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I adapt a numerical method, previously applied to investigate the Yang-Millsvacuum wavefunctional, to the problem of extracting the effective Polyakov lineaction from SU(N) lattice gauge theories, with or without matter fields. Themethod can be used to find the variation of the effective Polyakov line actionalong any trajectory in field configuration space; this information issufficient to determine the potential term in the action, and stronglyconstrains the possible form of the kinetic term. The technique is illustratedfor both pure and gauge-Higgs SU(2) lattice gauge theory at finite temperature.A surprise, in the pure gauge theory, is that the potential of thecorresponding Polyakov line action contains a non-analytic (yetcenter-symmetric) term proportional to |P|^3, where P is the trace of thePolyakov line at a given point, in addition to the expected analytic termsproportional to even powers of P.

The potential of the effective Polyakov line action from the underlying lattice gauge theory