TOWARDS Non-Commutative (=NC) integrable systems and soliton theories. Masashi HAMANAKA (Nagoya University, Dept. of Math. )

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1 TOWARDS No-Commtatve NC tegrable systems ad solto theores Masash HAMANAKA Nagoya Uversty Dept. o Math. Based o Math. Phys. Semar York o Oct th MH JMP [hep-th/006] MH PB [hep-th/0507] c. MH ``NC soltos ad tegrable systems Proc. o NCGP00 [hep-th/05000]

2 Sccessl pots NC theores Appearace o ew physcal obects Descrpto o real physcs Varos sccessl applcatos to D-brae dyamcs etc.

3 Itegrable eqatos dverse dmesos At-Sel-Dal Yag-Mlls eq. statos Bogomol y eq. moopoles KP eq. BCS eq. DS eq. ~ F µν F µν KdV eq. Bossesq eq. NS eq. Brgers eq. se-gordo eq. ae Toda eld eq.

4 Ward s observato: Almost all tegrable eqatos are redctos o the ASDYM eqs. ASDYM eq. Redctos KP eq. BCS eq. Ward s chral model KdV eq. Bossesq eq. NS eq. mkdv eq. se-gordo eq. Brgers eq. Almost all!?

5 NC Ward s observato: Almost all NC tegrable eqatos are redctos o the NC ASDYM eqs. NC ASDYM eq. NC Redctos Sccessl NC KP eq. NC BCS eq. NC Ward s chral model NC KdV eq. NC Bossesq eq. NC NS eq. NC mkdv eq. NC se-gordo eq. NC Brgers eq. Almost all!? Redctos Sccessl?

6 Program o NC eteso o solto theores Cormato o NC Ward s coectre NC twstor theory geometrcal org D-brae terpretatos applcatos to physcs Completo o NC Sato s theory Estece o ``herarches varos solto eqs. Estece o te coserved qattes te-dm. hdde symmetry Costrcto o mlt-solto soltos Theory o ta-ctos strctre o the solto spaces ad the symmetry

7 Bre otes o how to get NC eqatos NC spaces: θ ] [ θ NC eteso s realzed by replacg prodcts o elds wth star-prodcts: Star-prodcts: Some eamples o NC tegrable eqs. NC KP: NC Brgers: θ ] : [ ep : g g r s θ ] [ yy yy t t Orderg o o-lear terms ad addtoal terms are determed to preserve tegrable-lke propertes. We dscss later. g g θ t ] [ θ θ t y ] [ ] [

8 Pla o ths talk. Itrodcto. Revew o solto theores. NC Sato s theory. Coservato aws 5. Eact Soltos ad Ward s coectre 6. Coclso ad Dscsso

9 . Revew o Solto Theores KdV eqato : descrbe shallow water waves k / k k k cosh k k t & 6 0

10 Hrota s method [PR7979] et s solve t ow! & 6 0 log τ τ & τ τ & τ τ τ τ τ ττ 0 τ kωt e ω k k cosh k k t

11 -solto solto τ θ θ θ θ θ A e k k A e t B BA k k A e k k A determat o Wrosk matr geeral property o solto sols. ``ta-ctos

12 There are may other solto eqs. wth smlar terestg propertes KP eqato -dm. KdV eqato : descrbe -dm shallow water waves t 6 yy 0 : : d y 0 Sato s theorem: [M.Sato & Y.Sato 98] The solto space o KP eq. s a te-dm. Grassma md. determed by ta-cs. May other solto eqs. are obtaed rom KP.

13 . NC Sato s Theory Sato s Theory : oe o the most beatl theory o soltos Based o the estece o herarches ad ta-ctos Sato s theory reveals essetal aspects o soltos: Costrcto o eact soltos Strctre o solto spaces Ite coserved qattes Hdde te-dm. symmetry et s dscss NC eteso o Sato s theory

14 Dervato o solto eqatos Prepare a a operator whch s a psedoderetal operator Itrodce a deretal operator Dee NC KP herarchy: : ] [ B m m 0 : B m θ ] [ k k m tmes m m m m m m

15 Negatve powers o deretal operators : 0 o o o o o ep : g g r s θ θ ] : [

16 M M Closer look at NC KP herarchy [ ] [ ] 5 t yy [ yy ] : : d etc. y t

17 KP herarchy varos herarches. E. KdV herarchy Redcto codto B : gves rse to NC KdV herarchy whch cldes -dm. NC KdV eq.: t N 0 N y t t

18 No-redcto NC KP -redcto NC KdV l-redcto o NC KP herarchy yelds wde class o other NC herarches -redcto NC Bossesq -redcto NC Copled KdV 5-redcto -redcto o BKP NC Sawada-Kotera -redcto o mkp NC mkdv Specal -redcto o mkp NC Brgers y t t t

19 Coservato laws: Q Q : t Q space dσ. Coservato aws tσ J σ dtσ spatal dsj space ty Coservato laws or the herarches res r t m : mres J θ r Ξ 0

20 Ite coserved destes or the NC solto eqs. σ m k m k k l res θ res l l k 0 l 0 res k m t m resr g : s 0 : r s s r θ s! s g Ths sggests te-dmesoal symmetres wold be hdde.

21 We ca calclate the eplct orms o coserved destes or the wde class o NC solto eqatos. Space-Space ocommtatvty: NC deormato s slght: voltve tegrable ovlle s sese Space-tme ocommtatvty NC deormato s drastcal: Eample: NC KP ad KdV eqatos σ σ θ res res res res [ t ] θ

22 5. Eact Soltos ad Ward s coectre We have od eact N-solto soltos or the wde class o NC herarches. -solto soltos are all the same as commtatve oes becase o vt* g vt vt g vt Mlt-solto soltos behave almost the same way as commtatve oes ecept or phase shts. Nocommtatvty aects the phase shts

23 Eact mlt-solto soltos o the NC solto eqs. ΦΦ Φ y Q ep ω t k : W y... y N N N ep ξ α a ep θ ω k ep ω t ξ α α α α k ep ω ω t ep ξ β θ ω k k k Eactly solvable!

24 NC Brgers herarchy NC -dm. Brgers eqato: [ t ] θ & & τ τ θ 0 τ τ log τ τ N l e k l t e ± k l N l e kl θ e l k t± k l

25 6. Coclso ad Dscsso Cormato o NC Ward s coectre NC twstor theory geometrcal org D-brae terpretatos applcatos to physcs Work progress Completo o NC Sato s theory Estece o ``herarches Estece o te coserved qattes te-dm. hdde symmetry Costrcto o mlt-solto soltos Solved! Theory o ta-ctos descrpto o the symmetry ad the solto soltos Gog well Talk at th NBMPS Dham o Nov.5 Sccessl Sccessl Work progress