Time-Reversed Sectors: TimeReversed
TimeReversed{I} represents the time-reversed, or conjugate-braided, version of a sector type I. It keeps the same objects, fusion rules, dimensions, and associators, but reverses the braiding.
Sector type
TensorKitSectors.TimeReversed — Type
struct TimeReversed{I <: Sector}
TimeReversed(a::Sector)
timereversed(a::Sector)Represents the time-reversed version of the sector a, i.e. the sector with the same fusion rules and F-symbols, but with the inverse braiding. Time reversal acts trivially on sectors with symmetric braiding. In such cases, timereversed(a) simply returns a.
TensorKitSectors.timereversed — Function
timereversed(a::Sector)Return the time-reversed version of the sector a. If a is already a time-reversed sector, return the original sector.
Construct a wrapped sector with TimeReversed(a) or TimeReversed{I}(a). The helper timereversed(a) avoids unnecessary wrappers for symmetric braiding categories and unwraps an already time-reversed sector:
julia> a = IsingAnyon(:σ)
IsingAnyon(:σ)
julia> timereversed(a)
TimeReversed{IsingAnyon}(:σ)
julia> timereversed(timereversed(a))
IsingAnyon(:σ)TimeReversed is not defined for sectors with NoBraiding().
Fusion Rules
Fusion is inherited from the original sector type. The fusion multiplicities, fusion style, units, duals, and quantum dimensions are unchanged:
\[N_{\overline{c}}^{\overline{a}\,\overline{b}} = N_c^{ab},\qquad d_{\overline{a}} = d_a,\qquad \overline{a}^{\,*} = \overline{a^*}.\]
values(TimeReversed{I}) follows the same order as values(I), with every element wrapped.
Topological Data
The Fsymbol, Asymbol, and Bsymbol are inherited directly from the original sector. The Rsymbol is replaced by its adjoint:
\[R_{\overline{c}}^{\overline{a}\,\overline{b}} = \left(R_c^{ab}\right)^\dagger.\]
For scalar braiding phases this is complex conjugation, so anyonic spins change sign:
\[θ_{\overline{a}} = \overline{θ_a}.\]
For example, in the conventions of this package,
\[θ_σ = e^{π i/8} \quad⟹\quad θ_{\overline{σ}} = e^{-π i/8}\]
for the Ising σ anyon.
The braiding style and scalar types are reported as those of the original sector type. For bosonic or fermionic symmetric categories, time reversal is physically trivial; timereversed(a) therefore returns a directly.