Vacua, Symmetries, and Higgsing of Chern-Simons Matter Theories

TL;DR

The paper develops a comprehensive framework to analyze vacua and RG flows of 3d Chern–Simons matter theories with $\mathcal{N}=3$ or enhanced $\mathcal{N}=4$ supersymmetry using Type IIB brane constructions and magnetic quivers. For $|\kappa|=1$ it employs explicit $SL(2,\mathbb{Z})$ dualities to map CS theories to Lagrangian $\mathcal{N}=4$ quivers, enabling complete Higgs and Coulomb branch descriptions via quiver methods and index/Hilbert-space checks. For $|\kappa|>1$ the authors introduce two magnetic quivers, $\mathsf{MQ}_{\text{A}}$ and $\mathsf{MQ}_{\text{B}}$, whose Coulomb branches encode the maximal A- and B- branches, with RG flows captured by the decay and fission algorithm and index/Hilbert-series cross-checks. They extend the framework to $\mathcal{N}=3$ theories by deriving a magnetic quiver for each maximal branch from brane systems with $(p,q)$ 5-branes, including non-Lagrangian $(p,q)$-brane theories, thereby enabling systematic predictions for maximal-branch geometry and Higgsing patterns. The work provides a versatile, unified method to study moduli spaces of 3d CSM theories, connecting brane intuition, quiver technology, and exact invariants to reveal rich branch structures and IR dynamics across a broad class of theories.

Abstract

Three-dimensional supersymmetric Chern-Simons Matter (CSM) theories typically preserve $ \mathcal{N}=3$ supersymmetry but can exhibit enhanced $\mathcal{N}=4$ supersymmetry under special conditions. A detailed understanding of the moduli space of CSM theories, however, has remained elusive. This paper addresses this gap by systematically analysing the maximal branches of the moduli space of $\mathcal{N}=3$ and $\mathcal{N}=4$ CSM realised via Type IIB brane constructions. Firstly, for $\mathcal{N}=4$ theories with Chern-Simons levels equal $1$, the $\mathrm{SL}(2,\mathbb{Z})$ dualisation algorithm is employed to construct dual Lagrangian 3d $\mathcal{N}=4$ theories without CS terms. This allows the full moduli space to be determined using quiver algorithms that compute Higgs and Coulomb branch Hasse diagrams and associated RG flows. Secondly, for $\mathcal{N}=4$ theories with CS-levels greater $1$, where $\mathrm{SL}(2,\mathbb{Z})$ dualisation does not yield CS-free Lagrangians, a new prescription is introduced to derive two magnetic quivers, $\mathsf{MQ}_A $ and $\mathsf{MQ}_B$, whose Coulomb branches capture the maximal A and B branches of the original $\mathcal{N}=4$ CSM theory. Applying the decay and fission algorithm to $ \mathsf{MQ}_{A/B}$ then enables the systematic analysis of A/B branch RG flows and their geometric structures. Thirdly, for $\mathcal{N}=3$ CSM theories, one magnetic quiver for each maximal (hyper-Kähler) branch is derived from the brane system. This provides an efficient and comprehensive characterisation of these previously scarcely studied features.

Figures (46)

• Figure 1: Chern--Simons matter theories $\mathrm{CSM}_{\kappa}$ and their $\mathcal{T}^T$-duals or magnetic quivers. \ref{['fig:aim_CS=1']}: for CS-levels $|\kappa_i|=1$, there is a $\mathrm{SL}(2,\mathbb{Z})$ duality web at play. The $\mathcal{T}^T$ map yields standard $3d$$\mathcal{N}=4$ quiver theory $\mathsf{Q}$, on which another $\mathcal{S}$ yields the mirror dual $\mathsf{Q}^\vee$. The original $\mathrm{CSM}_{|\kappa|=1}$ theory is mapped via $\mathcal{T}$ into another $\mathrm{CSM}^\prime_{|\kappa|=1}$ theory whose A and B branch are swapped with respect to those of $\mathrm{CSM}_{|\kappa|=1}$. Applying another $(\mathcal{T}^T)^{-1}$ transformation completes a closed $\mathrm{SL}(2,\mathbb{Z})$ web into $\mathsf{Q}^\vee$. \ref{['fig:aim_CS>1']}: for CS-levels $|\kappa_i|>1$, there is no $\mathrm{SL}(2,\mathbb{Z})$ transformation into Lagrangian non-CS theories. Instead, one observes that a $(\mathcal{T})^{|\kappa|}$ transformation maps $\mathrm{CSM}_{|\kappa|>1}$ into $\mathrm{CSM}^\prime_{|\kappa|>1}$; again, both theories are dual and have swapped A/B branches. The proposal is to map the A/B branches of both $\mathrm{CSM}_{|\kappa|>1}$ and $\mathrm{CSM}^\prime_{|\kappa|>1}$ into a magnetic quiver, called $\mathsf{MQ}_{\text{A}}$ and $\mathsf{MQ}_{\text{B}}$, respectively. The two magnetic quivers are sufficient to analyse the moduli space and the RG-flows of $\mathrm{CSM}_{|\kappa|>1}$.
• Figure 2: The $\mathrm{SL}(2,\mathbb{Z})$ duality web employed to study $\mathrm{CSM}_{|\kappa|=1}$ theories. All the computations carried out in this section involve exclusively the leftmost part of the diagram, namely the starting theory $\mathrm{CSM}_{|\kappa|=1}$ and its $\mathcal{T}^T$-dual $\mathsf{Q}$, as the rightmost sector of the web is equivalent. To match the index of these theories one has to properly transform of the $\mathrm{U}(1)_{\mathrm{axial}}$ fugacity $t$ (see eq. \ref{['eq:match_ind']}).
• Figure 3: \ref{['fig:abelian6nodes_lv0_branes_and_charges']}: The electric theory, namely an Abelian $\mathrm{CSM}_{|\kappa|=1}$ quiver with 6 gauge nodes. For the quiver diagram the 3d $\mathcal{N}=2$ language is employed. In particular, hypermultiplets (red) and twisted hypermultiplets (blue) are represented using pairs of arrows (namely chiral multiplets). The CS-levels are denoted in blue above the corresponding gauge nodes, while the topological fugacities are denoted in green. Below the quiver, the corresponding brane system is shown. Brane configurations are drawn by using coloured lines of different angles, see Appendix \ref{['app:branes']}. \ref{['fig:abelian6nodes_lv0_branes_and_charges_GK']}: The result of the GK duality on the starting electric theory above. The arcs on the gauge nodes represent the $\mathcal{N}=2$ massless adjoint chiral multiplets (they are present only on gauge nodes without CS-level since the CS term induces a mass for the adjoint chiral multiplet, see Appendix \ref{['app:3d_CSM']}). Below the quiver, the corresponding colour coded brane system is shown. \ref{['fig:abelian6nodes_lv0_branes_and_charges_dual']}: The $\mathcal{T}^T$-dual theory, where the flavour Cartans have been explicitly written in black to show the parameters map across the duality. Below the quiver, the corresponding colour coded brane system is shown.
• Figure 4: \ref{['fig:nonabelian8nodes_branes_and_charges']}: The electric theory, together with the associate brane system. \ref{['fig:nonabelian8nodes_branes_and_charges_GK']}: The result of the GK duality on the starting electric theory above, together with the associate brane system. \ref{['fig:nonabelian8nodes_branes_and_charges_dual']}: The $\mathcal{T}^T$-dual theory, together with the associate brane system.
• Figure 5: The Higgs branch transitions for the $\mathcal{T}^T$-dual theory (Figure \ref{['fig:nonabelian8nodes_branes_and_charges_dual']}). In each step the field names have noting to do with the names of the fields of the prior/subsequent step.