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S^3/Z_n partition function and dualities

Yosuke Imamura, Daisuke Yokoyama

TL;DR

This work addresses the problem of computing ${\bm S}^3/{\mathbb{Z}}_n$ partition functions for ${\cal N}=2$ gauge theories, where the nontrivial fundamental group induces degenerate vacua labeled by holonomies and requires careful phase choices in the holonomy sum. The authors develop the orbifolded double sine framework via $s_{b,h}(z)$, determine relative phases by comparing gauge theories to dual non-gauge theories, and show that for odd $n$ these phases can be absorbed into a redefinition $\widehat{s}_{b,h}(z)$. They extend these insights to multiple dual pairs, including ${\cal N}=4$ SQED, quiver theories, and the ABJM model, deriving derived dualities on the orbifold and highlighting the universality of the phase-absorption mechanism. The results provide concrete prescriptions for preserving dualities on ${\bm S}^3/{\mathbb{Z}}_n$ and have implications for precise tests of 3d dualities and AdS/CFT setups at both leading and subleading orders. The work suggests a coherent, potentially universal rule for phase choices in orbifold partition functions and motivates further principled derivations beyond duality-based fixes.

Abstract

We investigate S^3/Z_n partition function of N = 2 supersymmetric gauge theories. A gauge theory on the orbifold has degenerate vacua specified by the holonomy. The partition function is obtained by summing up the contributions of saddle points with different holonomies. An appropriate choice of the phase of each contribution is essential to obtain the partition function. We determine the relative phases in the holonomy sum in a few examples by using duality to non-gauge theories. In the case of odd n the phase factors can be absorbed by modifying a single function appearing in the partition function.

S^3/Z_n partition function and dualities

TL;DR

This work addresses the problem of computing partition functions for gauge theories, where the nontrivial fundamental group induces degenerate vacua labeled by holonomies and requires careful phase choices in the holonomy sum. The authors develop the orbifolded double sine framework via , determine relative phases by comparing gauge theories to dual non-gauge theories, and show that for odd these phases can be absorbed into a redefinition . They extend these insights to multiple dual pairs, including SQED, quiver theories, and the ABJM model, deriving derived dualities on the orbifold and highlighting the universality of the phase-absorption mechanism. The results provide concrete prescriptions for preserving dualities on and have implications for precise tests of 3d dualities and AdS/CFT setups at both leading and subleading orders. The work suggests a coherent, potentially universal rule for phase choices in orbifold partition functions and motivates further principled derivations beyond duality-based fixes.

Abstract

We investigate S^3/Z_n partition function of N = 2 supersymmetric gauge theories. A gauge theory on the orbifold has degenerate vacua specified by the holonomy. The partition function is obtained by summing up the contributions of saddle points with different holonomies. An appropriate choice of the phase of each contribution is essential to obtain the partition function. We determine the relative phases in the holonomy sum in a few examples by using duality to non-gauge theories. In the case of odd n the phase factors can be absorbed by modifying a single function appearing in the partition function.

Paper Structure

This paper contains 13 sections, 99 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The ${\bm S}^3/{\mathbb{Z}}_n$ partition function
  3. The ${\bm S}^3$ partition function
  4. ${\mathbb{Z}}_n$ orbifolding
  5. Dualities in ${\bm S}^3/{\mathbb{Z}}_n$
  6. ${\cal N}=2$ SQED and XYZ model
  7. $SU(2)$ gauge theory and a chiral multiplet
  8. ${\bm S}^3/\mathbb{Z}_{2k+1}$
  9. Derived dualities
  10. ${\cal N}=4$ SQED and hypermultiplet
  11. ${\cal N}=2$ SQED with $N_f\geq2$ and quiver gauge theory
  12. ABJM model and its mirror
  13. Conclusions and discussions

Figures (2)

  • Figure 1: Quiver diagram of the mirror of $\mathcal{N}=2$ SQED with $N_f$ flavors. Circles and squares represent the $U(1)$ gauge groups and the $U(1)$ flavor symmetry, respectively. $q_i$ and $\widetilde{q}_i$ represented by arrows are bifundamental fields, and $\widetilde{S}_i$ are neutral fields.
  • Figure 2: Quiver diagram of the mirror ABJM