on moduli and e ective theory · killing spinors: polyforms:, they contain complete information...

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


w


what is the 4D effective physics?


w

Furthermore fluxes generically generate a non-trivial warping:

with




w


with

Neglecting back-reaction: ,

4D effective theory:

(using standard

CY tools) (fluxless) CY spectrum

flux induced potential





w

with

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



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: ,


SUSY conditions Graña, Minasian, Petrini & Tomasiello `05

,,

Fluxes and SUSY

w Killing spinors:

Polyforms: ,


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)



Hitchin `02

e.g.

complex (IIB)

symplectic (IIA)

(F-flatness)



Hitchin `02

Induced polyform decomposition Gualtieri `04

6!

n=0

!nT !M =3!

k="3

Uk

(F-flatness)



Hitchin `02


6!

n=0

!nT !M =3!

k="3

Uk

complex case

e.g.

(F-flatness)



Hitchin `02;


6!

n=0

!nT !M =3!

k="3

Uk

Integrability GC structure

with

(F-flatness)



Hitchin `02;


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


The full closed string information is stored in

,

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

Koerber & L.M.`07


`half’ of NS degrees of freedom


,


Koerber & L.M.`07



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


,


Koerber & L.M.`07




RR degrees of freedom


,


Koerber & L.M.`07




RR degrees of freedom

The and moduli are associated to twisted cohomology classes of:

,


,


Koerber & L.M.`07

Moduli and 4D fields


moduli space ofHitchin `02;



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



assuming( )for minimal SUSY






,

-moduli RR axionic shift





,

-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


D-flatness condition



Expand: ,

bosonic components of linear multiplets dual to



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


complex structure moduli, lif ted up to conformal compensator




Chiral fields:





removed axion-dilaton

Chiral fields:





removed axion-dilaton

Chiral fields:

Dual linear multiplets:




Kähler potential

Kähler potential

Going to the Einstein frame, one gets the Kähler potential

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?


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

Benmachiche and Grimm `06

Kähler potential


does not seem topological! However

topologically well defined &in agreement with 4D

interpretation

Lindstrøm & Rocek; Ferrara, Girardello,

Kugo & Van Proeyen `83


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)


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



These equations can be integrated



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


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




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



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


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




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

redefining unwarped Kähler potential Grimm & Louis`04



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


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


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

on moduli and e ective theory · killing spinors: polyforms:, they contain complete information...

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