, S (see e.g. [7], [36] or [37]).

Definition 3.4.12 (A new complex field) Define a new complex field in terms of the fieldsφ,D and C:

S := e−2φ + 2iD+ CC. (3.69)

30

Next I will define what will turn out to be the Kahler potential.

Definition 3.4.13 (Kahler potential for this situation) Define the Kahler potential corre-sponding to the metric from this action as:

K := −log(S + S − 2CC). (3.70)

Note that this makes sense since S + S − 2CC = 2e−2φ > 0. From this we also see that:

K = −log(2e−2φ) = 2φ− log2. (3.71)

Since φ has its origins in the ten dimensional dilaton, often K is called the (four dimensional)dilaton.

Exactly what is meant by the metric from this action will be explained in section 3.5. Therewill be given an explanation why this is indeed the Kahler potential of this metric as well. For nowonly the definition of K in terms of S, S, C and C is of direct importance.

Remark: φ is relevant in perturbation theory since the expansion there is done in powers ofeφ.

The action from corollary 3.4.9 can now be written in terms of the two independent complexfields S and C and their complex conjugates S and C.

Corollary 3.4.14 (Four dimensional action for two complex scalar fields) The four dimen-sional action for the complex scalar fields S, S, C and C is (where ∗ = ∗(4)):

S4[S, S, C, C] = −16VolCY3

2α′4(2π)7

∫

R1,3

e2K(dS ∧ ∗dS − 2CdS ∧ ∗dC

−2CdS ∧ ∗dC + 2(S + S)dC ∧ ∗dC). (3.72)

Proof: If we insert the definitions for S and K into this action, we retrieve the action of equation(3.62). I will not present this derivation in detail here, I will only provide some intermediaryresults:

e−K = S + S − 2CC

= 2e−2φ;

dS ∧ ∗dS = 4e−4φdφ ∧ ∗dφ+ 4dD ∧ ∗dD + 2|C|2dC ∧ ∗dC−4Ce−2φdφ ∧ ∗dC − 4Ce−2φdφ ∧ ∗dC+CCdC ∧ ∗dC + CCdC ∧ ∗dC;

dS ∧ ∗dC = −2e−2φdφ ∧ ∗dC + 2idD ∧ ∗dC+CdC ∧ ∗dC + CdC ∧ ∗dC;

dC ∧ ∗dS = −2e−2φdC ∧ ∗dφ− 2idC ∧ ∗dD+CdC ∧ ∗dC + CdC ∧ ∗dC. (3.73)

2

3.5 Metrics

In this section we will see that the action I derived in section 3.4 expressed in different fields canbe used to construct a metric. The key notion here is that of a sigma model. The metric we getis the key object we are after and will provide the link between this chapter on physics and thechapters on mathematics that will follow in this paper.

31

Definition 3.5.1 (Sigma model) If we are dealing with a physical theory described by a Lan-grangian (density) of the form

L[φ1, . . . , φn] = gAB(φ)dφA ∧ ∗dφB , (3.74)

where A,B vary over 1, . . . , n, then we call this a sigma model. Depending on how gAB(φ) dependson φ we can call it a linear or a non-linear sigma model.

Lemma 3.5.2 (Sigma model leads to manifold with metric) If we have asigma model with invertible fields and coefficients gAB(φ) such that for every φ the matrix g(φ)(with gAB(φ) as entry in the Ath row and Bth column) is invertible, then, using the notation fromdefinition 3.5.1, we identify the coefficients gAB with the metric as follows:

ds2 = gAB(y)dyAdyB , (3.75)

where ds2 is the length of the “infinitesimal line element” squared on a manifold M and where they are local coordinates on M , induced by the fields in the following way (assume for the momentthat the fields φA are defined on a space X, in our specific case at hand X is the four dimensionalMinkowski space R1,3):

φ : X →M : x 7→(

φ1(x), . . . , φn(x))

=:(

y1, . . . , yn)

= y. (3.76)

The fields φ are thus the inverse charts on the manifold (remember that charts on a manifold arerequired to have an inverse).

Proof: There are not a lot of things that need proving here. We must only check that gAB issymmetric and non-degenerate, so we can use it as a metric. Symmetry follows from lemma 2.1.17.Futhermore, since for every φ, g is invertible, we have that for a vector x, gx = 0 if and only ifx = 0 and thus xT gx = 0 if and only if x = 0. So g is non-degenerate. 2

The obvious next thing to do is construct the metrics from the actions we derived in section3.4.

Corollary 3.5.3 (Constructing metrics) From the action from corollary 3.4.11 we get the fol-lowing line element:

ds2 = dφ2 + e2φ(dχ2 + dϕ2) + e4φ (χdϕ+ dσ)2. (3.77)

From the action from corollary 3.4.9 we get:

ds2 = dφ2 + e2φdCdC + e4φ(

i

2C←→d C + dD

)2

. (3.78)

Finally from corollary 3.4.14 we get:

ds2 = e2K(dSdS − 2CdSdC − 2CdSdC + 2(S + S)dCdC). (3.79)

Remember that these metric are connected through the following coordinate changes:

C =1√2(χ+ iϕ); (3.80)

C =1√2(χ− iϕ); (3.81)

σ = −D − 1

2ϕχ; (3.82)

S = e−2φ + 2iD + CC; (3.83)

K = −log(S + S − 2CC). (3.84)

32

Proof: We want to apply lemma 3.5.2. Remember from definition 3.2.1 that we are dealingwith invertible fields. Furthermore calculating the determinant of g (in the notation of lemma3.5.2) of equation (3.79) we get detg = e6K 6= 0 and thus g is invertible. The other metricsare related to this one via coordinate transformations. We conclude that we can apply lemma3.5.2), in particular to the action in corollary 3.4.11 with S4[φ, σ, χ, ϕ] =

∫

R1,3 L[φ, σ, χ, ϕ] and(

φ1, φ2, φ3, φ4)

= (φ, σ, χ, ϕ). So note that the φ from lemma 3.5.2 consists of all four scalar fieldsfrom the action of corollary 3.4.11 and so is not equal to the φ from this action. I have left outoverall prefactors since this is only a metric rescaling.

To get the other two results follow an analogous reasoning, starting with corollary 3.4.11,respectively corollary 3.4.14. 2

In definition 3.4.13 I called K the Kahler potential. In the next lemma we will see that this isjustified, but first I will introduce a convenient notation.

Notation: Let f(x) be a function dependent on variables (xµ)µ with 1 ≤ µ ≤ n. Then define:

f,µ :=∂f

∂xµ;

f,µ1µ2 :=∂2f

∂xµ1∂xµ2;

f,µ1µ2···µn:=

∂nf

∂xµ1∂xµ2 · · · ∂xµn. (3.85)

♦

Lemma 3.5.4 (Kahler metric) Let:

K(S, S, C, C) = −log(S + S − 2CC), (3.86)

then K is the Kahler potential for the metric in equation (3.79) and thus this metric is a Kahlermetric.

Proof: From corollary 2.2.10 we see that we have to show that metric (3.79) can be written as:

ds2 = K,SSdSdS +K,SCdSdC +K,CSdCdS +K,CCdCdC, (3.87)

where we view K as a function of the variables S, S, C and C. Simple calculations show that:

K,S = −1S+S−2CC

; (3.88)

K,SS = 1(S+S−2CC)2

= e2K ; (3.89)

K,SC = −2C(S+S−2CC)2

= −2Ce2K ; (3.90)

K,C = 2CS+S−2CC

; (3.91)

K,CS = −2C(S+S−2CC)2

= −2Ce2K ; (3.92)

K,CC = 2(S+S)(S+S−2CC)2

= 2(S + S)e2K . (3.93)

Inserting these results into equation (3.87) we see that we indeed get metric (3.79). 2

Now we have extracted a metric from physics, we can take a closer look at its properties andinvestigate a manifold on which we can place this metric. The next two chapters on mathematicswill be devoted to the study of the manifold SU(1, 2)/U(2) which will turn out to be the manifoldon which the metric which we found can be placed. This metric expressed in the coordinates Sand C as in equation (3.79) will play an important role in chapter 5.

33

Chapter 4

The manifold SU(1, 2)/U(2)

4.1 Why study SU(1, 2)/U(2)?

In chapter 3 I compactified ten dimensional type II A string theory on a CY3 manifold. Thisresulted eventually in section 3.5 in a four dimensional metric. The next step is to investigate forwhich manifold this is a metric. In fact it is known that the manifold we are after is SU(1, 2)/U(2),see e.g. [1], [2], [7], [14], [15], [19], [20], [36], [37] and [50]. In this chapter and the next I will studythis manifold and show that it is indeed compatible with the metric. In this chapter the main focuswill be on the properties that SU(1, 2)/U(2) has as a consequence of U(2) and SU(1, 2) being Liegroups. The next chapter will predominantly deal with the geometrical aspects of this manifold.These geometrical aspects will also clearly show why the metric from the previous chapter belongsto the manifold SU(1, 2)/U(2).

My approach is to study SU(1, 2)/U(2) and show its relation to the metric found. I use theknowledge that SU(1, 2)/U(2) is a candidate manifold for the metric we found, but I will not gointo details about how people found this manifold to be a likely candidate. There are two mainingredients to that reasoning. One is that it was found by e.g. De Wit that the infinitesimalisometries of the metric form the Lie algebra of SU(1, 2) (more about that in section 5.4), theother is the result of Bagger and Witten (see [5]) that if we have a sigma model and we demandso called local N = 2 supersymmetry, then the resulting metric has to be quaternion-Kahler.The qualification of Aleksievski then shows us which options we have left for our manifold. AlsoFerrara has done a lot of work in this area (see e.g. [19] and [20]). But, like I said, I will not takethis approach in my paper.

Troughout this paper I have assumed the reader to have some basic knowledge of linear algebra,(differential) geometry and group theory. Information about these subjects can be found in e.g.[21] (linear algebra), [22] and [27] (differential geometry) or [3] and [39] (group theory).

4.2 Some matrix groups

In our study of SU(1, 2)/U(2) we will encounter many matrix groups (i.e. groups that can berepresented with matrices as elements), most notobably of course SU(1, 2) and U(2). That is whyI will begin by introducing the groups we will need and give some extra attention to SU(1, 2). Butfirst I will introduce the quaternions.

Definition 4.2.1 (Quaternions) Define three different “numbers” i, j and k which have thefollowing properties: i2 = j2 = k2 = −1, ij = −ji and k = ji. Then the set of quaternions, H,is defined as the set of all elements of the form q = a + bi + cj + dk, where a, b, c, d ∈ R. Thequaternions can be represented by complex 2× 2 matrices in the following way:

q =

(

z w−w z

)

, (4.1)

34

where z = a+ ib ∈ C and w = c+ id ∈ C.

Remark: Note that quaternion multiplication in general is not commutative. The groups I will discuss are are all subgroups of the general linear group, so I will first

introduce that group.

Definition 4.2.2 (The general linear groups GL(n,K)) Let K be a either R, C or H and letn ∈ N, then the general linear group GL(n,K) consists of all invertible n×n matrices with elementsof K as entries.

The group operation is the usual matrix multiplication, the unit element is the n × n identitymatrix (notation: In) regardless of K and the unique inverse of a given matrix is the inverse ofthat matrix. Since the following groups are all subgroups of GL(n,K) for a given K, they will begroups under the same operation as above, with the same identity and inverses.

Definition 4.2.3 (The special linear groups SL(n,K)) Let K be a either R or C, then forn ∈ N the special linear group SL(n,K) is defined as follows:

SL(n,K) := A ∈ GL(n,K)| detA = 1 . (4.2)

Definition 4.2.4 (The orthogonal groups O(n)) The orthogonal groups O(n) are subgroupsof GL(n,R), for every n ∈ N, and defined as follows:

O(n) :=

A ∈ GL(n,R)| ATA = In

. (4.3)

Definition 4.2.5 (The special orthogonal groups SO(n)) The special orthogonal groups SO(n)are subgroups of GL(n,R), for every n ∈ N, and defined as follows:

SO(n) := A ∈ O(n)| detA = 1 . (4.4)

Definition 4.2.6 (The unitary groups U(n)) The unitary groups U(n) are subgroups of GL(n,C),for every n ∈ N, and defined as follows:

U(n) :=

A ∈ GL(n,C)| ATA = In

. (4.5)

Definition 4.2.7 (The special unitary groups SU(n)) The special unitary groups SU(n) aresubgroups of GL(n,C), for every n ∈ N, and defined as follows:

SU(n) := A ∈ U(n)| detA = 1 . (4.6)

As can be seen from the definition the elements of the (special) unitary groups (S)U(n) arematrices, A, with complex entries, that preserve the identity:

AT

1 0 . . . 00 1 . . . 0...

.... . .

...0 0 0 1

A =

1 0 . . . 00 1 . . . 0...

.... . .

...0 0 0 1

. (4.7)

The identity can here be viewed as the metric in Euclidean space. Analogously there are matrixgroups of matrices which preserve other metrics, like the following groups.

Definition 4.2.8 (The special unitary groups SU(n−, n+)) Let, for n−, n+ ∈ N, n := n− +n+, then the special unitary groups SU(n−, n+) are subgroups of GL(n,C), defined as:

SU(n−, n+) :=

A ∈ GL(n,C)| AT

−1 0 . . . 00 −1 . . . 0...

.... . .

...0 0 0 1

A =

−1 0 . . . 00 −1 . . . 0...

.... . .

...0 0 0 1

, detA = 1

,(4.8)

where the preserved metric is an n×n matrix with the first n− diagonal elements −1, the last n+

diagonal elements +1 and all other elements 0.

35

The special unitary group we will encounter most often in this paper is SU(1, 2). As the nextlemma shows we can give another definition of this group, equivalent with definition 4.2.8. Insome cases this definition is more useful as we will see.

Lemma 4.2.9 (Another way to represent SU(1, 2)) We can represent SU(1, 2) as the set ofall matrices A ∈ GL(3,C) that satisfy:

detA = 1; (4.9)

ATV A = V, (4.10)

where:

V :=

0 0 −10 2 0−1 0 0

. (4.11)

Proof: We already know from definition 4.2.8 that we can represent SU(1, 2) as the matricesA ∈ GL(3,C) with detA = 1 and ATWA = W , where:

W :=

−1 0 00 1 00 0 1

. (4.12)

If we can find a change of coordinates matrix C such that CTV C = W , then we are done. It iseasily checked that the following matrix has this property:

C :=1√2

1 0 10 1 01 0 −1

. (4.13)

2

It will be useful to have conditions on the entries of SU(1, 2) matrices.

Corollary 4.2.10 (Conditions on the entries of SU(1, 2)) Let A ∈ SU(1, 2) according to thedescription given in lemma 4.2.9 with:

A =

a11 a12 a13

a21 a22 a23

a31 a32 a33

. (4.14)

Then the entries of A have to satisfy the following conditions:

−a11a31 + 2|a21|2 − a31a11 = 0 (4.15)

−a11a32 + 2a21a22 − a31a12 = 0 (4.16)

−a12a33 + 2a22a23 − a32a13 = 0 (4.17)

−a13a33 + 2|a23|2 − a33a13 = 0 (4.18)

−a11a33 + 2a21a23 − a31a13 = −1 (4.19)

−a12a32 + 2|a22|2 − a32a12 = 2; (4.20)

detA = 1. (4.21)

Proof: If we write out the conditions from lemma 4.2.9 (or the equivalent ATV A = V ) wefind these conditions. Actually we find three extra conditions, but these turn out to be complexconjugates of conditions we already have. 2

These conditions seem very awkward, but as the next lemma shows, in special cases theybecome workable.

36

Lemma 4.2.11 (Three subgroups of SU(1, 2)) The following sets are all subgroups of SU(1, 2)in the representation of lemma 4.2.9.

The diagonal matrices:

D :=

A ∈ SU(1, 2)| A =

reiθ 0 00 e−2iθ 00 0 1

reiθ

with r, θ ∈ R, r > 0

. (4.22)

The unipotent lower triangular matrices:

L :=

A ∈ SU(1, 2)| A =

1 0 0z 1 0

ix+ |z|2 2z 1

with x ∈ R, z ∈ C

. (4.23)

The unipotent upper triangular matrices:

U :=

A ∈ SU(1, 2)| A =

1 2z ix+ |z|20 1 z0 0 1

with x ∈ R, z ∈ C

. (4.24)

Proof: There are a couple of things that have to be proven here. First of all why the matrices inthe sets D,L and U have the form they do. Then we need to see that these sets are indeed groups.Note that the existence of a unity element, I3, is trivial. Futhermore all the matrices in these setshave determinant equal to one, because they are also SU(1, 2) matrices, so they have an inverse andan easy calculation shows that these inverse matrices are of the desired form (diagonal, unipotentupper triangular or unipotent lower triangular). Thus to prove that these sets are groups we onlyneed to prove that the product of two matrices is again in the set. I will now look at each caseseperately.

• Diagonal matrices are matrices with all non-diagonal entries equal to zero, so using thenotation of corollary 4.2.10 we have a12 = a13 = a21 = a23 = a31 = a32 = 0. Inserting thisin the conditions of corollary 4.2.10) we are left with:

2|a22|2 = 2

−a11a33 = −1

detA = 1. (4.25)

Simple calculations now show that the matrices in D must be of the form as given in thislemma. To see that D is indeed a group we note that (with r, s, θ, φ ∈ R, r, s > 0):

reiθ 0 00 e−2iθ 00 0 1

reiθ

seiφ 0 00 e−2iφ 00 0 1

seiφ

=

rsei(θ+φ) 0 0

0 e−2i(θ+φ) 00 0 1

rsei(θ+φ)

, (4.26)

where rs, θ + φ ∈ R, rs > 0.

• Unipotent lower triangular matrices have a11 = a22 = a33 = 1 and a12 = a13 = a23 = 0.Inserting this into corollary 4.2.10) we get:

−a31 + 2|a21|2 − a31 = 0

−a32 + 2a21 = 0

detA = 1. (4.27)

37

The last equation is already satisfied no matter what the precise expressions for a21, a31 anda32 are. The first two conditions give:

2Rea31 = 2|a21|2

a32 = 2a21. (4.28)

This leads to the form given in the lemma. L also is a group, since (with x, y ∈ R andw, z ∈ C):

1 0 0z 1 0

ix+ |z|2 2z 1

1 0 0w 1 0

iy + |w|2 2w 1

=

1 0 0z + w 1 0

i(x+ y) + |z|2 + |w|2 + 2zw 2(w + z) 1

. (4.29)

Since:|w + z|2 = |z|2 + |w|2 + zw + wz, (4.30)

andzw − wz = 2i (Re(z)Im(w) − Im(z)Re(w)) , (4.31)

we see that:

i(x+ y) + |z|2 + |w|2 + 2zw = i (x+ y + 2 (Re(z)Im(w) − Im(z)Re(w))) + |w + z|2. (4.32)

Thus the right hand side of equation (4.29) is of the desired form with new parametersx+ y + 2 (Re(z)Im(w) − Im(z)Re(w)) ∈ R and w + z ∈ C.

• The proof for the unipotent upper triangular matrices goes along the same lines as in thecase of the unipotent lower triangular matrices, only now we start with the conditions a11 =a22 = a33 = 1 and a21 = a31 = a32 = 0.

2

Remark: It is noteworthy to remark that the unipotent upper triangular matrices from theprevious lemma are also known as (a representation of) the Heisenberg group1. Information aboutthe Heisenberg Lie algebra follows in lemma 5.7.3.

Before turning to another type of group, I will give one more way of representing SU(1, 2).This will be used in section 5.8.

Lemma 4.2.12 (New representation for SU(1, 2)) Define:

M :=

1 0 00 1 00 0 −1

. (4.33)

Then we can represent SU(1, 2) as the set of all matrices A satisftying:

detA = 1; (4.34)

ATMA = M. (4.35)

The change of coordinate matrix from the representation of definition 4.2.8 to this one is the onethat interchanges the first and third coordinate:

I :=

0 0 10 1 01 0 0

. (4.36)

1The Heisenberg algebra has it’s appearances in physics literature, e.g. [1], [7], [14] and [15].

38

The change of coordinate matrix from the representation of lemma 4.2.9 to the one in this lemmais:

Γ :=1√2

−1 0 10 2 01 0 1

. (4.37)

Proof: If we are able to give a coordinate transformation that changes the Gram matrix of thebilinear form of definition 4.2.8 or lemma 4.2.9, i.e. W respectively V , into M , then we have shownthat this representation is equivalent to these other two representaions. Simple calculation showsthat indeed ITMI = W and ΓTMΓ = V . 2

Remark: Conditions on the entries of matrices in SU(1, 2) like those given in corollarie 4.2.10for the representation in lemma 4.2.9 can of course also be given for the representations in definition4.2.8 and lemma 4.2.12. I will leave it to the reader to calculate these conditions when they areneeded.

I have introduced three different ways to represent SU(1, 2) in definition 4.2.8 and lemmas4.2.9 and 4.2.12. We see that these different representations imply different conditions on theentries of the matrices, so I will always make clear which representation of SU(1, 2) I am usingat a particular moment. After all this information on SU(1, 2) I will introduce one more type ofgroup, the compact symplectic group.

Definition 4.2.13 (The compact symplectic groups Sp(n)) Let H be the field of quaternions.For n ∈ N, the vector space Hn has the inner product:

〈v, w〉 :=

n∑

i=1

viwi, for v, w ∈ H. (4.38)

The compact symplectic groups Sp(n) are the subgroups of GL(n,H) consisting of inner productpreserving matrices:

Sp(n) :=

A ∈ GL(n,H)|ATA = I

. (4.39)

Note that the condition ATA = I is equivalent with 〈Av,Aw〉 = 〈v, w〉.

Proposition 4.2.14 (Alternative way of viewing Sp(n)) The compact symplectic group canalso be expressed as:

Sp(n) =

A ∈ U(2n)| AT JnA = Jn

, (4.40)

where Jn is a 2n× 2n symplectic form (i.e. a bilinear antisymmetric form):

Jn =

(

0 −In

In 0

)

. (4.41)

Proof: Since we can identify H with C + jC, where j is the second complex structure on thequaternions (remember that q ∈ H can be written as q = a+ bi+ cj+dk, with a, b, c, d ∈ R, ji = kand i2 = j2 = k2 = −1) an element of GL(n,H) can be identified with an element of GL(2n,C).Let’s first take a look at the case n = 1. If A ∈ GL(2,C) must also be an element of GL(1,H) it

must be of the form

(

z w−w z

)

, with w, z ∈ C (since every quaternion can be written in this

form). Notice that this is an element of U(2) since it cannot be zero. In general an element of Hn

can be written as an element of U(2n). It can easily be seen that this is equivalent to demandingthat:

AJ1 = J1A. (4.42)

It can be shown that by a transformation of basis Jn can be rewritten as:

J1 0 . . . 00 J1 . . . 0...

.... . .

...0 0 0 J1

, (4.43)

39

from which it follows that also for general n ∈ N the condition:

AJn = JnA, (4.44)

must hold. Since A is unitary, we also have that AT = A−1

, so we see that ATJnA = Jn musthold. 2

4.3 Lie groups and Lie algebras

The groups I introduced in section 4.2 are so called Lie groups. Let me first introduce this concept.In the following smooth will mean C∞.

Definition 4.3.1 (Lie group) A Lie group is a smooth manifold with a group structure, suchthat the group operation and taking the inverse are smooth.

I will now state without proof the following theorem.

Theorem 4.3.2 (The introduced groups are Lie groups) The groupsGL(n,R), GL(n,C), GL(n,H), SL(n,R), SL(n,C), O(n), SO(n), U(n), SU(n), SU(n−, n+) andSp(n) are Lie groups, for all n, n−, n+ ∈ N.

Proof: See e.g. [6]. 2

Next I will introduce the important concept of the Lie algebra of a Lie group, after which Iwill give some of the Lie algebras of the Lie groups I’ve discussed. To do this I will first have todefine the Lie bracket. There is a general definition for the Lie bracket for general Lie groups (seee.g. [6]), but since I’m only dealing with matrix Lie groups here, I will only give the definition ofthe special case of the Lie bracket applied to two matrices.

Definition 4.3.3 (Lie bracket of two matrices) Let X and Y be two n×n matrices, then theLie bracket of X and Y is again an n× n matrix, defined by:

[X,Y ] := XY − Y X. (4.45)

In other words for matrices applying the Lie bracket to two matrices is the same as taking thecommutator of the two matrices.

Before saying what a Lie algebra of a Lie group is, I will first define the general notion of Liealgebra.

Definition 4.3.4 (Lie algebra) A Lie algebra is a linear vector space, a, together with a bilinearmap [., .] : a× a→ a which satisfies, for all X, Y and Z ∈ a:

[X,Y ] = −[Y,X ] (4.46)

[X, [Y, Z]] + [Y, [Z,X ]] + [Z, [X,Y ]] = 0. (4.47)

These are the antisymmetry condition on the Lie bracket and the Jacobi identity.

Definition 4.3.5 (Lie subalgebra) Let a be a Lie algebra and b a subspace of a (i.e. b is asubset of a and is closed under the operations of vector addition and scalar multiplication whichexist on a). Then b is a Lie subalgebra of a if and only if it is closed under the Lie bracketoperation which exists on a, i.e. if for all b1, b2 ∈ b it holds that [b1, b2] ∈ b.

Now I come to the important notion of Lie algebra of a Lie group. Like the Lie brackets, thisconcept is also defined for a more general case, but I will only give the definition for a Lie groupthat can be represented by a group of matrices. Unless stated otherwise, these results also holdfor more general cases, but since I have not defined the general Lie bracket here, I will refer formore information about the general theory to e.g. [6].

40

Definition 4.3.6 (Lie algebra of a Lie group) Let G be a Lie group that can be representedby sets with matrices as entries, then the Lie algebra of G is the linear vector space TeG (i.e. thetangent space at the identity element of the Lie group G) denoted by g, together with the bilinearLie brackets [., .] : g × g → g. I will often refer to g (instead of g and [., .]) as the Lie algebra ofG. The existence of the Lie bracket is then implied.

Definition 4.3.7 (Structure constants) Let a be an n-dimensional (n ∈ N∪∞) Lie algebraand Xii (1 ≤ i ≤ n) a basis for a. Then the Lie bracket of two basis vectors Xi, Xj ∈ a is againa vector in the algebra, [Xi, Xj ] ∈ a, so we can write it as a linear combination of basis vectors:

[Xi, Xj ] =n∑

k=1

CkijXk. (4.48)

The constants Ckij are called the structure constants of the Lie algebra corresponding to the chosenbasis.

4.4 U(2) as subgroup of SU(1, 2)

My main goal in this section is to investigate U(2) as a subgroup of SU(1, 2). How can it beembedded? Is it a normal subgroup? First I will prove that U(2) is indeed a subgroup of SU(1, 2).

Theorem 4.4.1 (U(2) is a subgroup of SU(1, 2))

U(2) ⊆ SU(1, 2). (4.49)

Proof: The following map is an injective homomorphism (thus an injective map that preservesthe group structure: ψ(AB) = ψ(A)ψ(B)) and thus proves the theorem:

ψ : U(2)→ SU(1, 2) : A 7→(

e−iθ 00 A

)

, (4.50)

where eiθ = detA and elements of SU(1, 2) are expressed in the representation of definition 4.2.8.We conclude that there is at least one copy of U(2) in SU(1, 2). 2

Remark: The following lemma shows that if we have one embedding of U(2) in SU(1, 2), wecan create more by composing with automorhpisms.

Lemma 4.4.2 (Embeddings) Let G be a group and H a subgroup. Let ψ be the embedding ofH in G. If φ ∈ Aut(H) (i.e. φ is an automorphism on H) and χ ∈ Aut(G), then χψ φ is againan embedding of H in G.

In light of definition 4.4.7 and lemma 4.4.8 that follow later in this section note that since thisin particular holds for Lie groups, the statement with G and H replaced by their Lie algebras andthe group homomorphisms replaced by Lie algebra homomorphisms is also true.

Proof: This follows directly since composition of an embedding with an automorphism is againan embedding. 2

Definition 4.4.3 (Normal subgroup) Let G be a group and let H be a subgroup of G. ThenH is called a normal subgroup if for all g ∈ G it holds that gH = Hg, or equivalently thatgHg−1 = H. Here I have used the notation:

gH := k ∈ G| ∃h ∈ H such that k = g · h , (4.51)

and:Hg := k ∈ G| ∃h ∈ H such that k = h · g , (4.52)

where · is the group operation.

41

The natural question to ask is now: is U(2) a normal subgroup of SU(1, 2)? This will turn outnot to be the case, but before I can prove this, I need the following lemma.

Lemma 4.4.4 (Characterization of U(2) in SU(1, 2)) The embedded U(2) inSU(1, 2) in the representation of definition 4.2.8 by the map ψ from theorem 4.4.1 can be completelydescribed as follows:

U(2) =

B ∈ SU(1, 2)| ∀a ∈ C BTΩaB = Ωa

, (4.53)

where:

Ωa :=

a 0 00 1 00 0 1

. (4.54)

Proof: Write:

B :=

b11 b12 b13b21 b22 b23b31 b32 b33

. (4.55)

Then the condition in equation (4.53) written out, gives that for B to be in U(2) it must hold forall a ∈ C that:

a|b11|2 + |b21|2 + |b31|2 = a

a|b12|2 + |b22|2 + |b32|2 = 1

a|b13|2 + |b23|2 + |b33|2 = 1

ab11b12 + b21b22 + b31b32 = 0

ab12b11 + b22b21 + b32b31 = 0

ab11b13 + b21b23 + b31b33 = 0

ab13b11 + b23b21 + b33b31 = 0

ab12b13 + b22b23 + b32b33 = 0

ab13b12 + b23b22 + b33b32 = 0. (4.56)

Since this has to hold for all a we find that b12 = b13 = b21 = b31 = 0. We are thus left with theequations

b12 = b13 = b21 = b31 = 0

|b22|2 + |b32|2 = 1

|b23|2 + |b33|2 = 1

b22b23 + b32b33 = 0. (4.57)

(4.58)

Since B ∈ SU(1, 2) we must also have that detB = 1, which gives the final condition:

b11 =1

b22b33 − b23b32. (4.59)

We see that this gives B exactly the form of an embedded element of U(2).The reverse direction, i.e. an element of U(2) satisfies the condition in equation (4.53), can be

proved by straightforward calculation. 2

Lemma 4.4.5 (U(2) is not a normal subgroup of SU(1, 2)) U(2) is not a normal subgroupof SU(1, 2).

Proof: I will now prove that U(2) is not a normal subgroup of SU(1, 2) by showing that forarbitrary A ∈ SU(1, 2) and B ∈ U(2) the matrix ABA−1 does not satisfy the condition (4.53) andthus is no embedded element of U(2).

42

Assume that ABA−1 does satisfy (ABA−1)TΩa(ABA−1) = Ωa for all a, with Ωa as in lemma

4.4.4. Then ABTATΩaABAT = Ωa for all a. With A−1 = A

Tand B−1 = B

Twe can rewrite this

as ABTATΩa = ΩaABTAT for all a. This does not hold however, as can be verified for example

by computing the entry in the first row, second column of the lefthand side and righthand side ofabove equation. 2

The following results give some more insight in how we find U(2) in SU(1, 2).

Theorem 4.4.6 (Group homomorphism on SU(1, 2)) Define the following map:

α : SU(1, 2)→ SU(1, 2) : A 7→(

AT)−1

. (4.60)

Then α is a group homomorphism with the following property:

α2 = IdSU(1,2). (4.61)

Furthermore we get the following subgroup of SU(1, 2):

F := A ∈ SU(1, 2)| α(A) = A ∼= SU(1, 2) ∩ SU(3) ∼= U(2). (4.62)

We can define the set of all Hermitian matrices in SU(1, 2) with the help of α:

H :=

A ∈ SU(1, 2)| α(A) = A−1

. (4.63)

This set is not a subgroup of SU(1, 2).

Proof: α is a homomorphism because, for A,B ∈ SU(1, 2), we have that α(AB) =(

(AB)T)−1

=(AT )−1(BT )−1 = α(A)α(B). Property (4.61) is easily checked. Because α is a homomorphismwe see that F is a group. That H consists of Hermitean matrices follows from the definition ofα. Since I3 6∈ H it is not a group. Finally I have to show that FSU(1, 2) ∩ SU(3) ∼= U(2). Thefirst isomorphism is obvious from the definition of F and SU(3) (definition 4.2.7). For the secondisomorphism I use the embedding ψ of U(2) into SU(1, 2) from theorem 4.4.1. It is easy to see thatelements B ∈ ψ(U(2)) satisfy α(B) = B. To prove that a D ∈ F is also an element of ψ(U(2)),we use that F ∼= SU(1, 2) ∩ SU(3). Writing out part of the conditions for D (with entries dij) tobe in SU(1, 2) we find:

−|d11|2 + |d21|2 + |d31|2 = −1 (4.64)

−|d12|2 + |d22|2 + |d32|2 = 1 (4.65)

−|d13|2 + |d23|2 + |d33|2 = 1. (4.66)

Writing out the SU(3) counterparts of these condtions, we get:

|d11|2 + |d21|2 + |d31|2 = 1 (4.67)

|d12|2 + |d22|2 + |d32|2 = 1 (4.68)

|d13|2 + |d23|2 + |d33|2 = 1. (4.69)

Combining these two sets of equations we see that d12 = d13 = d21 = d31 = 0. From this thedesired form of D follows. 2

This map has consequences on the level of the Lie algebras, as the following results show.

Definition 4.4.7 (Lie algebra homomorphism) Let h and g be Lie algebras, then a linearmap κ : h→ g is a Lie algebra homomorphism if:

κ([x, y]h) = [κ(x), κ(y)]g, (4.70)

where x, y ∈ h and the subscript at the Lie brackets denotes in which algebra the Lie bracketoperation is performed.

43

Lemma 4.4.8 (Lie group and Lie algebra homomorphisms) Let H and G be Lie groups, h

and g their respective Lie algebras and ψ : H → G a Lie group homomorphism. Then the tangentmap at the identity element of H, dψ := Teψ : h→ g, is a Lie algebra homomorphism.

Proof: See [6]. 2

Corollary 4.4.9 (α on the level of Lie algebras) The map α from theorem 4.4.6 gives a Liegroup homomorphism:

dα : su(1, 2)→ su(1, 2) : A 7→ −(

AT)

. (4.71)

Furthermore α2 = Idsu(1,2). α has eigenvalues 1 and −1 and the corresponding eigenspaces consistof the Hermitian, respectively antihermitian, matrices in su(1, 2). The eigenspace correspondingto the value −1 is isomorphic to u(2).

Proof: Since α is a group homomorphism, lemma 4.4.8 says that dα is a Lie algebra homomor-phism. We can see how dα acts by writing A ∈ SU(1, 2) as A = I3 +εB, where ε is an infinitesimalparameter and B ∈ su(1, 2). Then we calculate:

α(A) =(

(I2 + εB)T)−1

= I2 − εBT

:= I2 + εdα(B). (4.72)

Thus dα acts as described in this corollary. That α2 = Idsu(1,2) is easily checked. Since dα acts asa linear map on a vectorspace we can calculate its eigenvalues. The previous property shows thatthese are either 1 or −1. Since there are both Hermitian and antihermitian matrices in su(1, 2) wesee that both eigenvalues occur (it’s trivially seen that the Hermitian matrices give an eigenvalueof 1 and the antihermitian ones give −1). Finally, since theorem 4.4.6 says that the set of allmatrices in SU(1, 2) that obey α(A) = A is isomorphic to U(2), we see that the eigenspace witheigenvalue 1 of dα is isomorpic to u(2). 2

4.5 Bases for u(2) and su(1, 2)

Since U(2) and SU(1, 2) are Lie groups we can try and find bases for their Lie algebras. In thissection I will present the bases I will be using throughout the rest of this paper. Since differentbases can be useful in different situations I will give more than one basis for su(1, 2).

4.5.1 The Lie algebra of U(2)

A general method to compute the Lie algebra of a Lie group is to introduce an infinitesimal (real)constant ε of which we demand that

ε 6= 0 (4.73)

ε2 = 0 (4.74)

ε = ε. (4.75)

This element we will use to investigate group elements in the neighbourhood of the identity. Afterall, what we are after is the tangent space at the identity. We will now write an element ofA ∈ U(n) near the identity as In + εB, where B is the n× n matrix in u(n) we are after. To findrestrictions on B I now demand that ATA = In as definition 4.2.6 prescribes. So, using equation(4.73), we get:

(In + εB)T (In + εB) = In (4.76)

(In + εBT )(In + εB) = In (4.77)

In + ε(BT +B) = In (4.78)

BT = −B. (4.79)

44

We see that:u(n) =

B ∈ GL(n,C)| BT = −B

. (4.80)

Lemma 4.5.1 (A basis for u(2)) u(2) is a four dimensional Lie algebra and a basis for it is:

(X1, X2, X3, X4) :=

((

i 00 0

)

,

(

0 00 i

)

,

(

0 1−1 0

)

,

(

0 ii 0

))

. (4.81)

Proof: For a matrix B ∈ u(2), B =

(

b11 b12b21 b22

)

, we have found in equation (4.80) that the

following equations must hold:

b11 = −b11, b22 = −b22 and b21 = −b12. (4.82)

If we write this out in terms of the real and imaginary parts of the bij we find:

Re(b11) = 0, Re(b22) = 0,Re(b21) = −Re(b12), Im(b21) = Im(b12).

We see that this gives four conditions on the eight real entries for a 2 × 2 complex matrix, sodimu(2) = 8− 4 = 4. It can easily be checked that the four 2× 2 matrices from equation (4.81)satisfy these four conditions and are linearly independent over R, so they form a basis for u(2).2

4.5.2 The Lie algebra of SU(1, 2)

In the following I will denote by In−,n+ the (n−+n+)×(n−+n+) matrix with as the first (startingin the upper left corner of the matrix) n− diagonal elements −1, as the other n+ diagonal elements1 and the other entries 0. E.g. the matrix used in definition 4.2.8 will be denoted I1,2.

Analogous to what we have done in section 4.5.1, we find, by writing A ∈ SU(n−, n+) (seedefinition 4.2.8) as (In + εB), that:

su(n−, n+) =

B ∈ GL(n,C)| BT In−,n+ = −In−,n+B,TrB = 0

. (4.83)

From section 4.4 we know that U(2) is a Lie subgroup of SU(1, 2) so we expect u(2) to be asubalgebra of su(1, 2). This means that we must be able to choose a basis for su(1, 2) in which thebasis for u(2) is easily recognizable. To this end I will first define a map to embed u(2) in su(1, 2).

Definition 4.5.2 (Lie algebra elements of the ψ-image of U(2)) Let ψ be as defined in the-orem 4.4.1 and su(1, 2) constructed according to equation (4.83). Then the matrix:

F :=

(

−TrF ′ 00 F ′

)

∈ su(1, 2), (4.84)

corresponding to F ′ ∈ u(2), is Teψ(F ′), the image of F ′ under the tangent map of ψ in the identityelement. In the following I will use the notation dψ := Teψ.

This map I defined I can use to identify basis elements of dψ(u(2)) among my basis elementsfor su(1, 2).

45

Lemma 4.5.3 (A basis for su(1, 2)) su(1, 2) is an eight dimensional Lie algebra and using thedescription from equation (4.83) a basis for it is:

(Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8) :=

−i 0 00 i 00 0 0

,

−i 0 00 0 00 0 i

,

0 0 00 0 10 −1 0

,

0 0 00 0 i0 i 0

,

0 1 01 0 00 0 0

,

0 −i 0i 0 00 0 0

,

0 0 10 0 01 0 0

,

0 0 −i0 0 0i 0 0

.

(4.85)

(Y1, Y2, Y3, Y4) is a basis of dψ(u(2)).

Proof: From equation (4.83) we have 10 conditions for the 18 real entries, so dimsu(1, 2) =18− 10 = 8. Written out, the conditions are:

Re(b11) = 0, Re(b22) = 0, Re(b33) = 0Re(b21) = Re(b12), Im(b21) = −Im(b12), Re(b31) = Re(b13)Im(b31) = −Im(b13), Re(b32) = −Re(b23), Im(b33) = Im(b23),

and TrB = Im(b11) + Im(b22) + Im(b33) = 0. The eight linearly independent matrices fromequation (4.85) satisfy these conditions and thus form a basis for su(1, 2). Futhermore we seethat, for i ∈ 1, 2, 3, 4, Yi = dψ(Xi), where the Xi are the basis elements of u(2) from equation(4.81). 2

Remark: Note that the antihermiticity of the basis elements of dψ(u(2)) in su(1, 2) was to beexpected from corollary 4.4.9

What I have done above with bases for su(1, 2) was based on the representation of SU(1, 2)as in definition 4.2.8. We can also choose a basis that follows from the representation in lemma4.2.9. This will now be explained.

Lemma 4.5.4 (A new basis for su(1, 2)) The following set of matrices is a basis for su(1, 2)based on the representation found in lemma 4.2.9:

(N1, N2, N3, N4, N5, N6, N7, N8) :=

−1

2

i 0 i0 −2i 0i 0 i

,

0 0 −i0 0 0−i 0 0

,

1

2

0 −2 01 0 −10 2 0

,1

2

0 2i 0i 0 −i0 −2i 0

,1

2

0 2 01 0 10 2 0

,

1

2

0 −2i 0i 0 i0 −2i 0

,

1 0 00 0 00 0 −1

,

0 0 i0 0 0−i 0 0

. (4.86)

(N1, N2, N3, N4) is a basis for an embedded u(2).

46

Proof: I can use the change of coordinates matrix C from equation (4.13) and the basis (Yi)i fromlemma 4.5.3. Then remember from linear algebra (e.g. [21]), that if I have a change of coordinatesmatrix C, then, for i ∈ 1, · · · , 8, the matrices Ni change as follows: Ni = CYiC

−1. This alsogives that (N1, N2, N3, N4) is a basis for an embedded u(2), since (Y1, Y2, Y3, Y4) was a basis fordψ(u(2)). 2

Remark: If I write, for A ∈ SU(1, 2), A = I3 + εB, with ε an infinitesimal quantity, then I getby demanding that A preserves V from lemma 4.2.9 and that detA = 1 the following descriptionfor su(1, 2) based on the representation of SU(1, 2) from lemma 4.2.9:

su(1, 2) =

B ∈ GL(3,C)| BTV = −V B,TrB = 0

. (4.87)

We see that the basis from lemma 4.5.4 is compatible with this description.

4.6 Homogeneous spaces

4.6.1 Homogeneous spaces

Definition 4.6.1 (Group action) Let G be a group and X a set, then a (group) action of G onX is a homomorphism from G to the group of all permutations of elements of X.

Definition 4.6.2 (Lie group action on a manifold) Let G be a Lie group and X a manifold.Then the Lie group action of G on the manifold X is a differentiable map σ : G×X → X whichsatisfies (with e the identity in G), ∀x ∈ X and ∀g1, g2 ∈ G:

σ(e, x) = x (4.88)

σ(g1, σ(g2, x)) = σ(g1g2, x). (4.89)

This is called a left action, since the group elements act from the left on the points of the manifold(see also the part on the notation directly below this definition). A right action would have hadequation (4.89) replaced with

σ(g1, σ(g2, x)) = σ(g2g1, x). (4.90)

In the following, if left or right is not specified, then I will always mean left action.

Notation: Often σ(g, x), for g ∈ G and x ∈ X , will be denoted by gx (not only if G is a Liegroup and X a manifold, but also in the general case where G is a group and X a set). Note thatin the case of a right action, the notation would have been xg. ♦

Definition 4.6.3 (Transitive action) Let G be a group and X a set. Then an action of G onX is transitive if ∀x1, x2 ∈ X, ∃g ∈ G such that gx1 = x2.

Definition 4.6.4 (Orbit) The orbit of the point x ∈ X under an action of a group G on the setX is the set:

G(x) := y ∈ X | ∃g ∈ G such that gx = y . (4.91)

Note that for a transitive action G(x) = X for all x ∈ X .

Definition 4.6.5 (Isotropy group) Let G be a group with an action on the set X. Then theisotropy group of x ∈ X, also called stabilizer group or little group is the set:

Gx := g ∈ G| gx = x . (4.92)

If g, h ∈ Gx then gh, g−1 ∈ Gx and e ∈ Gx, with e ∈ G the identity element. So Gx is indeed agroup.

47

Theorem 4.6.6 (Stabilizers of points in the same orbit are conjugate) Let G be a groupwith an action on the set X. Let x ∈ X, y ∈ G(x) and let g ∈ G be such that gx = y. Then:

gGxg−1 = Gy . (4.93)

Proof: First note that always x ∈ G(x), since ex = x. So x and y are in the same orbit, hence thetitle of this theorem. Now let h ∈ Gx, then ghg−1y = ghg−1gx = ghx = gx = y. So gGxg

−1 ⊆ Gy.For the reverse statement let k ∈ Gy , then x = g−1gx = g−1y = g−1ky = g−1kgx. So g−1kg ∈ Gx,or equivalently k ∈ gGxg−1. 2

Definition 4.6.7 (Homogeneous space) A space with a transitive group action by a Lie groupis called a homogeneous space.

Definition 4.6.8 (Quotient space of groups and vectorspaces) Let G be a group(vectorspace) and H a subgroup (subspace) of G. If g ∈ G then:

gH := k ∈ G| ∃h ∈ H such that k = g · h , (4.94)

where · is the group operation (vectorspace addition) in G, is called a left coset. (In the case ofvectorspaces gH is often written as g +H.) The quotient space G/H is the set of all left cosetsgH. An equivalent way to look at this is to view G/H as G with elements g, k ∈ G identifiedthrough the equivalence relation g ∼ k if and only if k−1 · g ∈ H, where k−1 is the inverse of kwith respect to the group operation (vectorspace addition). For this reason cosets are also calledequivalence classes. In general for an equivalence relation the notation for the set of points in Gidentified through the equivalence relation is G/ ∼.

Analogously there are right cosets:

Hg := k ∈ G| ∃h ∈ H such that k = h · g , (4.95)

and the set of all right cosets is denoted by H\G. Since vectorspace addition is commutativegH = Hg in the case of vectorspaces; in the case of groups this is so if and only if H is a normalsubgroup (see definition 4.4.3).

In the following I will mean by coset a left coset, unless stated otherwise.

Definition 4.6.9 (Representative) Let G be a group (vectorspace), H a subgroup (subspace)and g ∈ G, then an element h0 of the coset gH is called a representative of that coset. The samegoes for an element in a right coset Hg.

Notation: Often in the situation of definition 4.6.9, in an abuse of notation, the notationh0 ∈ G/H will be used for a representative of the coset [h0], the equivalence class of h0 in G/H .Strictly speaking [h0] ∈ G/H and h0 ∈ G. ♦

Theorem 4.6.10 (A homogeneous space is isomorphic to a quotient space) Let X be ahomogeneous space with a transitive group action by the Lie group G and let x0 ∈ X. Let H be asubgroup of G and the isotropy group of x0, then X is isomorphic to G/H, where the isomorphismis an isomorphism of manifolds, thus a bijection.

Proof: H is the isotropy group of x0 ∈ X , with x0 a fixed point in X , so H = Gx0 . Define:

φ : X → G/H : x 7→ gH, (4.96)

where g ∈ G/H is such that x = gx0. First we have to check if this is well defined.Let g, g′ ∈ G be such that, for some x ∈ X , gx0 = x = g′x0. Then x0 = g−1g′x0, thus

g−1g′ ∈ H . From this it follows that g′ ∈ gH , so φ is well defined. Now I will prove that φ is anisomorphism.

Onto: Let g ∈ G/H , then φ(gx0) = g, where gx0 ∈ X , due to the transitivity of the action.1-1 : If φ(x) = φ(y) = g ∈ G/H , then x = gx0 = y. 2

48

4.6.2 The homogeneous space SU(1, 2)/U(2) on Lie algebra level

That SU(1, 2)/U(2) is homogeneous follows immediately from definition 4.6.7 since SU(1, 2) actstransitively on this space. Lemma 4.4.5 tells us U(2) is not normal in SU(1, 2) and thus SU(1, 2)/U(2)is not a group. This has also consequences on the level of the Lie algebras: su(1, 2)/u(2) is not aLie algebra. In the theory of Lie algebras this can be understood by saying that u(2) is not anideal in su(1, 2), as explained below.

Definition 4.6.11 (Ideal) Let a be a Lie algebra and b a Lie subalgebra. Then b is called anideal in a if, for all A ∈ a and all B ∈ b, [B,A] ∈ b.

Lemma 4.6.12 (Ideals give Lie algebras) Let a be a Lie algebra and b a Lie subalgebra of a.Then a/b is a Lie algebra if and only if b is an ideal in a.

Proof: What must be checked is if the operation of taking the Lie bracket of two elements in thequotient vector space, inherited from the Lie bracket operation on a, is well defined. Let Xi ∈ a

for i ∈ 1, 2, 3, 4 and let Yj ∈ b for j ∈ 1, 2 be such that X2 = X1 + Y1 and X4 = X3 + Y2. Fora/b to be a Lie algebra it must hold that [X1, X3] = [X2, X4] as elements of a/b. The bilinearityof the Lie brackets gives:

[X2, X4] = [X1, X3] + [X1, Y2] + [Y1, X3] + [Y1, Y2]. (4.97)

[Y1, Y2] is an element of b as desired, but [X1, Y2] and [Y1, X3] are in b if and only if b is an idealin a. 2

To see that the Lie bracket operation is not well defined on su(1, 2)/u(2) we have to show thatu(2) is not an ideal in su(1, 2).

Lemma 4.6.13 (u(2) is not an ideal in su(1, 2)) u(2) is not an ideal in su(1, 2).

Proof: It’s enough to show that there are elements A ∈ su(1, 2) (see equation (4.83)) and B ∈ u(2)

(see equation (4.80)) such that [B,A] 6∈ u(2). Take B =

−i 0 00 i 00 0 0

and A =

0 1 01 0 00 0 0

.

Then [B,A] = 2

0 −i 0i 0 00 0 0

6∈ u(2). 2

Notation: Let a, b be two subalgebras of a Lie algebra g. Then [a, b] is the subspace of g

spanned by the elements [A,B], with A ∈ a and B ∈ b. ♦The next lemma will show that U(2) not being a normal subgroup of SU(1, 2) (and thus

SU(1, 2)/U(2) not being a group) is closely related to u(2) not being an ideal in su(1, 2) (and thusTI3 (SU(1, 2)/U(2)) not being a Lie algebra).

Lemma 4.6.14 (Relation between normal subgroups and ideals) Let G be a Lie group, Ha Liegroup and subgroup of G and let g and h be the associated Lie algebras. Then h is an idealin g if H is a normal subgroup of G.

Proof: Let H be a normal subgroup of G, then K := G/H is again a Lie group. We can writeG = K ⊗H and consequently for the Lie algebras g = k ⊕ h. Then (see e.g. [6]) [k, h] = 0, thus[g, h] ⊂ h. So h is an ideal in g. 2

Remark: The reverse statement of lemma 4.6.14 is in general not true. For more informationsee e.g. [6]. The idea is that, since only the so called component of the identity of G is generatedby exponentiating g the reverse statement only holds for the components of the identity of H andG and not a priori for the whole groups H and G.

We have discovered a lot about SU(1, 2)/U(2) in these last few section, but we have not yetmade a connection with the metric from chapter 3. This connection will be made in the nextchapter.

49

Chapter 5

Geometry and isometries

5.1 Two important geometrical pictures

In this chapter we will take a look at some geometrical views of the manifold SU(1, 2)/U(2). Thefirst one, which I named the geometrical picture in the (S,C)-coordinates for reasons that willbecome clear, will be introduced in section 5.2 and provides a perfect point of view to clarifythe connection between SU(1, 2)/U(2) and the metric in section 3.5, in particular in the form ofequation (3.79). The other geometrical point of view is that of a complex two dimensional ballwith the so called Bergman metric defined on it. This will be explained in section 5.8.

5.2 The (S, C)-coordinates

The geometrical interpretation we will derive in this section is inspired by lemma 4.2.9. If SU(1, 2)is represented as matrices that preserve the bilinear form V from that lemma they will in particularpreserve level sets of that bilinear form and act as internal transformations on those level sets.The following lemma makes this explicit.

Theorem 5.2.1 (A geometrical representation of SU(1, 2)/U(2)) Startwith C3 equipped with the bilinear form 〈v, w〉 := −v1w3 − v3w1 + 2v2w2. Now define:

P :=

z ∈ C3| 〈z, z〉 < 0

. (5.1)

Then:SU(1, 2)/U(2) ∼= P/C∗ := B. (5.2)

Proof: From theorem 4.6.10 we see that there are two things we must prove, namely that SU(1, 2)acts transitively on B and that U(2) is the isotropy group of a point in B. To prove the firststatement start by defining for negative c ∈ R:

Pc :=

z ∈ C3| 〈z, z〉 = c

. (5.3)

We then see that P =⋃

c<0 Pc. Furthermore since all A ∈ SU(1, 2) preserve the bilinear form Vfrom lemma 4.2.9, we have that for every A ∈ SU(1, 2), for every c < 0 and every z ∈ Pc it holdsthat Az ∈ Pc. This implies that SU(1, 2) acts as an internal transformation on Pc for every c < 0and in particular on P . To see if this action is transitive on B note the following. For every c < 0and every v ∈ Pc we have a vector w = 2√

−cv ∈ P−2 such that v and w are in the same equivalence

class in B. Thus it suffices to prove that SU(1, 2) acts transitively on P−2. Define:

w :=

101

∈ P−2. (5.4)

50

We have to show that for every w ∈ P−2 there is an A ∈ SU(1, 2) such that Aw = w. Writing thisout in conditions on the matrix and vector entries, we get:

a11 + a13 = w1;

a21 + a23 = w2;

a31 + a33 = w3. (5.5)

Since w ∈ P−2 this gives:

−2 = 〈w,w〉= −(a11 + a13)(a31 + a33)− (a31 + a33)(a11 + a13) + 2(a21 + a23)(a21 + a23)

= (−a11a31 + 2|a21|2 − a31a11) + (−a13a33 + 2|a23|2 − a33a13)

+(−a11a33 + 2a21a23 − a31a13) + (−a13a31 + 2a23a21 − a33a11). (5.6)

If we take a look at the conditions (4.15), (4.18), (4.19) and the complex conjugate of (4.19) formatrices in SU(1, 2) then we see that these are compatible with equation (5.6). We can thusconstruct a matrix A that satisfies the conditions for being in SU(1, 2) as well as Aw = w. ThusSU(1, 2) acts transitively on P−2 and thus on B.

Next we have to prove that U(2) is the isotropy group of a point in B. Take z =

101

∈ P .

Let A ∈ SU(1, 2) be in the isotropy group of the line spanned by z in P (that is the stabilizer ofthe equivalence class of z in B). Then A has to preserve P⊥, where:

P⊥ :=

z ∈ C3| 〈z, z〉 = 0

;

=

z ∈ C3| z =

z1z2−z1

with z1, z2 ∈ C

. (5.7)

Since A ∈ SU(1, 2), A preserves the bilinear form 〈., .〉 and since A also preserves P⊥ it preservesin particular this bilinear form restricted to P⊥. For w, z ∈ P⊥ the bilinear form becomes:

〈w, z〉 = 2(w1z1 + w2z2). (5.8)

The group that preserves this form is U(2), so this is indeed the isotropy group of z ∈ B. 2

To see why this geometrical picture clarifies the connection between SU(1, 2)/U(2) and themetric from equation (3.79), the following corollary gives coordinates on SU(1, 2)/U(2).

Corollary 5.2.2 (More insight into the geometrical situation) With thenotation of theorem 5.2.1 we have:

SU(1, 2)/U(2) ∼=

(C, S) ∈ C2| S + S − 2CC > 0

, (5.9)

where the isometry is between manifolds. So there exists a bijection.

Proof: From theorem 5.2.1 we know that (using the notation from that theorem):

SU(1, 2)/U(2) ∼= P/C∗ = B. (5.10)

P consists of those z ∈ C3 that obey:

−z1z3 − z3z1 + 2|z2|2 < 0. (5.11)

Now let z ∈ P . Then z3 6= 0 (and also z1 6= 0). Using the action of C∗ that we still have to divideout, we can define coordinates on B:

S :=z1z3

(5.12)

C :=z2z3. (5.13)

51

If we divide the condition from equation (5.11) by z3z3, then the condition on this new coordinatesbecomes:

S + S − 2CC > 0. (5.14)

Note that I could have equally well divided by z1 instead of z3. In my computations however Iwill use the convention to divide by z3. 2

Notation: Corollary 5.2.2 tells us that as sets B and

(C, S) ∈ C2| S + S − 2CC > 0

are isomorphic. Therefore I will use B as notation for both thisset and P/C∗ in the following. ♦

5.3 SU(1, 2) as isometry group

In corollary 5.2.2 we already see the condition that was necessary for the definition of the Kahlerpotential in definition 3.4.13. The next thing to do is check if we can define an action of SU(1, 2)on B from theorem 5.2.1 written in the new coordinates from corollary 5.2.2. If this is the casethen we can investigate if SU(1, 2) is an isometry group of the metric (3.79). It turns out thatwe can define such an action in terms of fractional linear transformations. In the “proof” of thefollowing definition I will explain why the action given is a natural action.

Definition 5.3.1 (Fractional linear transformations) Let

A =

a11 a12 a13

a21 a22 a23

a31 a32 a33

∈ SU(1, 2), (5.15)

in the representation of lemma 4.2.9. Then we define an action of SU(1, 2) on B from theorem5.2.1 written in the coordinates from corollary 5.2.2 by:

A

(

SC

)

=

a11S + a12C + a13

a31S + a32C + a33

a21S + a22C + a23

a31S + a32C + a33

. (5.16)

A transformation of this form is called a fractional linear transformation.

Proof: We can show that this is a legitimate action by starting from the standard matrix multi-plication action of SU(1, 2) on C

3:

Az =

a11z1 + a12z2 + a13z3a21z1 + a22z2 + a23z3a31z1 + a32z2 + a33z3

. (5.17)

Since we pass from coordinates

z1z2z3

to coordinates

SC1

by dividing by z3, we arrive at

the desired result by calculating A

SC1

and then dividing by a31S + a32C + a33. That this is

possible (i.e. that a31S + a32C + a33 6= 0) comes from the fact that:(

SC

)

6= 0, (5.18)

because of equation (5.14), the fact that a31, a32 and a33 cannot all be zero, because detA = 1and the fact that the action of C

∗ on P (from B = P/C∗) commutes with the action of SU(1, 2)on P . 2

We can now investigate what this action of SU(1, 2) does with the metric (3.79). To this endfirst a useful lemma.

52

Lemma 5.3.2 (Kahler metric and potential) Let g be a Kahler metric on a n-dimensionalmanifold X defined in terms of a Kahler potential K as explained in corollary 2.2.10. Let z becomplex coordinates on X with corresponding complex conjugates. Now let f be a holomorphictransformation that changes z and z into w and w:

f(z, z) = (w, w) = (w(z), w(z)). (5.19)

Theng(z, z) := g(f(z, z)) = g(z, z) (5.20)

if there is a holomorphic function h(z) such that

K(z, z) := K(f(z, z)) = K(z, z) + h(z) + h(z). (5.21)

Proof: Since the mixed terms are the only non-zero ones (see corollary 2.2.10) we have that g = gif and only if gij = gij , so if and only if:

∂2K

∂zi∂z j=

∂2K

∂zi∂z j. (5.22)

We see that this is satisfied if equation (5.21) holds. 2

Theorem 5.3.3 (SU(1, 2) is an isometry group of the metric (3.79))SU(1, 2) is an isometry group of the metric (3.79).

Proof: According to lemma 5.3.2 if we can show that acting with SU(1, 2) on B from theorem5.2.1 in the coordinates from corollary 5.2.2 (which is a holomorphic transformation) changes theKahler potential K from definition 3.4.13 only by adding a holomorphic function and its complexconjugate, then we have proven the desired statement. So we act with SU(1, 2) on (S,C)T via theaction in definiton 5.3.1 and investigate what it does with K:

S + S − 2CC 7→ a11S + a12C + a13

a31S + a32C + a33+a11S + a12C + a13

a31S + a32C + a33

−2a21S + a22C + a23

a31S + a32C + a33

a21S + a22C + a23

a31S + a32C + a33

=1

|a31S + a32C + a33|2(

(a11S + a12C + a13)(a31S + a32C + a33)

+(a31S + a32C + a33)(a11S + a12C + a13)

−2(a21S + a22C + a23)(a21S + a22C + a23))

=1

|a31S + a32C + a33

(

(a13a33 + a31a13 − 2|a23|2)

+(a12a33 + a32a13 − 2a22a23)C + (a13a32 + a33a12 − 2a23a22)C

+(a11a33 + a31a13 − 2a21a23)S + (a13a31 + a33a11 − 2a23a21)S

+(a12a32 + a32a12 − 2|a22|2)|C|2 + (a11a31 + a31a11 − 2|a21|2)|S|2

+(a12a31 + a32a11 − 2a22a21)CS + (a11a32 + a31a12 − 2a21a22)CS)

=1

|a31S + a32C + a33|2(S + S − 2|C|2), (5.23)

where for the last identity I used the relations from corollary 4.2.10 which the entries of A haveto obey. If we define ζ(S,C) := a31S + a32C + a33, then we have:

S + S − 2CC 7→ (S + S − 2CC)1

ζ(S,C)

1

ζ(S,C), (5.24)

thus:K 7→ K + logζ + logζ. (5.25)

53

2

From corollary 5.2.2 we saw that as sets SU(1, 2)/U(2) and B are isomorphic. From theorem5.3.3 we see that SU(1, 2) is an isometry group of the metric (3.79) which can be defined on B.

5.4 Generators of isometries in physics literature

In this section I will take a look at the isometries of the action (3.79) as given in [7] (I will describethese later in this section) and what their connection is with the linear fractional transformationsgiven in definition 5.3.1. We will encounter again the three subgroups of SU(1, 2) I gave in lemma4.2.11.

In definition 5.3.1 I have given an action of SU(1, 2) on B. I can now take a look at the actionof the subgroups of lemma 4.2.11 we get in this way.

Corollary 5.4.1 (Action of the subgroups from lemma 4.2.11) Let matrices A ∈ SU(1, 2)

in the representation of lemma 4.2.9 act on

(

SC

)

as defined in definition 5.3.1. Then the

subgroups of SU(1, 2) defined in lemma 4.2.11 act as follows.

• The diagonal matrices act as:

reiθ 0 00 e−2iθ 00 0 1

reiθ

(

SC

)

=

(

r2Sre−3iθC

)

. (5.26)

• The unipotent lower triangular matrices act as:

1 0 0z 1 0

ix+ |z|2 2z 1

(

SC

)

=

S

1 + 2zC + (ix+ |z|2)S

C + zS

1 + 2zC + (ix+ |z|2)S

. (5.27)

• The unipotent upper triangular matrices act as:

1 2z ix+ |z|20 1 z0 0 1

(

SC

)

=

(

ix+ |z|2 + 2zC + Sz + C

)

. (5.28)

Proof: Just act with a matrix of the specified type on B according to the action of definition5.3.1. 2

Before we turn to the relation between these linear transformations and the isometries given in[7] I will identify the unipotent upper triangular matrices with the stabilizer group of the Kahlerpotential K.

Lemma 5.4.2 (The unipotent upper diagonal matrices and K) LetK := −log(S+S−2CC) as in definition 3.4.13 and let U be the group of unipotent upper triangularmatrices as defined in lemma 4.2.11. Then U is the stabilizer group of K.

Proof: From equation (5.25) we see that K remains unchanged under the action of SU(1, 2) ifand only if ζ = 1, where ζ(S,C) := a31S + a32C + a33 as defined in theorem 5.3.3. Since we wantζ = 1 to hold for all S and C, we see that a31 = a32 = 0 and a33 = 1. Inserting this into theconditions for elements of SU(1, 2) of corollary 4.2.10 we get among others the conditions: a21 = 0and a11 = a22 = 1. We see from the proof of lemma 4.2.11 that we are dealing with the unipotentupper triangular matrices. 2

54

Remark: Using a different terminology, lemma 5.4.2 can also be worded as: the Heisenberggroup is the stabilizer group of the dilaton.

With corollary 5.4.1 we are in a position to say what the isometries of the metric (3.79) as givenin [7] have to do with these linear fractional transformations. First let me quote the isometries [7]gives:

1. With the epsilons real parameters:

S → S − ε0S −i

4ε1S

2 − 1

2(ε3 + iε4)CS,

C → C − ε02C − i

4ε1CS −

1

2ε3(C

2 − S

2)− i

2ε4(C

2 +S

2). (5.29)

2. Exchange of Re(C) and Im(C).

3. With real parameters α, β and γ1:

S → S + iα+ 2(γ + iβ)C + γ2 + β2,

C → C + γ − iβ. (5.30)

First note that, with z := γ − iβ, we see from corollary 5.4.1 that the action of the unipotentupper triangular matrices gives the isometries of item 3 above2. Furthermore, if we take theparameters x and z in the action of the lower triangular matrices in corollary 5.4.1 as infinitesimalquantities (i.e. neglect terms of order x2 and z2 and higher), we see that the result of this actionbecomes:

C + zS

1 + 2zC + (ix+ |z|2)SS

1 + 2zC + (ix+ |z|2)S

=

(

(C + zS)(1− 2zC − ixS)S(1− 2zC − ixS)

)

=

(

C − 2zC2 − ixCS + zSS − ixS2 − 2zCS

)

. (5.31)

With ε0 = 0, z = 14 (ε3 − iε4) and x = 1

4ε1 we recognize the isometries from item 1 above. Thisimplies that the epsilons in item 1 must be taken as infinitesimal quantities contrary to the α, βand γ in item 3 which are finite, non-infinitesimal quantities. At least, this is so for ε1, ε3 and ε4,we have not yet found the ε0 isometry in our actions. Assume for the moment that ε0 from item1 must be taken to be infinitesimal. This means the Lie group isometry, i.e. with ε0 finite, is:

S → ε0S

C → √ε0C. (5.32)

We recognize this scale transformation as corresponding to the action of the diagonal matrices incorollary 5.4.1 with θ = 0 and r = 1√

ε0. Finally we see that the isometry in item 2 above is only a

small part of a larger isometry. Namely if we take in the action of the diagonal matrices in corollary5.4.1 r = 1 we get the isometry that changes C into e−3iθC and leaves S invariant. We see indeedthat this is an isometry and we get the special case from item 2 if we take θ to be either in −π

2 +2πZ

(in which case C → Im(C) − iRe(C)) or in π2 + 2πZ (in which case C → −Im(C) + iRe(C)). As

an extra check we can use lemma 5.3.2 to see that if we take the epsilons to be infinitesimalquantities and the α, β, γ and θ to be finite, then inserting the transformations above into theKahler potential from definition 3.4.13 shows that these transformations are indeed isometries ofthe metric (3.79). Obviously this had to be the case, because of theorem 5.3.3.

1These isometries are related to the Peccei-Quinn isometries that are discussed in e.g. [24] and [50].2It is expected in e.g. [7] that a discrete subgroup of these isometries of the unipotent upper triangular matrices,

or equivalently of the Heisenberg group, remains a symmetry group in the quantum theory. In [1] and [50] it isexpected that the Heisenberg symmetries are unmodified in string perturbation theory.

55

5.5 Special boundary points

Some of the subgroups described in lemma 4.2.11 can be recognized in the stabilizer groups ofsome special points on the boundary of B from theorem 5.2.1. First remember what complexprojective spaces are.

Definition 5.5.1 (Complex projective spaces) For every n ∈ N the complex projective spaceCPn is defined as the set of lines through the origin in C

n+1.

CPn =(

Cn+1 − 0

)

/ ∼, (5.33)

where the equivalence relation is defined as z ∼ w, for z, w ∈ Cn+1 − 0 if and only if w = az,for some non-zero complex a.

Now let’s make explicit what we mean by the boundary of B.

Definition 5.5.2 (The boundary of B) Let B and the bilinear form 〈., .〉 be as defined in the-orem 5.2.1. Then define the boundary of B as follows:

∂B := ∂P/C∗ ⊂ CP 2 where:

∂P :=

z ∈ C3| 〈z, z〉 = 0, z 6= 0

. (5.34)

Now note that the two points in the following definition are boundary points.

Definition 5.5.3 (Two boundary points of B) Define the boundary point ∞ ∈ ∂B to be theequivalence class in ∂B of the point ∞ ∈ ∂P and define the boundary point O ∈ ∂B to be theequivalence class in ∂B of the point o ∈ ∂P where:

∞ :=

100

;

o :=

001

. (5.35)

Definition 5.5.4 (Action of SU(1, 2) on ∂B) Let ∂P and ∂B as in definition 5.5.2, then wedefine an action of SU(1, 2) on ∂B as follows. Let b ∈ ∂B be an equivalence class in ∂B and letp ∈ ∂P be in the equivalence class b. Let A ∈ SU(1, 2), then the action is defined in the followingway:

if Ap = p where the action is ordinary matrix multiplication, then:

Ab = b where b is the equivalence class of p in ∂B. (5.36)

This is well defined since the action of C∗ and matrix multiplication commute.

Lemma 5.5.5 (The stabilizer groups of ∞ and O) Let ∞ and O be as in definition 5.5.3.Then the stabilizer group of∞ is the group consisting of products of matrices XY where X ∈ D andY ∈ U , with D the group of diagonal matrices from lemma 4.2.11 and U the group of the unipotentupper triangular matrices from lemma 4.2.11 The stabilizer group of O consists of products XZwith again X ∈ D and Z ∈ L, where L is the group of the unipotent lower triangular matricesfrom lemma 4.2.11.

Proof: Let A ∈ SU(1, 2) be an element of the stabilizer group of ∞, then according to definition5.5.4 we have in the notation of corollary 4.2.10 that:

A∞ =

a11

a21

a31

= c∞, (5.37)

56

for some c ∈ C∗. This leads to a11 = c and a21 = a31 = 0. Inserting this in the conditions ofcorollary 4.2.10) leads to the following exra conditions on A:

a32 = 0;

−a12a33 + 2a22a23 − a32a13 = 0;

−a13a33 + 2|a23|2 − a33a13 = 0;

ca33 = 1;

−a12a32 + 2|a22|2 − a32a12 = 2;

detA = 1. (5.38)

It is easily checked that the matrix:

A =

c 0 00 c

c0

0 0 1c

1 2z |z|2 + ix0 1 z0 0 1

, (5.39)

with x ∈ R and z ∈ C satisfies these conditions, stabilizes ∞ and is of the form as stated in thelemma.

An argument along the same lines shows that if A ∈ SU(1, 2) is in the stabilizer group of O,then we have for d ∈ C∗:

a13 = 0;

a23 = 0;

a33 = d;

−a11a31 + 2|a21|2 − a31a11 = 0;

−a11a32 + 2a21a22 − a31a12 = 0;

a12 = 0;

da11 = 1;

−a12a32 + 2|a22|2 − a32a12 = 2;

detA = 1. (5.40)

Again we can check that the matrix:

A =

1d

0 0

0 dd

00 0 d

1 0 0z 1 0

|z|2 + ix 2z 1

, (5.41)

with x ∈ R and z ∈ C satisfies the necessary conditions, stabilizes O and is of the form as statedin the lemma. 2

As the next lemma shows the group consisting of matrices XY with X ∈ D and Y ∈ U (inthe notation of lemma 5.5.5) is actually the group of all upper triangular matrices in SU(1, 2).

Lemma 5.5.6 (All upper triangular matrices in SU(1, 2)) Use the notation of lemma 4.2.11.Then the group consisting of all matrices of the form XY where X ∈ D and Y ∈ U is the groupof all upper triangular matrices. Call this group Υ.

Proof: It is trivial to see that matrices of the form XY as described in the lemma are uppertriangular matrices and we know from lemma 4.2.11 that they are elements of SU(1, 2) so what isleft to prove is that if a matrix A ∈ SU(1, 2) in the representation from lemma 4.2.9 (with entriesaij) is upper triangular, then it is of the desired form. That A is upper triangular gives thata21 = a31 = a32 = 0. Inserting this into the conditions from corollary 4.2.10 gives the following

57

set of equations:

−a12a33 + 2a22a23 = 0

−a13a33 + 2|a23|2 − a33a13 = 0

−a11a33 = −1

|a22|2 = 1

detA = 1. (5.42)

The last three equations show that a11 = reiθ, a22 = e−2iθ and a33 = r−1eiθ, for r, θ ∈ R, r > 0.Now using the first two equations gives the desired results for a12, a13 and a23. 2

This lemma has a direct result that is worth mentioning and that will be used in section 5.7.

Corollary 5.5.7 (Upper triangular matrices as stabilizer) In the notation of lemmas 5.5.5and 5.5.6, Υ is the stabilizer group of ∞.

Proof: This follows directly from lemmas 5.5.5 and 5.5.6. 2

5.6 The complex upper half plane

In this section I will show that if we restrict ourselves to certain subsets of the set B from theorem5.2.1 we are dealing with the complex upper half-plane. To do this I will begin by defining somenew sets.

Definition 5.6.1 (C3 and disjoint unions) Let 〈., .〉 be the bilinear form from theorem 5.2.1,let P and B be the sets defined in the same theorem and let ∂P and ∂B be as defined in definition5.5.2. The set C3 is the union of the following four disjoint sets:

0 ; (5.43)

P− := P =

z ∈ C3| 〈z, z〉 < 0

;

P 0 := ∂P =

z ∈ C3| 〈z, z〉 = 0, z 6= 0

;

P+ :=

z ∈ C3| 〈z, z〉 > 0

. (5.44)

It follows that the complex projective plane CP 2 is a disjoint union of the following three sets:

B− := B = P/C∗ = P−/C∗,

B0 := ∂B = P 0/C∗,

B+ := P+/C∗. (5.45)

From theorem 5.2.1 we remember that SU(1, 2) acts transitively on B and that the stabilizerof a point in B is isomorphic to U(2). In an analogous way it can be seen that SU(1, 2) alsoactss transitively on B+ and the stabilizer of a point in B+ is isomophic to U(1, 1). From thediscussion in section 5.5 we deduce that the points ∞ and o are elements of P 0. This means thatfor SU(1, 2) to act transitively on B0 there must be, for every point z ∈ B0, an A ∈ SU(1, 2) suchthat A∞ = z. This is in accordance with the condition of equation (4.15). Note that if I wouldhave computed Ao = z I would have found condition (4.18). So SU(1, 2) acts transitively on B0

and from lemma 5.5.5 we see that the stabilizer of a point is isomorphic to the group of matricesthat are the product of a diagonal SU(1, 2) matrix and an unipotent upper (or equivalently lower)triangular matrix.

Definition 5.6.2 (Complex upper half-plane) The complex upper half-plane is defined as:

H := z ∈ C| Imz > 0 . (5.46)

Some more definitions of sets are needed before we can turn to the interesting results.

58

Definition 5.6.3 (Some more sets) Let w ∈ C3, w 6= 0 and let [w] be the equivalence class ofw in CP 2. Then define:

L[w] :=

z ∈ C3| 〈z, w〉 = 0

. (5.47)

Note that this definition indeed depends only on the equivalence class of w. Now for [w] ∈ B+

define:H[w] :=

(

L[w] ∩ P)

/C∗. (5.48)

Note that H[w] is a non-empty real two dimensional subset of B.

Since, for [w] ∈ B+, H[w] is a subset of B we can restrict the action of g ∈ SU(1, 2) toH[w]. Thisgives an isometry from H[w] to Hg[w] (remember that for x, y ∈ C we have that 〈gx, gy〉 = 〈x, y〉).Since furtermore SU(1, 2) acts transitively on B we can understand the geometry of a general H[w]

up to isometry by focussing on the special case of H[w]0 , where [w]0 is the equivalence class in B+

of w0 ∈ P+, with:

w0 :=

010

. (5.49)

Lemma 5.6.4 (H[w]0 is isomorphic to the upper half-plane) In the notation of definitions5.6.2 and 5.6.3 we have that H[w]0

∼= H, with [w]0 as defined above. The isomorphism is anisomorphism between manifolds, so a bijection.

Proof: We have that:

H[w]0 =

(

SC

)

∈ P | C = 0

= τ ∈ C| Imτ > 0 , (5.50)

where the second identity follows via the isomorphism

(

S0

)

7→ −iτ . After all, since

(

S0

)

∈ P

we have that S + S = 2ReS > 0. 2

Lemma 5.6.5 (The stabilizer of [w]0 in SU(1, 2)) Let [w]0 be as defined above, then its stabi-lizer in SU(1, 2) in the representation of lemma 4.2.9 consists of the matrices of the form:

a 0 −ib0 1 0ic 0 d

, (5.51)

with a, b, c, d ∈ R and ad− bc = 1.

Proof: Demanding that a matrix A stabilizes [w]0 ∈ B+ gives us the following form for A:

a11 0 a13

a21 a22 a23

a31 0 a33

. (5.52)

If A ∈ SU(1, 2) in the representation of lemma 4.2.9, then ATV A = V and detA = 1. We canrewrite thislatter equation as V −1ATV = A−1 since V is invertible. With detA = 1 we can write:

A−1 =

a22a33 − a23a32 a13a32 − a12a33 a12a23 − a22a13

a23a31 − a21a33 a11a33 − a13a31 a21a13 − a11a23

a32a21 − a22a31 a31a12 − a11a32 a11a22 − a12a21

. (5.53)

Using equation (5.52) then gives:

a33 −2a23 a13

0 a22 0a31 −2a21 a11

=

a22a33 0 −a22a13

a23a31 − a21a33 a11a33 − a13a31 a21a13 − a11a23

−a22a31 0 a11a22

. (5.54)

59

This shows a21 = a23 = 0 and a22 = a11a33 − a13a31 and thus:(

a33 a13

a31 a11

)

= a11a33 − a13a31

(

a33 −a13

−a31 a11

)

. (5.55)

If we take the determinant of both sides of this equation, we get that a22 = a11a33 − a13a31 = 1and thus:

(

a33 a13

a31 a11

)

=

(

a33 −a13

−a31 a11

)

. (5.56)

This leads for A to the form as stated in the lemma. 2

Corollary 5.6.6 (Fractional linear transformations on H) Use the notation from lemma 5.6.4.Since the matrices as in lemma 5.6.5 stabilize [w]0 they act as internal transformations on H[w]0 =H. This action is given by the following fractional linear transformation (with τ ∈ H):

a 0 −ib0 1 0ic 0 d

τ =aτ + b

cτ + d. (5.57)

Proof: Writing out the fractional linear transformation we get:

a 0 −ib0 1 0ic 0 d

(

S0

)

=

(

aS−ibicS+d

0

)

, (5.58)

where

(

S0

)

∈ H[w]0 . With

(

S0

)

7→ −iτ , where τ ∈ H as in the proof of lemma 5.6.4, we find

that for τ ∈ H we have:

a 0 −ib0 1 0ic 0 d

τ =

(

iaS−ibicS+d

0

)

=aτ + b

cτ + d. (5.59)

2

This is the well known action of SL(2,R) on H (with τ ∈ H):

(

a bc d

)

τ =aτ + b

cτ + d. (5.60)

More information on this transformation can be found in e.g. [17] or [26].

Lemma 5.6.7 (The upper half plane is isomorphic to SL(2,R)/SO(2)) Use the notation ofdefinitions 5.6.2 and 5.6.3. Then:

H ∼= H[w]0∼= SL(2,R)/SO(2), (5.61)

where the isomorphism are between manifolds, so they are bijections.

Proof: From:(

a b0 1

)

i = ai+ b, (5.62)

with a, b ∈ R, we see that SL(2,R) acts transitively on H . If we calculate the stabilizer of i ∈ Hwe find:

ai+ b = i(ci+ d) = −c+ di, (5.63)

thus a = d and b = −c, with a, b, c, d ∈ R. We find thus as stabilizer of i:(

a −cc a

)

, (5.64)

60

with a2 + c2 = 1. These are exactly the elements of SO(2) ∼= U(1) (this isomorphism is of course(a, c) 7→ ai+ c. 2

Remember that we found the complex upper half plane by setting C = 0. Translating this intothe coordinates of the metric (3.77) we get χ = ϕ = 0. Inserting this into that metric we are leftwith a metric on the upper half plane, namely:

ds2 = dφ2 + e4φdσ2. (5.65)

This metric is in accordance with the following well known metric on the upper half plane (seee.g. [17]):

ds2 =dτdτ

(τ − τ )2 . (5.66)

We see that this agrees with equation (5.65) in the following way. From corollary 3.5.3 we see thatif C = 0 then S = e−2φ− 2iσ. With τ = iS we then find that τ = 2σ+ ie−2φ if C = 0. Using thistranslation the metrics in equations (5.65) and (5.66) can be seen to be equivalent.

This same metric we find even if we just take χ and ϕ constant, not necessarily zero or evenequal to each other. So restricting ourselves to subspaces of SU(1, 2)/U(2) given by constant χand ϕ we find copies of the complex upper half plane H . We are left with a similar metric if weset ϕ and σ equal to a constant:

ds2 = dφ2 + e2φdχ2. (5.67)

This is probably what is meant in [1], [14] and [15] by the statement that φ and χ paramatrizeSL(2,R)/O(2), although we are dealing with SL(2,R)/SO(2) as shown in lemma 5.6.7. Note thatif we restrict χ and σ it does make a difference if we set both equal to zero:

ds2 = dφ2 + e2φdϕ2, (5.68)

or if we set them equal to some constants:

ds2 = dφ2 + e2φdϕ2 + e4φχ2dϕ2. (5.69)

5.7 The dilaton and the Heisenberg group

In this section I will find a new interesting way to look at SU(1, 2)/U(2) using the (S,C)-coordinates. The Heisenberg group or group of unipotent diagonal matrices in SU(1, 2) (U fromlemma 4.2.11) will play an important role in this section as will the dilaton φ, which is defined interms of the S and C coordinates as:

φ := −1

2log

(

S + S − 2CC

2

)

, (5.70)

as can be seen in definition 3.4.13. All through this section I will be using the notation that wasintroduced in section 5.2 and definition 5.6.1 and when I am talking about matrices in SU(1, 2) Iwill mean in the representation from lemma 4.2.9. Furthermore, remember from lemma 5.5.6 thatΥ was the subgroup of all upper triangular matrices in SU(1, 2).

Theorem 5.7.1 (Simple transitivity) For every point

(

SC

)

∈ B there is a unique upper

triangular matrix γS,C with positive real diagonal entries which maps the point

(

10

)

to

(

SC

)

.

This is called simple transitivity. The matrix γS,C is given by:

γS,C :=

1 2C 12 (S − S) + CC

0 1 C0 0 1

e−φ 0 00 1 00 0 eφ

, (5.71)

where φ is the dilaton from equation (5.70).

61

Proof: Let

(

SC

)

∈ B. That the matrix γS,C indeed maps

(

10

)

to

(

SC

)

is easily checked

by working out the fractional linear transformation by which γS,C acts. We still have to prove

uniqueness, so assume that we have two matrices A (with entries aij) and B (with entries bij), both

with the desired properties. Since both map

(

10

)

to

(

SC

)

we find the condition (remember

that we can set the entries a21, a31 and a32 to zero and similarly for B):

1

a33

(

a11 + a13

a23

)

=1

b33

(

b11 + b13b23

)

=

(

SC

)

. (5.72)

From lemma 5.5.6 we know that we can write A as:

A =

reiθ 0 00 e−2iθ 00 0 1

reiθ

1 2z ix+ |z|20 1 z0 0 1

, (5.73)

where r, θ, x ∈ R, r > 0 and z ∈ C. Since A has positive real diagonal entries we can set θ = 0. Wecan write a similar expression for B, where I will use s instead of r, y instead of x and w insteadof z to distinguish A and B. Inserting this into equation (5.72) gives:

r2(1 + |z|2 + ix) = s2(1 + |w|2 + iy) = Srz = sw = C.

(5.74)

Inserting the absolute values squared of the second line into the first we find:

r2(1 + ix) = s2(1 + iy) = S − CC. (5.75)

Thus r = s and x = y. Inserting this back into equation (5.74) we find z = w and thus A = B.2

This theorem has some interesting consequences. First note that γS,C is the product of twomatrices: γS,C = hS,CdS,C , where hS,C is an element of the Heisenberg group and dS,C is adiagonal matrix in SU(1, 2) with, in the notation of lemma 4.2.11, r = e−φ and θ = 0. Moreover,

when we vary

(

SC

)

over all of B we see that hS,C varies over all of the Heisenberg group and

dS,C varies over all diagonal matrices in SU(1, 2) with postive real diagonal entries. This means

that as soon as we choose a point of origin in B, like we did here by choosing

(

10

)

, we can

identify every point in B with the unique matrix γS,C which sends the origin to that point andin doing so we get the whole set of upper triangular matrices with positive real diagonal elementsin SU(1, 2) (call this set Υ>0). This also means we can define a group operation now on B bydefining:

(

SC

)

·(

S′

C ′

)

=

(

S′′

C ′′

)

, (5.76)

if:γS,CγS′,C′ = γS′′,C′′ . (5.77)

Checking that this indeed turns B into a group is the same as checking that Υ>0 is a subgroup ofΥ with matrix multiplication as group operation (since we already know that Υ is a group undermatrix multiplication). This is the case, since Υ>0 is closed under matrix multiplication as we seefrom writing γS,CγS′,C′ = γS′′,C′′ = hS′′,C′′dS′′,C′′ , with:

hS′′,C′′ =

1 2(C + C ′e−φ) `0 1 C + C ′e−φ

0 0 1

dS′′,C′′ =

e−(φ+φ′) 0 00 1 0

0 0 e(φ+φ′)

, (5.78)

62

where φ′ is the dilaton corresponding to the (S ′, C ′)-coordinates and I wrote for notational con-venience:

` :=1

2

(

S − S + (S′ − S′)e−2φ)

+ (CC ′ − CC ′)e−φ + (C + C ′eφ)(C + C ′eφ). (5.79)

We have now turned B and thus SU(1, 2)/U(2) into a group by choosing an origin!We have seen that Υ>0 acts simply transitively on B and there is only one orbit for Υ>0.

Futhermore, since we can identify every point in B with exactly one γS,C we can use these matricesas coordinates on B and in this way we see that the factor hS,C corresponds to fibering B by orbitsof the Heisenberg group, exactly as the fibering of Υ>0 by the cosets of the Heisenberg group.

We see furthermore that hS,C depends on C, C and 12 (S − S) = ImS, whereas dS,C depends

on φ, which in turn depends on 12 (S + S) − CC = ReS − CC. This means we can look at the

situation where C and ImS are fixed and φ runs from −∞ to∞ (this∞ is not the boundary pointfrom definition 5.5.3; where confusion can arise I will always say if I mean the boundary point orinfinity in the “normal” mathematical sense). Now look at the curve in B which is described by

cφ := γS,C

101

if φ runs (note that we still have to divide out the C∗ to get a point in B for

every φ). First we take a look at the limit φ→ −∞. Since:

cφ =

e−φ +

12 (S − S) + CC

eφ

eφCeφ

∈ C3, (5.80)

we can still divide out the action of C∗ to arrive at

1 +

12 (S − S) + CC

e2φ

e2φCe2φ

which gives in

the limit φ → −∞:

100

. We see that this gives the boundary point ∞ ∈ ∂B from definition

5.5.3.For the limiting case φ → ∞ we use the action of C∗ to normalize the third coordinate:

e−2φ + 12 (S − S) + CCC1

. This gives in the limit φ → ∞ the point

12 (S − S) + CC

C1

,

which can easily be checked to give a boundary point. Furthermore, as is easily checked, everyboundary point except the aforementioned∞ is of this form. We thus have that for every boundarypoint not equal to ∞ we can construct a curve in B which runs from the boundary point ∞ tothat particular boundary point. Moreover, the dilaton φ, or better eφ, is the natural coordinatealong these lines. It is tempting to call these lines geodesics in analogoy with the situation on thetwo dimensional complex disc, but since I have not proven here that these curves are geodesic Iwill not use this terminology.

The orbits of the Heisenberg group in B (i.e. let C and ImS run and keep φ fixed) intersecteach of these lines exactly once (since on these lines φ runs and C and ImS are fixed). We also seethat these lines provide a natural bijection between the Heisenberg group and the boundary minus∞, ∂B\∞. The full boundary ∂B is a topological 3-sphere (this will become clear when we lookat the boundary in the Bergman coordinates from section 5.8, only not using the Bergman metricthat will be introduced in that same section, but the Euclidean metric the Bergman ball inheritsfrom the four dimensional space in which it is embedded). Thus the geometrical picture we havehere is that of a 1-parameter family of 3-spheres through the boundary point∞, but without thatsame point. The parameter is of course φ and in the limit φ → −∞ the 3-sphere shrinks to theboundary point ∞ where in the limit φ → ∞ the 3-sphere expands to ∂B\∞. We see indeedfrom lemma 5.4.2 that φ is constant on each 3-sphere, since the Heisenberg group leaves K, andthus also φ, invariant. A two dimensional representation of this geometrical situation (an artistimpression if you like) is shown on the cover of this paper.

63

It is also interesting to see what the matrix γS,C

does to the dilaton φ. In other words, how

does the function φ(S,C) change under the action of γS,C

. Since the Heisenberg group leaves φinvariant, we only have to calculate the effect of d

S,Con the dilaton:

φ((γS,C

(S,C)T )T ) = φ((dS,C

(S,C)T )T )

= −1

2log

(

e−2φ(S,C)(S + S − 2CC)

2

)

= φ(S, C) + φ(S,C). (5.81)

We see that the dilaton is shifted with exactly the value it has in the point

(

S

C

)

.

We can now also take a look at the Maurer-Cartan form. First I will define this form andexplain why we want to study it.

Definition 5.7.2 (Maurer-Cartan form) Let G be a Lie group with Lie algebra g. Let Eiibe a basis of g. Let, for every g ∈ G, Xg

i i be a basis of TgG and let, for every g ∈ G, σigi be abasis of T ∗

gG dual to Xgi i. Then, for every g ∈ G, the Maurer-Cartan 1-form on TgG is defined

as:Ωg := Ei ⊗ σig . (5.82)

If it is clear on which tangent space Ωg acts we write Ω.

Let’s take a closer look at the Maurer-Cartan form and what it does. If we have a g ∈ G andwe take a vector V ∈ TgG, then we can write V = V iXg

i . Now:

Ω(V ) = Eiσig(V ) = Eiσ

ig(V

iXgi ) = EiV

i. (5.83)

So we see that Ω takes a vector V from the tangent space at g and creates from it a vector inthe tangent space at the unit element, i.e. in g, that has the same coefficients with respect to thechosen basis of the Lie algebra as V had with respect to the chosen basis of TgG.

Notation: Sometimes the notation g−1dg is used for Ω. This notation comes from Cartanand can be explained as follows. If X is a manifold and p ∈ X , then Cartan would write:

dp :=∂

∂xi⊗ dxi =

∂

∂xi⊗ δijdxj . (5.84)

So dp takes a vector from TpX into itself. Furthermore he would write:

g−1 := (Lg−1)∗, (5.85)

where (Lg−1)∗ is the differential map of the left translation. So by writing:

Ω = g−1dg = (Lg−1)∗ dg, (5.86)

we have a map that first takes a vector at g ∈ G into itself and then left translates it to g. ♦We see that in our case at hand since we are able to get a group structure once we have chosen

an origin, we also have a Maurer-Cartan form (where we must distinguish the operator d and thematrix dS,C)3:

−γ−1S,CdγS,C = −d−1

S,Ch−1S,C ((dhS,C)dS,C + hS,C(ddS,C))

= −d−1S,C(ddS,C)− d−1

S,Ch−1S,C(dhS,C)dS,C (5.87)

= −

−dφ 2eφdC e2φ(

12

(

dS − dS)

+ CdC − CdC)

0 0 eφdC0 0 dφ

.

3The reason for the −-sign in front of γ−1S,C

dγS,C is that the connection with connection matrix −γ−1S,C

dγS,C has

zero curvature, as will be shown in lemma 5.7.6.

64

Notation: If A and B are matrices with 1-forms aij and bkl as entries, then A∧B is the matrixwith as entries the 2-forms (A ∧ B)ik :=

∑

j aij ∧ bjk. This can be straightforwardly generalizedfor matrices with higher order forms as entries. ♦

Remark: If Ω is the Maurer-Cartan form, then it satisfies the Maurer-Cartan equation dΩ−Ω ∧ Ω = 0 (see e.g. [22] or [28]). In our situation with Ω = −γ−1

S,CdγS,C we calculate that thisholds indeed.

For a better understanding of this Maurer-Cartan form let me give a basis for the Lie algebraof Υ>0. This Lie algebra I will denote by LieΥ>0.

Lemma 5.7.3 (Basis for Lie algebra of Υ>0) Define:

X+ :=

0 2 00 0 10 0 0

, X− :=

0 −2i 00 0 i0 0 0

,

X0 :=

1 0 00 0 00 0 −1

, Y :=

0 0 2i0 0 00 0 0

.

(5.88)

Then these matrices form a basis of the Lie algebra of Υ>0. They satisfy the following commutationrelations:

[X+, X−] = 2Y, [X+, Y ] = [X−, Y ] = 0,[X0, X+] = X+, [X0, X−] = X−, [X0, Y ] = 2Y.

(5.89)

The matrices X+, X− and Y generate the Heisenberg Lie algebra (i.e. form a basis for that Liealgebra).

Proof: From lemma 4.5.4 we see that the given matrices are in su(1, 2). If we write down a matrixA = I3 + εB and demand that A is upper triangular with real positive diagonal entries, this givesconditions on B which the given matrices also satisfy. The commutation relations can now becalculated and from these we see that X+, X− and Y are a basis for the Heisenberg Lie algebra.2

I now define two 1-forms, following [50].

Definition 5.7.4 (Two complex 1-forms) Define the following complex 1-forms:

u = eφ (5.90)

v = e2φ(

1

2dS − CdC

)

. (5.91)

Remembering equation (5.70), we note that:

dφ = −v + v

2. (5.92)

Now going back to the Maurer-Cartan form in equation (5.87) we see that:

−γ−1S,CdγS,C = −v + v

2X0 −

u+ u

2X+ −

u− u2i

X− −v − v

2iY. (5.93)

We also see that all the coefficients in front of the matrices are real, so the Maurer-Cartanform is in the Lie algebra of Υ>0. We can rewrite equation (5.93) as:

−γ−1S,CdγS,C = v

−X0 + iY

2+ v−X0 − iY

2+ u−X+ + iX−

2− uX+ + iX−

2. (5.94)

Remark: Note that in equation (5.94) we recognize the vierbein from [50]:

uvv−u

. (5.95)

65

This gives rise to some mappings, but first I will introduce some more terminology and notation.

Definition 5.7.5 (Adjoint representation) Let g be a Lie algebra equipped with Lie brackets[., .], then the map ad is defined as follows

ad : g→ End(g) : X 7→ [X, .], (5.96)

where End(g) are the endomorphisms on g. In other words (adX)Y = [X,Y ] for X,Y ∈ g. Thismap is called the adjoint representation of a Lie algebra.

Lemma 5.7.6 (Curvature of the Maurer-Cartan connection) Define the connection derivedfrom the Maurer-Cartan form:

∇ : TB → LieΥ>0, (5.97)

with, for every section ξ of the tangent bundle TB:

∇ξ := −v(ξ) + v(ξ)

2X0 −

u(ξ) + u(ξ)

2− u(ξ)− u(ξ)

2i− v(ξ)− v(ξ)

2i. (5.98)

Then this is a globally defined LieΥ>0-valued map on the tangent bundle TB of B which is fibrewiseR-linear.

The alternating bilinear map:

2⊗

TB → LieΥ>0 : (ξ, η) 7→ ∇[ξ,η], (5.99)

corresponds with the following matrix of 2-forms:

−dv + dv

2X0 −

du+ du

2X+ −

du− du2i

X− −dv − dv

2iY. (5.100)

Furthermore the alternating bilinear map:

2⊗

TB → LieΥ>0 : (ξ, η) 7→ ∇ξ ∇η −∇η ∇ξ , (5.101)

corresponds to the following matrix of 2-forms:

1

4(v + v) ∧ (u+ u)[X0, X+] +

1

4i(v + v) ∧ (u− u)[X0, X−]

+1

4i(v + v) ∧ (v − v)[X0, Y ] +

1

4i(u+ u) ∧ (u− u)[X+, X−]

+1

4i(u+ u) ∧ (v − v)[X+, Y ] − 1

4(u− u) ∧ (v − v)[X−, Y ]. (5.102)

We also have the relations:

dv = v ∧ v + u ∧ u,

du =1

2u ∧ (v + v). (5.103)

Finally we have that the curvature form of the connection, that is the alternating bilinear map:

2⊗

TB → LieΥ>0 : (ξ, η) 7→ ∇[ξ,η] − (∇ξ ∇η −∇η ∇ξ) , (5.104)

vanishes. Formulated differently:

∇[ξ,η] = ∇ξ ∇η −∇η ∇ξ, (5.105)

and thus the connection ∇ derived from the Maurer-Cartan form is a homomorphism of Lie alge-bras.

66

Proof: Since the 1-forms u, u, v and v are globally defined, fibrewise R-linear, C-valued mapson the tangent bundle of B, equation (5.97) gives a globally defined LieΥ>0-valued map on thetangent bundle TB of B which is fibrewise R-linear. Because for a C∞ 1-form ω, viewed as linearmap TB → R, the 2-form dω is, by definition, the alternating bilinear map:

2⊗

TB → R, dω(ξ, η) := ω([ξ, η]), (5.106)

equations (5.99) and (5.100) follow. Furthermore since, for two C∞ 1-forms ω1, ω2, viewed aslinear maps TB → R, the 2-form ω1 ∧ ω2 is, by definition, the alternating bilinear map:

2⊗

TB → R, ω1 ∧ ω2(ξ, η) := ω1(ξ)ω2(η) − ω1(η)ω2(ξ), (5.107)

equations (5.101) and (5.102) follow. The relations in equation (5.103) follow through straight-forward calculation. Finally subtracting the matrix from equation (5.102) from that in equation(5.100) and rewriting the result with help of definition 5.7.5, I get for the curvature 2-form:

−dv + dv

2X0 −

du+ du

2X+ −

du− du2i

X− −dv − dv

2iY

−1

2

(

v + v

2adX0 +

u+ u

2adX+ +

u− u2i

adX− +v − v

2iadY

)

∧(

v + v

2X0 +

u+ u

2X+ +

u− u2i

X− +v − v

2iY

)

. (5.108)

Writing this out with respect to the basis (X0, X+, X−, Y ) of LieΥ>0 using the commutationrelations from lemma 5.7.3 we get:

−1

2

(dv + dv)(du+ du)−i(du− du)−i(dv − dv)

(5.109)

−1

8

0 0 0 0−(u+ u) (v + v) 0i(u− u) 0 (v + v) 02i(v − v) 2i ∗ u− baru) 2(u+ u) 2(v + v)

∧

(v + v)(u+ u)−i(u− u)−i(v − v)

.

Note that this formula has the same format as formulas (5.9) and (5.10) from [50], but it is notthe same. To show how this compuation goes, I will calculate one term:

−1

8(v + v) adX0 ∧ (u+ u)X+ = −1

8(v + v) ∧ (u+ u)(adX0)X+

= −1

8(v + v) ∧ (u+ u)X+. (5.110)

If we now insert the relations (5.103) into equation (5.109) we see that the curvature vanishes. 2

Remark: I will use ∇ from lemma 5.7.6 as a trivilization of the tangent bundle (i.e. a globalcoordinatization of the tangent bundle).

5.8 The Bergman metric

In this section I will discuss another geometrical point of view from which we can study SU(1, 2)/U(2).I will begin by introducing the two-complexdimensional open ball with the Bergman metric (seealso e.g. [36] and [37]). Then I will show that as a set SU(1, 2)/U(2) is isomorphic to this openball and that the Bergman metric has a connection to the metric of equation (3.79).

67

Definition 5.8.1 (The Bergman metric on the open ball) Define the open ball B4 as fol-lows:

B4 :=

z ∈ C2| |z1|2 + |z2|2 < 1

. (5.111)

Then on this ball we can define a Kahler potential:

KB := −log(1− |z1|2 − |z2|2). (5.112)

Note that this makes sense since 0 < 1− |z1|2 − |z2|2 ≤ 1. The metric we get from this is:

ds2 =dz1dz1 + dz2dz21− |z1|2 − |z2|2

+(z1dz1 + z2dz2)(z1dz1 + z2dz2)

(1− |z1|2 − |z2|2)2. (5.113)

as can be shown by straightforward calculation as explained in corollary 2.2.10. This metric isthus a Kahler metric. This metric is called the Bergman metric.

Remark: Note that instead of the Bergman metric we can also use the Euclidean metric thatB4 inherits from the C2 in which it is embedded: ds2 = dz1dz1 + dz2dz2.

Now I will show that as a set the open ballB4 from definition 5.8.1 is isomorphic to SU(1, 2)/U(2).Before I come to that result however I will first give a theorem which is analogous to theorem5.2.1, only here I will use the SU(1, 2) representation from lemma 4.2.12.

Theorem 5.8.2 (Another geometrical representation of SU(1, 2)/U(2))Define for v, w ∈ C3 the following inner product: 〈v, w〉 := v1w1 + v2w2− v3w3 and define the set:

N− := z ∈ C3| 〈z, z〉 < 0. (5.114)

Then:SU(1, 2)/U(2) ∼= N−/C

∗, (5.115)

where the isomorphism is an isomorphism of manifolds, thus a bijection.

Proof: Define for c ∈ R:

Nc := z ∈ C3||z1|2 + |z2|2 − |z3|2 = c. (5.116)

Notice that N− =⋃

c<0Nc. Futhermore since elements of SU(1, 2) in the representation of lemma4.2.12 preserve the bilinear form W defined in the proof of lemma 4.2.9, they preserve the levelsets Nc in particular. We thus see that the action of SU(1, 2) on Nc is an internal tranformationof Nc for every c < 0 and so the action on N− is also an internal transformation. What we haveto prove according to theorem 4.6.10 is that this action is transitive and that the isotropy groupof a point in N− is isomorphic to U(2). This is proven in an exactly analogous way to theorem5.2.1, where I use the change of coordinates given by matrix Γ from equation (4.37). 2

Now I can give the following lemma which shows the relation between the open ball B4 andSU(1, 2)/U(2).

Lemma 5.8.3 (A relation between B4 and SU(1, 2)/U(2)) We have that

SU(1, 2)/U(2) ∼= B4, (5.117)

where the isomorphism is an isomorphism of manifolds, thus a bijection.

Proof: From theorem 5.8.2 we see that SU(1, 2)/U(2) is isomorphic to:

z ∈ C3| |z1|2 + |z2|2 − |z3|2 < 0

/C∗, (5.118)

in the aforementioned sense. Since we see that z3 cannot be zero, we can use the action of C∗ thathas to be divided out, to divide the coordinates by z1:

Z1 :=z1z3

(5.119)

Z2 :=z2z3. (5.120)

68

Inserting this in the condition on the coordinates, we get:

|Z1|2 + |Z2|2 < 1. (5.121)

2

We have seen that the open ball and SU(1, 2)/U(2) are isomorphic. Now we will see that underthe proper coordinate transformations the Bergman metric turns into the metric from equation(3.79) of which we have shown in section 5.3 that it is a metric on SU(1, 2)/U(2).

Lemma 5.8.4 (The Bergman metric and metric (3.79)) Let the notation be as in defini-tion 5.8.1. Then changing the coordinates as follows:

z1 =1− S1 + S

, (5.122)

z2 =2C

1 + S, (5.123)

and inserting this into the Bergman metric leads to the metric (3.79). The transformation ofvariables from z1 and z2 to S and C is given by:

S =1− z11 + z1

, (5.124)

C =z2

1 + z1. (5.125)

Proof: For the proof of this statement I will invoke lemma 5.3.2. That is, I will show that underthis holomorphic change of coordinates the Kahler potential KB will change into the Kahlerpotential K from definition 3.4.13 plus a holomorphic function plus its complex conjugate:

1− |z1|2 − |z2|2 7→ 1− 1− (S + S) + |S|21 + |1 + S|2 − 4|C|2

1 + |1 + S|2

=2(S + S − 2CC)

|1 + S|2 . (5.126)

Thus:

KB 7→ K + log

(

1 + S√2

)

+ log

(

1 + S√2

)

. (5.127)

Also note from equation (5.126) that the condition 1 − |z1|2 − |z2|2 > 0 on the z-coordinates isequivalent to the condition S + S − 2CC > 0 on the (S,C)-coordinates.

The transformation from z1 and z2 to S and C can be calculated directly from equations(5.122) and (5.123). 2

Remark: Note that we have not suprisingly:

(

z1z2

)

= Γ

(

SC

)

(

SC

)

= Γ−1

(

z1z2

)

, (5.128)

where Γ is the change of coordinates matrix defined in lemma 4.2.12 acting via a linear fractonaltransformation (normalized on the third coordinate).

Let me make explicit the action of SU(1, 2) on B4. This action goes of course via a fractionallinear transformation.

69

Lemma 5.8.5 (Action of SU(1, 2) on B4) Let B4 =

z ∈ C2| |z1|2 + |z2|2 < 1

and let A ∈SU(1, 2) in the representation from lemma 4.2.12 with entries aij . Then A acts on z ∈ B4

according to:

Az :=

a11z1 + a12z2 + a13

a31z1 + a32z2 + a33

a21z1 + a22z2 + a32

a31z1 + a32z2 + a33

. (5.129)

Proof: This follows analogously to what was done in definition 5.3.1. We normalize on the thirdcoordinate after acting with A on z ∈ C3. 2

Now we know the action of SU(1, 2) on B4, we can explicitly check how the metric fromdefinition 5.8.1 changes under this action. We expect it to be invariant of course.

Theorem 5.8.6 (SU(1, 2) as isometry group of metric (5.113)) LetSU(1, 2) act on B4 as in lemma 5.8.5. Then SU(1, 2) in the representation from lemma 4.2.12 isan isometry group of the metric (5.113).

Proof: This proof is analogous of the proof of theorem 5.3.3, that is, I will show that under theaction of SU(1, 2) the Kahler potential KB , which was defined in equation (5.112), will changeat most as KB(z1, z1, z2, z2) 7→ KB(z1, z1, z2, z2) + h(z1, z2) + h(z1, z2), where h is a holomorphicfunction and h is its complex conjugate. If I can prove this, lemma 5.3.2 gives the desired result.

Now from lemma 5.8.5 we see that if a matrix A ∈ SU(1, 2) in the representation of lemma

4.2.12 with entries aij acts on

(

z1z2

)

∈ B4, then 1− |z1|2 − |z2|2 changes as follows:

1− |z1|2 − |z2|2 7→ 1

|a31z1 + a32z1 + a33|2(

(|a33|2 − |a13|2 − |a23|2)

+ (a31a33 − a11a13 − a21a23)z1 + (a33a31 − a13a11 − a23a21)z1

+ (a32a33 − a12a13 − a22a23)z2 + (a33a32 − a13a12 − a23a22)z2

+(a31a32 − a11a12 − a21a22)z1z2 + (a32a31 − a12a11 − a22a21)z2z1

+ (|a31|2 − |a11|2 − |a21|2)|z1|2 + (|a32|2 − |a12|2 − |a22|2)|z2|2)

=1

|a31z1 + a32z2 + a33|2(

1− |z1|2 − |z2|2)

. (5.130)

The last identity follows from the condition that A ∈ SU(1, 2) in the representation of lemma4.2.12. We see now that if we define ℵ(z1, z2) := a31z1 + a32z2 + a33, then under the action ofSU(1, 2) on B4 we have that:

KB 7→ KB + logℵ+ logℵ. (5.131)

2

From theorem 5.8.2 and lemma 5.8.3 we know that SU(1, 2) acts transitively on B4. The nexttheorem shows that we can systematically choose for every point z ∈ B4 an element of SU(1, 2)that sends 0 ∈ B4 to z. This is done by demanding Hermiticity.

Theorem 5.8.7 (Unique Hermitian matrix) Use the notation from lemma 5.8.5 and let 0, z ∈B4. Then there is a unique Hermtitian matrix hz ∈ SU(1, 2) (in the representation from lemma4.2.12 such that Hz0 = z. This matrix is given by:

Hz := M + r−1z (1 + rz)

−1ZzZTz , (5.132)

where M is as defined in lemma 4.2.12 and furthermore:

rz :=√

1− |z1|2 − |z2|2 > 0; (5.133)

Zz =

z1z2

1 + rz

; (5.134)

ZTz = (z1, z2, 1 + rz) . (5.135)

70

Proof: First I will prove that, given a z ∈ B4, there is a unique Hermitian matrix g ∈ SU(1, 2)such that g0 = z. Assume there is another such matrix g′, then we have that:

g

(

00

)

= g′(

00

)

=

(

z1z2

)

. (5.136)

This means that g′ = gh, where h is an element of the stabilizer group of 0 in SU(1, 2). So letus take a look at this stabilizer, like was done in the proof of theorem 5.8.2 (only more explicitthis time). Let B be a matrix (with entries bij) in this stabilizer group, then the condition

B

001

=

00c

for a c ∈ C∗ shows that b33 = c and b13 = b23 = 0. So we see that B has the

following form:

B =

v w 0−w v 00 0 1

|v|2+|w|2

, (5.137)

where v, w ∈ C such that c = 1|v|2+|w|2 . We see that this has the form of an embedded element of

U(2).We know that g′ = gh. If we now also demand that g and g′ are Hermitian, so gT = g and

(gh)T

= g′T

= g′ = gh, we get the following equality: gTh = gh = (gh)T

= hT gT = hT g andthus:

g = hT gh−1. (5.138)

Writing out this condition in terms of matrices, where h is of the same form as B above, we findamong others the conditions (where gij are the entries of g):

g13 = g13v − wg23g23 = g13w + g23v. (5.139)

Remember now also that g0 = z, thus (for c ∈ C∗):

g13 = cz1

g23 = cz2

g33 = c. (5.140)

Using this in equation (5.139) we find v = 1 and w = 0. Inserting this in h we find hT gh−1 = gand thus h = I3. We conclude g = g′.

Now we know that for every z ∈ B4, if there is a matrix in SU(1, 2) that sends 0 to z and thatis Hermitian, then it is unique. With the definitions from the theorem and using the identities:

ZTz Zz = (1 + rz)2 + |z1|2 + |z2|2 = 2(1 + rz)

ZTz MZz = −(1 + rz)2 + |z1|2 + |z2|2 = −2rz(1 + rz), (5.141)

we calculate that, for every z ∈ B4, Hz satifies the necessary conditions:

HTz = Hz

HTz MHz = M

Hz0 = z. (5.142)

This proves the theorem. 2

71

5.9 The quaternion-Kahler structure

In this section I try to understand the quaternionic-Kahler structure from the point of view ofthe (S,C)-coordinates. Andrew Strominger proves in [50] that SU(1, 2)/U(2) is a quaternionic-Kahler manifold. I will try to reconstruct his calculations starting from the geometrical knowledgeabout the manifold I have acquired in this paper. I will not fully succeed in this, but many otherinteresting properties related to the quaternions will be given in this section. How to completelyunderstand the quaternion-Kahler structure in terms of the geometry is still an open question forme. At the end of this section I will give some possible suggestions for future research on thismatter. I have not persued these suggestions myself, because I ran out of time at the end of myundergraduate thesis project. I will begin by incorporating the quaternions in my geometricalunderstanding of the situation.

5.9.1 Quaternions and geometry

When I use the notation from the previous sections we see that both B with the (S,C)-coordinatesas well as the complex open ball B4 with the Bergman coordinates are subsets of the complexprojective plane CP 2. In both cases we have normalized on the third coordinate, so both sets arecontained in a region of CP 2 where we can use those homogeneous coordinates where we divideby the third coordinate. Remember from equation (4.37) that we used:

Γ :=1√2

−1 0 10 2 01 0 1

, (5.143)

as the change of coordinate matrix to get from the representation of SU(1, 2) from lemma 4.2.9 tothe representation in lemma 4.2.12. Remember furthermore from equation (5.128) that if we letΓ act on CP 2 via a fractional linear transformation normalized on the third coordinate, we findthat:

Γ

(

SC

)

=

(

z1z2

)

. (5.144)

Thus Γ maps B bijectively onto B4 and it maps the boundary ∂B bijectively onto the boundary:

∂B4 :=

z ∈ C2| |z1|2 + |z2|2 = 1

. (5.145)

In particular it maps some special points as described in the next lemma.

Lemma 5.9.1 (The mapping of some points under Γ) In the notation of definition 5.5.3we have that the boundary points ∞ ∈ ∂B and O ∈ ∂B are mapped to ∂B4 as follows by Γfrom equation (5.144):

Γ∞ =

(

−10

)

(5.146)

ΓO =

(

10

)

. (5.147)

Furthermore the point we have chosen as origin in B maps to the centre of the ball B4:

Γ

(

10

)

=

(

00

)

. (5.148)

Finally we find the following useful result for a φ ∈ R:

Γ

(

e−2φ

0

)

=

(

tanhφ0

)

. (5.149)

72

Proof: This is simple calculation. I will only explicitly calculate the last result from the lemma.The rest goes in an analogous way.

Γ

e−2φ

01

=

1− e−2φ

01 + e−2φ

(5.150)

Normalizing on the third coordinate gives:

Γ

(

e−2φ

0

)

=

(

sinhφcoshφ

0

)

. (5.151)

2

The next lemma will give an isomorphism between C2 and the space of quaternions H.

Lemma 5.9.2 (Isomorphism between C2 and H) Define:

C : C2 → H : z 7→

(

z1 z2−z2 z1

)

. (5.152)

Then C is an isomorphism of manifolds, thus a bijection.

Proof: Since all quaternions can be written in the matrixform from this lemma, C is an isomor-phism. 2

We can restrict the map C to B4. This gives the following corollary.

Corollary 5.9.3 (B4 and quaternions) Let B4 be as defined in definition 5.8.1. Then:

B4 ∼=

q ∈ H| |q|2 < 1

, (5.153)

where the isomorphism is an isomorphism of manifolds, thus a bijection.

Proof: This is an immediate result from restricting the map C from lemma 5.9.2 to B4. 2

We can also use the map C in combination with Γ.

Definition 5.9.4 (Map from B to H) Define:

Q := C Γ : B → H :

(

SC

)

7→(

z1 z2−z2 z1

)

. (5.154)

Note that the map Q maps the line between ∞ and O given by cφ in equation (5.80) onto theinterval of the real quaternions between −1 and 1.

When working with quaternions the Pauli matrices are often useful.

Definition 5.9.5 (Pauli matrices) The Pauli matrices are defined as follows:

σ1 :=

(

0 11 0

)

, σ2 :=

(

0 −ii 0

)

, σ3 :=

(

1 00 −1

)

. (5.155)

For uniformity of the notation the identity matrix I2 is often written as:

σ0 :=

(

1 00 1

)

. (5.156)

The Pauli matrices have some interesting properties.

73

Lemma 5.9.6 (Properties of the Pauli matrices and quaternions) Let thenotation be as in definition 5.9.5. Then we have:

σ21 = σ2

2 = σ23 = σ0;

σ1σ2 = −σ2σ1 = iσ3;σ2σ3 = −σ3σ2 = iσ1;σ3σ1 = −σ1σ3 = iσ2.

(5.157)

In particular we get the following commutation relations:

[σ1, σ2] = 2iσ3, [σ2, σ3] = 2iσ1, [σ3, σ1] = 2iσ2,

[σ0, σ1] = [σ0, σ2] = [σ0, σ3] = 0. (5.158)

Any quaternion q ∈ H can be written as:

q = a0σ0 + i(a1σ1 + a2σ2 + a3σ3), (5.159)

with a0, a1, a2, a3 ∈ R. In particular we have (for z1, z2 ∈ C):

(

z1 z2−z2 z1

)

=z1 + z1

2σ0 +

z2 − z22

σ1 +z2 + z2

2iσ2 +

z1 − z12

σ3. (5.160)

The conjugate of q is then defined to be:

q := a0σ0 − i(a1σ1 + a2σ2 + a3σ3), (5.161)

and the norm of q, |q| ≥ 0, is then given by:

|q|2 := a20 + a2

1 + a22 + a2

3. (5.162)

A quaternion q is said to be a real quaternion if and only if q = q and it is said to be a purelyimaginary quaternion if and only if q = −q.

Every non-zero quaternion can uniquely be written as a product of a quaternion with norm oneand a positive real number: q = |q| q|q| . In particular every non-zero quaternion with norm < 1 can

uniquely be written as a product of a quaternion of norm one and a real number between 0 and 1.Let in the final part of this lemma Q ∈ H be such that |Q|2 = 1. Then we can write:

Q = cos2πt+ q′sin2πt, (5.163)

where t ∈ R, 0 ≤ t ≤ 12 , and q′ ∈ H purely imaginary with |q′| = 1. If Q 6= −1 we can also write:

Q = e2πq , (5.164)

where q ∈ H purely imaginary with |q| < 12 .

Proof: The relations between the Pauli matrices can be calculated straightforwardly. Since weknow how every quaternion can be written in matrixform equations (5.159) and (5.160) followimmediately. The norm for q follows from |q|2 = qq. Equation (5.163) can easily be seen. Itremains to prove equation (5.163).

To prove this, first note that (q′)2 = −1. This means that we can write (by comparing Taylorseries of the trigonometric functions and the exponential function):

Q = e2πtq′

. (5.165)

We see that if Q 6= ±1, then t and q′ are uniquely determined by Q and tq′ is a purely imaginaryquaternion with |tq′| < 1

2 . If Q = 1 then t = 0 and thus q′ is not uniquely determined by Q.However, we still have uniquely that tq′ = 0 and thus in particular |tq′| = 0 < 1

2 . If Q = −1

74

however we have that t = 12 and so q′ is not uniquely determined by Q and neither is tq′. This is

why equation (5.164) does not hold for Q = −1. 2

The group SU(2) acts on B4 via matrix multiplication:

(

v w−w v

)(

z1z2

)

=

(

vz1 + wz2−wz1 + vz2

)

, (5.166)

where v, w ∈ C and z =

(

z1z2

)

∈ B4. We immediately see that the centre of B4,

(

00

)

,

is preserved by this action. With a bit more calculation we also see that the Euclidean metricds2 = dz1dz1 + dz2dz2 is preserved. Furthermore 1 − |z1|2 − |z2|2 is preserved by this action,so also the Kahler potential of the Bergman metric and thus the Bergman metric are preserved.The orbits of this action consist of 3-spheres of points at equal Euclidean distance and Bergmandistance from the centre of the ball B4. Each orbit thus has a unique point of intersection with thereal line segment between (0, 0)T and (1, 0)T . Furthermore the map C from lemma 5.9.2 tells usthat this line segment corresponds with the real quaternions between 0 and 1 and that the groupSU(2) corresponds to the group of quaternions with norm 1.

Let us focus on the real numbers between 0 and 1 for a moment. Every r ∈]0, 1[ can be writtenas tanhφ for a uniquely determined φ ∈ R, φ > 0. It is then the image under Q of the point withcoordinate e−2φ on the line in B given by cφ which runs from (1, 0)T ∈ B to the boundary pointO ∈ ∂B. By this I mean the following. Let S = e−2φ and C = 0, then the curve described by cφwhen φ runs from −∞ to ∞ begins in the boundary point ∞ ∈ ∂B, runs through (1, 0)T ∈ B forφ = 0 and ends in O as can be see from:

γe−2φ,0

101

=

e−φ

0e−φ

. (5.167)

We then also see from equation (5.149) that:

Q(

e−2φ

0

)

= tanhφ

(

1 00 1

)

. (5.168)

We have now the necessary information to state and prove the following lemma.

Lemma 5.9.7 (The Bergman distance from the centre of the ball) TheBergman distance from (0, 0)T ∈ B4, the centre of the ball, to (tanhφ, 0)T ∈ B4 is φ. The Bergmandistance between the quaternions 0 ∈ H and q ∈ H (with |q| < 1) is:

φ = −1

2log

(

1− |q|1 + |q|

)

. (5.169)

Proof: If we take for z ∈ B4 that z1 = x ∈ R and z2 = 0 and insert this into the Bergman metricfrom equation (5.113), then we get for the Bergman metric on the real line segment in B4 between(0, 0)T and (1, 0)T :

ds2 =dx2

(1− x2)2. (5.170)

Integrating this along the geodesic gives for the Bergman distance between (0, 0)T ∈ B4 and(tanhφ, 0)T ∈ B4:

∫ tanhφ

0

dx2

(1− x2)2= φ. (5.171)

Since according to the discussion preceding this lemma all quaternions on a 3-sphere orbit of SU(2)around the centre of B4 have the same Bergman distance to the centre, to compute the distancefrom 0 to q ∈ H (with |q| < 1) I can compute this distance for a quaternion with the same norm

75

as q, so I take q = tanhφ ∈ R (note that then indeed |q| < 1). Inverting this expression givesequation 5.169. 2

The preceding discussion now gives the following two dimensional interpretation of the situ-

ation, where the de circles represent the orbits of a point

(

z1z2

)

∈ B4 under the action of the

quaternions with norm 1 (i.e. left or right matrix multiplication with a matrix in SU(2)) and theradii give represent the distance from the origin measured by the dilaton:

.

5.9.2 Trying to understand the quaternion-Kahler structure

Now I have some understanding of how the quaternions fit into my geometrical picture, I will makea beginning with trying to understand the quaternion-Kahler structure in geometrical terms. I willnot complete this task (in this paper), but hopefully I can still give a nudge in the right direction.The focus here is on making connections between my paper and the short discussion about thequaternion-Kahler structure of SU(1, 2)/U(2) found in [50]. Using the notation from the previoussubsection, I will begin by looking at the map Γγ−1

a,b for a point (a, b)T ∈ B.

Lemma 5.9.8 (The map Γγ−1a,b) Let (a, b)T ∈ B. Then the map on CP 2 which corresponds to

the matrix Γγ−1a,b maps B bijectively onto B4. (a, b)T is send to the centre of the ball B4, (0, 0)T .

If we define the dilaton in (a, b)T :

ψ := −1

2log

(

a+ a− 2bb

2

)

, (5.172)

then the explicit matrix is:

Γγ−1a,b =

eψ√2

−1 2b 2e−2ψ − a0 2e−ψ −2be−ψ

1 −2b a

. (5.173)

This matrix acts as a linear transformation in the following way:

Γγ−1a,b

(

SC

)

=

(

z1z2

)

=

−S + 2bC + 2e−2ψ − aS − 2bC + a

2e−ψC − 2be−ψ

S − 2bC + a

. (5.174)

The derivative at the point (a, b)T of this fractional linear transformation is:

(

dz1dz2

)∣

∣

∣

∣

(a,b)T

=

(

− 12e

2ψ e2ψb0 eψ

)(

dSdC

)

. (5.175)

76

Proof: Calculating the matrix Γγ−1a,b is straightforward matrix multiplication, where we use that:

γ−1a,b =

eψ 0 00 1 00 0 e−ψ

1 −2b − 12 (a− a) + bb

0 1 −b0 0 1

. (5.176)

The fractional linear transformation now follows by normalizing on the third coordinate. Thederivative is calculated in the following way. Calculate:

Γγ−1a,b

(

S + ε1C + ε2

)

= Γγ−1a,b

(

SC

)

+D(

ε1ε2

)

+R(ε1, ε2), (5.177)

where the epsilons in R occur in order two or higher. The matrix D is then the derivative in thepoint (S,C)T . Another way to put this is saying that the epsilons are infinitesimal quantities andsquare to zero (and ε1ε2 = 0). We can use then:

1

S + ε1 − 2bC − 2bε2 + a=

1

S − 2bC + a

1

1 + ε1S−2bC+a

− 2bε2S−2bC+a

(5.178)

≈ 1

S − 2bC + a

(

1− ε1

S − 2bC + a+

2bε2

S − 2bC + a

)

,

where I have indicated by ≈ that I have taken the epsilons to be infinitesimal. In this way we getfor the derivative in the point (S,C)T :

D =1

(S − 2bC + a)2

(

−2e−2ψε1 + 4e−2ψbε22e−ψ(b− C)ε1 + 2e−ψ(S + a− 2bb)ε2

)

. (5.179)

We want the derivative at the point (a, b)T . If we insert S = a and C = b in D and remember thedefinition of ψ then we find the desired result. 2

Corollary 5.9.9 (The derivative written differently) If we use the 1-forms u and v of defi-nition 5.7.4, we can write:

(

dz1dz2

)∣

∣

∣

∣

(a,b)T

=

(

−vu

)∣

∣

∣

∣

(a,b)T

. (5.180)

Corollary 5.9.10 (The metric in u and v) In the notation of definition 5.7.4 the metric onB is:

ds2 = uu+ vv. (5.181)

Proof: Since γ−1a,b is an element of SU(1, 2) in the representation of lemma 4.2.9, the map Γγ−1

a,b

is an isometry from B onto B4. We already know that this map sends the point (a, b)T ∈ B to(0, 0)T ∈ B4, so we can use the derivative of this map to translate the metric on the tangent spaceat (0, 0)T to the metric on the tangent space at (a, b)T . Explicitly, the metric on the tangent spaceat (0, 0)T is (insert z1 = z2 = 0 in equation (5.113)):

ds2 = dz1dz1 + dz2dz2. (5.182)

Pulling this back along the derivative of Γγ−1a,b , thus using corollary 5.9.9, we find for the metric

on the tangent space at (a, b)T :ds2 = uu+ vv. (5.183)

Since we know how to express u and v in either S and C or z1 and z2 we straightforwardly calculatethat equation (5.183) is equivalent to the metrics in equations (3.79) and (5.113). This is to beexpected because equation (5.183) gives the metric in an arbitrary point (a, b)T , so that equationholds at all points (a, b)T ∈ B. 2

We want to have a map that gives a quaternionic coordinate on B centered at the point (a, b)T .The following lemma gives us what we want.

77

Lemma 5.9.11 (Quaternionic coordinates on B) The map:

Qγ−1a,b : B → H, (5.184)

gives a quaternionic coordinate on B centered at the point (a, b)T . The derivative of this map at(a, b)T is:

(

d(

Qγ−1a,b

))∣

∣

∣

(a,b)T=

(

−v u−u −v

)∣

∣

∣

∣

(a,b)T

. (5.185)

Proof: The map γ−1a,b maps B onto B4 and in particular the point (a, b)T ∈ B onto (0, 0) ∈ B4

and the map Q gives a quaternionic coordinate on B4, so the map Qγ−1a,b does what the lemma

claims it does. The derivative follows immediately from definition 5.9.4 and corollary 5.9.9. 2

Remark: Note the resemblence between the matrix in equation (5.185) and the vierbein from[50] that I gave in equation (5.95).

In light of this remark I give the following definition.

Definition 5.9.12 (Vierbein) Define the following matrix of 1-forms:

V :=

(

−v u−u −v

)

. (5.186)

Lemma 5.9.13 (Vierbein map) Define:

∇ : TB → H, ∇ξ :=

(

−v(ξ) u(ξ)−u(ξ) −v(ξ)

)

. (5.187)

Then this is a globally defined H-valued map on the tangent bundle TB of B which is fibrewiseR-linear. In the notation of definition 5.9.5 we expand V from definition 5.9.12 on the basis(σ0, iσ1, iσ2, iσ3):

V = −v + v

2σ0 − i

u− u2

(iσ1) +u+ u

2(iσ2) + i

v − v2

(iσ3), (5.188)

where we note that the coefficients are real. Rewriting this, gives:

V = v−σ0 − σ3

2+ v−σ0 + σ3

2+ u

σ1 + iσ2

2+ u

σ1 − iσ2

2. (5.189)

Proof: Since the 1-forms u, u, v and v are globally defined, fibrewise R-linear, C-valued maps onthe tangent bundle of B, the matrix V gives the globally defined H-valued map on the tangentbundle TB of B which is fibrewise R-linear, which is given in equation (5.187). Writing out V isa straightforward calculation. 2

Remark: Note the analogy between equation (5.93) and equation (5.188). The quaternions act on H by left multiplication. Via ∇ this pulls back to an action of the

quaternions on the tangent space to B in the point (a, b)T . Thus for q ∈ H and a tangent vectorξ at (a, b)T we have:

(

−v(qa,b · ξ) u(qa,b · ξ)−u(qa,b · ξ) −v(qa,b · ξ)

)

= q ·(

−v(ξ) u(ξ)−u(ξ) −v(ξ)

)

. (5.190)

This can also be written as:(

−v u−u −v

)∣

∣

∣

∣

(a,b)T

· qa,b = q ·(

−v u−u −v

)∣

∣

∣

∣

(a,b)T

. (5.191)

I will stop my attempts to understand the quaternion-Kahler structure in a geometrical wayhere. I have given quaternionic charts on B as well as a way in which quaternions act on tangentspaces on B, but I have neither revealed a relation between all this and the holonomy group of

78

SU(1, 2)/U(2) or connections on this manifold, like those found in [50], nor have I given somerelation to the definitions and discussions in [51].

Hopefully future research will clear these things up. One line of reasoning that might possiblygive answers is the following. If a result analogous to that in theorem 5.7.1 can be formulatedfor lower triangular matrices with positive real diagonal entries instead of upper triangular ones,then we can also construct a Maurer-Cartan form with corresponding curvature from these matri-ces. We then have two different Maurer-Cartan forms and although both have zero curvature, aconvex combination (i.e. a linear combination with coefficients that sum to one) of them doesn’tnecessarily have zero curvature. In this way it might just be possible to construct the connectionwith non-zero curvature [50] gives.

Another property that may be worth some more study is the analogy between equation (5.93)and equation (5.188) I noted in the remark above. This may also lead to more insight in thesituation.

All in all, although I have not proven in a geometrical way that SU(1, 2)/U(2) is a quaternionic-Kahler manifold, it is clear that the quaternions provide a lot of interesting structure for thismanifold.

79

Appendix A

Transparancies

In this appendix I have incorporated the text1 of the transparancies I used during my undergrad-uate thesis talk on June 22 2004 in Utrecht. Although it does not go into much detail, it doesprovide a nice overview of some of the highlights of this paper.

Mathematical and physical aspects of the quaternion-Kahler manifold that arises intype II A string compactification on rigid Calabi-Yau manifolds

Yves van GennipSupervisors: dr. Stefan Vandoren and dr. Jan Stienstra

June 22 2004

Physics: string theory type II A

• Type II A string theory

• Compactification from ten to four dimensions

• Getting a metric

• Important for mirror symmetry: IIA/CY3 ↔ IIB/mirrorCY3

Mathematics: the manifold SU(1, 2)/U(2)

• SU(1, 2)/U(2) as the manifold for our metric

• Two different geometrical pictures and their relation to the metric

• The Heisenberg group and the dilaton

• The quaternionic structure

Quick review of physics background

• Action: S[φ] =∫

L[φ] ∈ C

• Path integral:∫

DφeiS[φ]

• Variational principle δS = 0 gives equations of motion

1After correction of typos. Footnotes are also added afterwards.

80

• Expand S around an extremum S0 = S[φ0]:

S[φ0 + δφ] = S0 +δ2S

δφ2(δφ)2

Einstein summation conventionSum over repeated indices:

Aµdxµ :=

∑

µ

Aµdxµ

Ingredients from type II A string theory

• Action for the 10d fields

• 10d massless spectrum: solutions of the equations of motion HF = 0

• Expanding the fields: part lives on a rigid 6d Calabi-Yau manifold (holonomy group inSU(3)), part on 4d Minkowski space; important reason for chosing CY3 to compactify on:low (N = 2) supersymmetry

• Integration 10d action over the Calabi-Yau manifold to end up with 4d action for the uni-versal hypermulitplet

• Getting a metric from the 4d action

Bosonic part of the massless spectrum

• a symmetric (2, 0)-tensorfield gµν , related to the graviton;

• an antisymmetric (2, 0)-tensorfield Bµν , called the NS 2-form, with corresponding field

strength H(3)10d := 16dB

(2)10d;

• a real scalar field φ, called the dilaton;

• a vector field Aµ with corresponding field strength, the antisymmetric (2, 0)-tensorfieldF (2) := dA(1);

• an antisymmetric (3, 0)-tensorfieldAµνσ with corresponding field strength, the antisymmetric(4, 0)-tensorfield F (4) := dA(3).

Example of expansionµ, ν: components on the ten dimensional manifold

µ, ν: components on 4d Minkowski spacei, j: components on CY3

The NS 2-form can be expanded as:

Bµνdxµ ∧ dxν = Bµνdx

µ ∧ dxν + 2Bµidxµ ∧ dyi +Bijdy

i ∧ dyj .

Now expand all the fields on basis of harmonic fields on CY3:

ξ(x, y) =∑

m

ζm(x)fm(y).

• Bµν : 0-form on CY3; Hodge numbers tell us the space of harmonic 0-forms on CY3 is 1-dimensional, so this gives us one field Bµν on Minkowski space.

81

• Bµi: 1-form on CY3; Hodge numbers tell harmonic 1-forms don’t exist on CY3, so this givesus no field on Minkowski space.

• Bij : 2-form on CY3; Hodge numbers tell us there are no harmonic (2, 0)- or (0, 2)-formson CY3 and the space of harmonic (1, 1)-forms is h1,1 dimensional, so we find a field bA onMinkowski space, where A = 1, . . . , h1,1.

Spectrum in four dimensionsAfter expanding all the bosonic fields in ten dimensions we are left with fields in four dimen-

sions. I will concentrate on the so called universal hypermultiplet consisting of the real scalar fieldφ (the dilaton in four dimensions), the complex scalar field C and its complex conjugate C (thesefields follow from compactifying the ten dimensional (3, 0)-tensorfield Aµνσ) and the field Bµν .

Compactifying the actionTen dimensional action:

S10[g, φ, F (2), F (4), B(2)10d] =

1

α′4(2π)7

∫ √gR d4x d6y − 1

2α′4(2π)7

∫

(

e−φH(3)10d ∧ ∗H

(3)10d + dφ ∧ ∗dφ

+e3φ2 F (2) ∧ ∗F (2) + e

φ2 F (4) ∧ ∗F (4) +B

(2)10d ∧ F (4) ∧ F (4)

)

,

Inserting the field expansions and integrating over CY3 gives the following 4d action for theuniversal hypermultiplet:

S4[S, S, C, C] = −16VolCY3

2α′4(2π)7

∫

R1,3

e2K(dS ∧ ∗dS − 2CdS ∧ ∗dC

−2CdS ∧ ∗dC + 2(S + S)dC ∧ ∗dC).

Some fields are redefined or combined to go from φ,C, C, Bµν to S, S, C, C . The transformationfrom 2-form Bµν into a scalar field is done via a dualtiy transformation. Also:

S + S − 2CC = e−K = 2e−2φ.

Sigma model and metricIf we have a sigma model:

L[φ1, . . . , φn] = gAB(φ)dφA ∧ ∗dφB ,

with:φ : 4d Minkowski→ some target manifold,

we get from that a metric:ds2 = gAB(φ)dφAdφB .

In our case we get:

ds2 = e2K(dSdS − 2CdSdC − 2CdSdC + 2(S + S)dCdC),

where K := −log(S + S − 2CC).K is the Kahler potential for the metric, which means:

gAB =∂2K

∂φA∂φB.

82

Take now C, C, S, S as coordinates on a manifold.

A manifold for the metricPhysics literature tells us that the metric we found is a metric on SU(1, 2)/U(2). In other

words: this is the target manifold of the fields in our sigma model. I will investigate what thismeans and discuss some geometrical properties of SU(1, 2)/U(2).

The Lie group SU(1, 2)Three possible ways to characterize matrices A ∈ SU(1, 2), related via basis transformations:

SU(1, 2)i :=

A ∈ GL(3,C)| ATMiA = Mi, detA = 1

,

where:

• M1 :=

0 0 −10 2 0−1 0 0

,

• M2 :=

1 0 00 1 00 0 −1

.

The change of coordinates to go from M1 to M2 is given by:

Γ :=1√2

−1 0 10 2 01 0 1

.

This means: ΓTM2Γ = M1.

The Lie group U(2)

U(2) :=

B ∈ GL(2,C)| BT I2B = I2

.

One possible embedding in SU(1, 2)2 is given by:

B 7→(

B 00 1

detB

)

.

U(2) is not a normal subgroup of SU(1, 2), so SU(1, 2)/U(2) is not a group.

A useful theoremLet X be a space with a transitive group action by the Lie group G and let x0 ∈ X . Let H be

a subgroup of G and the isotropy group of x0, then there is a bijection between X and G/H :

x 7→ gH, where x = gx0.

Geometrical interpretation 1: (S,C)-coordinatesStart with C3 equipped with the bilinear form:

〈v, w〉1 := −v1w3 − v3w1 + 2v2w2.

Now define:P :=

z ∈ C3| 〈z, z〉1 < 0

.

83

SU(1, 2)1 acts transitively on P/C∗ and U(2) is the isotropy group of a point in P/C∗, thus thetheorem above gives SU(1, 2)1/U(2) ∼= P/C∗ =: B. Since a z ∈ P satisfies −z1z3−z3z1+2z2z2 < 0we have that z3 6= 0. This means we can introduce the (S,C)-coordinates on B:

(

SC

)

:= the equivalence class in B of

SC1

∈ B.

Then we can write B as:

B =

(

SC

)

∈ C2| S + S − 2CC > 0

.

In these coordinates SU(1, 2)1 acts on B via a linear fractional transformation:

a11 a12 a13

a21 a22 a23

a31 a32 a33

(

SC

)

=

a11S + a12C + a13

a31S + a32C + a33

a21S + a22C + a23

a31S + a32C + a33

Acting in this way SU(1, 2)1 is an isometry group of the metric:

ds2 = e2K(dSdS − 2CdSdC − 2CdSdC + 2(S + S)dCdC).

C = 0, S = −iτ ⇒ Imτ > 0. Recognize the complex upper half plane as subset.

Geometrical interpretation 2: Bergman coordinatesStart with C3 equipped with the bilinear form:

〈v, w〉2 := v1w1 + v2w2 − v3w3.

Now define:N :=

z ∈ C3| 〈z, z〉2 < 0

.

SU(1, 2)2 acts transitvely on N/C∗ and U(2) is the isotropy group of a point in N/C∗, thus thetheorem tells us that SU(1, 2)2/U(2) ∼= N/C∗ := B4. Since a z ∈ N satisfies z1z1+z2z2−z3z3 < 0,we have that z3 6= 0. So we can introduce the following coordinates on B4:

(

z1z2

)

:= the equivalence class in B4 of

z1z21

∈ B4.

Then we can write B4 as:B4 =

z ∈ C2| |z1|2 + |z2|2 < 1

.

In these coordinates SU(1, 2)2 acts on B4 via a linear fractional transformation:

a11 a12 a13

a21 a22 a23

a31 a32 a33

(

z1z2

)

=

a11z1 + a12z2 + a13

a31z1 + a32z2 + a33

a21z1 + a22z2 + a23

a31z1 + a32z2 + a33

Acting in this way SU(1, 2)2 is an isometry group of the Bergman metric:

ds2 =dz1dz1 + dz2dz21− |z1|2 − |z2|2

+(z1dz1 + z2dz2)(z1dz1 + z2dz2)

(1− |z1|2 − |z2|2)2.

84

Relation between (S,C) and Bergman

(

z1z2

)

= Γ

(

SC

)

.

This also turns the (S,C)-metric into the Bergman metric.

The Heisenberg groupThe unipotent upper triangular matrices in SU(1, 2)1:

U :=

A ∈ SU(1, 2)| A =

1 2z ix+ |z|20 1 z0 0 1

with x ∈ R, z ∈ C

,

form the Heisenberg group. Properties of the Heisenberg group:

• It is the stabilizer group of the dilaton φ = − 12 log

(

S+S−2CC2

)

.

• It is expected that a discrete subgroup of the Heisenberg group is still a isometry groupwhen quantum corrections are calculated2.

• For every point

(

SC

)

∈ B there is a unique upper triangular matrix γS,C with positive

real diagonal entries which maps the point

(

10

)

to

(

SC

)

, given by:

γS,C :=

1 2C 12 (S − S) + CC

0 1 C0 0 1

e−φ 0 00 1 00 0 eφ

.

• Choosing an origin gives a group structure on B, with coordinates elements from U and eφ.

Heisenberg group continued

• Lines in B: γS,C

101

, where C, C and S − S are fixed and φ runs from −∞ to ∞.

• Let ∞ be the equivalence class of

100

in ∂B, where:

∂B := ∂P/C∗ ⊂ CP 2 with:

∂P :=

z ∈ C3| 〈z, z〉1 = 0, z 6= 0

.

The lines run from ∞ to

12 (S − S) + CC

C1

.

• These lines give a bijection between the Heisenberg group and ∂B\∞.2Replace this by: “It is expected that the Heisenberg group is still a isometry group when perturbative corrections

are taken into account.”

85

• The boundary is a topological 3-sphere as can be most easily seen in the Bergman coordi-nates: |z1|2 + |z2|2 = 1.

As a two dimensional “intuitive picture” of this situation, we get:

Note the analogy with the case of the complex upper half plane. Acting with the Heisenberg groupis the analogue of translating: only ∞ is fixed.

Some words on the quaternion structure

• Quaternionic-Kahler means holonomy group in Sp(1,H) · Sp(n,H), where n = dimension4 .

• The map:(

z1z2

)

7→(

z1 z2−z2 z1

)

,

identifies B4 with the quaternions of norm < 1.

• The Bergman distance from 0 ∈ H to q ∈ H with |q| < 1 is φ (the dilaton).

• The group SU(2) (quaternions of norm 1) acts on B4 via matrix multiplication:(

v w−w v

)(

z1z2

)

=

(

vz1 + wz2−wz1 + vz2

)

.

This gives the following two dimensional picture of the situation where the de circles represent the

orbits of a point

(

z1z2

)

∈ B4 under the action of SU(2) and the radii give represent the distance

from the origin measured by the dilaton:

Acting with the unit quaternions gives a generalisation of the angle.

If you want an electronic copy of my master thesis, please email me: Y.vanGennip@phys.uu.nl.

86

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Index

∗, 10B, 50B−, B0, B+, 58B4, 68CYn, 17D, 37Delta, 22H , 58H[w], 59K, 31K3 surface, 17KB , 68L, 37L[w], 59N−, 68P , 50P−, P 0, P+, 58TX , 7U , 37XIIA, 21[., .], 402, 5Γ, 39Ωq , 8Ωqx, 8Υ, 57Υ>0, 62δji , 9♦, 5εj1j2···jn , 10γS,C , 61GL(n,K, 35O(n), 35SL(n,K), 35SO(n), 35SU(n), 35SU(n−, n+), 35Sp(n), 39U(n), 35∞, ∞, 56C, 73Q, 73O, o, 56Tqr,x, 8LieΥ>0, 65

ad, 66∇, 14, 5∂B, 56∂P , 56∂, ∂, 12φ, 61εj1···jn , 9CPn, 56H, 34R1,3, 19cφ, 63d, d∗, 11dS,C , 62hS,C , 62

adjoint representationof a Lie algebra, 66

almost complex structureintegrable, 11on a manifold, 11on a vectorspace, 11

anti holomorphic 3-formon CY3, 25

ballopen, 68

base manifoldof a fibre bundle, 7

Berger’sclassification, 17list, 17

boundary, 56branes, 18

chartfor a fibre bundle, 7

Christoffel symbols, 15codifferential operator, 11complex projective space, 56complex upper half-plane, 58complexification

of a form, 8of a tensor(field), 8of a vectorspace, 6

91

conjugateof quaternion, 74

connectionG-, 16Christoffel, 15coefficients, 14Levi-Cevita, 15matrices, 14Riemannian, 15

coordinatespace, 20time, 20

cosetleft, 48right, 48

cosmological constantfour dimensional, 23

covariant derivativeon endomorphisms, 15on tensorfields, 14

covariantly constant, 14curvature form

of the connection, 66

dilatonfour dimensional, 31ten dimensional, 20

Dolbeault operators, 12

Einstein summation convention, 8elliptic curve, 17equivalence class, 48

fibre, 6bundle, 6

fieldreal scalar, 8tensor, 8vector, 8

field strength, 25form

complex, 12harmonic, 22real, 8

fractional linear transformation, 52frame, 14

graviphotonfour dimensional, 23

gravitonfour dimensional, 23ten dimensional, 20

groupcompact symplectic, 39

general linear, 35Heisenberg, 38orthogonal, 35special linear, 35special orthogonal, 35special unitary, 35unitary, 35

group action, 47Lie, 47transitive, 47

Heisenberg Lie algebra, 65Hodge number, 22Hodge star operator, 10holomorphic 3-form

on CY3, 25holonomy group, 16homeomorphism, 6homogeneous space, 48homomorphism, 41

Lie algebra, 43hypermultiplet of complex structure moduli

four dimensional, 23

ideal, 49isotropy group, 47

Kahler form, 12Kahler potential, 12Kronecker delta, 9

Lagrangian (density), 32Laplace operator, 22Levi-Cita epsilon

tensor, 10Levi-Civita epsilon

symbol, 9Lie algebra, 40

of a Lie group, 41Lie bracket, 40Lie group, 40

manifoldCalabi-Yau, 17complex, 11Hermitian, 12hyper Kahler, 17Kahler, 12, 17pseudo-Riemannian, 9quaternionic-Kahler, 17Riemannian, 9, 17

matrices in SU(1, 2)upper triangular with postive real diag-

onal entries, 62matrices in SU(1, 2)/U(2)

92

all upper triangular, 57diagonal, 37unipotent lower triangular, 37unipotent upper triangular, 37

Maurer-Cartan connection, 66Maurer-Cartan equation, 65Maurer-Cartan form, 64metric

Bergman, 68Hermitian, 12on a manifold, 9

mirror symmetry, 5

Neveu-Schwarz, 20norm

of quaternion, 74

orbit, 47

parallel transport, 16Pauli matrices, 73Peccei-Quinn isometries, 55projection, 6

quaternionpurely imaginary, 74real, 74

quaternions, 34quotient space, 48

Ramond, 20rank

of a vector bundle, 7representative, 48Ricci flat, 16Ricci scalar, 16Riemannian structure, 9rigid CY3 manifold, 22

sectionof a fibre bundle, 7

sigma model, 32non-linear, 32

simple transitivity, 61smooth, 40space-time

Minkowski, 19spectrum

massless, 19strings, 18structure constants, 41subalgebra

of Lie algebra, 40subgroup

normal, 41

tangent bundle, 7tensor, 7

curvature, 16Nijenhuis, 11Ricci, 16Riemann, 16torsion, 15

tensor multipletfour dimensional, 23

torsion, 11torsion free connection, 15total space

of a fibre bundle, 7transition map

on a fibre bundle, 7trivialization

of the tangent bundle, 67trvialization

localfor a fibre bundle, 7

type II A string theory, 18

universal hypermultiplet, 25

vector bundle, 7vectormultiplet of Kahler moduli

four dimensional, 23vierbein, 65, 78volume element

invariant, 11volume form

on CY3, 26

93