Spectral flow for 1st order selfadjoint elliptic operators on compact surface

The spectral flow of 1-parameter family of selfadjoint elliptic operators is the algebraic number of operator's eigenvalues intersecting 0. Let $A$ be a 1st order selfadjoint elliptic operator on vector bundle $E$ over compact surface $X$, $B$ be suitable boundary conditions for $A$, $g$ be a scalar gauge transformation of $E$. $g$ transforms $A$ to the operator $gA$ with the same symbol and leave $B$ unchanged. The goal of this talk is to compute the spectral flow along the path $(A(t), B)$ where $A(t)$ connects $A$ with $gA$ in the space of operators with the same symbol. This spectral flow does not change at the deformations of $g$ and $(A, B)$ so it defines homomorphism s from $G(X) \otimes H^1 (X, Z)$ to $Z$ where $G(X)$ is Grothendieck group constructed from the triples $(E, A, B)$ taken up to isomorphism and homotopy. I find group $G(X)$ and homomorphism s for arbitrary compact surface $X$. Another problem is the computation of the spectral flow along the closed path $(A(t), B(t))$ where symbol of $A(t)$ and boundary conditions are not constant. It defines homomorphism from some group $G_1 (X)$ to $Z$. I find $G_1 (X)$ and partially compute this homomorphism.