We determine the topological susceptibility χ at T = 0 in pure SU (3) gauge theory and its behaviour at finite T across the deconfining transition. We use an improved topological charge density operator. χ drops sharply by one order of magnitude at the deconfining temperature T c .

We compute the running QCD coupling on the lattice by evaluating two-point and three-point off-shell gluon Green's functions in a fixed gauge and imposing non-perturbative renormalisation conditions on them. Our exploratory study is performed in the quenched approximation at β = 6.0 on 16 4 and 24 4 lattices. We show that, for momenta in the range 1.8 − 2.3 GeV, our coupling runs according to the two-loop asymptotic formula, allowing a precise determination of the corresponding Λ parameter. The role of lattice artifacts and finite-volume effects is carefully analysed and these appear to be under control in the momentum range of interest. Our renormalisation procedure corresponds to a momentum subtraction scheme in continuum field theory, and therefore lattice perturbation theory is not needed in order to match our results to the M S scheme, thus eliminating a major source of uncertainty in the determination of α M S . Our method can be applied directly to the unquenched case.2

We investigate the performance of the hybrid Monte Carlo algorithm, the standard algorithm used for lattice QCD simulations involving fermions, in updating non-trivial global topological structures. We find that the hybrid Monte Carlo algorithm has serious problems decorrelating the global topological charge. This represents a warning which must be seriously considered when simulating full QCD by hybrid Monte Carlo.

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The O(N)-symmetric generalization of h94 theory is studied in the Gaussian-effective-potential (GEP) approach. It is shown that the GEP encompasses and transcends the leading-order 1 / N expansion results. We find two distinct, nontrivial versions of the (3 + 1)-dimensional theory: (i) "precarious 94 theory," with negative, infinitesimal AB, which coincides with the 1 / N result; and (ii) "autonomous b4 theory," with positive, infinitesimal hB and an infinite wave-function renormalization, which is inaccessible to perturbative or 1 / N methods, and could well have eluded lattice-based approaches. "Autonomous d4 theory" can exhibit spontaneous symmetry breaking, and we speculate on its relevance for the Higgs mechanism.