Modular tensor category

A finite semisimple ribbon braided monoidal category with nondegenerate modular S-matrix. For a unitary anyon theory, this means that the only transparent anyon is the vacuum anyon. Such data define a three-dimensional topological quantum field theory.

The net chiral central charge of an edge theory. For a bosonic unitary modular tensor category, its residue modulo eight is determined by the topological spins and quantum dimensions through $e^{2\pi ic/8}={\mathcal{D}}^{-1}{\sum }_a{d}_a^2{\theta }_a$. This phase corrects the signature dependence in the Witten-Reshetikhin-Turaev invariant.

Cutting one nonseparating circle on each handle of a closed orientable surface gives a sphere with $g$ pairs of oppositely oriented boundary circles. The topological quantum field theory gluing rule sums over the paired anyon labels $a,\bar{a}$. The resulting anyon fusion matrix formula becomes $\dim V({\Sigma }_g)={\mathcal{D}}^{2g-2}{\sum }_a{d}_a^{2-2g}$ by the Verlinde formula.

The matrix of Hopf link evaluations divided by the total quantum dimension $\mathcal{D}$. In a unitary modular tensor category it is a symmetric unitary matrix, with $S_{1a}=d_a/\mathcal{D}$ and $S_{\bar{a}b}=S_{ab}^{\ast }$. Its columns diagonalize the anyon fusion matrices through the Verlinde formula.

The simultaneous diagonalization formula for the anyon fusion matrices of a modular tensor category. The eigenvalue of $N_b$ in column $x$ of the modular S-matrix is $S_{bx}/S_{1x}$.