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

Abstract

Let Dt, \({0\;\leqslant\;t\;\leqslant\;1}\) be a 1-parameter family of Dirac type operators on a two-dimensional disk with m − 1 holes. Suppose that all operators Dt have the same symbol, and that D1 is conjugate to D0 by a scalar gauge transformation. Suppose that all operators Dt 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).

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Correspondence to Marina Prokhorova.

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Communicated by B. Simon

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Prokhorova, M. The Spectral Flow for Dirac Operators on Compact Planar Domains with Local Boundary Conditions. Commun. Math. Phys. 322, 385–414 (2013). https://doi.org/10.1007/s00220-013-1701-6

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Keywords

  • Gauge Transformation
  • Dirac Operator
  • Boundary Component
  • Fredholm Operator
  • Time Reversal Symmetry