Parallel transport in the determinant line bundle: The non-zero index case
The authors consider a compact spin manifold X of even dimension with spin bundle \(S=S_+\oplus S_- \to X.\) For a Hermitian vector bundle \(E\to X\) over X, the Weyl operators are defined by \(\partial_ y:L^ 2(X,S_+\otimes E)\mapsto L^ 2(X,S_-\otimes E)\) coupled to the connection \(\nabla^ E_ y\) and the (y-independent) metric on X. With these structures a smooth determinant line bundle \({\mathcal L}\) over \({\mathbb{R}}\) is constructed, and a natural Hermitian metric and a compatible connection are introduced. The parallel transport in \({\mathcal L}\) is investigated in the case that index \(\partial_ y\) is not necessarily zero. It is expressed in terms of the spectral asymmetry of a Dirac operator on \({\mathbb{R}}\times X\). The pull-backs of the inner product and connection on \({\mathcal L}\) are calculated. These results are used then to obtain the formulas for curvature in \({\mathcal L}\) and parallel transport in \({\mathcal L}\). The calculus of the resolvent of the Dirac operator and the notational conventions are given in two appendices.
scientific article; zbMATH DE number 4024213