Asymptotics of quantum representations of surface groups

TL;DR

This paper proves the convergence of quantum invariants of surface links to a polynomial function, relates this to the AMU conjecture on curve orders, and confirms the Witten asymptotic conjecture for surface links.

Contribution

It establishes the asymptotic behavior of quantum invariants, proves the Witten asymptotic conjecture for surface links, and provides evidence for the AMU conjecture.

Findings

Normalized invariants converge to a polynomial as roots of unity approach a limit.

Certain curves satisfy the AMU conjecture predicting infinite order in quantum representations.

The Witten asymptotic conjecture is proven for links in surface times circle.

Abstract

For a banded link $L$ in a surface times a circle, the Witten-Reshetikhin-Turaev invariants are topological invariants depending on a sequence of complex $2p$-th roots of unity $(A_p)_{p\in 2\mathbb{N}}$. We show that there exists a polynomial $P_L$ such that these normalized invariants converge to $P_L(u)$ when $A_p$ converges to $u$, for all but a finite number of $u$'s in $S^1$. This is related to the AMU conjecture which predicts that non-simple curves have infinite order under quantum representations (for big enough levels). Estimating the degree of $P_L$, we exhibit particular types of curves which satisfy this conjecture. Along the way we prove the Witten asymptotic conjecture for links in a surface times a circle.