Vortices and 3 dimensional dualities

TL;DR

The paper develops a vortex partition function (index) for 3d $\mathcal{N}=4$ and $\mathcal{N}=3$ gauge theories on $\mathbb{R}^2\times S^1$ and uses it to test Seiberg-like dualities under FI deformations. Through localization, it derives 1d vortex quantum mechanics with $U(k)$ gauge symmetry, whose saddle points are labeled by $N$-colored 1d Young diagrams, and computes indices that capture BPS vortex spectra. The authors confirm known $\mathcal{N}=4$ dualities (and propose broader IR fixed-point scenarios) by showing the vortex indices of dual pairs match after accounting for decoupled sectors, notably at $N_f=2N-1$ where a free twisted hypermultiplet emerges. For $\mathcal{N}=3$ CS-matter theories, they demonstrate that both topological and non-topological vortices must be considered to maintain duality invariance, and they discuss angular-momentum bounds via tensionless domain-wall pictures. Overall, the work links vortex dynamics to Seiberg duality in 3d and suggests rich IR structures and symmetry enhancements, aligning with and extending results from 3d sphere partition functions and monopole indices.

Abstract

We study a supersymmetric partition function of topological vortices in 3d N=4,3 gauge theories on R^2 x S^1, and use it to explore Seiberg-like dualities with Fayet-Iliopoulos deformations. We provide a detailed support of these dualities and also clarify the roles of vortices. The N=4 partition function confirms the proposed Seiberg duality and suggests nontrivial extensions, presumably at novel IR fixed points with enhanced symmetries. The N=3 theories with nonzero Chern-Simons term also have non-topological vortices in the partially broken phases, which are essential for the Seiberg duality invariance of the spectrum. We use our partition function to confirm some properties of non-topological vortices via Seiberg duality in a simple case.

Figures (2)

• Figure 1: The brane construction of $\mathcal{N}\!=\!4$ Seiberg-dual pairs with nonzero FI parameter $\zeta$. The red lines denote D1-brane vortices.
• Figure 2: The brane construction of $\mathcal{N}\!=\!3$ Seiberg-dual pairs and vortices with unbroken $U(n)_\kappa$ or $U(k-n)_{-\kappa}$ gauge symmetry.