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Smeared versus localised sources in flux compactifications

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. Introduction BPS solutions with Ricci-flat internal space BPS solutions with negatively curved twisted tori

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Smeared versus localised sources in flux compactifications

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  1. 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

  2. Overview Introduction BPS solutions with Ricci-flat internal space BPS solutions with negatively curved twisted tori Non-BPS solutions Discussion Smeared versus localised sources in flux compactifications

  3. Consider classical vacuum solutions to type II supergravity with D -branes ( ) or O -planes ( ): → delta functions will show up in e.o.m. (Einstein, dilaton, RR fields) Introduction: What is the issue? Smeared versus localised sources in flux compactifications

  4. Consider classical vacuum solutions to type II supergravity with 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? D-brane (loc.) "smearing" transv. space D-brane (smeared) … … transv. space Smeared versus localised sources in flux compactifications

  5. Consider classical vacuum solutions to type II supergravity with 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? D-brane (loc.) "smearing" transv. space D-brane (smeared) … … transv. space Smeared versus localised sources in flux compactifications

  6. 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 Smeared versus localised sources in flux compactifications

  7. 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! Smeared versus localised sources in flux compactifications

  8. 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 uplifting potential negative tension sources (O-planes) supporting this seem to be problematic, if localised (contribution only on submanifold) [M. Douglas, R. Kallosh 2010] Smeared versus localised sources in flux compactifications

  9. 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 uplifting potential negative tension sources (O-planes) supporting this seem to be problematic, if localised (contribution only on submanifold) → need to understand localisation effects on internal curvature! [M. Douglas, R. Kallosh 2010] Smeared versus localised sources in flux compactifications

  10. BPS solutions (1): smeared limit Ansatz: 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: [S. Giddings, S. Kachru, J. Polchinski 2001] Smeared versus localised sources in flux compactifications

  11. BPS solutions (1): smeared limit Ansatz: 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 [S. Giddings, S. Kachru, J. Polchinski 2001] "BPS condition" Smeared versus localised sources in flux compactifications

  12. BPS solutions (1): smeared limit Ansatz: 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 [S. Giddings, S. Kachru, J. Polchinski 2001] "BPS condition" Smeared versus localised sources in flux compactifications

  13. Ansatz: non-zero fields: warped metric: BPS solutions (1): localisation Smeared versus localised sources in flux compactifications

  14. Ansatz: non-zero fields: new warped metric: BPS solutions (1): localisation may vary over internal space Smeared versus localised sources in flux compactifications

  15. Ansatz: non-zero fields: new warped metric: All e.o.m. remain solved with conditions BPS solutions (1): localisation may vary over internal space BPS condition Smeared versus localised sources in flux compactifications

  16. 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 may vary over internal space BPS condition Smeared versus localised sources in flux compactifications

  17. BPS solutions (2): smeared limit Ansatz: 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: [S. Kachru, M. Schulz, P. Tripathy, S. Trivedi 2003] Smeared versus localised sources in flux compactifications

  18. BPS solutions (2): smeared limit Ansatz: 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: [S. Kachru, M. Schulz, P. Tripathy, S. Trivedi 2003] Smeared versus localised sources in flux compactifications

  19. BPS solutions (2): smeared limit Ansatz: 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: [S. Kachru, M. Schulz, P. Tripathy, S. Trivedi 2003] Smeared versus localised sources in flux compactifications

  20. BPS solutions (2): smeared limit Ansatz: 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 [S. Kachru, M. Schulz, P. Tripathy, S. Trivedi 2003] Smeared versus localised sources in flux compactifications

  21. Ansatz: non-zero fields: metric: generalisation of localisation discussed in BPS solutions (2): localisation [M. Schulz 2004; M. Graña, R. Minasian, M. Petrini, A. Tomasiello 2007] Smeared versus localised sources in flux compactifications

  22. BPS solutions (2): localisation Ansatz: non-zero fields: metric: generalisation of localisation discussed in may vary over internal space allow new term [M. Schulz 2004; M. Graña, R. Minasian, M. Petrini, A. Tomasiello 2007] Smeared versus localised sources in flux compactifications

  23. BPS solutions (2): localisation Ansatz: 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 may vary over internal space allow new term [M. Schulz 2004; M. Graña, R. Minasian, M. Petrini, A. Tomasiello 2007] Smeared versus localised sources in flux compactifications

  24. BPS solutions (2): localisation Ansatz: 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 and allow a new term in may vary over internal space allow new term [M. Schulz 2004; M. Graña, R. Minasian, M. Petrini, A. Tomasiello 2007] Smeared versus localised sources in flux compactifications

  25. 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 Smeared versus localised sources in flux compactifications

  26. 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 Smeared versus localised sources in flux compactifications

  27. 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 Smeared versus localised sources in flux compactifications

  28. 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 Smeared versus localised sources in flux compactifications

  29. 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 Smeared versus localised sources in flux compactifications

  30. Discussion When 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 Smeared versus localised sources in flux compactifications

  31. Discussion When 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 [M. Douglas, R. Kallosh 2010] Smeared versus localised sources in flux compactifications

  32. Thank you! Smeared versus localised sources in flux compactifications

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