Vacua, Symmetries, and Higgsing of Chern-Simons Matter 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-K\"ahler) branch is derived from the brane system. This provides an efficient and comprehensive characterisation of these previously scarcely studied features.

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

The paper focuses on three-dimensional supersymmetric Chern-Simons Matter (CSM) theories, which are significant in theoretical physics due to their rich structure and applications ranging from field theory to string and M-theory. While typical CSM theories exhibit $\mathcal{N}=3$ supersymmetry, they can showcase enhanced $\mathcal{N}=4$ supersymmetry under specific conditions, making the study of their moduli space particularly intriguing yet complex.

Main Contributions

The paper addresses the elusive nature of the moduli space for both $\mathcal{N}=3$ and $\mathcal{N}=4$ CSM theories via Type IIB brane constructions, presenting a comprehensive analysis of the maximal branches of these spaces. The authors systematically approach the problem through the following methodologies:

1. Semi-Independent Lagrangian Construction:

   • For $\mathcal{N}=4$ theories with CS levels equal to one, $\mathrm{SL}(2,\mathbb{Z})$ dualisation is employed to transition these theories into three-dimensional $\mathcal{N}=4$ theories free of CS terms. This transformation enables a complete determination of the moduli space, employing quiver algorithms for computing Higgs and Coulomb branch Hasse diagrams along with associated Renormalization Group (RG) flows.

2. Innovative Quiver Constructs:

   • For scenarios where CS levels are greater than one, the study introduces two magnetic quivers, $\mathsf{MQ}_A$ and $\mathsf{MQ}_B$. These quivers play a crucial role in encapsulating the maximal A and B branches of the original theory, and through the decay and fission algorithm, they allow a structural analysis of the RG flows and geometric transitions in the moduli space.

3. Efficient Characterization:

   • For $\mathcal{N}=3$ theories, engaging with magnetic quivers for hyper-Kähler branches derived from the brane system proves to be efficient. This approach significantly characterizes and comprehends those features which remain under-considered in the literature.

Implications and Observations

The research has profound implications both theoretically and practically. The systematic examination of CS theories' behaviors under various configurations provides insights into the fine structural dynamics of these fascinating systems.

• Numerical and Analytical Consistency: The analytical approach of employing quivers and magnetic quivers verifies meaningful relationships through numerical validation, such as index computations and Hilbert series evaluations, ensuring robust theoretical substantiation.
• Theory Verification: Complete symmetry considerations and RG flow mappings contribute immensely to verifying the theoretical predictions concerning the hyper-Kähler structuring in $\mathcal{N}=3$ theories, alongside confirming Higgs-Coulomb branch duality relationships in enhanced $\mathcal{N}=4$ variants.
• Future Developments: The methodologies developed herein pave the way for future research into more complex or higher-dimensional CSM theories, potentially expanding into realms of $\mathcal{N}=5$ and $\mathcal{N}=6$ theories and their broader duality properties.

Conclusion

The paper brings a structured, methodical approach to unresolved complexities in understanding vacua and symmetries in CSM theories. Through innovative uses of brane constructions and quiver applications, it untangles the mesh of relations between these theories’ moduli spaces and their symmetries. This study stands as a significant stride towards fully characterizing the nuanced landscape of Chern-Simons Matter Theories, establishing a solid groundwork for subsequent explorations into higher-order supersymmetries and their manifestations.