Title of the course: Introduction to Geometry of Quantum Field Theory: Fiber Bundles, Instantons, and the Dirac Operator
Instructor: Dr. Eyüp Yalçınkaya
Institution: TÜBİTAK
Dates: 17-23 August 2026
Prerequisites: Differential Geometry
Level: Graduate, advanced undergraduate
Abstract: This course aims to explore the elegant intersection of Yang-Mills theory and spin geometry within the framework of modern mathematical physics. The program begins by reconstructing classical gauge potentials through the language of differential geometry, specifically focusing on Fiber Bundles and Connections.
We will investigate the topological properties of the Yang-Mills functional on Euclidean 4-manifolds. Instead of relying on standard variational calculus, we will derive Instanton solutions geometrically via Self-Duality equations . Subsequently, the structure of spinor bundles and Clifford algebras will be introduced to analyze the elliptic nature of the Dirac Operator governing massless fermions.
The course concludes with an in-depth discussion of the Atiyah-Singer Index Theorem, establishing the profound link between the analytical zero modes of the Dirac operator and the topological charge (Chern class) of the background gauge field. This connection will be used to elucidate the geometric Quantum Field Theory.
Language: TR