Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theory

TL;DR

This lecture notes collection provides a comprehensive overview of symplectic and Poisson geometry, deformation quantization, and their connections to quantum field theory, covering foundational concepts and advanced mathematical techniques.

Contribution

It offers a detailed, pedagogical exposition of deformation quantization and its relation to quantum field theory, including new insights into Kontsevich's formality theorem and path integral approaches.

Findings

Introduction of the Moyal product and Kontsevich's star product construction.

Application of functional integral methods to deformation quantization.

Extension of quantization techniques to gauge theories and Poisson sigma models.

Abstract

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.