Quantizations of Character Varieties and Quantum Knot Invariants
Abstract
Let G be a simple complex algebraic group and g its Lie algebra. We show that the g-Witten-Reshetikhin-Turaev quantum invariants determine a deformation-quantization, Cq[XG(torus)], of the coordinate ring of the G-character variety of the torus. We prove that this deformation is in the direction of the Goldman's bracket. Furthermore, we show that every knot K defines an ideal IK in Cq[XG(torus)]. We conjecture that the homomorphism Cq[XG(torus)] -> C[XG(torus)], q -> 1, maps IK to the ideal whose radical is the kernel of the map C[XG(torus)] -> C[XG(S3 K)]. This conjecture is related to AJ-conjecture for sl(2,). The results of this paper are inspired by the theory of q-holonomic relations between quantum invariants of Garoufalidis and Le. Along the way, we disprove Conjecture 2 in Le's "The Colored Jones and the A-polynomial of Two-Bridge knots".