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We discuss a discrete analogue of the Dirac–Kähler equation in which chiral properties of the continuum counterpart are captured. We pay special attention to a discrete Hodge star operator. To build such an operator combinatorial construction of a double complex is used. We describe discrete exterior calculus operations on a double complex and obtain the discrete Dirac–Kähler equation using these tools. Self-dual and anti-self-dual discrete inhomogeneous forms are presented. The chiral invariance of the massless discrete Dirac–Kähler equation is shown. Moreover, in the massive case we prove that a discrete Dirac–Kähler operator flips the chirality. [ABSTRACT FROM AUTHOR]

We construct a new discrete analogue of the Dirac–Kähler equation in which some key geometric aspects of the continuum counterpart are captured. We describe a discrete Dirac–Kähler equation in the intrinsic notation as a set of difference equations and prove several statements about its decomposition into difference equations of Duffin type. We study an analogue of gauge transformations for the massless discrete Dirac–Kähler equations. [ABSTRACT FROM AUTHOR]