Factorization of S^3/Z_n partition function

Abstract

We investigate S^3/Z_n partition function of 3d N = 2 supersymmetric field theories. In a gauge theory the partition function is the sum of the contributions of sectors specified by holonomies, and we should carefully choose the relative signs among the contributions. We argue that the factorization to holomorphic blocks is a useful criterion to determine the signs and propose a formula for them. We show that the orbifold partition function of a general non-gauge theory is correctly factorized provided that we take appropriate relative signs. We also present a few examples of gauge theories. We point out that the sign factor for the orbifold partition function is closely related to a similar sign factor in the lens space index and the 3d index.

25 pages; v3: explanations improved, typos corrected, accepted for publication in Physical Review D

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• Localization of 4d $\mathcal{N}=1$ theories on ${\mathbb{D}}^2\times {\mathbb{T}}^2$
• $q$-Virasoro modular double and 3d partition functions
• Perturbative and nonperturbative aspects of complex Chern-Simons Theory
• Four-dimensional Spinfoam Quantum Gravity with Cosmological Constant: Finiteness and Semiclassical Limit
• Factorization of 4d N=1 superconformal index
• Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence
• Supersymmetric index on T^2 x S^2 and elliptic genus
• The joy of factorization at large $N$: five-dimensional indices and AdS black holes
• Abelian 3d mirror symmetry on ${\mathbb{RP}}^2\times {\mathbb{S}}^1$ with $N_f=1$
• U(1) spin Chern-Simons theory and Arf invariants in two dimensions
• Exact WKB Analysis of ${\mathbb{CP}}^1$ Holomorphic Blocks
• Lens partition function, pentagon identity and star-triangle relation
• General modular quantum dilogarithm and beta integrals
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$
• Five-dimensional fermionic Chern-Simons theory
• Stokes Phenomena in 3d $\mathcal{N}=2$ SQED${}_2$ and ${\mathbb{CP}}^1$ Models
• Topological phase, spin Chern-Simons theory and level rank duality on lens space
• Partition functions for supersymmetric gauge theories on spheres