M. Wijnholt. LMU MunichOverview

 Heterotic compactification

 Perturbative IIB
 F-Theory

Part I: Heterotic compactification

Heterotic string

 Bosonic fields in ten dimensions: Lots of symmetry: 10d super-Poincare, SO(32) or E8 x E8 gauge symmetry
 Break the symmetry to something smaller Kaluza-Klein Ansatz:

smallEffective four-dimensional theory

Narain compactification
 Simplest possibility: Often referred to as Narain compactification
 Admits an elegant formulation using even, self-dual momentum lattice
Moduli space:

Describes deformations ofalong However: N=1 supersymmetry in 10d reduces to N=4 supersymmetry in 4d
Very restrictive Want to consider more general compactifications with N=1 in 4d

Supersymmetry variations

The 10d supersymmetry variations of the heterotic string are given by:

We want to find non-trivial solutions towith 4d Poincare invariance
AssumeandWe are left with:

Assumefor now

constant

Covariantly constant spinors
The spinor

lives in(Majorana-Weyl)

Decompose underThenimplies
i.e. X6 admits a covariantly constant spinor

This is a very strong condition on X6. Discuss structure implied by thisrequirement.

Consequences (I)Covariantly constantRescale and set norm to one

Then also
Consider bilinear:

Then we haveorThis is an ``almost complex structure’’May not necessarily come from a complex structure
Integrability condition: vanishing of the Nijenhuis tensor:

Consequences (I)Fromand

we have also

So the Nijenhuis tensor vanishes and the complex structure is integrable

X6 is a complex manifold

Convenient to use complex coordinates
Holomorphic tangent bundle
Holomorphic co-tangent bundle

spanned byspanned by

Consequences (II)Fromalso find thati.e. the metric is Hermitian.
Using complex coordinates in which
We get

Then we can construct an associated two-form (the ``Kahler form’’):It is closed due to covariance constancy.
A complex manifold with closed associated two-form is called a Kahler manifold.
Such manifolds have very nice properties, some of which we discuss later

Consequences (III)Fromalso findWith a little manipulation, see that the Ricci curvature vanishes
So X6 is a Ricci-flat Kahler manifold
On any Kahler manifold, construct two-form from Ricci tensor:

It is closed (by Bianchi) and defines a cohomology class, the first Chern class of X
Ricci flatness implies that the first Chern class vanishes
Conversely, it turns out that this is sufficient to solve
first discuss holonomy.

, butHolonomy

Parallel transport a vector around a loopv
v’All loops

subgroup of SO(d) called the holonomy group

Furthermore, close connection with curvature:
For small loops enclosing area

: infinitesimal rotation

Similarly parallel transport spinors or forms around loop

Holonomy
We have equivalenceandLet’s parallel transport
Sinceand restrict toaround a loop
, just take globally defined

just comes back to itself
Holonomy group is subgroup of SU(4) that preserves
Use SU(4) transformation to put
Subgroup that preservesin the form

is SU(3)Covariantly constant spinor means that X6 has SU(3) holonomy

Holonomy
Kahler manifold has U(3) holonomy (subgroup of SO(6) that preserves Kahler form)

Determinant comes from parallel transporting (3,0) forms
So SU(3) holonomy

is trivial line bundle

Another way to see this: we have a global nowhere vanishing section:``holomorphic volume form’’

Again impliesConversely, the curvature of
So Ricci flat & Kahler

is trivial line bundle

is the Ricci form
(3, 0) forms are parallelSU(3) holonomy/covariantly constant spinor

Calabi-Yau manifolds
We define a Calabi-Yau manifold to be a Kahler manifold withWe have seen that Ricci-flat & Kahler implies
But converse is not obvious, since Ricci-flatness is differential geometric,andis merely topological. Nevertheless ….
Calabi conjecture/Yau’s theorem:
A compact Kahler manifold with vanishing first Chern class admits a Ricciflat metric. This metric is uniquely determined by the complex structure andthe Kahler class.

This simplifies life enormously. We don’t need to actually write down aRicci flat metric or a covariantly constant spinor.

We can get away with checking

Calabi-Yau manifolds

Thus the following conditions on X are all equivalent: X admits a covariantly constant spinor () X admits a metric of SU(3) holonomy
 X is Kahler and admits a Ricci flat metric X is a Calabi-Yau manifold (Kahler and)

Examples
A canonical example of a Kahler manifold is complex projective space

It is not Ricci flat.
But holomorphic submanifolds of complex projective spaces are also Kahler.
Can write explicit polynomial equations.
By choosing the degree carefully, we can makeOne can show that degree n+1 equation in Quintic three-fold: K3 surface: Elliptic curve:

Examples

Another famous example of a Calabi-Yau three-fold: the Tian-Yau manifold

Coordinates:

Equations:Properties of Calabi-Yau manifolds

We are now going to discuss some general properties of Calabi-Yau manifolds.

These properties will also tell us about the structure of the effective fourdimensional theory obtained by compactifying on the Calabi-Yau.

We quickly review some properties of differentiable manifolds, and thenstate their complex analogues.

Hodge decomposition

Let’s review some facts about manifolds.
Complex of differential forms:

De Rham cohomology:

Adjoint of exterior derivative:

Hodge Laplacian:Hodge decomposition
The Hodge decomposition is the following orthogonal decomposition:“Harmonic + exact + co-exact”
In particular for closed forms:
Every cohomology class has a unique harmonic representative
On a complex manifold: decomposition by (holomorphic, anti-holomorphic) indicesHodge numbers

Since, get a complex

Define Dolbeault cohomology:“Hodge numbers”

Dolbeault Laplacian:Hodge decomposition for:Every class in

has a unique harmonic representativeHodge numbers on a Kahler manifold
On a Kahler manifold it turns out that
Harmonic forms agree andpreserves (p,q) type

Further symmetries:

Complex conjugation:
Hodge *-isomorphism :On a Calabi-Yau with SU(n) holonomy, further have

Then we are only left with two independent Hodge numbers:andHodge diamond of a Calabi-Yau
Hodge diamond for a Calabi-Yau three-fold with SU(3) holonomy:

By uniqueness of CY metric, metric deformations are complex structure andKahler deformations.

counts complex structure deformations
Eg. For quintic three-fold: deform to general degree five polynomial
There are 101 such deformations
counts Kahler deformations (deformations of Kahler form J )Eg for quintic: only “breathing mode” inherited from

Moduli space of a Calabi-Yau

B-field deformations give further

real parameters:

We can economically think of this as deformations of a complexified Kahler form:

Thus adding B-field deformations has the effect of complexifying

BundlesWe’ve discussed the consequences ofin quite some detail.

Now it is time to return to the remaining condition:

This equation can be written as:

These are called the Hermitian Yang-Mills equations.
We first discuss the meaning of
Split the connection intoSo. We have:is pure gauge,

We can locally set

by a (complexified) gauge transformation.

Holomorphic bundles

Now consider the gauge field on two different patches:

On the overlap we have

But we saw that on each patch we could set
With this choice we will havei.e. the transition functions can be chosen holomorphic.
Thus the gauge field is a connection on a holomorphic bundle.

Stability
It turns out that curvature always decreases along holomorphic sub-bundles
If the connection satisfies, thensub-bundle U is negative (or possibly zero).

along holomorphic

Then if U is a holomorphic sub-bundle of our Hermitian YM bundle V,
it is not hard to see that the degree of U must be negative, whereA bundle V is said to be stable if

where slope = degree/rank. A bundle is poly-stable if it is a sum of stable bundlesof the same slope.
Hermitian-Yang-Mills (HYM) implies poly-stability.

Stability
Hermitian-YM implies poly-stability. Remarkably the converse is also true:
Donaldson-Uhlenbeck-Yau:
If a vector bundle V on a Kahler manifold is holomorphic and poly-stable,
then there exists a unique solution to the Hermitian Yang-Mills equations.
Finding explicit solutions of the HYM equations is practically impossible.
The above says that we can get away with writing down holomorphic bundles
and checking the slope of holomorphic sub-bundles.

Writing down interesting holomorphic bundles requires some moretechnology, which unfortunately we do not have time for. But we can give one
canonical example of a Hermitian-Yang-Mills bundle.

Tangent bundle

A simple example of a HYM bundle is the holomorphic tangent bundle of aCalabi-Yau.
We have already seen that

But this is precisely
We didn’t need Donaldson-Uhlenbeck-Yau for that.

Virtually every other interesting example does require DUY.

Bundle-valued Dolbeault cohomology

Since we had, for holomorphic bundles we can define a generalizationof the Dolbeault complex:

These cohomology groups are just what we need to understand the KaluzaKlein reduction of the gauge sector. They have the following interpretation:
Counts unbroken gauge generators that survivein effective 4d theory
Counts charged matter fields in effective 4d theory

Let’s illustrate this with an example.

Example: the standard embedding
We consider the E8 x E8 heterotic string.
Take the connection for the first E8 bundle to be the spin connection of X_6.
Under the maximal subgroupdecomposes as:, the adjoint representation

At the level of Dolbeault cohomology, this yields

The 4d spectrum is then:Gauge field in 78 , the adjoint of E_6
chirals in 27;
chirals in

E_6 Grand Unified Theory withgenerations