Noncommutative Solitons and Integrable Systems Masashi HAMANAKA University of Nagoya, Dept. of Math. Based on C.R.Gilson (Glasgow), MH and J.J.C.Nimmo (Glasgo w), ``Backlund tranformations for NC anti-self-dua l (ASD) Yang-Mills (YM) eqs.’’ arXiv:0709.2069 (&0 8mm.nnnn). MH, ``NC Ward's conjecture and integrable system s,’’ Nucl. Phys. B 741 (2006) 368, and others: JHE P 02 (07) 94, PLB 625 (05) 324, JMP46 (05) 052701…
Noncommutative Solitons and Integrable Systems. M asashi H AMANAKA University of Nagoya, Dept. of Math. Based on. C.R.Gilson (Glasgow), MH and J.J.C.Nimmo (Glasgow) , ``Backlund tranformations for NC anti-self-dual (ASD) Yang-Mills (YM) eqs. ’’ arXiv:0709.2069 (&08mm.nnnn). - PowerPoint PPT Presentation
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Noncommutative Solitons and Integrable Systems
Masashi HAMANAKA University of Nagoya, Dept. of Math.
Based on C.R.Gilson (Glasgow), MH and J.J.C.Nimmo (Glasgow),
MH, ``NC Ward's conjecture and integrable systems,’’ Nucl. Phys. B 741 (2006) 368, and others: JHEP 02 (07) 94, PLB 625 (05) 324, JMP46 (05) 052701…
Successful points in NC theories Appearance of new physical objects Description of real physics (in gauge theory) Various successful applications
to D-brane dynamics etc.
1. Introduction
Construction of exact solitons are important. (partially due to their integrablity)Final goal: NC extension of all soliton theories (Soliton eqs. can be embedded in gauge theories via Ward’s conjecture ! [R. Ward, 1985] )
Ward’s conjecture: Many (perhaps all?) integrable equations are reductions of the ASDYM eqs.
ASDYM
Ward’s chiral
(affine) Toda
NLS
KdV sine-Gordon
Liouville
Tzitzeica
KP DS
Boussinesq N-wave
CBS Zakharov
mKdV
pKdV
Yang’s form
gauge equiv.gauge equiv.
Infinite gauge group
ASDYM eq. is a master eq. !Solution Generating Techniques
Twistor Theory
NC Ward’s conjecture: Many (perhaps all?) NC integrable eqs are reductions of the NC ASDYM eqs.
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 CBS NC Zakharov
NC mKdV
NC pKdV
NC ASDYM eq. is a master eq. ?Solution Generating Techniques
NC Twistor Theory
Reductions
New physical objects Application to string theory
In gauge theory,NC magnetic fields
Plan of this talk
1. Introduction
2. Backlund Transform for the NC ASDYM eqs. (and NC Atiyah-Ward ansatz solutions in terms of quasideterminants )
3. Interpretation from NC twistor theory
4. Reduction of NC ASDYM to NC KdV
(an example of NC Ward’s conjecture)
5. Conclusion and Discussion
2. Backlund transform for NC ASDYM eqs. In this section, we derive (NC) ASDYM eq. from
the viewpoint of linear systems, which is suitable for discussion on integrable aspects.
We define NC Yang’s equations which is equivalent to NC ASDYM eq. and give a Backlund transformation for it.
The generated solutions are NC Atiyah-Ward ansatz solutions in terms of quasideterminants, which contain not only finite-action solutions (NC instantons) but also infinite-action solutions (non-linear plane waves and so on.)
Review of commutative ASDYM equations
Here 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
0],[],[
,0],[
,0],[
~~~~
~~~~
wwzzwwzz
wzwz
wzzw
DDDDFF
DDF
DDF
]),[:( AAAAF
.0)(
,0)(
~
~
wz
zw
DDM
DDL
1032
3210
2
1~
~
ixxixx
ixxixx
zw
wz
:ASDYM equation
e.g.
Yang’s form and Yang’s equation ASDYM eq. can be rewritten as follows
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
0],[],[
.),~~
(0~
,0~
,~
,0],[
.),(0,0,,0],[
~~~~
1~~~~~~~~
1
wwzzwwzz
zzwzwzwz
zzwzwzzw
DDDDFF
etchhAhDhDhDDF
etchhAhDhDhDDF
is gauge invariant. The decomposition into and corresponds to a gauge fixingJ h h
~
(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: (NC and associative)
)()()(2
)()()(2
exp)(:)()( 2
Oxgxfixgxfxg
ixfxgxf
�
ixxxxxx :],[
NC !
A deformed product
)()()()( xgxfxgxf
Presence of background magnetic fields
Note: coordinates and fields themselves are usual c-number functions. But commutator of coordinates becomes…
Here we discuss G=GL(N) NC ASDYM eq. from the viewpoint of linear systems with a spectral parameter .
Linear systems (NC case):
Compatibility condition of the linear system:
0],[)],[],([],[],[ ~~2
~~ wzwwzzzw DDDDDDDDML
0],[],[
,0],[
,0],[
~~~~
~~~~
wwzzwwzz
wzwz
wzzw
DDDDFF
DDF
DDF
)],[:( AAAAF
.0)(
,0)(
~
~
wz
zw
DDM
DDL
1032
3210
2
1~
~
ixxixx
ixxixx
zw
wz
:NC ASDYM equation
e.g.
(All products are star-products.)
0
00
0
2
2
1
1
O
O
Yang’s form and NC Yang’s equation NC ASDYM eq. can be rewritten as follows
0],[],[
0~
,0~
,~
,0],[
0,0,,0],[
~~~~
~~~~~~
wwzzwwzz
wzwzwz
wzwzzw
DDDDFF
hDhDhDDF
hDhDhDDF
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
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:
11
11
beb
bgebgfJ
.0)()(
,0)()(
,0)()(,0)()(
1~
11~
1~
1~
1
1~
11~
11~
1~
11~
11~
1~
11~
1
wwzzwwzz
wwzzwwzz
wwzzwwzz
ebgfebgfffff
bgfebgfebbbb
febfebbgfbgf
11
11~
11~
1~
11~
1
,
,,
,,
:
fbbf
febgfebg
bgfebgfe
newnewz
newww
newz
znew
wwnew
z
We could generate various (non-trivial) solutions
of NC Yang’s eq. from a (trivial) seed solution by using the previous Backlund trf. together with
a simple trf.
(Both trfs. are involutive ( ), but the combined trf. is non-trivial.)
For G=GL(2), we can present the transforms more explicitly and give an explicit form of a class of solutions (NC Atiyah-Ward ansatz).
01
10,: 1
0 CCJCJ new1
0 :
fg
eb
be
gfnewnew
newnew
idid 00,
Backlund trf. for NC ASDYM eq. Let’s consider the combined Backlund trf.
If we take a seed sol.,
the generated solutions are :
1][][
1][
1][][][
1][][][
][nnn
nnnnnnn beb
bgebgfJ
]3[]2[]1[
0 JJJ
NC Atiyah-Ward ansatz sols.
Quasideterminants !(a kind of NC determinants)
zwwz
DgDeDbDf
rrrr
nnnnnnnnnnnn
~,~
,,,
11
1
1][][
1
1][][
1
][][
1
11][][
0, 021
0]1[]1[]1[]1[ gebf
021
)2(01
)1(10
][
nn
n
n
nD
[Gelfand-Retakh]
Quasi-determinants Quasi-determinants are not just a NC 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
)( ijxX )( ijyY
some factor
11111
111111111
)()(
)()(
BCADCABCAD
BCADBACABCADBAAXY
DC
BAX
We can also define quasi-determinants recursively
ijX : the matrix obtained from X deleting i-th row and j-th column
Quasi-determinants Defined inductively as follows
ijji
jjij
ijiiij
jijjji
ijiiijij
xxXxxxXxxX
,
1
,
1 )())((
31
123
12232331331
133
132222312
211
221
23333213211
321
3323221211
31
21
1
3332
2322131211
333231
232221
131211
11
121
1121222221
1211
22111
1222212221
1211
21
221
2111122221
1211
12211
2212112221
1211
11
)()(
)()(
),(:3
,,
,,:2
:1
xxxxxxxxxxxx
xxxxxxxxxxxxx
x
x
xx
xxxxx
xxx
xxx
xxx
Xn
xxxxxx
xxXxxxx
xx
xxX
xxxxxx
xxXxxxx
xx
xxXn
xXn ijij
[For a review, see Gelfand et al.,math.QA/0208146]
convenient notation
Explicit Atiyah-Ward ansatz solutions of NC Yang’s eq. G=GL(2)
zwwz
ge
bf
rrrr
n
n
n
n
n
n
n
n
n
n
n
n
~,~
,
,,
11
1
01
)1(0
][
1
01
)1(0
][
1
01
)1(0
][
1
01
)1(0
][
0~11~00~11~0
1
01
10]1[
1
01
10]1[
1
01
10]2[
1
01
10]2[
,,,
,,,,
zwzwwzwz
gebf
Yang’s matrix J is also beautiful. [Gilson-MH-Nimmo. arXiv:0709.2069]
0, 021
0]1[]1[]1[]1[ gebf
We could generate various solutions of NC ASDYM eq. from a simple seed solution by using the previous Backlund trf.
Proof is made simply by using special identities of quasideterminants (NC Jacobi’s or Sylvester’s identities and a homological relation and so on.),
in other words, ``NC Backlund transformations are identities of quasideterminants.’’
0
)~,,~,exp(
''~~1
``1
0
0
wwzzoflinearwwzz
A seed solution:
NC instantons
NC Non-Linear plane-waves
0
3. Interpretation from NC Twistor theory In this section, we give an origin of the Backlund
trfs. from the viewpoint of NC twistor theory. NC twistor theory has been developed by several
authors What we need here is NC Penrose-Ward corresp
ondence between ASD connections and ``NC holomorphic vector bundle’’ on twistor sp.
Strategy from twistor side to ASDYM side:– (i) Solve Birkhoff factorization problem
Origin of NC Atiyah-Ward ansatz solutions n-th Atiyah-Ward ansatz for the Patching matrix
The Birkoff factorization leads to:
Under a gauge ( ), the solution leads to the quasideterminants solutions:
i
iin
n
n xxx
P
)();(
);(
0][
1
1,1]1[21
1
1,1]1[2221
1
1,1]1[21
1
1,1]1[2222
1
1,1]1[11
1
1,1]1[1211
1
1,1]1[11
1
1,1]1[1212
~
~~
~
~~
nnnnn
nnn
nnnnn
nnn
DhDhh
DhDhh
DhDhh
DhDhh
1][][
1][
1][][][
1][][][
1][
~
nnn
nnnnnn
n
beb
bgebgf
hhJ
1
1][][
1
1][][
1
][][
1
11][][ ,,,
nnnnnnnnnnnn DgDeDbDf
.,0~
2211 etchh
1~P
)()( Oxh )()(~ 1 Oxh
021
)2(01
)1(10
][
nn
n
n
nD
Origin of the Backlund trfs The Backlund trfs can be understood as the adjoi
nt actions for the Patching matrix:
actually:
The -trf. can be generalized as any constant matrix. Then this Backlund trf. would generate all solutions in some sense and reveal the hidden symmetry of NC ASDYM eq.
-trf. is also derived with a singular gauge trf.
01
10,
0
10,:,: 1
10
1 CBPCCPBPBP newnew
]1[1
)1(11
][0
000:
nn
n
n
n
n
n
n PCBBCP
0
0
0,:
1
1
f
bsBsnew
4. Reduction of NC ASDYM to NC KdV
Here, we show an example of reductions of NC ASDYM eq. on (2+2)-dimension to NC KdV eq.
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 and
exact multi-soliton solutions. (integrable-like)
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
AAAAAAii
AAi
01
00~zA
The reduced NC ASDYM is:
qqqqqqqf
qqqqAOA
qqqq
qA zww
2
1),,,(
2
1
,,1
~
We can get NC KdV eq. in such a miracle way ! ,
)(4
3
4
1)( uuuuuuiii qu 2 ixt ],[
)2()2(,, 0 slglCBA Note: U(1) part is necessary !
NOT traceless !
The NC KdV eq. has integrable-like properties:
possesses infinite conserved densities:
has exact N-soliton solutions:
))(2)((4
3211 uLresuLresLres nnn
n
:nrLres coefficient of in
rx nL
: Strachan’s product (commutative and non-associative))(