3-Manifolds and 3d Indices

TL;DR

The paper identifies a broad class of 3d ${\cal N}=2$ SCFTs (class ${\cal R}$) arising from abelian CS-matter UV descriptions and from M5-branes on 3-manifolds, with an equivalence under 2–3 moves. It develops a comprehensive framework for the 3d supersymmetric index, including its transformation rules under ${\rm Sp}(2N,\mathbb{Z})$ actions, affine shifts, superpotential deformations, and a discrete symmetry $\rho$, and shows how the index factors through line operators and 4d boundary data. The index of a 3-manifold theory $T_M$ is constructed from tetrahedron building blocks via explicit gluing rules, yielding a topological invariant that matches a non-holomorphic ${\rm SL}(2,\mathbb{C})$ Chern-Simons partition function on $M$ with a novel integration cycle. The work also connects 3d-4d dualities, mutation invariance, and amoeba/tentacle structures of Lagrangians, and embeds the construction in a six-dimensional and brane-based perspective, suggesting higher-rank generalizations and deep links to hyperbolic geometry and quantum topology.

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."

Figures (14)

• Figure 1: Leading powers of $q$ in the index. The graph interpolates smoothly between integer points.
• Figure 2: An ideal tetrahedron, with edge parameters.
• Figure 3: The bipyramid, in its two triangulations. The equatorial polarization $\Pi = (X_1,X_2;P_1,P_2)$ has positions $X_1$ and $X_2$ corresponding to the two equatorial edges in the front.
• Figure 4: Meridian and longitude cycles on the boundary of a knot complement.
• Figure 5: The standard triangulation of the figure-eight knot complement