A duality for Spin Verlinde spaces and Prym theta functions C Pauly and S Ramanan
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English

A duality for Spin Verlinde spaces and Prym theta functions C Pauly and S Ramanan

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Niveau: Supérieur, Master, Bac+4
A duality for Spin Verlinde spaces and Prym theta functions C. Pauly and S. Ramanan August 29, 2000 Abstract We prove canonical isomorphisms between Spin Verlinde spaces,i.e., spaces of global sec- tions of a determinant line bundle over the moduli space of semistable Spinn-bundles over a smooth projective curve C, and the dual spaces of theta functions over Prym varieties of unramified double covers of C. 1 Introduction To any smooth, projective curve C, one classically associates a collection of principally polarized abelian varieties: the Jacobian JC, parametrizing degree zero line bundles, as well as, for any unramified double cover C˜? ? C, depending on a non-zero 2-torsion point ? ? JC[2], a Prym variety P?. The projective geometry of the configuration JC ? ? ? P? has been much studied [M1], [vGP], [B1] and encodes e.g. the Schottky-Jung identities among theta-constants [M1]. Less classically, one can consider the moduli space M(G) of semistable principal G-bundles over the curve C, where G is a simple and simply-connected algebraic group. For some ample line bundle L over M(G), the vector space of global sections H0(M(G),L) has been identified to a space of conformal blocks arising in conformal field theory (see e.

  • theta

  • line bundle

  • divisors associated

  • associated half-spinor

  • any smooth

  • spinor vector

  • spin group

  • bundles over

  • prym varieties


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Nombre de lectures 150
Langue English
A duality for Spin Verlinde spaces and Prym theta functionsC. Pauly and S. RamananAugust 29, 2000
AbstractWe prove canonical isomorphisms between Spin Verlinde spaces,i.e., spaces of global sec-tions of a determinant line bundle over the moduli space of semistable Spinn-bundles overa smooth projective curveC, and the dual spaces of theta functions over Prym varieties ofunramied double covers ofC.
1 IntroductionTo any smooth, projective curveC, one classically associates a collection of principally polarizedabelian varieties: the JacobianJC, parametrizing degree zero line bundles, as well as, for any˜unramied double coverCC, depending on a non-zero 2-torsion pointJC[2], a PrymvarietyP. The projective geometry of the congurationJCSPhas been much studied [M1],[vGP], [B1] and encodes e.g. the Schottky-Jung identities among theta-constants [M1].Less classically, one can consider the moduli spaceM(G) of semistable principalG-bundlesover the curveC, whereGis a simple and simply-connected algebraic group. For some ampleline bundleLoverM(G), the vector space of global sectionsH0(M(G)L) has been identied toa space of conformal blocks arising in conformal eld theory (see e.g.[S] for a survey or [LS] fora proof), which made the computation of its dimension possible. This is the celebrated Verlindeformula.In this paper we are interested in dualities between Verlinde spacesH0(M(G)L) and spacesof abelian theta functions ,i.e.,sections of some multiple of a principal polarization overJCor,more generally,JCSP. Such dualities were rst proved for the structure groupG= SL2andthe line bundlesLlforl= 124 [B1], [B2], [OP] or, more generally,G= SLnand the line bundleL[BNR], whereLis the ample generator of the Picard group ofM(SLn).In the articles [O1] and [O2], W.M. Oxbury constructs linear maps between a Verlinde spacefor the complex spin group,i.e.,G= Spinn, and a space of abelian theta functions overJCSP.Our main theorem states that these linear maps are actually isomorphisms. More precisely weshow1.1. Theorem.For any curveCand any integerm1, we have canonical isomorphisms(1)Xs:XH0(Pev(2m+ 1))H0(M(Spin2m+1)(C2m+1)) (1.1)JC[2](2)Xs:X£H0(Pev(2m))H0(Podd(2m))¤H0(M(Spin2m)(C2m))JC[2](1.2)
1
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We refer to section 2 and 3 for the rather technical details encoded in the notation.We note that theorem 1.1 for the group Spin3coincides with the above mentioned duality forVerlinde spaceH0(M(SL )L4) under the exceptional isomorphism Spin = SL . In this case
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