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, C_q[X_G(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 I_K in C_q[X_G(torus)]. We conjecture that the homomorphism C_q[X_G(torus)] -> C[X_G(torus)], q -> 1, maps I_K to the ideal whose radical is the kernel of the map C[X_G(torus)] -> C[X_G(S^3 K)]. This conjecture is related to AJ-conjecture for sl(2,\C). 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".
Cite
@article{arxiv.0807.0943,
title = {Quantizations of Character Varieties and Quantum Knot Invariants},
author = {Adam S. Sikora},
journal= {arXiv preprint arXiv:0807.0943},
year = {2008}
}
Comments
17 pages, 1 picture