Relativity Seminar at Oxford Non-Commutative Solitons and Integrable...
Transcript of Relativity Seminar at Oxford Non-Commutative Solitons and Integrable...
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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 and Conservation Laws for NC Lax Hierarches,’’ [hep-th/0311206]cf. MH,``NC Solitons and D-branes,’’
Ph.D thesis (2003) [hep-th/0303256]
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1. Introduction• Non-Commutative (NC) spaces are defined by
noncommutativity of the coordinates:
This looks like CCR in QM:( ``space-space uncertainty relation’’ )
Resolution of singulality( New physical objects)
e.g. resolution of small instanton singularity( U(1) instantons)
ijji ixx θ=],[hipq =],[
θ~
Com. space NC space
ijθ : NC parameter
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NC gauge theories Com. gauge theories in background of
(real physics) magnetic fields
• Gauge theories are realized on D-branes which are solitons in string theories
• In this context, NC solitons are (lower-dim.) D-branesAnalysis of NC solitons Analysis of D-branes
(easy)
Various successful applicationse.g. confirmation of Sen’s conjecture on decay of D-branes
NC extension of soliton theories are worth studying !
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Soliton equations indiverse 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. Sawada-Kotera 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.Reductions
KP eq. BCS eq. KdV eq. Boussinesq eq.
NLS eq. mKdV eq. sine-Gordon eq. Burgers eq. …
(Almost all ! )e.g. [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 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 TheoryHere we discuss NC gauge theory of instantons.
(Ex.) 4-dim. (Euclidean) G=U(N) Yang-Mills theory• Action
• Eq. Of Motion:
• BPS eq. (=(A)SDYM eq.)
∫−= µνµν FFTrxdS 4
21 ( )
( )[ ]µνµνµνµν
µνµνµνµν
FFFFTrxd
FFFFTrxd
~2~41
~~41
24
4
±−=
+−=
∫
∫
m
0]],[,[ =µνν DDD
µνµν FF ~±= instantons
]),[:( νµµννµµν AAAAF +∂−∂=⇔= 0 BPS 2C↔
)0,0(212211==+⇔ zzzzzz FFF
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(Q) How we get NC version of the theories?(A) They are obtained from ordinary commutative
gauge theories by replacing products of fields 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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(Ex.) 4-dim. NC (Euclidean) G=U(N) Yang-Mills theory
(All products are star products)• Action
• Eq. Of Motion:
• BPS eq. (=NC (A)SDYM eq.)
∫ ∗−= µνµν FFTrxdS 4
21 ( )
( )[ ]µνµνµνµν
µνµνµνµν
FFFFTrxd
FFFFTrxd
~2~41
~~41
24
4
∗±−=
∗+∗−=
∗∫
∫
m
0]],[,[ =∗∗µνν DDD
µνµν FF ~±= NC instantons
Don’t omit even for G=U(1)
))()1(( ∞≅UUQ
)],[:( ∗+∂−∂= νµµννµµν AAAAF⇔= 0 BPS 2C↔
)0,0(212211==+⇔ zzzzzz FFF
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ADHM construction of instantons
0],[0],[],[
21
2211
=+=−++ ∗∗∗∗
IJBBJJIIBBBB
ADHM eq. (G=``U(k)’’): k times k matrix eq.
BPS
D-brane’sinterpretationDouglas, Witten
Atiyah-Drinfeld-Hitchin-Manin, PLA65(’78)185
k D0-branesADHM data kNNkkk JIB ××× :,:,:2,1
1:1
Instatntons NNA ×:µ
N D4-branesASD eq. (G=U(N), C2=-k): N times N PDE
0
0
21
2211
=
=+
zz
zzzz
F
FF BPS
String theory is a treasure house of dualities
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ADHM construction of BPST instanton (N=2,k=1)
Final remark: matrices B andcoords. z always appear in pair: z-B
ADHM eq. (G=``U(1)’’)
0],[0],[],[
21
2211
=+=−++ ∗∗∗∗
IJBBJJIIBBBB
0
0
21
2211
=
=+
zz
zzzz
F
FF
⎟⎠⎞
⎜⎝⎛
===ρ
ρα0
( ),0,,2,12,1 JIB
)(222
2
22
)(
))((2,
)()( −
−
+−=
+−−
= µνµνµν
ν
µ ηρ
ρρ
ηbx
iFbxbxi
A
ρ
iα
position
0→
size
ρsingular
ASD eq. (G=U(2), C2=-1)M
Small instanton singulality
0=ρ
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ADHM construction of NC BPST instanton(N=2,k=1)
0],[],[],[
21
2211
=+=−++ ∗∗∗∗
IJBBJJIIBBBB ζ
Nekrasov&Schwarz,CMP198(‘98)689[hep-th/9802068]
ADHM eq. (G=``U(1)’’) 1 times 1 matrix eq.
⎟⎠⎞
⎜⎝⎛
=+==ρ
ζρα02( ),0,,2,12,1 JIB
size slightly fat?position
µνµ FA , : something smooth Regular! (U(1) instanton!)0→ρ
ASD eq. (G=U(2), C2=-1)
0
0
21
2211
=
=+
zz
zzzz
F
FF
Resolution of the singulality
M
0=ρ
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NC monopoles are also interesting• Magnetic flux of Dirac monopoles ``Visible’’
Dirac string(regular !) z z
On commutative space On NC space
yx, yx,
Dirac string(singular)
ADHMN construction works well.Moduli space is the same as commutative one.
Gross&Nekrasov, JHEP[hep-th/0005204]
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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. SymmetryLet’s discuss NC extension of Sato’s theory
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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’’ (m=1,2,3,…)
0
34
23
12
)(::
≥
−−−
∗∗=+∂+∂+∂+∂=
LLBuuuL
m
xxxx
L
L
),,,( 321 Lxxxuu kk =)(33
2
23233
03
222
02
01
uuuLB
uLB
LB
xx
x
x
′++∂+∂==
+∂==
∂==
≥
≥
≥
Noncommutativity is introduced here: ijji ixx θ=],[
m times
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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
M
)1−∂ x 222243223 3333 uuuuuuuu ′∗+∗′+′′+′′+′′′=∂ etc.
(2+1)-dim.NC KP equation∗
−− ∂+∂+∗+∗+= ],[43
43)(
43
41 11
yyxyyxxxxxxt uuuuuuuuu
),,,( 321 Lxxxuu =
x y t
and other NC equations(NC hierarchy equations)
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(KP hierarchy) (various hierarchies.)• (Ex.) KdV hierarchy
Reduction 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 directions
l -reduction yields wide class of other NC hierarchies which include NC Boussinesq, coupled KdV, Sawada-Kotera, mKdV hierarchies and so on.
Nx2
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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 ! (Integrable?)
NC Cole-Hopf transformation
)log( 01 τττ θxu ∂⎯⎯ →⎯′∗= →−
ττ ′′=& : Linear & first order diff. eq. w.r.t. time
(Integrable !)
(NC) Diffusion equation:
NC Burgers eq. can be derived from G=U(1) NC ASDYM eq. (One example of NC Ward’s observation)
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NC Ward’s observation (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
A ,10
012
,0
0
NOT traceless0
0220
)( =⎟⎟⎠
⎞⎜⎜⎝
⎛∗∗+′′+
∗∗−′′−⇒
qqqqqiqqqqqi
ii&
&
qqqqqi ∗∗+′′= 2& : NC NLS eq.U(1) part isimportant),2(),2(,, 0 CslCglCBA ⎯⎯→⎯∈ →θ
Note:
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NC Ward’s observation (NC Burgers eq.)),( xtx →µ MH&K.Toda, JPA
[hep-th/0301213]• Reduced ASDYM eq.:
G=U(1)
0],[)(
0],[)(
=+′−
=+
∗
∗
CBBCii
ABAi&
&A, B, C: 1 times 1matrices (gauge fields)
should remainFurtherReduction: uCuuBA =−′== ,,0 2
uuuu ∗′+′′= 2&⇒)(ii : NC Burgers eq.
Note: Without the commutators [ , ], (ii) yields:
uuuuuu ′∗+∗′+′′=& : neither linearizable nor Lax formsymmetric
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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 soliton theories? YES !
Now we show the existence of infinite number of conserved quatities which suggests a hidden infinite-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
ijij
xxn
m JBAJLres Ξ∂+∂=+∂=∂ ∗− θ],[1
Then is a conserved quantity.
σ : Conserved density
Follwing G.Wilson’s approach, we have:
troublesome
I have succeeded in the evaluation explicitly !
spacetime
:nr Lres−
coefficient of inr
x−∂ nL
should be space or time derivativej∂
Noncommutativity should be introduced in space-time directions only.
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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 bBaL −∞
=−
−∞
=− ∂+∂=∂+∂= ∑∑
11,
◊ : Strachan’s product
mxt ≡
)(21
)!12()1()(:)()(
2
0
xgs
xfxgxfs
jiij
s
s
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ ∂∂
+−
=◊ ∑∞
=
rsθ
MH, [hep-th/0311206]
Example: NC KP and KdV equations )],([ θixt =
))()((3 22311 uLresuLresLres nnn ′◊+′◊−= −−− θσ : meaningful ?
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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
• What is space-time noncommutativity ?• What is the twistor descriptions ? NC twistor theory
There are many things to be seen.
τlog2xu ∂=
e.g. Kapustin&Kuznetsov&Orlov, Hannabuss, Hannover group,…