Non-Commutative Solitonsand Integrable Systems

Masashi HAMANAKA(Nagoya University, Dept. of Math. Sci.)

MH, ``Commuting Flows and Conservation Laws for NC Lax Hierarchies,’’[hep-th/0311206]cf. MH,``NC Solitons and D-branes,’’

Ph.D thesis (2003) [hep-th/0303256]URL: http://www2.yukawa.kyoto-u.ac.jp/~hamanaka

1. Introduction• Non-Commutative (NC) spaces are defined by

noncommutativity of the coordinates:

This looks like CCR in QM:( ``space-space uncertainty relation’’ )

Resolution of singularity( New physical objects)

e.g. resolution of small instanton singularity( U(1) instantons)

ijji ixx θ=],[hipq =],[

θ~

Com. space NC space

ijθ : NC parameter

NC gauge theories Com. gauge theories in background of

(real physics) magnetic fields

• Gauge theories are realized on D-branes which are solitons in string theories

• In this context, NC solitons are (lower-dim.) D-branesAnalysis of NC solitons Analysis of D-branes

(easy)

Various successful applicationse.g. confirmation of Sen’s conjecture on decay of D-branes

NC extension of soliton theories are worth studying !

Soliton equations indiverse 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. Sawada-Kotera 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.Reductions

KP eq. BCS eq. KdV eq. Boussinesq eq.

NLS eq. mKdV eq. sine-Gordon eq. Burgers eq. …

(Almost all ! )e.g. [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 KdV eq. NC Boussinesq eq.

NC NLS eq. NC mKdV eq. NC sine-Gordon eq. NC Burgers eq. …

(Almost all !?)

Reductions

Successful

Successful?Sato’s theory may answer

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

Reductions

Plan of this talk

1. Introduction2. NC Gauge Theory (Review)3. NC Sato’s Theory4. Conservation Laws5. Conclusion and Discussion

2. NC Gauge TheoryHere we discuss NC gauge theory of instantons.

(Ex.) 4-dim. (Euclidean) G=U(N) Yang-Mills theory• Action

• Eq. Of Motion:

• BPS eq. (=(A)SDYM eq.)

∫−= µνµν FFTrxdS 4

21 ( )

( )[ ]µνµνµνµν

µνµνµνµν

FFFFTrxd

FFFFTrxd

~2~41

~~41

24

4

±−=

+−=

∫

∫

m

0]],[,[ =µνν DDD

µνµν FF ~±= instantons

]),[:( νµµννµµν AAAAF +∂−∂=⇔= 0 BPS 2C↔

)0,0(212211==+⇔ zzzzzz FFF

(Q) How we get NC version of the theories?(A) They are obtained from ordinary commutative

gauge theories by replacing products of fields with star-products:

• The star product:

hgfhgf ∗∗=∗∗ )()(

)()()(2

)()()(2

exp)(:)()( 2θθθ Oxgxfixgxfxgixfxgxf ji

ij

jiij +∂∂+=⎟

⎠⎞

⎜⎝⎛ ∂∂=∗

rs

ijijjiji ixxxxxx θ=∗−∗=∗ :],[ NC !

Associative

)()()()( xgxfxgxf

A deformed product

∗→

Presence of background magnetic fields

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

(Ex.) 4-dim. NC (Euclidean) G=U(N) Yang-Mills theory

(All products are star products)• Action

• Eq. Of Motion:

• BPS eq. (=NC (A)SDYM eq.)

∫ ∗−= µνµν FFTrxdS 4

21 ( )

( )[ ]µνµνµνµν

µνµνµνµν

FFFFTrxd

FFFFTrxd

~2~41

~~41

24

4

∗±−=

∗+∗−=

∗∫

∫

m

0]],[,[ =∗∗µνν DDD

µνµν FF ~±= NC instantons

Don’t omit even for G=U(1)

))()1(( ∞≅UUQ

)],[:( ∗+∂−∂= νµµννµµν AAAAF⇔= 0 BPS 2C↔

)0,0(212211==+⇔ zzzzzz FFF

ADHM construction of instantonsAtiyah-Drinfeld-Hitchin-Manin, PLA65(’78)185

ADHM eq. (G=``U(k)’’): k times k matrix eq.

0],[0],[],[

21

2211

=+=−++ ∗∗∗∗

IJBBJJIIBBBB

ADHM data kNNkkk JIB ××× :,:,:2,1

1:1

Instantons NNA ×:µ

ASD eq. (G=U(N), C2=-k): N times N PDE

0

0

21

2211

=

=+

zz

zzzz

F

FF

ADHM construction of instantons

0],[0],[],[

21

2211

=+=−++ ∗∗∗∗

IJBBJJIIBBBB

ADHM eq. (G=``U(k)’’): k times k matrix eq.

BPS

D-brane’sinterpretationDouglas, Witten

Atiyah-Drinfeld-Hitchin-Manin, PLA65(’78)185

k D0-branesADHM data kNNkkk JIB ××× :,:,:2,1

1:1

Instantons NNA ×:µ

N D4-branesASD eq. (G=U(N), C2=-k): N times N PDE

0

0

21

2211

=

=+

zz

zzzz

F

FF BPS

String theory is a treasure house of dualities

ADHM construction of BPST instanton (N=2,k=1)

Final remark: matrices B andcoords. z always appear in pair: z-B

ADHM eq. (G=``U(1)’’)

0],[0],[],[

21

2211

=+=−++ ∗∗∗∗

IJBBJJIIBBBB

0

0

21

2211

=

=+

zz

zzzz

F

FF

⎟⎠⎞

⎜⎝⎛

===ρ

ρα0

( ),0,,2,12,1 JIB

)(222

2

22

)(

))((2,

)()( −

−

+−=

+−−

= µνµνµν

ν

µ ηρ

ρρ

ηbx

iFbxbxi

A

ρ

iα

position

0→

size

ρsingular

ASD eq. (G=U(2), C2=-1)M

Small instanton singularity

0=ρ

ADHM construction of NC BPST instanton(N=2,k=1)

0],[],[],[

21

2211

=+=−++ ∗∗∗∗

IJBBJJIIBBBB ζ

Nekrasov&Schwarz,CMP198(‘98)689[hep-th/9802068]

ADHM eq. (G=``U(1)’’) 1 times 1 matrix eq.

⎟⎠⎞

⎜⎝⎛

=+==ρ

ζρα02( ),0,,2,12,1 JIB

size slightly fat?position

µνµ FA , : something smooth Regular! (U(1) instanton!)0→ρ

ASD eq. (G=U(2), C2=-1)

0

0

21

2211

=

=+

zz

zzzz

F

FF

Resolution of the singularity

M

0=ρ

NC monopoles are also interesting• Magnetic flux of Dirac monopoles ``Visible’’

Dirac string(regular !) z z

On commutative space On NC space

yx, yx,

Dirac string(singular)

ADHMN construction works well.Moduli space is the same as commutative one.

Gross&Nekrasov, JHEP[hep-th/0005204]

3. NC Sato’s Theory• Sato’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 solutions– Structure of solution spaces– Infinite conserved quantities– Hidden infinite-dim. SymmetryLet’s discuss NC extension of Sato’s theory

NC (KP) Hierarchy:

∗=∂∂ ],[ LBxL

mm

L+∂∂

+∂∂

+∂∂

−

−

−

34

23

12

xm

xm

xm

u

u

u

L+∂

+∂

+∂

−

−

−

34

23

12

)(

)(

)(

xm

xm

xm

uF

uF

uF

Huge amount of ``NC evolution equations’’ (m=1,2,3,…)

0

34

23

12

)(::

≥

−−−

∗∗=+∂+∂+∂+∂=

LLBuuuL

m

xxxx

L

L

),,,( 321 Lxxxuu kk =)(33

2

23233

03

222

02

01

uuuLB

uLB

LB

xx

x

x

′++∂+∂==

+∂==

∂==

≥

≥

≥

Noncommutativity is introduced here: ijji ixx θ=],[

m times

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

M

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

(2+1)-dim.NC KP equation∗

−− ∂+∂+∗+∗+= ],[43

43)(

43

41 11

yyxyyxxxxxxt uuuuuuuuu

),,,( 321 Lxxxuu =

x y t

and other NC equations(NC 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 yields wide class of other NC hierarchies which include NC Boussinesq,

coupled KdV, Sawada-Kotera, mKdVhierarchies and so on.

• 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• …

NC Burgers hierarchyMH&K.Toda,JPA36(‘03)11981[hep-th/0301213]

• NC (1+1)-dim. Burgers equation:uuuu ′∗+′′= 2& : Non-linear &

Infinite order diff. eq. w.r.t. time ! (Integrable?)

NC Cole-Hopf transformation

)log( 01 τττ θxu ∂⎯⎯ →⎯′∗= →−

ττ ′′=& : Linear & first order diff. eq. w.r.t. time

(Integrable !)

(NC) Diffusion equation:

NC Burgers eq. can be derived from G=U(1) NC ASDYM eq. (One example of NC Ward’s observation)

NC Ward’s observation (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.U(1) part isimportant),2(),2(,, 0 CslCglCBA ⎯⎯→⎯∈ →θ

Note:

NC Ward’s observation (NC Burgers eq.)),( xtx →µ MH&K.Toda, JPA

[hep-th/0301213]• Reduced ASDYM eq.:

G=U(1)

0],[)(

0],[)(

=+′−

=+

∗

∗

CBBCii

ABAi&

&A, B, C: 1 times 1matrices (gauge fields)

should remainFurtherReduction: uCuuBA =−′== ,,0 2

uuuu ∗′+′′= 2&⇒)(ii : NC Burgers eq.

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

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

4. Conservation Laws• We have obtained wide class of NC hierarchies

and NC (soliton) equations.

• Are they integrable or specialfrom viewpoints of soliton theories?

Now we show the existence of infinite number of conserved quatities which suggests a hidden infinite-dimensional symmetry.

YES !

Conservation Laws• Conservation laws:

Conservation laws for the hierarchies

iit J∂=∂ σ

σ∫= spacedxQ :

∫∫ ==∂=∂inity

spatiali

ispace tt JdSdxQinf

0σQ

ijij

xxn

m JBAJLres Ξ∂+∂=+∂=∂ ∗− θ],[1

Then is a conserved quantity.

σ : Conserved density

Follwing G.Wilson’s approach, we have:

troublesome

I have succeeded in the evaluation explicitly !

spacetime

:nr Lres−

coefficient of inr

x−∂ nL

should be space or time derivativej∂

Noncommutativity should be introduced in space-time directions only.

Hot ResultsInfinite conserved densities for NC

hierarchy eqs. (n=1,2,…)

)1(

1

1

11 1

)1( +−−+−−

=

−++

+=

−−+− ◊∂⎟

⎠⎞

⎜⎝⎛−+= ∑ ∑ −−

− lknmkmiln

m

k

kmn

nl

lknmmin baLresnl

kmθσ

kmx

m

kkm

mxm

lnx

lln

nx

n bBaL −

=−

−∞

=− ∂+∂=∂+∂= ∑∑

11,

◊ : Strachan’s product

mxt ≡

)(21

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

2

0

xgs

xfxgxfs

jiij

s

s

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ∂∂

+−

=◊ ∑∞

=

rsθ

MH, [hep-th/0311206]

We can calculate the explicit forms of conserved densities for wide class of NC soliton equations.

• Space-Space noncommutativity: NC deformation is slight:

• Space-time noncommutativityNC deformation is drastical:– Example: NC KP and KdV equations

))()((3 22311 uLresuLresLres nnn ′◊+′◊−= −−− θσ

)],([ θixt =

meaningful ?

nLres 1−=σ

5. Conclusion and Discussion• In this talk, we discussed integrability and various

aspects of NC soliton eqs. in diverse dimensions.• In higher dimensions, we saw how resolutions of

singularity occur and new physical objects appear from the viewpoint of ADHM construction.

• In lower dimensions, we proved the existence of infinite conserved quantities for wide class of NC hierarchies and gave the infinite conserved densities explicitly from the viewpoint of Sato’s theory, which suggests that infinite-dim. symmetry would be hidden in the NC (soliton) equations.

• Of course, there are still many things to be seen.

Successful !

Going well !

Further directions• Completion of NC Sato’s theory

– Hirota’s bilinearization and 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 physical meanings• Foundation of Hamiltonian formalism with space-time

noncommutativity– Initial value problems, Liouville’s theorem, Noether’s thm,…

τlog2xu ∂=

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

Cf. Date-Jimbo-Kashiwara-Miwa,…

You are welcome !!!