Operator (biology)

Pure mathematics

Mathematics

Boundary (topology)

Differential operator

Transcription factor

Parametrix

Gene

Dirac operator

Mathematical analysis

Boundary value problem

Repressor

Semi-elliptic operator

Chemistry

Atiyah–Singer index theorem

Cauchy distribution

Biochemistry

American Journal of Mathematics

We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed.

An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary