Non-Commutative Solitonsand Integrable Systems
Masashi HAMANAKA(Nagoya University, Dept. of Math. Sci.)
MH, ``Commuting Flows and Conservation Laws for NC Lax Hierarchies,’’[hep-th/0311206]cf. MH,``NC Solitons and D-branes,’’
Ph.D thesis (2003) [hep-th/0303256]URL: http://www2.yukawa.kyoto-u.ac.jp/~hamanaka
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 singularity( 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
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 !
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)
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]
NC Ward’s observation: Almost all NC integrable equations are
reductions of the NC ASDYM eqs.
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
MH&K.Toda, PLA316(‘03)77[hep-th/0211148]
Reductions
Plan of this talk
1. Introduction2. NC Gauge Theory (Review)3. NC Sato’s Theory4. Conservation Laws5. Conclusion and Discussion
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
(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???)
(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
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
ADHM data kNNkkk JIB ××× :,:,:2,1
1:1
Instantons NNA ×:µ
ASD eq. (G=U(N), C2=-k): N times N PDE
0
0
21
2211
=
=+
zz
zzzz
F
FF
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
Instantons 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
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 singularity
0=ρ
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 singularity
M
0=ρ
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]
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
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
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=∗−∗=∗ :],[
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)
(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
l-reduction yields wide class of other NC hierarchies which include NC Boussinesq,
coupled KdV, Sawada-Kotera, mKdVhierarchies and so on.
• 2-reduction NC KdV• 3-reduction NC Boussinesq• 4-reduction NC Coupled KdV• 5-reduction …• 3-reduction of BKP NC Sawada-Kotera• 2-reduction of mKP NC mKdV• Special 1-reduction of mKP NC Burgers• …
NC Burgers hierarchyMH&K.Toda,JPA36(‘03)11981[hep-th/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)
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:
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
4. Conservation Laws• We have obtained wide class of NC hierarchies
and NC (soliton) equations.
• Are they integrable or specialfrom viewpoints of soliton theories?
Now we show the existence of infinite number of conserved quatities which suggests a hidden infinite-dimensional symmetry.
YES !
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.
Hot 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
m
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]
We can calculate the explicit forms of conserved densities for wide class of NC soliton equations.
• Space-Space noncommutativity: NC deformation is slight:
• Space-time noncommutativityNC deformation is drastical:– Example: NC KP and KdV equations
))()((3 22311 uLresuLresLres nnn ′◊+′◊−= −−− θσ
)],([ θixt =
meaningful ?
nLres 1−=σ
5. Conclusion and Discussion• In this talk, we discussed integrability and various
aspects of NC soliton eqs. in diverse dimensions.• In higher dimensions, we saw how resolutions of
singularity occur and new physical objects appear from the viewpoint of ADHM construction.
• In lower dimensions, we proved the existence of infinite conserved quantities for wide class of NC hierarchies and gave the infinite conserved densities explicitly from the viewpoint of Sato’s theory, which suggests that infinite-dim. symmetry would be hidden in the NC (soliton) equations.
• Of course, there are still many things to be seen.
Successful !
Going well !
Further directions• Completion of NC Sato’s theory
– Hirota’s bilinearization and tau-functionshidden symmetry (deformed affine Lie algebras?)
– Geometrical descriptions from NC extension of the theories of Krichever, Mulase and Segal-Wilson and so on.
• Confirmation of NC Ward’s conjecture – NC twistor theory
– D-brane interpretations physical meanings• Foundation of Hamiltonian formalism with space-time
noncommutativity– Initial value problems, Liouville’s theorem, Noether’s thm,…
τlog2xu ∂=
e.g. Kapustin&Kuznetsov&Orlov, Hannabuss, Hannover group,…
Cf. Date-Jimbo-Kashiwara-Miwa,…
You are welcome !!!