Mirror Symmetry for Quiver Algebroid Stacks

TL;DR

The paper develops a framework to glue noncommutative deformation spaces of Lagrangian immersions into quiver algebroid stacks, enabling a global mirror perspective in symplectic geometry. It extends the Fukaya category to NC families and constructs mirror functors to twisted complexes over these stacks, incorporating gerbe terms that arise when gluing quivers with different numbers of vertices. A central achievement is the NC mirror construction and the natural $A_0$-transformations linking commutative and NC mirrors, exemplified by the NC local projective plane $K_{ ext{P}^2}$, realized as a quiver algebroid stack glueing Seidel-type Lagrangians with a global middle agent. The approach provides a local-to-global description of NC crepant resolutions and noncommutative mirrors, with a concrete non-Archimedean realization, universal bundles, and a Fourier–Mukai-type correspondence, offering new avenues for interpreting derived equivalences in the NC setting. The results illuminate how NC deformation spaces, gerbes, and gluing data interweave with HMS/SYZ perspectives to yield explicit, computable NC mirrors of Calabi–Yau geometries.

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.

Figures (27)

• Figure 1: The quiver on the left corresponds to $\mathbb{C}^3$ and its noncommutative deformations. The quiver on the right is used as a noncommutative resolution of $\mathbb{C}^3/\mathbb{Z}_3$. These two quiver algebras with different numbers of vertices will be glued together in the context of quiver stacks.
• Figure 2: The left shows a pair-of-pants decomposition of the three-punctured elliptic curve and Seidel Lagrangians. The right shows a way to put Seidel Lagrangians so that they can be isomorphic to the 'middle Lagrangian' $\mathbb{L}$.
• Figure 3: An algebroid stack which is a noncommutative deformation of $K_{\mathbb{P}^2}$.
• Figure 4: Each transverse intersection point corresponds to two Floer generators.
• Figure 5: The left hand side shows the Seidel Lagrangian in a pair-of-pants. The right hand side shows a lifting to 3-to-1 cover by a three-punctured elliptic curve.

Theorems & Definitions (132)

• Theorem 1.1: Theorem \ref{['thm: alg- functor']} and Proposition \ref{['prop:inj']}
• Definition 1.2
• Remark 1.3
• Definition 1.4
• Example 1.5: Free projective space
• Theorem 1.6: Theorem \ref{['prop:isomorphism_KP2']}
• Theorem 1.7: Theorem \ref{['thm:nat-trans-X-A']}
• Theorem 1.8: Theorem \ref{['thm:loc inj']}
• Definition 2.1
• Definition 2.2