论文标题
谎言代数费米
Lie Algebra Fermions
论文作者
论文摘要
我们定义了在简单的谎言代数中采用值的费米子的超对称量子力学。我们总结了与Koszul差异d相对应的Laplacian的频谱和特征空间的了解。首先，我们专注于零特征值本特征空间，该特征植物与Lie代数同胞相吻合。我们提供了对有用工具的物理洞察力来计算共同体学，即莫尔斯理论和霍奇柴尔德 - 派光谱序列。我们列出了Lie代数共同体学环的明确发电机。其次，我们集中于在给定费用数字上具有最大特征值的超对称量子力学的特征空间。这些特征空间在简单谎言代数的Borel子代数的Abelian理想方面具有明确的描述。我们还引入了一个谎言代数的模型，在两个维度上估计费米子，其中最大特征值的空间获得了共同的解释。我们的工作提供了数学家对结果的物理解释，并简化了一些定理的证明。此外，我们回想起这些数学结果在四个维度中在纯的超对称量规理论中起作用，并观察到它们产生了四维手性环的规范表示。
We define a supersymmetric quantum mechanics of fermions that take values in a simple Lie algebra. We summarize what is known about the spectrum and eigenspaces of the Laplacian which corresponds to the Koszul differential d. Firstly, we concentrate on the zero eigenvalue eigenspace which coincides with the Lie algebra cohomology. We provide physical insight into useful tools to compute the cohomology, namely Morse theory and the Hochschild-Serre spectral sequence. We list explicit generators for the Lie algebra cohomology ring. Secondly, we concentrate on the eigenspaces of the supersymmetric quantum mechanics with maximal eigenvalue at given fermion number. These eigenspaces have an explicit description in terms of abelian ideals of a Borel subalgebra of the simple Lie algebra. We also introduce a model of Lie algebra valued fermions in two dimensions, where the spaces of maximal eigenvalue acquire a cohomological interpretation. Our work provides physical interpretations of results by mathematicians, and simplifies the proof of a few theorems. Moreover, we recall that these mathematical results play a role in pure supersymmetric gauge theory in four dimensions, and observe that they give rise to a canonical representation of the four-dimensional chiral ring.
