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Gauge invariant extremization

Recently, Duncan and Mawhinney introduced a method to find saddle points of the action in simulations of non-abelian lattice gauge theory. The idea, called `extremization', is to minimize $\int(δS/δA_μ)^2$ instead of the action $S$ itself as in conventional `cooling'. The method was implemented in an explicitly gauge variant way, however, and gauge dependence showed up in the results. Here we present a gauge invariant formulaton of extremization on the lattice, applicable to any gauge group and any lattice action. The procedure is worked out in detail for U(1) and SU(N) lattice gauge theory with the plaquette action.