Line Defects, Tropicalization, and Multi-Centered Quiver Quantum Mechanics

TL;DR

This paper investigates BPS line defects in N=2 supersymmetric theories, proposing a method to compute their BPS Hilbert space using infrared data and exploring the algebraic structure of line defects.

Contribution

It introduces a novel approach to determine the BPS Hilbert space of ultraviolet defects via infrared data and analyzes the algebraic structure of line defects in quiver theories.

Findings

Derived a formula for discontinuities in the defect RG map.

Showed the operator product algebra contains universal subalgebras.

Applied methods to explicit examples of quiver theories.

Abstract

We study BPS line defects in N=2 supersymmetric four-dimensional field theories. We focus on theories of "quiver type," those for which the BPS particle spectrum can be computed using quiver quantum mechanics. For a wide class of models, the renormalization group flow between defects defined in the ultraviolet and in the infrared is bijective. Using this fact, we propose a way to compute the BPS Hilbert space of a defect defined in the ultraviolet, using only infrared data. In some cases our proposal reduces to studying representations of a "framed" quiver, with one extra node representing the defect. In general, though, it is different. As applications, we derive a formula for the discontinuities in the defect renormalization group map under variations of moduli, and show that the operator product algebra of line defects contains distinguished subalgebras with universal multiplication rules. We illustrate our results in several explicit examples.