A 3d-3d appetizer

Abstract

We test the 3d-3d correspondence for theories that are labelled by Lens spaces. We find a full agreement between the index of the 3d $\mathcal{N}=2$ "Lens space theory" $T[L(p,1)]$ and the partition function of complex Chern-Simons theory on $L(p,1)$. In particular, for $p=1$, we show how the familiar $S^3$ partition function of Chern-Simons theory arises from the index of a free theory. For large $p$, we find that the index of $T[L(p,1)]$ becomes a constant independent of $p$. In addition, we study $T[L(p,1)]$ on the squashed three-sphere $S_b^3$. This enables us to see clearly, at the level of partition function, to what extent $G_{\mathbb{C}}$ complex Chern-Simons theory can be thought of as two copies of Chern-Simons theory with compact gauge group $G$.

27 pages. v2: misprints corrected, references added. v3: misprints corrected, a clarification added

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• Indices for Superconformal Field Theories in 3,5 and 6 Dimensions
• Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant
• A Duality Appetizer
• Complex Quantum Chern-Simons
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Cited by in corpus (16)

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• 3d Modularity
• Higher-Form Symmetries, Bethe Vacua, and the 3d-3d Correspondence
• Symmetry enhancement and closing of knots in 3d/3d correspondence
• A slow review of the AGT correspondence
• Precision Microstate Counting for the Entropy of Wrapped M5-branes
• 3d-3d correspondence for mapping tori
• Perturbative and nonperturbative aspects of complex Chern-Simons Theory
• Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality
• 3d $\mathcal{N}=2$ minimal SCFTs from Wrapped M5-branes
• On $\mathcal{N}=4$ supersymmetry enhancements in three dimensions
• Infrared phases of 3D Class R theories
• Chern-Simons Theory with Complex Gauge Group on Seifert Fibred 3-Manifolds
• Non-hyperbolic 3-manifolds and 3D field theories for 2D Virasoro minimal models
• 3D-3D Correspondence from Seifert Fibering Operators
• Wrapped $M5$-branes and complex saddle points