The Spectral Flow for Dirac Operators on Compact Planar Domains with Local Boundary Conditions

Let D_t, $${0\;\leqslant\;t\;\leqslant\;1}$$ be a 1-parameter family of Dirac type operators on a two-dimensional disk with m - 1 holes. Suppose that all operators D_t have the same symbol, and that D₁ is conjugate to D₀ by a scalar gauge transformation. Suppose that all operators D_t are considered with the same elliptic local boundary condition. Our main result is a computation of the spectral flow for such a family of operators. The answer is obtained up to multiplication by an integer constant depending only on the number of holes in the disk. This constant is calculated explicitly for the case of the annulus (m = 2).