Noncommutative Solitons and Integrable Systems

MH,``Conservation laws for NC Lax hierarchy,’’ JMP46(2005)052701[hep-th/0311206]

MH, ``NC Ward's conjecture and integrable systems,’’ NPB741 (2006) 368, [hep-th/0601209]

MH, ``Notes on exact multi-soliton solutions of NC integrable hierarchies ,’’ [hep-th/0610006]

Based on

Masashi HAMANAKA Nagoya University, Dept. of Math. (visiting Glasgow until Feb.18)

Successful points in NC theoriesAppearance of new physical objectsDescription of real physicsVarious successful applications

to D-brane dynamics etc.

1. Introduction

NC Solitons play important roles (Integrable!)

Final goal: NC extension of all soliton theories

Integrable equations in diverse dimensions

4 Anti-Self-Dual Yang-Mills eq.

(instantons)

3 Bogomol’nyi eq.

(monopoles)

2

(+1)

Kadomtsev-Petviashvili (KP) eq.

Davey-Stewartson (DS) eq. …

1

(+1)

KdV eq. Boussinesq eq.

NLS eq. Burgers eq.

sine-Gordon eq. (affine) Toda field eq. …

FF~

Dim. of space

NC extension (Successful)

NC extension (This talk)

NC extension(Successful)

NC extension (This talk)

Ward’s conjecture: Many (perhaps all?)

integrable equations are reductions of the ASDYM eqs.

ASDYM eq.

(KP eq.) (DS eq.) Ward’s chiral model KdV eq. Boussinesq eq. NLS eq. Toda field eq.

sine-Gordon eq. Liouville eq. Painleve eqs. Tops …

Reductions

R.Ward, Phil.Trans.Roy.Soc.Lond.A315(’85)451

Almost confirmed by explicit examples !!!

NC Ward’s conjecture: Many (perhaps all?) NC integrable equations are reductions of t

he NC ASDYM eqs.

NC ASDYM eq.

Many (perhaps all?)NC integrable eqs.

NC Reductions

Successful

Successful?

MH&K.Toda, PLA316(‘03)77 [hep-th/0211148]

Reductions

・ Existence of physical pictures ・ New physical objects・ Application to D-branes・ Classfication of NC integ. eqs.

NC Sato’s theory plays important roles in revealing integrable aspects of them

Program of NC extension of soliton theories

(i) Confirmation of NC Ward’s conjecture – NC twistor theory geometrical origin– D-brane interpretations applications to physics

(ii) Completion of NC Sato’s theory– Existence of ``hierarchies’’ various soliton eqs.– Existence of infinite conserved quantities

infinite-dim. hidden symmetry – Construction of multi-soliton solutions– Theory of tau-functions structure of the solution s

paces and the symmetry

(i),(ii) complete understanding of the NC soliton theories

Plan of this talk 1. Introduction

2. NC ASDYM equations (a master equation)

3. NC Ward’s conjecture

--- Reduction of NC ASDYM to

KdV, mKdV, Tzitzeica, …

4. Towards NC Sato’s theory (KP, …)

hierarchy, infinite conserved quantities,

exact multi-soliton solutions,…

5. Conclusion and Discussion

2. NC ASDYM equationsHere we discuss G=GL(N) (NC) ASDYM eq. from the viewpo

int 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

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 :Yang’s eq.

1~~

1~~

11 ~~,

~~,, hhAhhAhhAhhA wwzzwwzz

The solution reproduce the gauge fields asJ

(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:

hgfhgf )()(

)()()(2

)()()(2

exp)(:)()( 2 Oxgxfixgxfxgi

xfxgxf ji

ij

jiij

0

ijijjiji ixxxxxx :],[ NC !

Associative

A deformed product

)()()()( xgxfxgxf

Presence of background magnetic fields

In this way, we get NC-deformed theorieswith infinite derivatives in NC directions. (integrable???)

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.)

Don’t omit even for G=U(1) ))()1(( UU

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

(All products are star-products.)

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:

ABA

ABBABAJ ~~

~~~1

.0~~~~

)()(

,0~~~

)~~

(~

)~~

(

,0)~

()~

(,0)~~

()~~

(

~~1

~11

~1

~~1

~11

~1

~~~~

wwzzwwzz

wwzzwwzz

wwzzwwzz

BABBABAAAAAA

BABBABAAAAAA

ABAABAABAABA

11

~~

~~

~,

~,

~~,

~~,

~~,

~~

AAAA

ABABABAB

ABABABAB

newnew

znew

wwnew

z

znew

wwnew

z

We can generate new solutions from known (trivial) solutions

MH [hep-th/0601209, 0ymmnnn]The book of Mason-Woodhouse

3. NC Ward’s conjecture --- reduction to (1+1)-dim.

From now on, we discuss reductions of NC ASDYM on (2+2)-dimension, including NC KdV, mKdV, Tzitzeica...

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

0

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, [q-alg/9701008]MH, [hep-th/0610006]cf. Paniak, [hep-th/0105185]

Explicit!

Explicit!

ixt ],[

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

(2) Take a further reduction condition:

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

MH, NPB741, 368[hep-th/0601209]

0],[)(

0],[],[)(

0],[)(

~~~

~

zwwz

wwzzww

zw

AAAAiii

AAAAAAii

AAi

01

00~zA

The reduced NC ASDYM is:

d

bcA

aA

p

pA zww 0

,0

00,

0

1~

We get

],[4

1

2

1

4

1,],[

4

1

2

1

4

1

,2

1

2

1,

2

1

2

1

33

22

ppppdppppc

ppbppa

ixt ],[and )(

4

3

4

1)( ppppppppiii NC mKdV !

NOT traceless !

Relation between NC KdV and NC mKdV (1) Take a dimensional reduction and gauge fixing:

(2) Compare the further reduction conditions:

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

MH, NPB741, 368[hep-th/0601209]

01

00~zA

d

bcA

aA

p

pA zww 0

,0

00,

0

1~

qqqqqqqf

qqqqAOA

qqqq

qA zww

2

1),,,(

2

1

,,1

~NCKdV:

NCmKdV:

Note: There is a residual gauge symmetry:

1

01,11

ggggAgA

The gauge trf. 22, ppqpq

NC Miura map !

Gauge equivalent

Reduction to NC Tzitzeica eq. Start with NC Yang’s eq.

(1) Take a special reduction condition:

(2) Take a further reduction condition:

MH, NPB741, 368[hep-th/0601209]

We get (a set of) NC Tzitzeica eq.:

)1),exp(),(exp()exp( diagg

We get a reduced Yang’s eq.

)exp()~,()~exp( wEzzgwEJ

0)()( ~1

~1 JJJJ wwzz

010

001

100

001

100

010

E

E

0],[)( 1~

1

gEgEgg zz

0))exp()(exp(

),exp()2exp())exp()(exp())exp()(exp(

),2exp()exp())exp()(exp())exp()(exp(

~

~

~

zzz

zzz

zzz

V

V

V

))2exp()exp(( ~0

zz

In this way, we can obtain various NC integrable equations from NC ASDYM !!!

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.

Summarized in MH[hep-th/0601209]

Infinite gauge group

Almost all ?

MH[hep-th/0507112]

4. Towards NC Sato’s Theory Sato’s Theory : one of the most beautiful theory o

f solitons– Based on the exsitence of hierarchies and tau-functio

ns– Various integrable equations in (1+1)-dim. can be der

ived elegantly from (2+1)-dim. KP equation. Sato’s theory reveals essential aspects of solitons:

– Construction of exact solutions– Structure of solution spaces– Infinite conserved quantities– Hidden infinite-dim. symmetry

Let’s discuss NC extension of Sato’s theory

Derivation of soliton equations Prepare a Lax operator which is a pseudo-

differential operator

Introduce a differential operator

Define NC KP hierarchy:

321 )()(2: xxxx ugufuL

],[ LBx

Lmm

0)(: LLBm ijji ixx ],[

),,,( 321 xxxuu

Noncommutativity is introduced here:

m times

34

23

12

xm

xm

xm

u

u

u

34

23

12

)(

)(

)(

xm

xm

xm

uf

uf

uf

yields NC KP equations andother NC hierarchy eqs.

Find a suitable L which satisfies NC KP hierarchy ! solutions of NC KP eq.

Negative powers of differential operators jn

xjx

j

nx f

j

nf

)(:

0

1)2)(1(

))1(()2)(1(

jjj

jnnnn

ffff

fffff

xxx

xxxx

2

3322

1233

4322

3211

32 xxxx

xxxx

ffff

ffff

)(2

exp)(:)()( xgi

xfxgxf jiij

0

ijijjiji ixxxxxx :],[

Star product:

which makes theories``noncommutative’’:

: binomial coefficient which can be extended to negative n negative power of differential operator (well-defined !)

Closer look at NC KP hierarchy

)

)

)

3

2

1

x

x

x

)1x

2322 2 uuu

],[222 32223432 uuuuuuu

],[2242 4222234542 uuuuuuuuu

222243223 3333 uuuuuuuu

],[

4

3

4

3)(

4

3

4

1 11yyxyyxxxxxxt uuuuuuuuu

x

uux

:

For m=2

For m=3

Infinite kind of fields are represented in terms of one kind of field

(2+1)-dim.NC KP equation

etc.

),,,( 321 xxxuu

x y t

uu 2

x

x xd:1MH&K.Toda, [hep-th/0309265]

and other NC equations(NC KP hierarchy equations)

(KP hierarchy) (various hierarchies.)

(Ex.) KdV hierarchy

Reduction condition

gives rise to NC KdV hierarchy

which includes (1+1)-dim. NC KdV eq.:

):( 22

2 uBL x

)(4

3

4

1xxxxxt uuuuuu

02

Nx

uNote

: 2-reduction

: dimensional reduction in directions

Nx2

reductions

KP :

KdV :

...),,,,,( 54321 xxxxxu

...),,,( 531 xxxu

x y t : (2+1)-dim.

: (1+1)-dim.x t

l-reduction of NC KP hierarchy yields wide class of other NC hierarchies

No-reduction NC KP 2-reduction NC KdV 3-reduction NC Boussinesq 4-reduction NC Coupled KdV … 5-reduction … 3-reduction of BKP NC Sawada-Kotera 2-reduction of mKP NC mKdV Special 1-reduction of mKP NC Burgers …

),,(),,( 321 xxxtyx ),(),( 31 xxtx

),(),( 21 xxtx

Noncommutativity should be introduced into space-time coords

Exact N-soliton solutions of the NC KP hierarchy

1

1

2 xx

x

u

L

1,11 ),,...,(:

NNN fffWf

),(exp),(exp iiii xaxf

solves the NC KP hierarchy !

quasi-determinantof Wronski matrix

Etingof-Gelfand-Retakh, [q-alg/9701008]

33

221),( xxxx

),,(detlog2)(2 120

1

1Nx

N

iiixx ffWWWu

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

m

mx

mx

mx

mxxx

m

m

fff

fff

fff

fffW

12

11

1

21

21

21 ),,,(

Wronski matrix:

( l-reduction condition)li

li

Quasi-determinants Quasi-determinants are not just a 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

Quasi-determinants Defined inductively as follows

jijjij

ijiiij

jijjji

ijiiijij

xXxx

xXxxX

,

1

,

1

)(

))((

31

123

12232331331

133

132222312

211

221

23333213211

321

332322121111

121

11212222111

12222121

221

21111212211

22121111

)()(

)()(:3

,,

,,:2

:1

xxxxxxxxxxxx

xxxxxxxxxxxxxXn

xxxxXxxxxX

xxxxXxxxxXn

xXn ijij

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

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

We have found exact N-soliton solutions for the wide class of NC hierarchies.

Physical interpretations are non-trivial because when are real, is not in general. However, the solutions could be real in some cases.

– (i) 1-soliton solutions are all the same as commutative ones because of

– (ii) In asymptotic region, configurations of multi-soliton solutions could be real in soliton scatterings and the same as commutative ones.

)(),( xgxf

Interpretation of the exact N-soliton solutions

)(*)( xgxf

)()()(*)( vtxgvtxfvtxgvtxf

MH [hep-th/0610006]

2-soliton solution of KdV

each packet has the configuration:

Scattering process (commutative case)

The shape and velocity is preserved ! (stable)

The positions are shifted ! (Phase shift)

22322)( 2,4),4(cosh2 iiiiiiii khkvtkxkku

velocity height

2-soliton solution of NC KdV

each packet has the configuration:

Scattering process (NC case)

The shape and velocity is preserved ! (stable)

The positions are shifted ! (Phase shift)

22322)( 2,4),4(cosh2 iiiiiiii khkvtkxkku

velocity height

In general, complexUnknown in the middle region

Asymptotically realand the same as commutative configurations

Asymptotically

Infinite conserved densities for the NC KP hierarchy. (n=1,2,…, ∞)

)()( )1(

1

0 01

mki

nl

lkx

m

k

k

l

imnn LresLresLres

l

k

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

mxt

)(2

1

)!12(

)1()(:)()(

2

0

xgs

xfxgxfs

jiij

s

s

0

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

:nrLres coefficient of in

rx

nL

This suggests infinite-dimensionalsymmetries would be hidden.

We can calculate the explicit forms of conserved densities for the wide class

of NC soliton equations. Space-Space noncommutativity:

NC deformation is slight:

involutive (integrable in Liouville’s sense) Space-time noncommutativity

NC deformation is drastical:– Example: NC KP and KdV equations

))()((3 22311 uLresuLresLres nnnn

)],([ ixt

meaningful ?

nn Lres 1

5. Conclusion and Discussion

Confirmation of NC Ward’s conjecture – NC twistor theory geometrical origin – D-brane interpretations applications to physics

Completion of NC Sato’s theory– Existence of ``hierarchies’’ – Existence of infinite conserved quantities

infinite-dim. hidden symmetry? – Construction of multi-soliton solutions– Theory of tau-functions description of the symmetry

in terms of infinite-dim. algebras.

Solved!

Solved!

Successful

Successful

Near at hand ?

Work in progress [NC book of Mason&Woodhouse ?]