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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.

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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.

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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:

using TensorKitSectors

a = IsingAnyon(:σ)
timereversed(a)               # TimeReversed{IsingAnyon}(:σ)
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 adjointed:

\[R_{\overline{c}}^{\overline{a}\,\overline{b}} = \left(R_c^{ab}\right)^\dagger.\]

For scalar braiding phases this is complex conjugation, so anyonic spins change sign:

\[\theta_{\overline{a}} = \overline{\theta_a}.\]

For example, in the conventions of this package,

\[\theta_\sigma = e^{\pi i/8} \quad\Longrightarrow\quad \theta_{\overline{\sigma}} = e^{-\pi i/8}\]

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.

References