Boundary fixed-point conjecture for Coulomb branches

Let $Q$ be a quiver with corresponding Coulomb branch $\mathcal M(\lambda,\mu)$, let $\chi_i$ be a one-parameter subgroup lying on the boundary of the negative Weyl chamber, and let $\varpi$ be the factorization morphism. Write $\mu'=\langle\mu,h_i\rangle$.

Boundary fixed-point conjecture. The fixed-point set $\mathcal M(\lambda,\mu)^{\chi_i}$ is either empty or isomorphic to a Coulomb branch $\mathcal M_{A_1}(\lambda',\mu')$ of an $A_1$-type framed quiver gauge theory; its intersection with any stratum is either empty or a stratum of $\mathcal M_{A_1}(\lambda',\mu')$. Moreover, the restriction of the $i$-th component of $\varpi$ to this $A_1$ Coulomb branch equals its factorization morphism up to adding $0$.

This conjecture reduces boundary fixed-point geometry to the $A_1$ case and is intended to support the construction of the Kac–Moody module structure. The paper proves related results in affine type $A$.