Commutative exact superalgebras and super-modular tensor categories

@misc{zuevsky2026commutative,
  title = {{Commutative exact superalgebras and super-modular tensor categories}},
  author = {A. Zuevsky},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24568},
  eprint = {2610.24568},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24568},
  note = {CC0},
}

TY  - UNPB
TI  - Commutative exact superalgebras and super-modular tensor categories
AU  - A. Zuevsky
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.24568
UR  - https://arxiv.org/abs/2610.24568
AB  - We extend the theory of commutative exact algebras in braided finite tensor categories (Shimizu-Yadav) to superalgebras, motivated by extensions of vertex operator (super)algebras. For a braided category C let B=C^gr be the category of ℤ/2-graded objects with the Koszul-signed braiding. It is the underlying category of the supercategory of Creutzig-Kanade-McRae. A supercommutative superalgebra A=A_0⊕ A_1 in C is the same as a commutative algebra bA in B, and super-local modules are the local bA-modules. Hence rigidity, pivotality and ribbon statements are instances of the ordinary theorems, with the ribbon condition θ_A_0=id, θ_A_1=ε id, where ε=±1 is fixed by the statistics of the odd part. For C non-degenerate and bA haploid and exact we prove that the category of super-local modules is a braided finite tensor category whose Müger center is sVect. It is the centralizer of a fermion in a non-degenerate category obtained by condensing in Cboxtimes(toric code). If C is ribbon and bA is symmetric Frobenius it is ribbon. We construct examples from graded simple currents, treat the free fermion and the symplectic fermion, and compare with Creutzig-Kanade-McRae and Creutzig-McRae-Shimizu-Yadav.
ER  -