QCD lattice simulations with 2+1 flavours (when two quark flavours are mass degenerate) typically start at rather large up-down and strange quark masses and extrapolate first the strange quark mass and then the up-down quark mass to its respective physical value. Here we discuss an alternative method of tuning the quark masses, in which the singlet quark mass is kept fixed. Using group theory the possible quark mass polynomials for a Taylor expansion about the flavour symmetric line are found, first for the general 1 + 1 + 1 flavour case and then for the 2 + 1 flavour case. This ensures that the kaon always has mass less than the physical kaon mass. This method of tuning quark masses then enables highly constrained polynomial fits to be used in the extrapolation of hadron masses to their physical values. Numerical results for the 2 + 1 flavour case confirm the usefulness of this expansion and an extrapolation to the physical pion mass gives hadron mass values to within a few percent of their experimental values. Singlet quantities remain constant which allows the lattice spacing to be determined from hadron masses (without necessarily being at the physical point). Furthermore an extension of this programme to include partially quenched results is given.

We simulate a supersymmetric matrix model obtained from dimensional reduction of 4d SU(N ) super Yang-Mills theory. The model is well defined for finite N and it is found that the large N limit obtained by keeping g 2 N fixed gives rise to well defined operators which represent string amplitudes. The space-time structure which arises dynamically from the eigenvalues of the bosonic matrices is discussed, as well as the effect of supersymmetry on the dynamical properties of the model. Eguchi-Kawai equivalence of this model to ordinary gauge theory does hold within a finite range of scale. We report on new simulations of the bosonic model for N up to 768 that confirm this property, which comes as a surprise since no quenching or twist is introduced.

We use perturbation theory to construct perfect lattice actions for quarks and gluons. The renormalized trajectory for free massive quarks is identi ed by blocking directly from the continuum. We tune a parameter in the renormalization group transformation such that for 1-d con gurations the perfect action reduces to the nearest neighbor Wilson fermion action. The xed point action for free gluons is also obtained by blocking from the continuum. For 2-d con gurations it reduces to the standard plaquette action. Classically perfect quark and gluon elds, quark-gluon composite operators and vector and axial vector currents are constructed as well. Also the quark-antiquark potential is derived from the classically perfect Polyakov loop. The quark-gluon and 3-gluon perfect vertex functions are determined to leading order in the gauge coupling. We also construct a new block factor n renormalization group transformation for QCD that allows to extend our results beyond perturbation theory. For weak elds it leads to the same perfect action as blocking from the continuum. For arbitrarily strong 2-d Abelian gauge elds the Manton plaquette action is classically perfect for this transformation.

The 2D O(3) model with a 9 vacuum term is formulated in terms of Wolff clusters. Each cluster carries an integer or half-integer topological charge. The clusters with charge~1/2 are identified as merons. At 0 =~t he merons are bound in pairs inducing a second order phase transition at which the mass gap vanishes. The construction of an improved estimator for the topological charge distribution makes numerical simulations of the phase transition feasible. The measured critical exponents agree with those of the k = 1 Wess-Zumino-Novikov-Witten (WZNW) model. Our results are consistent with Haldane's conjecture fro 1D antiferromagnetic quantum spin chains. PACS numbers: 75.10.Jm Some time ago Haldane conjectured [1] that integer and half-integer 1D antiferromagnetic quantum spin chains behave qualitatively differently. While integer spin chains have a mass gap, half-integer chains should be gapless.This has been confirmed numerically for finite chains of spin 1 and spin 2 [2,3] and analytically for half-integer spins and for spin 1 [4]. The long-range physics of 1D quantum spin chains is described by an effective 2D classical O(3) model. Haldane argued that the effective action for a chain of spins S contains a topological term iOQ Here Q . is the topological charge and 9 = 2n5 is the vacuum angle. Since the physics is periodic in 0, i.e. , 0 E] -~, 7r], integer spins have 0 = 0 and half-integer spins have 0 = vr The st.andard O(3) model at 0 = 0 has a mass gap in agreement with Haldane's conjecture. On the other hand, Haldane's conjecture together with the (nonrigorous) mapping of spin chains on the O(3) model imply that the mass gap disappears at 0 = m. This corresponds to a phase transition in the vacuum angle. Because of the complex action it is notoriously difficult to simulate 0 vacua numerically. A previous numerical study that was limited to~0~( 0. 8m. found no phase transition in that region [5]. In fact, Haldane's conjecture has not yet been verified in the context of the O(3) model. In this paper we use the Wolff cluster algorithm [6] combined with a reweighting technique [7] to attack this problem. The construction of an improved estimator for the topological charge distribution enables us to simulate 0 vacua reliably for any value of 0. Affleck and Haldane have suggested a dynamical mechanism that explains why the mass gap disappears at 0 = vr [3]. In this picture pseudoparticles with topological charge~1/2 -so-called merons -are the relevant degrees of freedom. At 0 = 0 the merons form an ideal gas. They disorder the system and thereby give nonzero mass to the physical particles. At 0 =~, on the other hand, the merons are bound in pairs and thus do not generate mass. Affleck confirmed this picture in a model where the O(3) symmetry is explicitly broken to O(2). Then the merons behave like vortices, and the phase transition in 0 is analogous to the Kosterlitz-Thouless transition of the O(2) model. When the explicit O(3) breaking is switched off, it is unclear if this dynamical picture still holds. In fac...

We describe a number of aspects in our attempt to construct an approximately perfect lattice action for QCD. Free quarks are made optimally local on the whole renormalized trajectory and their couplings are then truncated by imposing 3-periodicity. The spectra of these short ranged fermions are excellent approximations to continuum spectra. The same is true for free gluons. We evaluate the corresponding perfect quark-gluon vertex function, identifying in particular the \perfect clover term". First simulations for heavy quarks show that the mass is strongly renormalized, but again the renormalized theory agrees very well with continuum physics. Furthermore we describe the multigrid formulation for the non-perturbative perfect action and we present the concept of an exactly (quantum) perfect topological charge on the lattice.