Seiberg-Witten Theory, Symplectic Forms

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1、UCLA/02/TEP/29Columbia/Math/02Seiberg-WittenTheory,SymplecticForms,andHamiltonianTheoryofSolitons∗EricD’Hokera,I.Kricheverb,andD.H.PhongbaDepartmentofPhysicsUniversityofCalifornia,LosAngeles,CA90095bDepartmentofMathematicsColumbiaUniversity,NewYork,NY10027AbstractThisis

2、anexpandedversionoflecturesgiveninHangzhouandBeijing,onthesymplecticformscommontoSeiberg-Wittentheoryandthetheoryofsolitons.Meth-odsforevaluatingtheprepotentialarediscussed.Theconstructionofnewintegrablemodelsarisingfromsupersymmetricgaugetheoriesarereviewed,includingtw

3、istedCalogero-Mosersystemsandspinchainmodelswithtwistedmonodromyconditions.ApracticalframeworkispresentedforevaluatingtheuniversalsymplecticformintermsofLaxpairs.AsubtledistinctionbetweenaLiealgebraandaLiegrouparXiv:hep-th/0212313v126Dec2002versionofthissymplecticformis

4、clarified,whichisnecessaryinchainmodels.∗ResearchsupportedinpartbytheNationalScienceFoundationundergrantsPHY-98-19686,PHY-0140151,DMS-98-00783,andDMS-01-04621.Contents1Introduction22Seiberg-WittenSolutionsofN=2SuperYang-MillsTheories32.1TheWilsoneffectiveaction...........

5、................32.2TheSeiberg-WittenAnsatzfortheeffectiveprepotential..........42.3Thelogarithmicsingularitiesoftheeffectiveprepotential..........52.3.1ThepureSU(N)Yang-Millstheory..................52.3.2TheSU(N)Yang-Millstheorywithantisymmetricmatter.....92.3.3TheSU(N)Y

6、ang-Millstheorywithsymmetricmatter.......133HypermultipletsintheAdjointRepresentationandTwistedCalogero-MoserSystems143.1TheN=2SUSYYang-Millstheorywithadjointhypermultiplet......143.2TheSU(N)theoryandHitchinSystems...................143.3TheSU(N)theoryandCalogero-Mosers

7、ystems...............153.4GCalogero-MosersystemsandG(1)affineTodasystems..........173.5GTwistedCalogero-Mosersystemsand(G(1))∨affineTodasystems....183.6LaxpairswithspectralparameterforCalogero-Mosersystems.......193.6.1ConstructionoftheLaxpairs.....................193.6.2Th

8、escalinglimitoftheLaxpair...................213.7ExactsolutionforsuperYang-Millswithadjointhypermultiplet.....