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This paper outlines the strategy for computing the theta-dependence in nonabelian gauge theories beyond a semiclassical or steepest descent approximation. It involves isolating the relevant degrees of freedom including the sphaleron configuration for tunnelling across a classical potential barrier. Two approaches are discussed in the context of spherical geometries. The first is based on a hamiltonian version of the streamline or valley equation. The second, which in our opinion is far more efficient, is based on implementing theta-dependence through appropriate boundary conditions in configuration space. In a good approximation these can be formulated at the level of 15 (+3 gauge) modes, that are degenerate to lowest order in perturbation theory, while keeping all other modes gaussian.
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