The local index density of the perturbed de Rham complex
A closed 1-form $\Theta$ on a manifold induces a perturbation $d_\Theta$ of the de~Rham complex. This perturbation was originally introduced Witten for exact $\Theta$, and later extended by Novikov to the case of arbitrary closed $\Theta$. Once a Riemannian metric is chosen, one obtains a perturbed Laplacian $\Delta_\Theta$ on a Riemannian manifold and a corresponding perturbed local index density for the de~Rham complex. Invariance theory is used to show that this local index density in fact does not depend on $\Theta$; it vanishes if the dimension $m$ is odd, and it is the Euler form if $m$ is even. (The first author, Kordyukov, and Leichtnam (2020) established this result previously using other methods). The higher order heat trace asymptotics of the twisted de~Rham complex are shown to exhibit non-trivial dependence on $\Theta$ so this rigidity result is specific to the local index density. This result is extended to the case of manifolds with boundary where suitable boundary conditions are imposed. An equivariant version giving a Lefschetz trace formula for $d_{\Theta}$ is also established; in neither instance does the twisting 1-form $\Theta$ enter. Let $\Phi$ be a $\bar\partial$ closed $1$-form of type $(0,1)$ on a Riemann surface. Analogously, one can use $\Phi$ to define a twisted Dolbeault complex. By contrast with the de~Rham setting, the local index density for the twisted Dolbeault complex does exhibit a non-trivial dependence upon the twisting $\bar\partial$-closed 1-form $\Phi$.