On the Classification of 6D SCFTs and Generalized ADE Orbifolds (1312.5746v3)

Abstract: We study (1,0) and (2,0) 6D superconformal field theories (SCFTs) that can be constructed in F-theory. Quite surprisingly, all of them involve an orbifold singularity C^2 / G with G a discrete subgroup of U(2). When G is a subgroup of SU(2), all discrete subgroups are allowed, and this leads to the familiar ADE classification of (2,0) SCFTs. For more general U(2) subgroups, the allowed possibilities for G are not arbitrary and are given by certain generalizations of the A- and D-series. These theories should be viewed as the minimal 6D SCFTs. We obtain all other SCFTs by bringing in a number of E-string theories and/or decorating curves in the base by non-minimal gauge algebras. In this way we obtain a vast number of new 6D SCFTs, and we conjecture that our construction provides a full list.

• The paper demonstrates that 6D SCFTs from F-theory are classified via orbifold singularities, establishing minimal models as foundational building blocks.
• The paper extends the classic ADE framework by generalizing A- and D-series through the analysis of discrete U(2) subgroups.
• The paper introduces a robust F-theory compactification methodology that resolves complex singularities and systematically identifies new SCFT configurations.

Classification of 6D Superconformal Field Theories in F-Theory Framework

The paper "On the Classification of 6D SCFTs and Generalized ADE Orbifolds" presents a comprehensive examination into the field of six-dimensional superconformal field theories (6D SCFTs) within the framework of F-theory. The work specifically addresses both $(1,0)$ and $(2,0)$ theories that can be realized through this mathematical structure. The central finding is that these SCFTs can be associated with orbifold singularities of the form $\mathbb{C}^2 / \Gamma$, where $\Gamma$ is a discrete subgroup of $U(2)$.

The classification revealed by this paper comprises an assortment of minimal 6D SCFTs, particularly those governed by discrete subgroups of $SU(2)$, leading to the traditional ADE classification. Beyond these, for subgroups under $U(2)$, the classification extends to what the paper describes as generalizations of the A- and D-series.

Key Contributions

1. Orbifold Singularities and Minimal Models: The authors successfully demonstrate that all $(1,0)$ and $(2,0)$ SCFTs emergent from F-theory can indeed be associated with singularities represented by orbifolds. This realization posits these orbifolds as foundational elements for 6D SCFT landscapes within F-theory, particularly stressing the minimal configurations as building blocks.
2. Generalized A- and D-type Series: While ADE classification for $(2,0)$ theories is cemented in the role of $SU(2)$ subgroups, the paper extends this to consider more complex interactions within $U(2)$ subgroups, producing a generalized series that broadens the classification spectrum for 6D SCFTs.
3. Novel Approach to Configuration: Utilizing F-theory's approach to compactification theory, the paper presents a methodology for resolving singularities and implementing systematically permissible configurations. This provides a robust pathway to identify not only existing but potentially new SCFTs, underpinning a conjecture for a full listing of these theories.

Detailed Analysis of Results

• Classification Scope: The method achieves an extensive sweep across a multitude of complex configurations by systematically exploring 'endpoints' in $U(2)$ classifications. This involves the examination of generalized A-type and D-type theories, detailed with meticulous derivation of their interactions and singularity resolutions.
• Practical Implications: Practically, this framework allows for the extrapolation of SCFT characteristics, which are pivotal in understanding the dynamics of higher-dimensional quantum field theories in string theory contexts. This has significant implications in theoretical physics, primarily through extended understanding of quantum gravity and gauge theory interactions.
• Resolution Methodology: By calculating how the orbifold group actions resolve into specific continued fraction expansions (and correspondingly how these relate to F-theory constructs like toric varieties and Kodaira fibers), the work lays a mathematically rigorous foundation that intertwines algebraic geometry and high-energy theoretical physics.

Future Prospects and Theoretical Implications

The potential of this classification and its systemic aggregation of SCFTs into catalogued minimal models opens potential research pathways such as:

• Exploration of Non-Minimal Models: Having set a foundation with minimal models, future endeavors can focus on building more complex structures by incorporating additional models such as E-string theories or non-minimal gauge decorations.
• Interdimensional Flows: Given these classifications, focus can shift to understanding dimensional reductions or compactifications of SCFTs, potentially providing insights into exotic theories in four dimensions and beyond.
• Consistency Checks and Duality: The identified configurations can be tested against predictions from disparate string theory models and dualities, thereby fortifying the theoretical architecture of contemporary physics.

In closing, this paper encapsulates a seminal exploration within the boundless domain of theoretical physics, offering a structured approach in understanding 6D SCFTs. Its mathematical rigueur is palatably aligned with the complexity of the field, and it presents a solid groundwork for further explorations into the higher-order structure of quantum field theories. The work stands as a significant contribution in the ongoing dialogue between geometry and field theory, setting the stage for future discoveries in the physics of the universe.