Weighted Reshetikhin–Turaev sewing, framing anomaly and state dimensions

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.

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.

Assume full AC and its inherited countable-choice consequences, and fix a weighted RT bordism $M$ with all its boundary data, ribbon graph, object Lagrangians, and coupons held fixed. If its integer weight is $n$, then $Z(M,n+1)=\kappa Z(M,n)$. If $\kappa \mathrm{\ne }1$ and $Z(M,n)$ is a nonzero linear map, the weighted RT assignment cannot factor through the operation that forgets integer weights while keeping the same maps. This obstruction concerns ordinary forgetting of the weight; an anomaly-canceling enlargement or a different normalization requires separate data and is not asserted here.

Weighted modular TQFT model and physical interpretation boundary

The weighted Reshetikhin–Turaev TQFT and modular-functor factorization

Modular ribbon fusion category and Hopf-link data

The Axiom of Choice

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