Chapter 1
Special geometry
1.1
Introduction
Geometrical structures are abundant in string theory and supergravity. Supergravity theories generically contain a lot of scalar fields φα , whose kinetic terms
in the Lagrangian take the form of a non-linear sigma model:
1
L = − gαβ (φ)∂µ φα ∂ µ φβ .
2
(1.1.1)
The scalars are then interpreted as coordinates on a target space, that is equipped with a metric gαβ . Supersymmetry usually places strong restrictions on
the possibilities for these target spaces. Generically, these are not ordinary Riemannian manifolds, but their geometry is restricted to lie in specific classes of
manifolds.
In these notes, we will focus on one of these classes, the so-called special
geometries. These correspond to the geometries described by the scalars of
theories with 8 supercharges in four dimensions (the so-called N = 2 theories).
Due to the fact that one can consider theories with rigid supersymmetry or
local supersymmetry (supergravity), we will encounter two different kinds of
special geometries. The so-called rigid special geometries describe the target
spaces, described by vector multiplet scalars in N = 2 rigid supersymmetry,
while the local special geometries are associated to the vector multiplet target
spaces in N = 2 supergravity. Special geometry also appears in the context of
Calabi-Yau manifolds. Upon compactification of type IIA or IIB string theory
on a Calabi-Yau threefold, one generically ends up with an N = 2 supergravity
theory in four dimensions. The vacuum expectation values of the scalars can
then be interpreted as describing continuous deformations (or moduli) of the
compactification manifold. The space spanned by these moduli comes with a
natural metric, that appears as the sigma model metric gαβ . For Calabi-Yau
threefolds, one thus expects that this moduli space exhibits special geometry,
an assertion which can indeed be proven.
The aim of these lectures is to provide the reader with some necessary definitions and concepts that appear in special geometry, and to give some intuition
as to how these structures appear in supergravity and string theory. The structure of this text is as follows. The first section gives a short review on Kähler
geometry. We start with some definitions regarding complex geometry and
1
Kähler geometry. After that, we will discuss some issues regarding isometries
and Killing vectors on Kähler manifolds. Next, we show how these structures
appear in the context of N = 1 supergravity theories. The reason for this
section is two-fold. First of all, it is meant to illustrate the importance of geometrical knowledge in the context of supergravity and string theory. Secondly,
as special geometries are specific instances of Kähler manifolds, everything that
is discussed in this first section in principle also holds for the special geometries. After the section on Kähler geometry, we present both rigid and local
special geometry. In both cases, we give several definitions whose equivalence
is discussed. We also introduce some important objects, that can be defined on
special geometries. The fourth section comments on the appearance of special
geometry in the context of N = 2 supergravity and shows how the different
objects defined, find their place in supergravity. The last section discusses the
appearance of special geometry as moduli spaces of Calabi-Yau manifolds.
These notes assume some familiarity with Kähler and complex geometry.
In the section on Calabi-Yau manifolds, we will also frequently use some facts
regarding Calabi-Yau geometry. For the reader who is not so familiar with
these concepts, we refer to for instance [1, 2, 3]. We also gladly refer to [4], that
appeared in the proceedings of the first Modave school. Concerning notations,
most of the notations regarding indices are explained throughout the text. For
the complex conjugate of a certain quantity (or to denote the complex conjugate
of an index), we will use a bar: so z̄ will denote the complex conjugate of z.
Finally, we refrained from giving too much references. The interested reader
might consult the references mentioned here, as well as the references therein.
1.2
1.2.1
Kähler geometry and supergravity
Kähler geometry
Let us start by recalling the definition of a complex manifold.
Definition 1.2.1 M is a complex manifold if the following axioms hold:
1. M is a topological space.
2. M is equipped with a family of pairs {(U(α) , φα )}, where {U(α) } is a family
of open sets that cover M and φα is a homeomorphism from U(α) to an
open subset U of Cn .
T
3. Given U(α) and
U(β) 6= ∅, the map φβα = φβ ◦ φ−1
(α)
α
T U(β) such that UT
from φα (U(α) U(β) ) to φβ (U(α) U(β) ) is holomorphic.
A complex manifold is thus a space that locally looks like Cn . The number
n is called the complex dimension of M and is also denoted by n = dimC M.
Note that when one views complex manifolds as real manifolds, they have real
dimension 2n.
The notion of complex manifold can also be stated differently in terms of
extra structure defined on the manifold. A good introduction to this can be
found in [1, 2, 3]. Suppose that a 2n-dimensional manifold admits a globally
defined (1, 1)-tensor J with local expression Jm n dxm ⊗ ∂n with the following
property:
r
Jm n Jn r = −δm
.
(1.2.1)
2
The manifold is then called an almost complex manifold and J is called an
almost complex structure.
It turns out that complex manifolds are always almost complex. The reverse
is not necessarily true : not every almost complex manifold is also a complex
manifold. In order to determine whether an almost complex manifold is also
complex, one defines the so-called Nijenhuis tensor Nmn r :
1 s
Jm ∂[s Jn] r − (m ↔ n) .
6
Nmn r =
(1.2.2)
The following theorem holds:
Theorem 1.2.1 An almost complex manifold is a complex manifold if and only
if the Nijenhuis tensor of the associated almost complex structure vanishes.
So, one can also say that a complex manifold is a 2n-dimensional real manifold
that is equipped with an almost complex structure whose Nijenhuis tensor vanishes. On an arbitrary almost complex manifold, it is always possible to find
complex coordinates {z i , z̄ ī } (i = 1, · · · , n) in a point p such that J assumes the
following canonical form:
Ji j = iδij ,
Jī j̄ = −iδī j̄ ,
Jī j = Ji j̄ = 0 .
(1.2.3)
When the Nijenhuis tensor vanishes, it is possible to find such holomorphic coordinates in an entire neighborhood around the point p. The transition functions
relating these coordinates in overlapping patches are moreover holomorphic.
When the (almost) complex manifold is endowed with a metric hmn , one
can construct a new metric gmn on the manifold in the following fashion:
gmn =
1
(hmn + Jm r Jn s hrs ) .
2
(1.2.4)
This metric is positive definite if h is and it moreover satisfies the property
gmn = Jm r Jn s grs .
(1.2.5)
Depending on whether the manifold is almost complex or complex, a metric
obeying the property (1.2.5) is called an almost hermitian metric or a hermitian
metric and the corresponding manifold is called almost hermitian or hermitian.
In terms of the holomorphic coordinates, a hermitian metric takes a form in
which the components that are pure in their indices are zero:
ds2 = gij̄ dz i dz̄ j̄ ,
gij = gīj̄ = 0 .
(1.2.6)
Note that if one defines
Kmn ≡ Jm r grn ,
(1.2.7)
the property (1.2.5) is equivalent to the antisymmetry of Kmn :
Kmn = −Kmn .
(1.2.8)
One thus sees that on an (almost) hermitian manifold, a natural two-form can be
defined using the (almost) complex structure. This two-form K = 12 Kmn dxm ∧
dxn is called the fundamental two-form.
3
For Riemannian manifolds, one can introduce a natural connection that is
torsionless and preserves the metric, namely the Levi-Civita connection. In a
similar manner, one can introduce a natural connection on hermitian manifolds,
defined by imposing that it preserves the metric and the complex structure. This
does not uniquely determine the connection yet. It leads to the conditions:
Γkij̄ = Γk̄ij = 0 ,
(1.2.9)
together with the complex conjugates of these constraints. If one furthermore
imposes that the torsion is pure in its lower indices, the connection is uniquely
determined and it is pure in all its indices. Explicitly it is given by:
Γkij = g kl̄ ∂i gj l̄ .
(1.2.10)
A Kähler manifold is a hermitian manifold that obeys an additional restriction:
Definition 1.2.2 A hermitian manifold is said to be Kähler if the fundamental
two-form K is closed:
dK = 0 .
(1.2.11)
When dealing with Kähler manifolds, one often refers to the fundamental twoform as the Kähler form. In the rest of this chapter, we will adopt a normalization for the Kähler form such that in holomorphic coordinates it is given
by:
i
g dz i ∧ dz̄ j̄ .
(1.2.12)
K=
2π ij̄
The closure of K has some interesting consequences. Indeed, if we write out the
condition (1.2.11) explicitly in terms of holomorphic indices, we get:
dK ∼ i∂i gj k̄ dz i ∧ dz j ∧ dz̄ k̄ + i∂ī gj k̄ dz ī ∧ dz j ∧ dz̄ k̄ .
(1.2.13)
Both terms should be separately zero, implying that
∂ī gj k̄ = ∂k̄ gj ī .
∂i gj k̄ = ∂j gik̄ ,
(1.2.14)
From this, it follows that in local coordinate patches the hermitian metric can be
expressed in terms of a real function K = K(z, z̄), called the Kähler potential:
gij̄ = ∂i ∂j̄ K .
(1.2.15)
On the overlap of two coordinate patches U(α) and U(β) , the respective Kähler
potentials K(α) and K(β) can in general be related by a Kähler transformation:
K(α) = K(β) + fαβ (z) + f¯αβ (z̄) ,
(1.2.16)
where fαβ (z) is holomorphic. A second important property of Kähler manifolds
is that due to (1.2.14), the hermitian connection (1.2.10) is symmetric in its
lower indices and hence coincides with the Christoffel connection. For Kähler
manifolds, one thus finds that the Levi-Civita connection also preserves the
complex structure.
This fact has implications for the holonomy group of Kähler manifolds. The
holonomy group of a manifold is defined by using the notion of parallel transport
4
of tangent vectors. Suppose that the manifold M is endowed with an affine
connection. Consider a point p ∈ M. Denoting the tangent space at p by
Tp M, we can parallel transport a vector X ∈ Tp M along a closed loop c going
through p. The resulting vector X 0 ∈ Tp M can be different from X, hence
parallel transport along closed loops generates an action on Tp M. Upon parallel
transport of X along every possible loop, this action on Tp M defines a group,
called the holonomy group of the connection. It turns out that this holonomy
group is generated by the curvature tensor of the connection Rmnr s (p), seen
as a two-form. We will always consider holonomy groups of the Levi-Civita
connection. In that case, as the Levi-Civita connection preserves the metric,
the length of a tangent vector is not changed upon parallel transport along a
closed loop. For an n-dimensional Riemannian manifold, one thus sees that
the holonomy group should be contained in SO(n). For a Kähler manifold,
the fact that the Levi-Civita connection also preserves the complex structure,
implies that the holonomy group of a (complex) n-dimensional Kähler manifold
is contained in U(n).
Kähler geometry occurs naturally in supergravity theories with 4 supercharges, as we will see in an explicit example in the next section. The Kähler
manifolds that appear in supergravity theories generically obey an extra condition, namely the Kähler form should be of even integer cohomology. By this,
we mean that its integral over an arbitrary 2-cycle 1 γ gives an even integer:
Z
Kmn dxm dxn = 2n ,
n ∈ Z.
(1.2.17)
γ
Kähler manifolds obeying (1.2.17) are then called Hodge-Kähler manifolds or
Kähler manifolds of the restricted type. A more mathematical definition of a
Hodge-Kähler manifold is given by 2 :
Definition 1.2.3 A Kähler manifold is Hodge-Kähler (or of the restricted type)
if there exists a line bundle L, whose first Chern class is equal to one half the
Kähler class:
1
(1.2.18)
c1 (L) = [K] ,
2
where [K] denotes the cohomology class of the Kähler form.
Due to the fact that the first Chern class is an integer cohomology class, meaning
that its integral over an arbitrary 2-cycle is an integer, this definition implies
the property (1.2.17). We will indicate later how this definition is realized in
supergravity.
1.2.2
Holomorphic Killing vectors and moment maps
In this section, we will consider some properties of holomorphic Killing vectors
on Kähler manifolds. Essentially, we want to study infinitesimal, holomorphic
coordinate transformations
i
δz i = εΛ kΛ
(z) ,
(1.2.19)
1 An m-cycle is an m-dimensional submanifold that has no boundary and is itself not the
boundary of an m + 1-dimensional submanifold.
2 Actually, mathematicians usually define a Hodge-Kähler manifold to be a manifold equipped with a line bundle whose first Chern class is exactly equal to the Kähler class. We will
however adopt the definition given here, as this is the appropriate one for supergravity, as will
be made more clear later on.
5
which leave the metric invariant and hence correspond to isometries of the manifold. Holomorphicity means that these transformations also do not change the
i
complex structure of the manifold. The kΛ
(z) that generate these isometries
then correspond to holomorphic Killing vectors:
i
ī
kΛ = kΛ
∂i + kΛ
∂ī .
(1.2.20)
Let kΛ be a basis of holomorphic Killing vectors for the metric gij̄ . Holomorphicity implies the following constraint:
i
∂j̄ kΛ
(z) = 0
ī
∂j kΛ
(z̄) = 0 .
⇔
(1.2.21)
Writing the generic Killing equation ∇(µ kν) = 0 in holomorphic indices, one
ends up with (suppressing the index Λ for the moment):
∇i k j + ∇j k i
=
0,
∇ī kj + ∇j kī
=
0,
(1.2.22)
where kj = gj ī k ī .
Just like the Kähler metric can be obtained by applying derivatives on the
Kähler potential, the holomorphic Killing vectors in a Kähler manifold are also
derivatives of another object. Indeed, the first equation in (1.2.22) is automatically satisfied for holomorphic vectors, while the second equation is satisfied
when the Killing vectors assume the following form:
with PΛ∗ = PΛ .
i
kΛ
= ig ij̄ ∂j̄ PΛ ,
(1.2.23)
Otherwise stated, if we can find a real function PΛ such that the expression
ig ij̄ ∂j̄ PΛ is holomorphic, then (1.2.23) defines a Killing vector. The PΛ are
known as Killing prepotentials or also as moment maps.
One can also define a Poisson bracket for the Killing prepotentials. The
Poisson bracket of PΛ with PΣ is defined as
i j̄
i j̄
{PΛ , PΣ } ≡ 4πK(kΛ , kΣ ) = igij̄ (kΛ
kΣ − kΣ
kΛ ) ,
(1.2.24)
where K(kΛ , kΣ ) denotes the Kähler form, evaluated on the Killing vectors kΛ
and kΣ .
The several Killing vectors kΛ are associated to isometries, that form an
isometry group G, with Lie algebra G. Denoting the structure constants of G by
Γ
fΛΣ
, we present the following property of the Poisson brackets of the associated
moment maps:
Lemma 1.2.1 The following identity holds
Γ
{PΛ , PΣ } = fΛΣ
PΓ + CΛΣ ,
(1.2.25)
where CΛΣ are constants that fulfill the following (cocycle) condition:
Γ
Γ
Γ
fΛΠ
CΓΣ + fΠΣ
CΓΛ + fΣΛ
CΓΠ = 0 .
(1.2.26)
Note that, when the Lie algebra G has a trivial second cohomology group
H 2 (G) = 0, then the cocycle CΛΣ is a coboundary, meaning that one can find
suitable constants CΓ such that
Γ
CΛΣ = fΛΣ
CΓ .
6
(1.2.27)
By absorbing these constants CΓ in the moment maps:
PΛ → PΛ + CΛ ,
(1.2.28)
we can thus obtain the following equation
Γ
{PΛ , PΣ } = fΛΣ
PΓ .
(1.2.29)
Note that the condition H 2 (G) = 0 is true for all semi-simple Lie algebras.
1.2.3
Applications to N = 1 supergravity
As an illustration of the importance of Kähler geometry in string theory and supergravity, we will now discuss how the above defined notions enter the bosonic
Lagrangian of matter coupled supergravity in four dimensions, with four supercharges (N = 1 supergravity). Some good papers concerning N = 1 mattercoupled supergravity are [5, 6]. A nice reference concerning Kähler geometry in
N = 1 theories is for instance given by [7].
The fields of N = 1 supergravity theories are organized in supermultiplets,
that form representations of the N = 1 supersymmetry algebra. For N = 1
supergravity, the following supermultiplets are relevant:
• the gravity multiplet, containing the space-time metric gµν and a spin-3/2
gravitino ψµ (µ, ν = 0, · · · , 3 denote the space-time coordinates here),
• the chiral multiplet, containing two real scalars and a Weyl fermion,
• the vector multiplet, containing a vector and a Majorana fermion.
In the following, we will concentrate on the bosonic sector of N = 1 supergravity,
coupled to n chiral multiplets and m vector multiplets. We will denote the
scalars of the chiral multiplets as Ai and B i (i = 1, · · · , n) and the vectors
of the vector multiplets as AΛ
µ (Λ = 1, · · · , m). Their (abelian) field-strengths
Λ
= 2∂[µ AΛ
will, throughout this text, be denoted by Fµν
ν] . The scalars of the
chiral multiplets appear in the action in the form of a non-linear sigma model.
Denoting all the scalars collectively by φα , the φα can thus be interpreted as
coordinates on a target space M. They are seen as maps from four-dimensional
space-time to the target space M, which is equipped with a metric gαβ 3 . The
kinetic terms of the scalars in the action then read
√
(1.2.30)
Lkin.scalars = − −ggαβ ∂µ φα ∂ν φβ g µν .
Supersymmetry highly restricts the possibilities for the target spaces M that
can appear in supergravity theories. For four-dimensional N = 1 supergravity,
it turns out that M has to be a (Hodge-)Kähler manifold. One can thus combine
the 2n real scalars Ai and B i to form n complex scalars z i , such that the kinetic
terms of the scalars are given by
√
(1.2.31)
Lkin.scalars = − −ggij̄ ∂µ z i ∂ν z̄ j̄ g µν .
3 This
scalar-dependent sigma-model metric should not be confused with the space-time
metric gµν .
7
Let us now give the complete bosonic part of the supergravity Lagrangian:
L
=
√
−g
hR
i
1
Λ
− gij̄ Dµ z i Dµ z̄ j̄ + ImfΛΣ (z)Fµν
F Σµν − V (z, z̄)
2
4
1
Λ
Σ µνρσ
Fρσ
ε
,
− RefΛΣ (z)Fµν
8
(1.2.32)
The first term in the above action is the usual Einstein-Hilbert term, the second
term denotes the kinetic terms of the scalars, as discussed above, while the
third term represents the kinetic terms of the vector fields. The fourth term
represents a potential for the scalar fields, whose structure we will discuss in due
course. The last term is often called the Peccei-Quinn term. The kinetic terms
of the vector fields and the Peccei-Quinn term are determined by holomorphic
functions fΛΣ (z), the so-called gauge kinetic function. The covariant derivatives
of the scalars z i are defined as follows:
i
Dµ z i = ∂µ z i + AΛ
µ kΛ (z) .
(1.2.33)
In order to explain why we have introduced these covariant derivatives, we
first note that isometries of the Kähler manifold represent global symmetries
of the term (1.2.31), as they leave the Kähler metric invariant. In some cases,
it is possible to promote these global symmetries of the Lagrangian to local
symmetries, in such a way that the resulting theory is still supersymmetric.
The procedure by which this is done is often called ’gauging’ and the resulting
theory is called a ’gauged supergravity’. The effect of such a gauging is to
replace ordinary derivatives and field strengths 4 by covariant ones, as well
as to introduce potential terms for the scalars in the theory. The covariant
derivative (1.2.33) indicates that we have gauged some holomorphic isometries
of the Kähler target space. Note that the corresponding Killing vectors explicitly
appear in these covariant derivatives.
Let us finally discuss the potential for the scalars. It is given by
1
V (z, z̄) = eG (g ij̄ ∂i G∂j̄ G − 3) + ([Ref ]−1 )ΛΣ PΛ PΣ ,
4
(1.2.34)
where
G(z, z̄) = K(z, z̄) + log|W (z)|2 .
(1.2.35)
The potential essentially consists of two terms. The first term is determined by
the Kähler potential K and a holomorphic function W (z), called the superpotential. One often rewrites this term more conventionally as
VF = eK (−3W W̄ + g ij̄ Di W Dī W̄ ) ,
(1.2.36)
Di W = ∂i W + (∂i K)W .
(1.2.37)
with
This part of the N = 1 scalar potential is also called the F -term potential and
can always be present. Note that the covariant derivative (1.2.37) shows that
W should actually be interpreted as a section of a line bundle LW constructed
over the Kähler manifold. The transition functions of this bundle are such that
4 Note that the field strengths F Λ in (1.2.32) are also the properly covariantized (nonµν
abelian) ones.)
8
on the overlap of two patches U(α) and U(β) , the section W and the Kähler
potentials are related by:
W(α) = e−fαβ (z) W(β) ,
K(α) = K(β) + fαβ (z) + f¯αβ (z̄) .
(1.2.38)
The appropriate covariant derivative on W is then indeed given by (1.2.37) (and
Dī = ∂ī ). Calculating the curvature FiW
of the bundle LW (by calculating a
j̄
commutator [Di , Dj̄ ]), one finds that the curvature of this bundle is given by:
FiW
j̄ = gij̄ .
(1.2.39)
The first Chern class is thus given by the Kähler class:
i
F dz i ∧ dz̄ j̄ = K .
(1.2.40)
2π ij̄
The existence of this superpotential hence implies that the manifold is HodgeKähler (in the mathematical sense of footnote 2). In supergravity however, the
presence of fermions implies a stronger constraint. It turns out that also the
fermions behave as a section of a line bundle over the Kähler manifold. More
specifically, a fermion Ω should be seen as a section of a line bundle LΩ whose
transition functions are given by:
c1 (LW ) =
Ω(α)
=
K(α)
=
¯
e− 4 (fαβ (z)−fαβ (z̄)) Ω(β) ,
K(β) + fαβ (z) + f¯αβ (z̄) .
1
(1.2.41)
The appropriate covariant derivatives are then given by:
1
Di Ω = ∂i Ω + (∂i K) Ω ,
4
1
Dī Ω = ∂ī Ω − (∂ī K) Ω ,
4
(1.2.42)
Calculating the curvature FiΩj̄ and first Chern class of this line bundle, one finds
that:
1
1
FiΩj̄ = gij̄ ,
c1 (LΩ ) = K .
(1.2.43)
2
2
The presence of the fermions thus implies that the target space of the scalars
is Hodge-Kähler according to the definition given above. The fermions then
appear in the Lagrangian, covered by covariant derivatives (1.2.42).
The other term in the potential is called the D-term potential and is determined by a possible gauging (in the sense described above). So, unlike the
F -term, the D-term is only present when some isometries of the Kähler manifold are gauged. It depends explicitly on the moment maps that correspond to
the isometries that are gauged. We also mention that, in order to show that
this potential is invariant under the gauged isometries, one needs to make use
of the identity (1.2.29).
The above discussion shows that Kähler geometry enters N = 1 supergravity
in a crucial way. Although we have shown this only at the level of the bosonic
Lagrangian, the correspondence goes much further than that. For instance, also
the fermionic terms are completely determined by the Kähler geometry. The
same moreover holds for the supersymmetry transformation rules.
We will end our discussion on Kähler geometry and its appearance in supergravity here. In the following, we will on the other hand focus on the geometrical
structures that appear in theories in four dimensions with 8 supercharges (the
so-called N = 2 theories).
9
1.3
Special geometry
Generically, one distinguishes between two kinds of special geometries. From a
physical point of view, the distinction arises because one can consider theories
exhibiting rigid or local supersymmetry. The geometries that are associated
with theories exhibiting N = 2 rigid supersymmetry are denoted as rigid special
geometries, while the geometries appearing in N = 2 supergravity theories are
denoted as local special geometries. In the mathematical literature, rigid special
geometry is often called ’affine special geometry’, while local special geometry
is called ’projective special geometry’. In the rest of this section, we will give
several equivalent definitions of both rigid and local special geometry. The
advantage of having several equivalent definitions will be made more clear later
on. The main reference for this section is given by [8]. Other references include
the original paper [9] and the more mathematically inclined papers [10, 11].
1.3.1
Rigid (affine) special geometry
Manifolds that exhibit rigid special geometry are essentially Kähler manifolds,
with some extra structure defined on it. This extra structure is captured by the
following definition.
Definition 1.3.1 An n-dimensional rigid (or affine) special Kähler manifold is
an n-dimensional Kähler manifold that satisfies the following conditions:
• On every chart, there exist n independent holomorphic functions X A (z)
(A = 1, · · · , n) and a holomorphic function F (X) such that the Kähler
potential can be expressed as
∂
A ∂
F̄
(
X̄)
−
X̄
F
(X)
.
(1.3.1)
K(z, z̄) = i X A
∂X A
∂ X̄ A
X
• Forming 2n-component vectors
, containing the n functions X A
∂F
in the upper part and the n derivatives ∂F/∂X A in the lower part, one
has that on overlaps of charts U(α) and U(β) , these vectors are related by
transition functions of the following form
X
X
= eicαβ Mαβ
+ bαβ ;
(1.3.2)
∂F (α)
∂F (β)
where cαβ ∈ R, Mαβ ∈ Sp(2n, R) and bαβ ∈ C2n .
• The above transition functions should satisfy the cocycle conditions on the
overlap of three coordinate charts, i.e. one should have
eicαβ eicβγ eicγα
=
Mαβ Mβγ Mγα
=
1,
2n
.
(1.3.3)
Note that Sp(2n, R) denotes the symplectic group over the real numbers of
degree 2n. This group is defined as the subgroup of G `(2n, R) that leave the
skew-symmetric 2n × 2n-matrix
0
Ω=
(1.3.4)
−
0
10
invariant. More explicity, the matrix S ∈ Sp(2n, R) when S T ΩS = Ω. Denoting
A B
S=
,
(1.3.5)
C D
with A, B, C, D n × n-dimensional submatrices, the condition for S to be a
symplectic matrix amounts to
AT C − C T A = 0 ,
B T D − DT B = 0 ,
AT D − C T B =
.
(1.3.6)
A 2n-component vector V which transforms under symplectic transformations
as V → SV , is called a symplectic vector. For symplectic vectors V, W , one can
define the following inner product
< V, W >≡ V T ΩW .
(1.3.7)
This inner product is then invariant under symplectic transformations of V, W .
Note that, if we denote
X
V ≡
,
(1.3.8)
∂F
the transition functions (1.3.2) indicate that V transforms as a symplectic vector, up to a phase factor eicαβ and a shift bαβ . The Kähler potential is in this
notation given by
K(z, z̄) = i < V, V̄ > .
(1.3.9)
It is thus invariant under the symplectic transformation Mαβ and under the
phase factor eicαβ . One can easily verify that the effect of the shift bαβ is that
of a Kähler transformation. The transition functions (1.3.2) hence do not change
the Kähler metric.
The function F that appears in the above definition is well-known in the
physics literature as ’the prepotential’. The above definition relies heavily on
the existence of this prepotential. The essential part of the above definition
however is the existence of a certain vector bundle on the manifold, such that
the Kähler potential can be written in terms of a section of this bundle. To
highlight this bundle structure, we give the following equivalent definition.
Definition 1.3.2 A rigid special Kähler manifold is an n-dimensional Kähler
manifold for which there exists a U(1) × ISp(2n, R) vector bundle over the manifold with constant transition functions as in (1.3.2), i.e. with a complex inhomogeneous part. This bundle should have a holomorphic section V such that the
Kähler potential is given by
K(z, z̄) = i < V, V̄ > .
(1.3.10)
< ∂i V, ∂j V >= 0 .
(1.3.11)
and such that
Note that we have used the notation ISp to denote the fact that V transforms
as a symplectic vector, up to an inhomogeneous part.
To establish the equivalence of both definitions, we note that the bundle
properties of the special Kähler manifold of definition 2 are also encoded in the
transition functions (1.3.2) and the cocycle conditions (1.3.3). Note furthermore
11
that for a section V of the form (1.3.8) the condition (1.3.11) is automatically
satisfied. This shows that a manifold that is rigid special Kähler according to
the first definition is also rigid special Kähler according to the second definition.
In order to show that definition 1 follows from definition 2, we introduce the
following notation:
A e i
Ui ≡
≡ ∂i V .
(1.3.12)
hAi
In terms of Ui , the Kähler metric can then be written as
gij̄ = ∂i ∂j̄ K = i < Ui , Ūj̄ > .
(1.3.13)
One can show that eA i is an invertible n × n-matrix, due to the positivity of
this Kähler metric (this can be shown by a simplification of the argument given
in lemma A.7). We will denote the inverse of eA i by ei A . For the section V , we
will adopt the notation
A X
V =
.
(1.3.14)
FA
Note that the FA are for the moment just functions of the coordinates z i . We will
now however show that they can be expressed as derivatives of a holomorphic
function F (X), thereby establishing equivalence between the two definitions
given above. Using the inverse function theorem and the invertibility of eA i ,
one can express the coordinates z i as functions of X A and therefore also the
FA (z) become functions of the X A . A short calculation then gives:
∂
FB = ei A hBi = ei A (hCi eC j )ej B .
∂X A
(1.3.15)
Noting that the condition (1.3.11) can be rewritten as eA i hAj = eA j hAi , we can
obtain from the above calculation that
∂
∂
FB =
FA .
∂X A
∂X B
(1.3.16)
This integrability condition ensures that the functions FA can locally be rewritten in terms of a prepotential function F :
FA =
∂F
.
∂X A
(1.3.17)
This argument completes the proof of the equivalence between the two definitions of rigid special geometry.
Matrix formulation. There also exists a handy matrix formulation, that
neatly captures useful formulae in rigid special geometry. Starting from n symplectic vectors (defined over a chart) Ui (z, z̄) and their complex conjugates, one
forms the 2n × 2n-matrix
T Ui
V(z, z̄) ≡
.
(1.3.18)
ŪīT
Defining
Âi = ∂i VV −1 ,
Âī = ∂ī VV −1 ,
12
(1.3.19)
one imposes the following constraints:
0
−igij̄
G(i,j) k
T
VΩV =
, Âi =
igj ī
0
0
0
0
Âī =
.
k
G(ī,j̄) k̄
C̄(ī,j̄)
C(i,j) k̄
0
,
(1.3.20)
We have used the notation G(i,j) k , C(i,j) k̄ to indicate the fact that the corresponding matrix elements are constrained to be symmetric in the lower indices.
The first of these constraints leads to (1.3.11) and (1.3.13). The last two constraints essentially lead to
∂ī Uj = 0 ,
∂[i Uj] = 0 ,
(1.3.21)
implying that Ui are holomorphic vectors that are locally the derivative of a
symplectic vector V : Ui = ∂i V . By combining the first two constraints of
(1.3.20), one can easily obtain the following equation:
iC(i,j)k − iC(i,k)j −iG(i,j)k̄
0
−igj k̄
= ∂i
.
(1.3.22)
igkj̄
0
iG(i,k)j̄
0
(All indices are lowered with the metric gij̄ .) This equation implies that C is
a three-index symmetric tensor. Note that for the Kähler metric gij̄ the LeviCivita connection does not have mixed indices, and the relevant part of it is
given by:
Γkij = g kl̄ ∂j gil̄ .
(1.3.23)
From (1.3.22), we can thus also infer that G is nothing else but the Levi-Civita
connection, with all indices lowered. Defining covariant derivatives with the
Levi-Civita connection
Di Uj = ∂i Uj − Γkij Uk ,
Dī Uj = ∂ī Uj ,
(1.3.24)
we can rewrite the last two constraints of (1.3.20) as
Di V = Ai V ,
Dī V = Aī V ,
k̄
0
0 Cij
Ai =
,
Aī =
C̄īj̄ k
0
0
0
0
.
(1.3.25)
By taking a commutator of two covariant derivatives, we can obtain the following
relation for the Riemann curvature tensor:
Rij̄kl̄ = −Cikm C̄j̄ l̄m̄ g mm̄ .
(1.3.26)
Finally, we note that one can also derive the following formula for the C-tensor:
Cijk = ieA i eB j eC k FABC ,
(1.3.27)
3
∂
where FABC = ∂X A ∂X
B ∂X C F .
Period matrix. A very important object, that can be defined in the context
of special geometry, is the so-called period matrix. For rigid special geometry,
this period matrix N is an n × n-matrix, defined in the following way:
NAB = h̄Aī ēī B =
∂2
F̄ ≡ F̄AB .
∂ X̄ A X̄ B
13
(1.3.28)
The period matrix is thus symmetric. Another noteworthy property of the
period matrix is given by its transformation properties under symplectic transformations of the symplectic vector V . Denoting the transformed quantities
with a tilde, the transformation rule Ṽ = SV , with V given as in (1.3.14) and
S given by (1.3.5), explicitly leads to:
X̃ A
= AA B X B + B AB FB ,
F̃A
= CAB X B + DA B FB .
(1.3.29)
From these transformation rules, one can easily check that N transforms as
follows:
Ñ = (C + DN )(A + BN )−1 .
(1.3.30)
Note that the definition of the period matrix implies the following identity:
gij̄ = ieA i (N − N † )AB ēB j̄ .
(1.3.31)
The positive definiteness of the Kähler metric and the invertibility of eA i then
imply that the imaginary part of N is negative definite. Using lemma A.2, we
then obtain the invertibility of (A + BN ), which is required for (1.3.30) to make
sense. The period matrix hence transforms in a fractional way under symplectic
transformations. The relevance of the period matrix and its transformation rule
under symplectic transformations for physics will become more clear later on.
1.3.2
Local (projective) special geometry
We will now discuss local (or projective) special geometry. Our discussion will
be essentially parallel to the discussion of rigid special geometry. We will again
give several definitions and motivate why they are equivalent. As we did in
the rigid case, we will first give a definition that depends on the notion of a
prepotential.
Definition 1.3.3 A local special Kähler manifold is an n-dimensional HodgeKähler manifold that obeys the following 3 properties:
• On every chart there exist n + 1 complex functions Z I (z), where I =
0, · · · , n and a holomorphic function F (Z I ) that is homogeneous of second
degree, such that the Kähler potential is given by
h
i
∂
I ∂
K(z, z̄) = −log iZ̄ I
F
(Z)
−
iZ
F̄
(
Z̄)
,
(1.3.32)
∂Z I
∂ Z̄ I
• On overlaps of charts U(α) and U(β) , the functions of the previous item
are connected by transition functions of the following form:
Z
X
= efαβ (z) Mαβ
,
(1.3.33)
∂F (α)
∂F (β)
where fαβ (z) are holomorphic, Mαβ ∈ Sp(2n + 2, R).
• The above transition functions should satisfy the cocycle conditions on the
overlap of three coordinate charts, i.e. one should have
efαβ (z) efβγ (z) efγα (z)
=
Mαβ Mβγ Mγα
=
14
1,
2n
.
(1.3.34)
Let us compare this definition with the corresponding definition of rigid
special geometry. A first difference is that there are now n + 1 functions Z I
defined on the manifold, in contrast to the n functions X A for rigid special
geometry. The prepotential F is now also restricted to be homogeneous of
second degree, i.e.:
F (λZ) = λ2 F (Z) .
(1.3.35)
Furthermore, the expression for the Kähler potential is different as well. A
further difference is that the transition functions now involve local holomorphic
transition functions as a multiplicative factor, vs. a constant phase factor in
the
rigid
case. The inhomogeneous part is now also missing. The 2n + 2-vector
Z
is a section of a line bundle L, with transition functions given by the
∂F
efαβ (z) -factor. This line bundle obeys the conditions required by the definition
of Hodge-Kähler manifolds. The transition functions mentioned above, leave
the Kähler potential invariant, up to a Kähler transformation.
The above definition again depends on the notion of a prepotential function
F . Again, the crucial notion is that of a certain vector bundle constructed over
the manifold. We will thus present a second definition, that relies less on the
existence of a prepotential, and in which this bundle structure is more manifest.
Definition 1.3.4 A special Kähler manifold is an n-dimensional Hodge-Kähler
manifold, with the following 2 properties:
• There exists a holomorphic Sp(2n + 2, R)-vector bundle H over the manifold and a holomorphic section v(z) of L⊗H, such that the Kähler potential
is given by
K = −log[i < v̄, v >] .
(1.3.36)
Note that L denotes the holomorphic line bundle over the manifold, of
which the first Chern class equals the cohomology class of the Kähler form.
• The section v(z) satisfies
< v, ∂i v >= 0 .
(1.3.37)
Note that equation (1.3.37) is a proper equation in the L-bundle, since, due
to the antisymmetry of the symplectic inner product, it can equally well be
replaced by
< v, Di v >= 0 ,
(1.3.38)
where Di v ≡ ∂i v + (∂i K)v, denotes the covariant derivative in the L-bundle
(the so-called Kähler covariant derivative). Indeed, under
v
K
→ ef (z) v ,
→ K − f (z) − f¯(z̄) ,
(1.3.39)
the Kähler covariant derivative transforms as Di v → ef (z) Di v. By taking an
extra covariant derivative of (1.3.38) and antisymmetrizing, one can obtain the
equivalent of (1.3.11):
< Di v, Dj v >= 0 .
(1.3.40)
Finally, we note that the reason why one does not impose the equivalent of
(1.3.37) in the case of rigid special geometry, is because that would restrict the
15
Kähler potential to be homogeneous of second degree, which is an unnecessary
restriction.
In the supergravity literature, one often formulates local special geometry
in terms of a section of a different bundle. This section is denoted as V and is
related to v by
V ≡ eK/2 v .
(1.3.41)
The Kähler covariant derivatives are then defined as follows:
1
Ui ≡ Di V ≡ ∂i V + (∂i K)V
2
1
Ūī ≡ Dī V̄ ≡ ∂ī V̄ + (∂ī K)V̄
2
1
Dī V ≡ ∂ī V − (∂ī K)V ,
2
1
Di V̄ ≡ ∂i V̄ − (∂i K)V̄ , (1.3.42)
2
,
,
One then imposes the following constraints on the section V :
< V, V̄ >= i ,
(1.3.43)
Dī V = 0 ,
(1.3.44)
< V, Ui >= 0 ,
(1.3.45)
< Ui , Uj >= 0 .
(1.3.46)
The first constraint leads to the expression for the Kähler potential, while the
second expression leads to the holomorphy of the original section v. As explained
above, given the first two constraints, the third constraint implies the fourth 5 .
We will now explain in which sense the two definitions are equivalent. Since
the conditions of the first definition clearly imply those of the second definition,
we only have to show how definition 1 can be obtained from definition 2. We will
formulate the proof in terms of the section V . Let us first make some preliminary
observations regarding the inner products of V , Ui and their complex conjugates.
By taking covariant derivatives of < V, V̄ >= i, one easily gets:
< Ui , V̄ >= 0 ,
< Ūī , V >= 0 .
(1.3.47)
Taking an extra covariant derivative of (1.3.47) and noticing that the curvature
in the L-bundle is essentially the Kähler form:
[Di , Dj̄ ]V = −gij̄ V ,
(1.3.48)
one obtains
< Ui , Ūj̄ >= −igij̄ .
(1.3.49)
I
X
that satisfies
FI
the conditions (1.3.43-1.3.46), there
exists
a Sp(2n + 2, R)-transformation which
X̃ I
transforms V into a vector Ṽ =
, such that F̃I is the derivative of a
F̃I
holomorphic function F̃ (X̃) that is homogeneous of second degree. Note that
we require Ṽ and V to be related by an Sp(2n + 2, R)-transformation, in order
that the transition functions that relate the section Ṽ in overlaps of different
patches are still of the form (1.3.33).
We will essentially show that given a section V =
5 One can show that the fourth constraint also implies the third, unless n = 1, in which
case the fourth constraint is empty.
16
Since the metric is non-degenerate, lemma (A.3) implies that
I
fi hiI
rank
= n + 1,
X I FI
(1.3.50)
where we have introduced the following notations for the components of Ui :
I
fi ≡ Di X I
Ui =
.
(1.3.51)
hiI ≡ Di FI
Lemma (A.1) then implies that there exists a symplectic transformation S ∈
Sp(2n + 2, R) such that
I X̃
SV ≡ Ṽ =
,
(1.3.52)
F̃I
with
det
f˜iI
X̃ I
6= 0 .
(1.3.53)
Using lemmas (A.5) and (A.6), one can find a function F̃ (X̃), such that F̃I =
∂ F̃
. Note that one has made use of the constraints (1.3.44-1.3.46). These
∂ X̃ I
constraints are symplectically invariant and are hence satisfied by Ṽ as well.
Note that, in contrast to the rigid case, the existence of a prepotential is not
guaranteed. What is however true is that one can always go to a formulation
in which a prepotential is at hand, by applying a symplectic transformation.
A simple example that shows that the notion of a prepotential is not invariant
under symplectic transformations, is given by considering the following prepotential:
F (Z) = −iZ 0 Z 1 ,
(1.3.54)
The corresponding special Kähler manifold has n = 1 and one can choose a
coordinate z such that the symplectic vector v is given by
 0  

Z
1
 Z1   z 
 

v=
(1.3.55)
 ∂F0  =  −iz  .
∂Z
∂F
−i
∂Z 1
The corresponding manifold corresponds to SU(1, 1)/ U(1). The Kähler potential and metric are determined by
e−K = 2(z + z̄) ,
∂z ∂z̄ K = (z + z̄)−2 .
(1.3.56)
Let us now see what happens to the symplectic vector v upon performing a
specific symplectic mapping:




1 0 0 0
1
 0 0 0 −1 
 i 



ṽ = Sv = 
(1.3.57)
 0 0 1 0  v =  −iz  .
0 1 0 0
z
The transformed vector ṽ can clearly no longer be written in terms of a prepotential. The last two components cannot be written as functions of the first
two, so no prepotential F̃ (Z̃ 0 , Z̃ 1 ) can be found.
17
Finally, let us mention that in the frame in which a prepotential is at hand,
due to (1.3.53) and lemma (A.4), one also has that
h X B i
6= 0 ,
det ∂i
X0
(1.3.58)
where B = 1, · · · , n. The inverse function theorem then shows that one can also
B
use the n functions X
X 0 as coordinates on the manifold. The coordinate choice
zA =
XA
,
X0
A = 1, · · · , n ,
(1.3.59)
is often referred to in the literature as ’special coordinates’. In the above example, in the case one is working with the section v, the coordinate z = Z 1 /Z 0 is
an example of such a special coordinate choice. The example also shows that
such a special choice is not necessarily possible. Indeed, for the section ṽ, Z̃ 1 /Z̃ 0
does not lead to a good coordinate choice on the manifold.
Matrix formulation. As in the case of rigid special geometry, the constraints and properties of local special geometry can be concisely summarized
in a matrix formulation. One now defines the 2(n + 1) × 2(n + 1)-matrix:
 T 
V̄
 UT 
i

(1.3.60)
V=
 VT  .
iT
Ū
The inner products and differential equations satisfied by the quantities in V
are then summarized by
VΩV T = iΩ ,
Di V = Ai V ,
where

0
 0
Ai = 
 0
δij
0
0
δik
0
0
0
0
0
Dī V = Aī V ,
(1.3.61)

0
Cijk 
.
0 
0
(1.3.62)
The covariant derivatives now contain the Kähler connection as well as the LeviCivita connection (when acting on Ui for instance). Again one can prove that
C is a symmetric tensor, which is now covariantly holomorphic, and one can
obtain the following curvature formula
i l
Ri jk l = 2δ(j
δk) − Cjkm C̄ ilm .
(1.3.63)
Period matrix. For local special geometry, one can also define a period
matrix NIJ . In this case, it corresponds to an (n + 1) × (n + 1)-matrix, which
is defined in the following way:
−1
f¯īJ X J
NIJ ≡ h̄īI FI
.
(1.3.64)
Note that this definition is completely general. In particular, it does not depend
on whether one has a prepotential at hand or not. The definition assumes the
invertibility of
fiI X̄ I ,
(1.3.65)
18
an assertion which is proven in corallary (A.8). One can easily check that this
period matrix transforms under Sp(2n + 2, R) as in (1.3.30).
The definition of N also implies the following equation:
I < Ui , Ūj̄ > < Ui , V >
fi
J
J
†
¯
f
X
i
=
i(N
−
N
)
.
IJ
j̄
< V̄ , Ūj̄ > < V̄ , V >
X̄ I
(1.3.66)
Due to the constraints (1.3.43-1.3.49) the matrix on the left hand side is positive
definite. The invertibility of (1.3.65) then implies that the anti-hermitian part
of N is negative definite. One can furthermore derive a similar equation
I fi
< Ui , Uj > < Ui , V̄ >
T
J
J
f
X̄
(N̄ − N̄ )IJ
=
. (1.3.67)
j
< V̄ , Uj >
0
X̄ I
The conditions mentioned in (1.3.43-1.3.49) now imply that the right-hand side
is zero. The invertibility of (1.3.65) now leads to the symmetry of N . Using
this symmetry and the negative-definiteness of the imaginary part of N , we
can again invoke lemma A.2 to show that under a symplectic transformation,
A + BN is invertible and the transformation (1.3.30) is well-defined.
Note that in case a prepotential is at hand, the period matrix is more explicitly given by
NIJ (Z) = F̄IJ (Z̄) + 2i
Im FIK (Z) ImFJL (Z) Z K Z L
,
Im FKL (Z) Z K Z L
(1.3.68)
where again FIJ denotes the matrix of second derivatives of F with respect to
the variables X I .
1.4
Appearance in N = 2 supersymmetry and
supergravity
Special geometry is associated to the geometry of the scalars of vector multiplets
in theories with 8 supercharges (so-called N = 2 theories in four dimensions).
Let us first see how this happens in rigid supersymmetry and discuss local
supersymmetry (supergravity) afterwards.
The field content of an N = 2 vector multiplet is given by one complex
scalar X, one vector Aµ , two Majorana fermions Ωi (i = 1, 2) and a triplet
of scalar fields Yij (symmetric in (ij)), subject to a certain reality condition.
This supersymmetry multiplet forms an off-shell representation of the N = 2
supersymmetry algebra, which means that the algebra is realized on the fields,
without using the equations of motion for the fields. In the following, we will
mainly concentrate on the terms involving the complex scalar X and the vector
Aµ .
Consider now n of these vector multiplets. There are thus n complex scalars
A
X A , (A = 1, · · · , n) and n vectors AA
µ . Instead of using the field strengths Fµν ,
we will use the (anti-)self-dual combinations:
±A
Fµν
=
1 A
A
(F ± F̃µν
),
2 µν
19
i
A
F̃µν
= − εµνρσ F Aρσ .
2
(1.4.1)
It turns out that the supersymmetric Lagrangian is determined by the choice of
a holomorphic function F (X A ). Focusing on the terms that involve the scalars
X A and the vector fields, the Lagrangian is given by:
L
=
i∂µ FA ∂ µ X̄ A − i∂µ F̄A ∂ µ X A
1
1
−A −Bµν
+A +Bµν
+ iFAB Fµν
F
− iF̄AB Fµν
F
+ ··· ,
4
4
(1.4.2)
where the · · · represent fermionic terms and terms involving Yij . We have used
the usual notation
FA1 ···Aq =
∂
∂
···
F (X) .
∂X A1
∂X Aq
(1.4.3)
From the Lagrangian (1.4.2), one can easily extract that the complex scalars
X A parametrize an n-dimensional non-linear sigma model with metric
gAB̄ = −i(FAB − F̄AB ) .
(1.4.4)
This is a Kähler space, as its metric can be written in terms of a Kähler potential:
gAB̄ =
∂
∂
K(X, X̄) ,
∂X A ∂ X̄ B
K(X, X̄) = iX A F̄A (X̄) − iX̄ A FA (X) . (1.4.5)
The form of the Kähler potential shows that the scalars in N = 2 rigid supersymmetric theories indeed lie in a rigid special Kähler space. Furthermore, also
the period matrix gets a natural interpretation in the Lagrangian (1.4.2). Since
the period matrix in rigid special geometry is essentially given by:
NAB = F̄AB ,
(1.4.6)
one sees that the period matrix determines some of the couplings between the
scalar fields and the vector fields. The period matrix is just the scalar dependent
matrix that multiplies the kinetic terms of the vector fields. The other quantities
that we defined in rigid special geometry, get a physical interpretation in the
Lagrangian and supersymmetry transformation rules as well. As an example,
we mention that the Cijk -tensor gets the interpretation of Yukawa couplings in
the Lagrangian.
In case one is considering local supersymmetry, similar results hold. In this
case, special geometry arises in coupling a number of vector multiplets (with
field content described above) to the N = 2 gravity multiplet. The relevant
field content of the latter multiplet consists of the graviton gµν , 2 gravitini ψµi
(i = 1, 2), and one vector field (the so-called graviphoton). When considering n
vector multiplets coupled to supergravity, one thus has n complex scalars and
n + 1 vector fields (1 vector coming from the graviphoton). In this case, the
bosonic part of the Lagrangian is given by:
√
R
1
I
L =
−g
− gij̄ ∂µ z i ∂ µ z̄ j̄ + (ImNIJ )Fµν
F Jµν
2
4
1
I
J
− (ReNIJ )εµνρσ Fµν
Fρσ
.
(1.4.7)
8
It turns out that in the supergravity case, the metric gij̄ is the metric on a
local special Kähler space, whereas the matrix NIJ corresponds to the period
20
matrix defined on that space. All quantities defined for local special geometry
find an interpretation in either the Lagrangian or the explicit supersymmetry
transformation rules. In particular, the tensor Cijk , that appeared in (1.3.62),
appears in the Yukawa couplings in the Lagrangian. The section v(z) also
appears explicitly in the supersymmetry transformation rules. Note that one
does not suppose the existence of a prepotential in writing down the above
Lagrangian (and supersymmetry transformation rules). Any section v(z) that
obeys the constraints discussed earlier, can be used to write down a valid N = 2
supergravity Lagrangian, even when this section cannot be written in terms of a
prepotential function. When no prepotential exists, the period matrix will not
be of the form (1.3.68), but one can still compute it using the general formula
(1.3.64).
To end this section, let us note that the symplectic transformations that
appear in the definition of special geometry, have a physical interpretation in
terms of electric-magnetic duality transformations in four dimensions. Focusing
on the supergravity case, we note that the kinetic terms of the vectors in (1.4.7)
can be rewritten in terms of (anti-)self-dual field strengths as:
√
−g
+I +Jµν
Im(NIJ Fµν
F
).
(1.4.8)
L1 =
2
Defining the dual field strengths G+I as follows:
Gµν
+I ≡ 2i
∂((−g)−1/2 L1 )
= NIJ F +Jµν ,
+I
∂Fµν
(1.4.9)
the set of field equations and Bianchi identities for the n + 1 vectors can be
written as:
+I
∂ µ Im Fµν
=
0,
∂ µ Im Gµν
+I
=
0,
(1.4.10)
where the first equation corresponds to the Bianchi identities and the second
equation gives the equations of motion for the vectors. Note that the set of
Bianchi identities and field equations is invariant under G `(2n + 2, R). Namely,
if one forms a 2n + 2-dimensional vector
+I F
,
(1.4.11)
G+I
and acts upon this vector with a matrix in G `(2n + 2, R):
+ + A B
F
F̃
=
,
C
D
G
G̃+
+
(1.4.12)
one still has that
+I
∂ µ Im F̃µν
=
0,
G̃µν
+I
=
0.
∂ µ Im
(1.4.13)
Note however that the Gµν are related to the Fµν as in (1.4.9). We will therefore limit the G `(2n + 2, R) transformations to those that preserve the relation
21
(1.4.9). We will thus require that there exists a (symmetric) matrix ÑIJ , such
that as in (1.4.9)
G̃+ = Ñ F̃ + .
(1.4.14)
In other words, we will limit the possible transformations in such a way as to
require that the transformed dual field strengths can still be derived from a
Lagrangian L̃1 , which has a similar expression as in (1.4.8), with all quantities
replaced by the transformed versions with a tilde. Since
G̃+ = (C + DN )F + = (C + DN )(A + BN )−1 F̃ + ,
(1.4.15)
we obtain that
Ñ = (C + DN )(A + BN )−1 .
(1.4.16)
Requiring that this transformed matrix Ñ is again a symmetric matrix, leads
to the conditions (1.3.6) and hence we should restrict the G `-transformations
to symplectic transformations. If this restriction is made, one can indeed write
down a dual Lagrangian L̃1 , whose set of Bianchi identities and equations of
motion is given by (1.4.13). Due to the fact that these symplectic transformations, according to (1.4.12), mix the field strengths F with their duals G,
as well as Bianchi identities with field strengths, they form an extension of
electric-magnetic duality of ordinary Maxwell theory. If one would add electric
and magnetic sources to the equations (1.4.10), the symplectic duality transformtions would mix electric and magnetic charges.
The conclusion of the above discussion is that electric-magnetic duality can
be implemented on a Lagrangian, when one is able to construct an object N
that transforms as in (1.4.16). In N = 2 supersymmetry, the vector multiplets
not only contain vectors, but also scalar fields. Electric-magnetic duality acts
on the vector field strengths and their duals and supersymmetry then implies
that it has a similar action on the other fields in the same multiplet. In particular, the symplectic duality transformations should have an action on the scalar
fields. In N = 2 supergravity, this is realized by the action of Sp(2n + 2, R)transformations on the symplectic section v(z) that can be constructed over
the special Kähler manifold. Indeed, as a consequence of the fact that v(z)
forms a symplectic vector, the period matrix (1.3.64) indeed has the correct
transformation law (1.4.16) that is consistent with electric-magnetic duality.
The conclusion is thus that the cases in which the local special geometry cannot be formulated in terms of a prepotential, should not be discarded, but they
rather are dual formulations of a supergravity theory, in which a prepotential
is at hand. This is the reason why we formulated several definitions of special
geometry. Furthermore, as was discussed in the context of N = 1 supergravity,
also in the case of N = 2 supergravity one can gauge isometries of the scalar
manifold. We will not enter into the details of this construction, but we suffice
here by mentioning that the choice of a symplectic section is important for this
gauging procedure. This choice determines for instance which isometries can be
made local. Since the gauging procedure introduces a potential for the scalars
(similar to the D-term potential in N = 1 supergravity) and since this is the
only way to introduce a potential term for the scalars in N = 2 supergravity 6 ,
the question whether a symplectic section can be derived from a prepotential or
6 Note
that this is different from N = 1, where one can also have the F -term potential.
22
not can be a very physical one. We refer to [12] for more details on N = 2 gaugings. For more information on the relation between electric-magnetic duality
and special geometry, we refer to [13, 14].
Let us finally note an important difference between rigid and local supersymmetry, from the point of view of electric-magnetic duality transformations.
In order to implement electric-magnetic duality in a theory of scalars coupled
to vectors, one should have that
A + BN ,
(1.4.17)
is invertible, as is evident from (1.4.16). For rigid supersymmetry this leads to
the invertibility of
∂ X̃ A
,
(1.4.18)
∂X B
due to the fact that the period matrix is given by the matrix of second derivatives
of the prepotential. Taking the X A as coordinates on the special manifold, this
is equivalent to the invertibility of ẽA i (the upper part of ∂i Ṽ , where Ṽ is
the transformed section). In the discussion on the equivalence of the different
definitions of rigid special geometry, it was noted that this condition ensures the
existence of a prepotential. In rigid supersymmetry, it is thus possible to find
a prepotential function F̃ in terms of which the dual theory can be formulated.
This is no longer true in supergravity, due to the more complicated structure of
the period matrix in local special geometry. The invertibility of (1.4.17) then
no longer leads to a condition that guarantees the existence of a prepotential.
1.5
Calabi-Yau moduli spaces
Calabi-Yau spaces are n-dimensional Kähler manifolds for which the holonomy
group is not given by U(n), but is instead restricted to be SU(n). As a consequence of this, one can show that Calabi-Yau n-folds admit a metric that
is Ricci-flat. A further property of Calabi-Yau manifolds that we will need
in the following, is that they admit a nowhere vanishing holomorphic n-form
Ω. Calabi-Yau manifolds are abundant in string theory, as compactification of
higher-dimensional string theories on a Calabi-Yau manifold generically leads
to lower-dimensional theories that exhibit less supersymmetry than the parent
theory. For instance, compactification of type IIB or IIA supergravity (with 32
supercharges) on a Calabi-Yau threefold generically leads to four-dimensional
N = 2 theories (invariant under 8 supercharges). As usual, these compactifications lead to the appearance of a number of scalar fields in the lower-dimensional
theory, whose vacuum expectation values are associated to continuous deformations of the compact manifold. These deformations describe the moduli space
of the compactification manifold. As Calabi-Yau manifolds naturally lead to
N = 2 supergravities, some of whose scalars lie in special geometries, it is natural to assume that the moduli space of Calabi-Yau manifolds exhibits (local)
special geometry. In the following, we will show that this is indeed the case.
We will consider Calabi-Yau n-folds, where we will in the end mostly concentrate on the case n = 3. The complex coordinates on the Calabi-Yau manifold
will be denoted by z α , z̄ ᾱ , with α = 1, · · · , n. In these local coordinates, the
holomorphic n-form Ω is then given by
Ω(z 1 , · · · , z n ) = f (z 1 , · · · , z n )dz 1 ∧ · · · ∧ dz n ,
23
(1.5.1)
where f is a (nowhere vanishing) holomorphic function. Some references for
this section include [15, 16, 17].
1.5.1
The moduli space of Calabi-Yau manifolds
Let m, n = 1, · · · , 2n be indices denoting real coordinates on a Calabi-Yau nfold and let gmn be the Ricci-flat Calabi-Yau metric. We will then consider
the possible deformations δgmn of the metric, such that the deformed metric
gmn + δgmn still obeys the Calabi-Yau conditions. In particular, both gmn and
gmn + δgmn should be Ricci-flat:
Rmn (g) = 0 ,
Rmn (g + δg) = 0 .
(1.5.2)
Expanding the second equation in (1.5.2) to linear order in the fluctuations δg
and using the Ricci-flatness of g leads to the following equation:
∇k ∇k δgmn + 2Rm p n q δgpq = 0 .
(1.5.3)
Note that in order to obtain this equation, we have also used the following
∇m δgmn =
1
m
∇n δgm
,
2
m
with δgm
= g mp δgmp .
(1.5.4)
This is a gauge fixing condition, which is used in order to eliminate the metric
deformations δg that correspond to coordinate changes and are hence not of
interest. The equation (1.5.3) is known as the Lichnerowicz equation. The
number of independent solutions of the Lichnerowicz equation describes the
number of ways in which a Calabi-Yau manifold can be continuously deformed.
Upon choosing complex coordinates z α , z̄ ᾱ and using the index structure of
the metric and Riemann tensor of Kähler manifolds, one finds that the equations for the mixed components δgαβ̄ and the pure components δgαβ decouple.
The interpretation of the fluctuations δgαβ̄ and δgαβ is that they correspond
to deformations of the Kähler class, respectively the complex structure of the
manifold. Indeed, the fluctuations δgαβ̄ lead to a change in the Kähler form of
the Calabi-Yau manifold. The fluctuations δgαβ on the other hand result in a
metric which is no longer hermitian. In order to write the deformed metric in a
hermitian form (i.e. with no purely holomorphic and anti-holomorphic components), one thus has to choose complex coordinates in a different way. In other
words, one has to change the complex structure of the manifold.
With the deformations δgαβ̄ , one can associate the infinitesimal (1, 1)-form
δgαβ̄ dz α ∧ dz̄ β̄ .
(1.5.5)
Similarly, using the deformations δgαβ , one constructs the (2, 1)-form
Ωαβγ g γ δ̄ δgδ̄¯ dz α ∧ dz β ∧ dz̄ ¯ .
(1.5.6)
Note that this last form is complex, so one should also consider its complex
conjugate, involving the deformations δgᾱβ̄ . One can then show that as a consequence of the Lichnerowicz equation, the forms (1.5.5) and (1.5.6) correspond
to harmonic forms on the Calabi-Yau manifold. The upshot of the above discussion is thus that the independent solutions of the Lichnerowicz equation are in
24
one-to-one correspondence with independent harmonic (1, 1)- and (2, 1)-forms
on the manifold.
This moduli space of fluctuations that correspond to (1, 1)- and (2, 1)-forms
has a natural metric defined on it. The line element of this metric takes the
form
Z
1
√
d6 z g g αβ̄ g γ δ̄ δgαγ δgβ̄ δ̄ + δgαδ̄ δgγ β̄ ,
(1.5.7)
ds2 =
2V
where V is the volume of the Calabi-Yau manifold. This metric is called the
Weil-Petersson metric. Note that it consists of two parts, indicating that the
moduli space M of Calabi-Yau manifolds locally has a direct product structure:
M = M(2,1) × M(1,1) ,
(1.5.8)
where M(2,1) corresponds to the moduli space of complex structure deformations, while M(1,1) denotes the moduli space of Kähler deformations. The form
of this Weil-Petersson metric appears rather naturally in dimensional reduction.
It appears after expanding the Einstein-Hilbert term of a higher-dimensional
(e.g. 10-dimensional supergravity) theory for small metric fluctuations δg. The
fluctuations δg correspond in the lower-dimensional theory to scalar fields that
span a non-linear sigma model with a metric, determined by the above WeilPetersson metric. In the following, we will try to obtain explicit expressions for
the metrics on M(2,1) and M(1,1) .
1.5.2
The complex structure moduli space
Let us first investigate the structure of the moduli space of complex structure
deformations. Since the above reasoning showed that this moduli space is in
one-to-one correspondence with the space of harmonic complex (2, 1)-forms on
the Calabi-Yau manifold, we will parametrize it by (complex) coordinates wi ,
with i = 1, · · · , dim(H (2,1) ). We can then define a set of (2, 1)-forms according
to
χi =
∂δgγ̄ δ̄
1
(χi )αβγ̄ = − Ωαβ δ̄
.
2
∂wi
1
(χi )αβγ̄ dz α ∧ dz̄ β̄ ∧ dz̄ γ̄ ,
2
(1.5.9)
Indices are always raised and lowered with the hermitian metric gαβ̄ . As explained above, these forms are harmonic. Inversion of these relations then shows
that under a deformation of the complex structure, the components of the metric
are deformed by
δgᾱβ̄ = −
1
Ω̄ᾱ γδ (χi )γδβ̄ δti ,
||Ω||2
with ||Ω||2 =
1
Ωαβγ Ω̄αβγ .
6
(1.5.10)
If we write the line element on the moduli space (given by the first term in
(1.5.7)) as
ds2 = 2Gij̄ δti δ t̄j̄ ,
(1.5.11)
we find, upon plugging the deformations (1.5.10) in the expression for the WeilPetersson metric, the following metric on the complex structure moduli space:
R
i χi ∧ χ̄j̄
i j̄
R
Gij̄ δt δ t̄ = −
δti δ t̄j̄ .
(1.5.12)
i Ω ∧ Ω̄
25
It turns out that this metric is Kähler with Kähler potential K 2,1 given by
Z
2,1
K = −log i Ω ∧ Ω̄ .
(1.5.13)
To see how this comes about, we note that Ω depends on the complex structure,
due to its dependence on the holomorphic coordinates z α and on the differentials
dz α . Due to the fact that under a deformation of the complex structure, dz
becomes a linear combination of dz and dz̄, one has that under a complex
structure deformation Ω becomes a combination of (3, 0)- and (2, 1)-forms. More
precisely, we can state the following result (without proof):
∂i Ω = Ki Ω + χi ,
(1.5.14)
where ∂i = ∂/∂wi and Ki depends only on the coordinates wi but not on
the coordinates of the Calabi-Yau manifold. Calculating ∂i ∂j̄ K 2,1 and using
(1.5.14), one indeed ends up with (1.5.12).
Although we focused mainly on Calabi-Yau threefolds so far, the above
result is actually more general. For a general Calabi-Yau n-fold, the moduli space of complex structure deformations is Kähler, with Kähler potential
given by (1.5.13). For n = 3 however, one can show that this moduli space
is not just Kähler, but that it is actually local special Kähler. In order to
describe this in more detail, we introduce a basis of three-cycles AI , BJ with
I, J = 0, · · · , dim(H (2,1) ), chosen such that their intersection numbers are
\
\
\
\
AI
BJ = −BJ
AI = δJI ,
AI
AJ = B I
BJ = 0 .
(1.5.15)
The dual cohomology basis will be denoted by (αI , β I ) and is defined by the
following relations
Z
Z
Z
Z
αI = αI ∧ β J = δIJ ,
β I = β I ∧ αJ = −δJI .
AJ
BJ
Z
Z
J
β =
αJ = 0 .
(1.5.16)
AI
BI
The so-called periods of the holomorphic 3-form are then defined as follows:
Z
Z
I
Z =
Ω = Ω ∧ βI ,
AI
Z
Z
FI =
Ω = Ω ∧ αI .
(1.5.17)
BI
I
Note that these periods Z and FI depend on the complex structure moduli, via
Ω. We can take the Z I as coordinates on the dim(H (2,1) )-dimensional space of
complex structure deformations. However as there are dim(H (2,1) ) + 1 of them,
the Z I actually form an overcomplete set and one should instead take:
wi =
Xi
,
X0
i = 1, · · · , dim(H (2,1) ) ,
(1.5.18)
as coordinates on the moduli space. The FI -periods can then also be thought
of as functions of the X I : FI = FI (X). The holomorphic three-form Ω can be
expanded as
Ω = X I αI − FI (X)β I .
(1.5.19)
26
Due to (1.5.14), one has
Z
Ω ∧ ∂I Ω = 0 ,
(1.5.20)
∂FJ
.
∂X I
(1.5.21)
which implies
FI = X J
This last equation then implies that
FI =
∂
∂X I
1 J
X FJ
2
.
(1.5.22)
In other words, the periods FI are obtained in terms of a prepotential function
F in the following way:
FI =
∂F
,
∂X I
F =
1 J
X FJ .
2
(1.5.23)
Since
∂F
,
(1.5.24)
∂X J
the prepotential F is also homogeneous of degree two. Inserting the expansion
(1.5.19) in the formula for the Kähler potential (1.5.13), one finally concludes
that
I¯ ∂
2,1
I ∂
F̄ (Z̄) − Z̄
K = −log i Z
F (Z) ,
(1.5.25)
∂Z I
∂ Z̄ I¯
2F = X J
and hence that the complex structure moduli space is local special Kähler according to the first definition given in section 1.3.2.
1.5.3
The Kähler moduli space
Let us consider the case of a Calabi-Yau manifold of complex dimension 3. Let
ωi = Ui|αβ̄ dz α ∧ dz̄ β̄ be a basis of (1, 1)-forms on the manifold. The WeilPetersson metric in the space of (1, 1)-forms is then defined by the following
components:
R 6 √
d z g Ui|αβ̄ Uj|γ δ̄ g αδ̄ g β̄γ
2
R
.
(1.5.26)
gij = π
√
d6 z g
The constant π 2 in front of this expression just amounts to a convention. We
can rewrite (1.5.26) as
R
Z
ωi ∧ ∗ωj
1
gij = π 2 R 6 √ = R 6 √
d6 z Ui|αβ̄ ∗ Uj|γδ¯ζ̄ αγδ β̄¯ζ̄ ,
(1.5.27)
d z g
d z g
where we have defined the Hodge-dual of the (1, 1)-forms ωi by
∗ωj = ∗Uj|αβγ̄ δ̄ dz α ∧ dz β ∧ dz̄ γ̄ ∧ dz̄ δ̄ ,
(1.5.28)
1√
g¯γ̄ δ̄ ζαβ g η̄ζ g ι¯ Uj|ιη̄ .
4
(1.5.29)
where
∗Uj|αβγ̄ δ̄ =
Using the identities
δ̄¯ζ̄ gαδ̄ gβ¯ gγ ζ̄ =
27
√
gαβγ ,
(1.5.30)
and
[δ̄
ζ̄]
ᾱβ̄γ̄ δ̄¯ζ̄ = 3! δ[ᾱ δβ̄¯ δγ̄] ,
(1.5.31)
one can easily derive that
∗Uj|αβγ̄ δ̄ =
1 ¯ζ
(g Uj|ζ¯ ) gαγ̄ gβ δ̄ − gβ δ̄ Uj|αγ̄ .
2
(1.5.32)
Written in a coordinate-independent way, we have thus found that (denoting
the Kähler 2-form on the Calabi-Yau manifold by K):
3
2π
1 2π
(K.ωj )K ∧ K +
K ∧ ωj ,
(1.5.33)
∗ωj = −
2
i
i
with
i ¯ζ
g Uj|ζ¯ .
(1.5.34)
2π
Note that, due to the fact that ωj is harmonic, one has that d(g ¯ζ Uj|ζ¯ ) = 0.
This implies that
2π
K.ωj = constant = cj .
(1.5.35)
i
From the definition (1.5.28,1.5.29), one easily derives that
Z
Z
Z
√
√
d6 z g g αβ̄ Uj|αβ̄ = cj d6 z g = −2πi K ∧ ∗ωj .
(1.5.36)
K.ωj =
Using this result, (1.5.33) and the equation
Z
Z
(2π)3 i
√
K ∧ K ∧ K = d6 z g ,
3!
(1.5.37)
one finally obtains
R
K ∧ K ∧ ωj
3
i
R
cj = K.ωj = −
.
2π
(2π)2 K ∧ K ∧ K
The final result for the moduli space metric then reads
R
R
R
ωi ∧ ωj ∧ K 3 ( ωi ∧ K ∧ K)( ωj ∧ K ∧ K)
3
R
R
gij =
−
.
2
2
K∧K∧K
( K ∧ K ∧ K)2
(1.5.38)
(1.5.39)
Since the Kähler form K is a (1, 1)-form, it can be written as a combination of
the ωi basis-forms:
K = Imti ωi .
(1.5.40)
We have parametrized the Kähler form in terms of the imaginary part of complex
parameters ti . These complex parameters ti thus form coordinates on the space
of Kähler deformations of the Calabi-Yau manifold. In string theory, the real
parts of ti find their origin in the NS-NS B-field. Upon compactification on the
Calabi-Yau manifold, this two-form B-field also gives rise to lower-dimensional
scalar fields, that correspond to (1, 1)-forms on the Calabi-Yau manifold. It
turns out that in string theory, it is more appropriate to consider deformations
of the so-called complexified Kähler form KC :
KC = B + iK ,
28
(1.5.41)
where B denotes the B-field along the internal Calabi-Yau directions. In terms
of the basis ωi , this complexified Kähler form can then be expanded as
KC = ti ωi .
(1.5.42)
We furthermore also note that on a Calabi-Yau manifold, one has that
Z
ωi ∧ ωj ∧ ωk = dijk ,
(1.5.43)
where dijk are constants, the so-called
immediately get the following:
Z
K∧K∧K =
Z
ωi ∧ K ∧ K =
Z
ωi ∧ ωj ∧ K =
intersection numbers. One can therefore
dijk Imti Imtj Imtk ,
dijk Imtj Imtk ,
dijk Imtk .
(1.5.44)
Defining the function
with
Y ≡
K 1,1 = −log i Y ,
(1.5.45)
(2i)3
dijk Imti Imtj Imtk ,
3!
(1.5.46)
one finds that the moduli space metric gij is Kähler with Kähler potential K 1,1 :
Z
∂ ∂ 1,1
∂ ∂
4
gij = i j̄ K = − i j̄ log
K ∧K ∧K.
(1.5.47)
∂t ∂ t̄
∂t ∂ t̄
3
It is thus more appropriate to rename gij to gij̄ , so that the line element on the
moduli space is given by
ds2 = gij̄ δti δ t̄j̄ ,
(1.5.48)
with δti an infinitesimal deformation of the complexified Kähler form. Moreover,
if we define
Z0 = 1 ,
Z i = ti ,
1
Z iZ j Z k
F (Z) = dijk
.
3!
Z0
we find that
(1.5.49)
∂
¯ ∂
F̄ (Z̄) − Z̄ I
F (Z) .
(1.5.50)
∂Z I
∂ Z̄ I¯
Note that F (Z) defined above is homogeneous of degree two. The net result
is thus that the moduli space of (complexified) Kähler deformations of CalabiYau manifolds is local special Kähler, according to the first definition of section
1.3.2.
Y = ZI
29
Appendix
A.
Useful lemma’s in special geometry.
In this appendix, we collect some lemma’s that are useful in for instance showing
the equivalence between the several definitions of special geometry.
I X
Lemma A.1 If the 2m × m-matrix V =
, where X I and YI are mYI
vectors, is of rank
m, then
there exists a symplectic rotation S ∈ Sp(2m, R) such
X̃
that Ṽ = SV =
has the property that X̃ is invertible.
Ỹ
Proof.
The proof of this lemma proceeds by direct construction of S and is rather long
and technical. We refer the reader to appendix A of [8] for a full proof of this
lemma.
Lemma A.2 If N is a symmetric
m × m-matrix with strictly negative imagi
A B
nary part and
∈ Sp(2m, R), then A + BN is invertible. FurtherC D
−1
more, (C + DN )(A + BN ) is also a symmetric matrix with strictly negative
imaginary part.
Proof.
The proof of this lemma is rather technical and we refer the reader to appendix
A of [8] for a full proof of this lemma.
Lemma A.3 If V satisfies the conditions (1.3.43-1.3.46), and gij̄ is not degenerate then
XI
FI
W≡
Di X I Di FI
(where I = 0, · · · , n and i = 1, · · · , n) has rank n + 1.
Proof.
Suppose rank(W) ≤ n, then there exist λi and λ0 , not all zero, such that
λi Ui + λ0 V = 0 .
(A.1)
Taking first an inner product with V̄ , and using (1.3.47) and condition (1.3.43),
gives λ0 = 0, and thus λi is not trivial. Then taking the inner product with Ūβ ,
and using (1.3.47) and (1.3.49), we get λi gij̄ = 0, which is in contradiction with
our assumption that gij̄ is non-degenerate.
Lemma A.4
det
Di X B
XB
Di X 0
X0
B X
6= 0 ⇔ det ∂i
6= 0
X0
if X 0 6= 0.
30
Proof.
The proof follows from the following equalities:
Di X B Di X 0
∂i X B ∂i X 0
det
=
det
XB
X0
XB
X0
B B
0
X
∂i X 0
∂i X B − X
0 n+1
X 0 ∂i X
=
(X
.(A.2)
= det
)
det
∂i
X0
XB − XB
X0
Di X I
Lemma A.5 If det
XI
F̃I (X J ) exist such that
6= 0 and Dī
XI
FI
= 0, then some functions
F̃I (X J (z i , z̄ ī )) = FI (z i , z̄ ī ) ,
(A.3)
F̃I = X J ∂J F̃I .
(A.4)
and
Proof.
B
XI
FI
i
= 0, X
0 and X 0 are holomorphic functions of the z .
X
FI
Lemma A.4 guarantees that one can express the z i in terms of the independent
B
variables X
X 0 . Now set e.g.
Because of Dī
J
0
F̃I (X ) ≡ X fI
with
fI (z i ) ≡
z
i
XB
X0
FI (z i , z̄ ī )
.
X 0 (z i , z̄ ī )
,
(A.5)
(A.6)
Equation (A.4) is then immediate since F̃I is homogeneous of first degree.
Di X I
Lemma A.6 If det
6= 0, conditions 1.3.46 and 1.3.45 hold, and (A.3)
XI
and (A.4) hold, then some function F̃ (X J ) exists such that
F̃I = ∂I F̃ (X I ) .
(A.7)
Proof.
The two conditions mentioned can, using the equations of lemma A.5, be written
respectively as
0
=
Di X I Dj X J ∂[I F̃ J]
X I ∂i FI − FI ∂i X I = ∂i X I X J ∂[I F̃ J] = Di X I X J ∂[I F̃ J] . (A.8)
Di X I
, these equations imply that ∂[I F̃ J] =
Because of the invertibility of
XI
0, which is the integrability condition for the existence of F̃ (X).
0
=
31
Lemma A.7 If the matrix
i
hUi , Ūj̄ i
hV̄ , Ūj̄ i
is positive definite, then the matrix
hUi , V i
hV̄ , V i
Di X I
X̄ I
(A.9)
is invertible.
Proof.
Suppose a linear combination of the columns of this matrix equals zero:
ai Di X I + bX̄ I = 0 .
(A.10)
Then the same statement applies to the complex conjugate equation. So it
follows immediately that
hai Ui + bV̄ , āj̄ Ūj̄ + b̄V i = 0 .
This can be written as follows:
hUi , Ūj̄ i
i
−i a b
hV̄ , Ūj̄ i
hUi , V i
hV̄ , V i
j̄ ā
=0.
b̄
(A.11)
(A.12)
The positivity of (A.9) implies that a = bi = 0.
Corollary A.8 If the conditions hV, V̄ i = i and hV, Ui i = 0 are satisfied and
the metric gij̄ ≡ ihUi , Ūj̄ i is positive definite then the matrix Di X I X̄ I is
invertible.
32
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