3-Manifolds and 3d Indices

TL;DR

This paper introduces a broad class of 3d N=2 superconformal theories linked to 3-manifolds, explores their supersymmetric indices, and reveals their connection to quantum invariants and Chern-Simons theory.

Contribution

It defines a new class of theories related to 3-manifolds, studies their indices, and establishes their mathematical and physical properties, including a novel topological invariant.

Findings

The index acts as a topological invariant for 3-manifold theories.

The index relates to non-holomorphic SL(2,C) Chern-Simons theory.

Theories in class R are connected via a quantum-field-theoretic '2-3 move'.

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 "2-3 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."