1. Heterotic strings and fluxes: status and prospects
In this lecture I will be talking about the construction of flux backgrounds for heterotic strings.
Katrin Becker
``New perspectives in string theory''
GGI, Florence, April 2009

2. Introduction and motivation
Constructing of flux backgrounds for heterotic strings
The background space is a torsional heterotic geometry.
Why are these geometries important?
1) Generality
2) Phenomenology: moduli stabilization, warp factors,
susy breaking,...

3. The heterotic string is a natural setting for building realistic
models of particle phenomenology. The approach taken
in the past is to specify a compact Kaehler six-dimensional
space together with a holomorphic gauge field. For
appropriate choices of gauge bundle it has been possible
to generate space time GUT gauge groups like E6, SO(10)
and SU(5).
Candelas, Horowitz, Strominger,Witten, '85
Braun,He, Ovrut, Pantev,
Bouchard, Donagi,
It is natural to expect that many of the interesting physics and consequences for string phenomenology which arise from type II fluxes (and their F-theory cousins) will also be found in generic heterotic compactifications and that moduli
can also be stabilized for heterotic strings.

4. 3) World-sheet description: flux backgrounds for heterotic
strings are more likely to admit a world-sheet description as
opposed to the type II or F-theory counterparts.
Type II string theory or F-theory: Fluxes stabilize moduli
but most of the time the size is not determined and
remains a tunable parameter. This is very useful since
we can take the large volume limit describe the background
in a supergravity approximation. However, these are RR
backgrounds and its not known how to quantize them.
Moreover, the string coupling is typically O(1). It is unlikely
that a perturbative expansion exists.

5. Heterotic strings: the string coupling constant is a tunable
parameter which can be chosen to be small. There is a
much better chance of being able to find a 2d world-sheet
description in terms of a CFT, to control quantum corrections.
Based on:
K. Becker, S. Sethi,
Torsional heterotic geometries,
K. Becker, C. Bertinato, Y-C. Chung, G. Guo,
Supersymmetry breaking, heterotic strings and fluxes,
to appear.

6. Overview: 1) Motivation and introduction
Overview: 1) Motivation and introduction. 2) Heterotic low-energy effective action in 10d. 3) Method of construction. 4) Torsional type I background. 5) The Bianchi identity. 6) K3 base. 7) Conclusion.

7. Heterotic low energy effective action in 10d

8. The curvature is defined using the connection
Bergshoeff, de Roo,
NPB328 (1989) 439
The curvature is defined using the connection
H includes a correction to

9. We will solve:

10. 1) If H=0 the solutions are CY3 or K3xT2
2) If H does not vanish the spaces are complex but
no longer Kaehler. There is still a globally defined
two-form J but it is not closed. The deviation from
`Kaehlerity' is measured by the flux
There are no algebraic geometry methods available
to describe spaces with non-vanishing H. So we need
a new way to describe the background geometry. Our
approach will be to describe them explicitly with a metric.

11. Method of construction
I will first describe the duality chain we will use to obtain torsional backgrounds using duality.
Then we will use the intuition gained from duality to make a guess for more general solutions.

12. M-theory In the presence of flux the space-time metric is no longer
Kaehler
form of CY4
In the presence of flux the space-time metric is no longer
a direct product
Let me start describing the method of construction of flux backgrounds for heterotic strings.
We will start with a consistent M-theory compactification with flux. So we pick a CY4 and
our favorite (2,2) flux which is primitive.
In the presence of flux the space-time metric is no longer a direct product. A warp factor delta
has to be included in the geometry.
coordinates
of the CY4
warp factor

13. fibred so that we can lift the 3d M-theory background to 4d...
The CY4 is elliptically
fibred so that we can lift the
3d M-theory background to 4d...
The G4 flux lifts to
together with gauge bundles on 7-branes. Susy requires
We are going to impose two additional restrictions.
First we are going to require that this background can be lifted to 4d.
Generic CY4's do not lead to a 4d compactification so we will require the CY4 to
be elliptically fibred. By shrinking the volume of the fiber to zero this can be
converted in a 4d compactification of type iib.
Next since we want this background to have an heterotic dual description we will
further require the CY4 to admit a K3 fibration. The heterotic string then appears
by wrapping the M5 brane on the K3 fibers.
Moreover, we will require that there is a non-singular orientifold locus
so that the elliptic fibration is locally constant. By shrinking the size of the fiber to
zero we obtain a type IIB description.

14. The orientifold locus Symmetry
At the orientifold locus the background is type IIB on B6 with constant coupling To describe B6 start with an elliptic CY3
base coordinates
Symmetry
In this way we obtain type IIB string theory compactified on a elliptic CY3 at constant coupling
The CY3 can be described by a Weierstrass eqn
y^2 = x^3 + f(u) x + g(u)
where f and g are appropriate polynomial depending on the base coordinates.
At each point of the base this describes a torus.
A property which is important for our construction is that there is a Z2 symmetry
which acts as an inversion of the elliptic fiber and which send
y -> -y.
The orientifold action is then given by the product of the operator which changes the sign
of left moving fermions, the world-sheet parity Omega and the inversion of the fiber coordinates.
By further T-dualizing along both circles of the torus fiber we can map the Z2 transformation to \Omega.
Since modding out the type IIB theory by Omega produces the type I theory we see that the calabi-yau
orientifold is dual to type I theory.
A further S-duality transforms this into a background for the heterotic string.
We will use this sequence of string dualities in the presence of flux to explicitly construct
new examples of flux backgrounds for heterotic strings.
So to perform these T-dualities we need an explicit expression for a metric of an
elliptic cy space

15. The metric for an elliptic CY space
In the semi-flat approximation:
We start with type iib compactified on a complex d-dimensional base with axio-dilaton varying
holomorphically over the base.
We will assume that this background satisfies
the 10d equations of motion in the presence of sources.
This is flat 10-d dimensional space-time. We make one of the space-time coordinates
periodic. T-dualize to obtain a IIA background and then lift to M-theory.
The result if M-theory compactified on an elliptic cy space with a metric
given by this result. Here tau is the holomorphic axio dilaton of the type IIB background we started with.
This metric has the properties that it is kaehler if tau is holomorphic and ricci flatness
requires this relation between the base metric and the complex structure of the elliptic fiber.
Two important facts about this metric are that there is the Z_2 symmetry which inverts the fiber
and changes the sign of the two coordinates w. Moreover, away from the locus where the fiber
degenerates this metric has two U(1) isometries related to the shifts in the torus coordinates
along which we can T-dualize to obtain the type I background.
Away from the singular fibers there is a U(1)xU(1)
isometry...

16. Flux In type II theories the fluxes are...
....and should be invariant under
Since the NS-NS and R-R 3-forms are odd under

17. Type IIB background 3-form flux 5-form flux
Now we have all the data we need to specify the type IIB background. So let me flash everything
on one slide.
There is a space-time metric which includes a warp factor.
We use the metric in the semi-flat approximation.
The axio-dilaton is constant.
There is 3-form flux which is primitive and ISD in order to preserve susy.
And there is 5-form flux which satisfies this bianchi id and is self-dual.
Unbroken susy requires the 5-form to be related to the warp factor and the bianchi
identity gives rise to a differential equation for the warp factor directly in type IIB.
I would like to point out two facts about this eqn. There is an alpha' in front of tr R/\R
and this equation only has non-trivial solutions on a compact space if this alpha' corrections
is taken into account.
As the M-theory counterpart this is an equation which can be solved in the large volume limit.
This means we expand all quantities in powers of 1/size and can solve order by order if we wish.
The only condition which has to be satisfied is that the total charge cancels. That the integrated
equation holds if you wish.
Now we want to dualize this to a type I or heterotic torsional solution.
As I am going to explain in a moment one step in the duality used to transform the type IIB background into a torsional space will
be to T-dualize along the two fiber directions. Since tau is a holo function of the base coordinates, in general, the
fibers will be singular. We do not want to T-dualize along singular fibers. The way to proceed is to
choose a CY3 which itself has a non-singular orientifold limit. Which means there is a point in moduli space in which
tau is locally constant up to sl(2,z) jumps. In this case the base is either K3 or or a Hirzebruch surface.
So the approach will be to dualize each of the cases that admit orientifold limits.
But again the original metric is valid even for those cases that cannot be dualized the fibers
may become singular.
5-form flux

18. Torsional type I background

19. In order to construct the type I solution we have to
perform two T-dualitites in the fiber directions. To apply
the dualitites rigorously we will assume that up to
jumps by SL(2,Z), the complex structure parameter of
of the elliptic fiber of the CY3 can be chosen to be
constant. With other words we assume that the CY3
itself has a non-singular orientifold locus. In this case
the base of the elliptic fibration is either K3 or one of
the Hirzebruch surfaces Fn (n=0,1,2,4).
For these cases the duality can be done rigorously.
But solutions also exists if the complex structure parameter
is not constant....

20. Type I background
We now have enough information to apply T-duality to both directions of the
elliptic fiber. Applying T-duality the metric becomes

21. Comments:
1) The spinor equations are solved after imposing
appropriate conditions on the flux
The SUSY conditions (spinor equations) are a
repackaging of the spinor conditions in type IIB.
2) The spinor equations are solved if
and the base is not K3 or one of the Hirzebruch
surfaces.
3) The scalar function is not determined by susy.
It is the Bianchi identity which gives rise to a differential
equation for

22. The Bianchi identity
Next we need to understand how to construct the flux on the heterotic side.
Our guiding principle will be the bianchi identity

23. The Bianchi identity in SUGRA
In type II SUGRA
After two T-dualities in the fiber directions the NS-NS
field vanishes as expected from a type I background
This is a differential equation for the scalar function
which turns out to be the warp factor equation of type IIB
which schematically takes the form

24. The Bianchi identity in a perturbative expansion
One can always solve the Bianchi identity in a perturbative
expansion in 1/L, where L is the parameter which is mirror
to the size in type IIB. On the type I side it corresponds
to a large base and small fiber.
curvature 2-form
of the base

25. K3 base

26. Torsional heterotic background with K3 base
The heterotic background is obtained from the type I
background by S-duality
The resulting metric is K3

27. Topology change
When the fiber is twisted the topology of the space changes.
A caricature is a 3-torus with NS-flux
Using the same type of argument one can show that
the twisting changes the betti numbers of the torsional space
compared to the CY3 we started with. Topology change is
the `mirror' statement to moduli stabilization in type IIB.
Some fields are lifted from the low-energy effective
action once flux is included....

28. The other fields are...

29. For solutions with N=2 susy in 4d
Next we want to solve...
To leading order
For solutions with N=2 susy in 4d
equations of motion

30. K3

31. Conclusion We have seen how to construct torsional spaces given
the data specifying an elliptic CY space.
There are many more torsional spaces than CY's.
Many properties which are generic for compactifications
of heterotic strings on Kaehler spaces may get modified
once generic backgrounds are considered.

32. The End
This is what I wanted to say.

33. College Station, Texas
June July 2, 2010