Noncommutative Integrable Systems and Quasideterminants. Masashi HAMANAKA University of Nagoya, Dept of Math.

Based on

Claire R.Gilson (Glasgow), MH and

Jonathan J.C.Nimmo (Glasgow), ``Backlund trfs and the Atiyah-Ward ansatz for NC anti-self-dual (ASD) Yang-Mills (YM) eqs.’’ Proc. Roy. Soc. A465 (2009) 2613 [arXiv:0812.1222].

MH, NPB 741 (2006) 368, JHEP 02 (07) 94, PLB 625 (05) 324, JMP46 (05) 052701…

NC extension of integrable systems Matrix generalization Quarternion-valued system Moyal deformation (=extension to NC spaces=presence of magnetic

flux)

plays important roles in QFT a master eq. of (lower-dim) integrable eqs. (Ward’s conjecture)

1. Introduction

4-dim. Anti-Self-Dual Yang-Mills Eq.

NC ASDYM eq. with G=GL(N)NC ASDYM eq. (real rep.)

.1203

,3102

2301 ,

FF

FF

FF

)],[:( AAAAF

0

00

0

2

2

1

1

O

O

(Spell:All products are Moyal products.)

)()()(2

)()(

)(2

exp)(:)()(

2

Oxgxfixgxf

xgi

xfxgxf

�

Under the spell, we geta theory on NC spaces:

ixxxxxx :],[

NC ASDYM eq. with G=GL(N)NC ASDYM eq. (real rep.)

.1203

,3102

2301 ,

FF

FF

FF

)],[:( AAAAF

0

00

0

2

2

1

1

O

O

(Spell:All products are Moyal products.)

0

,0

,0

~~

~~

wwzz

wz

zw

FF

F

F

1032

3210

2

1~

~

ixxixx

ixxixx

zw

wz

(complex rep.)

Reduction to NC KdV from NC ASDYM

qqqqqqqqqqqq

qqqqA

qqq

qA zw

2

1)(

2

1

4

12

1

,1

22

00

00,

01

00~~ wz AA

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

)(4

3

4

1uuuuuu qu 2

:NC ASDYM eq. G=GL(2)

:NC KdV eq.!

Reduction conditions

0

,0

,0

~~

~~

wwzz

wz

zw

FF

F

F

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]

cf. [Jon Nimmo san’s turorial] 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

Reduction to NC NLS from NC ASDYM

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

1,0

0zw AA

00

00,

10

011 ~~ wz AA

21

Reduction conditions

:NC ASDYM eq. G=GL(2)

:NC NLS eq.!

0

,0

,0

~~

~~

wwzz

wz

zw

FF

F

F

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 CBSNC Zakharov

NC mKdV

NC pKdV

Yang’s form

gauge equiv.gauge equiv.

Infinite gauge group

Solution Generating Techniques

NC Twistor Theory,

MH[hep-th/0507112]Summariized in [MH NPB 741(06) 368]

In gauge theory,NC magnetic fields

New physical objects Application to string theory

Plan of this talk

1. Introduction

2. Backlund Transforms for the NC

ASDYM eqs. (and NC Atiyah-Ward

ansatz solutions in terms of

quasideterminants )

3. Origin of the Backlund trfs

from NC twistor theory

4. 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 integrability.

We rewrite the NC ASDYM eq. into NC Yang’s equations and give Backlund transformations for it.

The generated solutions are represented 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.)

The proof is in fact very simple!

A derivation of NC ASDYM equations We discuss G=GL(N) NC ASDYM eq. from the viewpoint of NC linear systems with a (commutative)spectral parameter .

Linear systems:

Compatibility condition of the linear system:

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

~~ wzwwzzzw DDDDDDDDML

0],[],[

,0],[

,0],[

~~~~

~~~~

wwzzwwzz

wzwz

wzzw

DDDDFF

DDF

DDF

:NC ASDYM eq.

.0)(

,0)(

~

~

wz

zw

DDM

DDL

NC Yang’s form and Yang’s equationNC 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 :NC 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

Backlund trf. for NC Yang’s eq. G=GL(2)Yang’s J matrix can be reparametrized as fo

llows

Then NC Yang’s eq. becomes

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

11

11

beb

gbegbfJ

.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

Backlund transformation for NC Yang’s eq.Yang’s J matrix can be reparametrized as fo

llows

Then NC Yang’s eq. becomes

Another trf. also leaves NC Yang’s eq. as it is:

1111

11111

0)()(

)()(:

egbffbge

bfeggefb

fg

eb

be

gfnewnew

newnew

01

10, 00

10 CCJCJ new

11

11

beb

gbegbfJ

.0)()(

,0)()(

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

1~

11~

1~

1~

1

1~

11~

11~

1~

11~

11~

1~

11~

1

wwzzwwzz

wwzzwwzz

wwzzwwzz

ebgfebgfffff

bgfebgfebbbb

febfebbgfbgf

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

Then we could generate various (non-trivial) solutions of NC Yang’s eq. from a (trivial) seed solution (so called, NC Atiyah-Ward solutions)

idid 00,

]2[]1[]0[

0 JJJ

1][][

1][

1][][][

1][][][

][nnn

nnnnnnn beb

bgebgfJ

11

11~

11~

1~

11~

1

,

,,

,,

:

fbbf

febgfebg

bgfebgfe

newnewz

newww

newz

znew

wwnew

z

1

0 :

fg

eb

be

gfnewnew

newnew

0

Generated solutions (NC Atiyah-Ward sols.) Let’s consider the combined Backlund trf.

Then, the generated solutions are :

]2[]1[]0[

0 JJJ

Quasideterminants !(a kind of NC determinants)

zwwz

DgDeDfDb

iiii

nnnnnnnnnnnn

~,~

,,,

11

1

1][]1[

1

1][]1[

1

11][]1[

1

][]1[

021

)2(01

)1(10

][

nn

n

n

nD

[Gelfand-Retakh]

with a seed solution:

1][][

1][

1][][][

1][][][

][nnn

nnnnnnn beb

bgebgfJ

0, 021

0]0[]0[]0[]0[ gefb

Quasi-determinantsQuasi-determinants are not just a NC

generalization of commutative determinants, but rather related to inverse matrices.

[Def1] For an n by n matrix and the inverse of , quasi-determinant of is directly defined by

)( ijxX )( ijyY

proportional to the det. or ratio of dets.

ijX

X X

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

X

XyX

ij

ji

jiijdet

det

)1(1

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

Incommutative limit

Quasi-determinantsQuasi-determinants are not just a NC

generalization of commutative determinants, but rather related to inverse matrices.

[Def1] For an n by n matrix and the inverse of , quasi-determinant of is directly defined by

[Def2(Iterative definition)]

)( ijxX )( ijyY

ijX

ijji

jjij

ijiiij

jijjji

ijiiijij

xxXxxxXxxX

,

1

,

1 )())((

n × nn+1 × n+1 convenient notation

X X

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

X

XyX

ij

jilinmit

ecommutativ

jiijdet

det

)1(1

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

Examples of quasi-determinants

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

11111

111111111

)()(

)()(

BCADCABCAD

BCADBACABCADBAAXY

DC

BAX

Cf.

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

zwwz

ge

fb

iiii

n

n

n

n

n

n

n

n

n

n

n

n

~,~

,

,,

11

1

0

0

][

1

0

0

][

1

0

0

][

1

0

0

][

0~11~00~11~0

1

01

10]1[

1

01

10]1[

1

01

10]1[

1

01

10]1[

,,,

,,,,

zwzwwzwz

gefb

[Gilson-MH-Nimmo, arXiv:0709.2069]

0, 021

0]0[]0[]0[]0[ gefb

Yang’s matrix J (gauge invariant)

The Backlund trf. is not just a gauge trf. but a non-trivial one!

011

1

1

02

20

1

1

110

][

00

01

00010

nn

n

n

n

n

nn

nJ

The proof is in fact very simple!Proof is made simply by using only specia

l identities of quasideterminants. (NC Jacobi’s identities and a homological relation, Gilson-Nimmo’s derivative formula etc.)

In other words, ``NC Backlund trfs are identities of quasideterminants.’’ This is an analogue of the fact in lower-dim. commutative theory: ``Backlund trfs are identities of determinants.’’

Some exact solutions

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

0 0

A seed solution:

)~,,~,exp(

''~~1

``1

0

0

wwzzoflinearwwzz

NC instantons (special to NC spaces)

NC Non-Linear plane-waves (new solutions beyond ADHM)

3. Interpretation from NC twistor theory

In this section, we give an origin of all ingredients 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 correspondence between ASD connections and ``NC holomorphic vector bundle’’ on NC twistor space.

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

NC Penrose-Ward correspondenceLinear systems of ASDYM ``NC hol. Vec. bdl’’

.0)(

,0)(

~

~

wz

zw

DDM

DDL

.0~)~

(~~,0~)

~(~~

~

~

wz

zw

DDM

DDL

/1~

1~

),~,~(

wzzwP

),(~)(~

),0,()(

xxh

xxh

Patching matrix

1:1

ASDYM gauge fields are reproduced

We have only to factorize agiven patching matrix into and to get ASDYM fields. (Birkhoff factorization or Riemann-Hilbert problem)

)()();( Oxhx

)~

()(~

)~

;(~ Oxhx

~

[Takasaki]

Origin of NC Atiyah-Ward (AW) ansatz sols.

The n-th AW ansatz for the Patching matrix

The recursion relation is derived from:

i

iin

n

n xwzzwxx

P

)(),~,~();(

);(

0][

,0)();(

,0)();(

~

~

wz

zw

xm

xl

zwwziiii~,~

11

OK!

Origin of NC Atiyah-Ward (AW) ansatz sols.

The n-th AW ansatz for the Patching matrix

The Birkoff factorization leads to:

Under a gauge ( ), this solution coincides with the quasideterminants sols!

i

iin

n

n xwzzwxx

P

)(),~,~();(

);(

0][

1

1,1]1[21

1

1,1]1[2222

1

1,1]1[11

1

1,1]1[1212

1

1,1]1[21

1

1,1]1[2221

1

1,1]1[11

1

1,1]1[1211

~~

~~

~

~

nnnnn

nnnnn

nnn

nnn

DhDhh

DhDhh

DhDhh

DhDhh

1][][

1][

1][][][

1][][][

][

][

1

][

][1][ 1

0

0

1~

nnn

nnnnnn

n

n

n

nn

beb

bgebgf

e

f

b

ghhJ

1

1,1]1[][12

1

1,1]![][21

1

11]1[][11

1

1,1]1[][22

~,

,,~

nnnnnn

nnnnnn

DghDeh

DfhDbh

1~

,0~

1122

2112

hh

hh

1~ P

)()( Oxh )~

()(~ Oxh

021

)2(01

)1(10

][

nn

n

n

nD

OK!

Origin of the Backlund trfs

The Backlund trfs can be understood as the adjoint actions for the Patching matrix:

actually:

The -trf. leads to The -trf. is derived with a singular gauge tr

f.

01

10,

0

10,:,: 010

100

1 CBPCCPBPBP newnew

]1[1

)1(

011

0][0

000:

nn

n

n

n

n

n

n PCBBCP

0

0

0,:

1

1

f

bsBsnew

0

10 CJCJ new

1

0

1

0

121

1

kf

b

e

fh

new

newnew

OK!1~

112

1

1

)(

bgfkfe

bf

zwnew

w

new

)( 2 Okh

0)( ~ zw DDL1-2 component of

The previous -trf!

The previous -trf!0

0

4.Conclusion and Discussion

　 NC integrable eqs (ASDYM) in higher-dim. ADHM (OK) Twistor (OK) Backlund trf (OK), Symmetry (Next)

NC integrable eqs (KdV) in lower-dims. Hierarchy 　 (OK) 　 Infinite conserved quantities 　 (OK) Exact N-soliton solutions 　 (OK) Symmetry （ ``tau-fcn’’, Sato’s theory ）　 (Next) Behavior of solitons… 　 (Next)

Quasi-determinants are important !

Quasi-determinants are important !

Profound relation ?? (via Ward conjecture)

Very HOT！