The $\mathbb Z_N$-Tambara-Yamagami categories: TambaraYamagami
TambaraYamagami{N, K} represents the Tambara-Yamagami fusion category ${\rm TY}(\mathbb Z_N, K)$ based on the cyclic group of order $N$.
The $N+1$ simple objects coincide with the group elements of $\mathbb Z_N$ supplemented with one non-invertible object $m$. The type parameter K specifies the Frobenius-Schur indicator of $m$ as $\varkappa_m = (-1)^K = \pm 1$, i.e. K = false corresponds to $\varkappa_m = 1$ and K = true to $\varkappa_m = -1$.
Sector type
TensorKitSectors.TambaraYamagami — Type
struct TambaraYamagami{N, K} <: Sector
TambaraYamagami{N, K}(n::Integer)Represents the Tambara-Yamagami fusion category built from the cyclic group $ℤ_N$. The simple objects are the group elements 0, 1, …, N - 1 of $ℤ_N$, together with a single non-invertible object m.
The non-trivial fusion rules are given by
\[g ⊗ h = g + h \mod N, \qquad g ⊗ m = m ⊗ g = m, \qquad m ⊗ m = \bigoplus_{g ∈ ℤ_N} g.\]
The F-symbols are constructed from a non-degenerate symmetric bicharacter $χ(g, h) = \exp(2π i g h / N)$, together with a Frobenius-Schur sign for the non-invertible object; the latter is fixed as κ = (-1)^K through the type parameter K::Bool. For fixed N, the two choices K = false (κ = 1) and K = true (κ = -1) generally give distinct fusion categories.
Only the case N == 2 and K == false admits a braiding, as this case coincides with Ising, but this is not currently implemented.
Fields
n::UInt8: a group element for0 <= n < N, or the non-invertible objectmforn == N.
References
[1] D. Tambara and S. Yamagami, Tensor categories with fusion rules of self-duality for finite abelian groups, J. Algebra 209, 692-707 (1998). [2] M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, Symmetry Fractionalization, Defects, and Gauging of Topological Phases, Phys. Rev. B 100, 115147 (2019), arXiv:1410.4540.
Here, the type parameters N and K correspond respectively to the order of the underlying cyclic group and the Frobenius-Schur sign $(-1)^K$ of the non-invertible object.
Fusion Rules
The non-trivial fusion rules read
\[g ⊗ h = g + h \mod N, \qquad g ⊗ m = m ⊗ g = m, \qquad m ⊗ m = \bigoplus_{g ∈ ℤ_N} g,\]
for all group elements $g, h \in \mathbb Z_N$.
Hence FusionStyle(::Type{<:TambaraYamagami}) = SimpleFusion().
The quantum dimensions are
\[d_g = 1, \quad \forall g\in\mathbb Z_N, \quad {\rm and} \quad d_m = \sqrt{N}.\]
Topological Data
We write $χ(g, h) = \exp(2π i g h / N)$ for a normalised non-degenerate symmetric bicharacter on $\mathbb Z_N$. The nontrivial F-symbols are then given by
\[F^{g\,m\,h}_{m} = χ(g, h), \qquad F^{m\,g\,m}_{h} = χ(g, h), \qquad \left[F^{m\,m\,m}_{m}\right]_g^h = \frac{\varkappa_m}{\sqrt{N}}\,\overline{χ(g, h)},\]
for all $g, h \in \mathbb Z_N$.
Crucially, there exists no braiding on this fusion category, except when $N=2$, in which case it coincides with IsingAnyon.
Iteration and basis conventions
values(TambaraYamagami{N,K}) iterates the labels 0, 1, …, N-1,:m in increasing order.
References
[1] D. Tambara and S. Yamagami, Tensor categories with fusion rules of self-duality for finite abelian groups, J. Algebra 209, 692-707 (1998).
[2] M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, Symmetry Fractionalization, Defects, and Gauging of Topological Phases, Phys. Rev. B 100, 115147 (2019), arXiv:1410.4540.