Clifford bundles over mapping spaces

Akira Asada


Abstract. Let $X$ be compact spin manifold, $E$ an Hermitian vector bundle over $X$, $D$ a non-degenerate 1-st order self-adjoint elliptic pseudo-differential operator. Fixing the Sobolev metric on $W^{k}(X)$, the $k$-Sobolev space of section of $E$, by $D$, we have developped calculus of differential forms, including $(\infty -p)$--forms, on an open set of $W^{k}(X)$ (\cite{asa3}, \cite{asa4}) and constructed Clifford algebra over $W^{-k}(X)$ with $\infty$--spinor ($\gamma_{5}$) together with its representation in algebra of bounded linear operators on $\wedge W^{-k}(X)\oplus \wedge W^{k}(X)$ (\cite{asa5}). By using these results, $(\infty -p)$--forms and Hodge operators on a mapping space Map$(X,M)$ (and some of its extensions) are investigated and Clifford bundle is constructed. If $D$ is positive definite, such construction is always possible, while $D$ is Dirac type, vanishing of string class Map$(X,M)$ is necessary to the construction of Clifford bundle. Difficulties with the construction of spinor bundle and definition of Dirac-Ramond operator are investigated.

AMSclassification. 58D15, 58G25, 15A63, 81T30

Keywords. Mapping space, Clifford bundle, Regularized dimension, Connections of differential operators