Weighted Lagrangian extension and root anomaly data

For the fixed root parameter, let ${\mathcal{C}}_p$ denote the semisimplified Temperley–Lieb category with labels $0,\dots ,r-2$, pivotal dimensions $d_j=[j+1]$, and Hopf-link data inherited from the Kauffman bracket. Put ${\mathcal{D}}_p^2={\sum }_{j=0}^{r-2}{d}_j^2$ and choose ${\mathcal{D}}_p\in \mathbb{C}$ with that square. Choose $\kappa \in \mathbb{C}$ satisfying ${\kappa }^6=A^{-6-(p+1)p\mathrm{/}2}$. The root-specific extended data are the weighted Lagrangian ribbon bordisms whose objects are closed oriented labeled surfaces with a Lagrangian subspace of real first homology and whose morphisms are collared oriented labeled ribbon bordisms with integer weights. If $M_1:{\mathrm{\Sigma }}_0\to {\mathrm{\Sigma }}_1$ and $M_2:{\mathrm{\Sigma }}_1\to {\mathrm{\Sigma }}_2$ have relations ${\mathrm{\Lambda }}_{M_1},{\mathrm{\Lambda }}_{M_2}$ and boundary Lagrangians ${\lambda }_0,{\lambda }_1,{\lambda }_2$, define

$K_1={\mathrm{\Lambda }}_{M_1}({\lambda }_0),K_0={\lambda }_1,K_2={\mathrm{\Lambda }}_{M_2}^t({\lambda }_2),n(M_2\circ M_1)=n_2+n_1+\mu (K_1,K_0,K_2),$

using the Wall-signature Maslov convention. The category composition, units, and anomaly weight are those specified by this formula and the weighted Lagrangian bordism definition; no ordinary oriented anomaly-free functor is part of this Definition.

Root and loop conventions

Root category

Fusion, ribbon, and Hopf-link data

Weighted Lagrangian ribbon-bordism category and composition convention

Blanchet, Habegger, Masbaum, and Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §§1–2 and Appendix B

Pohorence, 3-manifold Invariants and TQFTs, §3.1

Walker, On Witten's 3-Manifold Invariants, §§1 and 18

Full choice assumption inherited by the weighted category

The BHMV source uses $p_1$-structures, while the registered prerequisite map specifies Lagrangian-weighted data. This Definition supplies the latter using the existing local category carrier; the equality between its Maslov correction and the BHMV $p_1$/signature anomaly remains a theorem obligation and is not asserted here.