Relativity Seminar at Oxford

Non-commutative Solitonsand Integrable Equations

Masashi HAMANAKA(Tokyo U. present)

(Nagoya U. from Feb. 2004)

MH, ``Commuting Flows andConservation Laws for NC Lax Hierarches,’’ [hep-th/0311206]cf. MH,``Noncommutative Solitonsand D-branes,’’ Ph.D thesis [hepth/0303256]

1. Introduction• Noncommutative spaces are

defined by noncommutativityof spatial coordinates:

This looks like CCR in QM:

``space-space uncertainly relation’’

Resolution of singulality( New physical objects)

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

ijji ixx θ=],[

hipq =],[

NC gauge theories(real physics)

Commutative gauge theories in background of magnetic fields

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

• In this contect, NC solitons are (lower-dim.) D-branesAnalysis of NC solitons (easy)

Analysis of D-branesVarious successful applicationse.g. confirmation of Sen’sconjecture on decay of D-branes

NC soliton theories are worthwhile !

Soliton equations indiverse dimensions

4 Anti-Self-Dual Yang-Mills eq.(instantons)

2(+1)

KP eq. BCS eq. 2-dim. AKNS system …

3 Bogomol’nyi eq.(monopoles)

1(+1)

KdV eq. mKdV eqBoussinesq eq.NLS eq. Burgers eq. sine-Gordon eq.Sawada-Kotera 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 integrableequations are reductions of the ASDYM eqs.

ASDYM eq.

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

NLS eq. mKdV eq. sine-Gordon eq.

Burgers eq. …(Almost all ! )

Reductions

e.g. [Mason＆Woodhouse]

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

NC Ward’s observation: Almost all NC integrableequations are reductions of the NC ASDYM eqs.

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

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

Plan of this talk

1. Introduction2. NC Gauge Theory3. NC Sato’s Theory4. Conservation Laws5. Conclusion and Discussion

2. NC Gauge Theory• NC gauge theories are equivalent

to ordinary gauge theories in background of magnetic fields.

• They are obtained from ordinary commutative gauge theories by replacing products of fields with star-products.

• The star product:

)(2

exp)(:)()( xgixfxgxf jiij ⎟

⎠⎞

⎜⎝⎛ ∂∂=∗

rsθ

)()()(2

)()( 2θθ Oxgxfixgxf jiij +∂∂+=

A deformed product

hgfhgf ∗∗=∗∗ )()( Associative

ijijjiji ixxxxxx θ=∗−∗=∗ :],[ NC !

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

• Action

• Eq. Of Motion:

• BPS eq. (=(A)SD eq.)

∫= µνµν FFTrxdS 4

( )( )[ ]µνµνµνµν

µνµνµνµν

FFFFTrxd

FFFFTrxd~2~

~~

24

4

±=

+=

∫∫

m

]),[:( νµµννµµν AAAAF +∂−∂=

0]],[,[ =µνν DDD

µνµν FF ~±= instantons

0,0212211==+ zzzzzz FFF

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

(All products are star products)

• Action

• Eq. Of Motion:

• BPS eq. (=(A)SD eq.)

∫ ∗= µνµν FFTrxdS 4

( )( )[ ]µνµνµνµν

µνµνµνµν

FFFFTrxd

FFFFTrxd~2~

~~

24

4

∗±=

∗+∗=

∗∫∫

m

0]],[,[ =∗∗µνν DDD

µνµν FF ~±= NC instantons

0,0212211==+ zzzzzz FFF

)],[:( ∗+∂−∂= νµµννµµν AAAAF

Don’t omit evenfor G=U(1)

))()1(( ∞≅UUQ

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

kNNk

kk

JIB

××

×

:,:,:2,1

NNA ×:µ

0

0

21

2211

=

=+

zz

zzzz

F

FF

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

ADHM data

Instatntons

1:1 k D0-branes

BPS

N D4-branes

BPS

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

ADHM eq. (G=``U(1)’’)

0],[0],[],[

21

2211

=+=−++ ∗∗∗∗

IJBBJJIIBBBB

⎟⎠⎞

⎜⎝⎛

===ρ

ρα0

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

position size

)(222

2

22

)(

))((2,

)()( −

−

+−=

+−−

= µνµνµν

ν

µ ηρ

ρρ

ηbx

iFbxbxi

A

0→ρASD eq. (G=U(2), C2=-1)

singular

0

0

21

2211

=

=+

zz

zzzz

F

FFM

0=ρ

Small instanton singulality

ADHM construction ofNC BPST instantons (N=2,k=1)

ADHM eq. (G=``U(1)’’)

0],[],[],[

21

2211

=+=−++ ∗∗∗∗

IJBBJJIIBBBB ζ

⎟⎠⎞

⎜⎝⎛

=+==ρ

ζρα02( ),0,,2,12,1 JIB

Size slightly fat?position

µνµ FA , : something smooth0→ρ

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

0

0

21

2211

=

=+

zz

zzzz

F

FFM

0=ρ

Resolution of the singulality

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. symmetry

Is it possible to extend it

to NC spaces ? YES!

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’’

0

34

23

12

)(::

≥

−−−

∗∗=+∂+∂+∂+∂=

LLBuuuL

m

xxxx

L

L

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

2

2323

3

22

2

1

uuuB

uB

B

xx

x

x

′++∂+∂=

+∂=

∂=

Noncommutativity is introduced here

ijji ixx θ=],[

m times

Negative powers of differential operators

jnx

jx

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’’:

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 uu ≡2

MH&K.Toda, [hep-th/0309265]For m=3

M

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

∗−− ∂+∂+∗+= ],[

43

43)(

43

41 11

yyxyyxxxxxt uuuuuuu

xuux ∂∂

=:

NC KP equation

),,,( 321 Lxxxuu =

x y t∫ ′=∂− x

x xd:1 etc.

(KP hie.) (various hies.)

• KdV hierarchyReduction condition

gives rise to NC KdV hierarchywhich includes NC KdV eq.:

):( 22

2 uBL x +∂==

xxxxt uuuu )(43

41

∗+=

02

=∂∂

lxu

Note

: 2-reduction

: dimensional reduction

l -reduction yields other NC hierarchies which include NC Boussinesq, coupled KdV, Sawada-Kotera, mKdV hierarchies and so on.

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

• NC (1+1)-dim. Burgers equation:

uuuu ′∗+′′= 2& : Non-linear&

Infinite order diff. eq. w.r.t. time !

NC Cole-Hopf transformation

)log( 01 τττ θxu ∂⎯⎯→⎯′∗= →−

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

Integrable !

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

4. Conservation Laws

• We have obtained wide class of NC hierarchies and NC (soliton) equations.

• Are they integrable or specialfrom viewpoints of solitontheories? YES !

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

Conservation Laws• Conservation laws:

Conservation laws for the hierarchies

iit J∂=∂ σ

σ∫= spacedxQ :

∫∫ ==∂=∂inity

spatiali

ispace tt JdSdxQinf

0σQ

σ : Conserved density

Then is a conserved quantity.

Follwing G.Wilson’s approach, we have:

ijij

xxn

m JBAJLres Ξ∂+∂=+∂=∂ ∗− θ],[1

troublesome

I have succeeded in the evaluation explicitly !

Noncommutativity should be introduced in space-time directions only.

:nr Lres− coefficient of in

rx−∂ nL

Main 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

kkm

mxm

lnx

lln

nx

n

bB

aL

−∞

=−

−∞

=−

∂+∂=

∂+∂=

∑

∑

1

1

◊ : Strachan’s product

mxt ≡

)(21

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

2

0xg

sxfxgxf

s

jiij

s

s

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ∂∂

+−

=◊ ∑∞

=

rsθ

MH, [hep-th/0311206]

)],([ θixt =Example: NC KP and KdV equations

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

5. Conclusion and Discussion

• We proved the existence of infinite conserved quantities for wide class of NC hierarchies and gave the infinite conserved densities explicitly.

• Our results strongly suggest that infinite-dim. symmetry would be hidden in NC (soliton) equations.What is it ?

theories of tau-functions are needed(via e.g. Hirota’s bilinearization)the completion of NC Sato’s theory

• The interpretation of space-time noncommutativity should be clarified.

• What is the twistor descriptions ?

There are many things to be seen.

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