  We consider the lattice formulation of SO(10) chiral gauge theory with
left-handed Weyl fermions in the sixteen dimensional spinor representation
($\underline{16}$) within the framework of the Overlap fermion/the
Ginsparg-Wilson relation. We define a manifestly gauge-invariant path-integral
measure for the left-handed Weyl field using all the components of the Dirac
field, but the right-handed part of which is just saturated completely by
inserting a suitable product of the SO(10)-invariant 't Hooft vertices in terms
of the right-handed field. The definition of the measure applies to all
possible topological sectors. The measure possesses all required transformation
properties under lattice symmetries and the induced effective action is CP
invariant. The global U(1) symmetry of the left-handed field is anomalous due
to the non-trivial transformation of the measure, while that of the
right-handed field is explicitly broken by the 't Hooft vertices. There remains
the issue of locality in the gauge-field dependence of the Weyl fermion
measure, but the question can be addressed in the weak gauge-coupling expansion
at least using Monte Carlo methods without encountering the sign problem. We
also discuss the relations of our formulation to other approaches/proposals to
decouple the species-doubling/mirror degrees of freedom. Those include
Eichten-Preskill model, Ginsparg-Wilson Mirror-fermion model, Domain wall
fermion model with the boundary Eichten-Preskill term, 4D Topological
Insulator/Superconductor with gapped boundary phase, and the recent studies on
the PMS phase/"Mass without symmetry breaking". We clarify the similarity and
the difference in technical detail and show that our proposal is a well-defined
testing ground for that basic question.
