  We localize the loop equation of large-N YM theory in the ASD variables on a
critical equation for an effective action by means of homological methods as
opposed to the cohomological localization of equivariantly closed forms in
local field theory. Our localization occurs for some special simple quasi BPS
Wilson loops, that have no perimeter divergence and no cusp anomaly for
backtracking cusps, in a partial Eguchi-Kawai reduction from four to two
dimensions of the non-commutative theory in the limit of infinite
non-commutativity and in a lattice regularization in which the ASD integration
variables live at the points of the lattice, thus implying an embedding of
parabolic Higgs bundles in the YM functional integral. We find that the beta
function of the effective action is saturated by the non-commutative ASD
vortices of the EK reduction. An exact canonical beta function of NSVZ type
that reproduces the universal first and second perturbative coefficients
follows by the localization on vortices. Finally we argue that a scheme can be
found in which the canonical coupling coincides with the physical charge
between static quark sources in the large-N limit and we compare our
theoretical calculation with some numerical lattice result.
