## Volume conjecture: refined and categorified.(English)Zbl 1282.57016

Let $$J_n(K;q)$$ be the colored Jones polynomial of a knot $$K$$ in the three-sphere associated with the $$n$$-dimensional irreducible representation of the Lie algebra $$sl(2;\mathbb{C})$$. The volume conjecture ([R. M. Kashaev, Lett. Math. Phys. 39, No. 3, 269–275 (1997; Zbl 0876.57007); H. Murakami and J. Murakami, Acta Math. 186, No. 1, 85–104 (2001; Zbl 0983.57009)]) states that the limit $$\lim_{n\to\infty}\frac{2\pi\log|J_n(K;e^{2\pi\sqrt{-1}/n})|}{n}$$ would coincide with the volume of the knot complement. Given a complex parameter $$u$$, this can be generalized to the generalized volume conjecture $$J_n(K;e^{\hbar})\underset{{n\to\infty}\atop{\hbar\to0}}{\sim}\exp\left(\frac{1}{\hbar}S_0(u)+\sum_{k\geq0}S_{k+1}(u)\hbar^k\right)$$, where $$\hbar$$ satisfies $$e^{n\hbar}=e^{u}$$ and $$S_0(u)$$ is expected to define the Chern–Simons invariant of a representation of the fundamental group of the knot complement to $$SL(2;\mathbb{C})$$ parameterized by $$u$$. ([S. Gukov, Commun. Math. Phys. 255, No. 3, 577–627 (2005; Zbl 1115.57009); H. Murakami, Adv. Math. 211, No. 2, 678–683 (2007; Zbl 1121.57006); S. Gukov and H. Murakami, Fields Institute Communications 54, 261–277 (2008; Zbl 1171.57012)]). When $$u=0$$, it is a restatement of the (complexified) volume conjecture [H. Murakami et al., Exp. Math. 11, No. 3, 427–435 (2002; Zbl 1117.57300)]. The parameter $$u$$ is regarded as a deformation parameter in the case of a hyperbolic knot. The eigenvalues of the image of the meridian by the associated representation are $$e^{\pm u}$$. If we denote the eigenvalues of the image of the longitude by $$e^{\pm v}$$, then $$v$$ coincides with $$d\,S_{0}(u)/d\,u$$ (see [W. D. Neumann and D. Zagier, Topology 24, 307–332 (1985; Zbl 0589.57015)]). Therefore $$S_{0}(u)$$ is expected to be given as the integral $$\int v\,du=\int\log(y)\frac{dx}{x}$$, where $$(x,y):=(e^u,e^v)$$ is in the zero locus of the $$A$$-polynomial $$A_K(x,y)$$ [D. Cooper et al., Invent. Math. 118, No. 1, 47–84 (1994; Zbl 0842.57013)].
It was observed in [S. Gukov, loc. cit.] that the generalized volume conjecture is closely related to the AJ-conjecture, S. Garoufalidis [Coventry: Geometry and Topology Publications. Geometry and Topology Monographs 7, 291–309 (2004; Zbl 1080.57014)], which states that the colored Jones polynomial is annihilated by a quantum version of the $$A$$-polynomial $$\hat{A}_K(\hat{x},\hat{y};q)$$. Precisely speaking, for any knot $$K$$ there exists a non-commutative polynomial $$\hat{A}_K(\hat{x},\hat{y};q)$$ such that $$\hat{A}_K(\hat{x},\hat{y};q)J_n(K;q)=0$$ and that $$A_K(x,y):=\hat{A}_K(\hat{x},\hat{y};q)\big|_{q=1,\hat{x}=x,\hat{y}=y}$$ coincides with the $$A$$-polynomial, where $$\hat{x}$$ acts on $$J_n(K;q)$$ by multiplication by $$q^n$$ and $$\hat{y}$$ replaces $$n$$ with $$n+1$$. Note the commuting relation $$\hat{y}\hat{x}=q\hat{x}\hat{y}$$. This conjecture is called the quantum volume conjecture in this paper.
In the paper under review the authors propose further generalizations of the above conjectures. Let $$P_{n}(K;q,t)$$ be the Poincaré polynomial of a categorification $$\mathcal{H}^{n}_{i,j}(K)$$ of the colored Jones polynomial $$J_n(K;q)$$, that is, $$P_{n}(K;q,t)$$ is defined as $$\sum_{i,j}q^it^j\dim\mathcal{H}^{n}_{i,j}(K)$$ and $$P_{n}(K;q,-1)$$ coincides with $$J_n(K;q)$$. Then the refined (or categorified) quantum volume conjecture states that there would exist a non-commutative polynomial $$\hat{A}^{\roman{ref}}_K(\hat{x},\hat{y};q,t)$$ such that $$\hat{A}^{\roman{ref}}_K(\hat{x},\hat{y};q,t)P_{n}(K;q,t)$$ vanishes and that $$\hat{A}_K(\hat{x},\hat{y};q)=\hat{A}^{\roman{ref}}_K(\hat{x},\hat{y};q,-1)$$. The refined generalized volume conjecture states that the asymptotic expansion of $$P_n(K;e^{\hbar},t)$$ with respect to $$\hbar$$ would become $$P_n(K;e^{\hbar},t)\underset{n\to\infty}{\sim}\exp\left(\frac{1}{\hbar}S_0(u,t)+\sum_{k\geq0}S_{k+1}(u,t)\hbar^k\right)$$ such that $$S_0(u,t)$$ is given by the integral $$\int\log(y)\frac{dx}{x}$$, where $$(x,y)$$ is in the zero locus of a deformation (or a categorification) $$A^{\roman{ref}}_K(x,y;t):=A^{\roman{ref}}_K(x,y;1,t)$$ of the $$A$$-polynomial.
The authors calculate $$\hat{A}^{\roman{ref}}_K(\hat{x},\hat{y};q,t)$$ and $$A^{\roman{ref}}_K(x,y;t)$$ for the unknot and the torus knots of type $$(2,2p+1)$$ by using various techniques from physics.

### MSC:

 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 81T13 Yang-Mills and other gauge theories in quantum field theory 58J28 Eta-invariants, Chern-Simons invariants
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