Quasi Bps Wilson Loops, Localization of Loop Equation by Homology and Exact Beta Function in the Large-n Limit of Su(n) Yang-mills Theory

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 noncommutativity 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. Homological localization is based on an analogy with cohomological localization. The analog of the invariance of the cohomological class of a closed form for the addition of a co-boundary is the zig-zag symmetry, i.e. the invariance of the holonomy class of a quasi BPS Wilson loop for the addition of the boundary of a tiny strip, of size of the cut-off. The analog of the action being a closed form is the invariance of the RG flow of the Wilsonean renormalized effective action in the ASD variables, Γq, under the local conformal transformation that generates the holonomically trivial deformation of the quasi BPS Wilson loops. Finally, the analog of the rescaling by a divergent factor of the cohomological trivial deformation that localizes the exponential of a closed form is the local conformal rescaling that maps any (lattice) marked point of a quasi BPS Wilson loop to a cusp at infinity. The homological reason for localization is that the contact term that occurs at the marked points of the loop, that is the obstruction to localization in the loop equation, vanishes at the cusps because of the absence of cusp anomaly for the backtracking cusps of the quasi BPS Wilson loop and the manifest zig-zag invariant regularization of the loop equation in the ASD variables. Yet, a posteriori, we find a simple reason for homological localization to occur: Γq flows to the ultraviolet by the localizing conformal transformation and thus, since it is AF , to vanishing coupling. We find that the beta function of Γq is saturated by the non-commutative ASD vortices of the EK reduction. An exact canonical beta function of NSV Z type that reproduces the universal first and second perturbative coefficients follows by the localization of Γq 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.