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.

We use field theory and brane diamond techniques to demonstrate that Toric Duality is Seiberg duality for N = 1 theories with toric moduli spaces. This resolves the puzzle concerning the physical meaning of Toric Duality as proposed in our earlier work. Furthermore, using this strong connection we arrive at three new phases which can not be thus far obtained by the so-called "Inverse Algorithm" applied to partial resolution of C 3 /(Z Z 3 × Z Z 3 ). The standing proposals of Seiberg duality as diamond duality in the work by Aganagic-Karch-Lüst-Miemiec are strongly supported and new diamond configurations for these singularities are obtained as a byproduct. We also make some remarks about the relationships between Seiberg duality and Picard-Lefschetz monodromy.

Techniques are presented for computing the cohomology of stable, holomorphic vector bundles over elliptically fibered Calabi-Yau threefolds. These cohomology groups explicitly determine the spectrum of the low energy, four-dimensional theory. Generic points in vector bundle moduli space manifest an identical spectrum. However, it is shown that on subsets of moduli space of co-dimension one or higher, the spectrum can abruptly jump to many different values. Both analytic and numerical data illustrating this phenomenon are presented. This result opens the possibility of tunneling or phase transitions between different particle spectra in the same heterotic compactification.In the course of this discussion, a classification of SU (5) GUT theories within a specific context is presented. *

Harnessing the unimodular degree of freedom in the definition of any toric diagram, we present a method of constructing inequivalent gauge theories which are world-volume theories of D-branes probing the same toric singularity. These theories are various phases in partial resolution of Abelian orbifolds. As examples, two phases are constructed for both the zeroth Hirzebruch and the second del Pezzo surfaces. We show that such a phenomenon is a special case of "Toric Duality" proposed in hep-th/0003085. Furthermore, we investigate the general conditions that distinguish these different gauge theories with the same (toric) moduli space.

We study orbifolds of N = 4 U (n) super-Yang-Mills theory given by discrete subgroups of SU (2) and SU (3). We have reached many interesting observations that have graph-theoretic interpretations. For the subgroups of SU (2), we have shown how the matter content agrees with current quiver theories and have offered a possible explanation. In the case of SU (3) we have constructed a catalogue of candidates for finite (chiral) N = 1 theories, giving the gauge group and matter content. Finally, we conjecture a McKay-type correspondence for Gorenstein singularities in dimension 3 with modular invariants of WZW conformal models. This implies a connection between a class of finite N = 1 supersymmetric gauge theories in four dimensions and the classification of affine SU (3) modular invariant partition functions in two dimensions.