Noncommutative Solitons and Integrable Systems

Masashi HAMANAKA University of Nagoya, Dept. of Math.

Based on C.R.Gilson (Glasgow), MH and J.J.C.Nimmo (Glasgow),

``Backlund tranformations for NC anti-self-dual (ASD) Yang-Mills (YM) eqs.’’ arXiv:0709.2069 (&08mm.nnnn).

MH, ``NC Ward's conjecture and integrable systems,’’ Nucl. Phys. B 741 (2006) 368, and others: JHEP 02 (07) 94, PLB 625 (05) 324, JMP46 (05) 052701…

Successful points in NC theories Appearance of new physical objects Description of real physics (in gauge theory) Various successful applications

to D-brane dynamics etc.

1. Introduction

Construction of exact solitons are important. (partially due to their integrablity)Final goal: NC extension of all soliton theories (Soliton eqs. can be embedded in gauge theories via Ward’s conjecture ! [R. Ward, 1985] )

Ward’s conjecture: Many (perhaps all?) integrable equations are reductions of the ASDYM eqs.

ASDYM

Ward’s chiral

(affine) Toda

NLS

KdV sine-Gordon

Liouville

Tzitzeica

KP DS

Boussinesq N-wave

CBS Zakharov

mKdV

pKdV

Yang’s form

gauge equiv.gauge equiv.

Infinite gauge group

ASDYM eq. is a master eq. !Solution Generating Techniques

Twistor Theory

NC Ward’s conjecture: Many (perhaps all?) NC integrable eqs are reductions of the NC ASDYM eqs.

NC ASDYM

NC Ward’s chiral

NC (affine) Toda

NC NLS

NC KdV NC sine-Gordon

NC Liouville

NC Tzitzeica

NC KPNC DS

NC Boussinesq NC N-wave

NC CBS NC Zakharov

NC mKdV

NC pKdV

NC ASDYM eq. is a master eq. ?Solution Generating Techniques

NC Twistor Theory

Reductions

New physical objects Application to string theory

In gauge theory,NC magnetic fields

Plan of this talk

1. Introduction

2. Backlund Transform for the NC ASDYM eqs. (and NC Atiyah-Ward ansatz solutions in terms of quasideterminants )

3. Interpretation from NC twistor theory

4. Reduction of NC ASDYM to NC KdV

(an example of NC Ward’s conjecture)

5. Conclusion and Discussion

2. Backlund transform for NC ASDYM eqs. In this section, we derive (NC) ASDYM eq. from

the viewpoint of linear systems, which is suitable for discussion on integrable aspects.

We define NC Yang’s equations which is equivalent to NC ASDYM eq. and give a Backlund transformation for it.

The generated solutions are NC Atiyah-Ward ansatz solutions in terms of quasideterminants, which contain not only finite-action solutions (NC instantons) but also infinite-action solutions (non-linear plane waves and so on.)

Review of commutative ASDYM equations

Here we discuss G=GL(N) (NC) ASDYM eq. from the viewpoint of linear systems with a spectral parameter . Linear systems (commutative case):

Compatibility condition of the linear system:

0],[]),[],([],[],[ ~~2

~~ wzwwzzzw DDDDDDDDML

0],[],[

,0],[

,0],[

~~~~

~~~~

wwzzwwzz

wzwz

wzzw

DDDDFF

DDF

DDF

]),[:( AAAAF

.0)(

,0)(

~

~

wz

zw

DDM

DDL

1032

3210

2

1~

~

ixxixx

ixxixx

zw

wz

:ASDYM equation

e.g.

Yang’s form and Yang’s equation ASDYM eq. can be rewritten as follows

If we define Yang’s matrix:then we obtain from the third eq.:

hhJ 1~:

0)()( ~1

~1 JJJJ wwzz :Yang’s eq.

1~~

1~~

11 ~~,

~~,, hhAhhAhhAhhA wwzzwwzz

The solution reproduce the gauge fields asJ

0],[],[

.),~~

(0~

,0~

,~

,0],[

.),(0,0,,0],[

~~~~

1~~~~~~~~

1

wwzzwwzz

zzwzwzwz

zzwzwzzw

DDDDFF

etchhAhDhDhDDF

etchhAhDhDhDDF

is gauge invariant. The decomposition into and corresponds to a gauge fixingJ h h

~

(Q) How we get NC version of the theories?

(A) We have only to replace all products of fields in ordinary commutative gauge theories

with star-products: The star product: (NC and associative)

)()()(2

)()()(2

exp)(:)()( 2

Oxgxfixgxfxg

ixfxgxf

�

ixxxxxx :],[

NC !

A deformed product

)()()()( xgxfxgxf

Presence of background magnetic fields

Note: coordinates and fields themselves are usual c-number functions. But commutator of coordinates becomes…

Here we discuss G=GL(N) NC ASDYM eq. from the viewpoint of linear systems with a spectral parameter .

Linear systems (NC case):

Compatibility condition of the linear system:

0],[)],[],([],[],[ ~~2

~~ wzwwzzzw DDDDDDDDML

0],[],[

,0],[

,0],[

~~~~

~~~~

wwzzwwzz

wzwz

wzzw

DDDDFF

DDF

DDF

)],[:( AAAAF

.0)(

,0)(

~

~

wz

zw

DDM

DDL

1032

3210

2

1~

~

ixxixx

ixxixx

zw

wz

:NC ASDYM equation

e.g.

(All products are star-products.)

0

00

0

2

2

1

1

O

O

Yang’s form and NC Yang’s equation NC ASDYM eq. can be rewritten as follows

0],[],[

0~

,0~

,~

,0],[

0,0,,0],[

~~~~

~~~~~~

wwzzwwzz

wzwzwz

wzwzzw

DDDDFF

hDhDhDDF

hDhDhDDF

If we define Yang’s matrix:then we obtain from the third eq.:

hhJ 1~:

0)()( ~1

~1 JJJJ wwzz :NC Yang’s eq.

1~~

1~~

11 ~~,

~~,, hhAhhAhhAhhA wwzzwwzz

The solution reproduces the gauge fields asJ

Backlund transformation for NC Yang’s eq. Yang’s J matrix can be decomposed as follows

Then NC Yang’s eq. becomes

The following trf. leaves NC Yang’s eq. as it is:

11

11

beb

bgebgfJ

.0)()(

,0)()(

,0)()(,0)()(

1~

11~

1~

1~

1

1~

11~

11~

1~

11~

11~

1~

11~

1

wwzzwwzz

wwzzwwzz

wwzzwwzz

ebgfebgfffff

bgfebgfebbbb

febfebbgfbgf

11

11~

11~

1~

11~

1

,

,,

,,

:

fbbf

febgfebg

bgfebgfe

newnewz

newww

newz

znew

wwnew

z

We could generate various (non-trivial) solutions

of NC Yang’s eq. from a (trivial) seed solution by using the previous Backlund trf. together with

a simple trf.

(Both trfs. are involutive ( ), but the combined trf. is non-trivial.)

For G=GL(2), we can present the transforms more explicitly and give an explicit form of a class of solutions (NC Atiyah-Ward ansatz).

01

10,: 1

0 CCJCJ new1

0 :

fg

eb

be

gfnewnew

newnew

idid 00,

Backlund trf. for NC ASDYM eq. Let’s consider the combined Backlund trf.

If we take a seed sol.,

the generated solutions are :

1][][

1][

1][][][

1][][][

][nnn

nnnnnnn beb

bgebgfJ

]3[]2[]1[

0 JJJ

NC Atiyah-Ward ansatz sols.

Quasideterminants !(a kind of NC determinants)

zwwz

DgDeDbDf

rrrr

nnnnnnnnnnnn

~,~

,,,

11

1

1][][

1

1][][

1

][][

1

11][][

0, 021

0]1[]1[]1[]1[ gebf

021

)2(01

)1(10

][

nn

n

n

nD

[Gelfand-Retakh]

Quasi-determinants Quasi-determinants are not just a NC generalization of

commutative determinants, but rather related to inverse matrices.

For an n by n matrix and the inverse of X, quasi-determinant of X is directly defined by

Recall that

X

XyX

ij

ji

jiijdet

det

)1(01

)( ijxX )( ijyY

some factor

11111

111111111

)()(

)()(

BCADCABCAD

BCADBACABCADBAAXY

DC

BAX

We can also define quasi-determinants recursively

ijX : the matrix obtained from X deleting i-th row and j-th column

Quasi-determinants Defined inductively as follows

ijji

jjij

ijiiij

jijjji

ijiiijij

xxXxxxXxxX

,

1

,

1 )())((

31

123

12232331331

133

132222312

211

221

23333213211

321

3323221211

31

21

1

3332

2322131211

333231

232221

131211

11

121

1121222221

1211

22111

1222212221

1211

21

221

2111122221

1211

12211

2212112221

1211

11

)()(

)()(

),(:3

,,

,,:2

:1

xxxxxxxxxxxx

xxxxxxxxxxxxx

x

x

xx

xxxxx

xxx

xxx

xxx

Xn

xxxxxx

xxXxxxx

xx

xxX

xxxxxx

xxXxxxx

xx

xxXn

xXn ijij

[For a review, see Gelfand et al.,math.QA/0208146]

convenient notation

Explicit Atiyah-Ward ansatz solutions of NC Yang’s eq. G=GL(2)

zwwz

ge

bf

rrrr

n

n

n

n

n

n

n

n

n

n

n

n

~,~

,

,,

11

1

01

)1(0

][

1

01

)1(0

][

1

01

)1(0

][

1

01

)1(0

][

0~11~00~11~0

1

01

10]1[

1

01

10]1[

1

01

10]2[

1

01

10]2[

,,,

,,,,

zwzwwzwz

gebf

Yang’s matrix J is also beautiful. [Gilson-MH-Nimmo. arXiv:0709.2069]

0, 021

0]1[]1[]1[]1[ gebf

We could generate various solutions of NC ASDYM eq. from a simple seed solution by using the previous Backlund trf.

Proof is made simply by using special identities of quasideterminants (NC Jacobi’s or Sylvester’s identities and a homological relation and so on.),

in other words, ``NC Backlund transformations are identities of quasideterminants.’’

0

)~,,~,exp(

''~~1

``1

0

0

wwzzoflinearwwzz

A seed solution:

NC instantons

NC Non-Linear plane-waves

0

3. Interpretation from NC Twistor theory In this section, we give an origin of the Backlund

trfs. from the viewpoint of NC twistor theory. NC twistor theory has been developed by several

authors What we need here is NC Penrose-Ward corresp

ondence between ASD connections and ``NC holomorphic vector bundle’’ on twistor sp.

Strategy from twistor side to ASDYM side:– (i) Solve Birkhoff factorization problem

– (ii) obtain the ASD connections from

1~),~,~( wzzwP

[Kapustin-Kuznetsov-Orlov, Takasaki, Hannabuss, Lechtenfeld-Popov, Brain-Majid…]

),(~)(~

),0,()(

xxh

xxh

Origin of NC Atiyah-Ward ansatz solutions n-th Atiyah-Ward ansatz for the Patching matrix

The Birkoff factorization leads to:

Under a gauge ( ), the solution leads to the quasideterminants solutions:

i

iin

n

n xxx

P

)();(

);(

0][

1

1,1]1[21

1

1,1]1[2221

1

1,1]1[21

1

1,1]1[2222

1

1,1]1[11

1

1,1]1[1211

1

1,1]1[11

1

1,1]1[1212

~

~~

~

~~

nnnnn

nnn

nnnnn

nnn

DhDhh

DhDhh

DhDhh

DhDhh

1][][

1][

1][][][

1][][][

1][

~

nnn

nnnnnn

n

beb

bgebgf

hhJ

1

1][][

1

1][][

1

][][

1

11][][ ,,,

nnnnnnnnnnnn DgDeDbDf

.,0~

2211 etchh

1~P

)()( Oxh )()(~ 1 Oxh

021

)2(01

)1(10

][

nn

n

n

nD

Origin of the Backlund trfs The Backlund trfs can be understood as the adjoi

nt actions for the Patching matrix:

actually:

The -trf. can be generalized as any constant matrix. Then this Backlund trf. would generate all solutions in some sense and reveal the hidden symmetry of NC ASDYM eq.

-trf. is also derived with a singular gauge trf.

01

10,

0

10,:,: 1

10

1 CBPCCPBPBP newnew

]1[1

)1(11

][0

000:

nn

n

n

n

n

n

n PCBBCP

0

0

0,:

1

1

f

bsBsnew

4. Reduction of NC ASDYM to NC KdV

Here, we show an example of reductions of NC ASDYM eq. on (2+2)-dimension to NC KdV eq.

Reduction steps are as follows:

(1) take a simple dimensional reduction

with a gauge fixing.

(2) put further reduction condition on gauge field. The reduced eqs. coincides with those obtained in

the framework of NC KP and GD hierarchies,

which possess infinite conserved quantities and

exact multi-soliton solutions. (integrable-like)

Reduction to NC KdV eq. (1) Take a dimensional reduction and gauge fixing:

(2) Take a further reduction condition:

),~,(),()~,,~,( wwzxtwwzz

MH, PLB625, 324[hep-th/0507112]

0],[)(

0],[],[)(

0],[)(

~~~

~

zwwz

wwzzww

zw

AAAAiii

AAAAAAii

AAi

01

00~zA

The reduced NC ASDYM is:

qqqqqqqf

qqqqAOA

qqqq

qA zww

2

1),,,(

2

1

,,1

~

We can get NC KdV eq. in such a miracle way ! ,

)(4

3

4

1)( uuuuuuiii qu 2 ixt ],[

)2()2(,, 0 slglCBA Note: U(1) part is necessary !

NOT traceless !

The NC KdV eq. has integrable-like properties:

possesses infinite conserved densities:

has exact N-soliton solutions:

))(2)((4

3211 uLresuLresLres nnn

n

:nrLres coefficient of in

rx nL

: Strachan’s product (commutative and non-associative))(

2

1

)!12(

)1()(:)()(

2

0

xgs

xfxgxfs

jiij

s

s

�

N

iiixx WWu

1

1)(2

iiii ffWW,1 ),...,(:

)),((exp),(exp iiii xaxf 3),( txx

:quasi-determinant of Wronski matrix

MH, JMP46 (2005) [hep-th/0311206]

Etingof-Gelfand-Retakh, MRL [q-alg/9701008]MH, JHEP [hep-th/0610006]cf. Paniak, [hep-th/0105185]

ixt ],[

2)('1 )(2 xxx ufu

L : a pseudo-diff. operator

5. Conclusion and Discussion

NC ASDYM

NC Ward’s chiral

NC (affine) Toda

NC NLS

NC KdV NC sine-Gordon

NC Liouville

NC Tzitzeica

NC KPNC DS

NC Boussinesq NC N-wave

NC CBSNC Zakharov

NC mKdV

NC pKdV

Yang’s form

gauge equiv.gauge equiv.

Infinite gauge group

NC ASDYM eq. is a master eq. !

Solution Generating Techniques

NC Twistor Theory,

Going well!

Going well! Going well!

[Brain, Hannabuss, Majid, Takasaki, Kapustin et al]

MH[hep-th/0507112]

[Gilson- MH- Nimmo,…]

[MH NPB 741(06) 368]