AI 中文总结
该研究开发计算通用量子维度的方法，将其从与γ无关部分扩展到相关因子，推导E通用多重态的贡献并首次得到其量子维度，还扩展猜想关系、给出算法，能为陈 - 西蒙斯可观测量提供统一描述。
AI 中文摘要
我们开发了一种计算通用量子维度的方法。先前工作提出了计算经典代数通用多重态中与γ无关部分的方法，本文将该方法扩展到与γ相关的因子。具体推导了伴随表示四次幂通用分解中出现的E通用多重态的这些贡献，首次得到其通用量子维度。分析需多重态经典和E8成员的量子维度知识，对其余例外代数，相应量子维度会自动恢复。还部分扩展了关于sl和so、sp成员的猜想关系到例外情况。最后给出实现该方法的显式算法，能推导高维表示的通用公式。通用量子维度在某些情况下为陈 - 西蒙斯可观测量提供统一描述。
英文摘要
We develop an approach for computing the universal, in Vogel's sense, quantum dimensions. In a previous work a method for calculating the $γ$-independent part of these dimensions for universal multiplets with nonzero associated members for classical algebras was proposed. In the present paper, this approach is extended to the $γ$-dependent factors. In particular, we derive these contributions for the $E$ universal multiplet, which appears in the universal decomposition of the fourth power of the adjoint representation, thereby obtaining its universal quantum dimension for the first time. The analysis requires knowledge of the quantum dimensions of the classical and $E_8$ members of the multiplet; for the remaining exceptional algebras, the corresponding quantum dimensions are recovered automatically, providing an additional consistency check of the approach. We also partially extend a previously conjectured relation between the $sl$ and $so, sp$ members - formulated in terms of vertical and horizontal sums of Young diagrams - to the exceptional case. Finally, we present an explicit algorithm implementing the proposed method, which enables the derivation of universal formulae for higher-dimensional representations. The universal quantum dimensions provide, in some cases, a unified description of Chern-Simons observables, including knot invariants and partition functions, in a form valid simultaneously for all gauge algebras.
Comments17 pages, LaTeX, 15 March 2026
Journal refPhysics Letters B, 877, (2026), 140466
DOI：10.1016/j.physletb.2026.140466
