Noncommutative Integrable Systems and Quasideterminants.

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. 1. Introduction 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) 4-dim. Anti-Self-Dual Yang-Mills Eq.

3. NCASDYM 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) Reduction conditions :NC KdV eq.!

6. The NC KdV eq. has integrable-like properties: : a pseudo-diff. operator • possesses infinite conserved densities: • has exact N-soliton solutions: MH, JMP46 (2005) [hep-th/0311206] coefficient of in : Strachan’s product (commutative and non-associative) cf. [Jon Nimmo san’s turorial] Etingof-Gelfand-Retakh, MRL [q-alg/9701008] MH, JHEP [hep-th/0610006] cf. Paniak, [hep-th/0105185] :quasi-determinant of Wronski matrix

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

8. NC Ward’s conjecture: Many (perhaps all?) NC integrable eqs are reductions of the NC ASDYM eqs. In gauge theory, NC magnetic fields New physical objects Application to string theory Solution Generating Techniques NC Twistor Theory, NC ASDYM Infinite gauge group Yang’s form NC DS NC Ward’s chiral NC KP MH[hep-th/0507112] Summariized in [MH NPB 741(06) 368] NC CBS NC (affine) Toda NC Zakharov NC KdV NC mKdV NC sine-Gordon gauge equiv. gauge equiv. NC NLS NC pKdV NC Liouville NC Boussinesq NC N-wave NC Tzitzeica

9. Plan of this talk 1. Introduction 2. Backlund Transforms for the NC ASDYM eqs. (and NC Atiyah-Ward ansatz solutionsin 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 NCYang’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 NCYang’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 : with a seed solution: Quasideterminants ! (a kind of NC determinants) [Gelfand-Retakh]

17. [For a review, see Gelfand et al., math.QA/0208146] 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 In commutative limit : the matrix obtained from X deleting i-th row and j-th column proportional to the det. or ratio of dets.

18. [For a review, see Gelfand et al., math.QA/0208146] 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)] : the matrix obtained from X deleting i-th row and j-th column n+1 × n+1 n × n convenient notation

19. Examples of quasi-determinants Cf.

20. Explicit Atiyah-Ward ansatz solutions ofNC 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 [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

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. The previous -trf! 1-2 component of OK! 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！ Quasi-determinants are important !