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arXiv · 0708.3707

First order approach and index theorems for discrete and metric graphs

Olaf Post
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Abstract

The aim of the present paper is to introduce the notion of first order (supersymmetric) Dirac operators on discrete and metric (``quantum'') graphs. In order to cover all self-adjoint boundary conditions for the associated metric graph Laplacian, we develop systematically a new type of discrete graph operators acting on a decorated graph. The decoration at each vertex of degree-d is given by a subspace of $\C^d$, generalising the fact that a function on the standard vertex space has only a scalar value. We develop the notion of exterior derivative, differential forms, Dirac and Laplace operators in the discrete and metric case, using a supersymmetric framework. We calculate the (supersymmetric) index of the discrete Dirac operator generalising the standard index formula involving the Euler characteristic of a graph. Finally, we show that the corresponding index for the metric Dirac operator agrees with the discrete one.

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Olaf Post. 2007-09-03. First order approach and index theorems for discrete and metric graphs. https://arxiv.org/abs/0708.3707

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