Index theory for short-ranged fields in higher dimensions
Let \(M\) be a noncompact Riemannian spin manifold of even dimension with a warped end. This means that there exist a compact Riemannian manifold \((N,d\sigma)\) and a compact set \(K_ 0 \subset M\) such that \(M\setminus K_ 0\) is isometric to the product \((0,\infty) \times N\) which is equipped with the warped product metric \(dr^ 2 + f^ 2(r) d\sigma^ 2\), where \(f\) is a positive smooth function defined on \([0,\infty)\). Let \(E\) be a trivial Hermitian vector bundle over \(M\) and let \(\nabla_ 0\) be the connection on \(E\) which is defined by the ordinary exterior derivative \(d\), taken with respect to some fixed trivialization of \(E\). Any other metric connection on \(E\) is given by \(\nabla_ 0 + A\), for some \(A \in \Omega^ 1({\mathcal G}_ E)\), where \({\mathcal G}_ E = \{\gamma \in \text{Hom}_ \mathbb{C}(E)\mid \gamma +\gamma^* = 0\}\). \(F = dA + A \wedge A\) is called a short-ranged field, if \(F\) has a compact support. The author studies the twisted Dirac operator \(D_ A : C^ \infty(S \otimes E) \to C^ \infty(S \otimes E)\), where \(S\) is the spinor bundle and \(D_ A\) is constructed using a connection \(\nabla_ 0 + A\) on \(E\) such that the field \(F\) is short ranged. The author proves, that the restrictions \(D_ A^ \pm\) of \(D_ A\) to the spinors of positive resp. negative chirality have finite dimensional \(L^ 2\)-kernel, and he computes an index formula for the \(L^ 2\)-index of \(D^ +_ A\). In the proof the computation of the \(L^ 2\)-kernel is related to the boundary value problem studied by \textit{M. F. Atiyah, V. K. Patodi} and \textit{I. M. Singer} [Math. Proc. Camb. Philos. Soc. 77, 43-69 (1975; Zbl 0297.58008)].
scientific article; zbMATH DE number 599823
scientific article