M. Wijnholt. LMU Munich

Recap (I)Heterotic compactification to 4d with N=1 SUSY:
Calabi-Yau three-fold, holomorphic poly-stable bundle
Type IIB compactification to 4d with N=2 SUSY:
Calabi-Yau three-fold, RR fields with vanishing flux

Break to N=1

FluxesD-branes and O-planes
U(N), SO(N) or USp(N)Recap (II)F-theory
encodes IIB with varying axio-dilaton by elliptic fibration over IIB space-time
Weierstrass:F-theory compactification on Y to four dimensions with N=1 SUSY:
EF-theory torus

B3

Compactification manifold

R1,3Minkowski spaceY
Y = elliptic Calabi-Yau
four-foldSingular elliptic fibers over discriminant locus:

7-branes

Questions
F-theory somehow encodes 7-branes in the geometry of elliptic fibrations
In type IIB, the world-volume Yang-Mills theory was obtained by quantizing
open strings.
Quantizing open strings connecting parallel branes gave us non-abelian gauge
symmetry.
Although we didn’t discuss it, quantizing open strings on intersecting
branes gives us charged matter.

Now we would like to understand the F-theory analogues:
How do we see the worldvolume gauge fields come out?
How do we see the gauge symmetry enhancement when
D-branes collide?

Abelian gauge fields
M-theory perspective:
Bosonic fields:
Abelian gauge fields are obtained by expanding C3 in harmonic forms:

It turns out that to survive in the F-theory limit,onand one index on.

must have one index

This is precisely what we need to get a IIB interpretation:

B3

T2dz is (1,0) form on elliptic fiber

ADE singularities
To understand how non-abelian gauge symmetries arise, first need to discussADE singularities. Simplest example:

Two ways to desingularize.
Blow-up:
Replace xy=z^2 by pair of equations
Linear equations
 whennon-singularsolve forisomorphic to xy = z^2 whenunconstrained
singular point replaced by

Picture of blow-up

ADE singularities

Second way to desingularize:
Deformation:

In both cases we find a finite size S2.

ADE singularities
Part of a general class:
Dynkin typeequation

restrictions…

…

Many nice properties, eg. admit (non-compact) Calabi-Yau metric
After deformation or resolution:

ADE root latticeM-theory on ADE singularities
Main claim:
M-theory on

gives non-abelian gauge theory with ADE gauge group Abelian gauge fields from(& corresponding adjoint fields from metric moduli) Non-abelian W-bosons from quantizing M2-branes wrapped on

This should carry over to F-theory.

IIB perspective

Back to IIB/F-theory. From IIB we expect to get enhanced gauge symmetrywhen 7-branes collide.
What does this look like from the point of view of the elliptic fibration?

B

M2 –brane wrapped on S2 maps to string stretched between D7-branes
As 7-branes collide, get enhanced SU(2) gauge symmetry from A1 singularity.
Note that only the elliptic fibers are singular, IIB space-time is still smooth.

Kodaira classification
Kodaira classified all the singular elliptic fibers you can get (in complex codim 1).
From our perspective, these are all the singular fibers you get by colliding
multiple 7-branes
ord ( f ) ord ( g ) ord () fiber type sing . type0

00smooth00nInAn 1

112II

1

23

IIA1

224

IVA22

3n6

I n*Dn  423n6I n*

Dn  4348IV *E6359III *E74510II *E8

As you can see, the singularity type (and hence the enhanced gauge symmetry)
is of type ADE.

Multi-pronged strings
In F-theory we also got exceptional gauge symmetries. How do we get thesefrom open strings?
In general M2-brane maps tomulti-pronged (p,q)-string
in IIB space-time
Open strings for E8:

Dot = (1,0)-brane
Circle = (1,-1)-brane

Box = (1,1)-brane(DeWolfe et al.)

Splitting O-planes

In fact the (p,q) 7-branes we discussed so far did not include the O7-planes of
perturbative IIB. What happened with these?

monodromy for (p,q) 7-brane:monodromy for O7-plane:

The O7-plane splits into a (1,1) 7-brane and a (1,-1) 7-brane at finite string coupling.

Application
As an application. Let us consider the following (non-compact) four-fold:x,yz-plane

Copy of S at x = y = z = 0.
With help from Kodaira:S

SU(5) singularity
SO(10) singularity
SU(6) singularity
E6 singularity
SO(12) singularity
What does this mean??

Application

F-theory picture of SU(5)Grand Unified Theories

IIB limit
F-theory can reduce to perturbative IIB and heterotic in suitable limits.
Briefly discuss IIB limit, also known as the Sen limit.
Instead of Weierstrass form, write elliptic fibers as:

Now add a parameter t:

The IIB limit corresponds toCalabi-Yau 4-fold Y splits intotwo components

t=0t-plane

More precisely, the elliptic fibers
split into two components

IIB limit
WEYXWT NaiveWT Define. AlternativeWEIntersection:
IIB compactification manifoldwith O7-plane
Singular fibers of WE :D7-branes
Exactly the generic D7/O7 system that we saw in perturbative IIB

Turning on IIB string coupling corresponds to smoothing out Y

F-theory summary

Bookkeeping device for IIB vacua with varying axio-dilaton/finite g s

For 4d N=1 SUSY, require elliptically fibered Calabi-Yau four-fold7-branes geometrized in terms off elliptic fibration

Colliding 7-branes

ADE singularities

enhanced gauge symmetries

Perturbative IIB (and heterotic string) as special limits