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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]
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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
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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)
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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]
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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.
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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
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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
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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.
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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)
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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)
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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)]
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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
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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.
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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=∗−∗=∗ :],[
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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)
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(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
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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
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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 ≡
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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.
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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
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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 −−=−−
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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!
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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
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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 !
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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
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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:
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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
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6. Conclusion and Discussion
There are still manythings to be seen.
Welcome !