The Beta Ansatz: A Tale of Two Complex Structures

TL;DR

This paper explores the relationship between Belyi pairs, complex structures, and gauge theories derived from brane tilings, providing algorithms for constructing Belyi pairs and challenging previous conjectures about their complex structures.

Contribution

It introduces algorithms for constructing Belyi pairs from brane tilings and demonstrates a counterexample to a conjecture linking Belyi complex structures with R-charge complex structures.

Findings

Algorithms for Belyi pair construction are detailed.

A counterexample disproves the conjecture relating complex structures.

Connections between elliptic curve covers and orbifold constructions are established.

Abstract

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 of a-maximization, in the context of isoradial embeddings of the dimer, also associates a complex structure to the torus, determined by the R-charges in the SCFT, which can be compared with the Belyi complex structure. Algorithms for the explicit construction of the Belyi pairs are described in detail. In the case of orbifolds, these algorithms are related to the construction of covers of elliptic curves, which exploits the properties of Weierstrass elliptic functions. We present a counterexample to a previous conjecture identifying the complex structure of the Belyi curve to the complex structure associated with R-charges.