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Stated $SL_n$-skein modules, roots of unity, and TQFT

Zhihao Wang

Abstract

For a pb surface $Σ$, two positive integers $m,n$ with $m\mid n$, and two invertible elements $v,ε$ in a commutative domain $R$ with $ε^{2m} = 1$, we construct an $R$-linear isomorphism between the stated $SL_n$-skein algebras $S_n(Σ,v)$ and $S_n(Σ,εv)$, which restricts to an algebraic ismorphism between subalgebras of $S_n(Σ,v)$ and $S_n(Σ,εv)$. Using this linear isomorphism, we prove the splitting map $Θ_{c}:S_n(Σ,v)\rightarrow S_n(\text{Cut}_c(Σ),v)$ for the pb surface $Σ$ and the ideal arc $c$ is injective when $v^{2m} = 1$ and $m\mid n$. We generalize Barrett's work to the $SL_n$-skein space and stated $SL_n$-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 $\mathcal{N}\neq\emptyset$, and let $(M,\mathcal{N}')$ 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 $S_n(M,\mathcal{N}',v)$ induced by the embedding $(M,\mathcal{N})\rightarrow (M,\mathcal{N}')$ is injective and $S_n(M.\mathcal{N}',v) = S_n(M,\mathcal{N},v)\otimes_{R}O_{q_v}(SL_n)$, where $O_{q_v}(SL_n)$ is the quantization of the regular function ring of $SL_n$. This shows the splitting map for $S_n(M,\mathcal{N},v)$ is always injective. We formulate the stated $SL_n$-TQFT theory, which generalizes the Costantino and Lê's stated $SL_2$-TQFT theory.

Stated $SL_n$-skein modules, roots of unity, and TQFT

Abstract

For a pb surface , two positive integers with , and two invertible elements in a commutative domain with , we construct an -linear isomorphism between the stated -skein algebras and , which restricts to an algebraic ismorphism between subalgebras of and . Using this linear isomorphism, we prove the splitting map for the pb surface and the ideal arc is injective when and . We generalize Barrett's work to the -skein space and stated -skein space. As an application, we prove the splitting map for the marked 3-manifolds is always injective when the quantum parameter . Let be a connected marked 3-manifold with , and let be obtained from by adding one extra marking. When , we prove the -linear map from to induced by the embedding is injective and , where is the quantization of the regular function ring of . This shows the splitting map for is always injective. We formulate the stated -TQFT theory, which generalizes the Costantino and Lê's stated -TQFT theory.
Paper Structure (16 sections, 27 theorems, 74 equations, 4 figures)

This paper contains 16 sections, 27 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Stated $SL_n$-skein modules
  4. Functoriality
  5. The splitting map
  6. Punctured bordered surfaces and stated $SL_n$-skein algebras
  7. The splitting map for pb surfaces
  8. The $O_q(SL_n)$
  9. Stated $SL_n$-skein algebras skewed by roots of unity
  10. On $SL_n$-skein algebras
  11. On stated $SL_n$-skein algebras
  12. Spin structure and the $SL_n$-skein space
  13. The $R$-linear isomorphism between $S_n(M,\mathcal{N},v)$ and $S_n(M,\mathcal{N},-v)$
  14. Stated $SL_n$-TQFT
  15. Stated $SL_n$-skein modules at 4-th roots of unity
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\Sigma$ be a pb surface, let $m,n$ be two positive integers with $m\mid n$, and let $\epsilon\in R$ with $\epsilon^{2m} = 1$. Then there exists an $R$-linear isomorphism $\varphi_{\epsilon}:S_n(\Sigma,v)\rightarrow S_n(\Sigma,\epsilon v)$.

Figures (4)

  • Figure 1: Five moves for the (stated) $n$-web diagrams. The orientations for (stated) $n$-web diagrams are arbitrary.
  • Figure 2: The sign determined by the left (resp. the right) picture is "$+$" (resp. "$-$").
  • Figure 3: The faint green portions belong to $\partial M'$, and the blue sections pertain to $c\times \partial D$. The red curves have arbitrary orientations and represent parallel copies of stated arcs. The states for these parallel copies of arcs are illustrated by vectors $\vec{a},\vec{b},\vec{c},\vec{d}$ whose entries are integers between $1$ and $n$. In the same relation, the left-hand side and the right-hand side are compatible with each other regarding the orientations of the red curves. The black arrow in the left (resp. right) picture is a part of $e_1'$ (resp. $e_2'$). The left (resp. right) picture is intended for relations that generate $\text{Ker}(h_1)$ (resp. $\text{Ker}(h_2)$).
  • Figure 4: The picture illustrates the $R$-linear map $f_*$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • ...and 43 more