Hierarchy of modular graph identities
Hierarchy of modular graph identities
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模块化图身份的层次结构
DOI：
10.1007/jhep11(2016)051
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发表时间：
2016
影响因子：
5.4
通讯作者：
J. Kaidi
中科院分区：
文献类型：
--
作者：
E. D'hoker;J. Kaidi
摘要：属一处 II 型超弦振幅的低能展开是根据与环面上共形标量场的费曼图相关的模图函数来组织的。在早期的工作中，构建了所有权重的双环图之间以及权重四和五的更高环图之间令人惊讶的恒等式。在本论文中，这些结果被推广到两个互补的方向。首先，获得并证明权重六的所有恒等式和权重七的所有二面体恒等式。每当尖点处的洛朗多项式可用时，这些恒等式的形式就证实了洛朗多项式的消失控制完整模恒等式的模式。其次，模块化图函数族被扩展为包括所有具有导数耦合和世界表费米子的图。这些扩展的模块化图函数族被证明服从非齐次拉普拉斯特征值方程的层次结构。特征值是针对最简单的无限子族进行分析计算的，并通过 Maple 获得相继更复杂的子族的特征值。该谱仅由权重限制的正整数 s 的特征值 s(s − 1) 组成，其多重性表现出丰富的表示理论模式。
A bstractThe low energy expansion of Type II superstring amplitudes at genus one is organized in terms of modular graph functions associated with Feynman graphs of a conformal scalar field on the torus. In earlier work, surprising identities between two-loop graphs at all weights, and between higher-loop graphs of weights four and five were constructed. In the present paper, these results are generalized in two complementary directions. First, all identities at weight six and all dihedral identities at weight seven are obtained and proven. Whenever the Laurent polynomial at the cusp is available, the form of these identities confirms the pattern by which the vanishing of the Laurent polynomial governs the full modular identity. Second, the family of modular graph functions is extended to include all graphs with derivative couplings and worldsheet fermions. These extended families of modular graph functions are shown to obey a hierarchy of inhomogeneous Laplace eigenvalue equations. The eigenvalues are calculated analytically for the simplest infinite sub-families and obtained by Maple for successively more complicated sub-families. The spectrum is shown to consist solely of eigenvalues s(s − 1) for positive integers s bounded by the weight, with multiplicities which exhibit rich representation-theoretic patterns.