Smallest 3d hyperbolic manifolds via simple 3d theories

TL;DR

The paper proposes that the volumes of the three smallest hyperbolic 3-manifolds are encoded by simple 3d N=2 Chern-Simons-matter theories: a charged chiral multiplet coupled to a U(1) Chern-Simons field with levels -5/2, -7/2, and -3/2, via the 3d/3-manifold correspondence. It supports the claim with numerical evidence from the b to 0 limit of the squashed sphere partition function and associated saddle points, matching vol(M) and even geodesic lengths through supersymmetric loop operators. It provides an explicit derivation for the Weeks manifold using the Dimofte-Gaiotto-Gukov construction and known 3d dualities, reducing T(Weeks) to a single chiral with CS level -5/2. Together, these results strengthen the bridge between hyperbolic geometry and simple 3d QFTs, offering a concrete framework to study geometric invariants via field theory dualities and Dehn fillings.

Abstract

We provide strong pieces of evidence that the mathematics of the three-dimensional hyperbolic manifolds of the first, second and third smallest volume is captured by the physics of the three-dimensional theories composed of a complex boson and a Dirac fermion, both of unit charge, coupled to a U(1) gauge field with the Chern-Simons level $-5/2$, $-7/2$ and $-3/2$, respectively.

Figures (1)

• Figure 1: The whitehead link, from which three smallest hyperbolic manifolds can be obtained by the Dehn filling.