NC Solitons and Integrable Systems - Kyoto U
Transcript of NC Solitons and Integrable Systems - Kyoto U
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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
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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
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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)
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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 !!!
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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
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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
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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
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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 +∂−∂=
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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
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(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???)
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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
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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.)
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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
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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)
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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 !
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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 =],[
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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 !
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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
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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
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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
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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
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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+∂+∂+∂+∂= −−− 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.
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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:
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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
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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
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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]
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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)
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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)
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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]