Frobenius–Schur sign in a normalized anyonic vacuum channel

Let $\mathcal{C}$ be a unitary spherical fusion category over $\mathbb{C}$, with pivotal structure and spherical dimension as in Pivotal structure, Spherical structure, and The dimension of an object relative to a pivotal structure. Let $a$ be a self-dual simple object with $N_{aa}^{\mathbf{1}}=1$. Choose a unit vector $v:\mathbf{1}\to a\otimes a$ spanning the one-dimensional vacuum split space and let $d_a>0$ be its positive spherical dimension.

A normalized vacuum-channel self-duality datum is a supplied pair of maps $\mathrm{coev}:\mathbf{1}\to a\otimes a$ and $\mathrm{ev}:a\otimes a\to \mathbf{1}$ of the form

$\mathrm{coev}=\sqrt{d_a}v,\mathrm{ev}={\kappa }_a\sqrt{d_a}{v}^{†},$

where ${\kappa }_a\in \{+1,-1\}$ and the two duality zig-zag identities hold after the self-duality identification, using the left-dual convention of Left dual and right dual object. We call ${\kappa }_a$ the Frobenius–Schur sign in this normalized cup/cap convention. It is not a braiding or $R$-symbol phase.

This item records supplied normalized duality data; it does not assert that every self-dual simple object admits the displayed gauge or prove a trace, ribbon, or category-equivalence theorem. An example must supply the datum or compute the sign from coherent normalized F-data; fusion rules alone do not specify it.