June 2014

PITP 2014
String Compactification
Martijn Wijnholt
Arnold Sommerfeld Center, Ludwig-Maximilians Universität
Theresienstrasse 37
D-80333 München, Germany

An introduction to compactiﬁcations of heterotic strings, IIB, and F -theory.

Contents
1 Heterotic compactiﬁcation

3

1.1

Kaluza-Klein Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

BPS equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Consequences of ∇m ϵ6± = 0 . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Examples of Calabi-Yau manifolds . . . . . . . . . . . . . . . . . . . . . . .

9

1.5Moduli and Hodge diamond of a Calabi-Yau three-fold . . . . . . . . . . . 11

1.6

Vector bundles, stability and Dolbeault cohomology . . . . . . . . . . . . . 16

1.7

The standard embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 Perturbative IIB

23

2.1

Kaluza-Klein Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2

Orientifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3

D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4

S-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 F-Theory

31

3.1

Elliptic ﬁbrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2

The view from M -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3

Abelian gauge ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4

Digression on ADE singularities . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5

Non-abelian gauge ﬁelds and the Kodaira classiﬁcation . . . . . . . . . . . 40

3.6

Application to Grand Uniﬁcation . . . . . . . . . . . . . . . . . . . . . . . 42

3.7

Relation with perturbative IIB and heterotic strings . . . . . . . . . . . . . 44

2

1. Heterotic compactification
1.1. Kaluza-Klein Ansatz
Today we are going to focus on the heterotic string. All superstring theories, including
the heterotic string, naturally live in ten space-time dimensions. The ten-dimensionalﬁelds are as follows:
gravity multiplet :gauge multiplet :gM N , BM N , ϕ, ψM α , χβ̇AM , λα(1.1)

We adopted the convention of using capital letters M , N , P, . . . to denote ten-dimensional
space-time indices 0, . . . , 9. We use α to denote indices in the ten-dimensional positivechirality spinor representation 16R (or 16 if we want to omit the Majorana condition),and similarly we use β̇ for the negative chirality 16′ R . The eﬀective ten-dimensional
action is of the form
[
]
√
1 2
α′
−2ϕ
2
2
d x |g|eR − H + 4(∇ϕ) + Tr(F ) + SF ermi (ψ, λ, χ, A, g, B, ϕ)
3
30
(1.2)
and the ﬁeld strength H can be deﬁned at least locally as
1
S= 2
2κ

∫

10

H = dB + α′ ωL −α′
ωY M .
30(1.3)

Here ωL and ωY M denote the Chern-Simons forms for the spin and Yang-Mills connections,
respectively.
The symmetry group of this lagrangian is rather big: we have ten-dimensional superPoincaré invariance and an E8 × E8 or SO(32) gauge group. We are interested in more
realistic theories with only four-dimensional Poincaré invariance at large distance scales,
and a smaller gauge group. To break the symmetries we will turn on proﬁles for theten-dimensional ﬁelds.
A good Ansatz is to consider a ten-dimensional background of the form
M1,3 × X6(1.4)

where X6 is a compact six-dimensional real manifold, and M1,3 denotes four-dimensional
Minkowski space. As in the original Kaluza-Klein model, at scales large compared to theinverse radius of curvature of X6 , we eﬀectively reduce to a four-dimensional theory.
3

Which manifold should we take for X6 ? The ﬁrst possibility we might consider is astraightforward generalization of the classic Kaluza-Klein Ansatz: we simply take X6 =T 6 . One can do the Kaluza-Klein reduction very explicitly in this case. A toroidal
compactiﬁcation is often referred to as a Narain compactiﬁcation, and admits an elegant
formulation in terms of an even self-dual momentum lattice Γ. For the heterotic string
on T 6 , the compactiﬁcation is speciﬁed by a Lorentzian lattice Γ6,22 . The space of suchlattices (and hence the eﬀective four-dimensional theory) is parametrized by a coset
M = O(6, 22; Z)\O(6, 22; R)/O(6; R) × O(22; R)(1.5)

parametrizing deformations of internal components of gM N , BM N and AM . Such compactiﬁcations have a lot of interesting properties, but for our purposes they are a bit
too restrictive. To understand why let us look at the Kaluza-Klein reduction of theten-dimensional supersymmetry transformations.
Supersymmetry transformation are parametrized by a spinor ϵα in the 16R . Under
SO(1, 3) × SO(6) a Weyl spinor decomposes as
16R → (2, 4) + (2, 4)(1.6)

Since a torus is simply a quotient of ﬂat space by some translations, spinors which are
independent of the internal coordinates will be solutions, so we end up with four independent supersymmetry transformations in four dimensions. So the N = 1 supersymmetry
in ten dimensions reduces to N = 4 supersymmetry in four dimensions. One cannot geta chiral theory from breaking N ≥ 2 four-dimensional supersymmetry. This is perhaps alittle too much of a good thing.
In these lectures we want to focus on compactiﬁcations to four dimensions with onlyN = 1 supersymmetry. There is actually a simple way to get an N = 1 compactiﬁcationout of the above: we simply quotient out by a Z3 symmetry which kills three of the fourspinors and leaves one invariant. That is, we take X6 to be an orbifold T 6 /Z3 . This is
not the route we are going to take here, although there is a deﬁnite connection: one canresolve the ﬁxed points of the Z3 action to get a smooth manifold, of the general typethat we will consider.
1.2. BPS equations
To get a handle on this, we consider the supersymmetry variations of the ten-dimensionalﬁelds. The variation of the fermions is given by1δψM = ∇M ϵ − HM AB ΓAB ϵ + (F ermi)24MNδλ = FM N Γ ϵ + (F ermi)24δχ = ̸ ∇ϕϵ +1
HM N P ΓM N P ϵ + (F ermi)2
24
(1.7)

There are similar equations for the bosons, of the form δ(Bose) ∼ F ermi × ϵ. However
to preserve the four-dimensional Poincaré symmetry, the Fermi ﬁelds must vanish. Then
to get a four-dimensional supersymmetry generator, we have to ﬁnd a ﬁeld conﬁguration
for g, B, ϕ on X6 and a spinor ϵ such that δψ = δλ = δχ = 0. We will further assume
that H vanishes and ϕ is constant. The equations become∇M ϵ = 0,

FM N ΓM N ϵ = 0(1.8)

So let us analyze these equations.
We ﬁrst decompose ϵ according to (1.6) asϵ = ϵ4+ ⊗ ϵ6+ + ϵ4− ⊗ ϵ6−(1.9)

where ϵ4− = ϵ∗4+ and ϵ6− = ϵ∗6+ . (It would be more accurate to write ϵ1,3 instead of ϵ4 , but
we are saving notation here). Taking M to be an internal index and decomposing intochiral parts, we get the equation∇m ϵ6± = 0.
(1.10)
This equation means that X6 must admit a covariantly constant spinor. By Leibnitz’
rule, such a spinor will have constant norm, and we can normalize such that the norm is
one everywhere. We will assume this in the following.
1.3. Consequences of ∇m ϵ6± = 0
The existence of a covariantly constant spinor is a strong condition on X6 , and in thissubsection we will spell out the consequences. Suppose that we have such a spinor, andlet us consider
Jmn = i ϵ†6+ Γmn ϵ6+(1.11)
where Γmn = 21 (Γm Γn − Γn Γm ). With some algebra, one shows that Jmn squares to minuspone (i.e. Jmn Jnp = −δm). This suggests we should introduce a complex coordinate systemjj̄z , with conjugates z , j = 1, 2, 3, such that Jmn takes the standard form
Jj̄ k̄ = −i δj̄k̄

Jj k = i δjk ,(1.12)

But this is generally not possible. Although it can be done at any given point, we may
not be able to extend it to an open neighbourhood around that point. Basically if we
could construct such a coordinate system, then we would have a set of tangent vector
5

ﬁelds that have eigenvalue +i under J and whose Lie bracket preserves that condition.
This integrability condition corresponds to the vanishing of a tensor (the anti-holomorphic
projection of this Lie bracket), which may be written as the Nijenhuis tensor:
Nmnp = Jmq ∇q Jnp − Jmq ∇n Jq p − Jnq ∇q Jmp + Jnq ∇m Jq p

(1.13)

Conversely, the Newlander-Nirenberg theorem states that when this integrability condition is satisﬁed, then local holomorphic coordinate systems do exist and we have a complex
manifold.
Now our Jmn is actually covariantly constant (again by Leibnitz’ rule), so we see thatin our case the Nijenhuis tensor vanishes and Jmn deﬁnes an integrable complex structure.
In other words, X6 must be a complex manifold.
It will often be convenient to use complex notation. We will generally use the indicesi, j, k when we work in a complex coordinate system. The complexiﬁed tangent bundle
splits into a holomorphic tangent bundle spanned by ∂/∂z i and an anti-holomorphic
tangent bundle spanned by ∂/∂z ī . Similarly the complexiﬁed cotangent bundle splits intoa holomorphic version spanned by dz i and an anti-holomorphic version spanned by dz ī .
When X is a complex manifold, we will take T X to mean the holomorphic tangent bundle
and T ∗ X to mean the holomorphic cotangent bundle. By the same token, we use T X
and T ∗ X to denote their anti-holomorphic versions.
We further claim that gmn is a Hermitian metric with respect to the complex structure,in other words that Jmp Jnq gpq = gmn . To see this, ﬁrst note that from (1.11) we get
Jmp gpq = −Jq p gpm , and hence Jmp Jnq gpq = −Jq p Jnq gpm = δnp gpm = gnm . In holomorphic
coordinates and using (1.12) this is equivalent to
gij = gīj̄ = 0(1.14)

i.e. g only has mixed components of the form gij̄ and gīj . Next we consider the associatedtwo-form
Jmn = Jmq gqn(1.15)We already noted that anti-symmetry follows from (1.11). In complex notation, this issimply Jij̄ = igij̄ = −Jj̄i and Jij = Jīj̄ = 0. This two-form is closed, again due tocovariance constancy. It is called a Kähler form, and by deﬁnition a complex manifoldwith a Hermitian metric whose Kähler form is closed is called a Kähler manifold.
From ∇m ϵ = 0 it also follows that
1[∇m , ∇n ]ϵ = Rmnpq Γpq ϵ = 04(1.16)

Contracting with Γn and using Γn Γpq = Γnpq + g np Γq − g nq Γp , we get that Rmnpq Γnpq ϵ −2Rmq Γq ϵ = 0. But Rmnpq Γnpq = 0 by the ﬁrst Bianchi identity, so Rmq Γq ϵ = 0 and hence
Rmn = 0, and we ﬁnd that X6 must be Ricci ﬂat.
6

s n
On any Kähler manifold, from ∇J = 0 one ﬁnds that Jmn Rm
pqr = Jp R sqr . Contractingwith g qr and using (1.12) we see that in complex coordinates the only non-vanishingcomponents of the Ricci tensor are of type (1, 1). In fact with a bit more calculation one
ﬁnds that we can locally write

Rij̄ = −∂i ∂j̄ log det(g)(1.17)

Now just as we constructed a two-form from gij̄ , so too we construct a two-form from
Rij̄ , the Ricci form Rij̄ dz i ∧ dz j̄ . It is closed by virtue of the second Bianchi identity, andhence deﬁnes a cohomology class, the ﬁrst Chern class of our Kähler manifold:c1 (T X) =1
2
[Rij̄ dz i ∧ dz j̄ ] ∈ HdR(X6 , R)
2π

(1.18)

The ﬁrst Chern class is usually deﬁned slightly diﬀerently, but one can show it is equivalent
to the above. Since we found that X6 must be Ricci ﬂat, c1 (T X) vanishes. A Kähler
manifold with c1 (T X) = 0 is called a Calabi-Yau manifold. So we found that our X6should be a Calabi-Yau three-fold. Note that for complex manifolds we typically countthe complex rather than the real dimension.
The condition that X6 is Calabi-Yau is thus clearly necessary to solve the system∇m ϵ6± = 0, but could it also be suﬃcient? It turns out that it is, but before we getto it, we can gain a better understanding by looking at this from the point of view ofholonomies.
Recall that given a tangent vector v ∈ Tp M and a loop γ(t) based at p, we canparallel transport v around the loop and get another vector v ′ ∈ Tp M by solving the ODE∇γ̇(t) v(t) = 0 along γ(t). On a Riemannian manifold they are related as v ′ = Rv for arotation R ∈ SO(d). By considering general loops, these rotations generate a subgroup
H ⊂ SO(d), the holonomy group. In general, if there is no extra structure, then H is allof SO(d). If the loop is very small, enclosing an inﬁnitesimal area element δAmn , then bya local calculation one ﬁnds that the holonomy around the loop is δpq + δAmn Rmnpq , socurvature and holonomy are closely related.
We can similarly parallel transport spinors. We have Spin(6) = SU (4), and thepositive chirality spinor representation of SO(6) is identiﬁed with the fundamental representation of SU (4), usually denoted as 4. Now a covariantly constant spinor comes backto itself after parallel transport (since clearly ∇m ϵ = 0 implies ∇γ̇(t) ϵ = 0). Using an
SU (4) transformation, we can write this spinor in the form
(0, 0, 0, ∗) ∈ 4(1.19)

Thus the stabilizer, the subgroup of SU (4) that ﬁxes such a spinor, is SU (3) ⊂ SU (4),acting on the ﬁrst three components. So the existence of a covariantly constant spinor isthe statement that there must exist a metric with SU (3) holonomy on X6 .
7

On a Kähler manifold the holonomy reduces to U (3) = SU (3) ×Z3 U (1). This is the
subgroup of SO(6) that preserves the Kähler form. The U (1) is the determinant of U (3).
Now vectors in the cotangent bundle transform as the 3 of U (3), so the top exterior powerΛ3 T ∗ X, whose sections are locally of the formdz 1 ∧ dz 2 ∧ dz 3

(1.20)

transforms as the determinant. The top exterior power of T ∗ X appears so often that it
has a special name; it is also called the canonical bundle and denoted as KX . So theU (1) part of the U (3) holonomy on a Kähler manifold comes from parallel transporting(3, 0) forms, and by the relation between holonomy and curvature, the holonomy reduces
to SU (3) when the curvature of KX = Λ3 T ∗ X vanishes.
Indeed, we can use our covariantly constant spinor to write down a global section:TΩ3,0
ijk = ϵ6+ Γijk ϵ6+(1.21)

This is referred to as the ‘holomorphic three-form’ or sometimes the ’holomorphic volumeform’. It is nowhere vanishing, so it explicitly trivializes Λ3 T ∗ X, and hence c1 (Λ3 T ∗ X) =−c1 (T X) = 0, as we derived previously. In fact we can calculate the curvature of KX . It is
given by Rmnpq Jrq g pr , which yields the Ricci form Rij̄ dz i ∧ dz j̄ , tying everything together.
The conclusion is that we will have solved ∇m ϵ6± = 0 if we can show that the holonomy
is reduced to SU (3), which will be true if we can show that our Kähler manifold admits a
Ricci ﬂat metric. Now ﬁnding a Ricci ﬂat metric explicitly is extremely hard – eﬀectively
we have to solve a complicated non-linear PDE of the form Rij̄ ∼ ∂i ∂¯j̄ log det g = 0.
Checking the necessary conditions that X is a Kähler manifold with c1 (T X) = 0 would
clearly be much simpler, but it seems like a weaker condition, a topological condition
instead of a diﬀerential geometric condition. This is where a famous conjecture of E.
Calabi comes in: the claim is that the vanishing of c1 (T X) on a Kähler manifold is also
suﬃcient for the existence of Ricci ﬂat metrics. The conjecture was settled in a Fields
Medal-worthy theorem by S.T.Yau:
A compact Kähler manifold X with c1 (T X) = 0 admits a Ricci ﬂat metric.
The metric is uniquely determined by the complex structure and the Kählerclass.
In honour of this theorem, such Ricci ﬂat metrics are called Calabi-Yau metrics.
Let us summarize what we have seen. We were looking for compactiﬁcations of theheterotic string to four dimensions with N = 1 supersymmetry. Using the Kaluza-Klein
Ansatz M1,9 = M1,3 × X6 , we found that X6 should admit a covariantly constant spinor.
We then further saw that all the following statements about X6 are equivalent:
1. ∇m ϵ6+ = 0 (existence of a covariantly constant spinor);8

2. X6 admits a metric of SU (3) holonomy;
3. X6 is a Kähler manifold with a Ricci ﬂat metric;
4. X6 is a Calabi-Yau manifold (a Kähler manifold with c1 (T X) = 0).
In particular, to construct examples we can use formulation number four, which is the easiest to deal with. In order to satisfy our BPS equations we still have to solve FM N ΓM N ϵ = 0.
We will set F = 0 for now, and get back to this later.
1.4. Examples of Calabi-Yau manifolds
At this stage it is good to look at some examples. Everybody’s favourite example ofa Kähler manifold is complex projective space: CPn = (Cn \{0})/C∗ , where the C∗ actsas(z0 , . . . , zn ) ∼ (λz0 , . . . , λzn ),λ ∈ C∗(1.22)
There is a natural metric called the Fubini-Study metric. In the patch z0 ̸= 0 it is given
by
(
)
n
∑
|zk /z0 |2(1.23)gij̄ = ∂i ∂¯j̄ log 1 +k=1

with analogous expressions in the other patches zk ̸= 0. It’s straightforward to checkthat the associated Kähler form is closed. However the metric is not Ricci ﬂat. Indeed,Λn T ∗ CPn is a non-trivial line bundle, often denoted as OCPn (−n − 1). (This notation
indicates it has local sections of the form Qd (z)/Pd+n+1 (z), where Q and P are polynomialsof the indicated degree.)But submanifolds cut out by algebraic equations are also Kähler manifolds. They areclearly complex manifolds and the pull-back of the Fubini-Study form is still closed. So
subvarieties (solutions of polynomial equations) in projective spaces are a natural factory
to produce additional Kähler manifolds. By carefully choosing the degree of the equations,
we can make examples with c1 (T X) = 0.
One of the most famous examples of this kind the quintic hypersurface in CP4 . Its
equation is given by
P5 = z05 + z15 + z25 + z35 + z45 = 0(1.24)and deformations thereof. It’s not hard to show that c1 (T X) = 0, either by a standard
Chern class computation or by writing down the holomorphic (3, 0) form.1 More generally,an equation of degree n + 2 in CPn+1 yields a Calabi-Yau n-fold. For n = 2 we get theEg. consider the sequence 0 → T X → TCPn |X → NX → 0, and let c1 (OCPn (1)) = H denotethe hyperplane class. If X is given by an equation of degree d, then c1 (NX ) = dH|X , and we havec1 (TCPn ) = c1 (OCPn (n + 1)) = (n + 1)H. It follows that c1 (T X) = (n + 1 − d)H|X , which vanishes if
d = n + 1.
1

9

K3 surface (where ‘surface’ is used in the complex sense; it is a four-manifold). For n = 1
we get the elliptic curve (i.e. the two-torus T 2 ).
By analogy with the Narain story, we would like to understand deformations we cando that preserve the equations we have just studied, i.e. the Calabi-Yau conditions.
Such deformations are called moduli and manifest themselves as scalar ﬁelds in the fourdimensional eﬀective theory. The N = 1 supersymmetry pairs up these scalar ﬁelds withchiral fermions into chiral multiplets.
Let’s see in how many ways we can deform the equation of a quintic. Naively we can
write
(
)
4+5
= 126
(1.25)
5
diﬀerent monomials of degree ﬁve on CP4 . However, adding multiples of P5 yields thesame space, and we further have to mod out by the P GL(5, C) coordinate transformations,of which there are 24. So the equation of a quintic depends on 126 − 1 − 24 = 101 complex
parameters, or ‘moduli.’Counting the deformations of the complex structure this way can be a little hazardous.
It happens to work for the quintic, but not in general. It’s interesting to do an analogous
count for the K3 surface, realized as a quartic in CP3 :P4 = z04 + z14 + z24 + z34 = 0(1.26)The same type of count gives us 35 − 1 − 15 = 19 complex deformations of the quartic.
However using the results of the next subsection it is not hard to see that the K3 surface
actually lives in a 20-dimensional family, as apparently ﬁrst recognized by Kodaira. The
remaining deformation cannot be realized by the embedding.
Apart from this, the quintic has a “breathing mode” inherited from CP4 . This doesn’tchange the equation of the quintic, in other words it doesn’t change the complex structure,but it does change the metric since we are changing volumes. Such a parameter is calleda ‘Kähler modulus’ since we can view it as changing the Kähler form J. In the sameway, we can use this two-form not to deform the metric, but to deform the B-ﬁeld. It isnatural to combine the two in a single complex two-form
JC = J + iB

(1.27)

called the complexiﬁed Kähler form, it being understood that deformations of the imaginary part correspond to B-ﬁeld deformations.
Another famous example of a Calabi-Yau is the Tian-Yau manifold, which is a submanifold of CP3 × CP3 . Using coordinates zi for the ﬁrst CP3 and wj for the second10CP3 , the Tian-Yau manifold is deﬁned by the equations
3∑

zi wi = 0,3∑zi3 = 0,

i=0

i=0

3
∑

wi3 = 0(1.28)

i=0

and deformations thereof. It has 23 complex structure deformations and 14 Kähler deformations. Only two of the Kähler deformations can be realized in CP3 × CP3 .
1.5. Moduli and Hodge diamond of a Calabi-Yau three-fold
What we saw for the quintic is indicative of the general story. To see this, let us
describe some additional properties of Kähler manifolds.
For any manifold we can deﬁne the complex of diﬀerential formsdd

0 → Ω0 (M, R) → · · · → Ωd (M, R) → 0(1.29)

where d2 = 0. This allows us to deﬁne a set of topological invariants called de Rham
cohomology classes:k
HdR(M, R) = {closed k − forms/exact k − forms}(1.30)

where a form α is closed if dα = 0, and exact if α = dβ for some form β. The dimensionsof these cohomology groups are called the Betti numbers, and denoted by bk .
The next item we want to review is a fundamental result known as the Hodge decomposition. First, by contracting with the ε-tensor and the metric one can deﬁne an
isomorphism ∗ : Ωk (M, R) → Ωd−k (M, R). Explicitly it is given by(∗α)i1 ...id−k =1j1 ...jk
αj1 ...jkεk! i1 ...id−k(1.31)

Using the ∗-operator we can conveniently express the L2 -inner product on forms as∫⟨α, β⟩ =α ∧ ∗β.

(1.32)∫∫We deﬁne the adjoint of d by a ∧ ∗dβ = d∗ α ∧ ∗β for any forms α, β. Integrating by
parts and using ∗2 = (−1)k(n−k) one ﬁnds that d∗ = − ∗ d∗. The Hodge Laplacian on
diﬀerential forms is then deﬁned as∆d = dd∗ + d∗ d.
11(1.33)

Since ⟨α, ∆α⟩ = ⟨dα, dα⟩ + ⟨d∗ α, d∗ α⟩, a form α is harmonic if and only if dα = d∗ α = 0.
We denote by Hk the harmonic forms of degree k. Then the Hodge decomposition is givenby the following orthogonal decomposition:Ωk (M, R) = Hk ⊕ dΩk−1 (M, R) ⊕ d∗ Ωk+1 (M, R)(1.34)

This decomposition has the following important consequence. Let us consider a generalk-form ω and decompose it into its harmonic, exact and co-exact pieces:ω = αH + dβ + d∗ γ

(1.35)

Now suppose that we are interested in closed forms. Then 0 = dω = dd∗ γ. It follows that⟨d∗ γ, d∗ γ⟩ = ⟨γ, dd∗ γ⟩ = 0 and hence d∗ γ = 0. Thus for closed forms we have the simpler
decomposition
ω = αH + dβ
(1.36)
Descending to cohomology, we see that every cohomology class contains a unique harmonic representative. The spectrum of harmonic forms is completely determined by thecohomology of the underlying manifold. From d∗ = − ∗ d∗ we also see that the ∗-operator
commutes with the Laplacian and hence descends to harmonic forms and to cohomology.
Thus we have H k (M, R) ≃ H d−k (M, R), which is the manifestation of Poincaré duality
in this language.
Now we would like to discuss an analogue of this story for complex manifolds. On a
complex manifold it is natural to consider complex valued forms and complex valued deRham cohomology classes. We can then further try to decompose such forms into forms
which have a ﬁxed number of p holomorphic and q anti-holomorphic indices. We denotesuch forms by Ωp,q (X, C) and refer to (p, q) as the type of the form, which is a reﬁnementof the degree.
The exterior derivative can also be decomposed as
d = ∂ + ∂¯

(1.37)

where ∂ : (p, q) → (p + 1, q) and ∂¯ : (p, q) → (p, q + 1). From d2 = 0 and by preservationof degrees one ﬁnds that ∂¯2 = 0, so we could use ∂¯ to deﬁne cohomology groups. For eachp we consider the complex∂¯∂¯

0 → Ωp,0 (X, C) → · · · → Ωp,n (X, C) → 0(1.38)

p,q
and denote the corresponding cohomology by H∂p,q(X). The dimen¯ (X) or simply by Hp,qp,qsions of H as a vector spaces are denoted by the Hodge numbers h .

12On complex forms, the Hodge ∗-operator is deﬁned to include complex conjugation
after applying the ordinary ∗-operation. It is therefore sometimes denoted as ∗¯, but wewill leave this implicit. Analogous to the de Rham case, we can now deﬁne a Hermitian
metric on the space of complex forms and deﬁne an adjoint ∂¯† , and an associated Laplacian¯ There is a Hodge decomposition for the ∂-complex:¯∆∂¯ = ∂¯∂¯† + ∂¯† ∂.
¯ p,q−1 (X, C) + ∂¯† Ωp,q+1 (X, C)Ωp,q (X, C) = Hp,q + ∂Ω

(1.39)

and as before, every class in H p,q (X) has a unique harmonic representative.
On a Kähler manifold one gets some further nice properties, which imply additional¯
constraints on the hp,q . In particular one ﬁnds that the d-Laplacian and the ∂-Laplacianare essentially the same, namely we have ∆d = 2∆∂¯. It follows that the d-Laplacian
commutes with the projection on (p, q) components, and the resulting harmonic forms¯
are those of the ∂-Laplacian.
On a Kähler manifold then we get the decompositionk
HdR(X, C) =∑H p,q (X)

(1.40)

p+q=k

By complex conjugation we have hp,q = hq,p , and from the ∗-isomorphism we have hp,q =hn−p,n−q . By taking powers of the Kähler form one further ﬁnds that hp,p > 0 for any p.
Let us consider Calabi-Yau three-folds in particular, and try to eliminate the redundancy in the Hodge numbers using the above relations. We can also use that h3,0 = 1 ona Calabi-Yau three-fold since there is a unique holomorphic three-form. Further, one may
show that harmonic (i, 0) forms on a Kähler manifold are covariantly constant. Brieﬂy,
one uses a Bochner-Weitzenböck identity∆∂¯α = −∇† ∇α + Rīj̄ ıj̄ α(1.41)

where ∇† ∇α is the Laplacian constructed from the covariant derivative, ıj̄ denotes contraction with∂/∂z j̄⟩, and Rīj is the Ricci curvature. Then for harmonic (i, 0) forms we⟨have 0 = α, ∇† ∇α = ⟨∇α, ∇α⟩ and so such forms are covariantly constant as promised.
(On a Calabi-Yau this even works more generally since the Ricci curvature vanishes).
Since harmonic forms of type (1, 0) transform in the 3 of the SU (3) holonomy, if h1,0
were non-zero then we would have a state in the 3 which would need to be preserved, andso the holonomy would have to be reduced to SU (2) or even smaller. This would meanthat there is another covariantly constant spinor and hence an enhanced supersymmetry
in the compactiﬁed theory. By a Calabi-Yau three-fold we will usually mean a ‘proper’Calabi-Yau for which the holonomy group is strictly larger than SU (2). For such ‘proper’
three-folds then we must have h1,0 = 0, and by similar reasoning we also ﬁnd h2,0 = 0.
Putting all this information together, it follows that the Hodge diamond takes the13following form
h3,3
h
h

3,2

3,1

h3,0

h

2,2

h2,1
h

2,0

0
h

1,3

h1,2
h

h

1
h2,3

1,1

1,0

h
h

0
h0,3

=

0,2

0
h

1,1

h2,1

1
0

0,1

h

1,1

0

h0,0

0
h2,1

1

(1.42)

001After taking all the constraints into account, there are really only two independent numbers left, h1,1 (X) and h2,1 (X).
Next we want to study deformations of g and B that preserve the equations we havejust studied. We ﬁrst consider deformations of the metric, δgmn . We can split these into
two types of deformations according to the type of the indices, δgij̄ and δgij .
We may view deformations δgij̄ as inﬁnitesimal deformations of the Kähler form Jij̄ .
Recall that there exists a unique Calabi-Yau metric for a given Kähler class, so δgij̄ dz i ∧dz j̄deﬁnes itself a cohomology class of type (1, 1). Such deformations are called Kähler moduli
and their number is counted by h1,1 (X). Again we can complexify this by pairing with
deformations of the B-ﬁeld.
We can view δgīj̄ as deformations of the complex structure, because in the originalcomplex structure it gives a (0, 2) component to the deformed metric, so we will have todeform the complex structure to make the metric of type (1, 1) again. Alternatively let
us consider µjī = δgīk̄ g k̄j , which is a (0, 1) form valued in the tangent bundle T X. We canuse this to deform the Dolbeault operator∂¯ī → ∂¯ī′ = ∂¯ī + µjī ∂j(1.43)and hence the deﬁnition of complex coordinates. In order for this to be consistent we¯ = 0. Furthermore, deformations ofneed ∂¯′2 = 0, which for inﬁnitesimal µ gives us ∂µjj
the form µī = ∂¯ī v can be undone by a non-holomorphic coordinate transformation, so
do not really correspond to complex structure deformations. This means that µ deﬁnes a
Dolbeault cohomology class in H 1 (T X) (deﬁned in the next subsection).
Since I haven’t yet deﬁned Dolbeault cohomology of holomorphic bundles, let me map
it to something more recognizable. On a Calabi-Yau three-fold we can contract µjī with
3,0
the holomorphic (3, 0) form, to get the closed form ωklī = µjī Ω3,0
jkl of type (2, 1). Since Ωis holomorphic and nowhere vanishing, this gives an isomorphism between H 1 (T X) andH 2,1 (X). Thus the number of such deformations is counted by h2,1 (X).
A related way to think about this is as follows. As we deform the complex structure of
the Calabi-Yau, the holomorphic (3, 0) form will no longer remain of type (3, 0) but pick
14

up pieces of mixed type. Since tangent vectors to the complex structure moduli space are
given by µjī , to ﬁrst order the holomorphic (3, 0) form is going to deform asΩ3,0 → Ω3,0 + µ · Ω3,0

(1.44)

But as we saw, µ · Ω3,0 corresponds to (2, 1) forms which are counted by h2,1 (X).
The fact that inﬁnitesimal complex structure deformations of a complex manifold arecounted by H 1 (T X) is a general result. On a Calabi-Yau manifold these inﬁnitesimal
deformations can actually be promoted to ﬁnite deformations. This is not completely
obvious, eg. the ﬁrst obstruction [µjī , µlk̄ ] lies in H 2 (T X) ≃ H 2,2 (X) which is non-zeroon a Calabi-Yau, so one has to actually check that this and higher order obstructions all
vanish, and then appeal to results in deformation theory to argue that one gets a ﬁnite
deformation. This was shown independently by Tian and Todorov.
We see that the two types of deformations, complex and Kähler, precisely ‘use up’
the independent parameters in the Hodge diamond. For the quintic we have h1,1 (X) = 1,
counting the real dimension of the Kähler moduli space, and h2,1 (X) = 101, counting
the complex dimension of the complex structure moduli space. Thus for the metricdeformations the moduli space is given by
MCY = Mcomplex × MKaehler(1.45)To this we want to add deformations of the B-ﬁeld. Imposing the gauge condition d∗ B = 0,
the equations of motion for B simply reduce tod∗ dB = ∆d B = 0(1.46)

So deformations of B are described by harmonic two-forms, which are counted by h1,1 (X).
Adding the B-ﬁeld deformations has the eﬀect of complexifying MK .
Note that on a K3 surface we can contract a class in H 1 (T X) with the holomorphic
two-form to get a cohomology class in H 1,1 (K3). The Hodge diamond for the K3 surface
turns out to be
1
0
0
1
20
1
(1.47)
001
In particular we have h1,1 (K3) = 20, so we see that the K3 surface lives in a twentydimensional complex family, as promised previously, and not in a nineteen-dimensionalfamily.

15

We have arrived at a partial generalization of the Narain story, describing deformationsof the metric and B-ﬁeld. The Narain space also included deformations of the gauge ﬁeld;
we will get to these in the next subsection.
1.6. Vector bundles, stability and Dolbeault cohomology
Now we need to talk about the remaining equation Fmn Γmn ϵ = 0. Above we temporarily set F = 0, but if h1,0 = 0 (more precisely if π1 (X6 ) = 0) this means we won’t ﬁnd
any non-trivial solutions. So if we want to break the ten-dimensional E8 × E8 or SO(32)gauge symmetry, we have to allow for F ̸= 0. Not only that, setting F to zero is not evenconsistent with the tadpole cancellation condition, dH ∼ Tr(R ∧ R) − Tr(F ∧ F ).
On a Kähler manifold, the equation Fmn Γmn ϵ = 0 can be broken up into two equations:
F 0,2 = 0,g ij̄ Fij̄ = 0(1.48)

On a Calabi-Yau, a quick way to see this is to use ϵT6 Γī1 ...īn ϵ6 ∼ Ωī1 ...īn to get the ﬁrst
equation, and iϵ†6 Γµν ϵ6 ∼ Jµν to get the second. On a four-dimensional Kähler manifold
these equations say that the ﬁeld strength is anti-self dual (ASD), so we may think of thisas a generalization of the instanton equations. They are called the Hermitian Yang-Mills
equations. Let us discuss them in turn.
Since F 0,2 = 0, the (0, 1) part of the connection is pure gauge¯
A0,1 = Λ−1 · ∂Λ

(1.49)and can be set to zero by a gauge transformation. You must remember here that when we
switched to complex coordinates we made our gauge ﬁelds complex. Eg. if we started witha U (N ) gauge group, then A0,1 takes values in GL(N, C), and the gauge transformation
Λ used to set A0,1 to zero is also valued in GL(N, C). For otherwise we could apply thesame argument to A1,0 simultaneously and conclude that the bundle is trivial, which is
not true in general.
e0,1 on a diﬀerent patch. On the overlap we would
Now let’s consider the connection Ahave¯ · Λ−1
e0,1 = ΛA0,1 Λ−1 − ∂ΛA
(1.50)
e0,1 to zero by a suitable
We saw that in their own patches, we could set A0,1 and A(complexiﬁed) gauge transformation. So the condition F 0,2 = 0 says that it is possible to¯ = 0, i.e the transition functions are holomorphic.
choose transition functions such that ∂ΛA bundle which admits holomorphic transition functions is a holomorphic vector bundle.
By analogy with the Riemannian connection on the tangent bundle we will want theconnection to be compatible with a choice of Hermitian metric on our bundle V , in orderto reduce the structure group back from the complexiﬁed gauge group to the original
16

group (i.e. from GL(N, C) to U (N ) in the example given previously). Although notnecessary, as we saw above it is frequently convenient to further require the connection
to be “compatible with the holomorphic structure,” which means that we require that
A0,1 = 0. It’s not hard to show that given a complex structure and a Hermitian metricon V , these two conditions determine a unique connection on V (see [2], pg. 73). It issometimes called the Chern connection.
To better understand the second equation, let us ﬁrst consider a holomorphic subbundle U ⊂ V . The Hermitian metric on V induces a Hermitian metric on U , and one cancompute the corresponding curvatures (see [2], pg. 79). Interestingly one ﬁnds thatFU ≤ FV |U(1.51)

in the sense that when contracted with holomorphic tangent vectors v i , the diﬀerence(FV |U − FU )ij̄ v i v j̄ is positive semi-deﬁnite at every point on X. One gets equality only
when V splits as a direct sum of holomorphic bundles. In other words in contrast to realgeometry, in complex geometry curvature is always non-increasing along holomorphic
subbundles.
This is particularly interesting when the connection on V satisﬁes the Hermitian YangMills equations, since then the ‘Kähler trace’ g ij̄ FU ij̄ should be negative semi-deﬁnite at
every point on X, and similarly for then J ∧ J ∧ FU ∼ g ij̄ FU ij̄ J ∧ J ∧ J. Taking the traceover gauge indices and integrating, we ﬁnd that the degree of U , deﬁned as
1deg(U ) =2πvol(X)∫J ∧ J ∧ Tr(FU )(1.52)

X

is also negative. Since dJ = 0 the degree doesn’t depend on the full curvature but only
on the cohomology class of Tr(FU ). Up to a factor of 2π this is the ﬁrst Chern class c1 (U )of U , a topological quantity.
Using the Chern connection, the equation J ∧ J ∧ FV = 0 is really a complicated nonlinear PDE for the Hermitian metric on V . Apart from its dependance on Kähler moduli,
the degree of a bundle on the other hand is a topological quantity, and as such requiring
the degree of every holomorphic sub-bundle to be negative seems like a much weakercondition than the Hermitian Yang-Mills equations. Nevertheless, somewhat analogous
to the problem of ﬁnding Calabi-Yau metrics, it turns out that this condition is suﬃcient.
Let us make some deﬁnitions.
A stable vector bundle is deﬁned as a vector bundle for which any holomorphic subbundle has a smaller slope:U ⊂ V ⇒ µ(U ) < µ(V )(1.53)
where the slope is deﬁned asµJ (V ) =degree(V )rank(V )17(1.54)

A bundle is poly-stable if it is a sum of stable bundles of the same slope. Note that for abundle with a simple gauge group, the degree is always zero.
We’ve argued that a solution of the Hermitian Yang-Mills equations requires a holomorphic poly-stable bundle, at least for simple gauge groups (in the non-simple case theHermitian Yang-Mills equations should be slightly generalized by adding a term proportional to the identity matrix). As we hinted, the converse is also true, and this is a diﬃcult
theorem of Donaldson-Uhlenbeck-Yau:
If a vector bundle V on a compact Kähler manifold is holomorphic and polystable, then there exists a unique solution to the Hermitian Yang-Mills equations.
Of course by unique we mean unique up to conventional gauge transformations.
So why is this a big deal? The beauty of this theorem is that poly-stability is analgebro-geometric condition, so we can use purely algebraic methods to construct lots ofexamples. In spirit it is similar to Yau’s theorem on Calabi-Yau manifolds: it is very
hard to write down Ricci ﬂat metrics on a Kähler manifold explicitly, since it corresponds
to the solution of a highly non-linear PDE, but a simple criterion (vanishing of the ﬁrst
Chern class) guaranteed us the existence of a solution of this PDE. In the present case,
we can get away with checking the slope of holomorphic sub-bundles.
Furthermore we generally need to know very little about the actual solution; we will
see that the massless ﬁelds in the eﬀective four-dimensional theory can be studied withquasi-topological methods, much like we did for the moduli of the Calabi-Yau metric.
A canonical example of a stable bundle on a Calabi-Yau is the tangent bundle T X.
Rather than use the above result, in this special case we can show it more directly as
follows: since ∇m ϵ = 0 we also have 0 = [∇m , ∇n ]ϵ = 41 Rmnpq Γpq ϵ = 14 Rpqmn Γpq ϵ, which isprecisely Fpq Γpq ϵ = 0 for the tangent bundle. Not only that, it also automatically satisﬁes
the tadpole cancelation condition if we take the second E8 bundle to be trivial, since then
Tr(F ∧ F ) = Tr(R ∧ R) exactly. We will come back to the tangent bundle in a moment.
There are at least two concrete ways to discuss many more examples, monads andspectral cover constructions. They rely on Donaldson-Uhlenbeck-Yau to prove the existence of a solution to the Hermitian Yang-Mills equations. Unfortunately, lack of time
precludes us from discussing them.
Now given a solution to the Hermitian Yang-Mills equations, how do we derive theeﬀective four dimensional theory obtained by Kaluza-Klein reduction of the gauge sector?
Can we do this without knowing the explicit solution? Given what we have seen earlier,
you should expect that this is possible. Massless four-dimensional ﬁelds have some kindof harmonic form as internal wave-function, and harmonic forms tend to express quasitopological information. We will now turn to Dolbeault cohomology and see that thisexpectation is correct.
Since ∂¯2 = 0 (or ∂¯2 = F 0,2 = 0, but we already set A0,1 = 0 earlier), on holomorphic
A18

bundles we can deﬁne a complex which is analogous to the de Rham complex:∂¯∂¯

0 → Ω0,0 (X6 , V ) → · · · → Ω0,n (X6 , V ) → 0(1.55)

Concretely, these are forms with anti-holomorphic indices and which take values in V .
We can then consider the associated cohomology groups¯¯
H p (X6 , V ) = {∂−closed(0, p) forms}/{∂−exact(0, p) forms}(1.56)

which is called Dolbeault cohomology. Taking V to be Λq T ∗ X we get back the Hodge
numbers we discussed earlier
H q,p (X) = H p (X, Λq T ∗ X)

(1.57)

but Dolbeault cohomology exists more generally; it makes sense for any holomorphic
bundle. Note that these Dolbeault cohomology groups are independent of the Hermitian
metric on the bundle; they depend only on the holomorphic data.
As we will now see, these bundle valued Dolbeault cohomology groups are just what
we need to describe the Kaluza-Klein reduction of the gauge sector. We put a complex
structure on the whole 10d space-time (including the uncompactiﬁed dimensions) andfocus on the (0, 1) part A0,1 , since the (1, 0) part may be recovered by Hermitian con0,1
jugation. Let us decompose A0,1 into its classical proﬁle A0,1
cl and the ﬂuctuations δAaround it. We actually used gauge transformations to set the proﬁle A0,1
cl = 0 earlier but
we temporarily left it for clarity. Now we can decompose the ﬂuctuations into eigenmodesof the internal ∆∂¯-Laplacian, resulting in an expansion of the form
δA0,1 =∑AIµ̄ dxµ̄ ∧ ω(0),I +I∑

θJ ω(1),J(1.58)

J

Note that if ∂¯A ω(0) ̸= 0, then ω(1),I ≡ ∂¯A ω(0),I is also an eigenmode of the Laplacianwith the same eigen-value. This suggest that such pairs (ω(0),I , ω(1),I ) give rise to a single physical ﬁeld in four dimensions. Indeed let us now apply a ten-dimensional gaugetransformation A0,1 → A0,1 + ∂¯A Λ and decompose into eigenmodes, Λ = λI ω(0),I . Then
we ﬁndθ I → θ I + λI
(1.59)AIµ̄ → AIµ̄ + ∂µ̄ λI ,We learn a couple of things. The ten-dimensional gauge symmetry is spontaneously broken
by the compactiﬁcation, and the pseudo-scalars θJ describe the longitudinal components
of massive Kaluza-Klein gauge bosons AIµ . Only gauge transformations of the form Λ =λI ω(0),I with ∂¯A ω(0),I = 0 survive as unbroken gauge symmetries. But sections ω(0),I suchthat ∂¯A ω(0),I = 0 are by deﬁnition generators of degree zero Dolbeault cohomology. So
19

we conclude that generators of H 0 (X, V ) precisely count the unbroken four dimensional
gauge symmetries. Taking commutators of global holomorphic sections gives a natural
Lie algebra structure
H 0 (X, V ) × H 0 (X, V ) → H 0 (X, V )(1.60)
which yields the Lie algebra of the four-dimensional gauge group. From our expansion wealso see that the AIµ corresponding to a generator of H 0 (X, V ) do not have a longitudinalcomponent, as expected.
Now let’s try to deform the proﬁle of the internal components of the gauge ﬁeld,
0,1
0,1
0,2
A0,1
= 0,
cl → Acl + δA . In order for this deformation to preserve the condition F
0,1
0,1
¯
¯
we need ∂A δA = 0. Furthermore deformations of the form δA = ∂A λ are not hon-est deformations but (broken) gauge symmetry transformations. Therefore inﬁnitesimal
deformations of a holomorphic bundle are described precisely by H 1 (X, V ). As before deformations give rise to massless bosonic ﬁelds in the four-dimensional eﬀective theory, andthe unbroken four-dimensional N = 1 supersymmetry pairs them up with chiral fermions
into chiral multiplets. Again taking commutators there is a natural action
H 0 (X, V ) × H 1 (X, V ) → H 1 (X, V )(1.61)

which means that the four-dimensional chiral ﬁelds live in certain representations of theunbroken gauge group. Depending on whether they are charged under the unbroken gauge
symmetry, we call them either moduli or four-dimensional matter ﬁelds.
Let us brieﬂy list some additional things that one can try to understand. We mentioned
previously that inﬁnitesimal complex and Kähler moduli on a Calabi-Yau can be promoted
to ﬁnite deformations. For bundle deformations the situation is more complicated, but it
can be encoded in terms of the holomorphic Chern-Simons functional∫Ω3,0 ∧ ωY M (A)W =(1.62)

X6

which gives the superpotential of the eﬀective four-dimensional N = 1 theory. Anotherinteresting story is that although the individual Kaluza-Klein modes in (1.58) are hardto understand, a certain combination of them (the holomorphic Ray-Singer torsion) is
again quasi-topological and turns out to describe loop corrections to the four-dimensional
gauge couplings. Further, since the stability condition depends on Kähler moduli there
is an interesting interplay that results in the phenomenon of stability walls in the Kähler
moduli space, where solutions of the Hermitian Yang-Mills equations come into or go out
of existence. These are slightly more advanced topics that we cannot go into here.
To compare with the Narain story, we have now seen that inﬁnitesimal deformationsof g, B and A on X are described by the following cohomology groups:
H 1 (T X),

H 1 (T ∗ X),
20

H 1 (X, V )(1.63)

We can combine this all into a single formula by considering the bundle T X ⊕T ∗ X ⊕V . If
you think about how the vertex operators are built from world-volume ﬁelds, this perhaps
will not seem so surprising.
1.7. The standard embedding
As we saw above, the spin connection is a Hermitian Yang-Mills connection, so let us
illustrate all this wonderful technology for the case where we embed the tangent bundle
in our E8 bundle. The tangent bundle has SU (3) holonomy, so we embed the spin
connection of the Calabi-Yau metric in the E8 connection using the maximal subgroup
SU (3) × E6 ⊂ E8 . Under this subgroup, the adjoint representation of E8 decomposes as
248 = (3, 27) + (3, 27) + (1, 78) + (8, 1)(1.64)and therefore at the level of Dolbeault cohomology we get
H p (X, VE8 ) = H p (T X) ⊗ 27 + H p (T ∗ X) ⊗ 27
+H p (OX ) ⊗ 78 + H p (End0 (T X)) ⊗ 1(1.65)To relate this to (1.64), T X corresponds to the 3 representation of the SU (3) holonomy,
and T ∗ X corresponds to the 3. The trivial line bundle OX corresponds to the singlet 1 of
SU (3). Further, End(T X) = T ∗ X ⊗ T X corresponds to the 3 ⊗ 3 = 8 ⊕ 1, and End0 (T X)
corresponds to the projection on the 8.
We ﬁrst consider the four-dimensional gauge ﬁelds. According to our previous discussion, they come from H p (X, VE8 ) with p = 0. Since H 0 (OX ) = H 0,0 (X) is onedimensional, we ﬁnd a gauge ﬁeld in the 78, the adjoint of E6 . We claim that we do not
get anything from the other pieces for p = 0. Indeed a generator of H 0 (T X) corresponds
to a global holomorphic section s : OX → T X, which embeds OX as a holomorphic
subbundle of T X. But deg(OX ) = deg(T X) = 0, so this would violate the stability
condition. If a bundle associated to a faithful representation of the structure group isstable, then bundles associated to other faithful representations of the structure groupare also stable. In the present case this means T ∗ X and End0 (T X) are also stable, andby the same type of argument we see that for any higher rank stable bundle the degree
zero Dolbeault cohomology vanishes (’no symmetries’). Therefore as claimed that is all
we get from H 0 (X, VE8 ). We conclude that the eﬀective four-dimensional theory has an
E6 gauge group.
Next we consider the four-dimensional matter ﬁelds, which come from H p (X, VE8 )with p = 1. From H 1 (T X) ⊗ 27 we ﬁnd h1 (T X) = h2,1 (X) chirals charged in the 27 of
E6 . Similarly there are h1,1 (X) = h1 (T ∗ X) chirals charged in the 27 of E6 . We further
have H 1 (OX ) = H 0,1 (X) = 0. Finally, H 1 (End0 (T X)) counts the deformation moduli of
the bundle, which preserve the unbroken E6 gauge group (since they live in the singlet of
E6 ), but take the connection away from the standard embedding.
21

Putting this together, we see that the eﬀective four-dimensional theory is an E6 GrandUniﬁed Theory, plus a hidden sector with whatever descends from the second E8 . The
net number of chiral generations is given by
N27 − N27 = h1,2 (X) − h1,1 (X) = −χ(X)/2(1.66)

where χ(X) is the Euler character of X. For the Tian-Yau manifold this gives nine
generations, and one can actually mod out this manifold by a discrete symmetry to get
precisely three-generations.
We can make the model more realistic. By taking the E8 connection to be valued in
special SU (6) bundles that take the form of an S(U (5)×U (1))-bundle, we break the gauge
group precisely to the Standard Model gauge group, SU (3)c × SU (2)w × U (1)Y . Suchconﬁgurations can be written down using the general constructions that we mentioned
previously.

22

2. Perturbative IIB
I will spend relatively little time on type IIB, but there are a few things we should
discuss. This will be mostly set-up for F -theory, so I will focus on D7 and O7-planes.
There will likely be a more extended discussion in the lectures by H. Ooguri.
2.1. Kaluza-Klein Ansatz
The ten-dimensional ﬁelds of the IIB theory are:
(2)

(4) +

1
2
1
2
gM N , BM N , ϕ, C (0) , CM N , CM N P Q , ψM
α , ψM α , χβ̇ , χβ̇(2.1)

The C (i) are anti-symmetric tensor ﬁelds, with C (4) satisfying a self-duality condition.
1
2
There are now two ten-dimensional gravitinos of the same chirality, ψM
α and ψM α , andsimilarly two ten-dimensional dilatinos. There are no gauge ﬁelds in ten dimensions. In
the Einstein frame, the action may be written as[]√∂µ τ̄ ∂ µ τ
1
1
1
2
i j
2
d x |g| R −− |dC(0) | − Mij F3 F3 − |F̃(5) | + SCS + SF ermi
2 Im(τ )2 2
2
2
(2.2)
where τ = ie−ϕ + C(0) is the axio-dilaton, F3 = (dB, dC(2) ) combines the RR and NSNS
two-forms, F̃(5) = dC(4) − 21 C(2) ∧ dB + 12 B2 ∧ dC(2) is the self-dual version of the four-form
ﬁeld strength, and
(
)
1
|τ |2
−Re(τ )
Mij =
(2.3)
1
Im(τ ) −Re(τ )ϵ ∫The Chern-Simons term is given by SCS = − 4κij2 C4 ∧ F3i ∧ F3j .
1
S= 2
2κ

∫

10

The Kaluza-Klein reduction proceeds much like for the heterotic string. We make an
AnsatzM1,9 = M1,3 × X6
(2.4)
1
and set the ﬂuxes to zero. Then we are left with solving δψM= ∇M ϵ1 = 0 and2δψM= ∇M ϵ2 = 0, where ϵ1α and ϵ2α parametrize the two ten-dimensional supersymmetries. As before the supersymmetry variations tell us that X6 should admit a covariantly
constant spinor, and so X6 is Calabi-Yau. However our supersymmetry transformations
were parametrized by two ten-dimensional spinors, so compactifying type II strings on a
Calabi-Yau yields N = 2 supersymmetry in four dimensions.

The natural multiplets for four-dimensional N = 2 supersymmetry are the gravity
multiplet, the vector multiplet and the hypermultiplet. The bosonic ﬁelds in a vector
multiplet are (Aµ , φ) where Aµ is a vector ﬁeld and φ is a complex scalar ﬁeld. The
23

N = 2 multiplet

multiplicity

gravity1vectorh2,1 (X)

hyper

h1,1 (X) + 1

Table 1: Spectrum of type IIB compactiﬁed on a CalabiYau three-fold.

bosonic ﬁelds in a hypermultiplet are a pair of complex scalar ﬁelds (φ1 , φ2 ). The bosonicﬁelds of the gravity multiplet consist of (gµν , Aµ ), where gµν is the metric and Aµ is avector ﬁeld called the gravi-photon.
The story for the NS-NS ﬁelds is exactly the same as for the heterotic string. What’s
new is that their modes will pair up with modes of RR ﬁelds into N = 2 multiplets.
The compactiﬁcation works out as follows. As for the B-ﬁeld, the massless modes of
RR ﬁelds are obtained from expanding in harmonic forms. The only way to get masslessfour-dimensional vectors is by expanding C4+ in harmonic three-forms:
C4+ = AIµ dxµ ∧ ω3,I

(2.5)

From the Hodge diamond we see that there are 2h2,1 (X) + 2 such forms, however only
half of these are independent due to the self-duality condition. The vector coming fromh3,0 (X) is the gravi-photon which ends up in the gravity multiplet, the remaining h2,1pair with the complex structure moduli in h2,1 vector multiplets.
We further get 4 × h1,1 (X) scalars from expanding C4+ in four-forms, B2 and C2 in
two-forms, and Kähler moduli of the metric on X6 . These precisely ﬁll up h1,1 hypermultiplets. Two-forms B2 and C2 with all indices in the uncompactiﬁed directions canbe dualized to scalars. They combine with the remaining scalars (ϕ and C0 ) into an
additional hypermultiplet. We have summarized the spectrum in table 1.
There are two ways we could break further to N = 1 supersymmetry. One is to turnon some of the ﬂuxes. To get a solution, we have to allow a more general warped Ansatz
of the form
ds2 = Z −1/2 ηµν dxµ dxν + Z 1/2 gmn dxm dxn(2.6)
where ηµν is the four-dimensional Minkowski metric, gmn is the metric on the six-dimensional
internal space, and Z is a non-trivial function on the internal space. The other way tobreak to N = 1 supersymmetry is to add 1/2 BPS branes.
It turns out that if we take gmn to be a Calabi-Yau metric and impose certain conditions

24

on the ﬂuxes, then ﬁnding a solution with N = 1 supersymmetry is reduced to solving−∇2 Z = (2πα′ )2 ρD3(2.7)

Here ∇ refers to the covariant derivative for gmn , and ρD3 is the ‘D3-brane density.’ The
latter really refers to anything that sources C(4) , whether it is explicit D3-branes, H3 ∧ F3
(through the Chern-Simons term SCS ), or ﬂuxes and curvature terms on higher dimensional defects. Deriving the eﬀective four-dimensional theory for a warped background canbe a diﬃcult business. We are going to think of the warping as a higher order eﬀect andnot consider it further, but there are apparently situations where this is not correct andyou can miss an important mode, with the mode discovered by Gubser/Herzog/Klebanov
in the warped deformed conifold being an example.
2.2. Orientifolds
The perturbative IIB theory has two basic Z2 symmetries. The ﬁrst is world-sheet
parity, which interchanges the left- and right-movers. This changes the signs of C0 , B2
and C4+ , and interchanges the two gravitini (as well as ϵ1 and ϵ2 ). The other basic Z2
symmetry is the left-moving fermion number (−1)FL , which changes the sign if the leftmovers are in the Ramond sector. This therefore changes the sign of C0 , C2 and C4+ ,1
as well as the sign of one of the gravitini, say ψM(and similarly for ϵ1 ). We could
further combine such involutions with an involution σ of the IIB space-time manifold.
By deﬁnition, an orientifold is a quotient of the IIB theory that involves the world-sheet
parity transformation P .
We are going to consider orientifolds of the following form. We will quotient by(−1)FL · P · σ(2.8)

where σ is a holomorphic involution of the Calabi-Yau X6 that reverses the sign of theholomorphic (3, 0) form and has a ﬁxed point locus of complex codimension one. In
other words, locally it is of the form (z1 , z2 , z3 ) → (z1 , z2 , −z3 ). Such a quotient preserves
only one linear combination of the two unbroken spinors ϵ1 and ϵ2 , and so leads to fourdimensional N = 1 supersymmetry. The ﬁxed point locus is called an O7-plane. If wedenote the holomorphic submanifold that is ﬁxed by S, then we say that the O7-plane‘wraps’ S.
We can be very explicit. Let B3 = X/Z2 , which will be a Kähler manifold, and let b2
be a section of KB−23 , where KB3 is the canonical bundle KB3 ≡ Λ3 T ∗ X. Simple examples
would beB3 = CP3 ,

B3 = CP2 × CP1 ,
25

B3 = CP1 × CP1 × CP1 .

(2.9)

but there are many more. For example when B3 = CP3 , we would have that b2 is simply
a polynomial of degree 2 × 4 = 8. Given b2 , we can write an equation for X explicitly asξ 2 = b2 .

(2.10)

It has the involution ξ → −ξ and ﬁxed locus b2 = 0. The Calabi-Yau
condition on X√
−2
is precisely the condition that b2 lives in KB3 (since dξ ∼ db2 / b2 gives KX = π ∗ KB3 +1
N).
2 b2 =02.3. D-branes
The second type of objects we can add are D-branes. By deﬁnition, a D-brane isa defect where a fundamental string can end. By quantizing open strings, we ﬁnd newdegrees of freedom propagating along the D-brane. The low energy world-volume theory
on a single D-brane is the dimensional reduction of the ten-dimensional supersymmetric
Yang-Mills theory with U (1) gauge group. When N D-branes coincide, the gauge groupenhances from U (1)N to U (N ). The extra oﬀ-diagonal ﬁelds come from the ground statesof open strings stretched between the diﬀerent D-branes. The mass of these modes is
proportional to the distance between the branes, so they become massless when the branes
coincide.
A common case is for the branes to ﬁll the four uncompactiﬁed dimensions (as did theten-dimensional gauge ﬁelds in the heterotic string). Then the branes ‘wrap’ even dimensional submanifolds of the Calabi-Yau three-fold. To be 1/2 BPS, a D-brane should wrap
a holomorphic cycle and the world-volume ﬁelds should satisfy a dimensional reductionof the Hermitian Yang-Mills equations that we discussed in the context of the heterotic
string.
If we want to preserve the same N = 1 supersymmetry that is preserved by (2.8), thenwe can consider D7-branes wrapped on holomorphic four-cycles and D3-branes localizedat points of X6 . The bosonic world-volume ﬁelds on a D7-brane are a gauge ﬁeld AM anda complex adjoint ﬁeld Φ. When we wrap the D7-brane on a holomorphic cycle S, theseﬁelds may have a non-zero proﬁle along S, satisfying the dimensionally reduced versionof the Hermitian Yang-Mills equations. In the abelian case, these equations are simply¯ = 0,∂ΦF 0,2 = 0,J ∧F =0(2.11)

where J is the pull-back of the Kähler form of X6 to S. When multiple branes are wrappedon the same cycle, we get a non-abelian version of these equations, often referred to as
Hitchin’s equations.
There is something peculiar going on with the adjoint ﬁeld Φ, so let us say a few
more words about this. We take (z 1 , z 2 ) to be holomorphic coordinates along S, and z 3
for the normal direction. Then to carry out the dimensional reduction, in the Hermitian26

Yang-Mills equations we would like to replace gauge ﬁelds along the normal direction byadjoint ﬁelds and take all the ﬁelds to be independent of z 3 :A1̄ (z 1 , z 2 , z 3 ), A2̄ (z 1 , z 2 , z 3 ), A3̄ (z 1 , z 2 , z 3 ) → A1̄ (z 1 , z 2 ), A2̄ (z 1 , z 2 ), Φ3̄ (z 1 , z 2 )(2.12)and similarly for their Hermitian conjugates. Restricted to S, the tangent bundle splits
holomorphically as T X|S = T S ⊕ NS , where NS is the normal bundle to S. Similarly the∨anti-holomorphic cotangent bundle splits as T X|S = T ∗ S ⊕ N S , where NS∨ is the dualof the normal bundle. The gauge ﬁelds (with anti-holomorphic indices) are sections of
T ∗ S ⊗ Ad(P ), where P is a principal G-bundle and Ad(P ) is its associated adjoint vector∨
bundle. Then the adjoint scalar ﬁeld should be a section of N S ⊗ Ad(P ). Using theHermitian metric restricted to S, we can consider Φ3 = Φ3̄ g 3̄3 which takes values in NS .
We say the adjoint ﬁeld is “valued in” or “twisted by” the normal bundle.
We can go one step further. Using the holomorphic (3, 0) form, we can considerΦij = Φ3̄ g 3̄3 Ω3ij , which is a (2, 0) form on S, so we may also consider Φ as a section ofKS ⊗ Ad(P ). Similarly, the Hermitian conjugate Φ† may get mapped to a (0, 2) form on
S. Thus the U (1) R-symmetry of the Yang-Mills theory (under which Φ has charge one)is identiﬁed with a subgroup of the Lorentz group of the 7-brane. This is the phenomenonof topological twisting, so we found that the world-volume theory of a D7 brane wrappedon a non-trivial cycle in a Calabi-Yau is a topologically twisted Yang-Mills theory.
With these identiﬁcations, the dimensionally reduced Hermitian Yang-Mills equations
can be written as
F 0,2 = 0,J ∧ F + i[Φ† , Φ] = 0¯ = 0,∂Φ(2.13)(Here we used physicists’ conventions in which the gauge generators are Hermitian; inmath papers the factor of i would be dropped). The rest of the story is parallel to theheterotic string. The ﬁrst two equations are equivalent to saying that we need a Higgs
bundle, i.e. a holomorphic bundle together with a holomorphic section of the twisted
adjoint bundle. The last equation is then equivalent to saying that the Higgs bundle ispoly-stable.
In the literature you will often ﬁnd an alternative description of such a conﬁguration asa poly-stable coherent sheaf L. These two descriptions are complementary to each other,with diﬀerent regimes of validity. Unfortunately I do not have time to go into coherent
sheaves here.
In an orientifold, the D-brane conﬁguration should be invariant under the orientifold
symmetry. More precisely, the gauge ﬁeld on the brane transforms as
A → −γ −1 σ ∗ AT γ

(2.14)

where γ is taken to be a constant gauge transformation, which satisﬁes γ −1 γ T = 1 (up toa further gauge transformation) in order to get an involution. This leads to the following
27

D7

O7

Figure 1: Tilting a D-brane away from an O-plane by turning on
a position dependent proﬁle for the adjoint ﬁeld of the worldvolume
gauge theory.

possibilities. A stack of branes that gets mapped to itself, and hence coincides with an
O-plane, yields an SO(N ) or U Sp(N ) gauge group in the quotient theory, depending onwhether γ T = ±γ; a pair of stacks that gets interchanged yields a U (N ) gauge groupin the quotient theory. An O-plane is called an O− -plane if γ T = γ and an O+ -plane
if γ T = −γ. Note that unlike for the heterotic string, we do not get exceptional gauge
groups in this set-up.
We cannot add these defects arbitrarily. In the heterotic case we had the tadpole/generalized Gauss law constraint dH = Tr(R ∧ R) − Tr(F ∧ F ). In the type IIcase we get analogous constraints for all the Ramond-Ramond and NS-NS ﬂuxes.
A concrete and generic way to write down a conﬁguration is as follows. A D7-brane
generates one unit of RR ﬂux for F(1) = dC(0) , so we try to cancel this ﬂux by including
an O7− -plane, which generates −4 units of RR ﬂux. (O7+ -planes source positive RR
ﬂux, so they would not be much help). Recall we could write our geometry X with suchan O7-plane as ξ 2 = b2 , where b2 is a section of KB−23 , B3 = X/Z2 (eg. B3 = CP3 forsimplicity), and b2 = 0 is the O7-locus. To cancel the RR charge the D7 would then haveto be wrapped on a cycle of the form b8 = 0, where b8 is a section of KB−83 . This guarantees
that there are no net sources for dF(1) through any closed two-cycle.
However consistency imposes a further constraint: a D7 brane cannot wrap a genericcycle, because a D7-brane should self-intersect (i.e. have singularities of the form z 2 −w2 =0) when it intersects the O7-plane.
To see this, brieﬂy let us consider a coinciding O7/D7 system, leading to gauge group
SO(2). Now we tilt the D7-plane away from the O7-plane by turning on a position28

N = 1 multiplet

multiplicity

gravity1vectorh2,1+ (X)chiral

1,1
2,1
h1,1
+ (X) + h− (X) + h− (X) + 1

Table 2: Closed string spectrum of type IIB compactiﬁed on
a Calabi-Yau three-fold, with an orientifold involution of theform (−1)FL P σ.

dependent expectation value for the SO(2)-valued ﬁeld Φ,(Φ(z) ∼ z

0 1
−1 0

)
(2.15)

where z is a complex coordinate along the worldvolume of the O7-plane. Then the eigenvalues of Φ are ±z, indicating a pair of D7-branes meeting the orientifold plane in a
double intersection at z = 0, as claimed. This means that our D7-brane cannot wrap a
generic cycle of the form b8 = 0, but must wrap a cycle of the formξ 2 b6 − b24 = 0(2.16)in order to have double point singularities when the D7 and O7 intersect (at ξ = b4 = 0).
A more generic expression of the form b8 = 0 would not have this property. We conclude
that we need three sections, b2 , b4 and b6 to write down a generic D7/O7 conﬁguration.
With a bit more eﬀort we can also specify a gauge ﬁeld conﬁguration. The easiest way to
do this is by using the fact that that a quantized ﬁeld strength F 1,1 /2π of type (1, 1) canbe dually described by a linear combination of holomorphic curves on the D7-brane.
Given a IIB compactiﬁcation with branes, we would also like to know the eﬀective
four-dimensional theory obtained by Kaluza-Klein reduction. The reduction of the closed
string ﬁelds was already discussed above, with only the slight complication that a subset of
the ﬁelds will get projected out by the orientifold projection. Brieﬂy, since the involution
σ is holomorphic, the cohomology groups H p,q (X) split into eigenspaces with eigenvalue±1, H+p,q (X) and H−p,q (X). Since C4+ is even under (−1)FL P , its modes only survive inthe quotient theory if its internal wave-function is also even. So for example we only get
2,1h2,1+ (X) vector ﬁelds, with the remaining h+ (X) of the underlying N = 2 theory being
projected out. The complex structure moduli on the other hand, which where pairedwith these vector ﬁelds under N = 2 supersymmetry, came from expanding deformationsof the holomorphic (3, 0)-form. Since Ω3,0 is is odd under the involution, only h2,1
− (X)deformations survive in the quotient theory. Continuing in this way, Kähler moduli and29modes of C4+ with four internal indices give h1,1
+ (X) chirals, modes of (B2 , C2 ) with twointernal indices give h1,1(X)chirals,andtheaxio-dilaton gives one ﬁnal chiral. This−
spectrum is summarized in table 2.
But how do we reduce the degrees of freedom from the open string sector? It turns
out there’s a very simple general answer, but because I didn’t have time to talk aboutcoherent sheaves I will only mention it in passing.
Recall that in the heterotic string, the massless ﬁelds obtained from reduction of
the gauge sector were counted by H p (X, V ). For p = 0 we get four-dimensional gauge
ﬁelds and for p = 1 we get four-dimensional matter ﬁelds and moduli. In IIB the 1/2
BPS D-brane conﬁguration can be expressed as a coherent sheaf L, and the deformation
theory of sheaves then tell us that the massless ﬁelds are then counted by the Ext groups
ExtpX (L, L). For p = 0 we get gauge ﬁelds and for p = 1 we get matter ﬁelds and moduli.
In an orientifold we keep the odd generators and throw out the even generators.
2.4. S-duality
The type IIB theory also has a strong coupling duality, S-duality which takes gs →1/gs . It is convenient to formulate this in terms of the axio-dilaton
τ = i e−ϕ + C0

(2.17)

It naively takes values in the upper half plane (since Im(τ ) = e−ϕ > 0), but due tothe symmetries we have to make some identiﬁcations. The axion has a shift symmetry
C0 → C0 + const which is non-perturbatively broken to C0 → C0 + n, n ∈ Z. Together
these two symmetries act on τ asS : τ → −1/τ,T :τ →τ +1(2.18)and generate an SL(2, Z) duality group. The two-form ﬁelds (B2 , C2 ) transform as a
doublet under SL(2, Z), and the Einstein frame metric is invariant. The whole IIB supergravity action that we wrote in (2.2) is invariant under this duality group.
S-duality takes a fundamental IIB string to a D-string. Since a D7-brane is deﬁned asa locus where a fundamental string can end, the S-dual of a D7-brane is a defect wherea D-string can end. More generally, SL(2, Z) dualities can take a fundamental string toa (p, q) string, a bound state of p fundamental strings and q D-strings. We can deﬁne a(p, q) 7-brane as a defect of codimension two where a (p, q) string can end.
This concludes our lightning quick review of perturbative IIB compactiﬁcation.

30

3. F-Theory
Heterotic and IIB probably looked quite familiar. They are conventional string theories
with a small string coupling. F -theory is a diﬀerent beast, generalizing some of thestructures we have seen previously in the language of geometry. It can reduce to heterotic
or IIB in suitable weak coupling limits, but the way in which it does is not obvious.
3.1. Elliptic ﬁbrations
Let us brieﬂy recall some properties of elliptic curves. We have H1 (T 2 , Z) = Z2 ,generated by one-cycles a and b such that a ∩ a = b ∩ b = 0 and a ∩ b = 1. Denotethe holomorphic one-form by Ω. Then the complex structure of an elliptic curve can beparametrized as∫Ωτ = ∫b(3.1)Ω
aThe torus has large diﬀeomorphism or modular transformations which map the one-cycles
as (a, b) → M (a, b), where M ∈ SL(2, Z). The basic transformations are (a, b) → (a, b+a)and (a, b) → (−b, a), and the others are generated by these. On the complex structure
parameter these act asτ → τ + 1,
τ → −1/τ(3.2)
This looks exactly like the S-duality transformations for the axio-dilaton of type IIB. So
why not identify the axio-dilaton with the complex structure parameter of an auxiliary
torus? This is the basic idea that leads to F -theory [3].
The one-sentence description of F -theory is as follows. F -theory is basically a bookkeeping device to describe vacua of IIB string theory with a varying axio-dilaton. We do
this by identifying the axio-dilaton τ with the modular parameter of an auxiliary torus,
and imagining that this auxiliary torus is real in some sense, by formally attaching thistorus at each point in the IIB space-time. In this way we promote the ten-dimensional
space-time of the IIB theory to twelve dimensions, and we speak of twelve-dimensional
compactiﬁcations of F -theory.
You should keep in mind that this is nothing more than a very clever change of variable.
Specifying a varying τ directly would get rather complicated due to the branch cuts for τ
and the SL(2, Z) monodromies around them. It is much simpler to write down the torus
directly, and thereby specify τ implicitly. The T 2 is typically written in Weierstrass form,
i.e. described as a cubic equation of the form
y 2 = x3 + f x + g

31

(3.3)

Branch cut

ττ+1

Figure 2: Elliptic ﬁber degenerating over the discriminant locus,
due to one of its one-cycles pinching to zero size. Singular ﬁbers
generate monodromies for τ , and hence are associated to 7-branes.

which can be done globally on the IIB space-time. Instead of specifying τ , we specify f
and g. The area of the torus has no meaning in F -theory and should be taken zero.
Now 7-branes source the axio-dilaton, and supergravity solutions for 7-branes have anaxio-dilaton which varies non-trivially over the IIB space-time. The reformulation aboveis particulary eﬃcient for encoding such solutions, as we discuss next.
As above we label the generating one-cycles of the elliptic ﬁber as a and b, witha ∩ b = 1, a ∩ a = b ∩ b = 0. On a subset of real codimension two on the IIB spacetime, the elliptic ﬁber pinches due to a one-cycle γ = pa + qb shrinking to zero. One
can calculate that this happens when ∆ ≡ 4f 3 + 27g 2 = 0. As we go around this locus,
which is called the discriminant locus, the one-cycles undergo a monodromy following thePicard-Lefschetz formula:δ → δ + (δ ∩ γ)γ
(3.4)
∫
∫
Let us use our earlier expression τ = b Ω/ a Ω. With a little algebra, we see that themonodromy acts on τ as()
1 + pqp2
τ → K[p,q] τ,K[p,q] =(3.5)−q 2 1 − pq
We claim this identiﬁes the locus with a (p, q) 7-brane, i.e. a type of 7-brane on whicha (p, q) string can end. To see this, consider the case of a (1, 0) brane. In this case wehave τ → τ + 1 as we go around a 7-brane, i.e. C0 → C0 + 1 and e−ϕ invariant. This isprecisely the right monodromy for a single D7-brane, because it means that the 7-branesources one unit of RR ﬂux. To see this, integrate dF(1) around a small disk in the IIB
space-time containing the point where the discriminant vanishes. Then we have∫∫dC(0) = C(0) |2π − C(0) |0 = 1

dF(1) =
disk

S1

32

(3.6)

where we used the multi-valuedness of C(0) around the discriminant locus. By applying
SL(2, Z) duality transformations, we recover the other cases.
You may start to get suspicious here: we seem to ﬁnd only 7-branes on which strings
can end. But in perturbative IIB we also had O7-planes, and strings don’t end on O7planes, so what happened to them? It turns out that the O7-plane is not an elementary
object in F -theory, but is composed of two distinct (p, q) 7-branes. In fact the SL(2, Z)
monodromy around an O7-plane is given by
(
)
(
)(
)
−1 4
2 1
0 1
=
(3.7)
0 −1
−1 0
−1 2
so we see that it can be composed out of a (1, 1) brane and a (1, −1) brane. As we turnon the string coupling, the O7-plane splits into its two components.
3.2. The view from M -theory
There is another perspective on F -theory by starting with M -theory, as follows. M theory on T 2 , in the limit that the area goes to zero, is equivalent to type IIB on a circleof radius R = 1/A, with axio-dilaton given by the modular parameter τ of the T 2 , and A
is the area of the T 2 . Now let us ﬁber this duality over a base.
In particular, let us consider an M -theory compactiﬁcation to three dimensions with
N = 2 supersymmetry. (This will eventually lift to N = 1 supersymmetry in four
dimensions). We haven’t considered M -theory compactiﬁcations so far but the principles
are very similar to the other cases we discussed. We make an Ansatz
M1,10 = M1,2 × Y,

(3.8)

set the ﬂuxes to zero, and analyze the BPS equations. As usual we have to solve anequation of the form δψM = ∇M ϵ = 0. The main diﬀerence with the cases we discussed
previously is that to get N = 2 supersymmetry, we need two covariantly constant spinors
on Y of the same chirality. The positive chirality spinor representation of SO(8) is the 8s ,which we therefore want to decompose as 6 + 1 + 1, with the holonomy group preserving
the two singlets. This means that Y should have Spin(6) ≃ SU (4) holonomy, and so one
ﬁnds that Y should be a Calabi-Yau four-fold, henceforth denoted Y4 .
To make use of the M -theory/IIB duality in nine dimensions cited above, we now
further require that Y4 admits an elliptic ﬁbration, i.e. a ﬁbration
π : Y4 → B3

(3.9)

where the ﬁbers are elliptic curves. As usual the subscripts on Y4 and B3 denote thecomplex dimension. If we represent this ﬁbration in Weierstrass form
y 2 = x3 + f x + g
33(3.10)

then in order for the total space Y4 to be Calabi-Yau, we must have that f is a section of
KB−43 and g is a section of KB−63 . Applying the M -theory/IIB duality ﬁberwise, we deducethat
M − theory on R1,2 × Y4←→
IIB on R1,2 × SR1 × B3 ,(3.11)
with a varying axio-dilaton over B3 on the IIB side. Since R = 1/A, we then send A → 0.
In the limit that the elliptic ﬁber shrinks to zero, the S 1 decompactiﬁes and we recovertype IIB compactiﬁed on R1,3 × B3 , with varying axio-dilaton and N = 1 supersymmetry
in four dimensions. We say that this corresponds to an F -theory compactiﬁcation on Y4
to four dimensions.
3.3. Abelian gauge ﬁelds
Since our F -theory compactiﬁcation contains 7-branes, we expect to see eight-dimensional
Yang-Mills theory come out in some way. The way that this happens is quite unlike what
we have seen so far from the heterotic string and perturbative IIB, and it will occupy us
for the next few subsections.
Although we have some kind of 7-brane defect, the string coupling is not small andwe can’t go and quantize open strings. Indeed, what we are really doing is describing
the 7-brane as a solitonic solution of type IIB supergravity, the ‘stringy cosmic string’
solution (string here referring to the fact that the soliton is localized in codimension two,
which means it’s really a 7-brane in ten dimensions). This means that we should expect
the world-volume theory to arise from the collective coordinates of the soliton.
In particular, abelian tensor ﬁelds on a defect arise as zero modes of the ten-dimensional
tensor ﬁelds that are roughly localized on the soliton. Since the gauge symmetry we arelooking for exists already in eight dimensions, the eight-simensional gauge ﬁeld mustcome from zero modes of B2 and C2 . However B2 and C2 are not invariant under themonodromies; they form a doublet under the SL(2, Z) duality group.
In keeping with the philosophy of F -theory, we thus want to reformulate B2 and C2
in terms of an object that can be speciﬁed globally over the IIB space-time, and is not
subject to monodromies. This can be done by encoding the two-form ﬁelds in a three-form
ﬁeld:iC(3) ∼(C2 − τ B2 ) ∧ (dx + τ̄ dy) + c.c.
(3.12)Im(τ )
where x and y are the two coordinates on the T 2 ﬁber. Note this is not a completely
general three-form, since it has two indices on the IIB space-time and one index on
the elliptic ﬁber. Three-form ﬁelds with diﬀerent numbers of indices in the base andthe ﬁber do not exist in F -theory. This three-form ﬁeld C(3) is SL(2, Z) invariant andcan be deﬁned globally. By compactifying on S 1 and going to M -theory, it corresponds
to the usual C(3) ﬁeld of eleven-dimensional supergravity, except that components withdisallowed indices are frozen out in the F -theory limit. The four-form ﬂux of this tensor
ﬁeld is conventionally called the G-ﬂux, G = dC(3) .
34

Now to get a 7-brane gauge ﬁeld, we need to expand C(3) in terms of real harmonic
two-forms, with one index on the base and one index on the ﬁber (so that the gauge ﬁeld
index lives in the IIB space-time):
C(3) = AI ∧ ω(2),I(3.13)As a toy example, let us consider F -theory compactiﬁed on an elliptically ﬁbered K3surface to eight dimensions. We describe our K3 in Weierstrass form as an elliptic ﬁbration
over CP1 :y 2 = x3 + f8 (z)x + f12 (z)(3.14)
Here f and g are sections of KP−41 and KP−61 , so they correspond to polynomials of degree2eight and twelve respectively. Then the discriminant ∆ = 4f83 + 27g12= 0 is of degree
twenty-four, so a generic elliptic K3 has twenty-four singular ﬁbers. From the perspectiveof IIB this corresponds to a compactiﬁcation of the form R1,7 ×CP1 with twenty-four (p, q)
7-branes inserted at special positions on the CP1 . From the Hodge numbers discussed
earlier it follows that there are twenty-two harmonic two-forms on a K3 surface, but oneof these has two indices on the base and one has two indices on the ﬁber. Thus there
are twenty harmonic forms we can expand in, yielding twenty U (1) gauge ﬁelds in theeight-dimensional theory.
This is precisely the number of U (1) gauge ﬁelds you will ﬁnd in generic string theory
compactiﬁcations to eight-dimensions with half-maximal supersymmetry, for example ina heterotic compactiﬁcation on T 2 . Indeed it turns out that there is a duality
F − theory on R1,7 × CP1 n T 2←→Heterotic on R1,7 × T 2(3.15)(Here we used the imprecise notation CP1 nT 2 to denote an elliptically ﬁbered K3.) Thisis one of the fundamental dualities in the business and has been studied in great detail.
Unfortunately lack of time precludes us from discussing it further.
Note that the harmonic forms giving rise to U (1) gauge ﬁelds cannot be normalizable in
the local supergravity solution for a 7-brane: if so we would get an independent gauge ﬁeld
for each singular ﬁbre, but there are twenty-four singular ﬁbers in an elliptically ﬁbered
K3 and only twenty gauge ﬁeld from the 7-branes. Thus in contrast to perturbative IIB,
thinking about 7-branes in F -theory as being sharply localized at a codimension two locus(here the discriminant locus) is actually quite misleading: their energy density is quite
spread out, much like for other solitonic objects.
Let us say a few more words about F -theory compactiﬁcations to four dimensions
with N = 1 supersymmetry. We saw that if the ﬂux vanishes then F -theory needs tobe compactiﬁed on an elliptically ﬁbered Calabi-Yau four-fold Y4 . Now suppose we wantto consider compactiﬁcations with non-zero ﬂux. This situation was analyzed by Becker
and Becker. It turns out that the Kaluza-Klein Ansatz needs to be generalized slightly toa warped Ansatz, but when this is done one ﬁnds that Y4 is still a Calabi-Yau four-fold,
35

and the G-ﬂux on Y4 needs to satisfy the following conditions:G ∈ H 2,2 (CY4 )J ∧G = 0(F − term equation)(D − term equation)(3.16)

Here J is the Kähler form on the Calabi-Yau four-fold. Note that these conditions are
similar to the ASD equations on the internal worldvolume of a 7-brane, F 2,0 = 0 = J ∧F 1,1 .
It may seem somewhat strange that we get the simple condition J ∧ G = 0, whereas
in type IIB and heterotic compactiﬁcation we found stability conditions, which are evenqualitatively dissimilar. It’s not hard to see the problem: the equations for the G-ﬂux
above were obtained by extrapolating from M -theory. There is a non-renormalization theorem for F -terms but not for D-terms, so the D-terms should be taken with a grain of salt.
In fact one should regard even the F-terms with some suspicion; the non-renormalization
theorem does not guarantee that after extrapolation the F -terms are expressed in termsof the right degrees of freedom, and ignoring this can easily lead to puzzles.
3.4. Digression on ADE singularities
In order to understand where non-abelian gauge ﬁelds come from, we need to make adigression on ADE singularities, their resolutions and their deformations.
Let us consider an isolated singularity of a complex surface. Using complex coordinates
x, y, z, the simplest such singularity takes the local form
P (x, y, z) = xy + z 2 = 0(3.17)

It is clearly singular as ∂x P = ∂y P = ∂z P = 0 at x = y = z = 0. This singularity
goes under many names in the literature. Algebraic geometers tend to call it an ordinary
double point singularity.
Now we can think of P (x, y, z) as the determinant of the following matrix:
(
M=

x z
−z y

)
(3.18)The vanishing of P = det(M ) implies that the matrix M has an eigenvector with eigenvalue zero. We can parametrize the eigenvector as (λ1 , λ2 ). Then instead of the surface
P = 0, we can consider the new surface deﬁned by the pair of equations
(

x z
−z y

)(

λ1
λ2

)
= 0

(3.19)

More precisely, this pair of equations deﬁnes a surface (and not a three-fold) if we identify(λ1 , λ2 ) ≃ (cλ1 , cλ2 ) for any c ∈ C∗ . In other words, we consider the above as a pair ofequations on C3 × CP1 .
36

Figure 3: Pictures for the deformed singularity xy = P2 (z). (A):Fibration of conics over the z-plane, with the conic ﬁber degenerating over two special points. (B): Fibering the shrinking S 1 of theconic ﬁbers over a path in the z-plane.

When (x, y, z) are not all zero, we can solve for the ratio λ1 /λ2 in terms of (x, y, z),
and so we get back the same surface as before. But when (x, y, z) are all zero, the pair
(λ1 , λ2 ) is undetermined and parametrizes a CP1 , which we denote by E1 . Furthermore
it is easy to check that this new surface is non-singular. The surface deﬁned by (3.19) is
said to be the blow-up or resolution of the singular surface P = 0 at x = y = z = 0, andthe new CP1 is the exceptional divisor of the blow-up.
Note that since (λ1 , λ2 ) are never simultaneously zero, our eigenvalue equation still
implies that P = det(M ) = 0, so the process of blowing up is not a complex structure
deformation, but rather a Kähler deformation. By varying the Kähler form we can continuously change the size of the exceptional CP1 , making it large or shrinking it back toa singularity.
An alternative way to smooth the singularity is to change the equation to
xy + z 2 = µ,µ ̸= 0(3.20)

This is a complex structure deformation, and not a Kähler deformation. In this case thesmoothed geometry again contains a topological S 2 , but unlike for the resolution it is
not a holomorphic submanifold. To see it deﬁne x = u + iv, y = u − iv, in which casethe equation becomes u2 + v 2 + z 2 = µ. Restricting all the variables to be real, this isprecisely the equation of a (non-holomorphically embedded) two-sphere. Again as µ → 0
this two-sphere shrinks to zero size.
It will be useful to formulate the latter construction in a way that will generalize moreeasily. Let us write the deformation of the singularity as
xy = P2 (z)(3.21)

where P2 (z) is a quadratic polynomial with leading term z 2 . An equation of the form
xy = t is a conic, so we may think of our surface as a conic bundle (a ﬁbration of conics)
37

Figure 4: Picture for the deformed singularity xy = P3 (z). Fibering
the shrinking S 1 of the conic ﬁbers over diﬀerent paths in the zplane results in topologically distinct two-spheres.

over the z-plane. As t → 0 the conic degenerates to a pair of lines, given by x = 0 and
y = 0. The conic has a minimal S 1 , which can be seen by rewriting xy = t as u2 + v 2 = t
and taking the variables to be real. Note that the S 1 shrinks to zero as t → 0. There areexactly two points on the z-plane where this happens: the roots z1∗ and z2∗ of the quadratic
equation P2 (z) = 0. Now we take a path from z1∗ to z2∗ on the z-plane, and construct a
manifold by ﬁbering the minimal S 1 of the conic over this path. The resulting manifold
is topologically a sphere, see ﬁgure 3.
Let us make the singularity a bit more complicated. We consider
Q(x, y, z) = xy + z 3 = 0(3.22)

Again as a ﬁrst step we can consider replacing this by the following pair of equations:(x z−z 2 y

)(

λ1
λ2

)
= 0

(3.23)

However from the second equation −z 2 λ1 + yλ2 = 0 we see that our new surface is still
singular at z = y = λ2 = 0. But we can repeat the procedure, and replace this equationby the pair of equations
(
)(
)
λ2 z
µ1
= 0
(3.24)zλ1 yµ2This creates another exceptional CP1 , which we denote E2 , this time parametrized by(µ1 , µ2 ). Taking into account that (λ1 , λ2 ) and (µ1 , µ2 ) each live on a CP1 and cannot
vanish simultaneously, it’s not hard to see the surface resulting from our two blow-ups
is now non-singular. You can also check that the two exceptional CP1 ’s intersect at
precisely one point. Indeed when x = y = z = 0 the equations simply reduce to λ2 µ1 = 0,
which describes two CP1 ’s with a simple intersection at λ2 = µ1 = 0.
38

Dynkin typeequation

restrictions

An

xy + z n+1 = 0

n≥1

Dn

x2 + y 2 z + z n−1 = 0

n≥4

E6

x2 + y 3 + z 4 = 0

E7x2 + y 3 + yz 3 = 0

E8x2 + y 3 + z 5 = 0

Table 3: Canonical forms of the ADE surface singularities.

We can play a similar game with deforming the singularity, see ﬁgure 4. We write thedeformed equation as
xy = P3 (z)(3.25)
where P3 (z) is a generic degree three polynomial with leading term z 3 . Then the conic
ﬁbers degenerate when P3 (z) = 0, which happens at three special points on the z-plane,
say z1∗ , z2∗ and z3∗ . Then by ﬁbering the shrinking S 1 ’s over paths from z1∗ to z2∗ , and fromz2∗ to z3∗ , we get two topologically distinct two-spheres with intersection number one. We
can make a further two-sphere by ﬁbering the S 1 ’s over a path from z1∗ to z3∗ , but by
deforming this path to go through z2∗ , we see that the homology class of this two-sphereis the sum of the homology classes of the other two-spheres.
In general one can make some very nasty surface singularities. But there is a specialclass of surface singularities, of which the two cases above are the simplest examples, thathave the property that the blow-up does not aﬀect the topological type of the canonicalbundle. (In algebraic geometry parlance, the singularity should admit a crepant resolution). This class is particularly interesting for us, because we are interested in Calabi-Yau
spaces, and we want the topological type of the canonical bundle on the resolution to betrivial.
This class of surface singularities goes under many names: Kleinian singularities, DuVal singularities, etc. They turn out to have many nice properties, for example they admita Calabi-Yau metric that is Aymptotically Locally Euclidean (meaning that asymptotically they look like ﬂat space modded out by a ﬁnite group), hence they are also called
ALE spaces. We will just call them ADE surface singularities, because they turn out toadmit an ADE classiﬁcation: the homology lattice of SADE , either after deformation or
resolution, is isomorphic to an ADE root lattice:
H2 (SADE , Z) = ΓADE

(3.26)

In particular, picking a set of two-cycles Ei corresponding to fundamental roots, their39

intersections are given the corresponding Cartan matrix:
CijADE = Ei · Ej .

(3.27)The only diﬀerence with conventional root lattices is that we take the Cartan matrix to be
negative-deﬁnite instead of positive-deﬁnite, by including a minus sign. Up to coordinatetransformations, the ADE surface singularities ﬁt in the list shown in table 3.
The appearance of an ADE classiﬁcation suggests a connection with Lie groups, andhence with gauge theories. Indeed let us consider M -theory compactiﬁed on a resolved
ADE surface. As before we can recover U (1) gauge ﬁelds by expanding C(3) in harmonic
two-forms. Since we are considering compactifying on a non-compact manifold, in orderto get dynamical modes we should expand in normalizable harmonic forms. We have
H2 (SADE , Z) ≃ Hc2 (SADE , Z)∗ , where the subscript ‘c’ refers to compactly supported
cohomology, and it turns out that the image Hc2 (SADE , Z) → H 2 (SADE , Z) is generatedby L2 -harmonic forms. Thus the number of abelian U (1) gauge ﬁelds that we get matcheswith the number of Cartan generators of the corresponding ADE gauge group. These
modes are paired with metric modes that correspond to the scalar ﬁelds of the 7d YangMills theory. Namely we take the holomorphic (2, 0) form and the Kähler form and expand
these in the above harmonic forms, yielding three real scalars for each Cartan generator.
A big surprise in the ’90s was that we also recover the non-abelian W -bosons. M theory has solitonic M 2-branes which we ∑can wrap on the two-cycles. Suppose thatan M 2-brane is wrapped on a cycle α =ni Ei , yielding a particle-like object in the
low-energy theory. The intersection product is negative deﬁnite, since up to a minus
sign it is actually the Killing form on the ADE Lie algebra. There are no integer linear
combinations with α·α = −1, and M 2-branes wrapped on a cycle with α·α < −2 turn out
not to form bound states, in other words if they exist they correspond to multi-particle
states. So one is interested in M 2-branes wrapping cycles such that α · α = −2, i.e. suchthat α corresponds to a root. Quantizing the collective coordinates of such a wrapped
M 2-brane yields a massive vector multiplet, indeed that’s really the only multiplet we
could expect given that the resulting particle has no internal structure (the only bosoniccollective coordinates are translations) and the theory has sixteen supercharges. Almostby deﬁnition the non-abelian ﬁelds have the right U (1) Cartan charges. In this way werecover a non-abelian gauge theory with ADE gauge group.
The mass of each non-abelian W -boson is proportional to the volume of the cycle that
the M 2-brane is wrapped on. In the limit where we shrink the volume of these cycles
to zero size and the geometry becomes singular, the mass of the non-abelian W -bosons
goes to zero, and we recover the full unbroken non-abelian gauge symmetry. This is thegeneral mechanism by which string or M -theory relates ADE singularities to ADE gauge
theories. Unlike in perturbative IIB, where we could only get classical gauge groups, we
also get exceptional gauge symmetries this way.

40

(B)

(A)

Figure 5: An open fundamental string in type IIB lifts to a membrane wrapping an exceptional cycle in F -theory. Some W-bosons
may correspond to ground states of multi-pronged (p, q) strings .

3.5. Non-abelian gauge ﬁelds and the Kodaira classiﬁcation
Given what we said in the previous subsection, and considering that F -theory compactiﬁcations are closely related to M -theory, we would expect that non-abelian gauge
ﬁelds are closely related to ADE singularities. Let us now discuss how we can see thisfrom the point of view of (p, q) 7-branes and elliptic ﬁbrations.
Let us consider two parallel D7-branes. The non-abelian gauge bosons that enhance
the symmetry to SU (2) come from open strings stretched between the two 7-branes.
How is this seen in F -theory? Consider the path associated to an open string stretching
between the branes. On top of each point of this path we can associate a 1-cycle of theT 2 ﬁber, which we take to be the (1, 0) cycle. On the left and right ends of the path,
this (1, 0) cycle shrinks to zero. Altogether then we reconstruct a topological S 2 . Using
the M -theory perspective, we can wrap an M 2-brane on this S 2 , which turns into thefundamental open string as we go to F -theory. As the we let the 7-branes approach eachother, the S 2 shrinks to zero and the Calabi-Yau fourfold develops an A1 singularity. (TheIIb space-time is still perfectly smooth, only when we add the elliptic ﬁbration do we see
the singularity). As the S 2 shrinks to zero, the ground states of the wrapped M 2 braneor open fundamental string become massless, yielding the oﬀ-diagonal components of anSU (2) vector multiplet. We get both W + and W − by reversing the orientation of themembrane.
The type of singularities in an elliptic ﬁbration were classiﬁed by Kodaira (see table 4).
The elliptic ﬁbration may develop an ADE singularity by letting various branes approacheach other, and one would naturally expect that by wrapping M 2 branes on the vanishingcycles one gets an enhanced ADE gauge symmetry.
In perturbative IIB we didn’t see exceptional gauge symmetries. How do we understand these more general situations from the IIB space-time? It turns out that the
exceptional cycles of a resolved ADE singularity do not necessarily project to open strings
with two ends, but may yield so-called multi-pronged strings with multiple ends. This41

ord(f )ord (g)ord(∆)

ﬁber type singularity type

≥0

≥0

0

smooth

−

0

0

n

In

An−1

≥1

1

2

II

−

1

≥2

3

III

A1

≥2

2

4

IV

A2

2

≥3

n+6

In∗

Dn+4

≥2

3

n+6

In∗

Dn+4

≥3

4

8

IV ∗

E6

3

≥5

9

III ∗

E7

≥4

5

10

II ∗

E8

Table 4: Kodaira classiﬁcation of singularities of elliptic ﬁbrations, indicating the order of vanishing of ∆, f
and g.

is the key to getting the exceptional groups and can happen when the dilaton cannot betaken small. All the ADE Lie algebras have been reproduced from such (generally multipronged) strings. For instance the roots of E8 can be recovered from a conﬁguration ofseven A-branes, one B brane, and two C-branes, where A = (1, 0), B = (1, −1), C = (1, 1),
see ﬁgure 6. Conﬁgurations for the type D Lie algebras are similar except that they justuse one C brane instead of two. A B-brane and C-brane can be combined into an orientifold plane and yield weak coupling limits, but this is not possible for the exceptional
cases.
Another diﬀerence is that in perturbative IIB we found U (N ) gauge symmetry, but
in F -theory we ﬁnd SU (N ) gauge symmetry. A similar issue also appears in M -theory.
So what happened to the extra U (1) when we turn on the string coupling? It turns outthat this extra U (1) is typically already massive in perturbative IIB due to a Stückelberg
coupling to Ramond-Ramond ﬁelds, with a mass proportional to the string coupling.
When the string coupling is ﬁnite this mass is of order the Kaluza-Klein scale, so it mustcorrespond to a massive Kaluza-Klein mode.
3.6. Application to Grand Uniﬁcation
As an application of what we have learned, let us make a Calabi-Yau four-fold by
writing down a ﬁbration of deformed E8 surfaces over a compact Kähler manifold S. For
42

Figure 6: Representation of the fundamental roots of E8 from [4]. Here a cross denotes
an A-brane, a circle denotes a B-brane, and a box denotes a C-brane.
example we could take S to be CP2 or CP1 × CP1 , or a del Pezzo surface, i.e. a blow-up
of CP2 . We write our ﬁbration over S as the following equation:y 2 = x3 + a0 z 5 + a2 z 3 x + a3 z 2 y + a4 zx2 + a5 z 5(3.28)

Here ai is a section of KSi ⊗ L for some line bundle L on S. The equation has a term
linear in y and a term with x2 , so it is not in Weierstrass form, but can be rewritten as
such with a few coordinate changes.
The terms y 2 = x3 + a0 z 5 describe an E8 singularity (see table 3) and the remaining
terms are a partial deformation of it, making the ﬁbers less singular. The situation issomewhat analogous to what we saw in the heterotic string: there we started with an
E8 gauge theory, which was broken to a smaller gauge group by turning on a non-trivial
proﬁle for the gauge ﬁeld. Here we also see some kind of breaking of E8 , but it seems lessclear how this breaking comes about. You can actually translate this into the language ofeight-dimensional gauge theory and see that the breaking of E8 here is due to a non-trivial
Higgs bundle (A, Φ). In particular, the sections ai with i > 0 appearing in equation (3.28)
are essentially invariant polynomials in the adjoint ﬁeld Φ.
If you do the algebra for (3.28) you will see that one generically gets an SU (5) singularity along x = y = z = 0, which is a copy of S. So from a IIB space-time perspective
we have a stack of ﬁve D7-branes with SU (5) gauge symmetry wrapped on S. Inspecting
the discriminant, one ﬁnds that there are some additional 7-branes which intersect withthe 7-branes wrapped on S along two curves. The singularity type enhances there toeither SU (6) or SO(10), and one can show that charged matter ﬁelds propagate along
these 7-brane intersections, in a representation that depends on the type of enhancement. In particular, one ﬁnds matter ﬁelds in the 10 of SU (5) propagating along the
curve Σ10 = {a5 = 0} on S, and matter in the 5 of SU (5) propagating along the curveΣ5 = {a0 a25 − a2 a3 a5 + a23 a4 = 0}. These are precisely the representations that the describe the quarks and leptons (as well as the Higgses) in SU (5) Grand Uniﬁed models,
43

Br

Ε6

SU(7)

Σ10
SO(12)

Σ5

Σ5
Σ10

S

Figure 7: Schematic picture of SU (5) GUT models, for generic values of the complex
structure moduli. Indicated are the matter curves, branch locus (related to Weyl groupmonodromies that we did not discuss), and their intersections.
and the curves where they are localized are called the matter curves. We get additional
enhancements when the matter curves intersect, contributing to the Yukawa couplings.
The geometry (3.28) therefore describes SU (5) Grand Uniﬁed Models in F -theory.
3.7. Relation with perturbative IIB and heterotic strings
Finally we would like to brieﬂy describe how the ingredients of perturbative string
compactiﬁcations that we studied emerge from F -theory.
Let us start with the IIB limit, also known as the Sen limit of an F -theory compactiﬁcation, which is really the easier one of the known limits. We take our Calabi-Yau
four-fold to be an elliptic ﬁbration over a (non-Calabi-Yau) Kähler three-fold B3 . Recall
that the elliptic ﬁbration was presented in Weierstrass form as
y 2 = x3 + f x + g(3.29)

where f is a section of KB−43 , and g is a section of KB−63 . Now we are going to introduce
an alternative parametrization of the Weierstrass form, as follows. We consider three
sections b2 , b4 , b6 , where bi is a section of KB−i3 . Then we will write the equation of theelliptic ﬁbers in the form
1
(3.30)
y 2 = s3 + b2 s2 + 2b4 s + b6
3
44

This version has perhaps a bit more redundancy than the Weierstrass form, due to theextra quadratic term s2 on the right hand side, but it is clearly a perfectly legal way towrite a family of elliptic curves. One can relate it explicitly to the Weierstrass form byputting s = x − b2 to eliminate the quadratic term. The coeﬃcients were deliberately
denoted by b2 , b4 and b6 . As we will see shortly, they are closely related to their namesakesin the section on perturbative IIB compactiﬁcations.
Next we introduce a parameter t, as follows:1y 2 = s3 + b2 s2 + 2b4 st + b6 t2
3(3.31)The discriminant takes the form ∆ = 324 t2 b22 (b2 b6 − b24 ) + O(t3 ). We claim that the IIB
limit corresponds to t → 0, i.e. one can show that the string coupling eϕ = Im(τ )−1 goesto zero almost everywhere in this limit. So let’s see what happens to our geometry when
we take this limit.
If we take t → 0 in (3.31), then we are simply left with y 2 = s2 (b2 +s/3), or equivalentlyỹ = b2 + s/3 with ỹ = y/s. This is just a CP1 , so it might seem that our elliptic ﬁbration
has degenerated to a boring CP1 -ﬁbration over B3 . However, there is a slightly diﬀerent
way to take the limit. Let us deﬁne s = ts̃, y = tỹ, and then take the limit t → 0. In thiscase (3.31) becomesỹ 2 = b2 s̃2 + 2b4 s̃ + b6
(3.32)
2

which describes a ﬁbration of conics over B3 .
The overall picture then is that in the t → 0 limit, our elliptic ﬁbers have split into
two pieces, and hence the whole Calabi-Yau four-fold has split into two pieces. It’s not
hard to see that the intersection of these two pieces is precisely the Calabi-Yau three-foldξ 2 = b2 that we discussed in the context of IIB orientifolds. Our conics further degenerate
to a pair of lines when b2 b6 − b24 = 0, so there is an S 1 shrinking to zero in the ﬁbers
there, which should correspond to the D7-brane locus as we have seen before. Indeed it
has precisely the required form of the locus wrapped by the D7-branes discussed in thesection on perturbative IIB.
It is somewhat amazing to see what has happened here. We discussed perturbative
type IIB compactiﬁcations in terms of Calabi-Yau three-folds, orientifold planes and D7
branes. Now we see that much of this information can also be captured by Calabi-Yau
four-fold with slightly degenerate elliptic ﬁbers. Turning on the string coupling simply
corresponds to a smoothing of this degenerate four-fold. To complete the picture we also
have to discuss the ﬂuxes, which one can also do quite explicitly.
It is interesting to see how the perturbative IIB spectrum that we discussed previously emerges from the Calabi-Yau four-fold. A precise comparison requires logarithmic
cohomology on the degenerate Y4 , but there is a simple intuitive rule that works reasonably well: even cohomology groups H+p,q (X) on the IIB side lift directly to the four-fold,whereas odd cohomology groups H−p,q (X) only lift after wedging with a one-form dz on45

the elliptic ﬁbers. So the h2,1
− (X) complex structure moduli on the IIB side lift to (3, 1)
forms on Y4 , which do indeed describe complex structure moduli of Y4 . Similarly the
1,1
h1,1+ (X) Kähler moduli lift to Kähler moduli of Y4 , and the h− (X) periods of (B2 , C2 )lift to (2, 1)-forms, which describe periods of the F -theory three-form C(3) . The h2,1
+ (X)+vectors are a little harder to see. The four-form C4 of IIB lifts to C(6) on Y4 with two
indices on the elliptic ﬁber, where C(6) is the dual of C(3) in M -theory. The h2,1
+ (X) lifts
to (3, 2)-forms on Y4 , and expanding C(6) in such forms yields the expected vectors.
One can show that the open string spectrum of perturbative IIB, which we did not
discuss in detail, is also reproduced by the four-fold Y4 . Roughly they consist of gauge ﬁeld
deformations on the 7-brane worldvolume described by (0, 1)-forms, and deformations of
the adjoint ﬁeld described by (2, 0)-forms. By a (1, 1) shift they lift to (1, 2)-forms and(3, 1)-forms on Y4 , corresponding to additional deformations of C(3) and complex structure
moduli of Y4 respectively. Additional open string modes are obtained by quantizing
wrapped M 2-branes, as mentioned previously.
The heterotic limit is similar in spirit, but technically more complicated to discuss,essentially due to the more complicated group theory of E8 . Let us consider as a special
case a Weierstrass ﬁbration of the form
y 2 = x3 + z 5 + f xz 4 + gz 6 + z 7(3.33)

where we assumed that B3 admits a P1 -ﬁbration, z is a coordinate on that P1 , and f
and g are independent of z. This form can only be achieved in very special cases, because
in general all the coeﬃcients are non-trivial sections, but let us ignore that here. The
four-fold has an E8 singularity at y = x = z = 0 and another at z = ∞. We introduce aparameter t as
y 2 = x3 + tz 5 + f xz 4 + gz 6 + z 7(3.34)
Again as we take t → 0 the Calabi-Yau four-fold splits into two pieces. The ﬁrst pieceis given by taking t → 0 directly in the above equation, and the second is given by
deﬁning (x, y, z) = (t2 x̃, t3 ỹ, tz̃) and then taking t → 0. These two pieces intersect overa Calabi-Yau three-fold, which is identiﬁed with the heterotic Calabi-Yau. Since we havean E8 × E8 gauge symmetry in this case, the heterotic bundle is trivial.
References[1] Green/Schwarz/Witten, chapter 15.
[2] Griﬃths/Harris, “Principles of algebraic geometry.”[3] C. Vafa, “Evidence for F-Theory,” Nucl. Phys. B 469, 403 (1996) [arXiv:hepth/9602022].

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[4] O. DeWolfe and B. Zwiebach, “String junctions for arbitrary Lie algebra representations,” Nucl. Phys. B 541, 509 (1999) [arXiv:hep-th/9804210].

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