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Compositio Mathematica c Kluwer Academic Publishers Printed in the Netherlands

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18 Pages
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Compositio Mathematica 112: 183–216, 1998. 183 c 1998 Kluwer Academic Publishers. Printed in the Netherlands. The Picard group of the moduli of G-bundles on a curve ARNAUD BEAUVILLE1,? YVES LASZLO1? and CHRISTOPH SORGER2 ?? 1DMI – Ecole Normale Superieure, (URA 762 du CNRS), 45 rue d'Ulm, F-75230 Paris Cedex 05, France; e-mail: , 2Institut de Mathematiques de Jussieu, (UMR 9994 du CNRS), Univ. Paris 7 – Case Postale 7012, 2 place Jussieu, F-75251 Paris Cedex 05, France; e-mail: Received 30 January 1997; accepted in revised form 14 April 1997 Abstract. Let G be a complex semi-simple group, and X a compact Riemann surface. The moduli space of principal G-bundles on X , and in particular the holomorphic line bundles on this space and their global sections, play an important role in the recent applications of Conformal Field Theory to algebraic geometry. In this paper we determine the Picard group of this moduli space when G is of classical or G2 type (we consider both the coarse moduli space and the moduli stack). Mathematics Subject Classifications (1991): Primary: 14H60; Secondary: 14F05, 14L30.

  • standard represen- tation

  • canonical projec- tion

  • group

  • moduli space when

  • over

  • conformal field

  • connected


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Theπ-calculusasatheoryinlinearlogic:PreliminaryresultsDaleMillerComputerScienceDepartmentUniversityofPennsylvaniaPhiladelphia,PA19104-6389USAdale@cis.upenn.eduOctober29,1992AbstractTheagentexpressionsoftheπ-calculuscanbetranslatedintoatheoryoflinearlogicinsuchawaythatthereflectiveandtransitiveclosureofπ-calculus(unlabeled)reductionisidentifiedwith“entailed-by”.Underthistranslation,parallelcompositionismappedtothemultiplicativedisjunct(“par”)andrestrictionismappedtouniversalquantification.Prefixing,non-deterministicchoice(+),replication(!),andthematchguardareallrepresentedusingnon-logicalconstants,whicharespecifiedusingasimpleformofaxiom,calledhereaprocessclause.TheseprocessclausesresembleHornclausesexceptthattheymayhavemultipleconclu-sions;thatis,theirheadsmaybetheparofatomicformulas.Suchmultipleconclusionclausesareusedtoaxiomatizecommunicationsamongagents.Giventhistranslation,itisnaturetoasktowhatextentprooftheorycanbeusedtounderstandthemeta-theoryoftheπ-calculus.Wepresentsomepreliminaryresultsalongthislineforπ0,the“propositional”fragmentoftheπ-calculus,whichlacksrestrictionandvaluepassing(π0isasubsetofCCS).Usingideasfromproof-theory,weintroduceco-agentsandshowthattheycanspecifysometestingequivalencesforπ0.Ifnegation-as-failure-to-proveispermittedasaco-agentcombinator,thentestingequiv-alencebasedonco-agentsyieldsobservationalequivalenceforπ0.Thislatterresultfollowsfromobservingthatco-agentsdirectlyrepresentformulasintheHennessy-Milnermodallogic.1IntroductionInthispaperweaddressthequestion“Canweviewagivenprocesscalculusasalogic?”Thisisdifferent(althoughcertainlyrelated)tothequestion“Canlogicbeusedtocharacterizeagivenprocesscalculus?”Suchaquestionwouldviewlogicasanauxiliarylanguagetothatoftheprocesscalculus:forexample,theHennessy-MilnerlogichassucharelationshiptoCCS.Ourapproachherewillbetouselogicmoreimmediatelybytryingtomatchcombinatorsofthegivenprocesscalculusdirectlytologicalconnectivesand,ifacombinatorfailstomatch,tryingtoaxiomatizeditdirectlyanduniformlyinlogic.Forourpurposeshere,weshallconsideraformalsystemtobealogicifithasasequentcalculuspresentationthatadmitsacut-eliminationtheorem.Ofcourse,thisdefinitionoflogicisnotformalunlessformaldefinitionsofsequentcalculiandcut-eliminationareprovided.WeshallnotattemptThispaperappearsintheProceedingsofthe1992WorkshoponExtensionstoLogicProgramming,editedbyE.LammaandP.Mello,LectureNotesinComputerScience,Springer-Verlag.1