秩不超过13的积分模数据分类
Classification of integral modular data up to rank 13
AI总结：
本文通过改进的筛选方法，对秩不超过13的积分模数据进行了分类，并揭示了其在数学物理中的重要性。
摘要：
本文对所有积分模融合范畴的模数据进行了至秩13的分类。此外，还对所有至秩12的积分半Frobenius融合环进行了分类。我们发现，至秩13的所有完美积分模融合范畴，以及至秩12的所有完美积分半Frobenius融合环都是平凡的。我们还细化了低于25的非指向奇维模数据至三项，均为秩17，FPdim 225，类型[[1,3],[3,8],[5,6]]，填补了文献中的空白。对于秩25，我们已将完美情况缩小到3个开放类型。我们的初始关键见解是，通常用于列出可能类型的埃及分数可以选取平方分母。然后我们开发了几种类型标准作为初始过滤器。为了获得融合环，我们利用为本研究专门创建的Normaliz新功能来解决维度和结合性方程。S矩阵（如果存在）通过自转置特征表获得，而T矩阵则通过求解Anderson-Moore-Vafa方程得到。最后，我们验证了模数据的扩展公理。从秩13开始，类型进一步受到额外性质的限制，这些性质涉及模情况特有的普遍分级、模群的同余表示和伽罗瓦作用，导致关键算术约束。特别是，我们得出，在秩21以内，全局FPdim的素因子不超过秩，更强烈地在非指向情况下，不超过半秩。最终，我们将秩14的分类缩小到35种可能类型，其中8种是非完美的。
英文摘要：
This paper classifies the modular data of integral modular fusion categories up to rank 13, and integral half-Frobenius fusion rings up to rank 12. We establish that every perfect case within these bounds is trivial. Furthermore, we refine the non-pointed odd-dimensional modular data at ranks below 25 to exactly three items, all of rank 17, FPdim 225, and type [[1,3],[3,8],[5,6]], filling existing literature gaps. For rank 25, we narrow the perfect case to three open types. Our core insight is that Egyptian fractions, typically used to list possible types, can be chosen with squared denominators. We develop several type criteria as initial filters. To construct the fusion rings, we solve dimension and associativity equations utilizing custom-built features in Normaliz. S-matrices are generated by self-transposing the character table, and T-matrices are derived by solving the Anderson-Moore-Vafa equations, concluding with the verification of extended modular data axioms. From rank 13 onward, types are restricted by modular-specific properties involving universal grading, congruence representations of the modular group, and Galois action. This establishes critical arithmetic constraints: up to rank 21, a prime divisor of the global FPdim cannot exceed the rank, and up to rank 15 (non-pointed case), it cannot exceed half the rank. Ultimately, we reduce the rank 14 classification to 35 possible types, 8 of which are non-perfect.
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