Let's say we have an oriented compact 4-d Riemannian spin manifold $(M,g)$. Everybody who's anybody has heard about the index of the Dirac operator $D: S^+\rightarrow S-$; it's the $\hat{A}$-genus, which is $\displaystyle\frac{-1}{8}\tau(M)$ ($\tau$ is the signature). I don't know too much about the index theorem and it's inner workings, but I'm wondering what happens if you perturb the Dirac operator and consider $$D_{f,s}=D+s\textrm{grad}(f)\cdot$$ where s is a (let's say small) real parameter and $f\in C^{\infty}(M)$; "$\cdot$" indicates Clifford multiplication, and maps $S^+\rightarrow S^-$ since elements of $TM$ anticommute with the volume element in dimension 4.

Does the index stay the same? Change in a predictable way related to $f$? What if $f$ is special somehow? What about in the special case where $\tau=0$? Or the even-more-special case where $\tau=0$ because the dimension of the harmonic spinors is $0$, e.g. when $R>0$? How does the dimension of the kernel jump as $s$ changes?

Sorry for the avalanche of questions. I'm interested in any information people have about any subset of them.