Unitarity and modularity

Dagger structures and unitarity

For a category realized over $\mathbb C$, a dagger sends a morphism $f\colon X\to Y$ to $f^\dagger\colon Y\to X$, is conjugate-linear and involutive, reverses composition, and is compatible with tensor products. A unitary category has a compatible positive dagger structure.

The interface

The method dagger(f) is category-specific. In concrete matrix models it is implemented by conjugate transpose when the chosen bases carry the intended Hermitian structures. The predicate is_unitary(C) concerns the structure in those chosen coordinates; it does not search for a change of basis. Exact coefficients in an abstract number field do not by themselves choose complex conjugation or positivity, so a complex embedding or complex ball realization is needed for analytic unitarity questions.

Premodular and modular categories

A premodular category is a braided spherical fusion category. For ordered simple representatives $X_1,\ldots,X_r$, TensorCategories.jl uses the unnormalized matrix

\[\label{eq:categorical-s-matrix} S_{ij}=\operatorname{Tr}\!\left( c_{X_i,X_j}\circ c_{X_j,X_i} \right).\]

The displayed endomorphism acts on $X_j\otimes X_i$. Cyclicity of the spherical trace identifies this with the usual trace of the double braiding on $X_i\otimes X_j$. A premodular category is modular when this matrix is invertible (Etingof et al., 2015; Definitions 8.13.1, 8.13.2, and 8.13.4).

The interface

The public methods use the following conventions:

• smatrix(C) returns the unnormalized matrix in equation \eqref{eq:categorical-s-matrix}, ordered by simples(C);
• normalized_smatrix(C) returns $S/\sqrt{\dim(\mathcal C)}$, including the required choice of square root;
• tmatrix(C) is diagonal with entries twist_scalar(X) in the same order;
• the generic is_modular(C) requires is_fusion(C), is_braided(C), and is_spherical(C) and then tests whether smatrix(C) is invertible.

Thus the generic predicate presently imposes the split fusion condition. A non-split weak fusion category can satisfy an analogous nondegeneracy condition, but is_modular(C) does not report it as modular unless the category-specific implementation supplies a different method.

For ArbField and AcbField, invertibility is decided by checking that the determinant ball excludes zero. A successful result is rigorous for the input enclosures at the selected precision; failure to exclude zero is inconclusive and can require a higher-precision realization.

Continue with monoidal functors.