Torsional heterotic geometries

Katrin Becker

``14th Itzykson Meeting''IPHT, Saclay, June 19, 2009

Introduction and motivationConstructing of flux backgrounds for heterotic strings

, , φ, 0MN MNP MNg H F

The background space is a torsional heterotic geometry.

Why are these geometries important?

1) Generality

2) Phenomenology: moduli stabilization, warp factors, susy breaking,...

The heterotic string is a natural setting for building realistic models of particle phenomenology. The approach takenin the past is to specify a compact Kaehler six-dimensional space together with a holomorphic gauge field. Forappropriate 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, '85Braun,He, Ovrut, Pantev, 0501070Bouchard, 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 and that moduli can also be stabilized for heterotic strings.

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 modulibut 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 backgroundin 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.

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, 0903.3769.

K. Becker, C. Bertinato, Y-C. Chung, G. Guo, Supersymmetry breaking, heterotic strings and fluxes, 0904.2932  .

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.

Heterotic low energy effective action in 10d

10 2φ 2 22

2 2 3

1 1[ 4( ) | |

2κ 2'

(tr | | tr | | ) ( ' )]4

hetS d x ge R H

F R O

M M

1

2AB AB AB

MH

2

'[ ( ) ( )]

4hetH dB Cs Cs A

The curvature is defined using the connection

H includes a correction to ( ')O

Bergshoeff, de Roo, NPB328 (1989) 439

'[tr( ) tr( )]

4hetdH R R F F

2

2

2

1( ) ( ' )

41

( ) ( ' )2

( ' )

ABM M M AB O

H O

F O

We will solve:

( ) 0fermi '[tr( ) tr( )]

4hetdH R R F F

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

( )H i J

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.

Method of construction

M-theory

dsds

34

2 1 2 1/2 2CYR

( ) ( )ds y ds y ds

(2,2) 0J G

34CYR

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

warp factorcoordinates of the CY4

Kaehler form of CY4

The CY4 is elliptically fibred so that we can lift the 3d M-theory background to 4d...

0B

B C ie

3 3 3BG F ie H

together with gauge bundles on 7-branes. Susy requires

3 3G iG 3 0J G

The G4 flux lifts to

The orientifold locusAt the orientifold locus the background is type IIB on B6 with constant coupling . . To describe B6 start with an elliptic CY3

2 3 ( ) ( )y x f u x g u

6 3: , /I y y B CY I Symmetry

2 ( 1) LFZ I '2Z

base coordinates

1/R R

B(A. Sen9702165,9605150 )

The metric for an elliptic CY space

2 21 2

24

1| ( ) |

( )i j

CY ij

d base

ds g dy dy dw y dwy

: i iI w w 2

( ) 0

(log det log ) 0i

ij

y

g

Away from the singular fibers there is a U(1)xU(1) isometry...

In the semi-flat approximation:

FluxIn type II theories the fluxes are...

, , NS NS R RH F

....and should be invariant under

2 ( 1) LFZ I Since the NS-NS and R-R 3-forms are odd under ( 1) LF

1 2

1 2

3 1 2

3 1 2

w w

w w

H H dw H dw

F F dw F dw

5-form flux

5 3 3dF F H

24 4C dV

3

2 21 2

2

1| ( ) |

( )i j

CY ijds g dy dy dw y dwy

3

2 1 2CYds dx dx ds

3-form flux 3 3G iG 3 0J G

const.B

Type IIB background

( ) 0fermi

Torsional type I background

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....

2 1 2tords dx dx ds

2 22 1

2

1| ( ) |

( )i j

ij Hds g dy dy dw y dw Ay

logI IIB

2 1H w wA B B

2 1

23

2

' Im[( ( ) ) ]( )

ww w bF F y F E d

y

Type I background

one-bein in the fiber direction

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

.const

The Bianchi identity

The Bianchi identity in SUGRA

In type II SUGRA ( ) ( 1) ( 3)

3n n nF dC H C

After two T-dualities in the fiber directions the NS-NS field vanishes as expected from a type I background

' ' '3 2 3 0F dC dF

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

2 2( ) | | 0y H

In type IIB backgrounds the warp factor equation arises from

5 3 3dF F H 5 5

3 3

24 4

F F

iG G

C dV

one obtains

Add D7/O7 sources

4~ trC R RExample: K3xT2

2 2( ) | | 0y H

2 2 2[ ( ) | | ] 1 ' tr ...y H r r

ˆ( )ˆ( )

F B Tp

N

A RS C e

A R

In general

2 2( )

0

i jij

NS NS

ds g dy dy e d A

H

2 2i jij

NS NS

ds g dy dy e d

H dA d

4trC R R 3~ [tr ]C R R dHdH

Since the anomalous couplings are not compatible with T-duality we need to solve the Bianchi identity directly onthe type I/heterotic side

The Bianchi identity in a perturbative expansion

'3

'tr[ ( ) ( )]

4dF R R

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.

2 2 1'[ ( ) | | ] 1 tr[ ] ( )

4by H r r O L

curvature 2-form of the base

K3 base

1 1 2 2

2 4 ( )A y i jhet ij w w w wds dx dx e g dy dy E E E E

K3 k i k

k k

iw k y w

w w

E dw B dy

H dE

Torsional heterotic background with K3 baseThe heterotic background is obtained from the type Ibackground by S-duality

het Ie e 2 2Ihet Ids e ds

The resulting metric is

2

When the fiber is twisted the topology of the space changes. A caricature is a 3-torus with NS-flux

2 2 2 21 2 3

1 2 3

1, 1,2,3i i

NS

ds dx dx dx

x x i

B Nx dx dx

2 2 2 21 2 3 1 2

1 1 3 3 2

3 1 2 1 2

( )

1, 2,3

1,

( )

i i

ds dx dx dx Nx dx

x x i

x x x x Nx

d dx Nx dx Ndx dx

Topology change

1 2S S

1 1 2 2

4 ( )A yhet b b w w b w w

H de H E H E

2 ( )A y

1 2

(1,1) (2,0) (0,2) (1,1)1( ) , , ,

2w w wH H iH H H H H

0N 1N 2N

The other fields are...

'[ ( ) ( )]

4hetdH tr R R

Next we want to solve...

23 3

2 4 2 4

2 4 2 4

' '[ ( ) ( )] { ( ) 4 ( )

4 4

[( | | ) ]

+2 [( | | ) ]}

K K b

A Ab

A Ab

tr R R tr r r d d A

d d e H e

d e H de

equations of motion2( ' )O

For solutions with N=2 susy in 4d

To leading order4 ( ')

i i

Ahet b w b wdH d de H H O

1 1 2 2

2 4 '( )A y i jhet ij w w w wds dx dx e g dy dy E E E E

K3 k i k

k k

iw k y w

w w

E dw B dy

H dE

1 1 2 2

4 '( )A yhet b b w w b w w

H de H E H E

2 '( )A y

4 ' 4 2

( ) '( )

'A A

A y A y

e e A

Conclusion

We have seen how to construct torsional spaces giventhe 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.

The End