A general splitting formula for the spectral flow

Details A general splitting formula for the spectral flow A general splitting formula for the spectral flow

1999-02-24

arxiv.org/abs/math/9902142v1

Mathematics

Differential Geometry

27 pages, LaTeX. Appendix by K.P. Wojciechowski

Scientific paper

We derive a decomposition formula for the spectral flow of a 1-parameter family of self-adjoint Dirac operators on an odd-dimensional manifold $M$ split along a hypersurface $\Sigma$ ($M=X\cup_{\Sigma} Y$). No transversality or stretching hypotheses are assumed and the boundary conditions can be chosen arbitrarily. The formula takes the form $SF(D)= SF(D_{|X}, B_X) + SF(D_{|Y},B_Y) + \mu(B_Y,B_X) + S$ where $B_X$ and $B_Y$ are boundary conditions, $\mu$ denotes the Maslov index, and $S$ is a sum of explicitly defined Maslov indices coming from stretching and rotating boundary conditions. The derivation is a simple consequence of Nicolaescu's theorems and elementary properties of the Maslov index. We show how to use the formula and derive many of the splitting theorems in the literature as simple consequences.

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