Non-commutative Solitons and Integrable Equations

Transcription

1 Relatvty Sear at Oord No-cotatve Soltos ad Itegrable Eqatos Masash HAMANAKA Toyo U. preset Nagoya U. ro eb. MH ``Cotg lows ad Coservato aws or NC a Herarches [hep-th/6 c. MH``Nocotatve Soltos ad D-braes Ph.D thess [hepth/56

2 . Itrodcto Nocotatve spaces are deed by ocotatvty o spatal coordates: [ Ths loos le CCR QM: [ q p θ h ``space-space certaly relato Resolto o sglalty New physcal obects e.g. resolto o sall stato sglarty U statos

3 NC gage theores real physcs Cotatve gage theores bacgrod o agetc elds Gage theores are realed o D- braes whch are soltos strg theores I ths cotect NC soltos are lower-d. D-braes Aalyss o NC soltos easy Aalyss o D-braes Varos sccessl applcatos e.g. corato o Se s coectre o decay o D-braes NC solto theores are worthwhle!

4 Solto eqatos dverse desos At-Sel-Dal Yag-Mlls eq. statos ogool y eq. ~ oopoles KP eq. CS eq. -d. AKNS syste KdV eq. KdV eq ossesq eq. NS eq. rgers eq. se-gordo eq. Sawada-Kotera eq.

5 Ward s observato: Alost all tegrable eqatos are redctos o the ASDYM eqs. R.WardPhl.Tras.Roy.Soc.od.A5 855 ASDYM eq. Redctos KP eq. CS eq. KdV eq. ossesq eq. NS eq. KdV eq. se-gordo eq. rgers eq. Alost all! e.g. [Maso&Woodhose

6 NC Ward s observato: Alost all NC tegrable eqatos are redctos o the NC ASDYM eqs. MH&K.TodaPA6 77[hepth/8 NC ASDYM eq. Sccessl Redctos NC KP eq. NC CS eq. NC KdV eq. NC ossesq eq. NC NS eq. NC KdV eq. NC se-gordo eq. NC rgers eq. Alost all!? Sccessl?

7 Pla o ths tal. Itrodcto. NC Gage Theory. NC Sato s Theory. Coservato aws 5. Coclso ad Dscsso

8 . NC Gage Theory NC gage theores are eqvalet to ordary gage theores bacgrod o agetc elds. They are obtaed ro ordary cotatve gage theores by replacg prodcts o elds wth star-prodcts. The star prodct: ep : g g r s θ θ O θ g g h g h g θ : [

9 E. -d. Ecldea GUN Yag-Mlls theory Acto S d d d Eq. O Moto: Tr Tr Tr [ ~ ~ ± ~ ~ : µ Aν ν Aµ [ Aµ Aν ν [ D [ D D ν µ PS eq. ASD eq. ~ ±

10 E. -d. NC Ecldea GUN Yag-Mlls theory All prodcts are star prodcts Acto S d Tr d d Tr Tr Eq. O Moto: PS eq. ASD eq. [ ~ ~ ± ν [ D [ D D ν µ ~ ± ~ ~ : µ Aν ν Aµ [ Aµ Aν QU U

11 ADHM costrcto o statos [ [ [ IJ J J I I N N J I : : : N N A : µ

12 ADHM costrcto o PST statos N [ [ [ IJ J J I I ρ ρ α J I ν µ η ρ ρ ρ η b b b A ρ M ρ

13 ADHM costrcto o NC PST statos N [ [ [ IJ I I J J ζ α ρ I ζ J ρ A µ ρ M ρ

14 . NC Sato s Theory Sato s Theory : oe o the ost beatl theory o soltos ased 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-d. syetry Is t possble to eted t to NC spaces? YES!

15 NC KP Herarchy: [ : : θ [ tes

16 Negatve powers o deretal operators : o o o o o ep : g g r s θ θ : [ Star prodct

17 Closer loo at NC KP herarchy M [ [ 5 MH&K.Toda [hep-th/965 M [ yy yy t : y t d : etc.

18 KP he. varos hes. KdV herarchy Redcto codto : gves rse to NC KdV herarchy whch cldes NC KdV eq.: t l l

19 NC rgers herarchy MH&K.TodaJPA6 98[hepth/ NC -d. rgers eqato: & θ τ τ logτ & τ τ

20 . Coservato aws We have obtaed wde class o NC herarches ad NC solto eqatos. Are they tegrable or specal ro vewpots o solto theores? YES! Now we show the estece o te ber o coserved qattes whch sggests a hdde te-desoal syetry.

21 Coservato aws Coservato laws: σ t Q : J space dσ σ Q t Q dtσ spatal dsj space ty Coservato laws or the herarches res J [ A J θ Ξ res r : r

22 Ma Reslts Ite coserved destes or NC herarchy eqs. l l l l b a res l θ l l l b a t! : g s g s s s r s θ MH [hep-th/6 [ θ t res res res θ σ

23 5. Coclso ad Dscsso We proved the estece o te coserved qattes or wde class o NC herarches ad gave the te coserved destes eplctly. Or reslts strogly sggest that te-d. syetry wold be hdde NC solto eqatos. What s t? theores o ta-ctos are eeded va e.g. Hrota s blearato the copleto o NC Sato s theory The terpretato o space-te ocotatvty shold be clared. What s the twstor descrptos? There are ay thgs to be see.