Noncommutative Solitons and Integrable Systems

Transcription

1 Nocotatve Soltos ad Itegrable Systes Masash HAMANAKA Nagoya Uversty Dept. o Math. vstg Oord or oe year Based o GIST Sear at Glasgow o Oct th MH JMP [hep-th/6] MH PB65 5 [hep-th/57] c. MH ``NC soltos ad tegrable systes Proc. o NCGP [hep-th/5]

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

3 Itegrable eatos dverse desos At-Sel-Dal Yag-Mlls e. ~ statos F µν F µν Bogool y e. oopoles KP e. BCS e. DS e. KdV e. Bosses e. NS e. Brgers e. se-gordo e. ae Toda eld e.

4 Ward s observato: Alost all tegrable eatos are redctos o the ASDYM e. ASDYM es. Redctos KP e. BCS e. Ward s chral odel KdV e. Bosses e. NS e. Toda eld e. se-gordo e. Brgers e. Alost all!?

5 NC Ward s observato: Alost all NC tegrable eatos are redctos o the NC ASDYM es. NC ASDYM e. NC Redctos Sccessl Redctos NC KP e. NC BCS e. NC Ward s chral odel NC KdV e. NC Bosses e. NC NS e. NC Toda eld e. NC se-gordo e. NC Brgers e. Alost all!? Sccessl!!! Now t s te to stdy ro ore coprehesve raework.

6 Progra o NC eteso o solto theores Corato o NC Ward s coectre NC twstor theory geoetrcal org Gog well Talk at th NBMPS Dha o Nov.5 D-brae terpretatos applcatos to physcs Copleto o NC Sato s theory Estece o ``herarches Estece o te coserved attes te-d. hdde syetry Costrcto o lt-solto soltos Solved! Theory o ta-ctos descrpto o the syetry ad the solto soltos Sccessl Sccessl Work progress

7 Pla o ths talk. Itrodcto. Revew o solto theores. NC Sato s theory dervato o NC solto es.. Coservato aws te coserved attes 5. Eact Soltos ad Ward s coectre solvablty ad physcal pctres 6. Coclso ad Dscsso

8 . Revew o Solto Theores KdV eato : descrbe shallow water waves k / k k k cosh k k t & 6

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

10 -solto solto τ θ θ θ θ θ A e k k A e t B BA k k A e k k A deterat o Wrosk atr geeral property o solto sols. ``ta-ctos

11 . NC Sato s Theory Sato s Theory : oe o the ost beatl theory o soltos Based o the estece o herarches ad ta-ctos A set o te solto eatos A set o te blear eatos ters o log τ τ ters o Ite evolto es. whose lows are all cotg Ite coserved attes Plecker ebeddg aps whch dee a te-d. Grassa aold. the solto space Ite desoal syetry

12 Dervato o solto eatos Prepare a a operator whch s a psedoderetal operator Itrodce a deretal operator Dee NC KP herarchy: : ] [ B : B θ ] [ k k tes

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

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

15 KP herarchy varos herarches. E. KdV herarchy Redcto codto B : gves rse to NC KdV herarchy whch cldes -d. NC KdV e.: t N N y t t

16 l-redcto o NC KP herarchy yelds wde class o other NC GD herarches No-redcto NC KP y t -redcto NC KdV t -redcto NC Bosses t -redcto NC Copled KdV 5-redcto -redcto o BKP NC Sawada-Kotera -redcto o KP NC KdV Specal -redcto o KP NC Brgers

17 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 : res J θ r Ξ

18 Ite coserved destes or the NC solto es. σ k k k l res θ res l l k l res k t resr g : s : r s s r θ s! s g Ths sggests te-desoal syetres wold be hdde.

19 We ca calclate the eplct ors o coserved destes or the wde class o NC solto eatos. Space-Space ocotatvty: NC deorato s slght: σ res voltve tegrable ovlle s sese Space-te ocotatvty NC deorato s drastcal: Eaple: NC KP ad KdV eatos σ θ res res res [ t ] θ

20 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 sae as cotatve oes becase o vt* g vt vt g vt Mlt-solto soltos behave alost the sae way as cotatve oes ecept or phase shts. Nocotatvty aects the phase shts

21 Eact lt-solto soltos o the NC solto es. 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!

22 Qas-deterats Deed dctvely as ollows X X : : : X X X X X X W M O M M

23 NC Ward s coectre NC NS e. Redced ASDYM e.: t µ ] [ ] [ B C B A C A A C B & & C B A & & & s C B A θ

24 NC Ward s coectre NC Brgers e. Redced ASDYM e.: µ t A& [ B A] C& B [ B C] A B C & A B C &

25 NC Ward s coectre NC KdV e. Redced ASDYM e.: t µ ] [ ] [ B C B A C A A C B & & & C B A sl gl C B A θ

26 6. Coclso ad Dscsso We derved very wde class o NC solto es. the rae work o NC KP or GD herarches. We proved the estece o te coserved attes ad eact lt-solto sols. or the. We also gave soe eaples o NC Ward s coectre whch garatees physcal pctres. The reslts shows that they stll have very specal propertes thogh they clde te te! dervatves. O corse there are stll ay thgs to be see

27 Copleto o NC Sato s theory Frther drectos Theory o ta-ctos hdde syetry deored ae e algebras? Qas-deterats play crtal roles? Geoetrcal descrptos ro NC eteso o the theores o Krchever Mlase ad Segal-Wlso ad so o. Corato o NC Ward s coectre NC twstor theory D-brae terpretatos applcato to physcs Fodato o Haltoa orals or space-te ocotatvty ovlle s theore Noether s theore etc.