Higgs branch RG-flows via Decay and Fission

TL;DR

This work introduces a decay and fission algorithm for unitary magnetic quivers to map complete Higgs-branch phase diagrams (Hasse diagrams) and automatically extract transverse slice geometries between RG-fixed points. Built on convex linear algebra and 3d $ ext{N}=4$ Coulomb-Higgs dualities, the method yields both decays (rank-reducing) and fissions (splitting) without requiring exhaustive lists of elementary slices, enabling systematic tracing of Higgs branch RG-flows across $d=3$–$6$ theories, including SCFTs, instanton moduli spaces, and little string theories. The formal framework defines leaves, partial order, and elementary transitions with explicit rules for the transverse slices, while practical implementations and worked examples demonstrate consistency with existing approaches and reveal new RG-flow structures. The approach provides a versatile, implementable tool for classifying Higgs-branch RG flows, identifying potentially new SCFTs, and understanding transverse-slice geometry in a broad landscape of supersymmetric theories. It also highlights connections to 3d mirror symmetry, class $ ext{S}$ constructions, non-simply laced quivers, and higher-dimensional theories, offering a unified perspective on Higgs branch physics via magnetic quivers.

Abstract

Magnetic quivers have been an instrumental technique for advancing our understanding of Higgs branches of supersymmetric theories with 8 supercharges. In this work, we present the decay and fission algorithm for unitary magnetic quivers. It enables the derivation of the complete phase (Hasse) diagram and is characterised by the following key attributes: First and foremost, the algorithm is inherently simple; just relying on convex linear algebra. Second, any magnetic quiver can only undergo decay or fission processes; these reflect the possible Higgs branch RG-flows (Higgsings), and the quivers thereby generated are the magnetic quivers of the new RG fixed points. Third, the geometry of the decay or fission transition (i.e. the transverse slice) is simply read off. As a consequence, the algorithm does not rely on a complete list of minimal transitions, but rather outputs the transverse slice geometry automatically. As a proof of concept, its efficacy is showcased across various scenarios, encompassing SCFTs from dimensions 3 to 6, instanton moduli spaces, and little string theories.

Figures (16)

• Figure 1: Cartoon of decay and fission of quivers: The circle diameters symbolise the ranks. Upon a decay, one or several circles decrease in size; while a fission leads to the splitting into two.
• Figure 2: Hasse diagram obtained from decays of the quiver \ref{['eq:example1']} (left), and example of the subsequent computation of the geometry of an elementary transverse slice (right).
• Figure 3: Diagrams of decays from Figure \ref{['fig:hasseExample']} with the geometry of elementary transverse slices.
• Figure 4: Hasse diagram obtained from decays and fissions of the quiver \ref{['eq:example2']} (left), and the same diagram after the geometry of the transverse slices has been computed (right). The red line denotes the only fission transition, all other transitions are decays.
• Figure 5: (left) Shape of the allowed $(k'_1,k'_2)$ region in a situation where $a\geq 2$ and $b \geq 1$ (here $a=3$ and $b=2$). (right) Shape of the allowed $(k'_1,k'_2)$ region in a situation where $a=1$ and $b \geq 4$ (here $b=6$). In both cases, the unique minimal non-zero quiver is the red dot.