Heterotic strings and fluxes. Based on: K. Becker, S. Sethi, Torsional heterotic geometries, to appear. K. Becker, C. Bertinato, Y-C. Chung, G. Guo, Supersymmetry breaking, heterotic strings and fluxes, to appear.
K. Becker, S. Sethi,
Torsional heterotic geometries, to appear.
K. Becker, C. Bertinato, Y-C. Chung, G. Guo,
Supersymmetry breaking, heterotic strings and fluxes, to appear
Generic string compactifications involve both non-trivial metrics and non-trivial background fluxes.
Flux backgrounds have mostly been studied in the context oftype II string theory and generically involve NS-NS and R-R fluxes. The analysis of these backgrounds can be done in the supergravity approximation since the size is not stabilized by flux at the perturbative level.
However, quantum corrections are hard to describe since
the string coupling constant is typically of order 1....
In this lecture we will be discussing the construction of flux backgrounds for heterotic strings. Such compactifications
involve only NS fields: namely the metric and 3-form H. The
dilaton is never stabilized. Spaces with non-vanishing
H are called torsional spaces.
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)
Candelas, Horowitz, Strominger,Witten, '85
Braun,He, Ovrut, Pantev, 0501070
Bouchard, Donagi, 0512149
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.
Progress in this direction has been slow. On the one side
we do not have algebraic geometry techniques available
to describe torsional space. On the other there has
been a lack of concrete examples.
orientifolds of K3xT2
quotients of nilmanifolds
Dasgupta, Rajesh, Sethi, 9908088
Fernandez, Ivanov, Ugarte,
M. Becker, L-S. Tseng, Yau, 0807.0827
The number of flux backgrounds for heterotic strings is of the same order of magnitude than the number of flux backgrounds in type II theories...
M: Calabi-Yau 3-fold,
M: complex and
Complex manifold with hermitian metric
construct new classes of torsional heterotic geometries
using a generalization of string-string duality in the presence
of flux. We will work with a concrete metric.
use the intuition gained from duality to obtain a more general
a set of torsional spaces.
comments about phenomenology: susy breaking and
gauge symmetry breaking.
form of CY4
In the presence of flux the space-time metric is no longer
a direct product
of the CY4
Since X8 is conformal invariant (Chern+Simons, '74)
this is a Poisson equation.
is quartic in
A solution exists if
The data specifying this background are the choice of CY4 and the primitive flux. As long as the total charge vanishes
a solution exists.
the CY4 is elliptically fibred so that we can lift the
3d M-theory background to 4d. By shrinking the volume
of the fiber to zero this can be converted in a 4d
compactification of type IIB with coupling constant
determined by the complex structure of the elliptic fiber.
The G4 flux lifts to
together with gauge bundles on 7-branes. Susy requires
appears by wrapping the M5 brane on the K3 fibers.
The M5 brane supports a chiral 2-form. Since the
number of self-dual 2-forms of K3 is 3 there appear 3
compact scalars parameterizing a T3. In the F-theory
limit this T3 becomes a T2. G4 flux can be included
and the choice of flux will determine how the T2 varies
over the base. Schematically the metric of the space
on which the heterotic string is propagating is
This is almost an existence proof for the heterotic
non-singular orientifold locus so that the elliptic
fibration is locally constant.
We are left with type IIB with constant coupling...
Type IIB on an elliptic CY3 with constant coupling
In the semi-flat approximation (stringy cosmic string):
Away from the singular fibers there is a U(1)xU(1)
Type IIB background
Quotient of type IIB on CY3 by
Decompose the forms
The forms which are invariant under Z2 are
Locally we can write use this connection to twist
the torus fiber of the CY space.
1-forms of the fiber
the 2-form transforms as
The dilaton is
The conditions for unbroken susy are a repackaging of
the susy conditions on the type IIB side. In particular
any elliptic CY3 spacegives rise to a torsional space..
Dictionary between type I and heterotic
type I flux
...giving rise to the type I Bianchi identity
Type IIB Bianchi identity
Using the type IIB
Bianchi identity becomes the differential equation for
the warp factor
which we can solve in the large radius expansion, which
means for a large base and a large fiber. So the existence
of a solution is guaranteed.
In the large radius limit on the het side, which corresponds
to large base and small fiber in type IIB, this is a non-linear
differential equation for the dilaton (of Monge-Ampere
type if the base is K3).
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....
Gauge fields with components
K. Becker, M. Becker, J-X. Fu,
L-S. Tseng, S-T Yau,0604137
All of these backgrounds
satisfy the e.o.m. in sugra.
To solve the Bianchi identity
gauge fields along the fiber
have to be included.
Here and is the B field in the two
fiber directions. The low energy effective action is invariant
We have seen how to construct a torsional space 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.