The quantum content of the gluing equations. (English) Zbl 1283.57017
Let \(\langle{K}\rangle_N\in\mathbb{C}\) be the Kashaev invariant of a knot \(K\) in the three-sphere \(S^3\) [R. M. Kashaev, Mod. Phys. Lett. A 10, No. 19, 1409–1418 (1995; Zbl 1022.81574)], which turned out to coincide with the colored Jones polynomial \(J_N(K;q)\) evaluated at \(q=\exp(2\pi\sqrt{-1}/N)\), cf. H. Murakami and J. Murakami [Acta Math. 186, No.1, 85–104 (2001; Zbl 0983.57009)]. Kashaev conjectured that for a hyperbolic knot (a knot whose complement possesses a complete hyperbolic structure with finite volume), the limit \(\lim_{N\to\infty}N^{-1}\log|\langle{K}\rangle_N|\) exists and equals \(\text{Vol}(M)/(2\pi)\), cf. R. M. Kashaev [Lett. Math. Phys. 39, No. 3, 269–275 (1997; Zbl 0876.57007)], where \(M:=S^3\setminus{K}\) and \(\text{Vol}\) is the hyperbolic volume. It is also expected that the Kashaev invariant has the following asymptotic expansion:
\[
\exp\left(\hbar^{-1}S_{M,0}-\frac{3}{2}\log\hbar+S_{M,1}+\sum_{n=2}^{\infty}\hbar^{n-1}S_{M,n}\right)
\]
with \(\hbar:=2\pi\sqrt{-1}/N\), S. Gukov [Commun. Math. Phys. 255, No. 3, 577–627 (2005; Zbl 1115.57009)]. Here \(S_{M,0}\) is the complexified volume, that is, \(S_{M,0}=\sqrt{-1}\bigl(\text{Vol}(M)+\sqrt{-1}\text{CS}(M)\bigr)\) with \(\text{CS}\) the Chern–Simons invariant. The constant term \(S_{M,1}\) is conjectured to be related to the non-Abelian Reidemeister–Ray–Singer torsion \(\tau^{\roman{R}}_M\) of the holonomy representation, which defines the complete hyperbolic structure, with respect to the meridian, J. Porti [Mem. Am. Math. Soc. 612, 139 p. (1997; Zbl 0881.57020)] in the following way: \(\tau^{\roman{R}}_M=4\pi^3\exp(-2S_{M,1})\). Note that \(\tau^{\roman{R}}_M\) belongs to the invariant trace field \(E_M\), [J. Dubois and S. Garoufalidis, “Rationality of the SL(2,\(\mathbb C\))-Reidemeister torsion in dimension 3”, arXiv:0908.1690]. It is also conjectured that for any \(n\geq2\), \(S_{M,n}\) is in \(E_M\), [T. Dimofte et al., Commun. Number Theory Phys. 3, No. 2, 363–443 (2009; Zbl 1214.81151)].
In the paper under review, the authors introduce a topological invariant \(\tau_{M}\) for a one-cusped hyperbolic three-manifold \(M\), and conjecture that \(\tau_{M}=\pm\tau^{\roman{R}}_{M}\) if \(M\) is the complement of a hyperbolic knot.
Given an ideal triangulation \(\mathcal{T}\), let \(V_{\mathcal{T}}\) denote the affine variety of solutions (in \(\mathbb{C}\setminus\{0,1\}\)) of the gluing equations of \(\mathcal{T}\). For a representation \(\rho:\pi_1(M)\to\text{PSL}(2;\mathbb{C})\), let \(\mathcal{X}_{\rho}\) be the set of \(\rho\)-regular ideal triangulations of \(M\), where an ideal triangulation \(\mathcal{T}\) is called \(\rho\)-regular if \(\rho\) is in the image of the developing map \(V_{\mathcal{T}}\to\text{Hom}(\pi_1(M),\text{PSL}(2,\mathbb{C}))/\text{PSL}(2,\mathbb{C})\). An ideal triangulation of a hyperbolic knot is called regular if it is \(\rho_0\)-regular for the holonomy representation \(\rho_0\).
To define \(\tau_{M}\), one needs to fix a regular ideal triangulation and an enhanced Neumann–Zagier datum. Here an enhanced Neumann–Zagier datum is a quadruple \((z,\mathbf{A},\mathbf{B},f)\), where \(z=(z_1,\dots,z_N)\) is a collection of shape parameters for ideal hyperbolic tetrahedra, \(\mathbf{A}\) and \(\mathbf{B}\) are \(N\times N\) matrices over the integers that define how to glue the tetrahedra, and \(f=(f_1,f'_1,f''_1,\dots,f_N,f'_N,f''_N)\) is a collection of \(3N\) integers called a combinatorial flattening of the triangulation introduced in [W. D. Neumann, Topology ’90, Contrib. Res. Semester Low Dimensional Topol., Columbus/OH (USA) 1990, Ohio State Univ. Math. Res. Inst. Publ. 1, 243–271 (1992; Zbl 0768.57006)] to calculate the Chern–Simons invariant combinatorially.
To prove the invariance of \(\tau_{M}\), the authors show that it is invariant under \(2\)-\(3\) moves. Then they observe that any two regular refinements of the canonical ideal cell decomposition by D. B. A. Epstein and R. C. Penner [J. Differ. Geom. 27, No.1, 67–80 (1988; Zbl 0611.53036)] can be connected by a finite sequence of \(2\)-\(3\) moves in \(\mathcal{X}_{\rho_0}\), proving the topological invariance of \(\tau_{M}\).
In the paper under review, the authors introduce a topological invariant \(\tau_{M}\) for a one-cusped hyperbolic three-manifold \(M\), and conjecture that \(\tau_{M}=\pm\tau^{\roman{R}}_{M}\) if \(M\) is the complement of a hyperbolic knot.
Given an ideal triangulation \(\mathcal{T}\), let \(V_{\mathcal{T}}\) denote the affine variety of solutions (in \(\mathbb{C}\setminus\{0,1\}\)) of the gluing equations of \(\mathcal{T}\). For a representation \(\rho:\pi_1(M)\to\text{PSL}(2;\mathbb{C})\), let \(\mathcal{X}_{\rho}\) be the set of \(\rho\)-regular ideal triangulations of \(M\), where an ideal triangulation \(\mathcal{T}\) is called \(\rho\)-regular if \(\rho\) is in the image of the developing map \(V_{\mathcal{T}}\to\text{Hom}(\pi_1(M),\text{PSL}(2,\mathbb{C}))/\text{PSL}(2,\mathbb{C})\). An ideal triangulation of a hyperbolic knot is called regular if it is \(\rho_0\)-regular for the holonomy representation \(\rho_0\).
To define \(\tau_{M}\), one needs to fix a regular ideal triangulation and an enhanced Neumann–Zagier datum. Here an enhanced Neumann–Zagier datum is a quadruple \((z,\mathbf{A},\mathbf{B},f)\), where \(z=(z_1,\dots,z_N)\) is a collection of shape parameters for ideal hyperbolic tetrahedra, \(\mathbf{A}\) and \(\mathbf{B}\) are \(N\times N\) matrices over the integers that define how to glue the tetrahedra, and \(f=(f_1,f'_1,f''_1,\dots,f_N,f'_N,f''_N)\) is a collection of \(3N\) integers called a combinatorial flattening of the triangulation introduced in [W. D. Neumann, Topology ’90, Contrib. Res. Semester Low Dimensional Topol., Columbus/OH (USA) 1990, Ohio State Univ. Math. Res. Inst. Publ. 1, 243–271 (1992; Zbl 0768.57006)] to calculate the Chern–Simons invariant combinatorially.
To prove the invariance of \(\tau_{M}\), the authors show that it is invariant under \(2\)-\(3\) moves. Then they observe that any two regular refinements of the canonical ideal cell decomposition by D. B. A. Epstein and R. C. Penner [J. Differ. Geom. 27, No.1, 67–80 (1988; Zbl 0611.53036)] can be connected by a finite sequence of \(2\)-\(3\) moves in \(\mathcal{X}_{\rho_0}\), proving the topological invariance of \(\tau_{M}\).
Reviewer: Hitoshi Murakami (Tokyo)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
Keywords:
volume; complex Chern–Simons theory; Kashaev invariant; gluing equations; Neumann–Zagier equations; Neumann–Zagier datum; hyperbolic geometry; ideal triangulations; 1-loop; torsion; quantum dilogarithm; state integral; perturbation theory; Feynman diagram; formal Gaussian integralCitations:
Zbl 1022.81574; Zbl 0983.57009; Zbl 0876.57007; Zbl 1115.57009; Zbl 0881.57020; Zbl 1214.81151; Zbl 0768.57006; Zbl 0611.53036Software:
SnapPyReferences:
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