Distribution of Chern-Simons invariants

2017/10/25 by Marché, Julien
#57M27 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1710.09258

Abstract

Let M be a 3-manifold with a finite set X(M) of conjugacy classes of representations ρ:π₁(M)→SU₂. We study here the distribution of the values of the Chern-Simons function CS:X(M)→ ℝ/2πℤ. 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.

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