N=4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets (0805.3662v2)

Abstract: We extend the N=4 superconformal Chern-Simons theories of Gaiotto and Witten to those with additional twisted hyper-multiplets. The new theories are generically linear quiver gauge theories with the two types of hyper-multiplets alternating between gauge groups. Our construction includes the Bagger-Lambert model of SO(4) gauge group. A family of abelian theories are identified with those proposed earlier in the context of the M-crystal model for M2-branes probing (C^2/Z_n)^2 orbifolds. Possible extension with non-abelian BF couplings and string/M-theory realization are briefly discussed.

• The paper extends Gaiotto-Witten models by incorporating twisted hyper-multiplets, broadening the framework to include novel quiver gauge theories.
• It identifies a special case with the Bagger-Lambert model for SO(4) gauge symmetry and derives corresponding Abelian theories with BF-type couplings.
• The study applies the extended theory to M2-brane dynamics in Type IIB and M-theory, providing insights into dualities and supersymmetry-preserving mass deformations.

Analysis of =4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets

The paper under examination extends the landscape of $\mathcal{N}=4$ superconformal Chern-Simons theories by integrating twisted hyper-multiplets into the established structure devised by Gaiotto and Witten. The authors present new linear quiver gauge theories, characterized by an alternation between hyper-multiplets and twisted hyper-multiplets among the gauge groups. This generalization includes the Bagger-Lambert (BL) model for the $SO(4)$ gauge group and is relevant in the context of Type IIB string theory and M-theory, particularly for theories describing multiple M2-branes.

Theoretical Framework and Key Contributions

Several key contributions to the field are made in this work:

1. Generalization to Include Twisted Hyper-Multiplets: The extension of the original Gaiotto-Witten models incorporates twisted hyper-multiplets, offering a broader framework for constructing $\mathcal{N}=4$ Chern-Simons theories. This includes a robust classification where these theories are characterized by quiver diagrams, which can be linear or circular with multiple nodes.
2. Identification with the Bagger-Lambert Model: The paper establishes the conditions under which the Bagger-Lambert theory, with $SO(4)$ gauge group and maximal $\mathcal{N}=8$ supersymmetry, may be viewed as a special case within the broader Gaiotto-Witten framework extended by twisting.
3. Application to M-Crystal Model: By embedding these theories into the M-crystal model, the authors present a theoretical dual to M2-branes probing $(\mathbb{C}^2/\mathbb{Z}_n)^2$ orbifolds. This contributes to understanding M2-brane dynamics in toric Calabi-Yau four-fold backgrounds.
4. Exploration of Abelian Theories: Beyond non-abelian contexts, the authors derive Abelian counterparts with specific BF-type couplings, enriching the catalog of superconformal Chern-Simons theories while preserving $\mathcal{N}=4$ supersymmetry.
5. Mass Deformation Investigation: The research includes a discussion of mass deformation within these extended theories, preserving supersymmetry and elucidating connections between massive deformations of BL theory and the twisted Gaiotto-Witten models.

Implications and Future Directions

The theoretical implications of this paper extent the framework for constructing and understanding 3D $\mathcal{N}=4$ superconformal field theories. Practically, these theories can provide a deeper understanding of the low-energy behavior of M2-branes, crucial for M-theory and Type IIB dualities. Moreover, the quiver representations introduced open up pathways for exploring higher-dimensional dualities.

The work's potential impacts include:

• Expansion of string/M-theory dualities by providing new models of Chern-Simons theories with extended supersymmetries.
• Influence on the formalisms used to describe BPS objects in brane contexts.
• Contributions to the understanding of defect conformal field theories via the interaction between M2-branes and 5-branes.
• Exploration of the application of BF-type couplings, addressing unique topological features in lower-dimensional theories.

Conclusion

This paper deepens the exploration of superconformal field theories by cross-pollinating concepts between twisted and ordinary hyper-multiplets within the established Chern-Simons framework. As the field advances, these results encourage further research into dual theories and their applications in quantum gravity and string phenomenology. The symmetry enhancements and classifications developed in this paper serve both to challenge existing paradigms and to provide a rich tapestry for future theoretical exploration.