Abstract
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 how their geometry is determined by the tessellations.
Highlights
(BFT) [8,9,10,11,12] to relations to Grothendieck’s dessin d’enfant and subsequent connections to number theory [13,14,15,16]; as well as to the vast and exciting subject of encoding scattering amplitudes in N = 4 super-Yang-Mills theory in terms of planar bipartite graphs and cells in the positive Grassmannian [17,18,19,20,21]
Special attention will be paid to the master space and the moduli space of vacua of the gauge theory and to how their geometry is determined by the tessellations
Our dimer model consists of a balanced bipartite graph drawn on a Riemann surface Σ, i.e. a finite graph embedded into Σ with an equal number of nodes coloured black and white, and with every black node only connected to white nodes and vice versa
Summary
Let us begin by collecting some rudiments of the requisite mathematics and physics. We will introduce the bipartite tiling of Riemann surfaces, emphasizing the approach from dessins d’enfants and permutation triples, and their physical realization of four-dimensional. The vacuum moduli space of the gauge theory will be an associated Calabi-Yau variety. For theories on the torus, the Calabi-Yau geometry is precisely the one which the branes probe in the string theoretic realization, geometrically engineered by configurations of brane tilings, or equivalently, as the dual world-volume theory of a D3-brane
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