A Geometric Quantisation view on the AJ-conjecture for the Teichmüller TQFT

We provide a Geometric Quantisation formulation of the AJ-conjecture for the Teichm\"{u}ller TQFT, and we prove it in detail in the case of the knot complements of \(4_{1}\) and \(5_2\). The conjecture states that the level-\(N\) Andersen-Kashaev invariant, \(J^{(\mathrm{b},N)}_{M,K}\), is anni...

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Bibliographic Details
Published in:arXiv.org 2024-04
Main Authors: Andersen, Jørgen Ellegaard, Malusà, Alessandro
Format: Article
Language:English
Subjects:
Mathematical analysis
Operators (mathematics)
Polynomials
Online Access:Get full text
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Summary:We provide a Geometric Quantisation formulation of the AJ-conjecture for the Teichm\"{u}ller TQFT, and we prove it in detail in the case of the knot complements of \(4_{1}\) and \(5_2\). The conjecture states that the level-\(N\) Andersen-Kashaev invariant, \(J^{(\mathrm{b},N)}_{M,K}\), is annihilated by the non-homogeneous \(\hat{A}\)-polynomial, evaluated at appropriate \(q\)-commutative operators. We obtained the latter via Geometric Quantisation on the moduli space of flat \(\operatorname{SL}(2,\mathbb{C})\)-connections on a genus-\(1\) surface, by considering the holonomy functions associated to a meridian and longitude. The construction depends on a parameter \(\sigma\) in the Teichm\"{u}ller space in a way measured by the Hitchin-Witten connection, but we show that the resulting quantum operators are covariantly constant. Their action on \(J^{(\mathrm{b},N)}_{M,K}\) is then defined via a trivialisation of the Hitchin-Witten connection and the Weil-Gel'Fand-Zak transform.
ISSN:2331-8422