  Borophene is a monolayer materials made of boron. A perfect planar boropehene
called $\beta_{12}$ borophene has Dirac cones and they are well reproduced by a
tight-binding model according to recent experimental and first-principles
calculation results. We explicitly derive a Dirac theory for them. Dirac cones
are gapless when the inversion symmetry exists, while they are gapped when it
is broken. In addition, three-band touching points emerge together with
pseudospin triplet fermions when all transfer energy is equal and all on-site
energy is equal. The three-band touching is slightly resolved otherwise. We
construct effective three-band theories for triplet fermions. We also study the
edge states of borophene nanoribbons, which show various behaviors depending on
the way of edge terminations.
