The spectral flow for Dirac operators on compact planar domains with local boundary conditions

Details The spectral flow for Dirac operators on compact planar domains with local boundary conditions The spectral flow for Dirac operators on compact planar domains with local boundary conditions

2011-08-03

arxiv.org/abs/1108.0806v2

Physics

Mathematical Physics

27 pages, 4 figures. Text is improved and extended. The error in sign is corrected; Max Planck Institute for Mathematics Prepr

Scientific paper

Let $D_t$, $t \in [0,1]$ be an arbitrary 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 $D_1$ is conjugate to $D_0$ by a scalar gauge transformation. Suppose that all operators $D_t$ are considered with the same local elliptic boundary condition, given by a vector bundle over the boundary. 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 the holes in the disk. This constant is calculated explicitly for the case of the annulus ($m=2$).

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