论文标题
关于符号几何，泊松几何，变形量化和量子场理论的讲座
Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theory
论文作者
论文摘要
这些是作者在2020年秋季学期在苏黎世大学期间给出的“泊松几何形状和变形量化”课程的讲义。第一章是差异几何学的介绍，我们涵盖了歧管，张量字段，在歧管上的集成，Stokes的定理，De Rham的定理和Frobenius定理。第二章涵盖了符合性几何形状的最重要概念，例如拉格朗日亚曼福尔德（Lagrangian Submanifolds），温斯坦（Weinstein）的管状邻居定理，哈密顿力学，力矩图和符号降低。第三章介绍了泊松几何形状，我们还涵盖了库兰特结构，狄拉克结构，局部分裂定理，象征性叶子和泊松地图。第四章是关于变形量化的，我们涵盖了Moyal产品，$ L_ \ infty $ -Algebras，Kontsevich的形式定理，Kontsevich的Star产品通过图形，Kontsevich的星星产品的全球化方法以及形式的经营方法。第五章是关于Kontsevich的变形量化的量子场理论方法，在该方法中，我们涵盖功能性积分方法，作为路径积分量化，Faddeev-popov和BRST方法的功能积分产品，用于计量量的量化理论，无限二维扩展，POISSON SIGMA模型，通过Kontsevich的构建量的构造，该量子的构建量Poisson Sigma模型用于仿射泊松结构和一般结构。
These are lecture notes for the course "Poisson geometry and deformation quantization" given by the author during the fall semester 2020 at the University of Zurich. The first chapter is an introduction to differential geometry, where we cover manifolds, tensor fields, integration on manifolds, Stokes' theorem, de Rham's theorem and Frobenius' theorem. The second chapter covers the most important notions of symplectic geometry such as Lagrangian submanifolds, Weinstein's tubular neighborhood theorem, Hamiltonian mechanics, moment maps and symplectic reduction. The third chapter gives an introduction to Poisson geometry where we also cover Courant structures, Dirac structures, the local splitting theorem, symplectic foliations and Poisson maps. The fourth chapter is about deformation quantization where we cover the Moyal product, $L_\infty$-algebras, Kontsevich's formality theorem, Kontsevich's star product construction through graphs, the globalization approach to Kontsevich's star product and the operadic approach to formality. The fifth chapter is about the quantum field theoretic approach to Kontsevich's deformation quantization where we cover functional integral methods, the Moyal product as a path integral quantization, the Faddeev-Popov and BRST method for gauge theories, infinite-dimensional extensions, the Poisson sigma model, the construction of Kontsevich's star product through a perturbative expansion of the functional integral quantization for the Poisson sigma model for affine Poisson structures and the general construction.
