Mirror symmetry for quiver algebroid stacks

Abstract

In this paper, we construct noncommutative algebroid stacks and the associated mirror functors for a symplectic manifold. First, we formulate a version of stack that is well adapted for gluing quiver algebras with different numbers of vertices.

Second, we develop a representation theory of A∞ categories by quiver stacks. A key step is constructing an extension of the A∞ category over a quiver stack of a collection of nc-deformed objects. The extension involves non-trivial gerbe terms, which play an important role for quiver algebroid stacks.

Third, we apply the theory to construct mirror quiver stacks of local Calabi-Yau manifolds. In this paper, we focus on nc local projective plane. This example has a compact divisor which gives rise to interesting monodromy and homotopy terms which can be found from mirror symmetry. Geometrically, we find a new method of mirror construction by gluing with a middle agent using Floer theory. The method makes crucial use of the extension of Fukaya category over quiver stacks.