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The spinor bundle on the free loop space

Presented by: 
S Stolz [Notre Dame]
Monday 9th December 2002 - 15:30 to 16:30
INI Seminar Room 1
Session Title: 
Elliptic cohomology and chromatic phenomena
The spinor bundle $S(E)$ associated to an even dimensional real vector bundle $E$ with spin structure has (at least) two roles in life: from a homotopy theory point of view it represents the $K$-theory Euler class of $E$; from a geometric/analytic point of view, the Dirac operator acts on the sections of the spinor bundle $S(TX)$ associated to the tangent bundle of a spin manifold $X$.

Analogously, it is believed that the {\it spinor bundle} or {\it Fock space bundle} $\mathcal F(E)\to LX$ over the free loop space $LX$ associated to an even dimensional vector bundle $E\to X$ with `string structure' plays a similar dual role: it should represent the Euler class of $E$ in $tmf^*(X)$, and there should be a `Dirac-Witten operator' acting on the sections of $\mathcal F(TX)$, whose $S^1$-equivariant index is the Witten genus of $X$.

The main result of this joint work with Peter Teichner is that the spinor bundle $\mathcal F(E)$ can be equipped with additional structures we call `conformal connection' and `fusion'. We speculate that vector bundles over $LX$ equipped with these two structures represent elements in $tmf^*(X)$.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons