Noncommutative Solitons and Integrable Systems

1. 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:0709.2069 (&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) 052701…

2. 1. Introduction 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. 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] )

3. Ward’s conjecture: Many (perhaps all?) integrable equations are reductions of the ASDYM eqs. ASDYM eq. is a master eq. ! Solution Generating Techniques Twistor Theory ASDYM Infinite gauge group Yang’s form DS Ward’s chiral KP CBS (affine) Toda Zakharov KdV mKdV sine-Gordon gauge equiv. gauge equiv. NLS pKdV Liouville Boussinesq N-wave Tzitzeica

4. NC Ward’s conjecture: Many (perhaps all?) NC integrable eqs are reductions of the NC ASDYM eqs. NC ASDYM eq. is a master eq. ? In gauge theory, NC magnetic fields Solution Generating Techniques NC Twistor Theory NC ASDYM New physical objects Application to string theory NC DS NC Ward’s chiral Reductions NC KP NC CBS NC (affine) Toda NC Zakharov NC KdV NC mKdV NC sine-Gordon NC NLS NC pKdV NC Liouville NC Boussinesq NC N-wave NC Tzitzeica

5. 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

6. 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.)

7. 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: e.g. :ASDYM equation

8. 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

9. (Q) How we get NC version of the theories? (A) We have only to replace all products of fields in ordinary commutative gauge theories withstar-products: • The star product: (NC and associative) A deformed product Note:coordinates and fields themselves are usual c-number functions. But commutator of coordinates becomes… Presence of background magneticfields NC !

10. Here we discuss G=GL(N) NC ASDYM eq. from the viewpoint of linear systems with a spectral parameter. (All products are star-products.) • Linear systems(NC case): • Compatibility condition of the linear system: e.g. :NC ASDYM equation

11. 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

12. 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:

13. 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 thecombined 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).

14. Backlund trf. forNC 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]

15. 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

16. [For a review, see Gelfand et al., math.QA/0208146] Quasi-determinants • Defined inductively as follows convenient notation

17. Explicit Atiyah-Ward ansatz solutions ofNC Yang’s eq. G=GL(2) Yang’s matrix J is also beautiful. [Gilson-MH-Nimmo. arXiv:0709.2069]

18. 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

19. 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…]

20. 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:

21. 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.

22. 4. Reduction 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)

23. MH, PLB625, 324 [hep-th/0507112] Reduction to NC KdV eq. The reduced NC ASDYM is: • (1) Take a dimensional reduction and gauge fixing: • (2) Take a further reduction condition: NOT traceless ! We can get NC KdV eq. in such a miracle way ! , Note: U(1) part is necessary !

24. 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) Etingof-Gelfand-Retakh, MRL [q-alg/9701008] MH, JHEP [hep-th/0610006] cf. Paniak, [hep-th/0105185] :quasi-determinant of Wronski matrix

25. 5. Conclusion and Discussion NC ASDYM eq. is a master eq. ! Going well! Going well! Solution Generating Techniques NC Twistor Theory, [Gilson- MH- Nimmo,…] NC ASDYM [Brain, Hannabuss, Majid, Takasaki, Kapustin et al] Infinite gauge group Yang’s form NC DS NC Ward’s chiral NC KP Going well! MH[hep-th/0507112] [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