Deformations of calibrated D-branes in flux generalized complex manifolds

Transcription

1 Deformations of calibrated D-branes in flux generalized complex manifolds hep-th/ (with Luca Martucci) Paul Koerber Max-Planck-Institut für Physik Föhringer Ring 6 D München Germany Paul Koerber, MPI p.1/22

2 Motivation Generalized complex geometry µ tailored to describe susy Å ½ Å background sugra solutions with fluxes In the same way: supersymmetric µ D-branes generalized calibrations Open string moduli µ deformations of generalized calibrations Paul Koerber, MPI p.2/22

3 Calibrations A way to find minimal volumes surface in a curved space Second-order equations µ first-order equations Analogous to self-duality solves Yang-Mills equations Or more generally BPS equations solve equations of motion Paul Koerber, MPI p.3/22

4 Calibrations Calibration form : ¼ (1) Bound: Ô ÌÔ ÌÔ (2) (bound must be such that it can be saturated) Paul Koerber, MPI p.4/22

5 Calibrations Calibration form : ¼ (1) Bound: Ô ÌÔ ÌÔ (2) (bound must be such that it can be saturated) Calibrated submanifold : Saturates bound: Ô ÌÔ ÌÔ (3) Paul Koerber, MPI p.4/22

6 Calibrations Calibration form : ¼ (1) Bound: Ô ÌÔ ÌÔ (2) (bound must be such that it can be saturated) Calibrated submanifold : Saturates bound: Ô ÌÔ ÌÔ (3) For ¾ ½ Paul Koerber, MPI p.4/22

7 ¾ Ô Calibrations Calibration form : ¼ (1) Bound: Ô ÌÔ ÌÔ (2) (bound must be such that it can be saturated) Calibrated submanifold : Saturates bound: Ô ÌÔ ÌÔ (3) For ¾ ½ Vol ¾ µ Paul Koerber, MPI p.4/22

8 ¾ Calibrations Calibration form : ¼ (1) Bound: Ô ÌÔ ÌÔ (2) (bound must be such that it can be saturated) Calibrated submanifold : Saturates bound: Ô ÌÔ ÌÔ (3) For ¾ ½ Vol ¾ µ Ô ¾µ ¾ Paul Koerber, MPI p.4/22

9 Calibrations Calibration form : ¼ (1) Bound: Ô ÌÔ ÌÔ (2) (bound must be such that it can be saturated) Calibrated submanifold : Saturates bound: Ô ÌÔ ÌÔ (3) For ¾ ½ Vol ¾ µ Ô ¾µ ¾ ½µ ¾ µ ½ ½ Ô Vol ½ µ Paul Koerber, MPI p.4/22

10 Calibrations Calibration form : ¼ (1) Bound: Ô ÌÔ ÌÔ (2) (bound must be such that it can be saturated) Calibrated submanifold : Saturates bound: Ô ÌÔ ÌÔ (3) Calibration forms from invariant spinors: e.g. Å ½ Paul Koerber, MPI p.4/22

11 Generalized calibrations Introduce bulk fields À and RR on the D-brane, where ¾«¼ such À that Paul Koerber, MPI p.5/22

12 Generalized calibrations Calibration polyform : À RR (1) Bound: Ô Ì Ô Ì Ô (2) (bound must be such that it can be saturated) Paul Koerber, MPI p.6/22

13 Generalized calibrations Calibration polyform : À RR (1) Bound: Ô Ì Ô Ì Ô (2) (bound must be such that it can be saturated) Papadopoulos and Gutowski Paul Koerber, MPI p.6/22

14 Generalized calibrations Calibration polyform : À RR (1) Bound: Ô Ì Ô Ì Ô (2) (bound must be such that it can be saturated) Generalized geometry Paul Koerber, MPI p.6/22

15 Generalized calibrations Calibration polyform : À RR (1) Bound: Ô Ì Ô Ì Ô (2) (bound must be such that it can be saturated) Calibrated D-brane µ: Saturates bound: Ô Ì Ô Ì Ô For À µ ¾ ¾ µ ½ ½ µ Paul Koerber, MPI p.6/22

16 ¾ Ô ¾ Generalized calibrations Calibration polyform : À RR (1) Bound: Ô Ì Ô Ì Ô (2) (bound must be such that it can be saturated) Calibrated D-brane µ: Saturates bound: Ô Ì Ô Ì Ô For À µ ¾ ¾ µ ½ ½ µ E ¾ ¾ µ Paul Koerber, MPI p.6/22

17 ¾ ¾µ ¾ Ô ¾ Generalized calibrations Calibration polyform : À RR (1) Bound: Ô Ì Ô Ì Ô (2) (bound must be such that it can be saturated) Calibrated D-brane µ: Saturates bound: Ô Ì Ô Ì Ô For À µ ¾ ¾ µ ½ ½ µ µ ¾ E ¾ ¾ µ Paul Koerber, MPI p.6/22

18 ¾ ¾µ ¾ Ô ¾ ½ Generalized calibrations Calibration polyform : À RR (1) Bound: Ô Ì Ô Ì Ô (2) (bound must be such that it can be saturated) Calibrated D-brane µ: Saturates bound: Ô Ì Ô Ì Ô For À µ ¾ ¾ µ ½ ½ µ µ ¾ E ¾ ¾ µ ½µ ½ µ E ½ ½ µ µ Paul Koerber, MPI p.6/22

19 Generalized calibrations Correspond to supersymmetric D-branes Calibration forms are the pure spinors Ê ½ ÁÑ ½ ¾ satisfying À Ê ½ À ÁÑ ½ ¼ À ¾ ¼ Paul Koerber, MPI p.7/22

20 Generalized calibrations Correspond to supersymmetric D-branes Calibration forms are the pure spinors Ê ½ ÁÑ ½ ¾ satisfying À Ê ½ À ÁÑ ½ ¼ À ¾ ¼ In the rest of the talk we will focus on space-filling D-branes Paul Koerber, MPI p.7/22

21 D-flatness and F-flatness conditions Saturating bound consists of two parts Ô «½, where «varying phase µ µ is generalized complex submanifold with respect to ¾ This becomes an F-flatness condition in the 4d-effective theory Paul Koerber, MPI p.8/22

22 D-flatness and F-flatness conditions Saturating bound consists of two parts Ô «½, where «varying phase µ µ is generalized complex submanifold with respect to ¾ This becomes an F-flatness condition in the 4d-effective theory ½ ¼: analogous to the special in special ÁÑ lagrangian This becomes a D-flatness condition in the 4d-effective theory Paul Koerber, MPI p.8/22

23 D-flatness and F-flatness conditions Saturating bound consists of two parts Ô «½, where «varying phase µ µ is generalized complex submanifold with respect to ¾ This becomes an F-flatness condition in the 4d-effective theory ½ ¼: analogous to the special in special ÁÑ lagrangian This becomes a D-flatness condition in the 4d-effective theory We will study the deformations of these conditions separately! Paul Koerber, MPI p.8/22

24 Some technology I Decomposition of forms Pure spinor: e.g. ¾ Ä ¾ : Null space or also -eigenspace of  ¾ Definition: forms in Í µ can be written as ½ ¾ with Ð ¾ Ä ¾ µ. They have -eigenvalue of  ¾ Paul Koerber, MPI p.9/22

25 µ PD µ Some technology II D-brane current µ : generalization of the Poincaré dual: Explicitly: µ Å Pure spinor and À µ ¼ Null space: generalized tangent bundle Ì µ Paul Koerber, MPI p.10/22

26 Ð ¾ Æ µ µ work on µ ½ µ Some technology III Generalized normal bundle: Æ µ Ì Å Ì Å µ Elements ¾ Æ µ look like Ì µ Æ µ Æ: a normal vector to µ geometric deformations Æ ¾ ½ µ µ deformations gauge field Paul Koerber, MPI p.11/22

27 Some technology IV Lie algebroid exterior derivative Pure spinor, null space Ä, natural Ç µ metric Á Paul Koerber, MPI p.12/22

28 Some technology IV Lie algebroid exterior derivative Pure spinor, null space Ä, natural Ç µ metric Á Isomorphism Ä ³ Ä: ¾ Ä Á µ Paul Koerber, MPI p.12/22

29 ½ can be viewed as element of «¾ Ä Some technology IV Lie algebroid exterior derivative Pure spinor, null space Ä, natural Ç µ metric Á Isomorphism Ä ³ Ä: ¾ Ä Á µ Paul Koerber, MPI p.12/22

30 ½ can be viewed as element of «¾ Ä À ½ µ ³ Ä «µ Some technology IV Lie algebroid exterior derivative Pure spinor, null space Ä, natural Ç µ metric Á Isomorphism Ä ³ Ä: ¾ Ä Á µ Paul Koerber, MPI p.12/22

31 ½ can be viewed as element of «¾ Ä À ½ µ ³ Ä «µ Ä : Lie algebroid exterior derivative: for Î Ð ¾ ĵ Some technology IV Lie algebroid exterior derivative Pure spinor, null space Ä, natural Ç µ metric Á Isomorphism Ä ³ Ä: ¾ Ä Á µ Ä «Î ½ Î ½ µ ½µ Î µ«î ½ Î Î µ ½µ «Î Î À Î ½ Î Î Î µ Paul Koerber, MPI p.12/22

32 Deformations of gc submanifold Generalized complex submanifold: µ ¾ Í ¼ µ Paul Koerber, MPI p.13/22

33 Ä µ Í ¾ ¼, with Ä À À Deformations of gc submanifold Generalized complex submanifold: µ ¾ Í ¼ µ Deformation ¾ Æ µ : Paul Koerber, MPI p.13/22

34 Deformations of gc submanifold Generalized complex submanifold: µ ¾ Í ¼ µ Deformation ¾ Æ µ : Í ¾ ¼, with Ä À À Ä µ Becomes ¼½ µ µ ¼ À Paul Koerber, MPI p.13/22

35 Deformations of gc submanifold Generalized complex submanifold: µ ¾ Í ¼ µ Deformation ¾ Æ µ : Í ¾ ¼, with Ä À À Ä µ Becomes ¼½ µ µ ¼ À is a section of both Ä ¼½ ¾ Æ µ and : it acts on Ä µ Ä ¾ Ì µ Å Paul Koerber, MPI p.13/22

36 Ä µ ¼½ ¼ Deformations of gc submanifold Generalized complex submanifold: µ ¾ Í ¼ µ Deformation ¾ Æ µ : Í ¾ ¼, with Ä À À Ä µ Becomes ¼½ µ µ ¼ À is a section of both Ä ¼½ ¾ Æ µ and : it acts on Ä µ Ä ¾ Ì µ Å deformation that transforms gc submanifold into gc submanifold: Paul Koerber, MPI p.13/22

37 À ½ Ä µ µ Cohomology Gauge symmetry: Æ generated by ¼ µ In fact: deformation equation Ä µ ¼½ ¼ µ enhanced gauge symmetry: and  ¾ Divide out by Ä ¼½ Deformations classified by Meaning: imaginary gauge transformation: equivalent D-branes in topological string theory Kapustin,Li Paul Koerber, MPI p.14/22

38 Deformations of D-flatness Second condition: ÁÑ ½ ¼ (depends ½ ) Paul Koerber, MPI p.15/22

39 À ÁÑ ½ µ µ ¼ Deformations of D-flatness Second condition: ÁÑ ½ ¼ (depends ½ ) Deformations that preserve this condition: Paul Koerber, MPI p.15/22

40 Deformations of D-flatness Second condition: ÁÑ ½ ¼ (depends ½ ) Deformations that preserve this condition: À ÁÑ ½ µ µ ¼ Provides gauge fixing imaginary gauge transformations Paul Koerber, MPI p.15/22

41 Â ¾ ÁÑ ½ Deformations of D-flatness Second condition: ÁÑ ½ ¼ (depends ½ ) Deformations that preserve this condition: À ÁÑ ½ µ µ ¼ Provides gauge fixing imaginary gauge transformations For calibration µ: natural metric on Æ µ : µ µ Paul Koerber, MPI p.15/22

42 µ define Ý Ä µ Â ¾ ÁÑ ½ Deformations of D-flatness Second condition: ÁÑ ½ ¼ (depends ½ ) Deformations that preserve this condition: À ÁÑ ½ µ µ ¼ Provides gauge fixing imaginary gauge transformations For calibration µ: natural metric on Æ µ : µ µ Paul Koerber, MPI p.15/22

43 Ý Ä µ ¼½ ¼ Deformations of D-flatness Deformations must keep D-flatness (+ gauge-fixing real gauge transformations): Paul Koerber, MPI p.16/22

44 Ý Ä µ ¼½ ¼ Ä µ ¼½ ¼ Deformations of D-flatness So for deformations to preserve total calibration condition: Paul Koerber, MPI p.17/22

45 Ý Ä µ ¼½ ¼ À ½ Ä µ µ Ä µ ¼½ ¼ Deformations of D-flatness So for deformations to preserve total calibration condition: The deformations are still classified by Paul Koerber, MPI p.17/22

46 Ý Ä µ ¼½ ¼ À ½ Ä µ µ Ä µ ¼½ ¼ Deformations of D-flatness So for deformations to preserve total calibration condition: The deformations are still classified by Depends only on the integrable ¾! Paul Koerber, MPI p.17/22

47 ÁÑÅ ¼ Ä µ ³ Ì Å, Ä µ Example I: deformations of SLag ½ Å, ¾, À ¼ in type IIA McLean µ ¾ Í ¼ µ µ ¼ Ä µ µ ¾ Ì Å ³ Result: À ½ µ Note: as opposed to McLean: also gauge deformations However, McLean also shows there are no obstructions Paul Koerber, MPI p.18/22

48 Example II: B-branes with fluxes ½, ¾ Å, À ¼ À ¼ ¼ in type IIB complex ¾¼ ¼¾ ¼ Ê ¼ Paul Koerber, MPI p.19/22

49 µ ¾ Ì ¼½ ¾ Æ ½¼ Example II: B-branes with fluxes ½, ¾ Å, À ¼ À ¼ ¼ in type IIB complex ¾¼ ¼¾ ¼ Ê ¼ Ä µ ¼ µ Paul Koerber, MPI p.19/22

50 µ ¾ Ì ¼½ ¾ Æ ½¼ Example II: B-branes with fluxes ½, ¾ Å, À ¼ À ¼ ¼ in type IIB complex ¾¼ ¼¾ ¼ Ê ¼ Ä µ ¼ µ ³ only as vector space Æ ½¼ ¼½ Ì Ä µ Paul Koerber, MPI p.19/22

51 ½¼ Å ³ ÓÐ Ì µ ¾ Ì ¼½ ½¼ Æ ½¼ Ì ¾ Æ ½¼ Example II: B-branes with fluxes ½, ¾ Å, À ¼ À ¼ ¼ in type IIB complex ¾¼ ¼¾ ¼ Ê ¼ Ä µ ¼ µ ³ only as vector space Æ ½¼ ¼½ Ì Ä µ Paul Koerber, MPI p.19/22

52 ½ Ä µ ³ ½ Ì ¼½ Æ ½¼ Ä µ ³ Æ µ È À ½¾ Ü Example II: B-branes with fluxes Paul Koerber, MPI p.20/22

53 ½ Ä µ ³ ½ Ì ¼½ Æ ½¼ Ä µ ³ Æ µ È À ½¾ Ü Example II: B-branes with fluxes Kapustin Marchesano,Gomis,Mateos Paul Koerber, MPI p.20/22

54 ½ Ä µ ³ ½ Ì ¼½ Æ ½¼ Ä µ ³ Æ µ È À ½¾ Ü ¼ À Æ ½¼ µ ÖÆÀ µ À ¼ ½¼ Æ µ À ¼¾ Ä µ Example II: B-branes with fluxes ½ Ä µ µ À ¼½ µ À ¼ À Æ ½¼ µ Ä µ µ Paul Koerber, MPI p.20/22

55 Example III: type-changing gcs Gcs: ½µ µ µ (type 1) ½µ ¼ at certain points µ local complex structure Susy D3-brane can only move on ½µ locus Analysis shows: deformations off the locus lifted Paul Koerber, MPI p.21/22

56 Future work Find more examples (non-ëí µ-structure case): depends also on non-trivial ËÍ µ ËÍ µ background examples Calibrated D-branes on Ë Å Instantons? Coinciding D-branes: hard problem! Paul Koerber, MPI p.22/22

57 The Future work Find more examples (non-ëí µ-structure case): depends also on non-trivial ËÍ µ ËÍ µ background examples Calibrated D-branes on Ë Å Instantons? Coinciding D-branes: hard problem! end The end T he end Paul Koerber, MPI p.22/22