Multiplicities of Toric Diagrams, Kastelayn matrices & generating functions.
Abstract for the series
When D branes probe a singular CY manifold the gauge theory on the world volume of the brane gets dramatically changed. The details depend on the nature of the singularity and in all known cases lead to a quiver gauge theory. For toric singularities the gauge fields and matter fields are relatively easy to compute and there exist few equivalent methods for the computation. It has been a long standing problem to compute the superpotential which specifies how the matter fields couple to each other. In these lectures I will present a new concept - dimers - and will show how they simplify and solve the problem of computing the superpotential. Along the way we will find a new set of integers associated with a toric diagram, denoting the multiplicities of linear sigma model fields. The quiver gauge theories flow in the infrared to a superconformal fixed point, the details of it are captured using the AdS/CFT correspondence - A dual Sasaki-Einstein manifold arises for which some of its properties are exactly derived using the gauge theory description.