Asymptotic spectral flow for Dirac operators
Let \(M\) denote a compact, oriented Riemannian manifold of odd dimension \(n\geq 3\). Suppose that \(F\rightarrow M\) is a principal bundle with structure group \(\text{Spin}(n)\times_{\pm 1} U(k)\) such that \(F/U(k)\) is the principal \(SO(n)\) bundle of orthonormal frames. A connection on the principal bundle \(F/\text{Spin}(n)\rightarrow M\) determines a self-adjoint Dirac operator on a certain associated Clifford bundle. For a fixed pair \((A_0,A_1)\) of such connections and any path joining them, the author considers the Dirac spectral flow, i.e., a weighted count of the connections on this path where the Dirac operator has a zero eigenvalue (the result is independent of the chosen path). The main theorem of this paper claims that the Dirac spectral flow differs from the integral over \(M\) of the wedge product of the relative Chern-Simons form (depending on \((A_0,A_1)\)) and the differential form representing the \(\widehat{A}\) genus of \(M\) by an error term which is mainly given by some constants and integrals over the derivatives of the path of connections. The result is used to derive an estimate for the spectral asymmetry (basically, the difference of the eta invariants of \(A_0\) and \(A_1\)). The note generalizes certain estimates from [\textit{C. H. Taubes}, Geom. Topol. 11, 2117--2202 (2007; Zbl 1135.57015)].
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