Relativity Seminar at Oxford Non-Commutative Solitons and 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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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]
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
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 !
(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 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
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
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
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.