General Argyres-Douglas Theory (1204.2270v2)

Abstract: We construct a large class of Argyres-Douglas type theories by compactifying six dimensional (2,0) A_N theory on a Riemann surface with irregular singularities. We give a complete classification for the choices of Riemann surface and the singularities. The Seiberg-Witten curve and scaling dimensions of the operator spectrum are worked out. Three dimensional mirror theory and the central charges a and c are also calculated for some subsets, etc. Our results greatly enlarge the landscape of N=2 superconformal field theory and in fact also include previous theories constructed using regular singularity on the sphere.

• The paper pioneers a classification framework for AD theories using irregular singularities, identifying distinct construction scenarios on Riemann surfaces.
• The paper derives the Seiberg-Witten curve and uncovers fractional scaling dimensions for Coulomb branch operators via integrable system methods.
• The paper computes 3D mirror theories and central charges, offering deep insights into strongly coupled dynamics without Lagrangian descriptions.

An Overview of the General Argyres-Douglas Theory

The academic paper by Dan Xie explores the construction of a vast class of Argyres-Douglas (AD) type theories within the field of $\mathcal{N}=2$ superconformal field theory (SCFT). By compactifying a six-dimensional $(2,0)$ $A_{N-1}$ theory on a Riemann surface with irregular singularities, this work significantly expands the landscape of known SCFTs.

Construction and Classification

The paper advances the understanding of AD theories by establishing irregular singularities as the key components in the compactification process. The classification framework intricately assesses different Riemann surfaces and their singularity structures, providing a foundational basis for constructing numerous AD theories. The author identifies two primary scenarios for obtaining AD theories: a single irregular singularity or a combination of one irregular and one regular singularity on a Riemann sphere.

Seiberg-Witten Curve and Operator Spectrum

The Seiberg-Witten curve, an instrumental tool in grasping the dynamics of the four-dimensional theory, is derived from the spectral curve of the associated Hitchin integrable system. The work offers an in-depth exploration of the scaling dimensions of operators within these theories' spectra. Notably, AD theories are characterized by possessing fractional scaling dimensions for their Coulomb branch operators, contrasting with the integral dimensions seen in other SCFTs.

Three-Dimensional Mirror and Central Charges

A noteworthy contribution of the paper is the computation of three-dimensional mirror theories and the central charges $a$ and $c$ for various subsets of the constructed AD theories. These calculations are facilitated by employing 3d mirror symmetry, enabling the derivation of central charges even for strongly coupled theories without Lagrangian descriptions.

Implications and Broader Impact

Theoretical implications of this research are profound, as it illuminates the profound versatility and generality of AD theories, previously perceived as less widespread. Practically, these frameworks offer new avenues for studying quantum field theories by providing fertile ground for exploring dualities, flow behaviors, and integrable systems. The classification of singularities and understanding of their geometric encoding is pivotal for string theory and mathematical physics applications, where AD points often surface.

Future Developments

The paper opens new questions and potential for future research, notably in extending these results to other six-dimensional theories such as $D_n$ and $E_n$, as well as exploring further implications in the context of geometric Langlands duality, monodromy problems, and beyond. Moreover, the discoveries prompt the reevaluation of theories previously modeled only by regular singularities, suggesting their seamless integration or potential equivalency with irregular singularity frameworks.

In summary, Xie's work not only enriches the theoretical apparatus available to physicists but also enhances the connectivity between various branches of mathematical physics, offering a robust platform for future explorations in quantum field theories.