Noncommutative Solitonsand Integrable Systems

MH, JMP46 (2005) 052701 [hep-th/0311206]MH, PLB625 (2005) 324 [hep-th/0507112]

cf. MH, ``NC solitons and integrable systems’’Proc. of NCGP2004, [hep-th/0504001]

Based on

Masashi HAMANAKANagoya University, Dept. of Math.

(visiting Oxford for one year)GIST Seminar at Glasgow on Oct 31th

Successful points in NC theories

Appearance of new physical objectsDescription of real physicsVarious successful applicationsto 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)

2(+1)

KP eq. BCS eq. DS eq. …

3 Bogomol’nyi eq.(monopoles)

1(+1)

KdV eq. Boussinesq eq.NLS eq. Burgers eq. sine-Gordon eq. (affine) Toda field eq. …

µνµν FF ~−=

Dim. of space

NC extension (Successful)

NC extension (This talk)

NC extension(Successful)

NC extension(This talk)

Ward’s observation: Almost all integrable equations are

reductions of the ASDYM eqs.

ASDYM eq.

KP eq. BCS eq. Ward’s chiral modelKdV eq. Boussinesq eq.NLS eq. Toda field eq.

sine-Gordon eq. Burgers eq. …(Almost all !?)

Reductions

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

e.g. [The book of Mason&Woodhouse]

NC Ward’s observation: Almost all NC integrable equations are

reductions of the NC ASDYM eqs.

NC ASDYM eq.

NC KP eq. NC BCS eq. NC Ward’s chiral modelNC KdV eq. NC Boussinesq eq.NC NLS eq. NC Toda field eq.

NC sine-Gordon eq. NC Burgers eq. …(Almost all !?)

NC Reductions

Successful

Successful!!!

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

Reductions

Now it is time to study from more comprehensive framework.

Program of NC extension of soliton theories

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

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

infinite-dim. hidden symmetryConstruction of multi-soliton solutionsTheory of tau-functions description of the symmetry and the soliton solutions

Going well

Solved!

Successful

Successful

Talk at 13th NBMPS in Duham on Nov.5

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

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

Work inprogress

Plan of this talk1. Introduction2. Review of soliton theories (fun)3. NC Sato’s theory

(derivation of NC soliton eqs.)4. Conservation Laws

(infinite conserved quantities)5. Exact Solutions and Ward’s conjecture

(solvability and physical pictures)6. Conclusion and Discussion

2. Review of Soliton TheoriesKdV equation : describe shallow water waves

water

24k22k

k/1Experiment by Scott-Russel,

1834

solitary wave = soliton

u

x

)4(cosh2 322 tkkxku −= −This configuration satisfies

06 =′+′′′+ uuuu& : KdV eq. [Korteweg-de Vries,1895]

This is a typical integrable equation.

water tank

Let’s solve it now !Hirota’s method [PRL27(1971)1192]

06 =′+′′′+ uuuu& : naively hard to solve

τlog2 2xu ∂=

043 =′′′′+′′′′−′′′′+′−′ ττττττττττ &&

Hirota’s bilinear relation : more complicated ?

A solution: 3)(2 4,1 ke tkx =+= − ωτ ω

)4(cosh2 322 tkkxku −=→ −: The solitary wave !(1-soliton solution)

2-soliton solution

2

21

213

)(221

22

21

,4

1 2121

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

=−=

+++= +

kkkkBtkxk

eABAeAeA

iiiθ

τ θθθθ

Scattering process

The shape and velocityis preserved ! (stable)

The positions are shifted ! (Phase shift)

= A determinant of Wronski matrix(general propertyof soliton sols.)``tau-functions’’

3. NC Sato’s TheorySato’s Theory : one of the most beautiful theory of solitons

Based on the exsitence of hierarchies and tau-functions

A set of infinitesoliton equations

(in terms of )

A set of infinitebilinear equations

(in terms of )u ττlog2 2

xu ∂=

Infinite evolution eqs. whose flows are all commuting

Pluecker embedding mapswhich define an infinite-dim.Grassmann manifold.

(=the solution space)

Infinite conserved quantities Infinite dimensional symmetry

Derivation of soliton equationsPrepare a Lax operator which is a pseudo-differential operator

Introduce a differential operator

Define NC (KP) hierarchy:

L+∂+∂+∂+∂= −−− 34

23

12: xxxx uuuL

∗=∂∂ ],[ LBxL

mm

0)(: ≥∗∗= LLBm Lijji ixx θ=],[

),,,( 321 Lxxxuu kk =

Noncommutativityis introduced here:

m times

Here all products are star product:

L+∂∂

+∂∂

+∂∂

−

−

−

34

23

12

xm

xm

xm

u

u

u

L+∂

+∂

+∂

−

−

−

34

23

12

)(

)(

)(

xm

xm

xm

uf

uf

uf

Each coefficient yieldsa differential equation.

Negative powers of differential operatorsjn

xjx

j

nx f

jn

f −∞

=

∂∂⎟⎟⎠

⎞⎜⎜⎝

⎛=∂ ∑ )(:

0o

1)2)(1())1(()2)(1(

L

L

−−−−−−

jjjjnnnn

ffff

fffff

xxx

xxxx

′′+∂′+∂=∂

′′′+∂′′+∂′+∂=∂

2

3322

1233

o

o

Lo

Lo

−∂′′+∂′−∂=∂

−∂′′+∂′−∂=∂−−−−

−−−−

4322

3211

32 xxxx

xxxx

ffff

ffff

)(2

exp)(:)()( xgixfxgxf jiij ⎟

⎠⎞

⎜⎝⎛ ∂∂=∗

rsθ

ijijjiji ixxxxxx θ=∗−∗=∗ :],[

Star product:

which makes theories``noncommutative’’:

: binomial coefficientwhich can be extendedto negative n

negative power of

differential operator(well-defined !)

Closer look at NC KP hierarchy

M

)

)

)

3

2

1

−

−

−

∂

∂

∂

x

x

x

M

)1−∂ x

2322 2 uuu ′′+′=∂

∗+′∗+′′+′=∂ ],[222 32223432 uuuuuuu

∗+′′∗−′∗+′′+′=∂ ],[2242 4222234542 uuuuuuuuu

222243223 3333 uuuuuuuu ′∗+∗′+′′+′′+′′′=∂

∗−− ∂+∂+∗+∗+= ],[

43

43)(

43

41 11

yyxyyxxxxxxt uuuuuuuuu

xuux ∂∂

=:

For m=2

For m=3

Infinite kind of fields are representedin terms of one kind of field

(2+1)-dim.NC KP equation

etc.

),,,( 321 Lxxxuu =

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 hierarchyReduction condition

gives rise to NC KdV hierarchywhich includes (1+1)-dim. NC KdV eq.:

):( 22

2 uBL x +∂==

)(43

41

xxxxxt uuuuuu ∗+∗+=

02

=∂∂

Nxu

Note

: 2-reduction

: dimensional reduction in directionsNx2

reductions

KP :

KdV :

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

...),,,( 531 xxxux y t : (2+1)-dim.

: (1+1)-dim.x t

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

No-reduction NC KP 2-reduction NC KdV3-reduction NC Boussinesq4-reduction NC Coupled KdV …5-reduction …3-reduction of BKP NC Sawada-Kotera2-reduction of mKP NC mKdVSpecial 1-reduction of mKP NC Burgers…

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

),(),( 21 xxtx =

Noncommutativity should be introduced into space-time coords

4. Conservation LawsConservation laws:

Conservation laws for the hierarchies

iit J∂=∂ σ

σ∫= spacedxQ :

∫∫ ==∂=∂inity

spatiali

ispace tt JdSdxQinf

0σQ

ijij

xn

m JLres Ξ∂+∂=∂ − θ1

Then is a conserved quantity.

σ : Conserved density

I have succeeded in the evaluation explicitly !

Noncommutativity should be introducedin space-time directions only.

:nr Lres− coefficient

of inrx−∂ nL

should be space or time derivativeordinary conservation laws !

j∂

spacetime

mxt ≡

time space

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

)()( )1(

1

0 01

mki

nl

lkx

m

k

k

l

imnn LresLresLres

lk ∂◊∂⎟⎠⎞

⎜⎝⎛+= +−

−−

= =− ∑∑θσ

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

mxt ≡

)(21

)!12()1()(:)()(

2

0

xgs

xfxgxfs

jiij

s

s

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ∂∂

+−

=◊ ∑∞

=

rsθ

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

:nr Lres coefficient of inr

x∂ 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 noncommutativityNC deformation is drastical:

Example: NC KP and KdV equations))()((3 22311 uLresuLresLres nnn

n ′◊+′◊−= −−− θσ

)],([ θixt =

meaningful ?

nn Lres 1−=σ

We have found exact N-soliton solutions for the wide class of NC hierarchies.1-soliton solutions are all the same as commutative ones because of

Multi-soliton solutions behave in almost the same way as commutative ones except for phase shifts.Noncommutativity affects the phase shifts

)()()(*)( vtxgvtxfvtxgvtxf −−=−−

5. Exact Solutions and Ward’s conjecture

Exact multi-soliton solutions of the NC soliton eqs.

1−Φ∂Φ= xL

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

=ΦNNN fyyWf

))()((exp)exp(

)(exp)(exp)

xkktiki

xktixkti

jijijiij

jjii

+−+−≅

−∗−

ωωωθ

ωωQ [MH, work in progress]

),(exp),(exp iiii xaxy βξαξ +=

solves the NC Lax hierarchy !

The exact solutions are actually N-soliton solutions !Noncommutativity might affect the phase shift by

quasi-determinantof Wronski matrix

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

jiij kωθ

L+++= 33

221),( ααααξ xxxx

Exactly solvable!

Quasi-determinantsDefined inductively as follows

∑′′

′−

′′′−=ji

jjji

ijiiijij

xXxxX,

1)(

Wronski matrix:

L

311

231

22323313311

331

32222312

211

221

23333213211

321

332322121111

121

11212222111

12222121

221

21111212211

22121111

)()(

)()(:3

,,

,,:2

:1

xxxxxxxxxxxx

xxxxxxxxxxxxxXn

xxxxXxxxxX

xxxxXxxxxXn

xXn ijij

⋅⋅⋅−⋅−⋅⋅⋅−⋅−

⋅⋅⋅−⋅−⋅⋅⋅−⋅−==

⋅⋅−=⋅⋅−=

⋅⋅−=⋅⋅−==

==

−−−−

−−−−

−−

−−

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

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∂∂∂

∂∂∂=

−−−m

mx

mx

mx

mxxx

m

m

fff

ffffff

fffW

12

11

1

21

21

21 ),,,(

L

MOMM

L

L

L

NC Ward’s conjecture (NC NLS eq.)Reduced ASDYM eq.: ),( xtx →µ

0],[)(

0],[)(0)(

=+−′

=+−′

=′

∗

∗

BCBAiii

CAACiiBi

&

&

FurtherReduction: ⎟⎟

⎠

⎞⎜⎜⎝

⎛∗−′′∗

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

=qqq

qqqiCiB

qq

A ,10

012

,0

0

00220

)( =⎟⎟⎠

⎞⎜⎜⎝

⎛∗∗−′′+

∗∗−′′−⇒

qqqqqiqqqqqi

ii&

&

qqqqqi ∗∗+′′= 2& : NC NLS eq. !!!

A, B, C: 2 times 2matrices (gauge fields)

Note: )2()2(,, 0 suuCBA ⎯⎯→⎯∈ →θU(1) part is necessary !

NOT traceless

Legare, [hep-th/0012077]

[MH, PLB625,324]

NC Ward’s conjecture (NC Burgers eq.)Reduced ASDYM eq.: ),( xtx →µ

)1(,,,0],[)(

0],[)(

uCBACBBCii

ABAi

∈=+′−

=+

∗

∗

&

&

FurtherReduction: uCuuBA =−′== ,,0 2

⇒)(ii uuuu ′∗+′′= 2&

: NC Burgers eq. !!!

Note: Without the commutators [ , ], (ii) yields:

should remain

MH & K.Toda, JPA36[hep-th/0301213]

uuuuuu ′∗+∗′+′′=& : neither linearizable nor Lax formSymmetric

)(∞≅u

NC Ward’s conjecture (NC KdV eq.)Reduced ASDYM eq.: ),( xtx →µ

0],[)(

0],[)(0)(

=+−′

=++′

=′

∗

∗

BCBAiii

CAACiiBi

&

&

FurtherReduction:

qqqqii ′∗′+′′′=⇒=⎟⎟⎠

⎞⎜⎜⎝

⎛⊕−⊗

⊕⇒

43

410

0)( &

A, B, C: 2 times 2matrices (gauge fields)

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

′∗−′′−′′′′′′

′−∗′+′′=

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛−+′−

=

qqqqqqqf

qqqqC

Bqqq

qA

21),,,(

21

,0100

,1

2

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

: NC pKdV eq. !!!

)2()2(,, 0 slglCBA ⎯⎯→⎯∈ →θ U(1) part is necessary !Note:

NOT Traceless !

qu ′= NC KdV

6. Conclusion and DiscussionWe derived very wide class of NC soliton eqs.in the frame work of NC KP or GD hierarchies.We proved the existence of infinite conserved quantities and exact multi-soliton sols. for them.We also gave some examples of NC Ward’sconjecture, which guarantees physical pictures.The results shows that they still havevery special properties though they includeinfinite (time!) derivatives.Of course there are still many things to be seen

Further directionsCompletion of NC Sato’s theory

Theory of tau-functionshidden symmetry (deformed affine Lie algebras?)

Geometrical descriptions from NC extension of the theories of Krichever, Mulase and Segal-Wilson and so on.

Confirmation of NC Ward’s conjecture NC twistor theory

D-brane interpretations application to physics

Foundation of Hamiltonian formalism for space-time noncommutativity

Liouville’s theorem, Noether’s theorem, etc.

e.g. Kapustin&Kuznetsov&Orlov, Hannabuss, Hannover group,…

Quasi-determinants play crutial roles ?