Slant sums of quiver gauge theories

Abstract

We define the slant sum of quiver gauge theories, a gluing on the underlying quivers that identifies a gauge vertex with a framing vertex. Under some mild assumptions, we relate torus fixed points on the corresponding Higgs branches, which are Nakajima quiver varieties. Then we prove a ``branching rule" relating the quasimap vertex functions before and after a slant sum and deduce a number of ``factorization" corollaries. Our construction is motivated by a factorization conjecture for the vertex functions of zero-dimensional quiver varieties, which can be approached inductively using the branching rule. In special cases, it also shows that vertex functions can be written as sums over reverse plane partitions, even outside ADE type. We make some conjectures for Coulomb branches reflecting what can be seen on the Higgs side and prove them in ADE type. In particular, we obtain refined character formulas for the so-called ``extremal'' irreducible modules over shifted Yangians. We also study slant sums of Coulomb branches and their quantizations. We observe that for one-dimensional framing, the slant sum of Coulomb branches is the same as the product.