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Two-vector bundles and elliptic objects

Presented by: 
J Rognes [Oslo]
Monday 9th December 2002 - 10:00 to 11:00
INI Seminar Room 1
Session Title: 
Elliptic cohomology and chromatic phenomena
We (Baas, Dundas, Rognes) define a two-vector bundle over a base space X as a kind of family of two-vector spaces (in the sense of Kapranov and Voevodsky) parametrized over X. The rank 1 case recovers the notion of a Dixmier-Douady gerbe over X. The equivalence classes of two-vector bundles over X form an abelian monoid, whose Grothendieck group completion is the zero-th generalized cohomology group of X represented by the algebraic K-theory of the symmetric bimonoidal category of two-vector spaces. We argue that this K-theory agrees with the algebraic K-theory of the S-algebra (= A-infinity ring spectrum) representing connective topological K-theory, which by explicit computation (Ausoni, Rognes) is a connective, integrally defined form of elliptic cohomology, i.e., has chromatic complexity two. Two-vector bundles are thus geometric objects over X which provide the "cycles" for an elliptic cohomology theory at X. We also wish to indicate how a two-vector bundle over X leads to a virtual (anomaly) vector bundle over the free loop space of X, and an associated action functional for compact oriented surfaces over X.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons