Brane quantization and SYZ mirror symmetry
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911663834267648 |
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| author | Chan, Kwokwai Leung, Naichung Conan Li, Qin Suen, Yat-Hin Yau, Yutung |
| author_facet | Chan, Kwokwai Leung, Naichung Conan Li, Qin Suen, Yat-Hin Yau, Yutung |
| contents | Coisotropic A-branes were introduced by Kapustin--Orlov to enlarge the Fukaya category of a symplectic manifold in a way that aligns with predictions from homological mirror symmetry. From a mathematical perspective, however, the categorical framework governing such branes remains largely undeveloped. On the other hand, Gukov--Witten's brane quantization suggests that a holomorphic deformation quantization of a holomorphic symplectic manifold $X$ arises from the endomorphism algebra $Hom_A(B_{cc},B_{cc})$ of a canonical coisotropic A-brane $B_{cc}$, which naturally acts on the morphism space $Hom_A(B,B_{cc})$ with a Lagrangian A-brane $B$ that in turn gives precisely the geometric quantization of $B$.
In this paper, we consider a holomorphic symplectic manifold $X$ which admits an SYZ fibration and apply SYZ mirror symmetry to study its brane quantization. Given any semi-affine, space-filling coisotropic A-brane $B_{cc}$ on $X$, we construct the mirror B-brane $\check{B}_{cc}$ on the mirror manifold $\check{X}$ by an SYZ transform. We then present a mathematical definition of the endomorphism algebra $Hom_A(B_{cc},B_{cc})$ by constructing a distinguished non-formal holomorphic deformation quantization of $X$. Using a twisted family Toeplitz construction, we transform $Hom_A(B_{cc},B_{cc})$ to the mirror B-side and prove that this induces an isomorphism $Hom_A(B_{cc},B_{cc})\cong Hom_B(\check{B}_{cc},\check{B}_{cc})$ between the endomorphism algebras. Furthermore, taking any torus fiber of $X$ as the Lagrangian A-brane $B$, we fully realize Gukov--Witten's proposal, namely, there is a natural action of $Hom_A(B_{cc},B_{cc})$ on $Hom_A(B,B_{cc})$ which is precisely mirror to the natural action on the mirror B-side. This provides a mathematical framework which is compatible with Gukov--Witten's brane quantization proposal, SYZ mirror symmetry as well as family Floer theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26292 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Brane quantization and SYZ mirror symmetry Chan, Kwokwai Leung, Naichung Conan Li, Qin Suen, Yat-Hin Yau, Yutung Differential Geometry Mathematical Physics Quantum Algebra Symplectic Geometry Coisotropic A-branes were introduced by Kapustin--Orlov to enlarge the Fukaya category of a symplectic manifold in a way that aligns with predictions from homological mirror symmetry. From a mathematical perspective, however, the categorical framework governing such branes remains largely undeveloped. On the other hand, Gukov--Witten's brane quantization suggests that a holomorphic deformation quantization of a holomorphic symplectic manifold $X$ arises from the endomorphism algebra $Hom_A(B_{cc},B_{cc})$ of a canonical coisotropic A-brane $B_{cc}$, which naturally acts on the morphism space $Hom_A(B,B_{cc})$ with a Lagrangian A-brane $B$ that in turn gives precisely the geometric quantization of $B$. In this paper, we consider a holomorphic symplectic manifold $X$ which admits an SYZ fibration and apply SYZ mirror symmetry to study its brane quantization. Given any semi-affine, space-filling coisotropic A-brane $B_{cc}$ on $X$, we construct the mirror B-brane $\check{B}_{cc}$ on the mirror manifold $\check{X}$ by an SYZ transform. We then present a mathematical definition of the endomorphism algebra $Hom_A(B_{cc},B_{cc})$ by constructing a distinguished non-formal holomorphic deformation quantization of $X$. Using a twisted family Toeplitz construction, we transform $Hom_A(B_{cc},B_{cc})$ to the mirror B-side and prove that this induces an isomorphism $Hom_A(B_{cc},B_{cc})\cong Hom_B(\check{B}_{cc},\check{B}_{cc})$ between the endomorphism algebras. Furthermore, taking any torus fiber of $X$ as the Lagrangian A-brane $B$, we fully realize Gukov--Witten's proposal, namely, there is a natural action of $Hom_A(B_{cc},B_{cc})$ on $Hom_A(B,B_{cc})$ which is precisely mirror to the natural action on the mirror B-side. This provides a mathematical framework which is compatible with Gukov--Witten's brane quantization proposal, SYZ mirror symmetry as well as family Floer theory. |
| title | Brane quantization and SYZ mirror symmetry |
| topic | Differential Geometry Mathematical Physics Quantum Algebra Symplectic Geometry |
| url | https://arxiv.org/abs/2604.26292 |