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PreprintCC BY 4.0arXiv

The Galois actions on braided fusion categories

  • Andrew Schopieray
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Abstract

This note harmonizes two classical topics: minimal modular extensions of braided fusion categories and the Galois action on modular fusion categories. Their common generalizations are Galois-modular extensions, which always exist when a braided fusion category ℬ is pseudounitary. Each extension induces a Galois action on the simple objects of ℬ. Up to integral equivalence, the Galois-modular extensions of ℬ form a torsor over an explicit group of homomorphisms from Gal(ℚ̅/ℚ) to the invertible objects of order dividing two in the symmetric center ℰ of ℬ. If ℰ is Tannakian and ℬ has a minimal modular extension, all minimal modular extensions induce one canonical Galois action. We prove that many facts about the modular data of modular fusion categories extend to pseudounitary premodular fusion categories as well via the Galois action.

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Cite

BibTeXRIS
BibTeX
@misc{schopieray2026galois,
  title = {{The Galois actions on braided fusion categories}},
  author = {Andrew Schopieray},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05677},
  eprint = {2611.05677},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05677},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Galois actions on braided fusion categories
AU  - Andrew Schopieray
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05677
UR  - https://arxiv.org/abs/2611.05677
AB  - This note harmonizes two classical topics: minimal modular extensions of braided fusion categories and the Galois action on modular fusion categories. Their common generalizations are Galois-modular extensions, which always exist when a braided fusion category ℬ is pseudounitary. Each extension induces a Galois action on the simple objects of ℬ. Up to integral equivalence, the Galois-modular extensions of ℬ form a torsor over an explicit group of homomorphisms from Gal(ℚ̅/ℚ) to the invertible objects of order dividing two in the symmetric center ℰ of ℬ. If ℰ is Tannakian and ℬ has a minimal modular extension, all minimal modular extensions induce one canonical Galois action. We prove that many facts about the modular data of modular fusion categories extend to pseudounitary premodular fusion categories as well via the Galois action.
ER  -