3-Manifolds and 3d indices

Abstract

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 to 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.