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), ``Backlund tranformations for NC anti-self-dual (ASD) Yang-Mills (YM) eqs. ’’ arXiv: (&08mm.nnnn).  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) …

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. 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 KP NC DS NC BoussinesqNC 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 objectsApplication 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: :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.: :Yang ’ s eq. The solution reproduce the gauge fields as is gauge invariant. The decomposition into and corresponds to a gauge fixing

(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) NC ! A deformed product 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: :NC ASDYM equation e.g. (All products are star-products.)

Yang ’ s form and NC Yang ’ s equation  NC ASDYM eq. can be rewritten as follows If we define Yang ’ s matrix: then we obtain from the third eq.: :NC Yang’s eq. The solution reproduces the gauge fields as

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:

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).

Backlund trf. for NC ASDYM eq.  Let’s consider the combined Backlund trf.  If we take a seed sol., the generated solutions are : NC Atiyah-Ward ansatz sols. Quasideterminants ! (a kind of NC determinants) [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 some factor  We can also define quasi-determinants recursively : the matrix obtained from X deleting i-th row and j-th column

Quasi-determinants  Defined inductively as follows [For a review, see Gelfand et al., math.QA/ ] convenient notation

Explicit Atiyah-Ward ansatz solutions of NC Yang ’ s eq. G=GL(2) Yang ’ s matrix J is also beautiful. [Gilson-MH-Nimmo. arXiv: ]

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. ’’ A seed solution:  NC instantons  NC Non-Linear plane-waves

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 ASD connections and ``NC holomorphic vector bundle ’’ on twistor sp.  Strategy from twistor side to ASDYM side: –(i) Solve Birkhoff factorization problem –(ii) obtain the ASD connections from [Kapustin-Kuznetsov-Orlov, Takasaki, Hannabuss, Lechtenfeld-Popov, Brain-Majid … ]

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:

Origin of the Backlund trfs  The Backlund trfs can be understood as the adjoint 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.

4. R eduction 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: MH, PLB625, 324 [hep-th/ ] The reduced NC ASDYM is: We can get NC KdV eq. in such a miracle way !, 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: coefficient of in : Strachan ’ s product (commutative and non-associative) :quasi-determinant of Wronski matrix MH, JMP46 (2005) [hep-th/ ] Etingof-Gelfand-Retakh, MRL [q- alg/ ] MH, JHEP [hep- th/ ] cf. Paniak, [hep- th/ ] : a pseudo-diff. operator

5. Conclusion and Discussion NC ASDYM NC Ward ’ s chiral NC (affine) Toda NC NLS NC KdV NC sine-Gordon NC Liouville NC Tzitzeica NC KP NC DS NC BoussinesqNC N-wave NC CBS NC Zakharov NC mKdV NC pKdV Yang ’ s form gauge equiv. Infinite gauge group NC ASDYM eq. is a master eq. ! Solution Generating Techniques NC Twistor Theory, Going well! [Brain, Hannabuss, Majid, Takasaki, Kapustin et al] MH[hep-th/ ] [Gilson- MH- Nimmo, … ] [MH NPB 741(06) 368]