| Abstract |
Coisotropic branes were introduced by Kapustin-Orlov in order to enlarge the Fukaya category of a symplectic manifold so that it matches the homological mirror symmetry prediction. However, how such branes can be defined as an object in the Fukaya category is wildly unknown. In this talk, I would like to present a mathematical rigorous definition for the self-hom of the so-called canonical coisotropic branes, which is a space-filling brane. We begin with a semi-flat SYZ fibration $X\to B$ that carries a semi-affine canonical coisotropic brane. Such brane carries a natural complex structure $I$. We define the mirror B-brane by taking fiberwise geometric quantization, and by using family Toeplitz construction with a gauge transformation, we obtain an chain map from the Dolbeault complex of certain I-holomorphic deformation quantization of $(X,I)$ to that of the endomorphism algebra of the mirror B-brane. Our main result states that this map is a chain isomorphism of algebras. This provides an intrinsic definition of the self-hom and also the first mirror theorem of such coisotropic branes. As an application of our construction, we provide a rigorous mathematical definition of brane quantization in the semi-flat SYZ setting. This is a joint work with Kwokwai Chan, Nai-Chung Conan Leung, Qin Li, and Yu-Tung Tony Yau. |
