Parallel transport in the determinant line bundle: The zero index case
Let X be a compact spin manifold of even dimension, \(S=S_+\oplus S_-\) the spinor bundle of X and E a hermitian vector bundle over X. Let \(\bar S\) and \(\bar E\) denote the pullback of S resp. E to \({\mathbb{R}}\times X\), and let \({\bar \nabla}^ E\) be a connection on \(\bar E.\) For any \(y\in {\mathbb{R}}\) the connection \(\nabla^ E_ y\) on E is induced by \({\bar \nabla}^ E\) and \(\partial_ y: L^ 2(X,S_+\otimes E)\to L^ 2(X,S_-\otimes E)\) is the Dirac-operator also called Weyl operator coupled to the connection \(\nabla^ E_ y\) on E and the metric on X. A general construction of \textit{J.-M. Bismut} and \textit{D. S. Freed} [Commun. Math. Phys. 106, 159-176 (1986) and 107, 103-163 (1986)] applied to this special case yields Hilbert bundles \(H_+\) and \(H_-\) over \({\mathbb{R}}\) and a bundle map \(\partial: H_+\to H_-\) given by the Weyl operators \(\partial_ y\), and a determinant line bundle L over \({\mathbb{R}}\) with an inner product and a compatible connection. The authors assume that \(\partial_ y\) is invertible for all \(y\in {\mathbb{R}}\) and study the parallel displacement in L from -\(\infty\) to \(+\infty\). This parallel transport is described in terms of the determinants det \(\partial_{+\infty}\partial^*_{\infty}\) and det \(\partial_{- \infty}\partial^*_{-\infty}\), the dimension of the kernel and the spectral asymmetry \(\eta\) (H) of a self adjoint Dirac operator H on \(\bar S\otimes \bar E\). Since \({\mathbb{R}}\times X\) is not compact \(\eta\) (H) cannot be defined in the usual way. The result is in a special case an extension of a formula of Bismut and Freed (loc. cit.) for the holonomy in the determinant line bundle over a loop.
scientific article; zbMATH DE number 4058489
scientific article