  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 $\Phi$.
