Noncommutative Integrable Systems and Quasideterminants. Masashi HAMANAKA University of Nagoya, Dept of Math.
Based on
Claire R.Gilson (Glasgow), MH and
Jonathan J.C.Nimmo (Glasgow), ``Backlund trfs and the Atiyah-Ward ansatz for NC anti-self-dual (ASD) Yang-Mills (YM) eqs.’’ Proc. Roy. Soc. A465 (2009) 2613 [arXiv:0812.1222].
MH, NPB 741 (2006) 368, JHEP 02 (07) 94, PLB 625 (05) 324, JMP46 (05) 052701…
NC extension of integrable systems Matrix generalization Quarternion-valued system Moyal deformation (=extension to NC spaces=presence of magnetic
flux)
plays important roles in QFT a master eq. of (lower-dim) integrable eqs. (Ward’s conjecture)
1. Introduction
4-dim. Anti-Self-Dual Yang-Mills Eq.
NC ASDYM eq. with G=GL(N)NC ASDYM eq. (real rep.)
.1203
,3102
2301 ,
FF
FF
FF
)],[:( AAAAF
0
00
0
2
2
1
1
O
O
(Spell:All products are Moyal products.)
)()()(2
)()(
)(2
exp)(:)()(
2
Oxgxfixgxf
xgi
xfxgxf
�
Under the spell, we geta theory on NC spaces:
ixxxxxx :],[
NC ASDYM eq. with G=GL(N)NC ASDYM eq. (real rep.)
.1203
,3102
2301 ,
FF
FF
FF
)],[:( AAAAF
0
00
0
2
2
1
1
O
O
(Spell:All products are Moyal products.)
0
,0
,0
~~
~~
wwzz
wz
zw
FF
F
F
1032
3210
2
1~
~
ixxixx
ixxixx
zw
wz
(complex rep.)
Reduction to NC KdV from NC ASDYM
qqqqqqqqqqqq
qqqqA
qqq
qA zw
2
1)(
2
1
4
12
1
,1
22
00
00,
01
00~~ wz AA
)~,(),()~,,~,( wwzxtwwzz
)(4
3
4
1uuuuuu qu 2
:NC ASDYM eq. G=GL(2)
:NC KdV eq.!
Reduction conditions
0
,0
,0
~~
~~
wwzz
wz
zw
FF
F
F
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))(
2
1
)!12(
)1()(:)()(
2
0
xgs
xfxgxfs
jiij
s
s
�
N
iiixx WWu
1
1)(2
iiii ffWW,1 ),...,(:
)),((exp),(exp iiii xaxf 3),( txx
:quasi-determinant of Wronski matrix
MH, JMP46 (2005) [hep-th/0311206]
cf. [Jon Nimmo san’s turorial] Etingof-Gelfand-Retakh, MRL [q-alg/9701008]MH, JHEP [hep-th/0610006]cf. Paniak, [hep-th/0105185]
ixt ],[
2)('1 )(2 xxx ufu
L : a pseudo-diff. operator
Reduction to NC NLS from NC ASDYM
)~,(),()~,,~,( wwzxtwwzz
1,0
0zw AA
00
00,
10
011 ~~ wz AA
21
Reduction conditions
:NC ASDYM eq. G=GL(2)
:NC NLS eq.!
0
,0
,0
~~
~~
wwzz
wz
zw
FF
F
F
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 CBSNC Zakharov
NC mKdV
NC pKdV
Yang’s form
gauge equiv.gauge equiv.
Infinite gauge group
Solution Generating Techniques
NC Twistor Theory,
MH[hep-th/0507112]Summariized in [MH NPB 741(06) 368]
In gauge theory,NC magnetic fields
New physical objects Application to string theory
Plan of this talk
1. Introduction
2. Backlund Transforms for the NC
ASDYM eqs. (and NC Atiyah-Ward
ansatz solutions in terms of
quasideterminants )
3. Origin of the Backlund trfs
from NC twistor theory
4. 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 integrability.
We rewrite the NC ASDYM eq. into NC Yang’s equations and give Backlund transformations for it.
The generated solutions are represented 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.)
The proof is in fact very simple!
A derivation of NC ASDYM equations We discuss G=GL(N) NC ASDYM eq. from the viewpoint of NC linear systems with a (commutative)spectral parameter .
Linear systems:
Compatibility condition of the linear system:
0],[]),[],([],[],[ ~~2
~~ wzwwzzzw DDDDDDDDML
0],[],[
,0],[
,0],[
~~~~
~~~~
wwzzwwzz
wzwz
wzzw
DDDDFF
DDF
DDF
:NC ASDYM eq.
.0)(
,0)(
~
~
wz
zw
DDM
DDL
NC Yang’s form and Yang’s equationNC 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 :NC 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
Backlund trf. for NC Yang’s eq. G=GL(2)Yang’s J matrix can be reparametrized as fo
llows
Then NC Yang’s eq. becomes
The following trf. leaves NC Yang’s eq. as it is:
11
11
beb
gbegbfJ
.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
Backlund transformation for NC Yang’s eq.Yang’s J matrix can be reparametrized as fo
llows
Then NC Yang’s eq. becomes
Another trf. also leaves NC Yang’s eq. as it is:
1111
11111
0)()(
)()(:
egbffbge
bfeggefb
fg
eb
be
gfnewnew
newnew
01
10, 00
10 CCJCJ new
11
11
beb
gbegbfJ
.0)()(
,0)()(
,0)()(,0)()(
1~
11~
1~
1~
1
1~
11~
11~
1~
11~
11~
1~
11~
1
wwzzwwzz
wwzzwwzz
wwzzwwzz
ebgfebgfffff
bgfebgfebbbb
febfebbgfbgf
Both trfs. are involutive ( ), but the combined trf. is non-trivial.)
Then we could generate various (non-trivial) solutions of NC Yang’s eq. from a (trivial) seed solution (so called, NC Atiyah-Ward solutions)
idid 00,
]2[]1[]0[
0 JJJ
1][][
1][
1][][][
1][][][
][nnn
nnnnnnn beb
bgebgfJ
11
11~
11~
1~
11~
1
,
,,
,,
:
fbbf
febgfebg
bgfebgfe
newnewz
newww
newz
znew
wwnew
z
1
0 :
fg
eb
be
gfnewnew
newnew
0
Generated solutions (NC Atiyah-Ward sols.) Let’s consider the combined Backlund trf.
Then, the generated solutions are :
]2[]1[]0[
0 JJJ
Quasideterminants !(a kind of NC determinants)
zwwz
DgDeDfDb
iiii
nnnnnnnnnnnn
~,~
,,,
11
1
1][]1[
1
1][]1[
1
11][]1[
1
][]1[
021
)2(01
)1(10
][
nn
n
n
nD
[Gelfand-Retakh]
with a seed solution:
1][][
1][
1][][][
1][][][
][nnn
nnnnnnn beb
bgebgfJ
0, 021
0]0[]0[]0[]0[ gefb
Quasi-determinantsQuasi-determinants are not just a NC
generalization of commutative determinants, but rather related to inverse matrices.
[Def1] For an n by n matrix and the inverse of , quasi-determinant of is directly defined by
)( ijxX )( ijyY
proportional to the det. or ratio of dets.
ijX
X X
: the matrix obtained from X deleting i-th row and j-th column
X
XyX
ij
ji
jiijdet
det
)1(1
[For a review, see Gelfand et al.,math.QA/0208146]
Incommutative limit
Quasi-determinantsQuasi-determinants are not just a NC
generalization of commutative determinants, but rather related to inverse matrices.
[Def1] For an n by n matrix and the inverse of , quasi-determinant of is directly defined by
[Def2(Iterative definition)]
)( ijxX )( ijyY
ijX
ijji
jjij
ijiiij
jijjji
ijiiijij
xxXxxxXxxX
,
1
,
1 )())((
n × nn+1 × n+1 convenient notation
X X
: the matrix obtained from X deleting i-th row and j-th column
X
XyX
ij
jilinmit
ecommutativ
jiijdet
det
)1(1
[For a review, see Gelfand et al.,math.QA/0208146]
Examples of quasi-determinants
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
11111
111111111
)()(
)()(
BCADCABCAD
BCADBACABCADBAAXY
DC
BAX
Cf.
Explicit Atiyah-Ward ansatz solutions of NC Yang’s eq. G=GL(2)
zwwz
ge
fb
iiii
n
n
n
n
n
n
n
n
n
n
n
n
~,~
,
,,
11
1
0
0
][
1
0
0
][
1
0
0
][
1
0
0
][
0~11~00~11~0
1
01
10]1[
1
01
10]1[
1
01
10]1[
1
01
10]1[
,,,
,,,,
zwzwwzwz
gefb
[Gilson-MH-Nimmo, arXiv:0709.2069]
0, 021
0]0[]0[]0[]0[ gefb
Yang’s matrix J (gauge invariant)
The Backlund trf. is not just a gauge trf. but a non-trivial one!
011
1
1
02
20
1
1
110
][
00
01
00010
nn
n
n
n
n
nn
nJ
The proof is in fact very simple!Proof is made simply by using only specia
l identities of quasideterminants. (NC Jacobi’s identities and a homological relation, Gilson-Nimmo’s derivative formula etc.)
In other words, ``NC Backlund trfs are identities of quasideterminants.’’ This is an analogue of the fact in lower-dim. commutative theory: ``Backlund trfs are identities of determinants.’’
Some exact solutions
We could generate various solutions of NC ASDYM eq. from a simple seed solution by using the previous Backlund trf.
0 0
A seed solution:
)~,,~,exp(
''~~1
``1
0
0
wwzzoflinearwwzz
NC instantons (special to NC spaces)
NC Non-Linear plane-waves (new solutions beyond ADHM)
3. Interpretation from NC twistor theory
In this section, we give an origin of all ingredients 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 correspondence between ASD connections and ``NC holomorphic vector bundle’’ on NC twistor space.
[Kapustin-Kuznetsov-Orlov, Takasaki, Hannabuss, Lechtenfeld-Popov, Brain-Majid…]
NC Penrose-Ward correspondenceLinear systems of ASDYM ``NC hol. Vec. bdl’’
.0)(
,0)(
~
~
wz
zw
DDM
DDL
.0~)~
(~~,0~)
~(~~
~
~
wz
zw
DDM
DDL
/1~
1~
),~,~(
wzzwP
),(~)(~
),0,()(
xxh
xxh
Patching matrix
1:1
ASDYM gauge fields are reproduced
We have only to factorize agiven patching matrix into and to get ASDYM fields. (Birkhoff factorization or Riemann-Hilbert problem)
)()();( Oxhx
)~
()(~
)~
;(~ Oxhx
~
[Takasaki]
Origin of NC Atiyah-Ward (AW) ansatz sols.
The n-th AW ansatz for the Patching matrix
The recursion relation is derived from:
i
iin
n
n xwzzwxx
P
)(),~,~();(
);(
0][
,0)();(
,0)();(
~
~
wz
zw
xm
xl
zwwziiii~,~
11
OK!
Origin of NC Atiyah-Ward (AW) ansatz sols.
The n-th AW ansatz for the Patching matrix
The Birkoff factorization leads to:
Under a gauge ( ), this solution coincides with the quasideterminants sols!
i
iin
n
n xwzzwxx
P
)(),~,~();(
);(
0][
1
1,1]1[21
1
1,1]1[2222
1
1,1]1[11
1
1,1]1[1212
1
1,1]1[21
1
1,1]1[2221
1
1,1]1[11
1
1,1]1[1211
~~
~~
~
~
nnnnn
nnnnn
nnn
nnn
DhDhh
DhDhh
DhDhh
DhDhh
1][][
1][
1][][][
1][][][
][
][
1
][
][1][ 1
0
0
1~
nnn
nnnnnn
n
n
n
nn
beb
bgebgf
e
f
b
ghhJ
1
1,1]1[][12
1
1,1]![][21
1
11]1[][11
1
1,1]1[][22
~,
,,~
nnnnnn
nnnnnn
DghDeh
DfhDbh
1~
,0~
1122
2112
hh
hh
1~ P
)()( Oxh )~
()(~ Oxh
021
)2(01
)1(10
][
nn
n
n
nD
OK!
Origin of the Backlund trfs
The Backlund trfs can be understood as the adjoint actions for the Patching matrix:
actually:
The -trf. leads to The -trf. is derived with a singular gauge tr
f.
01
10,
0
10,:,: 010
100
1 CBPCCPBPBP newnew
]1[1
)1(
011
0][0
000:
nn
n
n
n
n
n
n PCBBCP
0
0
0,:
1
1
f
bsBsnew
0
10 CJCJ new
1
0
1
0
121
1
kf
b
e
fh
new
newnew
OK!1~
112
1
1
)(
bgfkfe
bf
zwnew
w
new
)( 2 Okh
0)( ~ zw DDL1-2 component of
The previous -trf!
The previous -trf!0
0
4.Conclusion and Discussion
NC integrable eqs (ASDYM) in higher-dim. ADHM (OK) Twistor (OK) Backlund trf (OK), Symmetry (Next)
NC integrable eqs (KdV) in lower-dims. Hierarchy (OK) Infinite conserved quantities (OK) Exact N-soliton solutions (OK) Symmetry ( ``tau-fcn’’, Sato’s theory ) (Next) Behavior of solitons… (Next)
Quasi-determinants are important !
Quasi-determinants are important !
Profound relation ?? (via Ward conjecture)
Very HOT!