This work presents recent developments on brane tilings and their vacuum moduli
spaces.
Brane tilings are bipartite periodic graphs on the torus and represent 4d N = 1
supersymmetric worldvolume theories living on D3-branes probing Calabi-Yau 3-fold
singularities. The graph and combinatorial properties of brane tilings make the set
of supersymmetric quiver theories represented by them one of the largest and richest
known so far. The aim of this work is to give a concise pedagogical introduction to brane
tilings and a summary on recent exciting advancement on their classification, dualities
and construction.
At first, particular focus is given on counting distinct Abelian orbifolds of the form
C3/[gamma]. The presented counting of Abelian orbifolds of C3 and in more general of CD gives a first insight on the rich combinatorial nature of brane tilings. Following the classification theme, the work proceeds with the identification of all brane tilings whose
mesonic moduli spaces as toric Calabi-Yau 3-folds are represented by reflexive polygons.
There are 16 of these special convex lattice polygons. It is shown that 30 brane tilings
are associated with them. Some of these brane tilings are related by a correspondence
known as toric duality.
The classification of brane tilings with reflexive toric diagrams led to the discovery
of a new correspondence between brane tilings which we call specular duality. The
new correspondence identifies brane tilings with the same master space - the combined
mesonic and baryonic moduli space. As a by-product, the new correspondence paves
the way for constructing brane tilings which are not confined to the torus but are on
Riemann surfaces with arbitrary genus. We give the first classification of genus 2 brane
tilings, illustrate the corresponding supersymmetric quiver theories and analyse their
vacuum moduli spaces.Open Acces