The quantum UV-IR map for line defects in $\mathfrak{gl}(3)$-type class $S$ theories

TL;DR

This paper develops a geometric method to compute the quantum UV-IR map for line defects in $rak{gl}(3)$ class S theories, linking BPS states, supersymmetric physics, and knot invariants.

Contribution

It introduces a novel geometric approach to calculate the quantum UV-IR map and proposes a new method for computing a specialization of the HOMFLY polynomial via BPS webs.

Findings

Explicit computation of the quantum UV-IR map in examples

A new geometric method inspired by 5D supersymmetric Yang-Mills theory

A novel link between BPS webs and knot polynomial calculations

Abstract

We consider the quantum UV-IR map for line defects in class $S$ theories of $\mathfrak{gl}(3)$-type. This map computes the protected spin character which counts framed BPS states with spin for the bulk-defect system. We give a geometric method of computing this map motivated by the physics of five-dimensional $\mathcal{N}=2$ supersymmetric Yang-Mills theory, and compute it explicitly in various examples. As a spin-off we propose a new way of computing a certain specialization of the HOMFLY polynomial for links in $\mathbb{R}^3$, as a sum over BPS webs attached to the link.