We study a c=-2 conformal field theory coupled to two-dimensional quantum gravity by means of dynamical triangulations. We define the geodesic distance r on the triangulated surface with N triangles, and show that dim[r^{d_H}]= dim[N], where the fractal dimension d_H = 3.58 +/- 0.04. This result lends support to the conjecture d_H = -2\alpha_1/\alpha_{-1}, where \alpha_{-n} is the gravitational dressing exponent of a spin-less primary field of conformal weight (n+1,n+1), and it disfavors the alternative prediction d_H = -2/\gamma_{str}. On the other hand, we find dim[l] = dim[r^2] with good accuracy, where l is the length of one of the boundaries of a circle with (geodesic) radius r, i.e. the length l has an anomalous dimension relative to the area of the surface. It is further shown that the spectral dimension d_s = 1.980 +/- 0.014 for the ensemble of (triangulated) manifolds used. The results are derived using finite size scaling and a very efficient recursive sampling technique known previously to work well for c=-2.Comment: 12 pages, LaTeX, 4 figures using psfig.sty and epsf.st
Monte-Carlo simulations of abelian projection of T = 0 pure lattice QCD show that 1) Polyakov loops written in terms of abelian link fields alone play a role of an order parameter of deconfinement transition, 2) the abelian Polyakov loops are decomposed into contributions from Dirac strings of monopoles and from photons, 3) vanishing of the abelian Polyakov loops in the confinement phase is due to the Dirac strings alone and the photons give a finite contribution in both phases. Moreover, these results appear to hold good with any abelian projection as seen from the studies in the maximally abelian gauge and in various unitary gauges. *
We study the fractal structure of space-time of two-dimensional quantum gravity coupled to c = −2 conformal matter by means of computer simulations. We find that the intrinsic Hausdorff dimension d H = 3.58 ± 0.04. This result supports the conjecture d H = −2α 1 /α −1 , where α n is the gravitational dressing exponent of a spinless primary field of conformal weight (n + 1, n + 1), and it disfavours the alternative prediction d H = 2/|γ|. On the other hand l n ∼ r 2n for n > 1 with good accuracy, i.e. the the boundary length l has an anomalous dimension relative to the area of the surface.
We numerically investigate the fractal structure of two-dimensional quantum gravity coupled to matter central charge c for −2 ≤ c ≤ 1. We reformulate Q-state Potts model into the model which can be identified as a weighted percolation cluster model and can make continuous change of Q, which relates c, on the dynamically triangulated lattice. The c-dependence of the critical coupling is measured from the percolation probability and susceptibility. The c-dependence of the string susceptibility of the quantum surface is evaluated and has very good agreement with the theoretical predictions. The c-dependence of the fractal dimension based on the finite size scaling hypothesis is measured and has excellent agreement with one of the theoretical predictions previously proposed except for the region near c ≈ 1.
Monte-Carlo simulations of abelian projection in $T \neq 0$ pure lattice QCD show that 1)\ Polyakov loops written in terms of abelian link fields alone play a role of an order parameter of deconfinement transition 2)\ the abelian Polyakov loops are decomposed into contributions from Dirac strings of monopoles and from photons 3)\ vanishing of the abelian Polyakov loops in the confinement phase is due to the Dirac strings alone and the photons give a finite contribution in both phases. Moreover, these results appear to hold good in unitary gauges. This suggests that monopole condensation as the color confinement mechanism is gauge independent.Comment: 3 pages (4 figures). Contribution to Lattice '9
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