Has Witten's perturbation on de Rham complex been studied on other elliptic complexes? In his famous work, Supersymmetry and Morse theory, Witten perturbs de Rham complex by perturbing the exterior derivative
$$d_h=e^{-ht}de^{ht}.$$
And he proves Morse inequality using some spectral information of the Laplacians of the complex.
Has similar perturbation on the the other elliptic complexes been studied? Complexes like Dolbeault complex or signature complex, Or maybe in general, twisted spin complexes. Can one expect new information from such a perturbed complexes?     
 A: Hormander's approach to solving the $\bar \partial$ problem is basically this, and his paper is from 1965, predating Witten's work by a couple of decades! By varying the "weight" function $h$, you can get families of estimates on the solution of $\bar \partial$ problem.  Check out Hormander's 1965 ACTA paper for more details.  Really a fabulous paper.  He does not make explicit that he is "perturbing the $\bar \partial$-complex", but that is exactly what he is doing.  I am sure Witten was reading Hormander.
It is also possible to perturb this complex in more dramatic ways.  My advisor (Jeff McNeal) has done a lot of this work.  You might be interested in his survey paper "$L^2$ Estimates on Twisted Cauchy Riemann Complexes". 
My thesis work is basically applying such twisted complexes (which can sometimes give estimates which are unavailable without the twisting) to prove some new results on approximation by holomorphic functions in $\mathbb{C}^n$.
A: I'm not sure if this is what you're looking for, but in http://arxiv.org/abs/math/0108185 Dunkl and Opdam define a deformed differential $d(k) = d + \Omega(k)$ on the polynomial de Rham complex on a vector space with an action by a complex reflection group $G$. Here $k$ is a parameter, $d$ is the standard de Rham differential, and $\Omega(k)$ is a 1-form depending on the parameter $k$. The (nontrivial) proof (Thm 2.9) that $d(k)$ is actually a differential (i.e. that it squares to 0) is used to show that the Dunkl operators associated to the complex reflection group $G$ commute. Proving this commutativity directly would require difficult computations.
(The Dunkl operator $T_\epsilon(k)$ for $\epsilon \in V$ is a deformation of the directional derivative $\partial_\epsilon$ depending on the parameter $k$, see formula (5). Dunkl defined these operators for real reflection groups in 1989, which is why they have that name.)
