Smeared versus localised sources in flux compactifications

J. Blåbäck, U. H. Danielsson, D. Junghans, T. Van Riet, T. Wrase and M. Zagermann

arXiv:1009.1877

Overview

1. Introduction2. BPS solutions with Ricci-flat internal space3. BPS solutions with negatively curved twisted tori4. Non-BPS solutions5. Discussion

2/11Smeared versus localised sources in flux compactifications

Consider classical vacuum solutions to type II supergravitywith D -branes ( ) or O -planes ( ):

→ delta functions will show up in e.o.m. (Einstein, dilaton, RR fields)

Introduction: What is the issue?

3/11Smeared versus localised sources in flux compactifications

Consider classical vacuum solutions to type II supergravitywith D -branes ( ) or O -planes ( ):

→ delta functions will show up in e.o.m. (Einstein, dilaton, RR fields)

Popular idea: simplify computations by assuming

But: D-branes/O-planes are localised objects(defined by boundary conditions/involutions)!

Introduction: What is the issue?

3/11Smeared versus localised sources in flux compactifications

"smearing"

D-brane (loc.)

transv. space

… …D-brane(smeared)

transv. space

Consider classical vacuum solutions to type II supergravitywith D -branes ( ) or O -planes ( ):

→ delta functions will show up in e.o.m. (Einstein, dilaton, RR fields)

Popular idea: simplify computations by assuming

But: D-branes/O-planes are localised objects(defined by boundary conditions/involutions)!

→ Smearing justified?

Introduction: What is the issue?

3/11Smeared versus localised sources in flux compactifications

"smearing"

D-brane (loc.)

transv. space

… …D-brane(smeared)

transv. space

Introduction: Why is that important?

• string theory compactifications with D-branes or O-planes have interesting properties for phenomenology

• most solutions only known in the smeared limit

4/11Smeared versus localised sources in flux compactifications

Introduction: Why is that important?

• string theory compactifications with D-branes or O-planes have interesting properties for phenomenology

• most solutions only known in the smeared limit→ need to understand whether having a smeared solution implies having a

localised solution!

4/11Smeared versus localised sources in flux compactifications

Introduction: Why is that important?

• string theory compactifications with D-branes or O-planes have interesting properties for phenomenology

• most solutions only known in the smeared limit→ need to understand whether having a smeared solution implies having a

localised solution!

• compactifications with positively curved spacetime are important for cosmology

• spaces with negative internal curvature can give upliftingpotential

• negative tension sources (O-planes) supporting this seem tobe problematic, if localised (contribution only on submanifold)

4/11Smeared versus localised sources in flux compactifications

[M. Douglas,R. Kallosh 2010]

Introduction: Why is that important?

• string theory compactifications with D-branes or O-planes have interesting properties for phenomenology

• most solutions only known in the smeared limit→ need to understand whether having a smeared solution implies having a

localised solution!

• compactifications with positively curved spacetime are important for cosmology

• spaces with negative internal curvature can give upliftingpotential

• negative tension sources (O-planes) supporting this seem tobe problematic, if localised (contribution only on submanifold)

→ need to understand localisation effects on internal curvature!

4/11Smeared versus localised sources in flux compactifications

[M. Douglas,R. Kallosh 2010]

BPS solutions (1): smeared limitAnsatz:• compactifications down to dimensions with

spacetime-filling O -plane ( ) • generalisation of well-known GKP solution, related to it by T-duality• non-zero fields: • metric:

5/11Smeared versus localised sources in flux compactifications

[S. Giddings, S. Kachru,J. Polchinski 2001]

BPS solutions (1): smeared limitAnsatz:• compactifications down to dimensions with

spacetime-filling O -plane ( ) • generalisation of well-known GKP solution, related to it by T-duality• non-zero fields: • metric:

All e.o.m. (Einstein, dilaton, field strengths, Bianchi id‘s) solved with conditions

5/11Smeared versus localised sources in flux compactifications

"BPS condition"

[S. Giddings, S. Kachru,J. Polchinski 2001]

BPS solutions (1): smeared limitAnsatz:• compactifications down to dimensions with

spacetime-filling O -plane ( ) • generalisation of well-known GKP solution, related to it by T-duality• non-zero fields: • metric:

All e.o.m. (Einstein, dilaton, field strengths, Bianchi id‘s) solved with conditions

→ -dimensional Minkowski solutions with Ricci-flat internal space

5/11Smeared versus localised sources in flux compactifications

"BPS condition"

[S. Giddings, S. Kachru,J. Polchinski 2001]

Ansatz:• non-zero fields:

• warped metric:

BPS solutions (1): localisation

6/11Smeared versus localised sources in flux compactifications

Ansatz:• non-zero fields: new

• warped metric:

BPS solutions (1): localisation

6/11Smeared versus localised sources in flux compactifications

may vary over internal space

Ansatz:• non-zero fields: new

• warped metric:

All e.o.m. remain solved with conditions

BPS solutions (1): localisation

6/11Smeared versus localised sources in flux compactifications

may vary over internal space

BPS condition

Ansatz:• non-zero fields: new

• warped metric:

All e.o.m. remain solved with conditions

→ solutions can be localised, if we add warping, a varying dilaton and

BPS solutions (1): localisation

6/11Smeared versus localised sources in flux compactifications

may vary over internal space

BPS condition

BPS solutions (2): smeared limitAnsatz:• compactifications down to dimensions with O -plane

filling spacetime and non-closed internal direction• generalisation of , related to GKP by T-dualities• non-zero fields: • metric:

7/11Smeared versus localised sources in flux compactifications

[S. Kachru, M. Schulz,P. Tripathy, S. Trivedi 2003]

BPS solutions (2): smeared limitAnsatz:• compactifications down to dimensions with O -plane

filling spacetime and non-closed internal direction• generalisation of , related to GKP by T-dualities• non-zero fields: • metric:

not closed! Hence find:

7/11Smeared versus localised sources in flux compactifications

[S. Kachru, M. Schulz,P. Tripathy, S. Trivedi 2003]

BPS solutions (2): smeared limitAnsatz:• compactifications down to dimensions with O -plane

filling spacetime and non-closed internal direction• generalisation of , related to GKP by T-dualities• non-zero fields: • metric: All e.o.m. solved with conditions

not closed! Hence find:

7/11Smeared versus localised sources in flux compactifications

[S. Kachru, M. Schulz,P. Tripathy, S. Trivedi 2003]

BPS solutions (2): smeared limitAnsatz:• compactifications down to dimensions with O -plane

filling spacetime and non-closed internal direction• generalisation of , related to GKP by T-dualities• non-zero fields: • metric: All e.o.m. solved with conditions

not closed! Hence find:

→ -dimensional Minkowski solutions with negatively curved twisted tori

7/11Smeared versus localised sources in flux compactifications

[S. Kachru, M. Schulz,P. Tripathy, S. Trivedi 2003]

Ansatz:• non-zero fields:• metric:

• generalisation of localisation discussed in

BPS solutions (2): localisation

8/11Smeared versus localised sources in flux compactifications

[M. Schulz 2004; M. Graña, R. Minasian,M. Petrini, A. Tomasiello 2007]

BPS solutions (2): localisationAnsatz:• non-zero fields:• metric:

• generalisation of localisation discussed in

8/11Smeared versus localised sources in flux compactifications

may vary over internal space

allow new term

[M. Schulz 2004; M. Graña, R. Minasian,M. Petrini, A. Tomasiello 2007]

BPS solutions (2): localisationAnsatz:• non-zero fields:• metric:

• generalisation of localisation discussed in

All e.o.m. again remain solved with some conditions determining , and the Ricci curvature as well as the BPS condition

8/11Smeared versus localised sources in flux compactifications

may vary over internal space

allow new term

[M. Schulz 2004; M. Graña, R. Minasian,M. Petrini, A. Tomasiello 2007]

BPS solutions (2): localisationAnsatz:• non-zero fields:• metric:

• generalisation of localisation discussed in

All e.o.m. again remain solved with some conditions determining , and the Ricci curvature as well as the BPS condition

→ solutions can be localised, if we add warping, a varying dilaton andallow a new term in

8/11Smeared versus localised sources in flux compactifications

may vary over internal space

allow new term

[M. Schulz 2004; M. Graña, R. Minasian,M. Petrini, A. Tomasiello 2007]

Now consider again the same ansatz as above, i.e.• non-zero fields: • some warped metric:

But deviate from the BPS condition:

Non-BPS solutions

9/11Smeared versus localised sources in flux compactifications

Now consider again the same ansatz as above, i.e.• non-zero fields: • some warped metric:

But deviate from the BPS condition:

Non-BPS solutions

9/11Smeared versus localised sources in flux compactifications

Now consider again the same ansatz as above, i.e.• non-zero fields: • some warped metric:

But deviate from the BPS condition:

Check a simple example ( ) to see what happens...

Go again through the e.o.m. to find:→ -brane

Non-BPS solutions

9/11Smeared versus localised sources in flux compactifications

Now consider again the same ansatz as above, i.e.• non-zero fields: • some warped metric:

But deviate from the BPS condition:

Check a simple example ( ) to see what happens...

Go again through the e.o.m. to find: BPS ( ):→ -brane → -plane

Non-BPS solutions

9/11Smeared versus localised sources in flux compactifications

Now consider again the same ansatz as above, i.e.• non-zero fields: • some warped metric:

But deviate from the BPS condition:

Check a simple example ( ) to see what happens...

Go again through the e.o.m. to find: BPS ( ):→ -brane → -plane

→ smeared solution gives AdS, but we cannot make sense out of localisation

Non-BPS solutions

9/11Smeared versus localised sources in flux compactifications

DiscussionWhen does a smeared solution imply a localised solution?• possibility of promoting smeared solutions to localised ones appears to

rely on whether solutions are BPS or not• for more complicated examples (e.g. with intersecting sources),

localisation may be more involved or even impossible

10/11Smeared versus localised sources in flux compactifications

DiscussionWhen does a smeared solution imply a localised solution?• possibility of promoting smeared solutions to localised ones appears to

rely on whether solutions are BPS or not• for more complicated examples (e.g. with intersecting sources), localisation

may be more involved or even impossible

What about localisation effects on the internal curvature?• argued that uplifting potentials by means of localized sources

are problematic• localising our solutions keeps integrated internal scalar curvature negative→ localisation effects seem to be more involved than expected before

10/11Smeared versus localised sources in flux compactifications

[M. Douglas,R. Kallosh 2010]

Thank you!

11/11Smeared versus localised sources in flux compactifications