Solitons in Matrix model and DBI action

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. Introduction

3. 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 + = D2-brane A Bound state (D0-branes are smeared) Every dots are D0-branes

4. Equivalent (or Dual)! 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. ex. The bound state of a D2-brane and (infinitely many) D0-branes D0-branes D2-brane giving VEV to field strength giving VEV to scalars magnetic flux B DBI action Matrix model action

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. 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. 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. What we will show in this talk MD0-D0bar pairs Nontrivial Tachyon Condensation Dp-brane with some VEV

9. different gauge Equivalent! ND0-branes What we will show in this talk MD0-D0bar pairs Nontrivial Tachyon Condensation in a gauge choice Dp-brane with some VEV with some VEV

10. Equivalent! Application(1): the D2-D0 bound state ST For magnetic flux background, Infinitely many D0-D0bar pairs with tachyon condensation, But, [X,X]=0 N BPSD0-branes[X,X]=i/B =Noncommutative D-brane D2-brane with magnetic flux B (Commutative world volume)

11. Equivalent! Application(2): the Supertube ST infinitely manyD0-D0bar pairs located on a tube y z x tubular D2-brane with magnetic flux B and “critical” electric flux E=1 ND0-branes located on a tube with [(X+iY), Z]=(X+iY) / B

12. N D4-branes and k D0-branes Equivalence! ADHM ↔ Instanton Application(3): the Instanton and ADHM MD0-D0bar pairs N D4-branes with instanton

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

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

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

17. a D2-brane with magnetic flux B 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 via Seiberg-Witten map (Conversely, always Noncommutative from D0-branes)

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

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 2 2 Potential V(T) ≈ -|m| T When the tachyon condense, T≠0, the unstable D-brane disappears Sen

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. 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. 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. 3.1 D-brane from Tachyon Condensation

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

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

27. BPS Dp-brane as soliton in M D0-D0bar pairs 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. We found an Exact Soliton in M D0-D0bar pairs which represents BPS Dp-brane (without flux): ST equivalent! = Every dots are D0-D0bar pairs A Dp-brane This is an analogue of the decent relation found by Sen (and generalized by Witten)

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. = 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. 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. D2-D0 bound state as an example Thus, we can construct the D2-brane (i.e. p=2) with the background magnetic fields B: where 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)

32. 3.2 Diagonalized Tachyon Gauge

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. Annihilation of D0-D0 pairs only D0-branes Consider the “Hamiltonian” . Each eigen state of H corresponds to a D0-D0bar pair except zero-modes. Assuming the “Hamiltonian” H has a gap above the ground state, H=0 . 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 = - Tachyon condensation Every dots are D0-D0bar pairs ST D0-branes only c.f. Ellwood

35. Tachyon condense T=diagonal gauge Equivalent! N BPSD0-branes with 3 different descriptions for the bound state! MD0-D0bar pairs with T=uD,X Tachyon condense X=diagonal gauge Dp-brane with background gauge field A

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)=<k|X|k> Tachyon induce the NC!

37. D2-D0 bound state and Tachyon MD0-D0bar pairs with T=uD,X Tachyon condense T=diag. gauge Tachyon condense X=diag. gauge Equivalent! N BPSD0-branes[tildeX,X]=i/B Noncommutative D-brane D2-brane with magnetic flux B

38. 4. Application to the Supertubes

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

40. D0-anti-D0-brane picture

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

42. Equivalent! What we have shown ST infinitely manyD0-D0bar pairs located on a tube y z x supertube =D2-brane with magnetic flux B and “critical” electric flux E=1 ND0-branes located on a tube with [(X+iY), Z]=(X+iY) / B

43. 5. Index Theorem, ADHM and Tachyon

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. 3 different descriptions for a D-brane system implies the Index Theorem via D0-brane charge MD0-D0bar pairs with T=uD,X coupling to RR-fields of D0-D0bar T=diagonal gauge X=diagonal gauge Equivalent! N BPSD0-branes with D2-brane with background gauge field A

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

47. We know that N D4-branes =large M D0-D0bar MD0-D0bar pairs with Tachyon condense X=diag. gauge N D4-branes and k D0-branes N D4-branes with instanton Applying the previous method, i.e. diag. T instead of X

48. We know that N D4-branes =large M D0-D0bar MD0-D0bar pairs with Tachyon condense X=diag. gauge Tachyon condense T=diag. gauge N D4-branes and k D0-branes N D4-branes with instanton Equivalence! ADHM ↔ Instanton Applying the previous method, i.e. diag. T instead of X

49. ADHM construction of Instanton via Tachyon Following the previous procedure: • Solve the zero modes of the Dirac operator in the instanton background: • In this case, however, there are non-normalizable zero modes of the Laplacian , which corresponds to the surviving N D4-branes: • 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. c.f. Nahm Corrigan-Goddard

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,,,,