Generalized Geometry and Flux Compactifications (part...
Transcript of Generalized Geometry and Flux Compactifications (part...
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Luca Martucci (LMU München)
Generalized Geometry andFlux Compactifications (part II)
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4D viewpoint
In the first part, the pure spinors and have been used to characterize (on-shell) type II supersymmetric vacua and their (generalized) calibration structures
![Page 3: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/3.jpg)
4D viewpoint
In the first part, the pure spinors and have been used to characterize (on-shell) type II supersymmetric vacua and their (generalized) calibration structures
effective 4D description? Natural question:
![Page 4: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/4.jpg)
4D viewpoint
In the first part, the pure spinors and have been used to characterize (on-shell) type II supersymmetric vacua and their (generalized) calibration structures
effective 4D description? Natural question:
Answer difficult:
• No clear way to disentangle massive and light/massless modes (warping)
• Ignorance about moduli
![Page 5: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/5.jpg)
Alternative strategy
![Page 6: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/6.jpg)
Alternative strategy
Try to identify first 4D (supersymmetric) structures
of the untruncated theory
and,
![Page 7: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/7.jpg)
Alternative strategy
Try to identify first 4D (supersymmetric) structures
of the untruncated theory
and,Does it make
sense?
![Page 8: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/8.jpg)
Alternative strategy
Truncate the theory at a second step, hopefully in a consistent way
Try to identify first 4D (supersymmetric) structures
of the untruncated theory
and,Does it make
sense?
![Page 9: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/9.jpg)
Alternative strategy
Truncate the theory at a second step, hopefully in a consistent way
Obtain effective potentials: , &
Try to identify first 4D (supersymmetric) structures
of the untruncated theory
and,Does it make
sense?
![Page 10: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/10.jpg)
N=2, N=1 (or N=0)?
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N=2, N=1 (or N=0)? If : underlying (gauged) N=2 structure
Graña, Louis & Waldram `05 & `07,
Benmachiche & Grimm `06 Cassani & Bilal `07, Cassani `08
(N=1 by orientifold truncation)
![Page 12: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/12.jpg)
N=2, N=1 (or N=0)? If : underlying (gauged) N=2 structure
Graña, Louis & Waldram `05 & `07,
Benmachiche & Grimm `06 Cassani & Bilal `07, Cassani `08
(N=1 by orientifold truncation) Cassani’s and Kashani-Poor’s talks
![Page 13: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/13.jpg)
N=2, N=1 (or N=0)? If : underlying (gauged) N=2 structure
Graña, Louis & Waldram `05 & `07,
Benmachiche & Grimm `06 Cassani & Bilal `07, Cassani `08
(N=1 by orientifold truncation)
SUSY vacua with non-trivial warping:Koerber & L.M. `07
intrinsically N=1 structure
Cassani’s and Kashani-Poor’s talks
![Page 14: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/14.jpg)
N=2, N=1 (or N=0)? If : underlying (gauged) N=2 structure
Graña, Louis & Waldram `05 & `07,
Benmachiche & Grimm `06 Cassani & Bilal `07, Cassani `08
(N=1 by orientifold truncation)
SUSY vacua with non-trivial warping:Koerber & L.M. `07
intrinsically N=1 structure
Cassani’s and Kashani-Poor’s talks
for IIB warped CY
DeWolfe & Giddings, `02
Douglas, Shelton & Torroba, `07
cfr Giddings & Maharana, `02
Douglas & Torroba, `08
Shiu, Torroba, Underwood & Douglas `07
![Page 15: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/15.jpg)
N=2, N=1 (or N=0)? If : underlying (gauged) N=2 structure
Graña, Louis & Waldram `05 & `07,
Benmachiche & Grimm `06 Cassani & Bilal `07, Cassani `08
(N=1 by orientifold truncation)
SUSY vacua with non-trivial warping:Koerber & L.M. `07
intrinsically N=1 structure
Cassani’s and Kashani-Poor’s talks
for IIB warped CY
DeWolfe & Giddings, `02
Douglas, Shelton & Torroba, `07
cfr Giddings & Maharana, `02
Douglas & Torroba, `08
Shiu, Torroba, Underwood & Douglas `07
SUSY warped vacua:Lüst, Marchesano, L.M. & Tsimpis `08
![Page 16: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/16.jpg)
N=2, N=1 (or N=0)? If : underlying (gauged) N=2 structure
Graña, Louis & Waldram `05 & `07,
Benmachiche & Grimm `06 Cassani & Bilal `07, Cassani `08
(N=1 by orientifold truncation)
SUSY vacua with non-trivial warping:Koerber & L.M. `07
intrinsically N=1 structure
Cassani’s and Kashani-Poor’s talks
for IIB warped CY
DeWolfe & Giddings, `02
Douglas, Shelton & Torroba, `07
cfr Giddings & Maharana, `02
Douglas & Torroba, `08
Shiu, Torroba, Underwood & Douglas `07
• 4D SUSY structures ??? SUSY warped vacua:Lüst, Marchesano, L.M. & Tsimpis `08
![Page 17: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/17.jpg)
N=2, N=1 (or N=0)? If : underlying (gauged) N=2 structure
Graña, Louis & Waldram `05 & `07,
Benmachiche & Grimm `06 Cassani & Bilal `07, Cassani `08
(N=1 by orientifold truncation)
SUSY vacua with non-trivial warping:Koerber & L.M. `07
intrinsically N=1 structure
Cassani’s and Kashani-Poor’s talks
for IIB warped CY
DeWolfe & Giddings, `02
Douglas, Shelton & Torroba, `07
cfr Giddings & Maharana, `02
Douglas & Torroba, `08
Shiu, Torroba, Underwood & Douglas `07
• 4D SUSY structures ??? SUSY warped vacua:Lüst, Marchesano, L.M. & Tsimpis `08 • clearer if
Camara & Graña `07
& spectrum is truncated
![Page 18: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/18.jpg)
N=2, N=1 (or N=0)? If : underlying (gauged) N=2 structure
Graña, Louis & Waldram `05 & `07,
Benmachiche & Grimm `06 Cassani & Bilal `07, Cassani `08
(N=1 by orientifold truncation)
SUSY vacua with non-trivial warping:Koerber & L.M. `07
intrinsically N=1 structure
Cassani’s and Kashani-Poor’s talks
for IIB warped CY
DeWolfe & Giddings, `02
Douglas, Shelton & Torroba, `07
cfr Giddings & Maharana, `02
Douglas & Torroba, `08
Shiu, Torroba, Underwood & Douglas `07I’ll focus on these papers
• 4D SUSY structures ??? SUSY warped vacua:Lüst, Marchesano, L.M. & Tsimpis `08 • clearer if
Camara & Graña `07
& spectrum is truncated
![Page 19: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/19.jpg)
Plan of this talk
Off-shell 4D N=1 structures in flux compactifications
4D potential, calibrations and broken SUSY
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Pure spinors and 4D chiral fields
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Off-shell pure spinors Restrict to configurations of the form
(with )
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Off-shell pure spinors Restrict to configurations of the form
(with )
Ordinary spinors:
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Off-shell pure spinors Restrict to configurations of the form
(with )
Ordinary spinors: O(6,6) pure spinors
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Off-shell pure spinors
Rescaled twisted pure spinors:
Restrict to configurations of the form
(with )
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Off-shell pure spinors
Rescaled twisted pure spinors:
Restrict to configurations of the form
(with )
contain complete information about: and
, , and
Internal spinors: and
NS sector:
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Adding RR-fields
(Twisted) RR-sector:
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Adding RR-fields
(Twisted) RR-sector:
Note that: [Hitchin,`02]
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Adding RR-fields
Combine and into:
(Twisted) RR-sector:
Note that: [Hitchin,`02]
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Adding RR-fields
Combine and into:
(Twisted) RR-sector:
Note that: [Hitchin,`02]
For fixed flux cohomology classes, the complete information about the configuration is in:
and (our 4D chiral fields)
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Kähler potential & superpotential
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(Conformal) Kähler potential
By dimensional reduction and
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(Conformal) Kähler potential
By dimensional reduction and
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(Conformal) Kähler potential
By dimensional reduction and
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(Conformal) Kähler potential
Einstein frame gauge fixing :
By dimensional reduction and
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Superpotential
From D-brane domain wall calibration [L.M. & Smyth, `05; Evslin &
L.M., `07]
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Superpotential
From D-brane domain wall calibration [L.M. & Smyth, `05; Evslin &
L.M., `07]
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Superpotential
From D-brane domain wall calibration [L.M. & Smyth, `05; Evslin &
L.M., `07]
+ holomorphy
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4D SUSY conditions
F-flatness:
D-flatness: from RR gauge symmetry
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4D SUSY conditions
F-flatness:
D-flatness: from RR gauge symmetry
They reproduce all 10D susy equations!
(for both Minkowski and AdS vacua)
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Localized sources
[L.M. `06]
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Localized sources
[L.M. `06]
All expressions are consistent with orientifolds
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Localized sources
The open string spectrum is automatically taken into account by RR BI’s:
[L.M. `06]
All expressions are consistent with orientifolds
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Localized sources
The open string spectrum is automatically taken into account by RR BI’s:
[L.M. `06]
All expressions are consistent with orientifolds
• E.g. splitting
D-branesuperpotential
[L.M. `06]
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N=1 -> N=2 In N=2 language:
~vector multiplets
~hypermultiplets
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N=1 -> N=2
Unsplit : purely N=1 description
In N=2 language:~
vector multiplets
~hypermultiplets
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N=1 -> N=2
Unsplit : purely N=1 description
In N=2 language:~
vector multiplets
~hypermultiplets
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N=1 -> N=2
Unsplit : purely N=1 description
In N=2 language:~
vector multiplets
~hypermultiplets
![Page 48: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/48.jpg)
N=1 -> N=2
Unsplit : purely N=1 description
Special Kähler structure [Hitchin, `02]
[Grana, Waldram & Louis, `05-`06; Benmachiche & Grimm `06;]
UnderlyingN=2 structure
In N=2 language:~
vector multiplets
~hypermultiplets
![Page 49: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/49.jpg)
Simple examples
![Page 50: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/50.jpg)
Example: Superpotential and Kähler potential in IIB wCY
In this subcase: ,
![Page 51: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/51.jpg)
Example: Superpotential and Kähler potential in IIB wCY
In this subcase: ,
[DeWolfe & Giddings, `02]
[Gukov, Vafa & Witten, `99]
Insensible to warp factor!
![Page 52: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/52.jpg)
Example: Superpotential and Kähler potential in IIB wCY
In this subcase: ,
[DeWolfe & Giddings, `02]
[Gukov, Vafa & Witten, `99]
Insensible to warp factor!
![Page 53: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/53.jpg)
Example: Superpotential and Kähler potential in IIB wCY
In this subcase: ,
[DeWolfe & Giddings, `02]
[Gukov, Vafa & Witten, `99]
Insensible to warp factor!
![Page 54: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/54.jpg)
Example: Superpotential and Kähler potential in IIB wCY
agrees with [Grimm & Louis, `04] forconstant warp-factor
In this subcase: ,
[DeWolfe & Giddings, `02]
[Gukov, Vafa & Witten, `99]
Insensible to warp factor!
![Page 55: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/55.jpg)
Example: Superpotential and Kähler potential in IIB wCY
agrees with [Grimm & Louis, `04] forconstant warp-factor
Generically non-trivial
In this subcase: ,
[DeWolfe & Giddings, `02]
[Gukov, Vafa & Witten, `99]
Insensible to warp factor!
![Page 56: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/56.jpg)
Example: Superpotential and Kähler potential in IIB wCY
agrees with [Grimm & Louis, `04] forconstant warp-factor
Generically non-trivial
In this subcase: ,
[DeWolfe & Giddings, `02]
[Gukov, Vafa & Witten, `99]
Insensible to warp factor!
Purely N=1 description!
see also [Douglas, Shelton & Torroba, `07]
[Douglas & Torroba, `08]
![Page 57: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/57.jpg)
Other example: SU(3)-structure in IIA
In constant warp-factor approximation, and using CY inspired truncation, in agreement with previous results, e.g. [...; Gurrieri, Louis, Micu & Waldram, `02;
Derendinger, Kounnas, Petropoulos & Zwirner `04;Villadoro & Zwirner, `05;
DeWolfe, Giriavets, Kachru & Taylor, `05; ...]
![Page 58: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/58.jpg)
Remark It is not clear if and give a full standard N=1 theory for untruncated modes
![Page 59: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/59.jpg)
Remark It is not clear if and give a full standard N=1 theory for untruncated modes
Indeed:
Redundancy of degrees of freedom in and
![Page 60: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/60.jpg)
Remark It is not clear if and give a full standard N=1 theory for untruncated modes
Checked only under 1st order SUSY equations.
Indeed:
Redundancy of degrees of freedom in and
![Page 61: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/61.jpg)
Remark It is not clear if and give a full standard N=1 theory for untruncated modes
Checked only under 1st order SUSY equations.
Kinetic terms? further analysis required[Shiu, Torroba, Underwood & Douglas, `08]
[Douglas & Torroba `08]
Indeed:
Redundancy of degrees of freedom in and
![Page 62: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/62.jpg)
Remark It is not clear if and give a full standard N=1 theory for untruncated modes
Full potential not precisely of the form
(see later)
Checked only under 1st order SUSY equations.
Kinetic terms? further analysis required[Shiu, Torroba, Underwood & Douglas, `08]
[Douglas & Torroba `08]
Indeed:
Redundancy of degrees of freedom in and
![Page 63: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/63.jpg)
Nevertheless... After consistent truncation, and are expected to
give correct effective and
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Nevertheless... After consistent truncation, and are expected to
give correct effective and
Agreement with different proposals for flux compactifications with CY-inspired truncated spectrum
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Nevertheless...
If
exactly true in IIA SU(3)-structure AdS vacua
: there are proposals of truncation giving well defined 4D N=1 and N=2 supergravities
[Kashani-Poor `07]
[Cassani `08]
[Caviezel, Koerber, Körs, Lüst, Tsimpis & Zagermann, `08]
After consistent truncation, and are expected to give correct effective and
Agreement with different proposals for flux compactifications with CY-inspired truncated spectrum
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Summarizing so far Rescaled pure spinors: ,
andchiralfields
Then
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Summarizing so far Rescaled pure spinors: ,
andchiralfields
Then
10D SUSYconditions
4D SUSY conditions from and
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Summarizing so far Rescaled pure spinors: ,
andchiralfields
Then
Well-defined generalized calibrations (BPS bounds for D-branes)
10D SUSYconditions
4D SUSY conditions from and
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Summarizing so far
Equation D-brane BPSness 4D SUGRA int.
gauge BPSness
string BPSness
DW BPSness
Graña, Minasian, Petrini
& Tomasiello `05 L.M. & Smyth`05 Koerber & L.M. `07
E.g. in N=1 compactifications (to flat ) we have
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Summarizing so far
Equation D-brane BPSness 4D SUGRA int.
gauge BPSness
string BPSness
DW BPSness
Graña, Minasian, Petrini
& Tomasiello `05 L.M. & Smyth`05 Koerber & L.M. `07
E.g. in N=1 compactifications (to flat ) we have
How to break SUSY in a controlled way?
![Page 71: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/71.jpg)
Summarizing so far
Equation D-brane BPSness 4D SUGRA int.
gauge BPSness
string BPSness
DW BPSness
Graña, Minasian, Petrini
& Tomasiello `05 L.M. & Smyth`05 Koerber & L.M. `07
E.g. in N=1 compactifications (to flat ) we have
need for better understanding of 4D potential
How to break SUSY in a controlled way?
![Page 72: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/72.jpg)
4D potential, calibrations and SUSY-breaking
[Lüst, Marchesano, L.M. & Tsimpis, `08]
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10D e.o.m. from 4D potential
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10D e.o.m. from 4D potential
Restrict to configurations of the form
(with )
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10D e.o.m. from 4D potential
Restrict to configurations of the form
(with )
The full set of 10D e.o.m. can be obtained from
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4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
![Page 77: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/77.jpg)
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
(...)
![Page 78: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/78.jpg)
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
(...)
>- 0
>- 0
>- 0
<- 0
<- 0
![Page 79: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/79.jpg)
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
(...)
>- 0
>- 0
>- 0
<- 0
<- 0
(Space-filling D-branecalibration )
~
![Page 80: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/80.jpg)
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
(...)
>- 0
>- 0
>- 0
<- 0
<- 0
(Space-filling D-branecalibration )
~
(D-string calibration)~
![Page 81: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/81.jpg)
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
(...)
>- 0
>- 0
>- 0
<- 0
<- 0
(Space-filling D-branecalibration )
~
(D-string calibration)~
(DW calibration)~
![Page 82: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/82.jpg)
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
(...)
>- 0
>- 0
>- 0
<- 0
<- 0
(Space-filling D-branecalibration )
~
(D-string calibration)~
(DW calibration)~
D-brane calibration bound
![Page 83: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/83.jpg)
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
(...)
>- 0
>- 0
>- 0
<- 0
<- 0
(Space-filling D-branecalibration )
~
(D-string calibration)~
(DW calibration)~
D-brane calibration bound
(DW calibration)
![Page 84: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/84.jpg)
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
(...)
>- 0
>- 0
>- 0
<- 0
<- 0
(Space-filling D-branecalibration )
~
(D-string calibration)~
(DW calibration)~
D-brane calibration bound
(DW calibration)
~
![Page 85: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/85.jpg)
DW SUSY-breaking (DWSB)
Equation D-brane BPSness 4D SUGRA int.
gauge BPSness
string BPSness
DW BPSness
Graña, Minasian, Petrini
& Tomasiello `05 L.M. & Smyth`05 Koerber & L.M. `07
In N=1 compactifications (to flat ) we have
![Page 86: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/86.jpg)
DW SUSY-breaking (DWSB)
We break N=1 N=0 by violating DW BPSness:
Equation D-brane BPSness 4D SUGRA int.
gauge BPSness
string BPSness
DW (non)BPSness
![Page 87: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/87.jpg)
1-parameter DWSB
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1-parameter DWSB Take generalized fibration
(Dirac structure)
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1-parameter DWSB Take generalized fibration
(Dirac structure)
Consider DWSB of the form
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1-parameter DWSB Take generalized fibration
(Dirac structure)
SUSY-breaking parameter
Consider DWSB of the form
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1-parameter DWSB Take generalized fibration
(Dirac structure)
SUSY-breaking parameter
breaking ofGCS integrability
Consider DWSB of the form
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1-parameter DWSUSY-breaking
The potential reduces to
![Page 93: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/93.jpg)
1-parameter DWSUSY-breaking
The potential reduces to
>-0
>-0
>-0
>-0
![Page 94: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/94.jpg)
1-parameter DWSUSY-breaking
The potential reduces to
>-0
>-0
>-0
>-0
gauge BPSness ( )
![Page 95: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/95.jpg)
1-parameter DWSUSY-breaking
The potential reduces to
>-0
>-0
>-0
>-0
D-string BPSness ( )
gauge BPSness ( )
![Page 96: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/96.jpg)
1-parameter DWSUSY-breaking
The potential reduces to
>-0
>-0
>-0
>-0
D-string BPSness ( )
calibrated D-branes
gauge BPSness ( )
![Page 97: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/97.jpg)
1-parameter DWSUSY-breaking
The potential reduces to
>-0
>-0
>-0
>-0
D-string BPSness ( )
calibrated D-branes
calibrated generalized foliation
( )
gauge BPSness ( )
![Page 98: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/98.jpg)
1-parameter DWSUSY-breaking
The potential reduces to
>-0
>-0
>-0
>-0
D-string BPSness ( )
calibrated D-branes
calibrated generalized foliation
( )-dependence disappears: no-scale structure
gauge BPSness ( )
![Page 99: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/99.jpg)
Simplest example: wCY[Graña & Polchinski `00]
[Giddings, Kachru & Polchinski `01]
In this case:
![Page 100: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/100.jpg)
Simplest example: wCY[Graña & Polchinski `00]
[Giddings, Kachru & Polchinski `01]
In this case: (and thus )
![Page 101: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/101.jpg)
Simplest example: wCY[Graña & Polchinski `00]
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
![Page 102: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/102.jpg)
Simplest example: wCY[Graña & Polchinski `00]
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
![Page 103: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/103.jpg)
Simplest example: wCY[Graña & Polchinski `00]
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
• DWSB:
![Page 104: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/104.jpg)
Simplest example: wCY[Graña & Polchinski `00]
• DWSB:
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
![Page 105: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/105.jpg)
Simplest example: wCY[Graña & Polchinski `00]
• DWSB:
,
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
![Page 106: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/106.jpg)
Simplest example: wCY[Graña & Polchinski `00]
• DWSB:
,
• D-string BPSness: ,
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
![Page 107: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/107.jpg)
Simplest example: wCY[Graña & Polchinski `00]
• DWSB:
,
• D-string BPSness: ,
• Gauge BPSness:,
(ISD)
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
![Page 108: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/108.jpg)
Remarks about DWSB vacua
![Page 109: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/109.jpg)
Remarks about DWSB vacua
We have constructed several examples with SU(3) and SU(2)-structure (also with )
see also [Camara & Graña, `07]
![Page 110: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/110.jpg)
Remarks about DWSB vacua
We have constructed several examples with SU(3) and SU(2)-structure (also with )
see also [Camara & Graña, `07]
Actually, generically some additional e.o.m. must be imposed
trivial or easily satisfied in our examples
![Page 111: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/111.jpg)
Remarks about DWSB vacua
We have constructed several examples with SU(3) and SU(2)-structure (also with )
see also [Camara & Graña, `07]
Interpretation in untruncated N=1 formalism less clear.
![Page 112: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/112.jpg)
Remarks about DWSB vacua
We have constructed several examples with SU(3) and SU(2)-structure (also with )
see also [Camara & Graña, `07]
Interpretation in untruncated N=1 formalism less clear.
Clearer 4D SUSY structure if either:
![Page 113: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/113.jpg)
Remarks about DWSB vacua
We have constructed several examples with SU(3) and SU(2)-structure (also with )
see also [Camara & Graña, `07]
Interpretation in untruncated N=1 formalism less clear.
Clearer 4D SUSY structure if either:
(like in [Graña, Waldram & Louis, `05]) one uses densities
![Page 114: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/114.jpg)
Remarks about DWSB vacua
We have constructed several examples with SU(3) and SU(2)-structure (also with )
see also [Camara & Graña, `07]
Interpretation in untruncated N=1 formalism less clear.
Clearer 4D SUSY structure if either:
(like in [Graña, Waldram & Louis, `05]) one uses densities
N=1 no-scale structuregravitino
mass (density)
• •
![Page 115: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/115.jpg)
Remarks about DWSB vacua
We have constructed several examples with SU(3) and SU(2)-structure (also with )
see also [Camara & Graña, `07]
Interpretation in untruncated N=1 formalism less clear.
Clearer 4D SUSY structure if either:
(like in [Graña, Waldram & Louis, `05]) one uses densities
N=1 no-scale structuregravitino
mass (density)
• •
one approximates and truncates spectrumcfr. [Camara & Graña, `07]
or
![Page 116: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/116.jpg)
Conclusion
![Page 117: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/117.jpg)
Conclusion Generic type II flux compactifications and their 4D
description are naturally formulated using generalized geometry
![Page 118: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/118.jpg)
Conclusion Generic type II flux compactifications and their 4D
description are naturally formulated using generalized geometry
SUSY
• emergent N=1 4D effective structures
• integrable GC and calibration structures
![Page 119: Generalized Geometry and Flux Compactifications (part II)hep.itp.tuwien.ac.at/~kreuzer/ESI/MCSP/Martucci.pdf · Generalized Geometry and Flux Compactifications (part II) 4D viewpoint](https://reader031.fdocuments.net/reader031/viewer/2022022517/5b08aebe7f8b9a3d018c8ab6/html5/thumbnails/119.jpg)
Conclusion Generic type II flux compactifications and their 4D
description are naturally formulated using generalized geometry
SUSY
• emergent N=1 4D effective structures
• integrable GC and calibration structures
SUSY
• less clear (warped) N=1 4D structures
• integrable GC but still calibration structures