on moduli and e ective theory · killing spinors: polyforms:, they contain complete information...
TRANSCRIPT
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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
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In type II flux compactifications the internal space is not CY
w
Motivation: fluxes and 4D physics
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In type II flux compactifications the internal space is not CY
w
Motivation: fluxes and 4D physics
what is the 4D effective physics?
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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?
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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?
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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?
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Plan of the talk
Type II (generalized complex) flux vacua
Moduli, twisted cohomologies and 4D fields
Kähler potential
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Type II (generalized complex) flux vacua
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Fluxes and SUSY
w
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Fluxes and SUSY
w NS sector:
metric
dilaton
3-form ( locally)
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Fluxes and SUSY
w
RR sector:
NS sector:
metric
dilaton
3-form ( locally)
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Fluxes and SUSY
w
RR sector:
NS sector:
metric
dilaton
3-form ( locally)
( )
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Fluxes and SUSY
w
RR sector:
NS sector:
metric
dilaton
3-form ( locally)
( )
C =!
k
Ck!1, with
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Fluxes and SUSY
w Killing spinors:
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Fluxes and SUSY
w Killing spinors:
Polyforms: ,
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Fluxes and SUSY
w Killing spinors:
Polyforms: ,
IIA ,
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Fluxes and SUSY
w Killing spinors:
Polyforms: ,
,IIB
IIA ,
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Fluxes and SUSY
w Killing spinors:
Polyforms: ,
,IIB
IIA ,
and are O(6,6) pure spinors!
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Fluxes and SUSY
w Killing spinors:
Polyforms: ,
they contain complete information about NS sector and SUSY
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Fluxes and SUSY
w Killing spinors:
Polyforms: ,
they contain complete information about NS sector and SUSY
SUSY conditions Graña, Minasian, Petrini & Tomasiello `05
,,
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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
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SUSY and GC geometry
(F-flatness)
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SUSY and GC geometryintegrable
generalized complex structure
Hitchin `02
(F-flatness)
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SUSY and GC geometryintegrable
generalized complex structure
Hitchin `02
e.g.
complex (IIB)
symplectic (IIA)
(F-flatness)
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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)
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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)
![Page 27: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/27.jpg)
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)
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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)
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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)
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Moduli, twisted cohomologies and
4D fields
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Moduli and polyforms
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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
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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
![Page 34: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/34.jpg)
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
![Page 35: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/35.jpg)
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
![Page 36: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/36.jpg)
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
![Page 37: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/37.jpg)
Moduli and 4D fields
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Moduli and 4D fields
moduli space ofHitchin `02;
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Moduli and 4D fields
moduli space ofHitchin `02;
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Moduli and 4D fields
moduli space ofHitchin `02;
In principle, all -moduli can be lifted (up to rescaling)
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Moduli and 4D fields
moduli space ofHitchin `02;
assuming( )for minimal SUSY
In principle, all -moduli can be lifted (up to rescaling)
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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
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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
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Dual picture: linear multiplets
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Dual picture: linear multiplets
D-flatness condition
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Dual picture: linear multiplets
D-flatness condition
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Dual picture: linear multiplets
D-flatness condition
Expand: ,
bosonic components of linear multiplets dual to
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Dual picture: linear multiplets
D-flatness condition
Linear-chiral functional dependence
explicit form depends onmicroscopical details
Expand: ,
bosonic components of linear multiplets dual to
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Example: IIB warped CYGrañã & Polchinski;
Gubser `00Giddings, Kachru &
Polchinski `01
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Example: IIB warped CY
Grañã & Polchinski;Gubser `00
Giddings, Kachru & Polchinski `01
![Page 51: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/51.jpg)
Example: IIB warped CY
Grañã & Polchinski;Gubser `00
Giddings, Kachru & Polchinski `01
![Page 52: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/52.jpg)
Example: IIB warped CY
complex structure moduli, lif ted up to conformal compensator
Grañã & Polchinski;Gubser `00
Giddings, Kachru & Polchinski `01
![Page 53: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/53.jpg)
Example: IIB warped CY
complex structure moduli, lif ted up to conformal compensator
Grañã & Polchinski;Gubser `00
Giddings, Kachru & Polchinski `01
![Page 54: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/54.jpg)
Example: IIB warped CY
complex structure moduli, lif ted up to conformal compensator
Grañã & Polchinski;Gubser `00
Giddings, Kachru & Polchinski `01
![Page 55: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/55.jpg)
Example: IIB warped CY
Chiral fields:
complex structure moduli, lif ted up to conformal compensator
Grañã & Polchinski;Gubser `00
Giddings, Kachru & Polchinski `01
![Page 56: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/56.jpg)
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
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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
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Kähler potential
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Kähler potential
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Kähler potential
Going to the Einstein frame, one gets the Kähler potential
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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
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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
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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?
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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?
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In general, dependence of linear multiplets on chiral multiplets cumbersome!
Example: IIB warped CY Giddings, Kachru & Polchinski `01
Grañã & Polchinski;Gubser `00
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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
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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
![Page 68: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/68.jpg)
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
![Page 69: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/69.jpg)
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
![Page 70: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/70.jpg)
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
![Page 71: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/71.jpg)
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
![Page 72: On moduli and e ective theory · Killing spinors: Polyforms:, they contain complete information about NS sector and SUSY SUSY conditions Graña, Minasian, Petrini & Tomasiello `05,](https://reader033.vdocuments.site/reader033/viewer/2022050218/5f642874d164da5704315d8d/html5/thumbnails/72.jpg)
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