Quiver Subtractions
Santiago Cabrera, Amihay Hanany
TL;DR
The paper introduces quiver subtraction as a concise, physically meaningful operation between 3d $\mathcal{N}=4$ quivers to describe transverse slices of Coulomb branches and the appearance of new massless states at infinite gauge coupling. It formally defines the subtraction, demonstrates its link to Kraft-Procesi transitions, and applies it to both exceptional and classical Lie algebras to realize new transitions and mixed branches. The work extends the physical realization of nilpotent-orbit transitions beyond the classical, showing how to analyze Higgs/Coulomb structure in 5d $\mathcal{N}=1$ SQCD at infinite coupling and in 6d contexts, with explicit examples including $E_6$, $E_7$, and $F_4$. Overall, quiver subtraction emerges as a powerful, broadly applicable tool for mapping vacuum structure and phase transitions in higher-dimensional theories with eight supercharges.
Abstract
We study the vacuum structure of gauge theories with eight supercharges. It has been recently discovered that in the Higgs branch of $5d$ and $6d$ SQCD theories with eight supercharges, the new massless states, arising when the gauge coupling is taken to infinity, can be described in terms of Coulomb branches of $3d ~\mathcal N=4$ quiver gauge theories. The description of this new phenomenon draws from the ideas discovered in the analysis of nilpotent orbits as Higgs and Coulomb branches of $3d$ theories and promotes the Higgs mechanism known as the Kraft-Procesi transition to the status of a new operation between quivers. This is the quiver subtraction. This paper establishes this operation formally and examines some immediate consequences. One is the extension of the physical realization of Kraft-Procesi transitions from the classical to the exceptional Lie algebras. Another result is the extension from special nilpotent orbits to non-special ones. One further consequence is the analysis of the effect in $5d ~\mathcal N=1$ SQCD of integrating out a massive quark while the gauge coupling remains infinite. In general, the subtraction of quivers sheds light on the different types of singularities within the Coulomb branch and the structure of the massless states that arise at those singular points; including the nature of the new Higgs branches that open up. This allows for a systematic analysis of mixed branches of $3d ~\mathcal N=4$ quivers that do not necessarily have a simple embedding in string theory. The subtraction of two quivers is an extremely simple resource for the theoretical physicist interested in the vacuum structure of gauge theories, and yet its power is so remarkable that is bound to play a crucial role in the coming discoveries of new and exciting physics in $5$ and $6$ dimensions.