Dynamics of N=2 Supersymmetric Chern-Simons Theories

TL;DR

This work analyzes how three-dimensional $N=2$ gauge theories with chiral matter acquire Chern-Simons couplings when flowing from $N=4$ mirror pairs by gauging a combination of R-symmetries. It develops abelian and non-abelian constructions, detailing how axial masses induce CS terms and modify the moduli space, including a toric description of the abelian Coulomb branch and instanton-generated superpotentials in non-abelian cases. The results clarify the IR dualities and vacuum structures of CS-matter theories, revealing compact Coulomb branches, symmetry enhancements at boundaries, and potential supersymmetry breaking depending on mass and FI parameters. Overall, the paper maps out how CS interactions shape the low-energy dynamics and dualities of 3D supersymmetric gauge theories with reduced supersymmetry.

Abstract

We discuss several aspects of three dimensional N=2 supersymmetric gauge theories coupled to chiral multiplets. The generation of Chern-Simons couplings at low-energies results in novel behaviour including compact Coulomb branches, non-abelian gauge symmetry enhancement and interesting patterns of dynamically generated potentials. We further show how, given any pair of mirror theories with N=4 supersymmetry, one may flow to a pair of mirror theories with N=2 supersymmetry by gauging a suitable combination of the R-symmetries. The resulting theories again have interesting properties due to Chern-Simons couplings.

Figures (5)

• Figure 1: The Coulomb branch of Theory B. The scalar $\phi$ (plotted horizontally) is restricted to lie within the interval $-m<\phi<m$. The dual photon provides a further ${\bf S}^1$ which is fibered over this interval such that its radius vanishes at the end points, resulting in a Coulomb branch with topology of the sphere.
• Figure 2: The toric realisation of ${\bf CP}^2$. A torus ${\bf T}^2$ is fibered over the triangle such that a single cycle degenerates at each edge. Each of these edges is itself a copy of ${\bf CP}^1$ of the form shown in Figure 1.
• Figure 3: The Coulomb branch of Theory B. Outside the triangle, CS terms develop, lifting the Coulomb branch. The two dual photons provide a torus ${\bf T}^2$ which is fibered over the triangle to realise ${\bf CP}^2$ as shown in Figure 2.
• Figure 4: The Coulomb branch of the $SU(2)$ theory in the limit $M\rightarrow\infty$. Non-abelian gauge symmetry is restored at the origin, denoted by the vertical jagged line. Instantons, represented by the arrows, induce a potential driving the vacuum to the point marked by the solid dot. This theory has a unique supersymmetric vacuum.
• Figure 5: The Coulomb branch of the $SU(3)$ theory with $M\rightarrow\infty$. The horizontal and vertical lines are $v_1-v_2$ and $v_2-v_3$ respectively. The jagged lines lie at the edge of the Weyl chambers and denote non-abelian symmetry enhancement. Instantons carry the unique supersymmetric vacuum to the spot marked with a dot.