The chiral determinant and the eta invariant (Q1822113)

For \(\{\partial_ y\}\), \(y\in {\mathbb{R}}\), a one parameter family of invertible Weyl operators of possibly nonzero index acting on spinors over an even dimensional compact manifold X, we express the phase of the chiral determinant det \(\partial^{\dag}_{-\infty}\partial_{\infty}\) in terms of the \(\eta\) invariant of a Dirac operator acting on spinors over \({\mathbb{R}}\times X\).