A perturbation of the de Rham complex was introduced by Witten for an exact 1-form Θ and later extended by Novikov for a closed 1-form on a Riemannian manifold M. We use invariance theory to show that the perturbed index density is independent of Θ; this result was established previously by J. A. Álvarez López, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order heat trace asymptotics of the perturbed de Rham complex exhibit nontrivial dependence on Θ. We establish similar results for manifolds with boundary imposing suitable boundary conditions and give an equivariant version for the local Lefschetz trace density. In the setting of the Dolbeault complex, one requires Θ to be a \(\overline \partial \) closed 1-form to define a local index density; we show in contrast to the de Rham complex, that this exhibits a nontrivial dependence on Θ even in the setting of Riemann surfaces.

Witten为精确的1形式Θ引入了de Rham复数的扰动，随后由Novikov扩展了黎曼流形M上的封闭1形式。我们使用不变性理论来证明扰动的折射率密度与Θ无关；这个结果是由JAÁlvarezLópez，YA Kordyukov和E.Leichtnam（2020）以前使用其他方法确定的。我们还显示了扰动的de Rham络合物的高阶热迹线渐近线表现出对θ的非平凡依赖。我们为带有边界的歧管建立了相似的结果，并施加了适当的边界条件，并给出了局部Lefschetz迹线密度的等变版本。在Dolbeault复合体的设置中，要求Θ是\（\ overline \ partial \）封闭的1形式定义局部指数密度；与de Rham复数相反，我们表明，即使在Riemann曲面的设置中，它也表现出对Θ的非平凡依赖。