RT state spaces from a modular category

Fix a complex modular ribbon fusion category as in Modular ribbon fusion category and Hopf-link data, with finite simple-label set $I$, tensor unit $\mathbf{1}$, and dual label $i^{\ast }$ for each $i$. Define the handle object

$H={⨁}_{i\in I}i\otimes i^{\ast }\mathrm{.}$

For an oriented marked surface $\mathrm{\Sigma }$ of genus $g$ with ordered marked points labeled by objects $W_1,\dots ,W_n$, fix a parametrization and the standard handle decomposition used to read its labels. The associated algebraic state space is

$V(\mathrm{\Sigma };W_1,\dots ,W_n)={\mathrm{Hom}}_{\mathcal{C}}\text{\hspace{0.17em}⁣}(\mathbf{1},W_1\otimes \cdots \otimes W_n\otimes H^{\otimes g})\mathrm{.}$

The empty tensor product is $\mathbf{1}$ and $H^{\otimes 0}=\mathbf{1}$. Reversing the orientation of a marked boundary circle replaces its label $W$ by the left dual $W^{\ast }$. This formula defines a vector space only; mapping-class operators, bordism maps, independence of parametrization, and gluing are not asserted here.