Cyclic Group Elements: `ZNElement{N,p}`

ZNElement{N,p} represents the elements of the cyclic group $ℤ_N$ directly as a Sector, as opposed to ZNIrrep which represents its (one-dimensional) irreps. Fusion is given by the group multiplication itself, and the associator can carry a nontrivial phase determined by a level-p 3-cocycle on $ℤ_N$. For special values of p this also yields a consistent braiding, turning ZNElement{N,p} into a simple model of abelian anyons. As a fusion category, this is identified with $\text{Vec}_{ℤ_N}^{ω_p}$, the category of $ℤ_N$-graded vector spaces with associator the 3-cocycle $ω_p$.

Group elements as sectors

TensorKitSectors.AbstractGroupElement — Type

abstract type AbstractGroupElement{G <: Group} <: Sector

Abstract supertype for sectors which corresponds to group elements of a group G.

Actual concrete implementations of those irreps can be obtained as Element[G], or via their actual name, which generically takes the form (asciiG)Element, i.e. the ASCII spelling of the group name followed by Element.

All group elements have FusionStyle equal to UniqueFusion(). Furthermore, the BraidingStyle is set to NoBraiding(), although this can be overridden by a concrete implementation of AbstractGroupElement.

For the fusion structure, a specific SomeGroupElement <: AbstractGroupElement{SomeGroup} should only implement the following methods

Base.:*(c1::SomeGroupElement, c2::SomeGroupElement) -> SomeGroupElement
Base.one(::Type{SomeGroupElement}) -> SomeGroupElement
Base.inv(c::SomeGroupElement) -> SomeGroupElement
# and optionally
TensorKitSectors.cocycle(c1::SomeGroupElement, c2::SomeGroupElement, c3::SomeGroupElement) -> Number

The methods conj, dual, ⊗, Nsymbol, Fsymbol, dim, Asymbol, Bsymbol and frobenius_schur_phase will then be automatically defined. If no cocycle method is defined, the cocycle will be assumed to be trivial, i.e. equal to 1.

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ZNElement{N,p} is currently the only concrete subtype of AbstractGroupElement{Group}.

Sector type

TensorKitSectors.ZNElement — Type

struct ZNElement{N, p} <: AbstractGroupElement{ℤ{N}}
ZNElement{N, p}(n::Integer)
GroupElement[ℤ{N}, p](n::Integer)

Represents an element of the group $ℤ_N$ for some value of N < 64. (We need 2 * (N - 1) <= 127 in order for a ⊗ b to work correctly.) For N equals 2, 3 or 4, ℤ{N} can be replaced by ℤ₂, ℤ₃, ℤ₄. An arbitrary Integer n can be provided to the constructor, but only the value mod(n, N) is relevant. The second type parameter p should also be specified as an integer 0 <= p < N and specifies the 3-cocycle, which is then being given by

cocycle(a, b, c) = cispi(2 * p * a.n * (b.n + c.n - mod(b.n + c.n, N)) / N^2))

If p is not specified, it defaults to 0, i.e. the trivial cocycle.

Fields

n::Int8: the integer label of the element, modulo N.

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Elements are labeled by a single integer n, stored as Int8 and only meaningful modulo N (which must satisfy N < 64). The concrete type is obtained through GroupElement[ℤ{N}, p], or through the aliases Z2Element{p}, Z3Element{p}, and Z4Element{p} for N = 2, 3, 4. If p is omitted, it defaults to 0, i.e. the trivial cocycle.

The unit is ZNElement{N,p}(0), and duals are given by group inversion:

\[a^* = a^{-1} = -a \bmod N.\]

Fusion Rules

Fusion is literally the group operation, so there is a single fusion channel with multiplicity one:

\[a ⊗ b = (a + b) \bmod N,\]

and FusionStyle(ZNElement{N,p}) = UniqueFusion() for every N and p. All quantum dimensions are 1.

Associativity: the 3-cocycle and F-symbol

The Fsymbol is not assumed trivial: it is given by the 3-cocycle

\[ω_p(a,b,c) = \exp\left(\frac{2π i\, p\, a\,\big(b + c - [b+c]_N\big)}{N^2}\right), \qquad [x]_N := x \bmod N,\]

which is a representative of a class in $H^3(ℤ_N, U(1)) ≅ ℤ_N$, indexed by the level p ∈ 0:N-1. Concretely,

\[F^{abc}_{e,d,f} = ω_p(a,b,c) \quad \text{when } e = a+b,\ d = e+c,\ f = b + c \pmod N,\]

and zero otherwise (since fusion is unique, all the other symbols such as Asymbol, Bsymbol and frobenius_schur_phase are likewise built directly out of ω_p). For p = 0 the cocycle is trivial and ZNElement{N,0} reduces to the ordinary $ℤ_N$-graded vector spaces. The pentagon_equation holds for every N and every p ∈ 0:N-1, i.e. every ZNElement{N,p} is a consistent fusion category.

Topological Data (Braiding)

Braiding is only defined in two cases:

p = 0: BraidingStyle(ZNElement{N,0}) = Bosonic(), and the Rsymbol is trivial (Rsymbol(a,b,c) = Nsymbol(a,b,c)).
2p ≡ N: BraidingStyle(ZNElement{N,p}) = Anyonic(), and
\[R^{ab}_{a+b} = \exp\left(\frac{2\pi i\, p\, a\, b}{N^2}\right),\]

giving abelian anyonic braiding statistics for the $ℤ_N$ charges. For example, N = 4, p = 2 (used in the example below) yields topological spins $θ_a = \exp(iπ a^2 / 4)$.

For any other value of p, BraidingStyle(ZNElement{N,p}) = NoBraiding().

Iteration

values(ZNElement{N,p}) iterates the labels 0, 1, …, N-1 in increasing order, exactly as for ZNIrrep.

Examples

using TensorKitSectors

a, b = Z3Element(1), Z3Element(2)
only(a ⊗ b) == unit(typeof(a))

# output
true

I = Z4Element{2} # 2p == N, so this admits a braiding
BraidingStyle(I)

# output
Anyonic()