Backlund Transformations for NonCommutative NC Integrable Eqs.

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Backlund Transformations for Non-Commutative
(NC) Integrable Eqs.
Masashi HAMANAKA University of Nagoya (visiting
Glasgow until Feb. 13 and IHES Feb13 - Mar13
2009)
Based on

• Claire R.Gilson (Glasgow), MH and
• Jonathan J.C.Nimmo (Glasgow), Backlund trfs
  for NC anti-self-dual (ASD) Yang-Mills (YM)
  eqs. arXiv0709.2069 (to appear in GMJ)
  08mm.nnnn.
• MH, NPB 741 (2006) 368, JHEP 02 (07) 94, PLB 625
  (05) 324, JMP46 (05) 052701

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NC extension of integrable systems
1. Introduction
All variables belong to NC ring, which implies
associativity.

• 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.
  (Wards conjecture)

4-dim. Anti-Self-Dual Yang-Mills Eq.
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4-dim. NC ASDYM eq. (GGL(N))
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Reduction to NC KdV from NC ASDYM
NC ASDYM eq. GGL(2)
Reduction conditions
NC KdV eq.!
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Reduction to NC NLS from NC ASDYM
NC ASDYM eq. GGL(2)
Reduction conditions
NC NLS eq.!
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NC Wards conjecture Many (perhaps all?) NC
integrable eqs are reductions of the NC ASDYM
eqs.
MH K.Toda, PLA316(03)77
In gauge theory, NC magnetic
fields
New physical objects
Application to string theory
Solution Generating Techniques
NC ASDYM
NC Twistor Theory,
Yangs form
Infinite gauge group
NC DS
NC Wards chiral
NC KP
MHhep-th/0507112
Summariized in MH NPB 741(06) 368
NC (affine) Toda
NC CBS
NC Zakharov
NC KdV
NC mKdV
NC sine-Gordon
gauge equiv.
gauge equiv.
NC NLS
NC pKdV
NC Liouville
NC Tzitzeica
NC Boussinesq
NC N-wave
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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

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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 define NC Yangs 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.)

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A derivation of NC ASDYM equations We discuss
GGL(N) NC ASDYM eq. from the viewpoint of NC
linear systems with a (commutative) spectral
parameter .

• Linear systems
• Compatibility condition of the linear system

NC ASDYM eq.
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Yangs form and NC Yangs equation

• NC ASDYM eq. can be rewritten as follows

If we define Yangs J-matrix then we obtain from
the third eq.
NC Yangs eq.
The solution reproduce the gauge fields as
is gauge invariant. The decomposition into
and corresponds to a gauge fixing
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Backlund trf. for NC Yangs eq. GGL(2)

• Yangs J matrix can be reparametrized as follows
• Then NC Yangs eq. becomes
• The following trf. leaves NC Yangs eq. as it is
  

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Backlund transformation for NC Yangs eq.

• Yangs J matrix can be reparametrized as follows
• Then NC Yangs eq. becomes
• Another trf. also leaves NC Yangs eq. as it is

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• Both trfs. are involutive (
  ), but the combined trf. is
  non-trivial.)
• Then we could generate various (non-trivial)
  solutions of NC Yangs eq. from a (trivial) seed
  solution (so called, NC Atiyah-Ward solutions)

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Generated solutions (NC Atiyah-Ward sols.)

• Lets consider the combined Backlund trf.
• Then, the generated solutions are

with a seed solution
Quasideterminants ! (a kind of NC determinants)
Gelfand-Retakh
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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

the matrix obtained from X deleting i-th row
and j-th column
some factor
? We can also define quasi-determinants
recursively
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Quasi-determinants
For a review, see Gelfand et al., math.QA/020814
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• Defined inductively as follows

n1 by n1
n by n
convenient notation
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Explicit Atiyah-Ward ansatz solutions of NC
Yangs eq. GGL(2)
Gilson-MH-Nimmo. arXiv0709.2069
Yangs matrix J (gauge invariant)
Gilson-Gu, GHN
The Backlund trf. is not just a gauge trf. but a
non-trivial one!
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• 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 Jacobis identities and
  a homological relation, Gilson-Nimmos derivative
  formula etc.),
• in other words, NC Backlund trfs are
  identities of quasideterminants.

A seed solution
? NC instantons
? NC Non-Linear plane-waves
(common feature in commutative Backlund in
lower-dim.!)
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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
  correspondence between sol. sp. of ASDYM and NC
  holomorphic vector bundle on a NC twistor
  space.

Kapustin-Kuznetsov-Orlov, Takasaki, Hannabuss,
Lechtenfeld-Popov, Brain-Majid
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NC Penrose-Ward correspondence

• Linear systems of ASDYM NC hol. Vec. bdl

Patching matrix
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Takasaki
We have only to factorize a given patching matrix
into and to get ASDYM fields.
(Birkhoff factorization or Riemann-Hilbert
problem)
ASDYM gauge fields are reproduced
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Origin of NC Atiyah-Ward (AW) ansatz sols.

• n-th AW ansatz for the Patching matrix
• The chasing relation is derived from

OK!
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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!
  

OK!
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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
  trf.

The previous -trf!
1-2 component of
OK!
The previous -trf!
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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 (NC Satos theory) (Next)

Quasi-determinants are important !
cf, NC binary Darboux trf. Saleem-Hassan-Siddiq
Profound relation ?? (via Ward conjecture)

Quasideterminants
might be
a key
Quasi-determinants are important !
Etingof,Gelfand,Retakh, Gilson,Nimmo,Ohta,Li,Soom
an,Tamizhmani,MacFarlane, Dimakis,Muller-Hoissen,
MH,..