Luca Martucci

Flux vacua in String Theory, generalized calibrations and

supersymmetry breaking

Workshop on complex geometry and supersymmetry, IPMU January 4-9 2009

Plan of this talk

Generalized calibrations and fluxes

Calibrations, SUSY-breaking and 4D potential

Generalized calibrations and SUSY vacua

Generalized calibrations and fluxes

Ordinary calibrations

An ordinary calibration is a differential -form such that

Harvey & Lawson `82

1) Algebraic condition:

2) Differential condition:

for any-dim submanifold

Calibrated submanifolds:

Ordinary calibrations

Main property: calibrated cycles are volume minimizing

Harvey & Lawson `82

Ordinary calibrations

Main property: calibrated cycles are volume minimizing

Harvey & Lawson `82

Ordinary calibrations

algebraic condition

Main property: calibrated cycles are volume minimizing

Harvey & Lawson `82

Ordinary calibrations

algebraic condition

differential condition

Main property: calibrated cycles are volume minimizing

Harvey & Lawson `82

Ordinary calibrations

algebraic condition

calibration condition

differential condition

Main property: calibrated cycles are volume minimizing

Harvey & Lawson `82

Supersymmetric branes and calibrations (no fluxes)

Supersymmetric branes and calibrations (no fluxes)

In absence of fluxes, a static -brane wrapping has energy

Supersymmetric branes and calibrations (no fluxes)

We expect BPS bound for supersymmetric branes

if

In absence of fluxes, a static -brane wrapping has energy

Supersymmetric branes and calibrations (no fluxes)

We expect BPS bound for supersymmetric branes

if

In absence of fluxes, a static -brane wrapping has energy

Explicitly realized by calibrations

bulk SUSY

brane SUSY

-calibration

How to include (type II) String Theory fluxes?

How to include (type II) String Theory fluxes?

Type II bulk fields

• NS sector:

• RR sector:

(locally )

(locally )

IIA IIB

, ,

How to include (type II) String Theory fluxes?

Type II bulk fields

• NS sector:

• RR sector:

(locally )

(locally )

IIA IIB

, ,

D-brane world-volume field: flux such that

(locally )

How to include (type II) String Theory fluxes?

D-brane energy density:

Type II bulk fields

• NS sector:

• RR sector:

(locally )

(locally )

IIA IIB

, ,

D-brane world-volume field: flux such that

(locally )

How to include (type II) String Theory fluxes?

D-brane energy density:

Type II bulk fields

• NS sector:

• RR sector:

(locally )

(locally )

IIA IIB

, ,

D-brane world-volume field: flux such that

(locally )

How to include (type II) String Theory fluxes?

D-brane energy density:

Type II bulk fields

• NS sector:

• RR sector:

(locally )

(locally )

IIA IIB

, ,

D-brane world-volume field: flux such that

(locally )

Generalized calibrationsKoerber `05; L.M. & Smyth `05

Generalized calibrationsKoerber `05; L.M. & Smyth `05

A generalized calibration is a background polyform

Generalized calibrationsKoerber `05; L.M. & Smyth `05

A generalized calibration is a background polyform

such that: 1)

2)

Algebraic:

Differential:

Generalized calibrationsKoerber `05; L.M. & Smyth `05

Gutowski, Papadopoulos& Townsend `99

`ordinary’ generalized calibrations

A generalized calibration is a background polyform

such that: 1)

2)

Algebraic:

Differential:

Generalized calibrationsKoerber `05; L.M. & Smyth `05

Gutowski, Papadopoulos& Townsend `99

`ordinary’ generalized calibrations

generalizedgeometry

A generalized calibration is a background polyform

such that: 1)

2)

Algebraic:

Differential:

Generalized calibrationsKoerber `05; L.M. & Smyth `05

Calibrated (or BPS) D-branes:

Gutowski, Papadopoulos& Townsend `99

`ordinary’ generalized calibrations

generalizedgeometry

A generalized calibration is a background polyform

such that: 1)

2)

Algebraic:

Differential:

The stability argument• Main property: calibrated D-branes are stable!

The stability argument• Main property: calibrated D-branes are stable!

The stability argument• Main property: calibrated D-branes are stable!

algebraic condition

The stability argument• Main property: calibrated D-branes are stable!

algebraic condition

differential condition

The stability argument• Main property: calibrated D-branes are stable!

algebraic condition

differential condition

D-brane calibration condition

Generalized calibrations and type II SUSY vacua

N=1 vacua and pure spinors

w

(or )

,plus ,

,

General 4+6 backgrounds:

Ordinary Killing spinors:

N=1 vacua and pure spinors

w

(or )

,

Ordinary Killing spinors: S0(6,6) pure spinors

N=1 vacua and pure spinors

w

(or )

,

Ordinary Killing spinors:

in IIB

in IIA

S0(6,6) pure spinors

N=1 vacua and pure spinors

w

(or )

,

Ordinary Killing spinors: S0(6,6) pure spinors

N=1 vacua and pure spinors

w

(or )

contain completeinformation about:

and

, and

and

NS sector:( excluded)

Internal spinors:

,

N=1 vacua and calibrationsL.M. & Smyth `05

SUSY conditions

w

Graña, Minasian, Petrini & Tomasiello `05 ,

N=1 vacua and calibrationsL.M. & Smyth `05

SUSY conditions

w

Graña, Minasian, Petrini & Tomasiello `05 ,

N=1 vacua and calibrationsL.M. & Smyth `05

SUSY conditions

w

Graña, Minasian, Petrini & Tomasiello `05

defines integrable generalized CY structure!

,

N=1 vacua and calibrationsL.M. & Smyth `05

SUSY conditions

They have a natural interpretation in terms of D-brane generalized calibrations!

w

Graña, Minasian, Petrini & Tomasiello `05

defines integrable generalized CY structure!

,

N=1 vacua and calibrationsL.M. & Smyth `05

is a generalized calibration for space-filling D-branes

space-filling

N=1 vacua and calibrationsL.M. & Smyth `05

is a generalized calibration for space-filling D-branes

space-filling

For SUSY (BPS) space-filling D-branes, internal satisfies:

N=1 vacua and calibrationsL.M. & Smyth `05

is a generalized calibration for space-filling D-branes

space-filling

For SUSY (BPS) space-filling D-branes, internal satisfies:

is generalized complex submanifold

w.r.t. GCS defined by

satisfies the `speciality’ condition

(F-flatness)

(D-flatness)

BPS lower bound for D-brane energy

no open-string tachyons!

N=1 vacua and calibrationsL.M. & Smyth `05

BPS lower bound for D-brane energy

no open-string tachyons!

N=1 vacua and calibrationsL.M. & Smyth `05

GaugeBPSness

For space-filling D-branes, it originates in bulk SUSY condition

N=1 vacua and calibrationsL.M. & Smyth `05

is a generalized calibration for D-strings

stringsstring

BPSness

N=1 vacua and calibrationsL.M. & Smyth `05

is a generalized calibration for D-strings

stringsstring

BPSness

is a generalized calibration for domain walls

domain wallsDWBPSness

Summarizing

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

chiral fields:

(N=2 vect. mult.)

(N=2 hypermult.)

Pre-GG Example: type IIB warped CYGraña & Polchinski `00;

Giddings, Kachru & Polchinski`01

The internal metric is CY, up to warping:

Pre-GG Example: type IIB warped CYGraña & Polchinski `00;

Giddings, Kachru & Polchinski`01

The internal metric is CY, up to warping:

Pre-GG Example: type IIB warped CYGraña & Polchinski `00;

Giddings, Kachru & Polchinski`01

In this case: ,

The internal metric is CY, up to warping:

SUSY wCY and calibrations

SUSY wCY and calibrations

Gauge BPSness:calibrated

D3 & D7 branes

SUSY wCY and calibrations

String BPSness:calibrated

D3 & D7 branes

Gauge BPSness:calibrated

D3 & D7 branes

SUSY wCY and calibrations

String BPSness:calibrated

D3 & D7 branes

DW BPSness: calibratedD5 & D7 branes

Gauge BPSness:calibrated

D3 & D7 branes

Calibrations, SUSY-breakingand 4D potential

Lüst, Marchesano, L.M. & Tsimpis`07

SUSY breaking?

Pre-GG prototypical example: IIB wCYGraña & Polchinski `00;

Giddings, Kachru & Polchinski`01

SUSY breaking?

Pre-GG prototypical example: IIB wCYGraña & Polchinski `00;

Giddings, Kachru & Polchinski`01

with 03/O7 and D3/D7

N=1 and N=0 shear the same geometry

SUSY breaking?

Pre-GG prototypical example: IIB wCYGraña & Polchinski `00;

Giddings, Kachru & Polchinski`01

with 03/O7 and D3/D7

N=1 and N=0 shear the same geometry

flux induced SUSY-breaking:

SUSY breaking?

Pre-GG prototypical example: IIB wCYGraña & Polchinski `00;

Giddings, Kachru & Polchinski`01

with 03/O7 and D3/D7

N=1 and N=0 shear the same geometry

Many `phenomenological’ models are based on these classical vacua

Kachru, Kallosh,, Linde & Trivedi `03; + MacAllister & Maldacena `03

Balasubramanian,, Berglund,, Conlon & Quevedo `05

e.g.

flux induced SUSY-breaking:

SUSY wCY and calibrations

In this case: ,

Gauge BPSness:

String BPSness:

DW (non)BPSness:

calibratedD3 & D7 branes

calibratedD3 & D7 branes

calibratedD5 & D7 branes( )

Generalized SUSY-breaking

Generalized SUSY-breaking

We are led to consider backgrounds that fulfill gauge and string BPSness, but not DW BPSness

Generalized SUSY-breaking

We are led to consider backgrounds that fulfill gauge and string BPSness, but not DW BPSness

domain wallsstringsspace-filling

, ,

BPS bounds for

Generalized SUSY-breaking

We are led to consider backgrounds that fulfill gauge and string BPSness, but not DW BPSness

DWSB backgrounds

domain wallsstringsspace-filling

, ,

BPS bounds for

Generalized SUSY-breaking

DWSB backgrounds

Equation D-brane BPSness 4D SUGRA int.

gauge BPSness

string BPSness

DW (non)BPSness

Generalized SUSY-breaking

In wCY, the DWSB has rather specific form

crucial to solve10D e.o.m.

Generalized SUSY-breaking

In wCY, the DWSB has rather specific form

crucial to solve10D e.o.m.

Can we analogously restrict the generalized DWSB?

?

1-parameter DWSB

1-parameter DWSB Take generalized fibration

(Dirac structure)

1-parameter DWSB

spanned by mobile D-branes

Take generalized fibration(Dirac structure)

1-parameter DWSB

spanned by mobile D-branes

Take generalized fibration(Dirac structure)

Consider DWSB of the form

1-parameter DWSB

spanned by mobile D-branes

Take generalized fibration(Dirac structure)

SUSY-breaking parameter

Consider DWSB of the form

1-parameter DWSB Take generalized fibration

(Dirac structure)

Consider DWSB of the form

In wCY case:

SUSY-breaking parameter

spanned by mobile D-branes

1-parameter DWSB Take generalized fibration

(Dirac structure)

Consider DWSB of the form

In wCY case: (and thus )

SUSY-breaking parameter

spanned by mobile D-branes

1-parameter DWSB Take generalized fibration

(Dirac structure)

Consider DWSB of the form

In wCY case:

spanned by mobile D3

(and thus )

SUSY-breaking parameter

spanned by mobile D-branes

1-parameter DWSB Take generalized fibration

(Dirac structure)

Consider DWSB of the form

In wCY case:

spanned by mobile D3

(and thus )

SUSY-breaking parameter

spanned by mobile D-branes

1-parameter DWSB Take generalized fibration

(Dirac structure)

Consider DWSB of the form

In wCY case:

spanned by mobile D3

(and thus )

,

SUSY-breaking parameter

spanned by mobile D-branes

1-parameter DWSB Take generalized fibration

(Dirac structure)

How to check that 10D e.o.m. are satisfied?

Which conditions must fullfill?

SUSY-breaking parameter

Consider DWSB of the form

10D e.o.m. from 4D potential General configurations of the form

,plus ,D-branes & orientifolds

,

10D e.o.m. from 4D potential General configurations of the form

,plus ,D-branes & orientifolds

,

The full set of 10D e.o.m. can be obtained from

10D e.o.m. from 4D potential General configurations of the form

We need to express the potential in terms of and

,plus ,D-branes & orientifolds

,

The full set of 10D e.o.m. can be obtained from

Potential and pure spinorsWe found, schematically (see also Cassani `08)

(...)

Potential and pure spinorsWe found, schematically (see also Cassani `08)

(...)

>- 0

>- 0

>- 0

>- 0

<- 0

<- 0

Potential and pure spinorsWe found, schematically (see also Cassani `08)

(...)

>- 0

>- 0

>- 0

>- 0

<- 0

<- 0

(gauge BPSness)2

(string BPSness)2

(DW BPSness)2

(D-brane BPS bound)

(DW BPSness)2

(string + DW BPSness)2

~

~

~

~

~

>- 0

-

>- 0

>- 0

> 0

Potential for 1-param. DWSB

>- 0

>- 0

>- 0

>- 0

calibrated D-branes

Potential for 1-param. DWSB

>- 0

>- 0

>- 0

gauge BPSness ( )

Potential for 1-param. DWSB

>- 0

>- 0

>- 0

>- 0

string BPSness ( )

Potential for 1-param. DWSB

>- 0

>- 0

>- 0

>- 0

calibrated generalized fibration

( )

Potential for 1-param. DWSB

>- 0

>- 0

>- 0

>- 0

calibrated generalized fibration

( )-dependence disappears: no-scale structure

Potential for 1-param. DWSB

>- 0

An example

Example: SU(3)-structure with D5/O5[cfr. Camara & Graña `07]

Example: SU(3)-structure with D5/O5[cfr. Camara & Graña `07]

In this case: ,

Example: SU(3)-structure with D5/O5[cfr. Camara & Graña `07]

In this case: ,

Gauge + string BPSness:

non-Kähler

ISD

Example: SU(3)-structure with D5/O5[cfr. Camara & Graña `07]

In this case: ,

Gauge + string BPSness:

non-Kähler

ISD

DWSB:

Example: SU(3)-structure with D5/O5[cfr. Camara & Graña `07]

In this case: ,

Gauge + string BPSness:

non-Kähler

ISD

DWSB:

Notice that: non-integrable complex structure if

Explicit example on twisted torus

Explicit example on twisted torus

Explicit example on twisted torus

Twisted periodicity in 3 direction:rd

Explicit example on twisted torus

Twisted periodicity in 3 direction:rd

Metric and fields

,

Explicit example on twisted torus

Twisted periodicity in 3 direction:rd

Metric and fields

,

SU(3)-structure

Explicit example on twisted torus

Twisted periodicity in 3 direction:rd

Metric and fields

,

SU(3)-structure

twist-induced DWSB DWSB

Conclusion

Conclusion

Generic type II flux compactifications are naturally described by generalized calibrations

Conclusion

Generic type II flux compactifications are naturally described by generalized calibrations

SUSY

• emergent N=1 4D effective structures

• integrable GC and calibration structures

Conclusion

Generic type II flux compactifications are naturally described by generalized calibrations

SUSY

• emergent N=1 4D effective structures

• integrable GC and calibration structures

SUSY

• less clear N=1 4D structures

• integrable GC but still calibration structures