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Clifford bundles over mapping spaces

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*Akira Asada*

**E-mail:** asada@math.shinshu-u.ac.jp
**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