Dirac gauge theory for topological spinors in 3+1 dimensional networks

Abstract

Gauge theories on graphs and networks are attracting increasing attention not only as approaches to quantum gravity but also as models for performing quantum computation.
Here we propose a Dirac gauge theory for topological spinors in $3+1$ dimensional networks associated to an arbitrary metric. Topological spinors are the direct sum of $0$-cochains and $1$-cochains defined on a network and describe a matter field defined on both nodes and links of a network. Recently in Ref. \cite{bianconi2021topological} it has been shown that topological spinors obey the topological Dirac equation driven by the discrete Dirac operator. 
In this work we extend these results by formulating the Dirac equation on weighted and directed $3+1$ dimensional networks which allow for the treatment of a local theory. The commutators and anti-commutators of the Dirac operators are non vanishing an they define the curvature tensor and magnetic field of our theory respectively. This interpretation is confirmed by the non-relativistic limit of the proposed Dirac equation. In the non-relativistic limit of the proposed Dirac equation the sector of the spinor defined on links follows the Schr\"odinger equation with the correct giromagnetic moment, while the sector of the spinor defined on nodes follows the Klein-Gordon equation and is not negligible. The action associated to the proposed field theory comprises of a Dirac action and a metric action. 
We describe the gauge invariance of the action under both Abelian and non-Abelian transformations and we propose the equation of motion of the field theory of both Dirac and metric fields. This theory can be interpreted as a limiting case of a more general gauge theory valid on any arbitrary network in the limit of almost flat spaces.

The mass of simple and higher-order networks

“…Indeed this latter equation can be shown to hold because D [1] D [2] = D [2] D [1] = 0 as it can be easily shown given equation (67). It follows that the space of topological spinors be decomposed in a unique way into…”

“…where µ indicates any eigenvalue in the spectrum L. Let us define the two operators D [1] and D [2] acting only on nodes and links and only on links and triangles respectively as…”

“…Recently, growing attention is addressed to the discrete topological Dirac operator [1,2], originally introduced in non-commutative geometry [3][4][5][6][7], and then used in quantum graphs [8][9][10][11][12][13][14][15] and in network theory [16][17][18][19][20][21]. On a network, the discrete topological Dirac operator is defined over topological spinors, which are the direct sum of zero-cochain and one-cochains.…”

“…It follows that as soon as the network is not a closed chain, their degeneracy is different. The Dirac operator can also be coupled to an algebra [1,2] in order to provide a discrete topological Dirac equation that corresponds to the continuous (d + 1)-dimensional Dirac equation.…”

Dirac signal processing of higher-order topological signals

“…The Dirac operator D is a linear operator that acts on the topological spinor [23,24,50,51,57]. In the canonical basis of topological spinors of a d = 2 dimensional simplicial complex, the Dirac operator D has the block structure…”