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Handlebody pairing nondegeneracy and gluing

Statement

For the fixed root parameter, the normalized BHMV sesquilinear pairing between the relative skein modules of two complementary marked handlebodies induces a perfect pairing between their radical quotients, using the coefficient-conjugate space on the oppositely oriented side. The inverse pairing defines the sewing map for gluing along a closed marked surface, and these maps satisfy the stated weighted Lagrangian anomaly correction.

Statement references

State-space pairing

Finite spine spanning set

Weighted Lagrangian anomaly convention

Blanchet, Habegger, Masbaum, and Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §§1 and 4–5

Kauffman and Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 6–7

Pohorence, 3-manifold Invariants and TQFTs, §3

Facts & Assumptions

Given: The finite spanning set of admissible spine states and the normalized gluing pairing.

[F1]

Skein TQFT state space and handlebody pairing defines the radical quotient and pairing.

[F2]

Finite generation by admissible handlebody-spine states supplies the finite admissible spine spanning set via its explicitly labelled external BHMV input.

[F3]

Weighted Lagrangian extension and root anomaly data fixes the weight and Maslov convention.

[F4]

External input — perfect pairing and sewing. BHMV, Theorem1.4 p.887, property(F) on p.886, Proposition1.9 p.891, and Theorem4.11 p.909: in this even-root theory the radical quotient is finite free with unimodular dual pairing, the admissible spine states form a basis with unit diagonal norms in the double pairing, and the universal gluing maps satisfy the TQFT axioms. Proposition1.9 permits the complementary handlebody giving $S^{3}$ , rather than assuming the self-double and $S^{3}$ pairings have the same matrix. Their normalization is the $Z_{p}$ normalization explicitly fixed in [F1]. Only perfectness/basis/sewing at this actual convention is imported.

[F5]

External input — actual weighted variant. Gilmer–Masbaum, arXiv:0912.4706v3, §2 equation(1) pp.4–6 and §11 pp.23–25, states the colored TQFT with Lagrangians and integer weights and one-weight multiplier $ν = κ^{3}$ in BHMV notation. The finite sign comparison $μ_{G M} = - μ_{W a l l}$ is as recorded in the Kirby lemma, so its composition is exactly that of [F3]. The source states rational Lagrangians. For the real Lagrangians of [F3], take the rational handlebody-kernel reference on each marked surface and transport by the weight-zero identity cylinder to its specified real Lagrangian; the reverse cylinder is its inverse because $μ (L, L^{'}, L) = 0$ . Conjugating morphisms by these cylinders is a functor to the rational-reference category, with the integer weights computed by [F3], so the imported functor extends to these real objects. This is a finite categorical transport, not an assumption that every real Lagrangian is rational.

Proof

Given: A marked handlebody spine, its admissible colorings, and the gluing pairing.

1.1F1F2F3given

On a fixed finite spine, each admissible coloring gives a finite colored graph state. Glue the first state to the semilinear orientation reversal of the second state, as in [F1], and evaluate the resulting closed colored graph using the specified root bracket and anomaly normalization.

2.1F1F2F4step 1.1algebra

By [F4], the pairing descends to a perfect sesquilinear pairing $V \times W \to C$ , where $V$ is the quotient from $H$ and $W$ is the quotient from the second filling $H^{'}$ with the same oriented boundary, by the right radical. The semilinear reversal in [F1] identifies $\overline{W}$ linearly with the quotient for $- H^{'}$ . Thus $B (v, \overline{w}) : = B_{Σ} (v, w)$ is a perfect bilinear pairing between $V$ and $\overline{W}$ . This is stronger than left nondegeneracy by the definition of a radical alone. Choose bases $v_{i}$ and ${\overline{w}}_{j}$ , and let $B_{i j}$ be their bilinear pairing matrix. Its inverse gives the coevaluation tensor $\sum_{i, j} (B^{- 1})_{j i} v_{i} \otimes {\overline{w}}_{j}$ ; contraction with $B$ on either side is the identity. This finite linear-algebra identity is exactly the sewing contraction in the imported gluing theorem and is independent of the chosen dual bases.

3.1F3F4F5step 2.1algebra∎

The imported weighted gluing in [F5] gives the prescribed anomaly factor, with weight law [F3]. Its parameter is the cube of the selected sixth root, not an independently chosen square root with an unchecked sign. The real-Lagrangian transport in [F5] respects composition because the intervening reference cylinders cancel. Thus the inverse-pairing sewing maps satisfy precisely the registered weighted law. No unweighted anomaly cancellation or positivity of every complex-root Hermitian form is asserted.

Depends on

Used by

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Sources

C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §§1 and 4–5 (standard reference, not scraped)
Louis H. Kauffman and Sostenes L. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Chapters 6–7 (standard reference, not scraped)
Sean Pohorence, 3-manifold Invariants and TQFTs (January 2016), §3 (standard reference, not scraped)
BHMV, Topological Quantum Field Theories Derived from the Kauffman Bracket, Topology 34 (1995), exact external inputs (standard reference, not scraped)
Gilmer–Masbaum, Maslov index, Lagrangians, Mapping Class Groups and TQFT, arXiv:0912.4706v3 (standard reference, not scraped)