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


effective 4D description? Natural question:

4D viewpoint


effective 4D description? Natural question:

Answer difficult:

• No clear way to disentangle massive and light/massless modes (warping)

• Ignorance about moduli

Alternative strategy


Try to identify first 4D (supersymmetric) structures

of the untruncated theory

and,




and,Does it make

sense?


Truncate the theory at a second step, hopefully in a consistent way



and,Does it make

sense?


Truncate the theory at a second step, hopefully in a consistent way

Obtain effective potentials: , &



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=1 by orientifold truncation) Cassani’s and Kashani-Poor’s talks





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








for IIB warped CY






SUSY warped vacua:Lüst, Marchesano, L.M. & Tsimpis `08








for IIB warped CY






• 4D SUSY structures ??? SUSY warped vacua:Lüst, Marchesano, L.M. & Tsimpis `08








for IIB warped CY






• 4D SUSY structures ??? SUSY warped vacua:Lüst, Marchesano, L.M. & Tsimpis `08 • clearer if

Camara & Graña `07

& spectrum is truncated








for IIB warped CY





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 )


(with )

Ordinary spinors:


(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:


(with )

contain complete information about: and

, , and

Internal spinors: and

NS sector:

Adding RR-fields

(Twisted) RR-sector:

Adding RR-fields


Note that: [Hitchin,`02]

Adding RR-fields

Combine and into:



Adding RR-fields

Combine and into:



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

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]

+ 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]


Localized sources

The open string spectrum is automatically taken into account by RR BI’s:

[L.M. `06]


• 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

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: ,


In this subcase: ,

[DeWolfe & Giddings, `02]

[Gukov, Vafa & Witten, `99]

Insensible to warp factor!


agrees with [Grimm & Louis, `04] forconstant warp-factor

In this subcase: ,






Generically non-trivial

In this subcase: ,






Generically non-trivial

In this subcase: ,




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


Indeed:

Redundancy of degrees of freedom in and


Checked only under 1st order SUSY equations.

Indeed:




Kinetic terms? further analysis required[Shiu, Torroba, Underwood & Douglas, `08]

[Douglas & Torroba `08]

Indeed:



Full potential not precisely of the form

(see later)


Kinetic terms? further analysis required[Shiu, Torroba, Underwood & Douglas, `08]

[Douglas & Torroba `08]

Indeed:


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


andchiralfields

Then

10D SUSYconditions

4D SUSY conditions from and


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


gauge BPSness

string BPSness

DW BPSness




How to break SUSY in a controlled way?

Summarizing so far


gauge BPSness

string BPSness

DW BPSness




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



(with )



(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),


(...)


(...)

>- 0

>- 0

>- 0

<- 0

<- 0


(...)

>- 0

>- 0

>- 0

<- 0

<- 0

(Space-filling D-branecalibration )

~


(...)

>- 0

>- 0

>- 0

<- 0

<- 0


~

(D-string calibration)~


(...)

>- 0

>- 0

>- 0

<- 0

<- 0


~


(DW calibration)~


(...)

>- 0

>- 0

>- 0

<- 0

<- 0


~


(DW calibration)~

D-brane calibration bound


(...)

>- 0

>- 0

>- 0

<- 0

<- 0


~


(DW calibration)~


(DW calibration)


(...)

>- 0

>- 0

>- 0

<- 0

<- 0


~


(DW calibration)~


(DW calibration)

~

DW SUSY-breaking (DWSB)


gauge BPSness

string BPSness

DW BPSness



In N=1 compactifications (to flat ) we have

DW SUSY-breaking (DWSB)

We break N=1 N=0 by violating DW BPSness:


gauge BPSness

string BPSness

DW (non)BPSness

1-parameter DWSB

1-parameter DWSB Take generalized fibration

(Dirac structure)


(Dirac structure)

Consider DWSB of the form


(Dirac structure)

SUSY-breaking parameter



(Dirac structure)

SUSY-breaking parameter

breaking ofGCS integrability


1-parameter DWSUSY-breaking

The potential reduces to



>-0

>-0

>-0

>-0



>-0

>-0

>-0

>-0

gauge BPSness ( )



>-0

>-0

>-0

>-0

D-string BPSness ( )

gauge BPSness ( )



>-0

>-0

>-0

>-0


calibrated D-branes

gauge BPSness ( )



>-0

>-0

>-0

>-0


calibrated D-branes

calibrated generalized foliation

( )

gauge BPSness ( )



>-0

>-0

>-0

>-0


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:



In this case: (and thus )



In this case:

spanned by mobile D3

(and thus )



In this case:


(and thus )

• DWSB:


• DWSB:


In this case:


(and thus )


• DWSB:

,


In this case:


(and thus )


• DWSB:

,

• D-string BPSness: ,


In this case:


(and thus )


• DWSB:

,

• D-string BPSness: ,

• Gauge BPSness:,

(ISD)


In this case:


(and thus )

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




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)

• •







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



SUSY

• emergent N=1 4D effective structures

• integrable GC and calibration structures



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

generalized geometry and flux compactiﬁcations (part...

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