On moduli and e ff ective theory of N=1 warped compactifications - PowerPoint PPT Presentation

Luca Martucci On moduli and e ff ective theory of N=1 warped compactifications Based on: arXiv:0902.4031 15- ti European Workshop on S ts ing Ti eor y Zü rich, 7-11 Sep tf mber 2009

�������� ���� Motivation: fluxes and 4D physics In type II flux compactifications w the internal space is not CY

�������� ���� Motivation: fluxes and 4D physics In type II flux compactifications w the internal space is not CY what is the 4D effective physics?

�������� ���� Motivation: fluxes and 4D physics In type II flux compactifications w the internal space is not CY what is the 4D effective physics? Furthermore fluxes generically generate a non-trivial warping: with

�������� ���� Motivation: fluxes and 4D physics In type II flux compactifications w the internal space is not CY what is the 4D effective physics? Furthermore fluxes generically generate a non-trivial warping: with Neglecting back-reaction: , 4D effective theory: (fluxless) CY spectrum (using standard CY tools) flux induced potential

�������� ���� Motivation: fluxes and 4D physics In type II flux compactifications w the internal space is not CY what is the 4D effective physics? Furthermore fluxes generically generate a non-trivial warping: 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 NS sector: w metric dilaton 3-form ( locally)

�������� ���� Fluxes and SUSY NS sector: w metric dilaton 3-form ( locally) RR sector:

�������� ���� Fluxes and SUSY NS sector: w metric dilaton 3-form ( locally) ( ) RR sector:

�������� ���� Fluxes and SUSY NS sector: w metric dilaton 3-form ( locally) ( ) RR sector: � , with C = C k − 1 k

���� �������� Fluxes and SUSY Killing spinors: w

�������� ���� Fluxes and SUSY Killing spinors: w Polyforms: ,

�������� ���� Fluxes and SUSY Killing spinors: w Polyforms: , , IIA

�������� ���� Fluxes and SUSY Killing spinors: w Polyforms: , , IIA , IIB

�������� ���� Fluxes and SUSY Killing spinors: w Polyforms: , , IIA , IIB and are O(6,6) pure spinors!

�������� ���� Fluxes and SUSY Killing spinors: w Polyforms: , they contain complete information about NS sector and SUSY

�������� ���� Fluxes and SUSY Killing spinors: w Polyforms: , they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05 , ,

�������� ���� Fluxes and SUSY Killing spinors: w Polyforms: , they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05 , , L.M. & Smyth `05 generalized calibrations precise interpretation in terms of: Koerber & L.M.`07 F- and D- flatness

SUSY and GC geometry (F-flatness)

SUSY and GC geometry integrable Hitchin `02 generalized complex structure (F-flatness)

SUSY and GC geometry integrable Hitchin `02 generalized complex structure (F-flatness) symplectic (IIA) e.g. complex (IIB)

SUSY and GC geometry integrable Hitchin `02 generalized complex structure (F-flatness) Induced polyform decomposition Gualtieri `04 6 3 � � Λ n T ∗ U k M = n =0 k = − 3

SUSY and GC geometry integrable Hitchin `02 generalized complex structure (F-flatness) Induced polyform decomposition Gualtieri `04 6 3 � � Λ n T ∗ U k M = n =0 k = − 3

SUSY and GC geometry integrable Hitchin `02 generalized complex structure (F-flatness) Induced polyform decomposition Gualtieri `04 6 3 � � Λ n T ∗ U k M = n =0 k = − 3 e.g. complex case

SUSY and GC geometry integrable Hitchin `02; generalized complex structure (F-flatness) Induced polyform decomposition Gualtieri `04 6 3 � � Λ n T ∗ U k M = n =0 k = − 3 Integrability GC structure with

SUSY and GC geometry integrable Hitchin `02; generalized complex structure (F-flatness) Induced polyform decomposition Gualtieri `04 6 3 � � Λ n T ∗ U k M = n =0 k = − 3 Integrability GC structure with Generalized Hodge decomposition (assuming -lemma) Cavalcanti `05

Moduli, twisted cohomologies and 4D fields

Moduli and polyforms

Moduli and polyforms The full closed string information is stored in Koerber & L.M.`07 see also: Grañã, Louis & Waldram `05;`06 Benmachiche and Grimm `06 ,

Moduli and polyforms The full closed string information is stored in Koerber & L.M.`07 see also: Grañã, Louis & Waldram `05;`06 Benmachiche and Grimm `06 , `half’ of NS degrees of freedom

Moduli and polyforms The full closed string information is stored in Koerber & L.M.`07 see also: Grañã, Louis & Waldram `05;`06 Benmachiche and Grimm `06 , information encoded in `half’ of NS degrees of freedom (second `half’ of NS degrees of freedom)

Moduli and polyforms The full closed string information is stored in Koerber & L.M.`07 see also: Grañã, Louis & Waldram `05;`06 Benmachiche and Grimm `06 , information encoded in `half’ of NS degrees RR degrees of of freedom freedom (second `half’ of NS degrees of freedom)

Moduli and polyforms The full closed string information is stored in Koerber & L.M.`07 see also: Grañã, Louis & Waldram `05;`06 Benmachiche and Grimm `06 , information encoded in `half’ of NS degrees RR degrees of of freedom freedom (second `half’ of NS degrees of freedom) The and moduli are associated to twisted cohomology classes of: ,

Moduli and 4D fields

Moduli and 4D fields moduli space of Hitchin `02;

Moduli and 4D fields moduli space of Hitchin `02;

Moduli and 4D fields In principle, all -moduli can be lifted (up to rescaling) moduli space of Hitchin `02;

Moduli and 4D fields In principle, all -moduli can be lifted (up to rescaling) moduli space of Hitchin `02; ( ) assuming for minimal SUSY

Moduli and 4D fields In principle, all -moduli can be lifted (up to rescaling) moduli space of Hitchin `02; ( ) assuming for minimal SUSY , -moduli RR axionic shift

Moduli and 4D fields In principle, all -moduli can be lifted (up to rescaling) moduli space of Hitchin `02; ( ) assuming for minimal SUSY , -moduli RR axionic shift and will be 4D chiral fields of 4D superconformal theory see e.g.: Kallosh, Kofman, Linde & Van Proeyen`00 Weyl-chiral weights:

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 Expand: , bosonic components of linear multiplets dual to Linear-chiral functional dependence explicit form depends on microscopical details

Grañã & Polchinski; Example: IIB warped CY Gubser `00 Giddings, Kachru & Polchinski `01

Grañã & Polchinski; Example: IIB warped CY Gubser `00 Giddings, Kachru & Polchinski `01

Grañã & Polchinski; Example: IIB warped CY Gubser `00 Giddings, Kachru & Polchinski `01

Grañã & Polchinski; Example: IIB warped CY Gubser `00 Giddings, Kachru & Polchinski `01 complex structure moduli, lifted up to conformal compensator

Grañã & Polchinski; Example: IIB warped CY Gubser `00 Giddings, Kachru & Polchinski `01 complex structure moduli, lifted up to conformal compensator

Grañã & Polchinski; Example: IIB warped CY Gubser `00 Giddings, Kachru & Polchinski `01 complex structure moduli, lifted up to conformal compensator

Grañã & Polchinski; Example: IIB warped CY Gubser `00 Giddings, Kachru & Polchinski `01 complex structure moduli, lifted up to conformal compensator Chiral fields:

Grañã & Polchinski; Example: IIB warped CY Gubser `00 Giddings, Kachru & Polchinski `01 complex structure moduli, lifted up to conformal compensator Chiral fields: removed axion-dilaton