Smallest 3d hyperbolic manifolds via simple 3d theories

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.