发表机构
LPTMC, UMR 7600 CNRS, Sorbonne Université（巴黎索邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究超 kagome 晶格格林函数的闭式解问题，通过精确晶格矩得到三阶微分算子，将其与 Gamma_0(30)+ 对应的二阶单值化方程关联，证明该格林函数为 30 级模并给出其权二重周期。
AI 中文摘要
立方晶格的晶格格林函数是经典的：每个都是二阶算子的对称平方，具有完全椭圆积分的闭式形式。由自旋液体候选材料 Na4Ir3O8 的铱亚晶格实现的超 kagome 晶格，一直未能得到此类处理：Varma 和 Monien 将其态密度简化为三重积分，他们发现无法精确求解。我们从精确晶格矩得到一个不可约三阶线性微分（Picard-Fuchs）算子，该算子湮灭格林函数。它不是字面意义上的对称平方，但其对称平方带有有理解：微分伽罗瓦群是正交的，一个二阶微分互挽子使该算子投影为对称平方。我们的主要结果是，基础二阶算子是 Gamma_0(30)+ 的零亏格模曲线的单值化方程，Gamma_0(30)+ 是由 Gamma_0(30) 及其所有 Atkin-Lehner 对合生成的 30 级群，是参数化自然变量的显式 eta 商。该单值化方程本身并非新内容——它是 Lian-Yau 零亏格群表中的一行——因此此处的新内容是将晶格格林函数与其等同起来，并给出证明：通过先验的极点次数界而非数值检查来建立 Schwarzian 恒等式，这也证明了表中条目。因此超 kagome 格林函数是 30 级模的，提供了 Varma 和 Monien 所寻求的闭式形式。我们明确得到其权二重周期：一个深度 1 的拟模形式，我们证明它在任何权下都不是模形式乘以代数函数。
英文摘要
The lattice Green's functions of the cubic lattices are classical: each is a symmetric square of a second-order operator, with a closed form in complete elliptic integrals. The hyperkagome lattice, realized by the iridium sublattice of the spin-liquid candidate Na4Ir3O8, has resisted such a treatment: Varma and Monien reduced its density of states to a threefold integral they found no way to solve exactly. From exact lattice moments we obtain an irreducible third-order linear differential (Picard-Fuchs) operator annihilating the Green's function. It is not a literal symmetric square, yet its symmetric square carries a rational solution: the differential Galois group is orthogonal, and an order-two differential intertwiner makes the operator projectively a symmetric square. Our main result is that the underlying second-order operator is the uniformizing equation of the genus-zero modular curve of Gamma_0(30)+, the level-thirty group generated by Gamma_0(30) and all its Atkin-Lehner involutions, an explicit eta quotient parametrizing the natural variable. That uniformizing equation is itself not new - it is a row of the Lian-Yau table of genus-zero groups - so what is new here is the identification of a lattice Green's function with it, together with a proof: the Schwarzian identity is established by an a priori pole-degree bound rather than checked numerically, which also proves the tabulated entry. The hyperkagome Green's function is therefore modular at level thirty, supplying the closed form Varma and Monien sought. Its weight-two period is obtained explicitly: a depth-one quasimodular form which we prove is not a modular form times an algebraic function at any weight.
Comments25 pages
