Title:

The Discrete Dirac operator and the mass of simple and higher-order networks

Abstract:

We discuss the properties of the discrete Dirac operator on simple and higher-order and its relevance to capture the coupled dynamics topological signals defined on nodes, links of graphs and even higher dimensional simplices.

We will show how the Dirac operator can be coupled with the algebra of gamma matrices to define a Dirac field theory in discrete Lorentzian spacetime in  which the spinor is given a geometrical interpretation. The field theory also includes metric degree of freedom interpreted as the weights of the links for which we can define an action. 

We use the discrete topological Dirac operator to define an action for a massless self-interacting topological Dirac field inspired by the Nambu–Jona-Lasinio model. We propose a theoretical framework that explains how the mass of simple and higher-order networks emerges from their topology and geometry. The mass of the network is strictly speaking the mass of this topological Dirac field defined on the network; it results from the chiral symmetry breaking of the model and satisfies a self-consistent gap equation. Interestingly, it is shown that the mass of a network depends on its spectral properties, topology, and geometry.

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https://oist.zoom.us/j/93469044610?pwd=K0NwY0ZCQWNEb2NMWXFrZFBtaUNMZz09