Nilpotent orbits and codimension-two defects of 6d N=(2,0) theories (1203.2930v1)

Abstract: We study the local properties of a class of codimension-2 defects of the 6d N=(2,0) theories of type J=A,D,E labeled by nilpotent orbits of a Lie algebra \mathfrak{g}, where \mathfrak{g} is determined by J and the outer-automorphism twist around the defect. This class is a natural generalisation of the defects of the 6d theory of type SU(N) labeled by a Young diagram with N boxes. For any of these defects, we determine its contribution to the dimension of the Higgs branch, to the Coulomb branch operators and their scaling dimensions, to the 4d central charges a and c, and to the flavour central charge k.

• The paper demonstrates how nilpotent orbits classify codimension-2 defects, determining contributions to the Higgs and Coulomb branch dimensions.
• The study introduces algorithms linking nilpotent orbit structures with central charges and anomaly coefficients in 6d N=(2,0) theories.
• The work highlights the role of dualities and symmetries, offering a mathematical framework that connects 6d defects to emerging 4d physics.

Summary of "Nilpotent Orbits and Codimension-Two Defects of 6d $N{=}(2,0)$ Theories"

The paper by Chacaltana, Distler, and Tachikawa investigates the properties of codimension-2 defects in 6-dimensional $N{=}(2,0)$ theories of type $P = A, D, E$, emphasizing the interplay between these defects and nilpotent orbits of Lie algebras. This paper offers insights into the local properties of these defects by examining their contributions to the Higgs branch and Coulomb branch dimensions, as well as four-dimensional central charges.

Overview

The focus is on codimension-2 defects labeled by nilpotent orbits in $P$, which also consider an outer-automorphism twist around the defect. These are generalized from defects in $SU(N)$ theories labeled by Young diagrams.

Key concepts explored include:

• Nilpotent Orbits: Understanding these orbits in the context of Lie algebras allows classification up to conjugacy, useful for defining the defects.
• Spaltenstein Duality: Mapping the nilpotent orbit of a given Lie algebra to its Langlands dual is significant in the analysis of the Coulomb branch dimensions of these defects.
• Coulomb and Higgs Branches: The local contributions to these branches' dimensions due to defects are systematically explored using their relation to nilpotent orbits.

Main Results

1. Dimensions of Branches: The contribution of a defect to the Higgs or Coulomb branch is tied directly to the nature of its nilpotent orbit. The paper outlines algorithms to calculate these contributions, providing a deeper understanding of the local behavior at the defect.
2. Central Charges and Anomaly Coefficients: The paper maps contributions to central charges $a$ and $c$, further integrating with the anomaly polynomial of 6d superconformal theories and establishing a robust framework for determining these charges based on the type of defect.
3. Role of Symmetries: Through the paper of symmetries and dualities, especially S-duality and its manifestation in W-algebras, the paper illustrates how defects contribute to flavor symmetries and associated central charges at various points in the compactification of the theory.
4. Mathematical Formulation: The authors propose mathematical conjectures on polynomial subalgebras related to the Spaltenstein dual and elaborate on how these form pathways to determining additional relevant invariants.

Implications and Future Work

This research greatly impacts the broader understanding of non-Lagrangian theories in 4d, by elucidating how 6d $N{=}(2,0)$ theories via punctures on surfaces can engender rich 4-dimensional physics. By offering a clear connection between defects, their algebraic labeling, and physical quantities like branch dimensions and central charges, the paper supports theoretical advancements in the understanding of higher-dimensional quantum field theories.

Moving forward, implications of the paper extend to more complex gauge groups and understanding the phenomenon of dualities in supersymmetric theories could be explored further, potentially influencing approaches in mathematical physics, notably in the characterization and usage of symmetries inherent to higher-dimensional theories.