Solitons in Matrix model and DBI action
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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)
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1.Introduction
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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)
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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
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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
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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.)
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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)
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What we will show in this talk
Dp-brane
M D0-D0bar pairs
NontrivialTachyon Condensation
with some VEV
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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
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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
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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
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Application(3): the Instanton and ADHM
N D4-braneswith instanton
N D4-branesand k D0-branes
Equivalence!ADHM ↔ Instanton
M D0-D0bar pairs
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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
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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
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2. BPS D0-branes and Noncommutative plane
(as an example of the Duality)
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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
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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
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3. The Duality from Unstable D-brane System
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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
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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
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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!
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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
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3.1 D-brane from Tachyon Condensation
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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.
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Index Theorem
Every points represent eigen modes
=Integral on
p-dimensional space“Geometric” picture
Number of zero modes of Dirac operator
“Analytic” picture
=
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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
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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.
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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.
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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
=
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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
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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
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3.2 Diagonalized Tachyon Gauge
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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)
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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
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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
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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!
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D2-D0 bound state and Tachyon
D2-brane with magnetic flux BN BPS D0-branes [tildeX,X]=i/BNoncommutative D-brane
M D0-D0bar pairs with T=uD,X
Tachyon condenseX=diag. gauge
Tachyon condenseT=diag. gauge
Equivalent!
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4. Application to the Supertubes
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Circular Supertube in D2-brane picture
Mateos-Townsend
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D0-anti-D0-brane picture
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D0-brane picture
Bak-LeeBak-Ohta
This coinceides with the supertube in the matrix model!
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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
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5. Index Theorem, ADHM and Tachyon
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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!
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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 gaugeT=diagonal gauge
Equivalent!
coupling to RR-fields of D0-D0bar
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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
N D4-branesand k D0-branes
N D4-braneswith instanton
WittenDouglas
D-brane interpretation
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We know that N D4-branes =large M D0-D0bar
N D4-braneswith instanton
N D4-branesand k D0-branes
M D0-D0bar pairs with
Tachyon condenseX=diag. gauge
Applying the previous method, i.e. diag. T instead of X
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We know that N D4-branes =large M D0-D0bar
N D4-braneswith instanton
N D4-branesand k D0-branes
Applying the previous method, i.e. diag. T instead of X
M D0-D0bar pairs with
Tachyon condenseX=diag. gauge
Tachyon condenseT=diag. gauge
Equivalence!ADHM ↔ Instanton
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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
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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,,,,
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End of the talk