Topology and Confinement in SU(N) Gauge Theories

The large N limit of SU(N) gauge theories in 3+1 dimensions is investigated on the lattice by extrapolating results obtained for $2 \le N \le 5$. A numerical determination of the masses of the lowest-lying glueball states and of the topological susceptibility in the limit $N\to\infty$ is provided. Ratios of the tensions of stable k-strings over the tension of the fundamental string are investigated in various regimes and the results are compared with expectations based on several scenarios -- in particular MQCD and Casimir scaling. While not conclusive at zero temperature in D=3+1, in the other cases investigated our data seem to favour the latter.Comment: 3 pages, 2 figures; talk presented by B. Lucini at Lattice2001(confinement

On the Connection Between 2d Topological Gravity and the Reduced Hermitian Matrix Model

We discuss how concepts such as geodesic length and the volume of space-time can appear in 2d topological gravity. We then construct a detailed mapping between the reduced Hermitian matrix model and 2d topological gravity at genus zero. This leads to a complete solution of the counting problem for planar graphs with vertices of even coordination number. The connection between multi-critical matrix models and multi-critical topological gravity at genus zero is studied in some detail.Comment: 29 pages, LaTe

The geometry of dynamical triangulations

We discuss the geometry of dynamical triangulations associated with 3-dimensional and 4-dimensional simplicial quantum gravity. We provide analytical expressions for the canonical partition function in both cases, and study its large volume behavior. In the space of the coupling constants of the theory, we characterize the infinite volume line and the associated critical points. The results of this analysis are found to be in excellent agreement with the MonteCarlo simulations of simplicial quantum gravity. In particular, we provide an analytical proof that simply-connected dynamically triangulated 4-manifolds undergo a higher order phase transition at a value of the inverse gravitational coupling given by 1.387, and that the nature of this transition can be concealed by a bystable behavior. A similar analysis in the 3-dimensional case characterizes a value of the critical coupling (3.845) at which hysteresis effects are present.Comment: 166 pages, Revtex (latex) fil

Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results

We review some recent attempts to extract information about the nature of quantum gravity, with and without matter, by quantum field theoretical methods. More specifically, we work within a covariant lattice approach where the individual space-time geometries are constructed from fundamental simplicial building blocks, and the path integral over geometries is approximated by summing over a class of piece-wise linear geometries. This method of ``dynamical triangulations'' is very powerful in 2d, where the regularized theory can be solved explicitly, and gives us more insights into the quantum nature of 2d space-time than continuum methods are presently able to provide. It also allows us to establish an explicit relation between the Lorentzian- and Euclidean-signature quantum theories. Analogous regularized gravitational models can be set up in higher dimensions. Some analytic tools exist to study their state sums, but, unlike in 2d, no complete analytic solutions have yet been constructed. However, a great advantage of our approach is the fact that it is well-suited for numerical simulations. In the second part of this review we describe the relevant Monte Carlo techniques, as well as some of the physical results that have been obtained from the simulations of Euclidean gravity. We also explain why the Lorentzian version of dynamical triangulations is a promising candidate for a non-perturbative theory of quantum gravity.Comment: 69 pages, 16 figures, references adde

Projectors, matrix models and noncommutative monopoles

We study the interconnection between the finite projective modules for a fuzzy sphere, determined in a previous paper, and the matrix model approach, making clear the physical meaning of noncommutative topological configurations.Comment: 22pages, LaTeX, no figure

Shape transformations of a model of self-avoiding triangulated surfaces of sphere topology

We study a surface model with a self-avoiding (SA) interaction using the canonical Monte Carlo simulation technique on fixed-connectivity (FC) triangulated lattices of sphere topology. The model is defined by an area energy, a deficit angle energy, and the SA potential. A pressure term is also included in the Hamiltonian. The volume enclosed by the surface is well defined because of the self-avoidance. We focus on whether or not the interaction influences the phase structure of the FC model under two different conditions of pressure ${\it \Delta} p$; zero and small negative. The results are compared with the previous results of the self-intersecting model, which has a rich variety of phases; the smooth spherical phase, the tubular phase, the linear phase, and the collapsed phase. We find that the influence of the SA interaction on the multitude of phases is almost negligible except for the evidence that no crumpled surface appears under {\it \Delta} p\=\0 at least even in the limit of zero bending rigidity \alpha\to \0. The Hausdorff dimension is obtained in the limit of \alpha\to \0 and compared with previous results of SA models, which are different from the one in this paper.Comment: 9 figure

Higher Genus Correlators for the Complex Matrix Model

We describe an iterative scheme which allows us to calculate any multi-loop correlator for the complex matrix model to any genus using only the first in the chain of loop equations. The method works for a completely general potential and the results contain no explicit reference to the couplings. The genus $g$ contribution to the $m$--loop correlator depends on a finite number of parameters, namely at most $4g-2+m$. We find the generating functional explicitly up to genus three. We show as well that the model is equivalent to an external field problem for the complex matrix model with a logarithmic potential.Comment: 17 page

Kertesz on Fat Graphs?

The identification of phase transition points, beta_c, with the percolation thresholds of suitably defined clusters of spins has proved immensely fruitful in many areas of statistical mechanics. Some time ago Kertesz suggested that such percolation thresholds for models defined in field might also have measurable physical consequences for regions of the phase diagram below beta_c, giving rise to a ``Kertesz line'' running between beta_c and the bond percolation threshold, beta_p, in the M, beta plane. Although no thermodynamic singularities were associated with this line it could still be divined by looking for a change in the behaviour of high-field series for quantities such as the free energy or magnetisation. Adler and Stauffer did precisely this with some pre-existing series for the regular square lattice and simple cubic lattice Ising models and did, indeed, find evidence for such a change in high-field series around beta_p. Since there is a general dearth of high-field series there has been no other work along these lines. In this paper we use the solution of the Ising model in field on planar random graphs by Boulatov and Kazakov to carry out a similar exercise for the Ising model on random graphs (i.e. coupled to 2D quantum gravity). We generate a high-field series for the Ising model on $\Phi^4$ random graphs and examine its behaviour for evidence of a Kertesz line

Phase Structure of Dynamical Triangulation Models in Three Dimensions

The dynamical triangulation model of three-dimensional quantum gravity is shown to have a line of transitions in an expanded phase diagram which includes a coupling mu to the order of the vertices. Monte Carlo renormalization group and finite size scaling techniques are used to locate and characterize this line. Our results indicate that for mu < mu1 ~ -1.0 the model is always in a crumpled phase independent of the value of the curvature coupling. For mu < 0 the results are in agreement with an approximate mean field treatment. We find evidence that this line corresponds to first order transitions extending to positive mu. However, the behavior appears to change for mu > mu2 ~ 2-4. The simplest scenario that is consistent with the data is the existence of a critical end point

The Crumpling Transition Revisited

The ``crumpling" transition, between rigid and crumpled surfaces, has been object of much discussion over the past years. The common lore is that such transition should be of second order. However, some lattice versions of the rigidity term on fixed connectivity surfaces seem to suggest that the transition is of higher order instead. While some models exhibit what appear to be lattice artifacts, others are really indistiguishable from models where second order transitions have been reported and yet appear to have third order transitions.Comment: Contribution to Lattice 92. 4 pages. espcrc2.sty file included. 6 figures upon request. UB-ECM-92/30 and UAB-FT-29