Gauge Theories and Dessins d'Enfants: Beyond the Torus

Brane Tilings represent one of the largest classes of superconformal theories with known gravity duals in 3+1 and also 2+1 dimensions. They provide a useful link between a large class of quiver gauge theories and their moduli spaces, which are the toric Calabi-Yau (CY) singularities. This thesis includes a discussion of an algorithm that can be used to generate all brane tilings with any given number of superpotential terms. All tilings with at most 8 superpotential terms h...

We propose a novel and simple method of computing the volume of the moduli space of BPS solitons in supersymmetric gauge theory. We use a D-brane realization of vortices and T-duality relation to domain walls. We there use a special limit where domain walls reduce to gas of hard (soft) one-dimensional rods for the Abelian (non-Abelian) cases. In the simpler cases of the Abelian-Higgs model on a torus, our results agree with exact results which are geometrically derived by an ...

We review recent developments in the theory of brane tilings and four-dimensional N=1 supersymmetric quiver gauge theories. This review consists of two parts. In part I, we describe foundations of brane tilings, emphasizing the physical interpretation of brane tilings as fivebrane systems. In part II, we discuss application of brane tilings to AdS/CFT correspondence and homological mirror symmetry. More topics, such as orientifold of brane tilings, phenomenological model buil...

We survey some of the AGT relations between N=2 gauge theories in four dimensions and geometric representations of symmetry algebras of two-dimensional conformal field theory on the equivariant cohomology of their instanton moduli spaces. We treat the cases of gauge theories on both flat space and ALE spaces in some detail, and with emphasis on the implications arising from embedding them into supersymmetric theories in six dimensions. Along the way we construct new toric non...

We establish a correspondence between generalized quiver gauge theories in four dimensions and congruence subgroups of the modular group, hinging upon the trivalent graphs which arise in both. The gauge theories and the graphs are enumerated and their numbers are compared. The correspondence is particularly striking for genus zero torsion-free congruence subgroups as exemplified by those which arise in Moonshine. We analyze in detail the case of index 24, where modular ellipt...

We classify subgroups of $\textrm{SL}(2,\mathbb{Z})$ up to conjugacy, which occur as monodromy groups of elliptically fibered K3 surfaces following a general strategy proposed by Bogomolov and Tschinkel. The essential step is the factorisation of the functional invariant $j$ with second factor a Belyi function of maximal possible degree and the classification of the corresponding subgroups $\bar\Gamma$ of $\textrm{PSL}(2,\mathbb{Z})$ using dessins d'enfants.

We introduce Orbifold Reduction, a new method for generating $2d$ $(0,2)$ gauge theories associated to D1-branes probing singular toric Calabi-Yau 4-folds starting from $4d$ $\mathcal{N}=1$ gauge theories on D3-branes probing toric Calabi-Yau 3-folds. The new procedure generalizes dimensional reduction and orbifolding. In terms of T-dual configurations, it generates brane brick models starting from brane tilings. Orbifold reduction provides an agile approach for generating $2...

We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi-Yau singularity. Our method combines information from the geometry and topology of Sasaki-Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quantum numbers, and superpotential of a superconformal gauge theory on D3-branes probing a toric Calabi-Yau singularity can be deduced. The infinite family of toric singularities with ...

We study a new duality which pairs 4d N=1 supersymmetric quiver gauge theories. They are represented by brane tilings and are worldvolume theories of D3 branes at Calabi-Yau 3-fold singularities. The new duality identifies theories which have the same combined mesonic and baryonic moduli space, otherwise called the master space. We obtain the associated Hilbert series which encodes both the generators and defining relations of the moduli space. We illustrate our findings with...

We propose a unified scheme for finding the hyperelliptic curve of $N=2$ SUSY YM theory with any Lie gauge groups. Our general scheme gives the well known results for classical gauge groups and exceptional $G_2$ group. In particular, we present the curve for the exceptional gauge groups $F_4, E_{6,7,8}$ and check consistency condition for them. The exact monodromies and the dyon spectrum of these theories are determined. We note that for any Lie gauge groups, the exact monodr...