138 research outputs found
Almost Flat Planar Diagrams
We continue our study of matrix models of dually weighted graphs. Among the
attractive features of these models is the possibility to interpolate between
ensembles of regular and random two-dimensional lattices, relevant for the
study of the crossover from two-dimensional flat space to two-dimensional
quantum gravity. We further develop the formalism of large character
expansions. In particular, a general method for determining the large limit
of a character is derived. This method, aside from being potentially useful for
a far greater class of problems, allows us to exactly solve the matrix models
of dually weighted graphs, reducing them to a well-posed Cauchy-Riemann
problem. The power of the method is illustrated by explicitly solving a new
model in which only positive curvature defects are permitted on the surface, an
arbitrary amount of negative curvature being introduced at a single insertion.Comment: harvmac.tex and pictex.tex. Must be compiled "big". Diagrams are
written directly into the text in pictex command
Two-Matrix model with ABAB interaction
Using recently developed methods of character expansions we solve exactly in
the large N limit a new two-matrix model of hermitean matrices A and B with the
action S={1\over 2}(\tr A^2+\tr B^2)-{\alpha\over 4}(\tr A^4+\tr B^4)
-{\beta\over 2} \tr(AB)^2. This model can be mapped onto a special case of the
8-vertex model on dynamical planar graphs. The solution is parametrized in
terms of elliptic functions. A phase transition is found: the critical point is
a conformal field theory with central charge c=1 coupled to 2D quantum gravity.Comment: harvmac, 24 pages, 5 figures (1 color figure
Perturbative analysis of gauged matrix models
We analyze perturbative aspects of gauged matrix models, including those where classically the gauge symmetry is partially broken. Ghost fields play a crucial role in the Feynman rules for these vacua. We use this formalism to elucidate the fact that nonperturbative aspects of [script N] = 1 gauge theories can be computed systematically using perturbative techniques of matrix models, even if we do not possess an exact solution for the matrix model. As examples we show how the Seiberg-Witten solution for [script N] = 2 gauge theory, the Montonen-Olive modular invariance for [script N] = 1*, and the superpotential for the Leigh-Strassler deformation of [script N] = 4 can be systematically computed in perturbation theory of the matrix model or gauge theory (even though in some of these cases an exact answer can also be obtained by summing up planar diagrams of matrix models)
Advances in large N group theory and the solution of two-dimensional R gravity
We review the recent exact solution of a matrix model which interpolates between flat and random lattices. The importance of the results is twofold: Firstly, we have developed a new large N technique capable of treating a class of matrix models previously thought to be unsolvable. Secondly, we are able to make a first precise statement about two-dimensional R^2 gravity. These notes are based on a lecture given at the Cargese summer school 1995. They contain some previously unpublished results
Character Expansion Methods for Matrix Models of Dually Weighted Graphs
We consider generalized one-matrix models in which external fields allow
control over the coordination numbers on both the original and dual lattices.
We rederive in a simple fashion a character expansion formula for these models
originally due to Itzykson and Di Francesco, and then demonstrate how to take
the large N limit of this expansion. The relationship to the usual matrix model
resolvent is elucidated. Our methods give as a by-product an extremely simple
derivation of the Migdal integral equation describing the large limit of
the Itzykson-Zuber formula. We illustrate and check our methods by analyzing a
number of models solvable by traditional means. We then proceed to solve a new
model: a sum over planar graphs possessing even coordination numbers on both
the original and the dual lattice. We conclude by formulating equations for the
case of arbitrary sets of even, self-dual coupling constants. This opens the
way for studying the deep problem of phase transitions from random to flat
lattices.Comment: 22 pages, harvmac.tex, pictex.tex. All diagrams written directly into
the text in Pictex commands. (Two minor math typos corrected.
Acknowledgements added.
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