1. Solitons in Matrix model and DBI action Seiji Terashima (YITP, Kyoto U.) at KEK March 14, 2007 Based on hep-th/0505184, 0701179 and hep-th/0507078, 05121297 with Koji Hashimoto (Komaba)

2. 2 1.Introduction

3. 3 D2-brane Every dots are D0-branes Bound state of D-branes The D-branes are very important objects for the investigation of the string theory, especially for the non-perturbative aspects. Interestingly, two different kinds of D-branes can form a bound state. ex. The bound state of a D2-brane and (infinitely many) D0-branes + = A Bound state (D0-branes are smeared)

4. 4 Bound state as “ Soliton ” (~ giving VEV) The bound state can be considered as a “soliton” on the D- branes or a “soliton” on the other kind of the D-branes. Equivalent (or Dual)! ex. The bound state of a D2-brane and (infinitely many) D0-branes magnetic flux B Matrix model action D0-branes D2-brane giving VEV to scalars giving VEV to field strength DBI action

5. 5 There are many examples of such bound states and dualities. D0-D4 (Instanton ↔ ADHM) D1-D3 (Monopole ↔ Nahm data) D0-F1 (Supertube) F1-D3 (BIon) Noncommutative solitons and so on

6. 6 This strange duality is very interesting and has many applications in string theory. However, It is very difficult to prove the duality because the two kinds of D-branes have completely different world volume actions, i.e. DBI and matrix model actions. (even the dimension of the space are different). Moreover, there are many kinds of such bound states of D-branes, but we could not treat them in each case. (In other words, there was no unified way to find what is the dual of a bound state of D-branes.)

7. 7 Unified picture of the duality in D-brane-anti-D-brane system In this talk, we will show that This duality can be obtained from the D-brane-anti-D-brane system in a Unified way by Tachyon Condensation! Moreover, we can prove the duality by this! → Solitons in DBI and matrix model are indeed equivalent. (if we includes all higher derivative and higher order corrections)

8. 8 What we will show in this talk Dp-brane M D0-D0bar pairs Nontrivial Tachyon Condensation with some VEV

9. 9 What we will show in this talk Dp-brane M D0-D0bar pairs Equivalent! N D0-branes different gauge Nontrivial Tachyon Condensation in a gauge choice with some VEV

10. 10 Application(1): the D2-D0 bound state Infinitely many D0-D0bar pairs with tachyon condensation, But, [X,X]=0 For magnetic flux background, ST Equivalent! D2-brane with magnetic flux B (Commutative world volume) N BPS D0-branes [X,X]=i/B =Noncommutative D-brane

11. 11 Application(2): the Supertube ST Equivalent! tubular D2-brane with magnetic flux B and “critical” electric flux E=1 N D0-branes located on a tube with [(X+iY), Z]=(X+iY) / B infinitely many D0-D0bar pairs located on a tube y x z

12. 12 Application(3): the Instanton and ADHM N D4-branes with instanton N D4-branes and k D0-branes Equivalence! ADHM ↔ Instanton M D0-D0bar pairs

13. 13 Remarks We study the flat 10D spacetime (but generically curved world volume of the D- branes) the tree level in string coupling only set α’=1 or other specific value

14. 14 Plan of the talk 1. Introduction 2. BPS D0-branes and Noncommutative plane (as an example of the duality) 3. The Duality from Unstable D-brane System 1. D-brane from Tachyon Condensation 2. Diagonalized Tachyon Gauge 4. Application to the Supertubes 5. Index Theorem, ADHM and Tachyon (will be skipped) 6. Conclusion

15. 15 2. BPS D0-branes and Noncommutative plane (as an example of the Duality)

16. 16 D2-brane with Magnetic flux and N D0-branes The coordinate of N D-branes is not a number, but (N x N) Matrix. Witten → Noncommutativity! BPS D2-brane with magnetic flux from N D0-branes where (N x N matrix becomes operator) = Every dots are D0-branes a D2-brane with magnetic flux B DeWitt-Hoppe-Nicolai BFSS, IKKT, Ishibashi Connes-Douglas-Schwartz N D0-branes action a D2-brane action

17. 17 D0-brane charge and Noncommutativity The D2-brane should have D0-branes charge because of charge conservation Magnetic flux on the D2-brane induce the D0-brane charge on it D2-brane should have magnetic flux =Gauge theory on Noncommutative Plane (Conversely, always Noncommutative from D0-branes) via Seiberg-Witten map a D2-brane with magnetic flux B

18. 18 3. The Duality from Unstable D-brane System

19. 19 Unstable D-branes D-branes are important objects in string theory. Stable D-brane system (ex. BPS D-brane) Unstable D-brane systems ex. Bosonic D-branes, Dp-brane-anti D-brane, non BPS D-brane (anti D-brane=Dbar-brane) unstable → tachyons in perturbative spectrum Potential V(T) ≈ -|m| T When the tachyon condense, T≠0, the unstable D-brane disappears 22 Sen

20. 20 Why unstable D-branes? Why unstable D-branes are important? Any D-brane can be realized as a soliton in the unstable D-brane system. Sen SUSY breaking (ex. KKLT) Inflation (ex. D3-D7 model) Inclusion of anti-particles is the important idea for field theory → D-brane-anti D-brane also may be important Nonperturbative definition of String Theory at least for c=1 Matrix Model (= 2d string theory) McGreevy-Verlinde, Klebanov-Maldacena-Seiberg, Takayanagi-ST

21. 21 Matrix model based on Unstable D-branes (K-matrix) We proposed Matrix model based on the unstable D0-branes (K- matrix theory ) Asakawa-Sugimoto-ST Infinitely many unstable D0-branes Analogue of the BFSS matrix model which was based on BPS D0-branes No definite definition yet (e.g. the precise form of the action, how to take large N limit, etc). We will not study dynamical aspects of this “theory” in this talk. However, even at classical level, this leads interesting phenomena: duality between several D-branes systems!

22. 22 Fields on D0-brane-anti D0-brane pairs Consider N D0-brane-anti D0-brane pairs where a D0-brane and an anti-D0-brane in any pair coincide. Fields (~ open string spectrum) on them are X : Coordinate of the D0-brane (and the anti-D0-brane) in spacetime, (which becomes (N x N) matrices for N pairs.) T: (complex) Tachyon which also becomes (N x N) matrix There are U(N) gauge symmetry on the D0-branes and another U(N) gauge symmetry on the anti-D0-branes. → U(N) x U(N) gauge symmetry In a large N limit, the N x N matrices, X and T, will become operators acting on a Hilbert space, H μ

23. 23 3.1 D-brane from Tachyon Condensation

24. 24 Any D-brane can be obtained from the D9-brane-anti- D9-brane pairs by the tachyon condensation. We can construct any D-brane from the D0-brane-anti-D0-brane pairs (instead of D9) by the tachyon condensation. This can be regarded as a generalization of the Atiyah-Singer index theorem.

25. 25 Index Theorem Every points represent eigen modes = Integral on p-dimensional space “Geometric” picture Number of zero modes of Dirac operator “Analytic” picture =

26. 26 Exact Equivalence between two D-brane systems Every points represent the pairs = Dp-brane “Geometric” picture (p-dimensional object) Infinitely many D0-D0bar-branes pairs “Analytic” picture (0-dimensional=particle) = Not just numbers, but physical systems ST, Asakawa-Sugimoto-ST

27. 27 BPS Dp-brane as soliton in M D0-D0bar pairs We found an Exact Soliton in M D0-D0bar pairs which represents BPS Dp-brane (without flux): This is an analogue of the decent relation found by Sen (and generalized by Witten) equivalent! = Every dots are D0-D0bar pairs A Dp-brane ST Instead of just D0-branes, we will consider M D0-D0bar pairs in the boundary state or boundary SFT. We take a large M limit.

28. 28 Remarks Tachyon is Dirac operator! Inclusion of gauge fields on the Dp-brane Here, the number of the pairs, M, is much larger than the number of D0-branes for the previous noncommutative construction, N.

29. 29 Generalization to the Curved World Volume We can also construct curved Dp-branes from infinitely many D0-D0bar pairs T= uD X=X(x) : embedding of the p-dimensional world volume in to the 10D spacetime =

30. 30 Remarks The Equivalence is given in the Boundary state formalism which is exact in all order in α’ and the Boundary states includes any information about D- branes. Thus the equivalence implies equivalences between tensions effective actions couplings to closed string D-brane charges

31. 31 D2-D0 bound state as an example But, the world volume is apparently commutative: How the Non-commutativity (or the BPS D0-brane picture) appears in this setting? Answer: Different gauge choice! (or choice of basis of Chan-Paton index) Thus, we can construct the D2-brane (i.e. p=2) with the background magnetic fields B: where

32. 32 3.2 Diagonalized Tachyon Gauge

33. 33 We have seen that the D0-brane-anti-D0-brane pairs becomes the D2-brane by the tachyon condensation. Note that we implicitly used the gauge choice such that the coordinate X is diagonal. Instead of this, we can diagonalize T (~ diagonalize the momentum p) by the gauge transformation (=change of the basis of Chan-Paton bundle). In this gauge, we will see that only the zero-modes of the tachyon T (~Dirac operator) remain after the tachyon condensation. Only D0-branes (without D0-bar)

34. 34 Annihilation of D0-D0 pairs only D0-branes Assuming the “Hamiltonian” H has a gap above the ground state, H=0. Consider the “Hamiltonian”. Each eigen state of H corresponds to a D0-D0bar pair except zero-modes. Because T^2=u H and u=infty, the D0-D0bar pairs corresponding to nonzero eigen states disappear by the tachyon condensation Denoting the ground states as |a> (a=1,,,,n), we have n D0-branes with matrix coordinate, where Every dots are D0-D0bar pairs D0-branes only Tachyon condensation = - c.f. Ellwood ST

35. 35 3 different descriptions for the bound state! Dp-brane with background gauge field A M D0-D0bar pairs with T=uD,X Tachyon condense X=diagonal gauge Equivalent! N BPS D0-branes with Tachyon condense T=diagonal gauge

36. 36 D2-D0 bound state as an example Consider a D2-brane with magnetic flux (=NC D-brane) H=D^2 : the Hamiltonian of the “electron” in the constant magnetic field → Landau problem Ground state of H =Lowest Landau Level labeled by a continuous momentum k → infinitely many D0-branes survive |k>=,,,, (tildeX)= Tachyon induce the NC!

37. 37 D2-D0 bound state and Tachyon D2-brane with magnetic flux BN BPS D0-branes [tildeX,X]=i/B Noncommutative D-brane M D0-D0bar pairs with T=uD,X Tachyon condense X=diag. gauge Tachyon condense T=diag. gauge Equivalent!

38. 38 4. Application to the Supertubes

39. 39 Circular Supertube in D2-brane picture Mateos-Townsend

40. 40 D0-anti-D0-brane picture

41. 41 D0-brane picture Bak-Lee Bak-Ohta This coinceides with the supertube in the matrix model!

42. 42 What we have shown ST Equivalent! supertube =D2-brane with magnetic flux B and “critical” electric flux E=1 N D0-branes located on a tube with [(X+iY), Z]=(X+iY) / B infinitely many D0-D0bar pairs located on a tube y x z

43. 43 5. Index Theorem, ADHM and Tachyon

44. 44 D0-brane charges D0-brane charge in D0-D0bar picture n D0-brane +m D0bar brane → net D0-brane charge = n – m = Index T(=Index D) (Because the tachyon T is n x m matrix for this case.) D0-brane charge in Dp-brane picture Chern-Simon coupling to RR-fields These two should be same. This implies the Index Theorem!

45. 45 3 different descriptions for a D-brane system implies the Index Theorem via D0-brane charge D2-brane with background gauge field A N BPS D0-branes with M D0-D0bar pairs with T=uD,X X=diagonal gauge T=diagonal gauge Equivalent! coupling to RR-fields of D0-D0bar

46. 46 Instantons and D-branes Consider the Instantons on the 4D SU(N) gauge theory 4D theory gauge fields N x N matrix A_mu(x) 1 to 1 (up to gauge transformation) 0D theory ADHM data(=matrices) k x k N x 2k low energy limit N D4-branes and k D0-branes N D4-branes with instanton Witten Douglas D-brane interpretation

47. 47 We know that N D4-branes =large M D0-D0bar N D4-branes with instanton N D4-branes and k D0-branes M D0-D0bar pairs with Tachyon condense X=diag. gauge Applying the previous method, i.e. diag. T instead of X

48. 48 We know that N D4-branes =large M D0-D0bar N D4-branes with instanton N D4-branes and k D0-branes Applying the previous method, i.e. diag. T instead of X M D0-D0bar pairs with Tachyon condense X=diag. gauge Tachyon condense T=diag. gauge Equivalence! ADHM ↔ Instanton

49. 49 Following the previous procedure: 1. Solve the zero modes of the Dirac operator in the instanton background: 2. In this case, however, there are non-normalizable zero modes of the Laplacian, which corresponds to the surviving N D4- branes: 3. This is ADHM! We derive ADHM construction of Instanton valid in all order in α’ ! This is indeed inverse ADHM construction. We can derive ADHM construction of instanton in same way from D4-D4bar branes. ADHM construction of Instanton via Tachyon c.f. Nahm Corrigan-Goddard

50. 50 Conclusion 3 different, but, equivalent descriptions The noncommutativity is induced by the tachyon condensation from the unstable D0-brane viewpoint. Supertubes in the D2-brane picture and in the D0-brane picture are obtained. ADHM is Tachyon condensation We can also consider the Fuzzy Sphere in the same way. ST Supertube with arbitrary cross-section ST NC ADHM and Monopole-Nahm Hashimoto-ST Future problems Nahm transformation (Instanton on T^4) New duality between Solitons and ADHM data like objects Including fundamental strings and NS5-branes Applications to the Black hole physics, D1-D5? Define the Matrix model precisely and,,,,

51. 51 End of the talk