Construction of BPS Solitonsvia

Tachyon Condensation

So Matsuura @ RIKEN

based on the work withT. Asakawa and K. Ohta

hep-th/0603***

Introduction and

Motivation

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)

How to read offinformation of solitons

fromthe superstring theory?

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

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

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)

CONTENTS

1. Introduction

2. Tachyon Condensation in Boundary State Formalism

3. Soliton Construction in Tachyon Condensation

4. Conclusion and Future Works

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

: 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)

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)

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)

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

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;

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

(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)

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.

(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

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)

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

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.

The gauge transformation of the super-connection is

where

tachyon condensation

corresponding boundary state

・・・・・

・・

There appears a gauge field,

on the remaining N Dp-branes after the tachyon condensation.

formulae

if

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.

What have we done?

tachyon condensation tachyon condensation

gauge

equivalent

full string level

correspondence

at different low energy limit

ADHM construction

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

(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.

(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

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.

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.