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Dimer models from mirror symmetry and quivering amoebae
Dimer models are 2-dimensional combinatorial systems that have been shown to encode the gauge groups, matter content and tree-level superpotential of the world-volume quiver gauge theories obtained by placing D3-branes at the tip of a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the quiver graph. However, the string theoretic explanation of this was unclear. In this paper we use mirror symmetry to shed light on this: the dimer models live on a T^2 subspace of the T^3 fiber that is involved in mirror symmetry and is wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the singular point, and geometrically encode the same quiver theory on their world-volume
Global symmetries and 't Hooft anomalies in brane tilings
We investigate the relation between gauge theories and brane configurations
described by brane tilings. We identify U(1)_B (baryonic), U(1)_M (mesonic),
and U(1)_R global symmetries in gauge theories with gauge symmetries in the
brane configurations. We also show that U(1)_MU(1)_B^2 and U(1)_RU(1)_B^2 't
Hooft anomalies are reproduced as gauge transformations of the classical brane
action.Comment: 41 pages, 6 figure
Flat Surfaces
Various problems of geometry, topology and dynamical systems on surfaces as
well as some questions concerning one-dimensional dynamical systems lead to the
study of closed surfaces endowed with a flat metric with several cone-type
singularities. Such flat surfaces are naturally organized into families which
appear to be isomorphic to the moduli spaces of holomorphic one-forms.
One can obtain much information about the geometry and dynamics of an
individual flat surface by studying both its orbit under the Teichmuller
geodesic flow and under the linear group action. In particular, the Teichmuller
geodesic flow plays the role of a time acceleration machine (renormalization
procedure) which allows to study the asymptotic behavior of interval exchange
transformations and of surface foliations.
This long survey is an attempt to present some selected ideas, concepts and
facts in Teichmuller dynamics in a playful way.Comment: (152 pages; 51 figures) Based on the lectures given by the author at
the Les Houches School "Number Theory and Physics" in March of 2003 and at
the workshop on dynamical systems in ICTP, Trieste, in July 2004. See
"Frontiers in Number Theory, Physics and Geometry. Volume 1: On random
matrices, zeta functions and dynamical systems'', P.Cartier; B.Julia;
P.Moussa; P.Vanhove (Editors), Springer-Verlag (2006) for the entire
collection (including, in particular, the complementary lectures of J.-C.
Yoccoz). For a short version see the paper "Geodesics on Flat Surfaces",
arXiv.math.GT/060939
Scaling limit of isoradial dimer models and the case of triangular quadri-tilings
32 pages, 4 figures (arXiv version, title has changed)International audienceWe consider dimer models on graphs which are bipartite, periodic and satisfy a geometric condition called {\em isoradiality}, defined in \cite{Kenyon3}. We show that the scaling limit of the height function of any such dimer model is times a Gaussian free field. Triangular quadri-tilings were introduced in \cite{Bea}; they are dimer models on a family of isoradial graphs arising form rhombus tilings. By means of two height functions, they can be interpreted as random interfaces in dimension 2+2. We show that the scaling limit of each of the two height functions is times a Gaussian free field, and that the two Gaussian free fields are independent
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