Advances in Theoretical and Mathematical Physics

3-Manifolds and 3d indices

Tudor Dimofte, Davide Gaiotto, and Sergei Gukov

Full-text: Open access


We identify a large class $\mathcal{R}$ of three-dimensional $\mathcal{N} = 2$ superconformal field theories. This class includes the effective theories $T_M$ of M5-branes wrapped on 3-manifolds $\mathcal{M}$, 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 $\mathcal{R}$ might be viewed as an extreme quantum generalization of the Bloch group; in particular, the equivalence relation among theories in class $\mathcal{R}$ is a quantum-field-theoretic "2 to 3 move." We proceed to study the supersymmetric index of theories in class $\mathcal{R}$, 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 $SL(2,\mathbb{C})$ Chern-Simons theory on $\mathcal{M}$ with a previously unexplored "integration cycle."

Article information

Adv. Theor. Math. Phys., Volume 17, Number 5 (2013), 975-1076.

First available in Project Euclid: 21 August 2014

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Dimofte, Tudor; Gaiotto, Davide; Gukov, Sergei. 3-Manifolds and 3d indices. Adv. Theor. Math. Phys. 17 (2013), no. 5, 975--1076.

Export citation