A note on discrete dynamical systems in theories of class $S$

TL;DR

This paper explores the structure of line operators in theories of class S, revealing a natural discrete dynamical system linked to the BPS spectrum, with implications for global theory properties and algebraic structures.

Contribution

It introduces a novel perspective connecting line operators to a discrete dynamical system and explores its applications in understanding BPS spectra and algebraic relations.

Findings

Line operators form a discrete dynamical system in theories of class S.

Constraints on the spectrum generator are identified, especially for SU(2) N=2*.

Connections between line defects and Double Affine Hecke Algebras are established.

Abstract

In this note we consider the set of line operators in theories of class $S$. We show that this set carries the action of a natural discrete dynamical system associated with the BPS spectrum. We discuss several applications of this perspective; the relation with global properties of the theory; the set of constraints imposed on the spectrum generator, in particular for the case of SU(2) $\mathcal{N}=2^*$; and the relation between line defects and certain spherical Double Affine Hecke Algebras.