Past exam of the anyons and topological quantum field theories course of the University of Oxford / 2025 / 2 / b / Solution

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 $2g-2$.