Clifford algebras on Sobolev spaces (Q2770365)

Let \(X\) be a compact (spin)manifold, \(E\) an Hermitian vector bundle over \(X\) and \(D\) a non-degenerate 1-st order self-adjoint (pseudo)differential operator acting on the smooth cross-sections of \(E\). Fixing the \(L^2\)-norm \(\| f\| \) of a section \(f\) of \(E\), we fix the Sobolev \(k\)-norm \(\| f\| _k\) of \(f\) to be \(\| D^kf\| \). Then we say that the virtual dimension of the Sobolev \(k\)-space \(W^k(X)\) of sections of \(E\) is \(\nu =\zeta_{| D| }(0)\). The author constructs the Clifford algebra \(C(W^{-k})[e^{\infty}]\) over \(W^{-k}(X)\) with the infinite spinor \(e^{\infty}\) for an integer \(\nu\). Modifying \(e^{\infty}\) according to the \(\text{mod\,}8\) class of \(\nu\), the author represents \(e^{\infty}\) in terms of \(D\) and its Green operator \(G\). Moreover, the author defines the Grassmann map from \(C(W^{-k})[e^{\infty}]\) to \(\wedge W^{-k}+\wedge W^k,\) where \(\wedge W^k\) is regarded to be the model of \((\infty -p)\)-forms over \(W^k(X)\), which has been defined by the author.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00064].