qQCD2_2 and G/G model

The 2D lattice gauge theory with a quantum gauge group $SL_q(2)$ is considered. When $q=e^{i\frac{2\pi}{k+2}}$, its weak coupling partition function coincides with the one of the G/G coset model ({\em i.e.} equals the Verlinde numbers). However, despite such a remarkable coincidence, these models are not equivalent but, in some certain sense, dual to each other.Comment: 7pp, NBI-HE-93-27, revised. Small changes: several fixed inaccuracies + updated reference

Wilson loop on a sphere

We give the formula for a simple Wilson loop on a sphere which is valid for an arbitrary QCD$_2$ saddle-point $\rho(x)$: \mbox{$W(A_1,A_2)=\oint \frac{dx}{2\pi i} \exp(\int dy \frac{\rho(y)}{y-x}+A_2x)$}. The strong-coupling-phase solution is investigated.Comment: 10 pages, NBI-HE-93-5

Three-dimensional simplicial gravity and combinatorics of group presentations

We demonstrate how some problems arising in simplicial quantum gravity can be successfully addressed within the framework of combinatorial group theory. In particular, we argue that the number of simplicial 3-manifolds having a fixed homology type grows exponentially with the number of tetrahedra they are made of. We propose a model of 3D gravity interacting with scalar fermions, some restriction of which gives the 2-dimensional self-avoiding-loop-gas matrix model. We propose a qualitative picture of the phase structure of 3D simplicial gravity compatible with the numerical experiments and available analytical results.Comment: 24 page

Quantum Deformation of Lattice Gauge Theory

A quantum deformation of 3-dimensional lattice gauge theory is defined by applying the Reshetikhin-Turaev functor to a Heegaard diagram associated to a given cell complex. In the root-of-unity case, the construction is carried out with a modular Hopf algebra. In the topological (weak-coupling) limit, the gauge theory partition function gives a 3-fold invariant, coinciding in the simplicial case with the Turaev-Viro one. We discuss bounded manifolds as well as links in manifolds. By a dimensional reduction, we obtain a q-deformed gauge theory on Riemann surfaces and find a connection with the algebraic Alekseev-Grosse-Schomerus approach.Comment: 31 pp.; uses epic.sty and eepic.st

3D Gravity and Gauge Theories

I argue that the complete partition function of 3D quantum gravity is given by a path integral over gauge-inequivalent manifolds times the Chern-Simons partition function. In a discrete version, it gives a sum over simplicial complexes weighted with the Turaev-Viro invariant. Then, I discuss how this invariant can be included in the general framework of lattice gauge theory (qQCD$_3$). To make sense of it, one needs a quantum analog of the Peter-Weyl theorem and an invariant measure, which are introduced explicitly. The consideration here is limited to the simplest and most interesting case of $SL_q(2)$, $q=e^{i\frac{2\pi}{k+2}}$. At the end, I dwell on 3D generalizations of matrix models.Comment: 20 pp., NBI-HE-93-67 (Contribution to Proceedings of 1993 Cargese workshop

Orthogonal polynomial method and odd vertices in matrix models

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.Comment: 13 pages, 3 Postscript figure

Critical scaling in the matrix model on the Bethe tree

The matrix model with a Bethe-tree embedding space (coinciding at large $N$ with the Kazakov-Migdal ``induced QCD'' model \cite{KM}) is investigated. We further elaborate the Riemann-Hilbert approach of \rf{Mig1} assuming certain holomorphic properties of the solution. The critical scaling (an edge singularity of the density) is found to be \gamma_{str} = -\frac{1}{\pi} \arcos D, for $|D|<1$, and \gamma_{str} = -\frac{1}{\pi} \arcos \frac{D}{2D-1}, for $D>1$. Explicit solutions are constructed at $D=\frac{1}{2}$ and $D=\infty$.Comment: 12 pp., NBI-93-5