Factorization of S^3/Z_n partition function

TL;DR

The authors address the sign ambiguity in the ${\cal N}=2$ 3d partition function on ${\bm S}^3/\mathbb{Z}_n$ by requiring factorization into holomorphic blocks, which fixes relative signs across holonomy sectors. They derive a projection framework using a projection operator and identify a holonomy-dependent phase $\rho(h)$ that determines the sign, then show that using an improved double-sine function ${s_{b,h}^{\rm imp}}$ yields correct holomorphic-block factorization for general (non-gauge) theories and provides a consistent structure for gauge theories, as demonstrated in SQED (duality with XYZ) and a Chern-Simons ${\rm su}(2)$ theory with adjoint matter. The results connect to the lens space index and 3d index, where the same sign structure appears via an explicit factor ${\prod_I\rho(h_I)}$, and the improved 3d index aligns with Dimofte’s analytic form. While nontrivial sector-summation rules emerge in the gauged case, the sign-factor construction succeeds in capturing essential consistency across dualities and dimensional reductions, with open questions remaining about the global choices of holonomy sectors in certain gauge theories.

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.