1. 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…

2. 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.

3. NC ASDYM eq. with G=GL(N) NC ASDYM eq. (real rep.) (Spell:All products are Moyal products.) Under the spell, we get a theory on NC spaces:

4. NC ASDYM eq. with G=GL(N) NC ASDYM eq. (real rep.) (Spell:All products are Moyal products.) (complex rep.)

5. Reduction to NC KdV from NC ASDYM :NC ASDYM eq. G=GL(2) :NC KdV eq.! Reduction conditions

6. 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/0311206] cf. [Jon Nimmo san’s turorial] Etingof-Gelfand-Retakh, MRL [q- alg/9701008] MH, JHEP [hep-th/0610006] cf. Paniak, [hep-th/0105185] : a pseudo-diff. operator

7. Reduction to NC NLS from NC ASDYM Reduction conditions :NC ASDYM eq. G=GL(2) :NC NLS eq.!

8. 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 Yang ’ s form 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 objectsApplication to string theory

9. 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

10. 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!

11. 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: :NC ASDYM eq.

12. NC Yang’s form and 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 reproduce the gauge fields as

13. Backlund trf. for NC Yang’s eq. G=GL(2) Yang ’ s J matrix can be reparametrized as follows Then NC Yang’s eq. becomes The following trf. leaves NC Yang’s eq. as it is:

14. Backlund transformation for NC Yang’s eq. Yang ’ s J matrix can be reparametrized as follows Then NC Yang’s eq. becomes Another trf. also leaves NC Yang’s eq. as it is:

15. 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)

16. Generated solutions (NC Atiyah-Ward sols.) Let ’ s consider the combined Backlund trf. Then, the generated solutions are : Quasideterminants ! (a kind of NC determinants) [Gelfand-Retakh] with a seed solution:

17. Quasi-determinants Quasi-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 proportional to the det. or ratio of dets. : the matrix obtained from X deleting i-th row and j-th column [For a review, see Gelfand et al., math.QA/0208 146] In commutative limit

18. Quasi-determinants Quasi-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)] n × nn+1 × n+1 convenient notation : the matrix obtained from X deleting i-th row and j-th column [For a review, see Gelfand et al., math.QA/0208 146]

19. Examples of quasi-determinants Cf.

20. Explicit Atiyah-Ward ansatz solutions of NC Yang’s eq. G=GL(2) [Gilson-MH-Nimmo, arXiv:0709.2069] Yang ’ s matrix J (gauge invariant) The Backlund trf. is not just a gauge trf. but a non-trivial one!

21. The proof is in fact very simple! Proof is made simply by using only special 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.’’

22. Some exact solutions We could generate various solutions of NC ASDYM eq. from a simple seed solution by using the previous Backlund trf. A seed solution:  NC instantons (special to NC spaces)  NC Non-Linear plane-waves (new solutions beyond ADHM)

23. 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 … ]

24. NC Penrose-Ward correspondence Linear systems of ASDYM ``NC hol. Vec. bdl’’ Patching matrix 1:1 ASDYM gauge fields are reproduced We have only to factorize a given patching matrix into and to get ASDYM fields. (Birkhoff factorization or Riemann-Hilbert problem) [Takasaki]

25. Origin of NC Atiyah-Ward (AW) ansatz sols. The n-th AW ansatz for the Patching matrix The recursion relation is derived from: OK!

26. 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!

27. 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. OK! 1-2 component of The previous -trf!

28. 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 ! Profound relation ?? (via Ward conjecture) Very HOT ！