Bow varieties: Stable envelopes and their 3d mirror symmetry

TL;DR

This paper investigates elliptic stable envelopes of Cherkis bow varieties, revealing a duality inspired by 3d mirror symmetry, and introduces a geometric resolution process linked to algebraic fusion.

Contribution

It proves a duality property of elliptic stable envelopes for Cherkis bow varieties, connecting geometric resolutions with algebraic fusion in the context of 3d mirror symmetry.

Findings

Elliptic stable envelopes of Cherkis bow varieties exhibit a mirror symmetry-inspired duality.

Resolution of large charge branes into smaller ones corresponds to algebraic fusion.

New geometric features of bow varieties are discovered.

Abstract

In this paper we study the elliptic characteristic classes known as ''stable envelopes'', which were introduced by M. Aganagic and A. Okounkov. We prove that for a rich class of holomorphic symplectic varieties$\unicode{x2013}$called Cherkis bow varieties$\unicode{x2013}$their elliptic stable envelopes exhibit a duality property inspired by mirror symmetry in $d=3$, $\mathcal N=4$ quantum field theories. A crucial step of our proof involves the process of ''resolving'' large charge branes into multiple smaller charge branes. This phenomenon turns out to be the geometric counterpart of the algebraic fusion procedure. Along the way we discover various new features in the geometry of bow varieties.