Noncommutative Ward’s Conjecture and Integrable Systems

Masashi HAMANAKANagoya University, Dept. of Math.

(visiting here until December)Relativity Seminar in Oxford on Nov. 14th

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]MH, ``NC Backlund transforms,’’ [hep-th/0ymmnnn]

Based on

1. Introduction

Successful points in NC theoriesAppearance 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)

Kadomtsev-Petviashvili (KP) eq.Davey-Stewartson (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 conjecture: Many (perhaps all?) integrable equations are reductions of

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

ASDYM eq.

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

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

Reductions

Almost confirmed by explicit examples !!!

NC Ward’s conjecture: Many (perhaps all?) NC integrable equations are reductions of

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

NC ASDYM eq.

Many (perhaps all?)NC integrable eqs.

NC Reductions

Successful

Successful?

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 integrableaspects 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

spaces and the symmetry

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

Plan of this talk1. Introduction2. NC ASDYM equations (a master equation)3. NC Ward’s conjecture

--- Reduction of NC ASDYM to KdV, mKdV, Tziteica, …

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

viewpoint of linear systems with a spectral parameter .Linear systems (commutative case):

Compatibility condition of the linear system:0],[]),[],([],[],[ ~~

2~~ =+−+= wzwwzzzw DDDDDDDDML ζζ

ζ

⎪⎩

⎪⎨

⎧

=−=−====

⇔0],[],[

,0],[,0],[

~~~~

~~~~

wwzzwwzz

wzwz

wzzw

DDDDFFDDFDDF

.0)(,0)(

~

~

=−==−=

ψζψψζψ

wz

zw

DDMDDL

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛1032

3210

21

~~

ixxixxixxixx

zwwz

:ASDYM equation

e.g.

]),[:( νµµννµµν AAAAF +∂−∂=

Yang’s form and Yang’s equationASDYM eq. can be rewritten as follows

⎪⎩

⎪⎨

⎧

=−=−==∃⇒====∃⇒==

0],[],[0~,0~,~,0],[

0,0,,0],[

~~~~

~~~~~~

wwzzwwzz

wzwzwz

wzwzzw

DDDDFFhDhDhDDFhDhDhDDF

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)(:)()( 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???)

Here we discuss G=GL(N) NC ASDYM eq. from the viewpoint of linear systems with a spectral parameter .ζ

(All products are star-products.)

Linear systems (NC case):

Compatibility condition of the linear system:0],[)],[],([],[],[ ~~

2~~ =+−+= ∗∗∗∗∗ wzwwzzzw DDDDDDDDML ζζ

⎪⎩

⎪⎨

⎧

=−=−====

⇔

∗∗

∗

∗

0],[],[,0],[,0],[

~~~~

~~~~

wwzzwwzz

wzwz

wzzw

DDDDFFDDFDDF

.0)(,0)(

~

~

=∗−=∗=∗−=∗

ψζψψζψ

wz

zw

DDMDDL

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛1032

3210

21

~~

ixxixxixxixx

zwwz

:NC ASDYM equation

e.g.

)],[:( ∗+∂−∂= νµµννµµν AAAAF

Don’t omit even for G=U(1) ))()1(( ∞≅UUQ

Yang’s form and NC Yang’s equationNC ASDYM eq. can be rewritten as follows

⎪⎩

⎪⎨

⎧

=−=−=∗=∗∃⇒===∗=∗∃⇒==

∗∗

∗

∗

0],[],[0~,0~,~,0],[

0,0,,0],[

~~~~

~~~~~~

wwzzwwzz

wzwzwz

wzwzzw

DDDDFFhDhDhDDFhDhDhDDF

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:

⎟⎟⎠

⎞⎜⎜⎝

⎛

∗∗−∗∗−

=−

ABAABBABAJ ~~~~~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 andexact 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

AAAAAAiiAAi

&

⎟⎟⎠

⎞⎜⎜⎝

⎛=

0100

~zAThe reduced NC ASDYM is:

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

′∗−′′−′′′′′′

′−∗′+′′==⎟⎟

⎠

⎞⎜⎜⎝

⎛−∗+′−

=qqqqqqqf

qqqqAOA

qqqqq

A zww

21),,,(

21

,,1

~

We can get NC KdV eq. in such a miracle way ! ,

)(43

41)( uuuuuuiii ′∗+∗′+′′′=⇒ & qu ′= 2 θixt =],[

NOT traceless !

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

The NC KdV eq. has integrable-like properties:

possesses infinite conserved densities:

has exact N-soliton solutions:

))(2)((43

211 uLresuLresLres nnnn ′◊−′′◊+= −−− θσ

:nr Lres coefficient of inr

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

)(21

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

2

0

xgs

xfxgxfs

jiij

s

s

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ∂∂

+−

=◊ ∑∞

=

rsθ

∑=

−∗∂∂=N

iiixx WWu

1

1)(2

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

)),((exp),(exp iiii xaxf αξαξ −+= 3),( αααξ 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!

θixt =],[

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

AAAAAAiiAAi

&

⎟⎟⎠

⎞⎜⎜⎝

⎛=

0100

~zAThe reduced NC ASDYM is:

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

=dbc

Aa

Ap

pA zww 0

,000

,0

1~

We get

∗∗ ′−+′′−=′−−′′=

+′−=−′−=

],[41

21

41,],[

41

21

41

,21

21,

21

21

33

22

ppppdppppc

ppbppa

θixt =],[and )(

43

41)( ppppppppiii ′∗∗+∗∗′−′′′=⇒ & NC mKdV !

NOT traceless !

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

(2) Take a further reduction condition:

),~,(),()~,,~,( wwzxtwwzz +=→

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

⎟⎟⎠

⎞⎜⎜⎝

⎛=

0100

~zA

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

=dbc

Aa

Ap

pA zww 0

,000

,0

1~

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

′∗−′′−′′′′′′

′−∗′+′′==⎟⎟

⎠

⎞⎜⎜⎝

⎛−∗+′−

=qqqqqqqf

qqqqAOA

qqqqq

A zww

21),,,(

21

,,1

~NCKdV:

NCmKdV:

Note: There is a residual gauge symmetry:

⎟⎟⎠

⎞⎜⎜⎝

⎛=∂∗+∗∗→ −−

101

,11

βµµµ ggggAgA

The gauge trf. 22, ppqpq −′=′−=β

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( ωωρ −∗= diagg

We get a reduced Yang’s eq.)exp()~,()~exp( wEzzgwEJ +− ∗∗−=

0)()( ~1

~1 =∂∂−∂∂ −− JJJJ wwzz

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

−

+

010001100

001100010

E

E

0],[)( 1~

1 =∗−∂∗∂ ∗+−

−− gEgEgg zz

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

~

~

~

=∂∗−∂=∂−−=−∗∗∂+−∂∗∂

−−=∗∗−∂+∂∗−∂

ρρωωωωωωωωωωωω

zzz

zzz

zzz

VV

V

))2exp()exp(( ~0 ωωωθ −−=⎯⎯ →⎯ →

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 KPNC DS

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]

NC TzitzeicaNC Boussinesq NC N-wave

4. Towards NC Sato’s TheorySato’s Theory : one of the most beautiful theory of solitons– Based on the exsitence of hierarchies and tau-

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

derived 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. symmetryLet’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+∂+∂+∂+∂= −−− 321 )()(2: xxxx ugufuL

∗=∂∂ ],[ LBxL

mm

0)(: ≥∗∗= LLBm Lijji ixx θ=],[

),,,( 321 Lxxxuu =

Noncommutativityis introduced here:

m times

L+∂∂

+∂∂

+∂∂

−

−

−

34

23

12

xm

xm

xm

u

u

u

L+∂

+∂

+∂

−

−

−

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.

Exact N-soliton solutions of the NC KP hierarchy

L+∂+∂=

Φ∂∗Φ=

−

−

1

1

2 xx

x

uL

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

=ΦNNN fffWf

),(exp),(exp iiii xaxf βξαξ +=

solves the NC KP hierarchy !

quasi-determinantof Wronski matrix

L+++= 33

221),( ααααξ xxxx

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

),,(detlog2)(2 120

1

1Nx

N

iiixx ffWWWu L∂⎯⎯→⎯∗∂∂= →

=

−∑ θ

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

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∂∂∂

∂∂∂=

−−−m

mx

mx

mx

mxxx

m

m

fff

ffffff

fffW

12

11

1

21

21

21 ),,,(

L

MOMM

L

L

L

Wronski matrix:

Quasi-determinantsQuasi-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 θ

)( ijxX = )( ijyY =

some factor

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−−−−−+

==

⇒⎟⎟⎠

⎞⎜⎜⎝

⎛=

−−−−−

−−−−−−−−−

11111

111111111

)()()()(

BCADCABCADBCADBACABCADBAA

XY

DCBA

X

We can also define quasi-determinants recursively

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

Gelfand et al.,math.QA/0208146]

∑

∑

′′′

−

′′′

′′′′′

−′

−=

−=

jijjij

ijiiij

jijjji

ijiiijij

xXxx

xXxxX

,

1

,

1

)(

))((

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

⋅⋅⋅−⋅−⋅⋅⋅−⋅−

⋅⋅⋅−⋅−⋅⋅⋅−⋅−==

⋅⋅−=⋅⋅−=

⋅⋅−=⋅⋅−==

==

−−−−

−−−−

−−

−−

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

Interpretation of the exact N-soliton solutions

We have found exact N-soliton solutions for the wide class of NC hierarchies.Physical interpretations are non-trivial becausewhen 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 )(*)( xgxf

)()()(*)( vtxgvtxfvtxgvtxf −−=−−

MH [hep-th/0610006]

2-soliton solution of KdVeach packet has the configuration:

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

velocity height

Scattering process (commutative case)

The shape and velocityis preserved ! (stable)

The positions are shifted ! (Phase shift)

2-soliton solution of NC KdVeach packet has the configuration:

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

velocity height

Scattering process (NC case)

The shape and velocityis preserved ! (stable)

In general, complexUnknown in the middle region

Asymptotically realand the same as commutative configurations

Asymptotically

The positions are shifted ! (Phase shift)

5. Conclusion and Discussion

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.

Infinite gauge group

NC twistors and N=2 strings[Brain-Hannabuss]