CODE  44142 

ACADEMIC YEAR  2022/2023 
CREDITS 

SCIENTIFIC DISCIPLINARY SECTOR  MAT/07 
TEACHING LOCATION 

SEMESTER  2° Semester 
TEACHING MATERIALS  AULAWEB 
Language: English
The purpose of the course is to provide an introduction to gauge theories. Specifically, after introducing the necessary notions of differential geometry (the theory of connections on vector and principal fibrations, Hodge theory), we will address some salient aspects of YangMills theory on 4dimensional Riemannian varieties, studying the structure Of the space module space.
Teaching style: In presence
Geometric Methods in Mathematical Physics
Academic year 20132014; first semester
1. FIBRE BUNDLES, CONNECTIONS AND HOLONOMY GROUPS
•Vector bundles and their operations; vector bundles with metric structure.
• Linear connections on vector bundles; curvature 2form; Cartan’s strucure equations; Bianchi’s identity; generalized LeviCivita connection.
• Principle bundles; fundamental vector fields.
• Connections on principal bundles; from vector bundles to principle bundle and back; group of gauge transformations
• Holonomy group; intrnsic torsion
• Classification of Riemannian holonomy gropus (statement of Berger's theorem and examples)
2. TOPICS IN RIEMANNIAN GEOMETRY
• Geodesics and parallel transport
• Surfaces; "theorema egregium"; the GaussBonnet theorem
• HopfRinow's theorem
• Symmetric spaces
3. INTRODUCTION TO KÄHLER MANIFOLDS
• Introduction to complex manifolds
• Kähler manifold; the complex projective space
• Riemann surfaces; algebraic curves
4. INTRODUCTION TO HODGE THEORY
• Differential operators on Riemannian manifolds
• The de Rham cohomology
• The Hodge theorem
• The Hodge decomposition theorem on compact Kähler manifolds
• ASD equations; instantons on S4.
CLAUDIO BARTOCCI (President)
PIERRE OLIVIER MARTINETTI
MARCO BENINI (President Substitute)
The class will start according to the academic calendar.
Oral.