Atomic Classification of 6D SCFTs (1502.05405v3)

Abstract: We use F-theory to classify possibly all six-dimensional superconformal field theories (SCFTs). This involves a two step process: We first classify all possible tensor branches allowed in F-theory (which correspond to allowed collections of contractible spheres) and then classify all possible configurations of seven-branes wrapped over them. We describe the first step in terms of "atoms" joined into "radicals" and "molecules," using an analogy from chemistry. The second step has an interpretation via quiver-type gauge theories constrained by anomaly cancellation. A very surprising outcome of our analysis is that all of these tensor branches have the structure of a linear chain of intersecting spheres with a small amount of possible decoration at the two ends. The resulting structure of these SCFTs takes the form of a generalized quiver consisting of ADE-type nodes joined by conformal matter. A collection of highly non-trivial examples involving E8 small instantons probing an ADE singularity is shown to have an F-theory realization. This yields a classification of homomorphisms from ADE subgroups of SU(2) into E8 in purely geometric terms, largely matching results obtained in the mathematics literature from an intricate group theory analysis.

• The paper introduces a framework that classifies 6D SCFTs by leveraging tensor branches as foundational geometries.
• It utilizes elliptic fibration enhancements and anomaly cancellation to validate gauge groups and matter representations.
• The classification extends the landscape of 6D SCFTs, offering new insights for F-theory and M-theory applications.

Overview of the Atomic Classification of 6D SCFTs

The paper "Atomic Classification of 6D SCFTs" presents a classification framework for six-dimensional superconformal field theories (6D SCFTs) using F-theory. The proffered classification of these theories is novel, employing a two-step approach wherein tensor branches are first identified as the basis for classification followed by the examination of elliptic fibration configurations over these bases. The authors refer to discrete group actions and heavily leverage anomaly cancellation conditions to vet the theoretical consistency of their proposed 6D SCFTs.

Framework Summary

The authors outline a systematic strategy built around key concepts:

1. Classification via Tensor Branches: The tensor branches are modeled around base geometries acceptable in F-theory, labeled within a set of "non-Higgsable clusters" (NHCs) and additional configurations of $-2$ curves representing non-trivial gauge groups. Anomaly cancellation is crucial for retaining consistency, ensuring that the effective theories remain physically viable.
2. Fibers and Enhancements: The fiber enhancements involve the introduction of additional structure in the elliptic fibration, essentially constraining the possible gauge groups and matter representations. These enhancements are evocative of the histological cross-sections of identifying tissue structures through varied staining to visualize different components.
3. Geometric Constraints and Chemistry Analogies: Utilizing non-Higgsable clusters as "atoms," the authors construct elaborate "molecular" frameworks that dictate which combinations produce consistent 6D SCFTs. The analogy drawn to chemistry, where NHCs are atoms forming structures, simplifies the comprehension of complex theoretical interlinkages and formation rules.

Numerical Results and Bold Claims

The methodology yields a broad spectrum of new 6D SCFTs where the interplay between geometric conditions and anomaly constraints are meticulously documented. It explores the possibility of infinite-length linear chains forming a sequential ordering in base nodes, reinforced by mathematical rigor. In the classification scheme, specific sequences of gauge symmetries either affirm or disallow potential back-to-back placements, significantly constraining the allowable structures.

The authors further explore the link between certain F-theory models and geometric phases, positing that purely geometric data in F-theory capture both the unbroken and broken symmetries observable in field theory analogs.

Theoretical and Practical Implications

This paper amplifies theoretical understanding in several substantive ways:

• Extension of Existing Models: The SCFT landscape is extended beyond previously canonical models, allowing practitioners to anticipate multifaceted interactions among elements within the theoretical interplay of geometry and field theory.
• Application to Lower Dimensions: The work paves the way for lower-dimension compactifications using the 6D SCFTs explored, unlocking potential dynamics that are relevant in both historical contexts (such as compactifications) and applications within string theory frameworks.
• Implications within M-theory: These insights are pivotal within the broader understanding of M-theory, underpinning the foundational dynamics probed by six-dimensional stringy landscapes.

Future Speculations

The authors propose further exploration into the microfoundational aspects of M5-branes given the six-dimensional framework's proximity to this topic. They conjecture on the possible development of a platform that inherently accounts for all SCFT forms via dimensional compactification pathways.

Conclusion

The paper provides a coherent and mathematically robust classification of 6D SCFTs through meticulously leveraging F-theory's geometry, ensuring anomaly cancellation integrity, and establishing explicit connections to known theoretical physics frameworks. The construction of new theoretical models through this lens contributes significantly to the broader understanding of superconformal theories and their applicable quantum field theories. This expansion into more robust frameworks serves as a touchstone for future theoretical inquiries and practical applications across multiple domains of high-energy theoretical research.