Advances in Theoretical and Mathematical Physics

Volume 17 (2013)

Number 5

3-Manifolds and 3d indices

Pages: 975 – 1076



Tudor Dimofte (Institute for Advanced Study, Princeton, New Jersey, U.S.A.; Trinity College, University of Cambridge, United Kingdom)

Davide Gaiotto (Institute for Advanced Study, Princeton, New Jersey, U.S.A.)

Sergei Gukov (California Institute of Technology, Pasadena, Calif., U.S.A.; Max-Planck-Institut für Mathematik, Bonn, Germany)


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.”

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