带边角场论的Batalin-Vilkovisky量子化:从导几何到导"量子"几何

This project motivates from two amazing theories --- Batalin-Vilkovisky (BV) quantization as well as the holographic principle in String theory and quantum field theories. The applicant considers the derived geometrical aspects of classical BV theory, and generalizes it to the case where the world-sheet manifold carries boundaries and corners. Our approach shall identify a derived Poisson structure on the space of fields, and uses sheaf cohomology computation of orbifolds. BV quantization in this case is going to be defined, with quantization obstruction and quantum observables identified. The algebraic structure of the latter forms a derived BV structure, which will lead to deeper understanding on derived quantum geometry. The applicant plans to use the above theory to define and prove the generalised holographic principle rigorously, thereby providing strong mathematical support for this physical statement. Global quantum observables shall give us candidates for topological and geometrical invariants on manifold with corners, which in turn, will uncover key properties of derived quantum geometry. As an application, the applicant will study a four-dimensional field theory with corners defined by Witten and Kapustin in their geometric Langlands program. The applicant hopes that the relevant result shall bring nice contributions the theory of geometric Langlands duality.

本项目的研究动机源自于两个理论——Batalin-Vilkovisky（BV）量子化与量子场论和弦论中的全息原理。申请人从经典BV理论的导几何意义出发，运用导Poisson几何以及轨形（orbifold）的层上同调的计算，将经典BV理论推广到世界图（worldsheet）为带边角流形的情况，继而定义BV量子化理论并计算量子化障碍（obstruction）和观测量代数。量子化所得到的代数结构为一个导BV结构，将应用于探索量子意义下的导几何。申请人计划运用上述理论在数学上建立并证明广义的全息原理，从而为量子场论中的重要理论提供数学依据。量子全局观测量将给出带边角流形上的不变量。这些不变量将进一步揭示导“量子”几何的诸多几何，拓扑性质。同时，申请人将运用该项目的主体理论研究几何朗兰兹对偶中由Witten及Kapustin构造的一个四维带边角流形上的场论，从而推动与几何朗兰兹纲领相关的研究。

本项目研究带边角流形上BV场论的形变量子化问题。带边和带角流形的局部结构，特别是边界子流形的维数，与BV场论的局部量子观测量代数的结构有关。而在更抽象的层次上，worldsheet的维数对于导泊松结构的量子化问题有决定性的影响。在研究中，我们分别从具体的量子化实例以及抽象的场论构造的层面上进行探索。在具体实例方面，我们研究了带边Poisson sigma模型的全息现象以及Chern-Simons理论的一个2-3-4维的对应。在场论的构造与量子化手段方面，我们研究了Lie-infinity代数的一类几何形变，以及BV理论与紧化近似之关系，以期探索worldsheet维数与量子化的关系以及广义的全息对应。