While reading this M.SE thread, I was introduced to Twisted derivatives.
"I got motivation from twisted Cohomology where we twisted the derivative $d_\psi=d+\psi\wedge$ and find cohomology class $H_{\psi}^k(M)$ where $\psi$ is closed one form. I try to define twisted connection similarly, which lead me to define ..."
I know a little bit about how connections and curvatures relate to geometry (from my 3rd-year manifold course which follows Differential Geometry by Loring W. Tu, chapters 1-5). But I wondered what additional geometric things are happening when we add a twist (a closed $1$-form here). I want to explore it. So far, I have found out twisted cohomology class (which might be called MORSE-NOVIKOV COHOMOLOGY):
$$H_{\psi}^{k}(M) := \frac{Ker (d_{\psi}^{k}:\Omega^{k}(M)\rightarrow \Omega^{k+1}(M))}{Im(d_{\psi}^{k-1}:\Omega^{k-1}(M)\rightarrow\Omega^{k}(M))}$$
But couldn't find out any resources where I could find its importance.
In vector bundle, we choose $1$-form, why not a closed higher form? And how to bring the same twist in a principal bundle. As my current knowledge supports, "A connection is equivalently specified by a choice of horizontal subspace to the principal bundle." But twisting seems totally an algebraic manipulation to me. How can it capture any geometric things?
Any resource where geometric insight was discussed would be a great suggestion for me.
