SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial

TL;DR

This work refines the link between the asymptotics of the colored Jones polynomial and $SL(2,\mathbb{C})$ Chern-Simons theory by fixing normalization and polarization ambiguities and by validating a parameterized (generalized) volume conjecture with a semi-classical action $S'(u)$. It shows that the leading term of the expansion arises from $S'(u)$, while subleading contributions reflect a loop expansion and representation-dependent invariants, notably the Ray-Singer torsion $T_K(u)$. The authors develop precise formulas for the logarithmic term via $\delta^{\text{rep}}_K(\rho)$, and analyze Abelian, connected-sum, and satellite knots to derive specific delta values; they also connect the subleading torsion term to Reidemeister torsion via the Cheeger–Müller theorem. Together, these results support a coherent picture in which the colored Jones asymptotics match the loop expansion of $SL(2,\mathbb{C})$ Chern-Simons theory on knot complements, with concrete predictions for a broad class of knots and representations.

Abstract

We clarify and refine the relation between the asymptotic behavior of the colored Jones polynomial and Chern-Simons gauge theory with complex gauge group SL(2,C). The precise comparison requires a careful understanding of some delicate issues, such as normalization of the colored Jones polynomial and the choice of polarization in Chern-Simons theory. Addressing these issues allows us to go beyond the volume conjecture and to verify some predictions for the behavior of the subleading terms in the asymptotic expansion of the colored Jones polynomial.

Figures (7)

• Figure 1: $N=100$
• Figure 2: $N=200$
• Figure 3: $N=300$
• Figure 4: $N=400$
• Figure 5: $N=500$

Theorems & Definitions (18)

• Conjecture 1.1: Volume Conjecture
• Conjecture 1.2
• Conjecture 1.3
• Definition 3.1
• Proposition 3.2
• proof
• Corollary 3.3
• Lemma 3.4
• proof
• Lemma 3.5