论文标题
2D费米子和统计力学：背景量规场中的临界二聚体和狄拉克·费米斯
2D Fermions and Statistical Mechanics: Critical Dimers and Dirac Fermions in a background gauge field
论文作者
论文摘要
在晶格间距的限制为零的限制下，我们在存在奇异$ sl（n，\ mathbb {c}）的存在下考虑二聚体模型，$ gauge字段远离一组穿刺。我们考虑此扭曲二聚体分区函数的群集扩展表明，它与在该量规场的存在下匹配2D Dirac分区函数的类似群集扩展。后者通常称为tau函数。这重现并概括了Dubédat的各种计算（J.Eur。Math。Soc。21（2019），第1号，第1-54页）。特别是，双方的群集扩展是按学期匹配的，每个项被证明等于特定的全态积分及其共轭的总和。在二聚体一侧，我们使用离散指数函数的各种精确晶格级别的身份和逆Kasteleyn矩阵来评估扩展中的术语。在费米恩（Fermion）方面，群集的扩展使我们达到了两个新型的tau函数系列扩展，一种涉及fuschian代表，一种涉及单肌表示。
In the limit of the lattice spacing going to zero, we consider the dimer model on isoradial graphs in the presence of singular $SL(N,\mathbb{C})$ gauge fields flat away from a set of punctures. We consider the cluster expansion of this twisted dimer partition function show it matches an analogous cluster expansion of the 2D Dirac partition function in the presence of this gauge field. The latter is often referred to as a tau function. This reproduces and generalizes various computations of Dubédat (J. Eur. Math. Soc. 21 (2019), no. 1, pp. 1-54). In particular, both sides' cluster expansion are matched up term-by-term and each term is shown to equal a sum of a particular holomorphic integral and its conjugate. On the dimer side, we evaluate the terms in the expansion using various exact lattice-level identities of discrete exponential functions and the inverse Kasteleyn matrix. On the fermion side, the cluster expansion leads us to two novel series expansions of tau functions, one involving the Fuschian representation and one involving the monodromy representation.
