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An \(L^2\)-index theorem for Dirac operators on \(S^1\times\mathbb{R}^3\). (English) Zbl 0973.58012
This paper establishes a criterion for Fredholmness and an \(L^{2}\)-index formula for a certain kind of Dirac operator. The Dirac operator lives on \(S^{1} \times \mathbb{R} ^{3}\). It acts on sections of the spin bundles tensored with an auxiliary vector bundle. The auxiliary bundle is the pullback of a vector bundle on \(\mathbb{R} ^{3}\) that extends to the sphere at infinity. This bundle \(E_{\infty} \rightarrow S^{2}\) is framed (trivialized) over the sphere at infinity. The connection on the auxiliary bundle extends to take the following form on the sphere at infinity: it is the pullback of a connection on \(E_{\infty}\) plus a constant skew-adjoint endomorphism of \(E_{\infty}\) times \(dz\). Here \(z\) is the variable on \(S^{1}\).
The Fredholm criterion for the Dirac operator involves the eigenvalues of the skew-adjoint map. The index formula involves two types of terms. One term is a degree measuring the obstruction to extending the framing over all of \(\mathbb{R} ^{3}\). Other terms are Chern numbers, calculated on the sphere at infinity, of bundles defined by the spectrum of the skew-adjoint map.
The proof of the index theorem proceeds as follows. When the framing extends, Fourier analysis on the circle reduces the problem to the Callias-Anghel-Råde index theorem [see, e.g., J. Råde, Commun. Math. Phys. 161, 51-61 (1994; Zbl 0797.58081)]. The Gromov-Lawson relative index theorem finishes the proof [see, e.g., N. Anghel, Houston J. Math. 19, 223-237 (1993; Zbl 0790.58040)].
This paper points out two further sources of interest in the Dirac operators studied here. These operators can be viewed as examples of elliptic fibred cusp operators studied by R. Mazzeo and R. Melrose. These Dirac operators also play a role in studying self-dual calorons (periodic instantons), which sit between ordinary instantons and monopoles.

MSC:
58J20 Index theory and related fixed-point theorems on manifolds
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