solitons in matrix model and dbi action

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

1.Introduction

3

D2-brane Every dots areD0-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

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 VEVto scalars

giving VEV to field strength

DBI 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 dualitybecause 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

Dp-brane

M D0-D0bar pairs

NontrivialTachyon Condensation

with some VEV

9

What we will show in this talk

Dp-brane

M D0-D0bar pairs

Equivalent!

N D0-branes

different gauge NontrivialTachyon Condensationin a gauge choice

with some VEVwith some VEV

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

Application(2): the SupertubeST

Equivalent!

tubular D2-brane with magnetic flux B and “critical” electric flux E=1

N D0-branes located on a tubewith [(X+iY), Z]=(X+iY) / B

infinitely many D0-D0bar pairs located on a tube

y

x

z

12

Application(3): the Instanton and ADHM

N D4-braneswith instanton

N D4-branesand k D0-branes

Equivalence!ADHM ↔ Instanton

M D0-D0bar pairs

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

1. Introduction2. BPS D0-branes and Noncommutative plane

(as an example of the duality)

3. The Duality from Unstable D-brane System1. D-brane from Tachyon Condensation2. Diagonalized Tachyon Gauge

4. Application to the Supertubes5. Index Theorem, ADHM and Tachyon

(will be skipped)6. Conclusion

15

2. BPS D0-branes and Noncommutative plane

(as an example of the Duality)

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

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

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

Potential V(T) ≈ -|m| T

When the tachyon condense, T≠0, the unstable D-brane disappears

2 2

Sen

20

Why unstable D-branes?

Why unstable D-branes are important? Any D-brane can be realized as a soliton in the unstabl

e D-brane system. Sen

SUSY breaking (ex. KKLT)

Inflation (ex. D3-D7 model)

Inclusion of anti-particles is the important idea for field t

heory → 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

Index Theorem

Every points represent eigen modes

=Integral on

p-dimensional space“Geometric” picture

Number of zero modes of Dirac operator

“Analytic” picture

=

26

Exact Equivalence between two D-brane systems

Every points represent the pairs

=Dp-brane

“Geometric” picture(p-dimensional object)

Infinitely manyD0-D0bar-branes pairs

“Analytic” picture (0-dimensional=particle)

=

Not just numbers, but physical systems

ST, Asakawa-Sugimoto-ST

27

BPS Dp-brane as soliton in M D0-D0bar pairs

We found an Exact Soliton in M D0-D0bar pairswhich 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 pairsin the boundary state or boundary SFT. We take a large M limit.

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 v

olume 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

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

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

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 pairsD0-branes only

Tachyon condensation

= -

c.f. EllwoodST

35

3 different descriptions for the bound state!

Dp-brane with background gauge field A

M D0-D0bar pairs with T=uD,X

Tachyon condenseX=diagonal gauge

Equivalent!

N BPS D0-branes with

Tachyon condenseT=diagonal gauge

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

D2-brane with magnetic flux BN BPS D0-branes [tildeX,X]=i/BNoncommutative D-brane


Tachyon condenseX=diag. gauge

Tachyon condenseT=diag. gauge

Equivalent!

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

Bak-LeeBak-Ohta

This coinceides with the supertube in the matrix model!

42

What we have shownST

Equivalent!

supertube =D2-brane with magnetic flux B and “critical” electric flux E=1

N D0-branes located on a tubewith [(X+iY), Z]=(X+iY) / B

infinitely many D0-D0bar pairs located on a tube

y

x

z

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

D2-brane with background gauge field A

N BPS D0-branes with


X=diagonal gaugeT=diagonal gauge

Equivalent!

coupling to RR-fields of D0-D0bar

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 theoryADHM data(=matrices)k x k N x 2k

low energy limit



WittenDouglas

D-brane interpretation

47

We know that N D4-branes =large M D0-D0bar



M D0-D0bar pairs with


Applying the previous method, i.e. diag. T instead of X

48

We know that N D4-branes =large M D0-D0bar



Applying the previous method, i.e. diag. T instead of X

M D0-D0bar pairs with


Tachyon condenseT=diag. gauge

Equivalence!ADHM ↔ Instanton

49

Following the previous procedure:1. Solve the zero modes of the Dirac operator in the instanton bac

kground:

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

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

End of the talk

solitons in matrix model and dbi action

Documents