Nilpotent orbits and codimension-two defects of 6d N=(2,0) theories
Oscar Chacaltana, Jacques Distler, Yuji Tachikawa
TL;DR
The paper develops a universal, Lagrangian-independent framework to characterize codimension-2 defects in 6d $\mathcal{N}=(2,0)$ theories labeled by nilpotent orbits, including twisted cases. By relating 6d defects to 3d theories $T^{\rho}[{\mathfrak g}]$ and to Hitchin-system data, it provides algorithms to compute local contributions to the Higgs and Coulomb branches, the scaling dimensions of Coulomb operators, and the 4d central charges $a,c$ and flavor levels $k$. A central organizing principle is the Spaltenstein duality $d(O_{\rho})$ and the induction framework, together with the discrete Symmetry data $\mathcal{C}(O_{\rho})$ from Sommers–Achar, which determine the spectrum of local Coulomb-branch operators across classical and exceptional algebras. The work unifies untwisted and twisted defects, connects to Gukov-Witten surface operators and W-algebras, and includes extensive case studies (including $G_2$ and $E_8$) and free-field fixtures to validate the formalism. Overall, it provides a robust toolkit for extracting local defect data in a broad class of 6d theories, enabling systematic construction and comparison of 4d $\mathcal{N}=2$ theories arising from compactifications on Riemann surfaces with punctures.
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.