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