Stated $S{L}_n$-skein modules, roots of unity, and TQFT

Abstract

For a pb surface $\Sigma$, two positive integers $m,n$ with $m\mid n$, and two invertible elements $v,\epsilon$ in a commutative domain $R$ with ${\epsilon }^{2m}=1$, we construct an $R$-linear isomorphism between the stated $S{L}_n$-skein algebras $S_n(\Sigma ,v)$ and $S_n(\Sigma ,\epsilon v)$, which restricts to an algebraic ismorphism between subalgebras of $S_n(\Sigma ,v)$ and $S_n(\Sigma ,\epsilon v)$. Using this linear isomorphism, we prove the splitting map ${\Theta }_c:S_n(\Sigma ,v)\to S_n({\text{Cut}}_c(\Sigma ),v)$ for the pb surface $\Sigma$ and the ideal arc $c$ is injective when $v^{2m}=1$ and $m\mid n$. We generalize Barrett's work to the $S{L}_n$-skein space and stated $S{L}_n$-skein space. As an application, we prove the splitting map for the marked 3-manifolds is always injective when the quantum parameter $v=-1$. Let $(M,\mathcal{N})$ be a connected marked 3-manifold with $\mathcal{N}\ne \varnothing$, and let $(M,{\mathcal{N}}^{\prime })$ be obtained from $(M,\mathcal{N})$ by adding one extra marking. When $v^4=1$, we prove the $R$-linear map from $S_n(M,\mathcal{N},v)$ to $S_n(M,{\mathcal{N}}^{\prime },v)$ induced by the embedding $(M,\mathcal{N})\to (M,{\mathcal{N}}^{\prime })$ is injective and $S_n(M.{\mathcal{N}}^{\prime },v)=S_n(M,\mathcal{N},v){\otimes }_R{O}_{q_v}(S{L}_n)$, where $O_{q_v}(S{L}_n)$ is the quantization of the regular function ring of $S{L}_n$. This shows the splitting map for $S_n(M,\mathcal{N},v)$ is always injective. We formulate the stated $S{L}_n$-TQFT theory, which generalizes the Costantino and Lê's stated $S{L}_2$-TQFT theory.

29 pages, the update for the accepted version, to appear in Israel Journal of Mathematics