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

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.

76 pages, 10 figures; v2: minor revisions, correction to Coulomb branch calculation for defects in SU(2) SYM

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• Reading between the lines of four-dimensional gauge theories
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• Non-commutative Donaldson-Thomas theory and the conifold
• Crystal Melting and Toric Calabi-Yau Manifolds
• More on the N=2 superconformal systems of type $D_p(G)$
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• Line defects and (framed) BPS quivers

Cited by in corpus (22)

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• Surface Defects and Chiral Algebras
• Infrared Computations of Defect Schur Indices
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• A slow review of the AGT correspondence
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Vafa-Witten invariants from exceptional collections
• Surface Defect Indices and 2d-4d BPS States
• Semiclassical framed BPS states
• The Coulomb Branch Formula for Quiver Moduli Spaces
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Line defect Schur indices, Verlinde algebras and $U(1{)}_r$ fixed points
• BPS Hall Algebra of Scattering Hall States
• $q$-nonabelianization for line defects
• Regge Trajectories in $\mathcal{N}=2$ Supersymmetric Yang-Mills Theory
• Categorified canonical bases and framed BPS states
• An infrared bootstrap of the Schur index with surface defects
• Quantum Line Defects and Refined BPS Spectra
• Quivers, Line Defects and Framed BPS Invariants
• On Framed Quivers, BPS Invariants and Defects
• A note on discrete dynamical systems in theories of class $S$
• The quantum UV-IR map for line defects in $\mathfrak{gl}(3)$-type class $S$ theories