Beta Anonymous：两个复杂结构的故事
The Beta Ansatz: A Tale of Two Complex Structures
摘要：
膜镶嵌，有时也称为二聚体模型，是一类环面上的二分图，它编码了四维SCFT的规范理论数据，这些SCFT与探测环面Calabi-Yau三重的D3-膜对偶。一种编码这些信息的有效方法是利用了儿童死亡理论，用置换三元组来表示结构，而置换三元组又与贝利对相关，即从环面到具有三个标记点的P^1的全纯映射。在二聚体的等径嵌入的背景下，a-最大化的过程也将复杂结构与环面相关联，由SCFT中的R-电荷确定，这可以与Belyi复杂结构相比较。算法的显式建设的Belyi对进行了详细说明。在orbifolds的情况下，这些算法涉及到建设覆盖的椭圆曲线，利用性质的维尔斯特拉斯椭圆函数。我们提出了一个反例，以前的猜想确定复杂的结构的Belyi曲线的复杂结构与R-收费。
英文摘要：
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.
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