Computing spectral flow via cup products
The authors investigate the spectral flow of the ``Dirac'' operator \(D_ A = *d_ A - d_ A*\) acting on \(0 + 2\) forms on a 3-manifold \(M\) as \(A\) varies in the connections in an SU(2)-bundle over \(M\). The main result states that if \(A_ t\) is a path of flat connections such that \(\text{Ker }D_{A_ t} = 0\) for \(t = 0\), then the spectral flow at time \(t = 0\) (i.e. the slopes of the eigenvalues of \(D_{A_ t}\) that are crossing 0 at \(t = 0\)) can be expressed in terms of the eigenvalues of a bilinear form which is a composition of two cup products in the cohomology of \(M\) with twisted coefficients. In particular, the authors compute the spectral flow \(\bmod 4\) between two representations of the fundamental group of a 3-manifold which fibers over the circle with fiber a torus.
scientific article; zbMATH DE number 727340