  The low-energy theory of electrons interacting via repulsive short-range
interactions on graphene's honeycomb lattice at half filling is presented. The
exact symmetry of the Lagrangian with local quartic terms for the Dirac field
dictated by the lattice is D_2 x U_c(1) x (time reversal), where D_2 is the
dihedral group, and U_c(1) is a subgroup of the SU_c(2) "chiral" group of the
non-interacting Lagrangian, that represents translations in Dirac language. The
Lagrangian describing spinless particles respecting this symmetry is
parameterized by six independent coupling constants. We show how first imposing
the rotational, then Lorentz, and finally chiral symmetry to the quartic terms,
in conjunction with the Fierz transformations, eventually reduces the set of
couplings to just two, in the "maximally symmetric" local interacting theory.
We identify the two critical points in such a Lorentz and chirally symmetric
theory as describing metal-insulator transitions into the states with either
time-reversal or chiral symmetry being broken. In the site-localized limit of
the interacting Hamiltonian the low-energy theory describes the continuous
transitions into the insulator with either a finite Haldane's (circulating
currents) or Semenoff's (staggered density) masses, both in the universality
class of the Gross-Neveu model. The picture of the metal-insulator transition
on a honeycomb lattice emerges at which the residue of the quasiparticle pole
at the metallic and the mass-gap in the insulating phase both vanish
continuously as the critical point is approached. We argue that the Fermi
velocity is non-critical as a consequence of the dynamical exponent being fixed
to unity by the emergent Lorentz invariance. Effects of long-range interaction
and the critical behavior of specific heat and conductivity are discussed.
