超流形上的不变算子与标准模型
Invariant operators on supermanifolds and Standard Models
AI总结：
该研究继续列举15类例外单李超代数的微分不变算子，涵盖有限维和可数无限维纤维的广义张量，借助SuperLie包完成计算，验证了部分前期结果，且这些算子可能与基本粒子标准模型及大统一理论相关。
摘要：
本文继续列举关于15类例外单李超代数多项式向量场的微分不变算子。此前，Kochetkov已在15类情形中的2类里得到了作用于有限维纤维张量的算子列表的一部分，Kac与Rudakov则在另外1类情形中得到了相关结果。Broadhurst和Kac猜想，这些结构中的一部分与基本粒子标准模型及大统一理论（Grand Unified Theories，GUT）相关。因此，若大统一理论存在，将借助我们找到的算子、或待发现的其r元类似物来构建。计算借助基于Mathematica的Grozman的SuperLie软件包完成。当简并条件被违反（即不存在奇异向量）时，对应的张量场模是不可约的。我们考虑了对应可数无限维纤维的广义张量，而不仅限于有限维纤维的张量。我们还验证了部分此前的研究结果。
英文摘要：
Here we continue to list the differential operators invariant with respect to the 15 exceptional simple Lie superalgebras of polynomial vector fields. A part of the list (for operators acting on tensors with finite dimensional fibers) was earlier obtained in 2 of the 15 cases by Kochetkov and in one more instance by Kac and Rudakov. Broadhurst and Kac conjectured that some of these structures pertain to the Standard Models of elementary particles and the Grand Unified Theories. So, GUT, if any exists, will be formulated in terms of operators we found, or their r-nary analogs to be found. Calculations are performed with the aid of Grozman's Mathematica-based SuperLie package. When degeneracy conditions are violated (absence of singular vectors) the corresponding module of tensor fields is irreducible. We consider generalized tensors corresponding to fibers of countable infinite dimension, not only finite dimensional ones. We also verified some of the earlier findings.
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