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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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heterotic strings and fluxes

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


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)

and SU(5).

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,

Villacampa, 0804.1648

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...

Type IIB:

M: Calabi-Yau 3-fold,


Heterotic strings:

M: complex and


torsional background

Complex manifold with hermitian metric

Hull+Witten, '85

Strominger, '86


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.



form of CY4

In the presence of flux the space-time metric is no longer

a direct product


of the CY4

warp factor


The warp factor satisfies the differential equation

Since X8 is conformal invariant (Chern+Simons, '74)

this is a Poisson equation.

8-form which

is quartic in


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.


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


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...


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...


The orientifold locus

Type IIB on an elliptic CY3 with constant coupling


base coordinates


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)



3-form flux

5-form flux

Type IIB background

warp factor


Orientifold action

Quotient of type IIB on CY3 by

Decompose the forms

The forms which are invariant under Z2 are



Under an transformation

Locally we can write use this connection to twist

the torus fiber of the CY space.


integral harmonic

1-forms of the fiber

the 2-form transforms as


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 spacegives rise to a torsional space..


Torsionalheterotic geometry

Dictionary between type I and heterotic


T-duality maps

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).


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....


anti-self dual (1,1) form

Gauge fields with components

along K3

Breaking E8

K. Becker, M. Becker, J-X. Fu,

L-S. Tseng, S-T Yau,0604137







choices of

(1,1) ASD

(2,0) SD

(0,2) SD


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.


A further generalization is possible

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.