Weighted modular TQFT model and physical interpretation boundary

Assume full AC and its inherited countable-choice consequences. Let $\mathcal{C}$ be a complex modular ribbon fusion category with finite simple-label set $I$, unit label $0$, simple pivotal dimensions $d_i$, twists ${\theta }_i$, and Hopf-link matrix $s$. Choose a square root $D\in \mathbb{C}\setminus \{0\}$ of ${\sum }_{i\in I}{d}_i^2$ and set $S=s\mathrm{/}D$, $p_{+}={\sum }_{i\in I}{\theta }_i{d}_i^2$, and $\kappa =p_{+}\mathrm{/}D$. Modularity means that $s$ is invertible; no positivity of $d_i$ or $D$ is assumed.

Use the weighted Lagrangian ribbon-bordism category. An object is a closed oriented surface $\mathrm{\Sigma }$ with a finite collection of marked points carrying simple-object labels and disjoint oriented interval germs, together with a Lagrangian $\lambda \subset H_1(\mathrm{\Sigma };\mathbb{R})$ for the signed intersection form. A morphism is a compact oriented smooth three-bordism with fixed product collars and boundary parametrizations, a properly embedded colored ribbon graph with actual coupon morphisms and collar-product boundary ribbons, and an integer weight $n$. For a bordism $M:{\mathrm{\Sigma }}_0\to {\mathrm{\Sigma }}_1$, its homology relation is $R_M=\{(x_0,x_1):j_0{x}_0=j_1{x}_1\}$ under the boundary inclusion maps. For composable bordisms $M_1:{\mathrm{\Sigma }}_0\to {\mathrm{\Sigma }}_1$ and $M_2:{\mathrm{\Sigma }}_1\to {\mathrm{\Sigma }}_2$, let $R_1,R_2$ be their individual relations and let $R^t$ denote relation transpose. In the specified middle surface, set $K_1=R_1({\lambda }_0)$, $K_0={\lambda }_1$, and $K_2=R_2^t({\lambda }_2)$; composition assigns weight $n_2+n_1+\mu (K_1,K_0,K_2)$, with the Walker signature/Maslov sign and order. Here $\mu$ is the signature of the Wall form on the triple of middle Lagrangians. For a parametrized genus-$g$ surface with ordered labels $W_1,\dots ,W_n$, write $H={⨁}_{i\in I}i\otimes i^{\ast }$ and

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

The weighted RT model is this category, these finite-dimensional complex state spaces, and the exact surgery-defined symmetric monoidal functor $Z$ of the supplied theorem. It includes actual coupon morphisms, disconnected bordisms and empty ends; increasing an integer weight by one multiplies the map by $\kappa$. The model retains the full anomaly and asserts no ordinary anomaly-free functor unless cancellation data are supplied separately.

Modular ribbon fusion category and Hopf-link data

Weighted Lagrangian ribbon bordisms

RT state spaces from a modular category

The weighted Reshetikhin–Turaev TQFT and modular-functor factorization

The Axiom of Choice

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

This is a mathematical definition of an algebraic model. Modularity alone does not supply a positive Hermitian form or a unitary physical theory, and no Chern–Simons path integral is asserted or used.