OVERVIEW

The course introduces the essential geometric tools for modelling gauge theories, namely vector bundles, principal bundles and connections.

AIMS AND CONTENT

LEARNING OUTCOMES

The purpose of the teaching is to provide an introduction to gauge theories. In particular, after having introduced the necessary notions of differential geometry (theory of connections on vector and principal bundles, Hodge theory), we will deal with some salient aspects of Yang-Mills theories on 4-dimensional Riemannian manifolds, arriving at the study of the structure local of the instanton module space.

AIMS AND LEARNING OUTCOMES

The course aims to familiarise students with certain methods and techniques from differential geometry that underpin the formulation of gauge theories. In particular, students will learn to

• examine the main geometric objects involved, in particular vector bundles, principal bundles and connections,
• calculate the curvature of a connection, construct its parallel transport and the holonomies along closed paths,
• associate topological invariants to vector/principal bundles by choosing a connection,
• formulate the Yang-Mills equations.

PREREQUISITES

The course is structured in such a way that there are no strict prerequisites; however, some basic knowledge of differential geometry is particularly useful.

TEACHING METHODS

The course consists of lectures delivered by the lecturer, during which the theory will be presented and examples and applications will be examined.

SYLLABUS/CONTENT

The course starts with an introductory section recalling some basic concepts of differential geometry (smooth manifolds, tangent vectors and differential forms, de Rham cohomology, Stokes’ theorem, Lie groups, and the basics of semi-Riemannian geometry). Afterwards, the following topics will be addressed systematically:

1. fibre bundles, vector bundles, principal bundles and associated bundles;
2. connections, curvature, parallel transport and holonomy;
3. Chern-Weil homomorphism and characteristic classes;
4. Yang-Mills equations.

RECOMMENDED READING/BIBLIOGRAPHY

The literature on this subject is extensive. Below is a list of references that students may find helpful:

• Kobayashi S., Nomizu K. - Foundations of differential geometry Volume 1 - Wiley 1963
• Kobayashi S., Nomizu K. - Foundations of differential geometry Volume 2 - Wiley 1969
• Tu L.W. - Differential geometry - Springer 2017
• Lee J.M. - Introduction to smooth manifolds - Springer 2012
• O'Neill B. - Semi-Riemannian geometry - Academic Press 1983
• Isham C.J. - Modern differential geometry for physicists - World Scientific 1999
• Sontz S.B. - Principal bundles - Springer 2015
• Nakahara M. - Geometry, topology and physics - IOP Publishing 2003

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://corsi.unige.it/corsi/11907/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of an oral test.

ASSESSMENT METHODS

During the oral examination, students must demonstrate that they can introduce and explain the fundamental concepts covered in the lectures, with particular regard to the statements and proofs of the main theorems, and that they are able to apply them to the applications examined during the course.

FURTHER INFORMATION

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it