Non-invertible SPT phases from fusion rings via Tannaka duality
Abstract
In a (1+1)-dimensional QFT with non-invertible symmetry, the fusion rules of topological lines are often the most accessible data, yet they do not fully determine whether the symmetry is anomaly-free, i.e. whether it admits trivially gapped phases. Such phases are characterized by fiber functors of the corresponding fusion category. Building on Tannaka duality, we extract fiber functors from the fusion ring alone by solving simultaneously for monoidal structure maps and tensor automorphisms. This allows us to find categorifications admitting a fiber functor without solving the pentagon equations directly. We test this procedure on the known Tambara--Yamagami ring of ℤ2×ℤ2. Our main result concerns the rank-11 fusion ring shared by the representation categories of the two extraspecial groups of order 27, which has a multiplicity 3 channel and is relevant to self-triality. In both cases we find more than one fiber functor, i.e. several non-invertible SPT phases. We use the explicit data of these fiber functors to derive the F-symbols of the two representation categories in a compact form depending on a non-degenerate alternating bicharacter and an invertible object ε. Using the reconstructed data, we compute the anomalous algebra on the interface between two different SPT phases.
Related
- Bosonic SPT and invertible phases and its relation to Steenrod's problem
- Poincaré Duality and Quadratic Refinements over Laurent Rings
- Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases
- Classifying Grothendieck rings of pivotal fusion categories of rank four
- Galois-modular obstructions to the categorification of near-group fusion rings
BibTeX
@misc{santaniello2026noninvertible,
title = {{Non-invertible SPT phases from fusion rings via Tannaka duality}},
author = {Antonio Santaniello and Roberto Valandro and Massimo Zorzenon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07165},
eprint = {2610.07165},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07165},
}
RIS
TY - UNPB
TI - Non-invertible SPT phases from fusion rings via Tannaka duality
AU - Antonio Santaniello
AU - Roberto Valandro
AU - Massimo Zorzenon
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07165
UR - https://arxiv.org/abs/2610.07165
AB - In a (1+1)-dimensional QFT with non-invertible symmetry, the fusion rules of topological lines are often the most accessible data, yet they do not fully determine whether the symmetry is anomaly-free, i.e. whether it admits trivially gapped phases. Such phases are characterized by fiber functors of the corresponding fusion category. Building on Tannaka duality, we extract fiber functors from the fusion ring alone by solving simultaneously for monoidal structure maps and tensor automorphisms. This allows us to find categorifications admitting a fiber functor without solving the pentagon equations directly. We test this procedure on the known Tambara--Yamagami ring of ℤ_2×ℤ_2. Our main result concerns the rank-11 fusion ring shared by the representation categories of the two extraspecial groups of order 27, which has a multiplicity 3 channel and is relevant to self-triality. In both cases we find more than one fiber functor, i.e. several non-invertible SPT phases. We use the explicit data of these fiber functors to derive the F-symbols of the two representation categories in a compact form depending on a non-degenerate alternating bicharacter and an invertible object ε. Using the reconstructed data, we compute the anomalous algebra on the interface between two different SPT phases.
ER -