Heterotic strings and fluxes

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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
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Generic string compactifications involve both
non-trivial metrics and non-trivial background
fluxes. Flux backgrounds have mostly been
studied in the context of type 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....
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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) and SU(5).
Candelas, Horowitz, Strominger,Witten,
'85 Braun,He, Ovrut, Pantev, 0501070 Bouchard,
Donagi, 0512149
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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 orbifolds
Dasgupta, Rajesh, Sethi, 9908088
Fernandez, Ivanov, Ugarte, Villacampa, 0804.1648
M. Becker, L-S. Tseng, Yau, 0807.0827
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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...
Type IIB M Calabi-Yau 3-fold,


M
Heterotic strings M complex and
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(No Transcript)
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The goal of this talk 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.
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Method of Construction
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• M-theory

Kaehler form of CY4
In the presence of flux the space-time metric is
no longer a direct product
coordinates of the CY4
warp factor
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The warp factor satisfies the differential
equation
Since X8 is conformal invariant (ChernSimons,
'74) this is a Poisson equation.
8-form which is quartic in curvature
A solution exists if
Euler ch. of CY4
number of M2 branes
The data specifying this background are the
choice of CY4 and the primitive flux. As long as
the total charge vanishes a solution exists.
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Additional restrictions 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
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the CY4 admits a K3 fibration. The heterotic
string 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 torsional geometries...
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Moreover, we will assume that the CY4 has a
non-singular orientifold locus so that the
elliptic fibration is locally constant.
We are left with type IIB with constant
coupling...
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The orientifold locus
Type IIB on an elliptic CY3 with constant
coupling
Symmetry
base coordinates
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The metric for an elliptic CY space
In the semi-flat approximation (stringy cosmic
string)
Away from the singular fibers there is a
U(1)xU(1) isometry...
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Type IIB background
warp factor
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Orientifold action
Quotient of type IIB on CY3 by
Decompose the forms
The forms which are invariant under Z2 are
where
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Decompose
integral harmonic 1-forms of the fiber
the 2-form
transforms as
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Type I background
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 space
gives rise to a torsional space..
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Torsional heterotic geometry
Dictionary between type I and heterotic
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The Bianchi identity
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Type IIB Bianchi identity
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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).
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Topology change
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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....
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Breaking E8
K. Becker, M. Becker, J-X. Fu, L-S. Tseng, S-T
Yau,0604137



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SUSY
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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.
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Generalized solutions
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A further generalization is possible
Here and is
the B field in the two fiber directions. The low
energy effective action is invariant under
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Conclusion
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.
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The End