Ihsane Malass
We consider the perturbation of the exterior differential by a closed one-form \(a\), \(d_a := d + a \wedge \cdot\), on a manifold with boundary. This gives rise to the perturbed (or twisted) de Rham complex \((\Omega^{\bullet}, d_a)\). The Hodge star commutes with the perturbed Laplace operator \(\Delta_a\); however, this occurs at the expense of a sign change in the perturbation and a modification of the associated boundary conditions. This leads to interesting cohomological properties—such as Lefschetz-type dualities in the perturbed case—as well as spectral properties, which manifest at the level of the heat kernel trace. Applying these newly observed properties yields intriguing relations at the level of the perturbed zeta determinant and the analytic torsion of the perturbed Laplace operator.
