Generalized Geometry and Flux Compactifications (part...
Transcript of Generalized Geometry and Flux Compactifications (part...
Luca Martucci (LMU München)
Generalized Geometry andFlux Compactifications (part II)
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
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:
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
Alternative strategy
Alternative strategy
Try to identify first 4D (supersymmetric) structures
of the untruncated theory
and,
Alternative strategy
Try to identify first 4D (supersymmetric) structures
of the untruncated theory
and,Does it make
sense?
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?
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?
N=2, N=1 (or N=0)?
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)
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
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
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
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
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
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
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
Plan of this talk
Off-shell 4D N=1 structures in flux compactifications
4D potential, calibrations and broken SUSY
Pure spinors and 4D chiral fields
Off-shell pure spinors Restrict to configurations of the form
(with )
Off-shell pure spinors Restrict to configurations of the form
(with )
Ordinary spinors:
Off-shell pure spinors Restrict to configurations of the form
(with )
Ordinary spinors: O(6,6) pure spinors
Off-shell pure spinors
Rescaled twisted pure spinors:
Restrict to configurations of the form
(with )
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:
Adding RR-fields
(Twisted) RR-sector:
Adding RR-fields
(Twisted) RR-sector:
Note that: [Hitchin,`02]
Adding RR-fields
Combine and into:
(Twisted) RR-sector:
Note that: [Hitchin,`02]
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)
Kähler potential & superpotential
(Conformal) Kähler potential
By dimensional reduction and
(Conformal) Kähler potential
By dimensional reduction and
(Conformal) Kähler potential
By dimensional reduction and
(Conformal) Kähler potential
Einstein frame gauge fixing :
By dimensional reduction and
Superpotential
From D-brane domain wall calibration [L.M. & Smyth, `05; Evslin &
L.M., `07]
Superpotential
From D-brane domain wall calibration [L.M. & Smyth, `05; Evslin &
L.M., `07]
Superpotential
From D-brane domain wall calibration [L.M. & Smyth, `05; Evslin &
L.M., `07]
+ holomorphy
4D SUSY conditions
F-flatness:
D-flatness: from RR gauge symmetry
4D SUSY conditions
F-flatness:
D-flatness: from RR gauge symmetry
They reproduce all 10D susy equations!
(for both Minkowski and AdS vacua)
Localized sources
[L.M. `06]
Localized sources
[L.M. `06]
All expressions are consistent with orientifolds
Localized sources
The open string spectrum is automatically taken into account by RR BI’s:
[L.M. `06]
All expressions are consistent with orientifolds
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]
N=1 -> N=2 In N=2 language:
~vector multiplets
~hypermultiplets
N=1 -> N=2
Unsplit : purely N=1 description
In N=2 language:~
vector multiplets
~hypermultiplets
N=1 -> N=2
Unsplit : purely N=1 description
In N=2 language:~
vector multiplets
~hypermultiplets
N=1 -> N=2
Unsplit : purely N=1 description
In N=2 language:~
vector multiplets
~hypermultiplets
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
Simple examples
Example: Superpotential and Kähler potential in IIB wCY
In this subcase: ,
Example: Superpotential and Kähler potential in IIB wCY
In this subcase: ,
[DeWolfe & Giddings, `02]
[Gukov, Vafa & Witten, `99]
Insensible to warp factor!
Example: Superpotential and Kähler potential in IIB wCY
In this subcase: ,
[DeWolfe & Giddings, `02]
[Gukov, Vafa & Witten, `99]
Insensible to warp factor!
Example: Superpotential and Kähler potential in IIB wCY
In this subcase: ,
[DeWolfe & Giddings, `02]
[Gukov, Vafa & Witten, `99]
Insensible to warp factor!
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!
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!
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]
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; ...]
Remark It is not clear if and give a full standard N=1 theory for untruncated modes
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
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
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
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
Nevertheless... After consistent truncation, and are expected to
give correct effective and
Nevertheless... After consistent truncation, and are expected to
give correct effective and
Agreement with different proposals for flux compactifications with CY-inspired truncated spectrum
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
Summarizing so far Rescaled pure spinors: ,
andchiralfields
Then
Summarizing so far Rescaled pure spinors: ,
andchiralfields
Then
10D SUSYconditions
4D SUSY conditions from and
Summarizing so far Rescaled pure spinors: ,
andchiralfields
Then
Well-defined generalized calibrations (BPS bounds for D-branes)
10D SUSYconditions
4D SUSY conditions from and
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
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?
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?
4D potential, calibrations and SUSY-breaking
[Lüst, Marchesano, L.M. & Tsimpis, `08]
10D e.o.m. from 4D potential
10D e.o.m. from 4D potential
Restrict to configurations of the form
(with )
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
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
(...)
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
(...)
>- 0
>- 0
>- 0
<- 0
<- 0
4D potential and calibrationsBy expressing in terms of pure spinors (see also Cassani `08),
(...)
>- 0
>- 0
>- 0
<- 0
<- 0
(Space-filling D-branecalibration )
~
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)~
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)~
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
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)
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)
~
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
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
1-parameter DWSB
1-parameter DWSB Take generalized fibration
(Dirac structure)
1-parameter DWSB Take generalized fibration
(Dirac structure)
Consider DWSB of the form
1-parameter DWSB Take generalized fibration
(Dirac structure)
SUSY-breaking parameter
Consider DWSB of the form
1-parameter DWSB Take generalized fibration
(Dirac structure)
SUSY-breaking parameter
breaking ofGCS integrability
Consider DWSB of the form
1-parameter DWSUSY-breaking
The potential reduces to
1-parameter DWSUSY-breaking
The potential reduces to
>-0
>-0
>-0
>-0
1-parameter DWSUSY-breaking
The potential reduces to
>-0
>-0
>-0
>-0
gauge BPSness ( )
1-parameter DWSUSY-breaking
The potential reduces to
>-0
>-0
>-0
>-0
D-string BPSness ( )
gauge BPSness ( )
1-parameter DWSUSY-breaking
The potential reduces to
>-0
>-0
>-0
>-0
D-string BPSness ( )
calibrated D-branes
gauge BPSness ( )
1-parameter DWSUSY-breaking
The potential reduces to
>-0
>-0
>-0
>-0
D-string BPSness ( )
calibrated D-branes
calibrated generalized foliation
( )
gauge BPSness ( )
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 ( )
Simplest example: wCY[Graña & Polchinski `00]
[Giddings, Kachru & Polchinski `01]
In this case:
Simplest example: wCY[Graña & Polchinski `00]
[Giddings, Kachru & Polchinski `01]
In this case: (and thus )
Simplest example: wCY[Graña & Polchinski `00]
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
Simplest example: wCY[Graña & Polchinski `00]
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
Simplest example: wCY[Graña & Polchinski `00]
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
• DWSB:
Simplest example: wCY[Graña & Polchinski `00]
• DWSB:
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
Simplest example: wCY[Graña & Polchinski `00]
• DWSB:
,
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
Simplest example: wCY[Graña & Polchinski `00]
• DWSB:
,
• D-string BPSness: ,
[Giddings, Kachru & Polchinski `01]
In this case:
spanned by mobile D3
(and thus )
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 )
Remarks about DWSB vacua
Remarks about DWSB vacua
We have constructed several examples with SU(3) and SU(2)-structure (also with )
see also [Camara & Graña, `07]
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
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.
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:
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
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)
• •
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
Conclusion
Conclusion Generic type II flux compactifications and their 4D
description are naturally formulated using generalized geometry
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
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