Bipartita: Physics, Geometry & Number Theory

Ribbon graphs embedded on a Riemann surface provide a useful way to describe the double line Feynman diagrams of large N computations and a variety of other QFT correlator and scattering amplitude calculations, e.g in MHV rules for scattering amplitudes, as well as in ordinary QED. Their counting is a special case of the counting of bi-partite embedded graphs. We review and extend relevant mathematical literature and present results on the counting of some infinite classes of...

We study and classify regular and semi-regular tessellations of Riemann surfaces of various genera and investigate their corresponding supersymmetric gauge theories. These tessellations are generalizations of brane tilings, or bipartite graphs on the torus as well as the Platonic and Archimedean solids on the sphere. On higher genus they give rise to intricate patterns. Special attention will be paid to the master space and the moduli space of vacua of the gauge theory and to...

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...

Dessin d'Enfants on elliptic curves are a powerful way of encoding doubly-periodic brane tilings, and thus, of four-dimensional supersymmetric gauge theories whose vacuum moduli space is toric, providing an interesting interplay between physics, geometry, combinatorics and number theory. We discuss and provide a partial classification of the situation in genera other than one by computing explicit Belyi pairs associated to the gauge theories. Important also is the role of the...

The fundamental role of on-shell diagrams in quantum field theory has been recently recognized. On-shell diagrams, or equivalently bipartite graphs, provide a natural bridge connecting gauge theory to powerful mathematical structures such as the Grassmannian. We perform a detailed investigation of the combinatorial and geometric objects associated to these graphs. We mainly focus on their relation to polytopes and toric geometry, the Grassmannian and its stratification. Our w...

Brane tilings, sometimes called dimer models, are a class of bipartite graphs on a torus which encode the gauge theory data of four-dimensional SCFTs dual to D3-branes probing toric Calabi--Yau threefolds. An efficient way of encoding this information exploits the theory of dessin d'enfants, expressing the structure in terms of a permutation triple, which is in turn related to a Belyi pair, namely a holomorphic map from a torus to a P^1 with three marked points. The procedure...

We review and extend the progress made over the past few years in understanding the structure of toric quiver gauge theories; those which are induced on the world-volume of a stack of D3-branes placed at the tip of a toric Calabi-Yau cone, at an ``orbifold point'' in Kaehler moduli space. These provide an infinite class of four-dimensional N=1 superconformal field theories which may be studied in the context of the AdS/CFT correspondence. It is now understood that these gauge...

We study the algebraic structure of the mesonic moduli spaces of bipartite field theories by computing the Hilbert series. Bipartite field theories form a large family of 4d N=1 supersymmetric gauge theories that are defined by bipartite graphs on Riemann surfaces with boundaries. By calculating the Hilbert series, we are able to identify the generators and defining generator relations of the mesonic moduli spaces of these theories. Moreover, we show that certain bipartite fi...

We study D-brane instantons in systems of D3-branes at toric CY 3-fold singularities. The instanton effect can be described as a backreaction modifying the geometry of the mirror configuration, in which the breaking of $U(1)$ symmetries by the instanton translates into the recombination of gauge D-branes, which also directly generates the instanton-induced charged field theory operator. In this paper we describe the D-brane instanton backreaction in terms of a combinatorial o...

Brane tilings provide the most general framework in string and M-theory for matching toric Calabi-Yau singularities probed by branes with superconformal fixed points of quiver gauge theories. The brane tiling data consists of a bipartite tiling of the torus which encodes both the classical superpotential and gauge-matter couplings for the quiver gauge theory. We consider the class of tilings which contain only tiles bounded by exactly four edges and present a method for gener...