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cmr10hep-th/0312001 ~hFSU-TPI-12/03.s獑[N cmbx12Sp` ecial GeometryofEuclideanSupersymmetryI:V ectorMultiplets^ٓR cmr71;" Y&"V
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cmti10InstitutdeMath$ematiquesxElieCartan"Universit$eHenriPoinc}'ar$e-NancyI,B.P.239,F-54506V;anduvr}'e-l$es-Nancy,F;r}'ance 9OFChristophTMa9yer,ThomasMohauptandF
rankSaueressig}VThe}'oretisch-PhysikalischesInstitut,F;rie}'drich-Schil ler-UniversitatJena,Max-Wien-Platz1, D-07743Jena,GermanyE (ABSTRA9CT! W*eKconstructthegeneralactionforAbGelianvectormultipletsinrigid4-dimensionalEuclidean(insteadofMinkowskian)
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cmsy10Nr=52supGersymmetry*,li.e.,overspace-timeswithapGositivedeniteinsteadofaLorentzianEmetric.BThetargetmanifoldsforthescalareldsturnouttobGepara-complexmanifoldsendowedwithaparticularkindofspGecialgeometry*,fwhichwecallanespGecialpara-Kahlergeometry*.W*egiveaprecisedenitionanddevelopthemathematicaltheoryofsuchmanifolds.TTherelationtotheane-spGecialKahlermanifoldsappearinginMinkowskian-Nb=2supersymmetryisdiscussed.{PStart-ingD9fromthegeneral5-dimensionalvectormultipletactionweconsiderdimensionalreductionovertimeandfspaceinparallel,
providingadictionarybGetweentheresultingEuclideanandMinkowskiantheories.ThenwereanalyzesupGersymmetryinfourdimensionsandndthatany(para-)holomorphicprepGotentialdeneseasupGersymmetricLagrangian,iprovidedthatweaddaspGecicfour-fermionterm,iwhichcannotbGe̪obtainedbydimensionalreduction.W*eshowthattheEuclideanactionandsupGersymmetrytrans-formations,when
writtenintermsofpara-holomorphiccoGordinates,takeexactlythesameformastheirMinkowskiankcounterparts.fTheappGearanceofapara-complexandcomplexstructureintheEuclideanandMinkowskiantheory*,orespGectively,oistracedbacktopropGertiesoftheunderlyingR-symmetrygroups.Finally*,weindicatehowourworkwillbGeextendedtoothertypGesofmultipletsandtosupGergravityinthe futureandexplaintherelevqanceofthispro 8jectforthestudyofinstantons,solitonsandcosmologicalsolutionsUUinsupGergravityUUandM-theory*.0 ff J="5-:Aa cmr61L|{Y cmr8WJorkXsupp