Weyl Mutations in Quiver Yangians

TL;DR

The paper develops a framework in which Weyl mutations act as Seiberg-like dualities on ADHM-like quiver data for ${\mathsf{A}}_n$ quivers, showing that the associated BPS algebra remains the Yangian $Y(\mathfrak{sl}_{n+1})$ across wall-crossings. It constructs framed quivers $\mathfrak{Q}_{n,u,h}$ with cyclic-chamber representations $\Upsilon_{u,h}$ (rectangular $\mathfrak{sl}_{n+1}$-weights) and realizes the Yangian action via fixed points tied to these representations, using Hecke-like operators and atomic plots. The Donaldson-Thomas generating function in the cyclic chamber matches the Schur character of $\Upsilon_{u,h}$ and extends to wall-crossing-invariant quantities, such as boxed plane partitions in a $(n+1-u)\times u\times h$ box. The work provides explicit mutation rules and phase-by-phase examples, linking wall-crossing, Nakajima quiver varieties, and quiver Yangians, and suggests avenues to generalize to non-cyclic chambers and higher algebras.

Abstract

The problem of solving non-linear equations would be considerably simplified by a possibility to convert known solutions into the new ones. This could seem an element of art, but in the context of ADHM-like equations describing quiver varieties there is a systematic approach. In this note we study moduli spaces and dualities of quiver gauge theories associated to effective dynamics of D-branes compactified on Calabi-Yau resolutions. We concentrate on a subfamily of quivers $\mathfrak{Q}_{\mathfrak{g}}$ covering Dynkin diagrams for simple Lie algebras $\mathfrak{g}$, where the respective BPS algebra is expected to be the Yangian algebra $Y(\mathfrak{g})$. For Yangians labeled by quivers their representations are described by solutions of ADHM-like equations. As quivers substitute Dynkin diagrams a generalization of the Weyl group $\mathcal{W}_{\mathfrak{g}}$ acts on the ADHM solutions. Here we work with the case $\mathfrak{g}=\mathfrak{sl}_{n+1}$ and treat this group as a group of electro-magnetic Seiberg-like dualities (we call them Weyl mutations) on the respective quiver gauge theories. We lift it to the case of higher representations associated to rectangular Young diagrams. An action of Weyl mutations on the BPS Yangian algebra is also discussed.