Suppose $E$ is a $SU(2)$-bundle over a closed three manifold $M$ and $S$ is the spinor bundle over $M$. Also assume $D_{A(t)}:\Gamma(S\otimes_{\mathbb c} E)\to \Gamma(S\otimes_{\mathbb c} E)$ is a family of Dirac operators twisted by a family of connections $A(t)$ on $E$, $t\in[0,1]$.
We have also a family of curl-div operators $L_{A(t)}:\Omega^0(E)\oplus\Omega^1(E)\to \Omega^0(E)\oplus\Omega^1(E)$ where \begin{equation} L_{A(t)}= \begin{pmatrix} 0&d^*_{A(t)}\\ d_{A(t)}&*d_{A(t)}\\ \end{pmatrix} \end{equation} My question is:
Suppose $A(0)$ and $A(1)$ are two flat connections and $\mathbb A:=\{A(t):t\in[0,1]\}$ is an instanton on $M\times[0,1]$. Are there any relations between Spectral flow of $\{D_{A(t)}:t\in[0,1]\}$ and Spectral flow of $\{L_{A(t)}:t\in[0,1]\}$?
Note: From the work of Atiyah-Patodi-Singer we could relate them with expressions involving eta invariants and some characteristic classes. Are they getting simplified in our present situations? If the question above was not expressed properly, that's my fault. Please let me know, I would try to modify them.
