3:30 pm Thursday, April 5, 2007
Geometry Seminar : "Higher Clifford algebras" by Andre Henriques in RLM 9.166
Abstract: It is well known that Clifford algebras and topological K-theory are intimately related. For example, one can get a geometric description of the higher K-theory groups via bundles of Cliff(n)-modules, and the fact that Cliff(n+2) is Morita equivalent to Cliff(n) directly implies Bott periodicity. For many purposes, it is useful to consider the Clifford algebras as objects of a symmetric monoidal 2-category whose objects are Z/2-graded algebras, whose morphisms are bimodules and whose 2-morphisms are intertwiners between bimodules. For example, one can then characterize the Clifford algebras as the invertible objects in that 2-category. Let vN2 be the symmetric monoidal 2-category of von Neuman algebras and bimodules, composition being given by Connes fusion, and the monoidal structure given by tensor product. Stolz and Teichner conjectured the existence of an interesting symmetric monoidal 3-category C with the property that Hom_C(1,1)=vN2. The invertible objects in that 3-category category should then play the same role for Elliptic cohomology that the finite dimensional Clifford algebras play in K-theory. Namely, the group of invertible objects should be a small abelian group that is somehow related to the periodicity of Elliptic cohomology. Since elliptic cohomology is sometimes also called "higher K-theory", it is reasonable to call the invertible objects in that 3-category the "higher Clifford algebras". Conformal nets are the objects of our candidate symmetric monoidal 3-category. And we also have a candidate for the generator of the group of invertible objects: the net of local fermions. As said above, Stolz and Teichner predicted that, whatever these higher Clifford algebras are, they should satisfy a periodicity similar to the periodicity of Clifford algebras. We don't know yet whether the net of local fermions is of finite order or not, but we do have an argument that shows that it's at least of order at least 24. This is joint work with Arthur Bartels and Chris Douglas.
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