Mirror Symmetry for Quiver Algebroid Stacks

TL;DR

This paper introduces a new framework for constructing quiver algebroid stacks and mirror functors in symplectic geometry, connecting sheaf theory and representation theory through innovative gluing methods and $A_ abla$-category extensions.

Contribution

It develops a novel approach to building mirror quiver stacks using quiver stacks and a new gluing method for Lagrangians, advancing the understanding of mirror symmetry in symplectic geometry.

Findings

Constructed mirror quiver stacks for specific Lagrangians.

Extended $A_ abla$ categories over quiver stacks with gerbe terms.

Applied framework to a punctured elliptic curve, obtaining a mirror nc local projective plane.

Abstract

In this paper, we provide a new construction of quiver algebroid stacks and the associated mirror functors for symplectic manifolds. First, we formulate the concept of a quiver stack, which is a geometric structure formed by gluing multiple quiver algebras together. Next, we develop a representation theory of $A_\infty$ categories by quiver stacks. The main idea is to extend the $A_\infty$ category over a quiver stack of a collection of nc-deformed objects. The extension involves non-trivial gerbe terms. It gives an application of symplectic geometry that bridges the study of sheaves and representation theory through mirror symmetry. We provide a general framework for constructing mirror quiver stacks. In particular, we develop a novel method of gluing Lagrangians which are disjoint from each other by using quasi-isomorphisms with a `global middle agent', which is a Lagrangian immersion that produces a mirror quiver. The method relies fundamentally on the use of quiver stacks. We carry out this construction for compact immersed Lagrangians in a punctured elliptic curve, which results in a mirror nc local projective plane.