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The Dirac equation in $(2+1)$ dimensions on the toroidal surface is studiedfor a massless fermion particle under the action of external fields. Using thecovariant approach based on general relativity, the Dirac operator stemmingfrom a metric related to the strain tensor is discussed within thePseudo-Hermitian operator theory. Furthermore, analytical solutions areobtained for two cases, namely, constant and position-dependent Fermi velocity.

Pseudo-Hermitian Dirac operator on the torus for massless fermions under the action of external fields