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.



Index of Modified Dirac Operator
For this particular case the two operators are conjugate (though note the conjugacy is not unitary unless $s$ is imaginary) 


$$
D\_{f,s}=e^{-sf}D e^{sf}
$$
so that the dimension of both the kernel and cokernel are independent of $s$.
Note that Chris Gerig's comments on the other hand apply more generally. The **index** of 
$$
D\_{s,\theta}=D+s\theta⋅
$$
for any one-form $\theta$ is independent of $s$. 


