Once more on the Witten index of 3d supersymmetric YM-CS theory

The problem of counting the vacuum states in the supersymmetric 3d Yang-Mills-Chern-Simons theory is reconsidered. We resolve the controversy between its original calculation by Witten at large volumes and the calculation based on the evaluation of the effective Lagrangian in the small volume limit. We show that the latter calculation suffers from uncertainties associated with the singularities in the moduli space of classical vacua where the Born-Oppenheimer approximation breaks down. We also show that these singularities can be accurately treated in the Hamiltonian Born-Oppenheimer method, where one has to match carefully the effective wave functions on the Abelian valley and the wave functions of reduced non-Abelian QM theory near the singularities. This gives the same result as original Witten's calculation.Comment: 27 page

Noncomputability Arising In Dynamical Triangulation Model Of Four-Dimensional Quantum Gravity

Computations in Dynamical Triangulation Models of Four-Dimensional Quantum Gravity involve weighted averaging over sets of all distinct triangulations of compact four-dimensional manifolds. In order to be able to perform such computations one needs an algorithm which for any given $N$ and a given compact four-dimensional manifold $M$ constructs all possible triangulations of $M$ with $\leq N$ simplices. Our first result is that such algorithm does not exist. Then we discuss recursion-theoretic limitations of any algorithm designed to perform approximate calculations of sums over all possible triangulations of a compact four-dimensional manifold.Comment: 8 Pages, LaTex, PUPT-132

The critical behaviour of Ising spins on 2D Regge lattices

We performed a high statistics simulation of Ising spins coupled to 2D quantum gravity on toroidal geometries. The tori were triangulated using the Regge calculus approach and contained up to $512^2$ vertices. We used a constant area ensemble with an added $R^2$ interaction term, employing the $dl/l$ measure. We find clear evidence that the critical exponents of the Ising phase transition are consistent with the static critical exponents and do not depend on the coupling strength of the $R^2$ interaction term. We definitively can exclude for this type of model a behaviour as predicted by Boulatov and Kazakov [Phys. Lett. {\bf B186}, 379 (1987)] for Ising spins coupled to dynamically triangulated surfaces.Comment: 15 pages with 3 figures in form of an uudecoded compressed tar-ps-file. FUB-HEP 06/9

Dynamically Triangulated Ising Spins in Flat Space

A model describing Ising spins with short range interactions moving randomly in a plane is considered. In the presence of a hard core repulsion, which prevents the Ising spins from overlapping, the model is analogous to a dynamically triangulated Ising model with spins constrained to move on a flat surface. It is found that as a function of coupling strength and hard core repulsion the model exhibits multicritical behavior, with first and second order transition lines terminating at a tricritical point. The thermal and magnetic exponents computed at the tricritical point are consistent with the exact two-matrix model solution of the random Ising model, introduced previously to describe the effects of fluctuating geometries.Comment: (10 pages + 4 figures), CERN-Th-7577/9

String Effects in the Wilson Loop: a high precision numerical test

We test numerically the effective string description of the infrared limit of lattice gauge theories in the confining regime. We consider the 3d Z(2) lattice gauge theory, and we define ratios of Wilson loops such that the predictions of the effective string theory do not contain any adjustable parameters. In this way we are able to obtain a degree of accuracy high enough to show unambiguously that the flux--tube fluctuations are described, in the infrared limit, by an effective bosonic string theory.Comment: 19 pages, LaTeX file + two .eps figure

Measuring the string susceptibility in 2D simplicial quantum gravity using the Regge approach

We use Monte Carlo simulations to study pure 2D Euclidean quantum gravity with $R^2$-interaction on spherical topologies, employing Regge's formulation. We attempt to measure the string susceptibility exponent $\gamma_{\rm str}$ by using a finite-size scaling Ansatz in the expectation value of $R^2$, as has been done in a previous study by Bock and Vink ( hep-lat/9406018 ). By considerably extending the range and statistics of their study we find that this Ansatz is plagued by large systematic errors. The $R^2$ specific string susceptibility exponent \GS' is found to agree with theoretical predictions, but its determination also is subject to large systematic errors and the presence of finite-size scaling corrections. To circumvent this obstacle we suggest a new scaling Ansatz which in principle should be able to predict both, \GS and \GS'. First results indicate that this requires large system sizes to reduce the uncertainties in the finite-size scaling Ans\"atze. Nevertheless, our investigation shows that within the achievable accuracy the numerical estimates are still compatible with analytic predictions, contrary to the recent claim by Bock and Vink.Comment: 33 pages, self unpacking uuencoded PostScript file, including all the figures. Paper also available at http://www.physik.fu-berlin.de/~holm

Screening and Deconfinement of Sources in Finite Temperature SU(2) Lattice Gauge Theory

Deconfinement and screening of higher-representation sources in finite-temperature $SU(2)$ lattice gauge theory is investigated by both analytical and numerical means. The effective Polyakov-line action at strong coupling is simulated by an efficient cluster-updating Monte Carlo algorithm for the case of $d\!=\!4$ dimensions. The results compare very favourably with an improved mean-field solution. The limit $d\!\to\!\infty$ of the $SU(2)$ theory is shown to be highly singular as far as critical behaviour is concerned. In that limit the leading amplitudes of higher representation Polyakov lines vanish at strong coupling, and subleading exponents become dominant. Each of the higher-representation sources then effectively carry with them their own critical exponents.Comment: 13pages+7figures, CERN-TH-7222/94 One reference added, else unchange

Multiple Potts Models Coupled to Two-Dimensional Quantum Gravity

We perform Monte Carlo simulations using the Wolff cluster algorithm of {\it multiple} $q=2,3,4$ state Potts models on dynamical phi-cubed graphs of spherical topology in order to investigate the $c>1$ region of two-dimensional quantum gravity. Contrary to naive expectation we find no obvious signs of pathological behaviour for $c>1$. We discuss the results in the light of suggestions that have been made for a modified DDK ansatz for $c>1$.Comment: 9 page

The deconfinement transition of finite density QCD with heavy quarks from strong coupling series

Starting from Wilson's action, we calculate strong coupling series for the Polyakov loop susceptibility in lattice gauge theories for various small N_\tau in the thermodynamic limit. Analysing the series with Pad\'e approximants, we estimate critical couplings and exponents for the deconfinement phase transition. For SU(2) pure gauge theory our results agree with those from Monte-Carlo simulations within errors, which for the coarser N_\tau=1,2 lattices are at the percent level. For QCD we include dynamical fermions via a hopping parameter expansion. On a N_\tau=1 lattice with N_f=1,2,3, we locate the second order critical point where the deconfinement transition turns into a crossover. We furthermore determine the behaviour of the critical parameters with finite chemical potential and find the first order region to shrink with growing \mu. Our series moreover correctly reflects the known Z(N) transition at imaginary chemical potential.Comment: 18 pages, 7 figures, typos corrected, version published in JHE

F-Theorem without Supersymmetry

The conjectured F-theorem for three-dimensional field theories states that the finite part of the free energy on S^3 decreases along RG trajectories and is stationary at the fixed points. In previous work various successful tests of this proposal were carried out for theories with {\cal N}=2 supersymmetry. In this paper we perform more general tests that do not rely on supersymmetry. We study perturbatively the RG flows produced by weakly relevant operators and show that the free energy decreases monotonically. We also consider large N field theories perturbed by relevant double trace operators, free massive field theories, and some Chern-Simons gauge theories. In all cases the free energy in the IR is smaller than in the UV, consistent with the F-theorem. We discuss other odd-dimensional Euclidean theories on S^d and provide evidence that (-1)^{(d-1)/2} \log |Z| decreases along RG flow; in the particular case d=1 this is the well-known g-theorem.Comment: 34 pages, 2 figures; v2 refs added, minor improvements; v3 refs added, improved section 4.3; v4 minor improvement