On moduli and effective theory of N=1 warped compactifications

Luca Martucci

15-th European Workshop on String Theory Zürich, 7-11 September 2009

Based on: arXiv:0902.4031

In type II flux compactifications the internal space is not CY

w

Motivation: fluxes and 4D physics

In type II flux compactifications the internal space is not CY

w

Motivation: fluxes and 4D physics

what is the 4D effective physics?

In type II flux compactifications the internal space is not CY

w

Furthermore fluxes generically generate a non-trivial warping:

with

Motivation: fluxes and 4D physics

what is the 4D effective physics?

In type II flux compactifications the internal space is not CY

w

Furthermore fluxes generically generate a non-trivial warping:

with

Neglecting back-reaction: ,

4D effective theory:

(using standard

CY tools) (fluxless) CY spectrum

flux induced potential

Motivation: fluxes and 4D physics

what is the 4D effective physics?

Motivation: fluxes and 4D physics

Furthermore fluxes generically generate a non-trivial warping:

w

with

What can we say about 4D effective theory of fully back-reacted vacua?

In type II flux compactifications the internal space is not CY

what is the 4D effective physics?

Plan of the talk

Type II (generalized complex) flux vacua

Moduli, twisted cohomologies and 4D fields

Kähler potential

Type II (generalized complex) flux vacua

Fluxes and SUSY

w

Fluxes and SUSY

w NS sector:

metric

dilaton

3-form ( locally)

Fluxes and SUSY

w

RR sector:

NS sector:

metric

dilaton

3-form ( locally)

Fluxes and SUSY

w

RR sector:

NS sector:

metric

dilaton

3-form ( locally)

( )

Fluxes and SUSY

w

RR sector:

NS sector:

metric

dilaton

3-form ( locally)

( )

C =!

k

Ck!1, with

Fluxes and SUSY

w Killing spinors:

Fluxes and SUSY

w Killing spinors:

Polyforms: ,

Fluxes and SUSY

w Killing spinors:

Polyforms: ,

IIA ,

Fluxes and SUSY

w Killing spinors:

Polyforms: ,

,IIB

IIA ,

Fluxes and SUSY

w Killing spinors:

Polyforms: ,

,IIB

IIA ,

and are O(6,6) pure spinors!

Fluxes and SUSY

w Killing spinors:

Polyforms: ,

they contain complete information about NS sector and SUSY

Fluxes and SUSY

w Killing spinors:

Polyforms: ,

they contain complete information about NS sector and SUSY

SUSY conditions Graña, Minasian, Petrini & Tomasiello `05

,,

Fluxes and SUSY

w Killing spinors:

Polyforms: ,

they contain complete information about NS sector and SUSY

SUSY conditions Graña, Minasian, Petrini & Tomasiello `05

,,

precise interpretation in terms of: generalized calibrations L.M. & Smyth `05 F- and D- flatness Koerber & L.M.`07

SUSY and GC geometry

(F-flatness)

SUSY and GC geometryintegrable

generalized complex structure

Hitchin `02

(F-flatness)

SUSY and GC geometryintegrable

generalized complex structure

Hitchin `02

e.g.

complex (IIB)

symplectic (IIA)

(F-flatness)

SUSY and GC geometryintegrable

generalized complex structure

Hitchin `02

Induced polyform decomposition Gualtieri `04

6!

n=0

!nT !M =3!

k="3

Uk

(F-flatness)

SUSY and GC geometryintegrable

generalized complex structure

Hitchin `02

Induced polyform decomposition Gualtieri `04

6!

n=0

!nT !M =3!

k="3

Uk

(F-flatness)

SUSY and GC geometryintegrable

generalized complex structure

Hitchin `02

Induced polyform decomposition Gualtieri `04

6!

n=0

!nT !M =3!

k="3

Uk

complex case

e.g.

(F-flatness)

SUSY and GC geometryintegrable

generalized complex structure

Hitchin `02;

Induced polyform decomposition Gualtieri `04

6!

n=0

!nT !M =3!

k="3

Uk

Integrability GC structure

with

(F-flatness)

SUSY and GC geometryintegrable

generalized complex structure

Hitchin `02;

Induced polyform decomposition Gualtieri `04

6!

n=0

!nT !M =3!

k="3

Uk

Integrability GC structure

with

Generalized Hodge decomposition (assuming -lemma) Cavalcanti `05

(F-flatness)

Moduli, twisted cohomologies and

4D fields

Moduli and polyforms

Moduli and polyforms

The full closed string information is stored in

,

see also: Grañã, Louis & Waldram `05;`06Benmachiche and Grimm `06

Koerber & L.M.`07

Moduli and polyforms

`half’ of NS degrees of freedom

The full closed string information is stored in

,

see also: Grañã, Louis & Waldram `05;`06Benmachiche and Grimm `06

Koerber & L.M.`07

Moduli and polyforms

`half’ of NS degrees of freedom

information encoded in (second `half’ of NS degrees of freedom)

The full closed string information is stored in

,

see also: Grañã, Louis & Waldram `05;`06Benmachiche and Grimm `06

Koerber & L.M.`07

Moduli and polyforms

`half’ of NS degrees of freedom

information encoded in (second `half’ of NS degrees of freedom)

RR degrees of freedom

The full closed string information is stored in

,

see also: Grañã, Louis & Waldram `05;`06Benmachiche and Grimm `06

Koerber & L.M.`07

Moduli and polyforms

`half’ of NS degrees of freedom

information encoded in (second `half’ of NS degrees of freedom)

RR degrees of freedom

The and moduli are associated to twisted cohomology classes of:

,

The full closed string information is stored in

,

see also: Grañã, Louis & Waldram `05;`06Benmachiche and Grimm `06

Koerber & L.M.`07

Moduli and 4D fields

Moduli and 4D fields

moduli space ofHitchin `02;

Moduli and 4D fields

moduli space ofHitchin `02;

Moduli and 4D fields

moduli space ofHitchin `02;

In principle, all -moduli can be lifted (up to rescaling)

Moduli and 4D fields

moduli space ofHitchin `02;

assuming( )for minimal SUSY

In principle, all -moduli can be lifted (up to rescaling)

Moduli and 4D fields

moduli space ofHitchin `02;

assuming( )for minimal SUSY

In principle, all -moduli can be lifted (up to rescaling)

,

-moduli RR axionic shift

Moduli and 4D fields

moduli space ofHitchin `02;

assuming( )for minimal SUSY

In principle, all -moduli can be lifted (up to rescaling)

,

-moduli RR axionic shift

Weyl-chiral weights:

and will be 4D chiral fields of 4D superconformal theory see e.g.: Kallosh, Kofman,Linde & Van Proeyen`00

Dual picture: linear multiplets

Dual picture: linear multiplets

D-flatness condition

Dual picture: linear multiplets

D-flatness condition

Dual picture: linear multiplets

D-flatness condition

Expand: ,

bosonic components of linear multiplets dual to

Dual picture: linear multiplets

D-flatness condition

Linear-chiral functional dependence

explicit form depends onmicroscopical details

Expand: ,

bosonic components of linear multiplets dual to

Example: IIB warped CYGrañã & Polchinski;

Gubser `00Giddings, Kachru &

Polchinski `01

Example: IIB warped CY

Grañã & Polchinski;Gubser `00

Giddings, Kachru & Polchinski `01

Example: IIB warped CY

Grañã & Polchinski;Gubser `00

Giddings, Kachru & Polchinski `01

Example: IIB warped CY

complex structure moduli, lif ted up to conformal compensator

Grañã & Polchinski;Gubser `00

Giddings, Kachru & Polchinski `01

Example: IIB warped CY

complex structure moduli, lif ted up to conformal compensator

Grañã & Polchinski;Gubser `00

Giddings, Kachru & Polchinski `01

Example: IIB warped CY

complex structure moduli, lif ted up to conformal compensator

Grañã & Polchinski;Gubser `00

Giddings, Kachru & Polchinski `01

Example: IIB warped CY

Chiral fields:

complex structure moduli, lif ted up to conformal compensator

Grañã & Polchinski;Gubser `00

Giddings, Kachru & Polchinski `01

Example: IIB warped CY

removed axion-dilaton

Chiral fields:

complex structure moduli, lif ted up to conformal compensator

Grañã & Polchinski;Gubser `00

Giddings, Kachru & Polchinski `01

Example: IIB warped CY

removed axion-dilaton

Chiral fields:

Dual linear multiplets:

complex structure moduli, lif ted up to conformal compensator

Grañã & Polchinski;Gubser `00

Giddings, Kachru & Polchinski `01

Kähler potential

Kähler potential

Kähler potential

Going to the Einstein frame, one gets the Kähler potential

Kähler potential

Going to the Einstein frame, one gets the Kähler potential

for , it reduces to Kähler potential of Grañã, Louis & Waldram `05,`06

Benmachiche and Grimm `06

Kähler potential

what is its explicit form?

Going to the Einstein frame, one gets the Kähler potential

for , it reduces to Kähler potential of Grañã, Louis & Waldram `05,`06

Benmachiche and Grimm `06

Kähler potential

Going to the Einstein frame, one gets the Kähler potential

does not seem topological! However

topologically well defined &in agreement with 4D

interpretation

Lindstrøm & Rocek; Ferrara, Girardello,

Kugo & Van Proeyen `83

what is its explicit form?

Kähler potential

Going to the Einstein frame, one gets the Kähler potential

does not seem topological! However

topologically well defined &in agreement with 4D

interpretation

Lindstrøm & Rocek; Ferrara, Girardello,

Kugo & Van Proeyen `83

Freezing the -moduli, knowing one can obtain by integration

la(t + t̄)

what is its explicit form?

In general, dependence of linear multiplets on chiral multiplets cumbersome!

Example: IIB warped CY Giddings, Kachru & Polchinski `01

Grañã & Polchinski;Gubser `00

In general, dependence of linear multiplets on chiral multiplets cumbersome!

Example: IIB warped CY Giddings, Kachru & Polchinski `01

However, if ( ) universal modulus

,v ! 1 =! exp("K/3)

!Re"la ! Iab Re !b =

" exp("K/3)"Re !a

Grañã & Polchinski;Gubser `00

In general, dependence of linear multiplets on chiral multiplets cumbersome!

Example: IIB warped CY Giddings, Kachru & Polchinski `01

However, if ( ) universal modulus

,v ! 1 =! exp("K/3)

!Re"la ! Iab Re !b =

" exp("K/3)"Re !a

Grañã & Polchinski;Gubser `00

In general, dependence of linear multiplets on chiral multiplets cumbersome!

These equations can be integrated

Example: IIB warped CY Giddings, Kachru & Polchinski `01

However, if ( ) universal modulus

,v ! 1 =! exp("K/3)

!Re"la ! Iab Re !b =

" exp("K/3)"Re !a

Grañã & Polchinski;Gubser `00

In general, dependence of linear multiplets on chiral multiplets cumbersome!

These equations can be integrated

Example: IIB warped CY Giddings, Kachru & Polchinski `01

However, if ( ) universal modulus

,v ! 1 =! exp("K/3)

!Re"la ! Iab Re !b =

" exp("K/3)"Re !a

Grañã & Polchinski;Gubser `00

In general, dependence of linear multiplets on chiral multiplets cumbersome!

These equations can be integrated

if , in agreement with Frey,Torroba, Underwood & Douglas `08

Example: IIB warped CY Giddings, Kachru & Polchinski `01

However, if ( ) universal modulus

,v ! 1 =! exp("K/3)

!Re"la ! Iab Re !b =

" exp("K/3)"Re !a

Grañã & Polchinski;Gubser `00

In general, dependence of linear multiplets on chiral multiplets cumbersome!

These equations can be integrated

if , in agreement with Frey,Torroba, Underwood & Douglas `08

redefining unwarped Kähler potential Grimm & Louis`04

Example: IIB warped CY Giddings, Kachru & Polchinski `01

However, if ( ) universal modulus

,v ! 1 =! exp("K/3)

!Re"la ! Iab Re !b =

" exp("K/3)"Re !a

Grañã & Polchinski;Gubser `00

Conclusions

Under some assumptions (e.g. -lemma), the 4D spectrum has been identified with -twisted cohomologies

The Kähler potential determined only implicitly. However, 4D chiral-linear duality can help in reconstructing it.

The 4D couplings of probe D-branes (space-filling, instantons, DW’s and strings) depend only on the cohomology classes