SO(3)-Knot States and the Volume Conjecture

TL;DR

This paper explores the relationship between SO(3)-knot states, geometric quantization, and the volume conjecture, proposing a link between quantum invariants and the geometric topology of knot complements.

Contribution

It establishes an isomorphism between the holomorphic sections of a prequantum line bundle and projective representations of the mapping class group, and conjectures a connection to knot complement volume.

Findings

Isomorphism between holomorphic sections and projective representations

Conjecture linking asymptotics of knot states to knot complement volume

Proposes a geometric quantization approach to the volume conjecture

Abstract

We study the alternating subspace of holomorphic sections of a special prequantum line bundle over SU(2)-character variety of torus, and show that it is isomorphic to the projective representation of mapping class group of peripheral torus given by the SO(3) Witten-Chern-Simons theory. We conjecture that the large r asymptotics of $L^2$-norm of SO(3)-knot states via geometric quantization capture the simplicial volume of knot complements.