M. Wijnholt. LMU Munich

Brief recap
Heterotic compactification – the story so far….
Kaluza-Klein Ansatz:

BPS equations
X6 is Calabi-Yau

Complex structure and Kahler deformations

B-field complexifies
Hermitian Yang-Mills
holomorphic poly-stable bundles

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 X6 .
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 E6
chirals in 27;
chirals in

E6 Grand Unified Theory withgenerationsThe next two slides are from Kijk Magazine, August 1987.

“The lost world of Calabi-Yau”

Part II: perturbative IIB

Perturbative IIB
The bosonic fields of the 10d theory are:

RR anti-symmetric tensor fields
As for heterotic, consider Kaluza-Klein Ansatz of the form:

Assume fluxes vanish. Supersymmetry variations simplify:

Hereparametrize the two ten-dimensional SUSY variations.
As before, X6 should be Calabi-Yau. But since we started with
two spinors in 10d, end up with N=2 supersymmetry in 4d.

Closed string spectrum

The natural multiplets for 4d N=2 SUSY are gravity, vector, hyper.
multiplet

multiplicitygravity

1vector
hyper
Massless modes of anti-symmetric tensor fields

Harmonic forms on X6

Eg: vector multiplets:

There aresuch three-forms, but self-duality ofeliminates halfU(1) gauge fields pair up with complex structure moduli
in vector multiplet
Remaining U(1) gauge field pairs up within gravity multiplet

Closed string spectrum

Similarly you can find the remaining massless modes. We expand

Together with the Kahler moduli of the Calabi-Yau metric, this yields
hypermultiplets.
Finally we define the axio-dilaton:

Together with the 4d Hodge duals ofhypermultiplet.

, this gives one final

Towards N=1

We would now like to break further to N=1 4d supersymmetry.

There are basically two ways:

 fluxes (flux superpotentials)

 Defects (D-branes and O-planes)

We will only discuss some aspects of defects.

O-planes
Perturbative IIB has two basic Z_2 symmetries:
 world-sheet parity P (interchanges left- and right-movers)

: acts as -1 if left-movers are in Ramond sector

Can further combine with involutionof X6 .

An orientifold is an orbifold by involution that involves P.
Fixed planes are called O-planes

We will consider orientifolds of the form

This preserves N=1 SUSY and leads to O3/O7-planes. The other option would be:This would give O5/O9-planesO-planes
We can be quite explicit. PutLet

be a holomorphic section of(is called the canonical line bundle)

Then we can write X6 as a double cover of B3 :The involution isEg. ifand the fixed locus/O7-plane is atthenis simply a polynomial of degree eight.

D-branes

A D-brane is a defect on which fundamental strings can end.

The low energy world-volume theory is the dimensional reduction of 10d superYang-Mills (with U(1) gauge group).
When parallel N D-branes approach each other, the gauge symmetry enhances:

If we want to preserve the same SUSY as preserved by, Thencan consider D7-branes wrapped on holomorphic submanifolds of X6 , orD3-branes localized at points on X6 .

D-branes and Higgs bundles
Suppose we now wrap a stack of D7-branes on a holomorphic submanifold
S of X6 . Let’s study equations for worldvolume fields.
The bosonic fields of the eight-dimensional Yang-Mills theory on the brane are agauge field AM and a complex adjoint field F.
Dimensional reduction of Hermitian Yang-Mills equations:

F can be mapped to a (2,0) form on S (topological twisting).
The dimensionally reduced HYM equations can be written as the following
equations on S:

Story now very similar to HYM.

First two equations define a Higgs bundle. Third equation holds iff the Higgs
bundle is poly-stable.

D-branes
To be a bit more concrete, let’s write down the generic form of sub-manifold
on which brane is wrapped (ignoring gauge field).

Takewith O7-plane at

D7-brane has RR charge +1; O7-plane has RR charge -4.
To cancel Ramond-Ramond charge of O7-plane, D7 should wrap a holomorphic
submanifold of X6 of the form

This guarantees that for any 2-cycle A,
Actually due to a subtlety when the D7 and O7 intersect, b8 must take specialform:

Eg. ifthenare simply a polynomials of degree 8, 16, 24.

D-branes in orientifolds

Under orientifold action, the U(N) gauge field on a stack of D-branes transforms as

This leads to the following possibilities:
 Two stacks are interchangedU(N) Stack is fixed andSO(N) Stack is fixed andUSp(N)

Unlike the heterotic string, no exceptional gauge symmetry. This will bedifferent in F-theory.

S-duality
Type IIB has an exact strong-weak coupling duality, taking
In terms of the axio-dilaton, this is

The axionic shift symmetry

acts asTogether these generate anduality group:

The pairtransforms as a doublet under

S-duality takes a fundamental string to a D-string. A more generaltransformation takes a fundamental string to a (p,q)-string, a bound state ofp fundamental and q D-strings.

We define a (p,q) 7-brane to be a 7-brane where a (p,q)-string can end.

Part III: F-theory

Elliptic curves

Complex structure (“modular parameter”):

Symmetries:

The complex structure parameter
the axio-dilaton of type IIB.

of an elliptic curve behaves just like

Formally attach elliptic curve to every point of IIB space-time,interpreting modular parameter as value of axio-dilaton“Twelve-dimensional compactification of F-theory”

F-theory

In a picture:

F-theory torus

12 dimensions
IIB space-timeWeierstrass form
Write elliptic fibration over IIB space-time in “Weierstrass form”:

Cubic equation

elliptic curve

are sections line bundles over IIB space-timeSpecifyaxio-dilaton is specified implicitlyImportant:

non-trivial profile for t

string coupling of order one, not weakly coupled

Somewhat similar to M-theory versus type IIA

Weierstrass form

For special z, elliptic curve can be come singularThis happens when“discriminant locus” of the elliptic fibration

Interpretation in IIB space-time???

7-branes

Modular parameter t hast

monodromy around discriminant locus

at  bct  dB

(Picard-Lefschetz:

Simplest case:
Integrate over yellow disk:

Hence singular fiber corresponds to D7-brane in IIB picture
Monodromy1  pqp2 M   2q1
pq
(p, q) 7-brane)M-theory perspective
Consider M-theory compactification to 3d with N=2 SUSY(Will lift this to 4d N=1 SUSY momentarily)

Y is eight-dimensional real manifold.
Solve
To get 3d N=2 SUSY we need two covariantly constant spinors

Y has Spin(6) = SU(4) holonomy

Y is a Calabi-Yau four-foldM-theory perspective
Duality in nine dimensions:

Identifications:
Modular parameteraxio-dilaton

Take our Calabi-Yau 4-fold Y to be elliptically fibered and apply duality

Asshrinks to zero, get
and N=1 SUSY in 4d“F-theory compactification on Y to four dimensions”

Schematic picture“F-theory compactification on Y to four dimensions”

EF-theory torus

B3Compactification manifold

R1,3

Minkowski spaceY
Y = elliptic Calabi-Yau
four-fold