We identify a large class R of three-dimensional N=2 superconformal field
theories. This class includes the effective theories T_M of M5-branes wrapped
on 3-manifolds M, discussed in previous work by the authors, and more generally
comprises theories that admit a UV description as abelian Chern-Simons-matter
theories with (possibly non-perturbative) superpotential. Mathematically, class
R might be viewed as an extreme quantum generalization of the Bloch group; in
particular, the equivalence relation among theories in class R is a
quantum-field-theoretic "2-3 move." We proceed to study the supersymmetric
index of theories in class R, uncovering its physical and mathematical
properties, including relations to algebras of line operators and to 4d
indices. For 3-manifold theories T_M, the index is a new topological invariant,
which turns out to be equivalent to non-holomorphic SL(2,C) Chern-Simons theory
on M with a previously unexplored "integration cycle."