Distribution of Chern-Simons invariants

TL;DR

This paper investigates the distribution of Chern-Simons invariants on 3-manifolds, showing they tend to be equidistributed with fluctuations akin to white noise, especially in sequences of Dehn fillings.

Contribution

It demonstrates the equidistribution of Chern-Simons invariants for certain 3-manifolds and computes their fluctuations in the context of Dehn fillings.

Findings

Chern-Simons invariants tend to become equidistributed on the circle.

Distribution resembles quadratic residues in some examples.

Fluctuations of invariants are of order |X(M)|^{-1/2} in sequences of manifolds.

Abstract

Let $M$ be a 3-manifold with a finite set $X(M)$ of conjugacy classes of representations $\rho:\pi_1(M)\to$SU$_2$. We study here the distribution of the values of the Chern-Simons function CS$:X(M)\to \mathbb{R}/2\pi\mathbb{Z}$. We observe in some examples that it resembles the distribution of quadratic residues. In particular for specific sequences of $3$-manifolds, the invariants tends to become equidistributed on the circle with white noise fluctuations of order $|X(M)|^{-1/2}$. We prove that for a manifold with toric boundary the Chern-Simons invariants of the Dehn fillings $M_{p/q}$ have the same behaviour when $p$ and $q$ go to infinity and compute fluctuations at first order.