Mirror Symmetry for Quiver Algebroid Stacks
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916977412407296 |
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| author | Lau, Siu-Cheong Nan, Junzheng Tan, Ju |
| author_facet | Lau, Siu-Cheong Nan, Junzheng Tan, Ju |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_03028 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mirror Symmetry for Quiver Algebroid Stacks Lau, Siu-Cheong Nan, Junzheng Tan, Ju Algebraic Geometry Symplectic Geometry 14J33, 53D37 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. |
| title | Mirror Symmetry for Quiver Algebroid Stacks |
| topic | Algebraic Geometry Symplectic Geometry 14J33, 53D37 |
| url | https://arxiv.org/abs/2206.03028 |