Modular dimensions follow from exact sewing

Assume full AC and its inherited countable-choice consequences. Let $\mathcal{C}$ be a complex modular ribbon fusion category with simple-label set $I$, unit label $0$, simple dimensions $d_i$, Hopf-link matrix $s$, and a chosen square root $D$ of ${\sum }_{i\in I}{d}_i^2$. Put $S=s\mathrm{/}D$ and $H={⨁}_{i\in I}i\otimes i^{\ast }$. In the weighted RT model, define the closed unmarked genus-$g$ state space by $V({\mathrm{\Sigma }}_g)={\mathrm{Hom}}_{\mathcal{C}}(\mathbf{1},H^{\otimes g})$. For every integer $g\ge 0$,

${\dim }_{\mathbb{C}}V({\mathrm{\Sigma }}_g)={\sum }_{i\in I}{S}_{0i}^{2-2g},$

with no complex conjugation in this formula. The sphere state space has dimension one, and the unmarked torus has dimension $\mathrm{\mid }I\mathrm{\mid }$ with basis $b_i:\mathbf{1}\to i\otimes i^{\ast }\subset H$. For every compatible separating or nonseparating circle cut, the actual normalized RT tube map from the complete direct sum of dual-label channels to the glued state space is an isomorphism; its inverse is the complete cutting resolution, retaining all multiplicities, markings, tensor orders, and Lagrangian identifications.

For the torus convention, orient the meridian/longitude basis $(m,l)$ by $\omega (m,l)=1$ and choose reference and object Lagrangians $\alpha =\lambda =\mathbb{R}m$. The positive Hopf mapping-cylinder class and positive core-twist class have respective matrices

$s_{+}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),t=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right),$

with columns the images of $m,l$. Their operators are $S_{ji}=s_{ji}\mathrm{/}D$ and $T_{ji}={\delta }_{ji}{\theta }_i$, where $s$ is the Hopf-link matrix of $\mathcal{C}$. For this positive pair, $S^2=C$ and $(ST{)}^3=\kappa I$, with $C_{ji}={\delta }_{j,i^{\ast }}$. The inverse Hopf class $s_{-}=s_{+}^{-1}$ acts by $S^{-1}$ and satisfies $(S^{-1}T{)}^3=\kappa C$; when labels are not all self-dual, $C$ need not be $I$.

Weighted modular TQFT model and physical interpretation boundary

RT torus operators and the state-space dimension formula

The weighted Reshetikhin–Turaev TQFT and modular-functor factorization

Modular ribbon fusion category and Hopf-link data

Simple ribbon fusion dimensions are nonzero

The modular Hopf-link matrix squares to D² times duality

For a compatible cut $\gamma$, $G_{\gamma }$ is the actual normalized tube-sewing map and $F_{\gamma }$ is its complete cutting inverse. The displayed triangle records $F_{\gamma }{G}_{\gamma }=\mathrm{i}\mathrm{d}$ on the direct-sum channel space; the other inverse identity $G_{\gamma }{F}_{\gamma }=\mathrm{i}\mathrm{d}$ holds on the glued state space. All labels and markings remain those of the supplied cut convention. The diagram records these maps and is not used as a proof.