## The AJ-conjecture for the Teichmüller TQFT

Research output: Working paper/Preprint Working paperResearch

### Standard

The AJ-conjecture for the Teichmüller TQFT. / Andersen, Jørgen Ellegaard; Malusà, Alessandro.

ArXiv, 2017.

Research output: Working paper/Preprint Working paperResearch

### Harvard

Andersen, JE & Malusà, A 2017 'The AJ-conjecture for the Teichmüller TQFT' ArXiv.

### APA

Andersen, J. E., & Malusà, A. (2017). The AJ-conjecture for the Teichmüller TQFT. ArXiv.

### CBE

Andersen JE, Malusà A. 2017. The AJ-conjecture for the Teichmüller TQFT. ArXiv.

### MLA

Andersen, Jørgen Ellegaard and Alessandro Malusà The AJ-conjecture for the Teichmüller TQFT. ArXiv. 2017.,

### Vancouver

Andersen JE, Malusà A. The AJ-conjecture for the Teichmüller TQFT. ArXiv. 2017.

### Author

Andersen, Jørgen Ellegaard ; Malusà, Alessandro. / The AJ-conjecture for the Teichmüller TQFT. ArXiv, 2017.

### Bibtex

@techreport{1cd753af6afa4940a8807298b5e972de,
title = "The AJ-conjecture for the Teichm{\"u}ller TQFT",
abstract = "We formulate the AJ-conjecture for the Teichm\{"}{u}ller TQFT and we prove it in the case of the figure-eight knot complement and the $5_2$-knot complement. This states that the level-$N$ Andersen-Kashaev invariant, $J^{(\mathrm{b},N)}_{M,K}$, is annihilated by the non-homogeneous $\widehat{A}$-polynomial, evaluated at appropriate $q$-commutative operators. These are obtained via geometric quantisation on the moduli space of flat $\operatorname{SL}(2,\mathbb{C})$-connections on a genus-$1$ surface. The construction depends on a parameter $\sigma$ in the Teichm\{"}{u}ller space in a way measured by the Hitchin-Witten connection, and results in Hitchin-Witten covariantly constant quantum operators for the holonomy functions $m$ and $\ell$ along the meridian and longitude. 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.",
keywords = "math.DG, math.QA",
author = "Andersen, {J{\o}rgen Ellegaard} and Alessandro Malus{\`a}",
note = "40 pages, 2 figures",
year = "2017",
language = "English",
publisher = "ArXiv",
type = "WorkingPaper",
institution = "ArXiv",

}

### RIS

TY - UNPB

T1 - The AJ-conjecture for the Teichmüller TQFT

AU - Andersen, Jørgen Ellegaard

AU - Malusà, Alessandro

N1 - 40 pages, 2 figures

PY - 2017

Y1 - 2017

N2 - We formulate the AJ-conjecture for the Teichm\"{u}ller TQFT and we prove it in the case of the figure-eight knot complement and the $5_2$-knot complement. This states that the level-$N$ Andersen-Kashaev invariant, $J^{(\mathrm{b},N)}_{M,K}$, is annihilated by the non-homogeneous $\widehat{A}$-polynomial, evaluated at appropriate $q$-commutative operators. These are obtained via geometric quantisation on the moduli space of flat $\operatorname{SL}(2,\mathbb{C})$-connections on a genus-$1$ surface. The construction depends on a parameter $\sigma$ in the Teichm\"{u}ller space in a way measured by the Hitchin-Witten connection, and results in Hitchin-Witten covariantly constant quantum operators for the holonomy functions $m$ and $\ell$ along the meridian and longitude. 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.

AB - We formulate the AJ-conjecture for the Teichm\"{u}ller TQFT and we prove it in the case of the figure-eight knot complement and the $5_2$-knot complement. This states that the level-$N$ Andersen-Kashaev invariant, $J^{(\mathrm{b},N)}_{M,K}$, is annihilated by the non-homogeneous $\widehat{A}$-polynomial, evaluated at appropriate $q$-commutative operators. These are obtained via geometric quantisation on the moduli space of flat $\operatorname{SL}(2,\mathbb{C})$-connections on a genus-$1$ surface. The construction depends on a parameter $\sigma$ in the Teichm\"{u}ller space in a way measured by the Hitchin-Witten connection, and results in Hitchin-Witten covariantly constant quantum operators for the holonomy functions $m$ and $\ell$ along the meridian and longitude. 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.

KW - math.DG

KW - math.QA

M3 - Working paper

BT - The AJ-conjecture for the Teichmüller TQFT

PB - ArXiv

ER -