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In a modular tensor category, a loop with the normalized Kirby color encircling a charge-$x$ strand projects onto the vacuum anyon charge and multiplies it by $\mathcal{D}$. Indeed, the encircling scalar is ${\mathcal{D}}^{-1}{\sum }_a{d}_a{S}_{ax}/S_{1x}$, which equals $\mathcal{D}{\delta }_{x1}$ by unitary matrix orthogonality for the modular S-matrix.

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.

Put $K={\sum }_a{N}_a{N}_{\bar{a}}$. The Verlinde formula diagonalizes every anyon fusion matrix using the same modular S-matrix:Because $S_{\bar{a}x}=S_{ax}^{\ast }$, the eigenvalue of $K$ in column $x$ isThe last equality uses unitary matrix orthogonality. ThereforeFinally, insert $S_{1x}=d_x/\mathcal{D}$, where $d_x$ is the quantum dimension and ${\mathcal{D}}^2={\sum }_x{d}_x^2$ is the squared total quantum dimension:Although only quantum dimensions remain, the result is an integer because a counts an anyon fusion space.

The sphere value is ${\mathcal{D}}^{-2}{\sum }_x{d}_x^2=1$, and the torus value is ${\sum }_{x}1$. If all $n$ simple anyons have quantum dimension one, the answer is $n^g$, agreeing with the Abelian handle count. These checks also fix the exponent $2g-2$.

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