Noncommutative Solitonsand Integrable Systems

Masashi HAMANAKANagoya University, Dept. of Math.

(visiting Oxford for one year)EMPG Seminar at Heiriot-Watt on Oct 27th

Based on

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]

1. IntroductionSuccessful points in NC theories

Appearance of new physical objectsDescription of real physicsVarious successful applicationsto D-brane dynamics etc.

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(Successful)

NC extension(This talk)

NC extension (This talk)

Ward’s observation: Almost all integrable equations are

reductions of the ASDYM eqs.R.Ward, Phil.Trans.Roy.Soc.Lond.A315(’85)451

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

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

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

reductions of the NC ASDYM eqs.MH&K.Toda, PLA316(‘03)77[hep-th/0211148]

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!!!

Reductions

Now it is time to study from more comprehensive framework.

Program of NC extension of soliton theories

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

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

infinite-dim. hidden symmetryConstruction of multi-soliton solutionsTheory of tau-functions structure of the solution spaces and the symmetry

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

Plan of this talk

1. Introduction2. Review of soliton theories3. NC Sato’s theory4. Conservation Laws5. Exact Solutions and Ward’s conjecture6. Conclusion and Discussion

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

water

24k22k

k/1Experiment by Scott-Russel,

1834

u

xwater tank

solitary wave = solitonThis configuration satisfies

)4(cosh2 322 tkkxku −= −

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

This is a typical integrable equation.

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 ?

3)(2 4,1 ke tkx =+= − ωτ ωA solution:

)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

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

The shape and velocityis preserved ! (stable)

The positions are shifted ! (Phase shift)

There are many other soliton eqs.with (similar) interesting properties

KP equation (2-dim. KdV equation): describe 2-dim shallow water waves

Sato’s theorem: [M.Sato & Y.Sato, 1981]

The solution space of KP eq. is an infinite-dim.Grassmann mfd. (determined by tau-fcns.)

Many other soliton eqs. are obtained from KP.

06 1 =∂+++ −yyxxxxxt uuuuu

xuux ∂∂

=: ∫ ′=∂− x

x xd:1 etc.

KdVKP0=∂y

[See e.g. The book of Miwa-Jimbo-Date (Cambridge UP, 2000)]

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

Based on the exsitence of hierarchies and tau-functions

Sato’s theory reveals essential aspects of solitons:

Construction of exact solutionsStructure of solution spacesInfinite conserved quantitiesHidden infinite-dim. symmetry

Let’s discuss NC extension of Sato’s theory

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

: binomial coefficientwhich can be extendedto negative n

negative power of

differential operator(well-defined !)

ffff

fffff

xxx

xxxx

′′+∂′+∂=∂

′′′+∂′′+∂′+∂=∂

2

3322

1233

o

o

Lo

Lo

−∂′′+∂′−∂=∂

−∂′′+∂′−∂=∂−−−−

−−−−

4322

3211

32 xxxx

xxxx

ffff

ffff

)(2

exp)(:)()( xgixfxgxf jiij ⎟

⎠⎞

⎜⎝⎛ ∂∂=∗

rsθStar product:

which makes theories``noncommutative’’:ijijjiji ixxxxxx θ=∗−∗=∗ :],[

Closer look at NC KP hierarchyFor m=2

2322 2 uuu ′′+′=∂

M

)

)

)

3

2

1

−

−

−

∂

∂

∂

x

x

x

∗+′∗+′′+′=∂ ],[222 32223432 uuuuuuu

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

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

uux ∂∂

=:uu ≡2

MH&K.Toda, [hep-th/0309265]∫ ′=∂− x

x xd:1For m=3

etc.M

)1−∂ x 222243223 3333 uuuuuuuu ′∗+∗′+′′+′′+′′′=∂

∗−− ∂+∂+∗+∗+= ],[

43

43)(

43

41 11

yyxyyxxxxxxt uuuuuuuuu(2+1)-dim.NC KP equation

),,,( 321 Lxxxuu =

x y t

and other NC equations(NC KP hierarchy equations)

(KP hierarchy) (various hierarchies.)reductions

(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

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 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

spacetime

time space

should be space or time derivativeordinary conservation laws !

j∂mxt ≡

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

5. Exact Solutions and Ward’s conjecture

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 −−=−−

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 +=

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

solves the NC Lax hierarchy !quasi-determinantof Wronski matrix

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

jiij kωθ

L+++= 33

221),( ααααξ xxxx

Exactly solvable!

Quasi-determinantsDefined inductively as follows [For a review, see

Gelfand et al.,math.QA/0208146]∑

′′′

−

′′′−=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

⋅⋅⋅−⋅−⋅⋅⋅−⋅−

⋅⋅⋅−⋅−⋅⋅⋅−⋅−==

⋅⋅−=⋅⋅−=

⋅⋅−=⋅⋅−==

==

−−−−

−−−−

−−

−−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∂∂∂

∂∂∂=

−−−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

&

&

Legare, [hep-th/0012077]

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

FurtherReduction: ⎟⎟

⎠

⎞⎜⎜⎝

⎛∗−′′∗

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

=qqq

qqqiCiB

qq

A ,10

012

,0

0

NOT traceless0

0220

)( =⎟⎟⎠

⎞⎜⎜⎝

⎛∗∗−′′+

∗∗−′′−⇒

qqqqqiqqqqqi

ii&

&

qqqqqi ∗∗+′′= 2& : NC NLS eq. !!![MH, PLB625,324]

)2()2(,, 0 suuCBA ⎯⎯→⎯∈ →θNote: U(1) part is necessary !

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

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

0],[)(

uCBACBBCii

ABAi

∈=+′−

=+

∗

∗

&

&

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

)(∞≅ushould remainFurtherReduction: uCuuBA =−′== ,,0 2

⇒)(ii uuuu ′∗+′′= 2&

: NC Burgers eq. !!!

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

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

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

0],[)(

0],[)(0)(

=+−′

=++′

=′

∗

∗

BCBAiii

CAACiiBi

&

&

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

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

FurtherReduction:

qqqqii ′∗′+′′′=⇒=⎟⎟⎠

⎞⎜⎜⎝

⎛⊕−⊗

⊕⇒

43

410

0)( &

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

′∗−′′−′′′′′′

′−∗′+′′=

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛−+′−

=

qqqqqqqf

qqqqC

Bqqq

qA

21),,,(

21

,0100

,1

2

: NC pKdV eq. !!!

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

NOT Traceless !

qu ′= NC KdV

Note:

6. 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 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

Work in progress

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

Talk at 13th NBMPS in Duham on Nov.5

6. Conclusion and Discussion

There are still manythings to be seen.

Welcome !