1997/10/28 by Jørgen Ellegaard Andersen, Jorgen Ellegaard Andersen, Andersen, Jorgen Ellegaard +2
Mathematics · #14H60 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #alg-geom #math.AG #math.QA #msc:14H60 #q-alg
paper · pdf · doi:10.48550/arxiv.alg-geom/9710031
33 pages, Latex, minor modifications, to appear in Mathematische Annalen
openalex publication_date 1997/10/28 · arxiv created 1998/12/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The moduli space M of semi-stable rank 2 bundles with trivial determinant over a complex curve carries involutions naturally associated to 2-torsion points on the Jacobian of the curve. For every lift of a 2-torsion point to a 4-torsion point, we define a lift of the involution to the determinant line bundle Ł. We obtain an explicit presentation of the group generated by these lifts in terms of the order 4 Weil pairing. This is related to the triple intersections of the components of the fixed point sets in M, which we also determine completely using the order 4 Weil pairing. The lifted involutions act on the spaces of holomorphic sections of powers of Ł, whose dimensions are given by the Verlinde formula. We compute the characters of these vector spaces as representations of the group generated by our lifts, and we obtain an explicit isomorphism (as group representations) with the combinatorial-topological TQFT-vector spaces of [BHMV]. As an application, we describe a `brick decomposition', with explicit dimension formulas, of the Verlinde vector spaces. We also obtain similar results in the twisted (i.e., degree one) case.