Rank-finiteness for modular categories

Abstract

We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular category $\mathcal{C}$ with $N=ord(T)$, the order of the modular $T$-matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension $D^2$ in the Dedekind domain $\mathbb{Z}[e^{\frac{2\pi i}{N}}]$ is identical to that of $N$.

25 pages (last version). Version 2: removed weakly integral rank 6 and integral rank 7 section, improved rank 5 classification up to monoidal equivalence. Version 3: removed rank 5 classification (note title change)--this will be published separately. Significantly improved exposition

References in corpus (3)

• Congruence Subgroups and Generalized Frobenius-Schur Indicators
• Central Invariants and Frobenius-Schur Indicators for Semisimple Quasi-Hopf Algebras
• Fusion rules of equivariantizations of fusion categories

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• Higher central charges and Witt groups
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• On classification of extremal non-holomorphic conformal field theories
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• Congruence Subgroups and Super-Modular Categories
• Extracting higher central charge from a single wave function
• Braided zesting and its applications
• Chiral spin liquids with crystalline ${\mathbb{Z}}_2$ gauge order in a three-dimensional Kitaev model
• Reconstruction of modular data from $S{L}_2(\mathbb{Z})$ representations
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• Gauge invariants from the powers of antipodes
• Galois Conjugation and Multiboundary Entanglement Entropy
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• Bantay's trace in Unitary Modular Tensor Categories
• Classification of Metaplectic Fusion Categories
• Eigenvalues of rotations and braids in spherical fusion categories
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• Ordering the topological order in the fractional quantum Hall effect
• Higher Hall conductivity from a single wave function: Obstructions to symmetry-preserving gapped edge of (2+1)D topological order
• Classification of integral modular data up to rank 13
• A Kac-Weyl Character Identity