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Construction of BPS Solitonsvia
Tachyon Condensation
So Matsuura @ RIKEN
based on the work withT. Asakawa and K. Ohta
hep-th/0603***
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Introduction and
Motivation
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Solitons
(Supersymmetric) Gauge Theory
non-trivial solutions of non-linear field equation
non-perturbative feature of gauge theory
non-trivial structure of the moduli space
Superstring TheoryBPS bound states of D-branes
non-perturbative feature of string theory
gauge/gravity duality
relation to black hole entropy
Witten(1996), Douglas(1996)
Atiya-Hitchin-Drinfeld-Manin(1978),Nahm(1980)
Ishibashi-Kawai-Kitatzawa-Tsuchiya(1997), Banks-Fishler-Shenker-Susskind(1997)
Maldacena(1998)...
Strominger-Vafa(1996)
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How to read offinformation of solitons
fromthe superstring theory?
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Typical Example ~ instanton in string theory ~
some field configurationof
0-dim gauge theory
D3-branes + open stringsD3-branes + D(-1)-branes
with open strings
one-to-one
correspondence
ADHM Construction
(today’s talk)
equivalent
at the string level
instanton solution of
4D gauge theory
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ADHM Construction
(bosonic) ADHM data; N x k complex matrices
; k x k complex matrices
ADHM constraint
the degrees of freedom of open strings on k D(-1)-branes
F-term and D-term conditions of the 0D SUSY gauge theory on D(-1)-branes
Corresponding instanton gauge field ; N x (N+2k) matrix
self-dual field strength instanton number k
Atiyah-Hitchin-Drinfeld-Manin (1978)
Watamura-san’s talk
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Tachyon Condensation
complex tachyon
N Dp-branes + k D(-1)-branes
condensation
tachyon
Kraus-Larsen (2001)Asakawa-Sugimoto-Terashima (2002)
Can we complete the following picture?
tachyon
condensation
tachyon
condensation
cf) Hashimoto-Terashima(2005)
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CONTENTS
1. Introduction
2. Tachyon Condensation in Boundary State Formalism
3. Soliton Construction in Tachyon Condensation
4. Conclusion and Future Works
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Tachyon Condensation in Boundary State Formalism (review)
D-brane
boundary of open strings
condensed state of closed strings
Neumann directions
modular transformation
Neumann boundary condition
Dirichletboundary condition
In the closed string language,Dirichlet directions
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: boundary coordinate
: fermionic partner
Dp-brane as a boundary state
boundary (super) coordinate
string (super) coordinate
tension of a Dp-brane
Callan-Lovelace-Nappi-Yost (1989)
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Excitation of open strings
insertion of vertex operators at the boundary
A Wilson loop operator is added;
is called as the boundary interaction.
(massless excitations)
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A system of (N+M) Dp-branes and M anti-Dp-branes
The fate of this system depends on the tachyon profile .
We can introduce complex tachyon;
We call as the super-connection;
Kraus-Larsen (2001)Takayanagi-Terashima-Uesugi (2001)Asakawa-Sugimoto-Terashima (2002)
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Tachyon condensation (1) ~ pair annihilation of D-branes ~
We set
The boundary interaction in the NSNS sector becomes
NOTEThe RR sector is exactly same as the N Dp-brane because of the super-trace;
N
M
MNM
M
N
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Tachyon condensation (2) ~ creation of D-instantons ~
Let us set the tachyon profile as
where
; SO(4) gamma matrices
We can show that this system becomes N D3-branes and k D(-1)-branes at the origin;
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Technical preliminary
If we decompose as
we can carry out in the definition of the boundary interaction;
(1) Sometimes it is convenient to integrate out θ
supersymmetric path-ordered product
usual path-ordered product
example
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(2) Gauge transformation of the boundary interaction
is invariant under the gauge transformation,
Consider the system of (N+M) Dp-branes and M anti-Dp-branes.The boundary interaction,
or
where,
(ex)
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Soliton Construction as Tachyon Condensation
Summary of construction of solitons
① Consider D-branes that construct a soliton as a bound state.
② Realize individual D-branes by the tachyon condensation.
③ Add a fluctuation to the boundary interaction.
④ Carry out a gauge transformation and separate D-branes that vanish.
⑤ Read off the information of the moduli space of the soliton solution from the tachyon profile.
ex) D3-branes + D(-1)-branes 4D instanton
vanish
ADHM construction is obtained.
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(Example 1) Construction of 4D instantons
① Consider D-branes that construct a soliton as a bound state.
N D3-branes
k D(-1)-branes
② Realize individual D-branes by the tachyon condensation.
Consider k-instanton solution of U(N) gauge theory.
N D3-branes + k D(-1)-branes
Pauli matrices
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③ Add a fluctuation to the boundary interaction.
Note
which expresses N D3-branes and k D(-1)-branes at the origin.
2k
N
2k
are fluctuation from the profile,
corresponds to scalar fields on D(-1)-brane, thus, must be hermitian.
Akhmedov-Gerasimov-Shatashivili (2001)Hashimoto-Terashima (2005)
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Let us define
Then the tachyon profile can be rewritten as
This is nothing but the ADHM data.
k k
Nkk
corresponding boundary state
tachyon condensation
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Let us consider the gauge transformation by;
such that
If we assume that is strictly positive definite, we can define
and can be written as
where V is a (N+2k)×N matrix which is a collection of zero vectors of
vanish
N 2k
④ Carry out a gauge transformation and separate D-branes that vanish.
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The gauge transformation of the super-connection is
where
tachyon condensation
corresponding boundary state
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・・・・・
・・
There appears a gauge field,
on the remaining N Dp-branes after the tachyon condensation.
formulae
if
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⑤ Read off the information of the moduli space of the soliton solution from the tachyon profile.
self-dual part anti-self-dual part
In order that this is an instanton solution, we must impose
This is nothing but the ADHM condition.
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What have we done?
tachyon condensation tachyon condensation
gauge
equivalent
full string level
correspondence
at different low energy limit
ADHM construction
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Comments
① Tachyon configuration , corresponds to the small instanton singularity of the instanton moduli space.
② The ADHM constraint is not necessary for this procedure.
③ The ADHM equations are parts of the tachyon potential
④ Another part determines the feature of the tachyon condensation.
⑤ The gauge transformation here is a large gauge transformation.
D(-1)-branes appear at .
deviation from the ADHM condition
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(Example 2) Construction of 2D vortex
Let us consider a bound state of D1-branes and D(-1)-branes.
Tachyon condensation from (N+k) D1-branes and k anti-D1-branes with
For U(1) (N=1), the field strength becomes
Then the minimum of the Yang-Mills energy,
is realized when H→∞.
Well known result.
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(Example 3) Construction of higher dimensional instantons
For 2n dimensional Yang-Mills theory, we must impose the “self-duality”for a maximal subgroup H of SO(2n);
invariant tensor of H
Then the Yang-Mills equation is trivial as a result of the Bianchi identity.
For 8D Yang-Mills theory,
For H=SO(4)xSO(4), the instanton is an intersection of the 4D instantons.
Construction of other solutions is a future work.
I’m sorry, under construction m(__)m
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Conclusion
1. We proposed a systematic way to construct a gauge field on D-branes by the tachyon condensation.
2. In particular, we can examine the structure of the moduli space of solitons in principle.
3. We applied it to the tachyon condensation of D3-branes and anti-D3-branes and showed that the ADHM construction can be understood as a gauge equivalence of two pictures of D-brane bound state.
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Future Work• Techniques to be developed
• This procedure is quite general one.
• Relation to the supersymmetry
moduli space of non-trivial vortex solutions construction of higher-dimensional instantons What is the category of the gauge field that is
constructed by this procedure?
At this stage, the role of supersymmetry is not clear. Nekrasov’s formula by the tachyon condensation?
Usage of curved D-branes. We want to impose the BPS condition at the level of the
boundary state.