Punctures and Dynamical Systems

TL;DR

This paper explores the complex dynamical systems underlying supersymmetric punctures in 6D SCFTs derived from M5-branes, revealing intricate behaviors and classifications that extend to various orbifold singularities.

Contribution

It introduces a dynamical system framework for classifying supersymmetric punctures in class S_Gamma theories, especially for single M5-branes, and generalizes to multiple branes and different orbifold types.

Findings

Periodic orbits occur only at specific rational initial momenta.

Small perturbations cause vastly different late-time behaviors.

Continuous zero mode moduli appear in some puncture solutions.

Abstract

With the aim of better understanding the class of 4D theories generated by compactifications of 6D superconformal field theories (SCFTs), we study the structure of N = 1 supersymmetric punctures for class S_Gamma theories, namely the 6D SCFTs obtained from M5-branes probing an ADE singularity. For M5-branes probing a C^2 / Z_k singularity, the punctures are governed by a dynamical system in which evolution in time corresponds to motion to a neighboring node in an affine A-type quiver. Classification of punctures reduces to determining consistent initial conditions which produce periodic orbits. The study of this system is particularly tractable in the case of a single M5-brane. Even in this "simple" case, the solutions exhibit a remarkable level of complexity: Only specific rational values for the initial momenta lead to periodic orbits, and small perturbations in these values lead to vastly different late time behavior. Another difference from half BPS punctures of class S theories includes the appearance of a continuous complex "zero mode" modulus in some puncture solutions. The construction of punctures with higher order poles involves a related set of recursion relations. The resulting structures also generalize to systems with multiple M5-branes as well as probes of D- and E-type orbifold singularities.