Modular S-matrix ($S_{ab}$)

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 $S_{bx}/S_{1x}$.