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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]
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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 =],[
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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 !
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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)
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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
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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
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Plan of this talk
1. Introduction2. NC Gauge Theory3. NC Sato’s Theory4. Conservation Laws5. Conclusion and Discussion
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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 !
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(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
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(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
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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
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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
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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
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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!
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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
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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’’:
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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 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.
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(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.
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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)
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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.
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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
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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
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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,…