A Geometric Quantisation view on the AJ-conjecture for the Teichmüller TQFT
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arXiv
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| Natura: | Preprint |
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2017
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| _version_ | 1866909958822428672 |
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| author | Andersen, Jørgen Ellegaard Malusà, Alessandro |
| author_facet | Andersen, Jørgen Ellegaard Malusà, Alessandro |
| contents | We provide a Geometric Quantisation formulation of the AJ-conjecture for the Teichmüller 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 $σ$ in the Teichmüller 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1711_11522 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | A Geometric Quantisation view on the AJ-conjecture for the Teichmüller TQFT Andersen, Jørgen Ellegaard Malusà, Alessandro Differential Geometry Quantum Algebra We provide a Geometric Quantisation formulation of the AJ-conjecture for the Teichmüller 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 $σ$ in the Teichmüller 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. |
| title | A Geometric Quantisation view on the AJ-conjecture for the Teichmüller TQFT |
| topic | Differential Geometry Quantum Algebra |
| url | https://arxiv.org/abs/1711.11522 |